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+Planning Visual Inspection Tours for a 3D Dubins Airplane
+Model in an Urban Environment
+Collin Hague ∗, Andrew Willis †, Dipankar Maity ‡, Artur Wolek §
+University of North Carolina at Charlotte, Charlotte, North Carolina, 28223
+This paper investigates the problem of planning a minimum-length tour for a three-
+dimensional Dubins airplane model to visually inspect a series of targets located on the ground
+or exterior surface of objects in an urban environment. Objects are 2.5D extruded polygons
+representing buildings or other structures. A visibility volume defines the set of admissible
+(occlusion-free) viewing locations for each target that satisfy feasible airspace and imaging con-
+straints. The Dubins traveling salesperson problem with neighborhoods (DTSPN) is extended
+to three dimensions with visibility volumes that are approximated by triangular meshes. Four
+sampling algorithms are proposed for sampling vehicle configurations within each visibility
+volume to define vertices of the underlying DTSPN. Additionally, a heuristic approach is pro-
+posed to improve computation time by approximating edge costs of the 3D Dubins airplane
+with a lower bound that is used to solve for a sequence of viewing locations. The viewing
+locations are then assigned pitch and heading angles based on their relative geometry. The
+proposed sampling methods and heuristics are compared through a Monte-Carlo experiment
+that simulates view planning tours over a realistic urban environment.
+I. Introduction
+U
+nmanned aerial vehicles (UAVs) are routinely used in applications such as visual reconnaissance, infrastructure
+inspection, and aerial photography to image a series of points of interest (henceforth referred to as targets). In
+three-dimensional environments (e.g., an urban city, mountainous terrain) the targets must be imaged from particular
+vantage points to avoid occlusions from surrounding objects (e.g., buildings, trees). Additional requirements, such as
+airspace restrictions and image resolution, further constrain the three-dimensional visibility volume from which an image
+of a target may be obtained. This paper investigates the problem of planning a path to image a set of targets by flying
+through their corresponding visibility volumes in minimum time. The UAV is modeled as a Dubins airplane [1, 2] and
+the environment consists of extruded polygonal objects with targets located on the ground or on the surface of objects.
+A. Relation to Prior Work
+The view planning problem considered here is related to the Dubins traveling salesperson problem (DTSP [3]) of
+constructing a minimum-time tour for a constant-speed planar Dubins vehicle model [4] to travel through a series of
+planar points (with arbitrary heading). The set of points to visit can be generalized to arbitrary planar regions (e.g.,
+polygons) to give the DTSP with neighborhoods (DTSPN [5]) wherein the Dubins vehicle must visit at least one point in
+each region/neighborhood. One application of the DTSPN is to plan visual inspection tours for an airplane to visit
+planar polygonal regions at a constant altitude to image ground targets [6]. More recently, the Dubins airplane model
+[1, 2] that includes additional degrees of freedom (altitude and pitch angle) was used to extend the DTSPN to three
+dimensions. Planning three-dimensional Dubins tours have typically assumed that the desired viewing regions have
+relatively simple geometries, such as spheres [7] or cylinders [8]. In contrast, this work admits more complex target
+visibility volumes that are approximated as triangular meshes.
+B. Contributions
+This paper formulates a view planning problem for a 3D Dubins airplane model to observe a set of targets occluded
+by objects in an urban environment. The contributions of the paper are: (1) four sampling algorithms that extend
+∗Graduate student, Department of Mechanical Engineering and Engineering Science
+†Associate Professor, Department of Electrical and Computer Engineering
+‡Assistant Professor, Department of Electrical and Computer Engineering
+§Assistant Professor, Department of Mechanical Engineering and Engineering Science, Member AIAA
+1
+arXiv:2301.05309v1  [eess.SY]  12 Jan 2023
+
+two-dimensional Dubins-based view planning to three dimensions with visibility volumes that have an arbitrary geometry
+approximated by a triangular mesh, and (2) a heuristic approach that solves for a tour using a modified Euclidean
+distance TSP (METSP) with edge costs that are lower bounds for the 3D Dubins path length and using the geometry of
+consecutive viewing locations in the METSP tour to assign heading and pitch angles. The relative performance of the
+algorithms are characterized through a Monte-Carlo experiment.
+C. Paper Organization
+The remainder of the paper is organized as follows. Section II describes the airplane motion model, the environment
+model, the target visibility volumes, and states the view planning problem. Section III describes a method for
+approximately computing the target visibility volumes and path planning for constant-altitude 2D tours. Section IV
+introduces 3D path planning algorithms and proposes heuristics to reduce computation time. Section V describes the
+results of a Monte-Carlo experiment that compares the 2D and 3D algorithms. The paper is concluded in Sec. VI.
+II. Problem Formulation
+This section formulates the problem of planning a minimum time path for an unmanned airplane to visually inspect
+a set of targets in the presence of occluding structures. The vehicle motion model, environmental model, and target
+visibility volumes are introduced, and the view planning problem is formally stated.
+A. Airplane Motion Model
+This work considers the three-dimensional Dubins airplane model [9, 10]:
+������������
+�𝑥
+�𝑦
+�𝑧
+�𝜓
+�𝛾
+������������
+=
+������������
+𝑣 cos 𝜓 cos 𝛾
+𝑣 sin 𝜓 cos 𝛾
+𝑣 sin 𝛾
+𝑢𝜓
+𝑢𝛾
+������������
+,
+(1)
+where (𝑥, 𝑦, 𝑧) ∈ R3 is the inertial position of the airplane expressed in an east-north-up coordinate system, 𝑣 is the
+vehicle’s speed, 𝜓 is the heading angle, and 𝛾 is the pitch angle (see Fig. 1). The control inputs are the turn-rate 𝑢𝜓 and
+the pitch-angle-rate 𝑢𝛾. The Dubins airplane model travels in the direction it is pointed so that the pitch angle 𝛾 is
+Fig. 1
+The model for a Dubins airplane flying at speed 𝑣 where (𝑥, 𝑦, 𝑧) is the inertial position, 𝜓 is the heading
+angle, and 𝛾 is the pitch angle.
+equivalent to the flight path angle and is constrained between a minimum and maximum angle, 𝛾 ∈ [𝛾min, 𝛾max]. The
+controls are constrained such that the path curvature 𝜌min is bounded [11]:
+𝜌min ≤
+1
+√︃
+𝑢2
+𝜓 cos2 𝛾 + 𝑢2𝛾
+.
+(2)
+2
+
+Let the vehicle’s configuration be denoted 𝒒 = (𝑥, 𝑦, 𝑧, 𝜓, 𝛾) ∈ 𝑄 where 𝑄 = R3 × S2 is the configuration space.
+An example 2D Dubins path (modified with a constant pitch angle to join two altitudes) and a 3D Dubins path that
+join 𝒒𝑖 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗, 𝜓 𝑗, 𝛾 𝑗) and 𝒒 𝑗 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗, 𝜓 𝑗, 𝛾 𝑗) are shown in Fig. 2. The modified 2D Dubins path uses a
+constant pitch angle 𝛾𝑐 that is computed from the change in altitude and planar displacement between the start and end
+configurations. The modified 2D Dubins path does not satisfy the required pitch angle at the start/end configurations and
+may violate pitch angle constraints along the path when the change in altitude is large relative to the planar displacement.
+Instead, a 3D Dubins path can join two configurations while limiting the pitch angle along the path to within the
+allowable bounds. The 3D Dubins paths are generated according to [10] by decomposing the 3D path into two decoupled
+2D Dubins paths. First, a 2D horizontal Dubins path is constructed in the 𝑥𝑦 plane to join the 2D Dubins configurations
+(𝑥𝑖, 𝑦𝑖, 𝜓𝑖) and (𝑥 𝑗, 𝑦 𝑗, 𝜓 𝑗) using a horizontal turn radius that is twice the minimum turn radius 𝜌h = 2𝜌min. Next, a 2D
+vertical path is constructed, with vertical plane turn radius 𝜌v that is found from [10]
+𝜌−2
+min = 𝜌−2
+h + 𝜌−2
+v
+,
+(3)
+to join the 2D Dubins configurations (𝑠𝑖, 𝑧𝑖, 𝛾𝑖) and (𝑠 𝑗, 𝑧 𝑗, 𝛾 𝑗) where 𝑠𝑖 and 𝑠 𝑗 are the initial and final arc-lengths
+along the Dubins path in the 𝑥𝑦 plane (where 𝑠𝑖 = 0). The turn radii, 𝜌h and 𝜌v, are iteratively varied while satisfying
+(3) to meet the acceptable pitch angle constraint while minimizing the path length as described in [10]. The length of a
+3D Dubins path between two configurations, 𝒒𝑖, 𝒒 𝑗 ∈ 𝑄 is denoted 𝐷(𝒒𝑖, 𝒒 𝑗) : 𝑄2 → R.
+Start
+End
+Modified 2D 
+Dubins Path
+3D Dubins Path
+Fig. 2
+An example 3D Dubins airplane path (green) [10] joining configurations 𝒒1 = (0, 0, 0, 𝜋
+6 , 0) and 𝒒2 =
+(0, 300 m, 400 m, 0, 0) is compared to a modified 2D Dubins path (red) that join the same pair of locations and
+heading angles. The modified 2D Dubins path is shorter (523 m compared to 1184 m) but violates the pitch angle
+constraint since a large altitude change is required over a relatively short distance. The paths are constructed
+with the parameters: 𝜌min = 40 m, 𝛾min = −𝜋/12, and 𝛾max = 𝜋/9.
+B. Environment
+The airplane operates in an urban environment that consists of a ground plane and a collection of 2.5-dimensional
+objects representing buildings or other structures. Let 𝑂 = {𝑂0, . . . , 𝑂𝑁𝑂−1} be the set of 𝑁𝑂 objects, where 𝑂𝑖 ⊂ R3
+for each 𝑖 ∈ {0, . . . , 𝑁𝑂 − 1}. The 𝑖th object is an extruded polygon 𝑂𝑖 = {(𝑥, 𝑦, 𝑧) ∈ R3 | (𝑥, 𝑦) ∈ 𝐴𝑖 and 𝑧 ∈ [0, ℎ𝑖]}
+where 𝐴𝑖 ⊂ 𝑅2 is the object’s footprint and ℎ𝑖 is the height of the object. The set of points along the boundary of 𝐴𝑖 is a
+simple two-dimensional polygon denoted 𝜕𝐴𝑖 whose shape is defined by an ordered set of points with a positive signed
+area. Points on the interior of 𝐴𝑖 belong to the set denoted int(𝐴𝑖). The polygonal areas of each object do not intersect
+int(𝐴𝑖) ∩ int(𝐴 𝑗) = ∅ for all 𝑖 ≠ 𝑗 with 𝑖, 𝑗 ∈ {0, . . . , 𝑁𝑂 − 1}. The height of the tallest object in 𝑂 is denoted ℎmax,
+and the airplane is constrained to fly in a feasible airspace
+𝐹 = 𝐷 × [𝑧min, 𝑧max] − 𝑂 ,
+(4)
+where 𝐷 ⊂ R2 is the planar region containing the polygonal objects, i.e., 𝐴𝑖 ⊂ 𝐷 for all 𝑖 ∈ {0, . . . , 𝑁𝑂 − 1}, 𝑧min and
+𝑧max > 𝑧min are the minimum and maximum operating altitudes of the airplane. The union of all the objects is subtracted
+from the rectangular volume 𝐷 × [𝑧min, 𝑧max] in (4). To ensure that 3D Dubins paths joining two configurations does not
+exceed the feasible airspace or encounter obstacles, the feasible airspace and set of objects can be artificially contracted
+and inflated, respectively. This work assumes that the minimum altitude 𝑧min is constrained to be above the tallest
+3
+
+building, 𝑧min > ℎmax + 2𝜌min, such that the airplane’s feasible airspace is free of objects and there is enough vertical
+space to maneuver without collision.
+C. Target Visibility Volumes
+The airplane is assumed to be equipped with a gimbaled camera and is tasked with inspecting a set of 𝑀 targets
+located at the points 𝑃 = { 𝒑0, . . . , 𝒑𝑀−1}. Each target 𝒑 = (𝑝𝑥, 𝑝𝑦, 𝑝𝑧) ∈ 𝑃 is located in an unobstructed area of the
+ground plane or on the exposed surface of an object. That is, each target has planar location (𝑝𝑥, 𝑝𝑦) ∈ 𝐷 and altitude
+𝑝𝑧 satisfying the following cases: (i) if (𝑝𝑥, 𝑝𝑦) ∩ 𝐴𝑖 = ∅ for all 𝑖 ∈ {0, . . . , 𝑁𝑂 − 1} then the target is on the ground
+plane with 𝑝𝑧 = 0, (ii) if 𝑝𝑦 ∩ 𝜕𝐴𝑖 ≠ ∅ for some 𝑖 ∈ {0, . . . , 𝑁𝑂 − 1} then the target is located on the vertical wall of the
+𝑖th object and 𝑝𝑧 ∈ [0, ℎ𝑖], or (iii) if 𝑝𝑦 ∩ int(𝐴𝑖) ≠ ∅ then 𝑝𝑧 = ℎ𝑖 such that the target is on top of the 𝑖th object. For
+each target, a target visibility volume 𝑉𝑖 is defined as the set of points 𝒈 ∈ R3 that have a direct line-of-sight to the target
+(i.e., not obscured by buildings). Let
+𝐿(𝜏; 𝒈, 𝒑) = ( 𝒑 − 𝒈)𝜏 + 𝒈 for 𝜏 ∈ [0, 1]
+(5)
+denote a line segment that joints two points 𝒈, 𝒑 ∈ R3 where 𝜏 is a normalized arc-length. The visibility volume for a
+target located at 𝒑 = (𝑝𝑥, 𝑝𝑦, 𝑝𝑧) is the subset of the feasible airspace that is within direct line-of-sight to the target,
+within a maximum range 𝑑max relative to the target, and at least a distance ℎview above the target:
+𝑉( 𝒑; 𝐹, 𝑂, 𝑑max, ℎview) = {𝒈 = (𝑔𝑥, 𝑔𝑦, 𝑔𝑧) ∈ 𝐹 such that || 𝒑 − 𝒈|| ≤ 𝑑max, ℎview + 𝑝𝑧 ≤ 𝑔𝑧 and
+𝐿(𝜏; 𝒈, 𝒑) ∩ 𝑂 𝑗 = ∅ for all 𝜏 ∈ [0, 1] and 𝑗 ∈ {0, . . . , 𝑁𝑂 − 1}} .
+(6)
+For brevity, visibility volumes (6) are henceforth denoted 𝑉( 𝒑). The maximum range 𝑑max constraint models minimum
+image resolution requirements. The minimum height-above-target ℎview < 𝑑max constraint ensures images are captured
+with sufficient surrounding context (e.g., the point target may actually represent an extended body that should be
+contained in the image) or to reduce gimbal pointing speed and precision requirements. For the problem to be well
+posed, there should always exist at least one valid viewing point above each target. This condition may be satisfied by
+the following parameter constraints:
+𝑧min ≤ ℎview + ℎmax ≤ 𝑧max ,
+(7)
+𝑧min ≤ 𝑑max ,
+(8)
+2𝑑max < || 𝒑𝑖 − 𝒑 𝑗||
+for all 𝒑𝑖, 𝒑𝑖 ∈ 𝑃 with 𝒑𝑖 ≠ 𝒑 𝑗 .
+(9)
+If a target is located on top of the highest object, then constraint (7) ensures that a viewing point exists that is below the
+maximum feasible altitude and above the minimum feasible altitude. For targets that are located on the ground plane,
+constraint (8) ensures that the sensor range is sufficiently large to view the target from the minimum feasible altitude.
+Lastly, constraint (9) is a simplifying assumption that guarantees targets are spaced sufficiently far apart such that their
+visibility volumes do not intersect 𝑉( 𝒑𝑖) ∩ 𝑉( 𝒑 𝑗) = ∅ for all 𝑖, 𝑗 ∈ {0, . . . , 𝑀 − 1} with 𝑖 ≠ 𝑗.
+D. View-planning Problem Statement
+Let 𝐵(𝒒) be a mapping from a configuration 𝒒 = (𝑥, 𝑦, 𝑧, 𝜓, 𝛾) ∈ 𝑄 to an integer in the set {0, . . . , 𝑀 − 1} that
+identifies the visibility volume corresponding to 𝒒, i.e., the integer 𝐵(𝒒) corresponds to the target 𝒑𝐵(𝒒) ∈ 𝑃 for which
+(𝑥, 𝑦, 𝑧) ∈ 𝑉( 𝒑𝐵(𝒒)). If 𝒒 is not contained in any visibility volume then 𝐵(𝒒) = ∅. The optimization problem is to find
+the sequence of vehicle configurations 𝒒0, . . . , 𝒒𝑀−1 that
+minimize
+𝑀−1
+∑︁
+𝑖=0
+𝐷(𝒒𝑖, 𝒒𝑖+1) + 𝐷(𝒒𝑀−1, 𝒒0) ,
+(10)
+subject to
+𝐵(𝒒𝑖) ≠ 𝐵(𝒒 𝑗),
+for all 𝑖, 𝑗 ∈ {0, . . . , 𝑀 − 1} with 𝑖 ≠ 𝑗 ,
+(11)
+𝐵(𝒒0) ∪ · · · ∪ 𝐵(𝒒𝑀−1) = {0, . . . , 𝑀 − 1} ,
+(12)
+where the cost function (10) is the total length of the 3D Dubins paths in the tour, the constraint (11 ensures that each
+vehicle configuration lies within a unique visibility volume and the constraint 12) ensures that all visibility volumes
+are visited. The view planning problem (10)–(12) is a mixed continuous/combinatorial optimization problem with a
+nonlinear cost function and constraints. Since the vehicle travels at a constant speed the minimum-length tour is also the
+minimum-time tour.
+4
+
+III. 2D Algorithms
+In this section, a target visibility volume mesh approximation is described (Sec. III.A) followed by a description of
+two-dimensional algorithms (Sec. III.B and Sec. III.C) that solve the view planning problem (10)–(12). The algorithms
+discussed here include (i) traveling directly over each target (i.e., formulating a Dubins traveling salesperson problem
+(DTSP) [12]), and (ii) the DTSP with neighborhoods (DTSPN) to visit one point in a set of visibility polygons
+corresponding to the targets [6] that is modified to use an optimized altitude for defining the visibility polygons
+A. Target Visibility Volume Approximation
+Volumes in 3D are commonly approximated by a triangular mesh [13]. While many prior works on the DTSP
+have assumed simplified 3D geometries (e.g., spheres, cylinders), we propose to use triangular meshes since they can
+represent arbitrary geometries. The 𝑖th target visibility volume 𝑉𝑖 is approximated with 𝑁𝐹 triangular mesh elements
+resulting in the mesh ˆ𝑉𝑖. Let | ˆ𝑉𝑖| denote the total number of mesh elements. The 𝑗th mesh element in ˆ𝑉𝑖 is defined
+as a set of vectors ˆ𝑉𝑖 𝑗 = {𝒄0
+𝑖 𝑗, 𝒄1
+𝑖 𝑗, 𝒄2
+𝑖 𝑗, 𝒏𝑖 𝑗} where the vectors 𝒄0
+𝑖 𝑗, 𝒄1
+𝑖 𝑗, 𝒄2
+𝑖 𝑗 ∈ R3 are the positions of the vertices of a
+triangular mesh element, and 𝒏𝑖 𝑗 ∈ R3 is an outward pointing normal vector, as illustrated in Fig. 3. The mesh-based
+target visibility volumes ˆ𝑉𝑖 are computed using the painter’s algorithm [14]. A sphere centered on each target is
+decomposed into six mutually perpendicular views, and each view looks out from the target point location with a
+90-degree field-of-view thereby covering one of the six sides of a cube enclosing the point. OpenGL [15] and a special
+version of the geometric depth map, i.e., inverse depth, is used to capture the depth of scene objects in the direction of
+each view. After calculating the depth values, those that are less than or equal to 𝑑max are tessellated into a preliminary
+3D visibility volume mesh. This mesh is genus-0 [13], i.e., a deformation of the sphere, and is also a manifold surface
+amenable to constructive solid geometry (CSG) Boolean operations. Next, the mesh is intersected with the feasible
+airspace 𝐹 and the minimum viewing distance constraint ℎmin is imposed using CSG Boolean intersection operations.
+To reduce the number of vertices in the resulting mesh a decimation procedure is applied [13].
+Fig. 3
+Example target visibility region with a mesh element defined by three vertices 𝒄0, 𝒄1, 𝒄2 and outward
+pointing normal vector 𝒏.
+B. Baseline Algorithm: Dubins Traveling Salesperson Problem (DTSP)
+The DTSP is the problem of finding the shortest planar tour that visits all points in a graph once using points that are
+connected with 2D Dubins paths. Since the objects considered here are extruded polygons, there are no features that can
+block viewing targets from above (e.g., bridges or tunnels are not admissible). Consequently, the view planning problem
+(10)–(12) can be solved with the DTSP by flying at a fixed altitude directly over each target. All feasible altitudes (i.e.,
+that are common to all visibility volumes) lead to identical cost tours. To account for the different possible heading
+angles at each overhead location the heading-angle-discretized DTSP is adopted [16]. An example solution is shown in
+Fig. (4a).
+5
+
+C. Optimized Altitude DTSP with Neighborhoods (DTSPN)
+A more sophisticated approach developed by Obermeyer et al. [6] considers the fixed altitude slices of the target
+visibility volumes (i.e., planar visibility polygons). Vehicle configurations in each visibility polygon are sampled and a
+DTSPN [5, 6] is formulated to visit one configuration in each visibility polygon. In [6], two sampling algorithms were
+proposed: entry pose sampling—wherein samples are made along the edge of the polygon with heading angles that are
+tangent or inward pointing (Fig. 4b)—and interior pose sampling—wherein samples are placed uniformly in a grid on
+the interior of the visibility polygons with uniformly sampled heading angles. In [6], entry pose sampling gave lower
+cost solutions than interior pose sampling. Thus, the entry pose sampling method is adopted here. The constraints of
+the view planning problem (7)–(9) allow for visibility volumes to occupy disjoint segments of altitude. That is, there
+may not exist an altitude 𝑧∗ ∈ [𝑧min, 𝑧max] that is common to all visibility volumes. While this does not pose an issue
+for some of the 3D algorithms proposed later, these cases cannot be solved by the 2D (constant-altitude) algorithms
+described here. However, introducing the additional constraint
+ℎmax + ℎview ≤ 𝑑max
+(13)
+ensures that the visibility volumes for a target located on the ground plane and for a target located atop the highest
+object have at least one common altitude at 𝑧∗ = 𝑑max. In general, there is a range of admissible altitudes 𝑧∗ that may
+be chosen. The choice of altitude impacts the 2D DTSPN algorithm since visibility polygons change in shape and
+size as the altitude varies. Intuitively, larger polygons are preferred over smaller ones since this increases the set of
+candidate configurations. This work proposes to identify an optimal working altitude for the 2D algorithm as follows.
+First, 𝑛slice polygons are generated from each visibility volume mesh (i.e., for all targets) using the method described
+in [17]. Let P = polygonFromMesh( ˆ𝑉, 𝑧) denote the polygon that results from slicing mesh ˆ𝑉 at altitude 𝑧 and let
+polygonArea(P) denote the corresponding area. The optimal altitude 𝑧∗ is chosen as the one that maximizes the sum
+of polygonal areas across all visibility polygons:
+𝑧∗ = argmax
+𝑧∈𝑍
+𝑀−1
+∑︁
+𝑖=0
+polygonArea(polygonFromMesh( ˆ𝑉𝑖, 𝑧))
+(14)
+where 𝑍 = {𝑧0, . . . , 𝑧𝑛slice−1} is a set of altitudes at which the polygons are computed. Note that all altitudes 𝑧 ∈ 𝑍 are
+constrained such that 𝑧min ≤ 𝑧 ≤ 𝑧max. Also, note that 𝑧0 and 𝑧slice−1 are not 𝑧min and 𝑧max respectively, instead 𝑧0 and
+𝑧slice−1 are slightly offset into the body to avoid floating point error. The selection of 𝑧∗ is visualized in Fig 5a.
+(a) 2D Dubins traveling salesperson problem (DTSP)
+(b) 2D DTSP with neighborhoods (DTSPN)
+Fig. 4
+Example solutions to the view-planning problem using 2D constant-altitude algorithms. The green
+regions are visibility polygons for a chosen altitude 𝑧∗ while the black arrows represent the heading angle at
+sampling points. The multicolored line is the solution path of the TSPs with increasing path length represented
+by color changes from red to purple. The DTSP (a) is solved with entry pose sampling using eight headings
+samples directly over the targets, while the 2D DTSPN (b) uses eight sample locations around the perimeter of
+the polygon with four heading angles that are tangent or point into the corresponding visibility polygons.
+IV. 3D Algorithms
+Inspection tours that admit three-dimensional maneuvering can potentially lead to path length reductions when
+compared to two-dimensional (constant-altitude) tours. The solution techniques in this work all use a transformation
+6
+
+approach to solve the (2D or 3D) DTSPN according to the following steps: compute the approximation of the
+target visibility volume (Sec. III.A), sample the visibility volumes to create graph vertices corresponding to vehicle
+configurations, calculate edge costs between vertices using the 3D Dubins path planning algorithm, and solve for a
+DTSPN tour. Section IV.A details three algorithms to sample the target visibility volumes: random face sampling, 3D
+edge sampling, and global weighted face. Section IV.B.2 then introduces a heuristic approach that improves the edge
+cost computation time using a modified Euclidean distance edge cost and a geometric approach to assign heading and
+pitch angles at each configuration in the tour.
+(a) Optimized altitude with entry pose sampling
+(b) Random face sampling
+(c) 3D edge sampling
+(d) Global weighted face sampling
+Fig. 5
+Visualizations of the four different sampling algorithms: optimized altitude with entry pose sampling,
+random face sampling, 3D edge sampling, and global weighted face sampling. Two-dimensional representations
+of the visibility volumes are in gray, altitude slices are orange lines, and sampled configurations are blue circular
+markers.
+A. Sampling Strategies
+1. Random Face Sampling
+The random face sampling algorithm extends the 2D entry pose strategy from [6] to sample 3D vehicle configurations
+across the surface of the target visibility volume with a uniform distribution. The approach is detailed in Algorithm 1
+and visualized in Fig. 5b. The algorithm randomly finds 𝑛pts three-dimensional points on the faces of each triangular
+mesh in the set of triangular meshes ˆ𝑉0:𝑀−1 = { ˆ𝑉1, . . . , ˆ𝑉𝑀−1} and assigns to each point a set of configurations with
+𝑛𝜓 and 𝑛𝛾 unique heading and pitch angles, respectively. The sampling method returns a total of 𝑛pts𝑛𝜓𝑛𝛾 vehicle
+configurations per visibility volume. First, the set of configurations Q is initialized as an empty set, and the area of
+each face in the mesh is calculated (lines 3-6). The area of each triangular face element, 𝑎𝑖 𝑗, is calculated by the
+elementArea function using
+𝑎𝑖 𝑗 = elementArea( ˆ𝑉𝑖 𝑗) = 1
+2 ||(𝒄0
+𝑖 𝑗 − 𝒄1
+𝑖 𝑗) × (𝒄0
+𝑖 𝑗 − 𝒄2
+𝑖 𝑗)|| ,
+(15)
+where × is the vector cross product and 𝒄0
+𝑖 𝑗, 𝒄1
+𝑖 𝑗, and 𝒄2
+𝑖 𝑗 are the three vertices contained in the triangular face element
+ˆ𝑉𝑖 𝑗. Next, the proportion of each face area to the total surface area of the mesh ˆ𝑉𝑖 is calculated using element-wise
+division (line 7). The randomSetOfIndices function identifies 𝑛pts random faces by sampling faces with probability
+in proportion to the weights 𝒘𝐹 (line 8). The use of the proportional surface area during the random selection process
+gives every point on the surface of the target visibility region an equal chance of being selected. For each selected
+triangular face, a point on the face is randomly selected using a Barycentric coordinate system [18] (lines 10-12). The
+Barycentric coordinate system allows for the mapping of two random numbers 𝑟0 and 𝑟1 sampled uniformly from the
+interval [0, 1] onto a triangle, embedded in R3, with the weighted sum of its vertices [18]. The random numbers 𝑟0 and
+𝑟1 are first sampled (line 11) and then a position on the chosen triangular face is determined (line 12). For each position,
+𝑛𝜓 heading angles sampled uniformly between 0 and 2𝜋 as well as 𝑛𝛾 pitch angles between 𝛾min and 𝛾max are sampled
+uniformly then added to the set of vehicle configurations. The runtime of the algorithm is dominated by the nested for
+loops on lines 2 and 4 running 𝑀| ˆ𝑉|max times—where | ˆ𝑉|max = max𝑖∈{0,1,...,𝑀−1}(| ˆ𝑉𝑖|) is the maximum number of
+7
+
+faces in a single mesh—and the collections of nested loops on lines 13-14 which run 𝑀𝑛pts𝑛𝜓𝑛𝛾. When the number of
+mesh faces in a visibility volume is greater than the number of samples collected | ˆ𝑉|max > 𝑛pts𝑛𝜓𝑛𝛾 the runtime is
+𝑂(𝑀| ˆ𝑉|max).
+Algorithm 1 Random Face Sampling
+function: RandomFaceSampling( ˆ𝑉0:𝑀−1, 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝛾max, 𝛾min)
+input: target visibility volume mesh ˆ𝑉0:𝑀−1, number of points to sample 𝑛pts, number of heading angles 𝑛𝜓, number of
+pitch angles 𝑛𝛾, max pitch angle 𝛾max, min pitch angle 𝛾min
+output: a set of vehicle configurations for each target Q
+1: Q ← ∅
+2: for ˆ𝑉𝑖 ∈ ˆ𝑉0:𝑀−1 do
+3:
+Q𝑖 ← ∅, 𝒂𝑖 ← ∅
+4:
+for ˆ𝑉𝑖 𝑗 ∈ ˆ𝑉𝑖 do
+5:
+𝒂𝑖 ← 𝒂𝑖 ∪ elementArea( ˆ𝑉𝑖 𝑗)
+6:
+end for
+7:
+𝒘𝐹 = 𝒂𝑖/�|𝒂𝑖 |−1
+𝑗=0
+𝑎𝑖 𝑗
+8:
+𝐼 ← randomSetOfIndices(𝑛pts, 𝒘𝐹)
+9:
+for 𝑖 ∈ 𝐼 do
+10:
+𝒄0
+𝑖 𝑗, 𝒄1
+𝑖 𝑗, 𝒄2
+𝑖 𝑗 ← getVertices( ˆ𝑉𝑖 𝑗)
+11:
+𝑟0 ∼ U[0,1], 𝑟1 ∼ U[0,1]
+12:
+𝒔 ← 𝒄0
+𝑖 𝑗 (1 − √𝑟0) + 𝒄1
+𝑖 𝑗
+√𝑟0(1 − 𝑟1) + 𝒄2
+𝑖 𝑗
+√𝑟0𝑟1
+13:
+for 𝑗 ∈ {0, . . . , 𝑛𝜓 − 1} do
+14:
+for 𝑘 ∈ {0, . . . , 𝑛𝛾 − 1} do
+15:
+Q𝑖 ← Q𝑖 ∪ �𝒔, 2𝑗𝜋/𝑛𝜓, 𝛾min + 𝑘(𝛾max − 𝛾min)/max(𝑛𝛾 − 1, 1)�
+16:
+end for
+17:
+end for
+18:
+end for
+19:
+Q ← Q ∪ Q𝑖
+20: end for
+2. 3D Edge Sampling
+The second sampling strategy proposed is 3D edge sampling wherein the 2D entry pose strategy from [6] is extended
+to sample 3D vehicle configurations across the lowest feasible altitude. For the visibility volume shapes studied here this
+is also the altitude where the cross-sectional area is largest for each shape. The 3D edge sampling algorithm, detailed
+in Algorithm 2 and visualized in Fig. 5c, finds 𝑛pts three-dimensional points on the polygon created by slicing the
+triangular mesh along the lowest feasible altitude and distributing points uniformly along the perimeters. The algorithm
+then assigns a set of configurations to each point with 𝑛𝜓 and 𝑛𝛾 unique heading and pitch angles, respectively. The
+sampling method returns a total of 𝑛pts𝑛𝜓𝑛𝛾 vehicle configurations per visibility volume. First, the set of configurations
+Q is initialized as an empty set (line 1). Then, for each visibility volume a subset of points contained in that volume
+is initialized (line 3). Next, the 𝑧 minimum altitude for the triangular mesh is found by finding the minimum height
+coordinate in the set of vertices in the mesh ˆ𝑉 𝑧
+𝑖 (line 4). After, the polygonFromMesh algorithm takes the triangular
+mesh and the 𝑧min altitude and returns a polygonal slice of the mesh (line 5). A set of points, 𝝀 ∈ R2, placed uniformly
+along the edge of the polygon is found using the uniformPerimeterPoints which takes a polygon and the number
+of points desired as arguments (line 6). Note that lines 4–6 can be modified to produce samples at multiple altitude
+slices if desired. Next, the algorithm iterates through each sampled point and assigns heading and pitch angles. To
+ensure inward-pointing heading angles, the direction of the line segment containing the sample point is found using the
+tangentAngle function (line 8). The points in the polygon defined by polygonFromMesh have a positive signed area.
+Thus, the inward-pointing heading angles are the angles from [0, 𝜋] measured counter-clockwise from the tangent angle.
+For each position, 𝑛𝜓 heading angles between 𝜓𝑞 and 𝜓𝑞 + 𝜋 and 𝑛𝛾 pitch angles between 𝛾min and 𝛾max are sampled
+uniformly and returned as part of the vehicle configurations (lines 9-15). To achieve 𝑛 equally spaced angle samples,
+including the minimum and maximum angle, the range is divided into 𝑛 − 1 sub-sections. The max function, on lines 10
+and 12, ensures the range is never divided by zero (the case where 𝑛𝛾 or 𝑛𝜓 is one). The runtime complexity is dominated
+8
+
+by the for loops on lines 2-18 which have a worst-case running time complexity of 𝑂(𝑀𝑘), where 𝑘 = | ˆ𝑉|2
+max + 𝑛pts𝑛𝜓𝑛𝛾.
+| ˆ𝑉|2
+max is the runtime of the polygonFromMesh algorithm while 𝑛pts𝑛𝜓𝑛𝛾 is the runtimes for the nested for loops (lines
+9-15). For a typical choice of parameters, the number of faces in the target visibility volume mesh squared is greater
+than the total number of configurations returned, | ˆ𝑉|2
+max > 𝑛pts𝑛𝛾𝑛𝜓 and the overall time-complexity is 𝑂(𝑀| ˆ𝑉|2
+max).
+Algorithm 2 3D Edge Sampling
+function: 3DEdgeSampling( ˆ𝑉0:𝑀−1, 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝛾max, 𝛾min)
+input: target visibility volume mesh ˆ𝑉0:𝑀−1, number of points to sample 𝑛pts, number of heading angles 𝑛𝜓, number of
+pitch angles 𝑛𝛾, max pitch angle 𝛾max, min pitch angle 𝛾min
+output: a set of vehicle configurations for each target Q
+1: Q ← ∅
+2: for ˆ𝑉𝑖 ∈ ˆ𝑉0:𝑀−1 do
+3:
+Q𝑖 ← ∅
+4:
+𝑧min ← min( ˆ𝑉 𝑧
+𝑖 )
+5:
+P ← polygonFromMesh(𝑧min, ˆ𝑉𝑖)
+6:
+{𝝀0, . . . , 𝝀𝑛pts−1} ← uniformPerimeterPoints(P, 𝑛pts)
+7:
+for 𝑚 ∈ {0, . . . , 𝑛pts − 1} do
+8:
+𝜓𝑞 ← tangentAngle(𝝀𝑚, P)
+9:
+for 𝑗 ∈ {0, . . . , 𝑛𝜓 − 1} do
+10:
+𝜓 ← 𝜓𝑞 + 𝑗𝜋/max(𝑛𝜓 − 1, 1)
+11:
+for 𝑘 ∈ {0, . . . , 𝑛𝛾 − 1} do
+12:
+𝛾 ← 𝛾min + 𝑘(𝛾max − 𝛾min)/max(𝑛𝛾 − 1, 1)
+13:
+Q𝑖 ← Q𝑖 ∪ (𝝀𝑚, 𝑧min, 𝜓, 𝛾)
+14:
+end for
+15:
+end for
+16:
+end for
+17:
+Q ← Q𝑖 ∪ Q
+18: end for
+3. Global Weighted Face Sampling
+The third proposed sampling strategy is global weighted face sampling. Rather than sampling the visibility volumes
+at the lowest altitude, all target visibility volumes are sampled along a common set of altitude planes and the number of
+samples allocated to each plane is determined by the cross-sectional perimeter distribution of each altitude summed
+across all target visibility volumes. This approach places more samples at altitudes common to all targets that, on
+average, also have large cross-sectional areas. This sampling method is detailed in Algorithm 3 and visualized in Fig. 5d.
+The algorithm takes a set of target visibility meshes ˆ𝑉0:𝑀−1 and returns a set of vehicle configurations Q for each mesh
+given the parameters 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝑛slice, 𝛾max, and 𝛾min where 𝑛slice ≥ 2 is the number of altitude slices to consider. Let
+ˆ𝑉 𝑧
+0:𝑀−1 denote the set of all 𝑧 heights for every vertex contained across the 𝑀 meshes ˆ𝑉0:𝑀−1. First, the global minimum,
+the global maximum altitude, and the slicing altitude step size are found (lines 1-2). Then a vector 𝝁 is initialized
+with zeros, denoted as 0𝑛slice×1 (line 3), and later stores the total perimeter summed across all visibility polygons at the
+corresponding altitude slice. The target visibility volumes are sliced into polygons with fixed altitude (i.e. parallel to
+the 𝑥𝑦 plane) using the polygonFromMesh function, lines 4-9. The lowest 𝑧 plane is the visibility volumes’ global
+minimum 𝑧 height (𝜁min) and the highest 𝑧 plane is the visibility volumes’ global maximum 𝑧 height (𝜁max), line 1. The
+nominal set of altitude planes is then 𝑍 = {𝑧0, . . . , 𝑧𝑛slice−1} where 𝑧0 = 𝜁min, 𝑧𝑛slice−1 = 𝜁max and 𝑧𝑖+1 − 𝑧𝑖 = 𝐿. At each
+plane 𝑧 ∈ 𝑍, polygons are created from the target visibility volume and the polygons’ perimeters are accumulated, line 7.
+The sample points in each 𝑧 plane are then distributed in proportion to the accumulated perimeters, lines 11-29. The
+function iteratePerimeters takes six arguments: the mesh to iterate across, a perimeter distribution, the minimum
+altitude, the maximum altitude, the step size, and the total number of sample points. It returns a variable number of
+𝑛𝑧 ≤ 𝑛slice elements where each element is a pair consisting of a 𝑧𝑟 altitude and the number of points to sample at that
+altitude, 𝑛𝑟. An altitude slice 𝑧𝑟 is either an element of 𝑍 and/or an altitude located at the top or bottom of each visibility
+volume. At each altitude 𝑧𝑟 the corresponding value of 𝝁 is determined (or interpolated, in the special case that 𝑧𝑟 ∉ 𝑍)
+and the 𝑛pts are distributed to each 𝑧𝑟 in proportion to the result. In the event that no slices intersect the visibility mesh
+9
+
+then 𝑛𝑧 = 2 and the heights 𝑧𝑟 are returned corresponding to the top and bottom of the target visibility volume. Next,
+samples 𝝀 ∈ {𝝀0, ..., 𝝀𝑛𝑟−1} are placed uniformly around the perimeter of each polygon created by the intersection of
+the 𝑧𝑟 planes and the target visibility volume with the function uniformPerimeterPoints, line 16. The heading and
+pitch angles are sampled in the same way as entry pose sampling [6], pointing tangent or inward with respect to the
+polygon. The angle tangent to each point 𝝀 on the perimeter of the polygon is found with the tangentAngle function.
+The pitch angles are uniformly sampled within the pitch angle constraints. The runtime complexity is dominated by the
+for loops on lines 11-29 which have a worst-case running time complexity of 𝑂(𝑀𝑛slice𝑘), where 𝑘 = | ˆ𝑉|2
+max + 𝑛𝑟𝑛𝛾𝑛𝜓.
+For a typical choice of parameters, the number of faces in the target visibility volume mesh squared is greater than the
+total number of configurations returned, | ˆ𝑉|2
+max > 𝑛𝑟𝑛𝛾𝑛𝜓 and the overall time-complexity is 𝑂(𝑀𝑛slice| ˆ𝑉|2
+max).
+Algorithm 3 Global Weighted Face
+function: GlobalWeightedFace( ˆ𝑉0:𝑀−1, 𝑛pts, 𝑛𝜓, 𝑛𝛾, 𝑛slice, 𝛾max, 𝛾min)
+input: set of triangular meshes ˆ𝑉0:𝑀−1, number of points to sample 𝑛pts, number of heading angles 𝑛𝜓, number of pitch
+angles 𝑛𝛾, number of altitude slices 𝑛slice, max pitch angle 𝛾max, min pitch angle 𝛾min
+output: a set of vehicle configurations for each target Q
+1: 𝜁max ← max( ˆ𝑉 𝑧
+0:𝑀−1), 𝜁min ← min( ˆ𝑉 𝑧
+0:𝑀−1)
+2: 𝐿 ← (𝜁max − 𝜁min)/(𝑛slice − 1)
+3: 𝝁 ← 0𝑛slice×1
+4: for 𝑖 ∈ {0, . . . , 𝑛slice − 1} do
+5:
+for ˆ𝑉𝑖 ∈ ˆ𝑉0:𝑀−1 do
+6:
+P ← polygonFromMesh(𝜁min + 𝐿𝑖, ˆ𝑉𝑖)
+7:
+𝜇𝑖 ← 𝜇𝑖 + perimeter(P)
+8:
+end for
+9: end for
+10: Q ← ∅
+11: for ˆ𝑉𝑖 ∈ ˆ𝑉1:𝑀 do
+12:
+(𝑧𝑟, 𝑛𝑟)𝑛𝑧−1
+𝑟=0
+← iteratePerimeters( ˆ𝑉𝑖, 𝝁, 𝜁min, 𝜁max, 𝐿, 𝑛pts)
+13:
+Q𝑖 ← ∅
+14:
+for 𝑟 ∈ {0, . . . , 𝑛𝑧 − 1} do
+15:
+P ← polygonFromMesh(𝑧𝑟, ˆ𝑉𝑖)
+16:
+{𝝀0, . . . , 𝝀𝑛𝑟−1} ← uniformPerimeterPoints(P, 𝑛𝑟)
+17:
+for 𝑚 ∈ {0, . . . , 𝑛𝑟 − 1} do
+18:
+𝜓𝑞 ← tangentAngle(𝝀𝑚, ˆ𝑉𝑖)
+19:
+for 𝑗 ∈ {0, . . . , 𝑛𝜓 − 1} do
+20:
+for 𝑘 ∈ {0, . . . , 𝑛𝛾 − 1} do
+21:
+𝜓 ← 𝜓𝑞 + 𝑘𝜋/max(𝑛𝜓 − 1, 1)
+22:
+𝛾 ← 𝛾min + 𝑘(𝛾max − 𝛾min)/max(𝑛𝛾 − 1, 1)
+23:
+Q𝑖 ← Q𝑖 ∪ (𝝀𝑚, 𝑧𝑟, 𝜓, 𝛾)
+24:
+end for
+25:
+end for
+26:
+end for
+27:
+end for
+28:
+Q ← Q𝑖 ∪ Q
+29: end for
+B. Proposed Heuristics
+1. Modified Euclidean Distance Traveling Salesperson Problem with Neighborhoods (METSPN)
+A bottleneck in the 3D DTSPN algorithms is the computation of the edge costs that require solving for a 3D
+Dubins path between two configurations 𝒒𝑖 = (𝑥𝑖, 𝑦𝑖, 𝑧𝑖, 𝜓𝑖, 𝛾𝑖) and 𝒒 𝑗 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗, 𝜓 𝑗, 𝛾 𝑗). Since the Dubins path is
+asymmetric the corresponding edge cost must be computed for each direction. Here, we propose an approximation to
+10
+
+this edge cost
+ˆ𝐷(𝒒𝑖, 𝒒 𝑗) = max
+�
+|𝛿𝑧|
+sin 𝛾limit
+, ∥𝒔𝑖 − 𝒔 𝑗 ∥2
+�
+,
+(16)
+where 𝛿𝑧 = 𝑧 𝑗 − 𝑧𝑖, 𝛾limit = 𝛾max if 𝛿𝑧 > 0 and 𝛾limit = 𝛾min otherwise, 𝒔𝑖 = (𝑥𝑖, 𝑦𝑖, 𝑧𝑖) and 𝒔 𝑗 = (𝑥 𝑗, 𝑦 𝑗, 𝑧 𝑗). The
+calculation is visualized in Fig. 6. The distance (16) is a lower bound on the actual 3D Dubins path length, i.e.,
+Fig. 6
+Visualization of the modified Euclidean distance. The Euclidean distance shown in blue with a pitch
+angle 𝛾 > 𝛾limit is modified by extending the distance traveled in the 𝑥𝑦 plane resulting in the red line with pitch
+angle 𝛾limit. ˆ𝐷𝑥𝑦 refers to the length of the Dubins path projected onto the 𝑥𝑦 plane.
+ˆ𝐷(𝒒0, 𝒒1) ≤ ���(𝒒0, 𝒒1), and is significantly faster to compute than solving for the Dubins path. Using this edge cost
+leads to a variant of the DTSPN we refer to as the modified Euclidean distance traveling salesperson problem (METSPN).
+Solving the METSPN gives a tour of 3D locations to visit. Once a tour is found for the METSPN it is converted into a
+feasible sequence of Dubins paths by assigning heading and pitch angles as follows.
+2. Bisecting Angle Approximation
+To assign heading and pitch angles a heuristic is adopted that extends the mean angle algorithm developed in [19] to
+three dimensions. The approach is summarized in Algorithm 4. The proposed bisecting angle approximation takes
+as parameters: 𝑽 a 𝑀 × 3 matrix corresponding to the sequence of vertices in the METSPN tour and the problem
+parameters: 𝜌min, 𝛾min, and 𝛾max. The algorithm returns a set of vehicle configurations Q at each point in 𝑽 with
+heading and pitch angles defined as the angle bisector of each consecutive triplet of vertices (for points spaced far apart)
+or as a straight segment (for points spaced close together).
+To obtain the angle bisector at each vertex, calculate vectors from the preceding vertex 𝒖 = 𝑽𝑖 − 𝑽𝑖−1 = (𝑢𝑥, 𝑢𝑦, 𝑢𝑧)
+and to the following vertex 𝒘 = 𝑽𝑖+1 − 𝑽𝑖 = (𝑤𝑥, 𝑤𝑦, 𝑤𝑧) (line 3). The vector 𝒃 = 𝒘 + 𝒖 = (𝑏𝑥, 𝑏𝑦, 𝑏𝑧) determines the
+heading angle 𝜓 in the 𝑥𝑦 plane computed with the four-quadrant arctangent function (line 4). A visualization of the
+calculation can be seen in Fig. 7. The circular indexing of 𝑽, a 𝑀 by 3 matrix, allows for the index −1 to refer to the last
+Fig. 7
+The notation used to determine the bisector vector for a triplet of three points: 𝑽𝑖−1, 𝑽𝑖, 𝑽𝑖+1. The
+orientation of the vectors 𝒖 = 𝑽𝑖 − 𝑽𝑖−1 and 𝒘 = 𝑽𝑖+1 − 𝑽𝑖 are summed and normalized resulting in the vector
+𝒗. The heading angle 𝜓 is the component of 𝒃 in the 𝑥𝑦 plane while the pitch angle 𝛾 is measured from the 𝑥𝑦
+plane.
+column of 𝑽 and the index 𝑛 to refer to the first element of 𝑽. The pitch angle bisector is the angle between the vector 𝒃
+11
+
+and the 𝑥𝑦 plane (line 5). The resulting angle is saturated to be within the pitch angle bounds on line 6. If vertices are
+close together then curve-curve-curve (CCC) Dubins paths may be created. This should be avoided because the cost of
+(CCC) Dubins paths is much greater than the Euclidean distance. The likelihood of CCC paths occurring is reduced by
+setting the heading and pitch in the direction of the line between two vertices. If the distance between two vertices is
+small (less than the long path case in [20]), then heading and pitch angles are aligned with the while loop on lines 10-21.
+To align the headings of two configurations, the vector between the internal coordinates is found. The angle of this
+vector, 𝒘, about the 𝑧 axis is used as the heading angle. Then, the angle between the 𝑥𝑦 plane and the vector 𝒘 is found
+and saturated between 𝛾min and 𝛾max to set the pitch angle. Inside the loop, the index is advanced once but it is also
+advanced a second time if the current vertex and the next vertex are within 4𝜌min units of each other (worst case for
+the long path case [20]). The second index advance is required to pass over the next configuration because it was just
+modified.
+Algorithm 4 Bisect Angle Approximation
+function: BisectAngleApprox(𝑽, 𝜌min, 𝛾min, 𝛾max)
+input: 𝑽 is a 𝑛 by 3 matrix of vertices that solve the METSPN, minimum turn radius 𝜌min, minimum pitch angle 𝛾min,
+maximum pitch angle 𝛾max
+output: set of configurations solving a DTSP Q
+1: Q ← ∅
+2: for 𝑖 ∈ {0, 1, 2 . . . 𝑀 − 1} do
+3:
+𝒃 ← 𝑽𝑖+1 + 𝑽𝑖−1// indexing into 𝑽 is circular
+4:
+𝜓 ← atan2(𝑏𝑥, 𝑏𝑦)
+5:
+𝛾 ← atan2(𝑏𝑧,
+√︃
+𝑏2𝑥 + 𝑏2𝑦)
+6:
+𝛾 ← max(min(𝛾, 𝛾max), 𝛾min)
+7:
+Q ← Q ∪ (𝑽𝑖, 𝜓, 𝛾)
+8: end for
+9: 𝑖 ← 0
+10: while 𝑖 < |𝑽| do
+11:
+if ||𝑽𝑖 − 𝑽𝑖+1|| < 4𝜌min then
+12:
+𝒘 ← 𝑽𝑖+1 − 𝑽𝑖
+13:
+𝜓 ← atan2(𝑢𝑥, 𝑢𝑦,)
+14:
+𝛾 ← atan2(𝑢𝑧,
+√︃
+𝑢2𝑥 + 𝑢2𝑦)
+15:
+𝛾 ← max(min(𝛾, 𝛾max), 𝛾min)
+16:
+Q𝑖𝜓 ← 𝜓, Q(𝑖+1) 𝜓 ← 𝜓
+17:
+Q𝑖𝛾 ← 𝛾, Q(𝑖+1)𝛾 ← 𝛾
+18:
+𝑖 ← 𝑖 + 1
+19:
+end if
+20:
+𝑖 ← 𝑖 + 1
+21: end while
+C. Illustrative Examples
+An example of a view planning solution for five targets scattered around a city model of Charlotte, North Carolina
+is shown in Fig. 8a. The example was constructed assuming a Dubins airplane model having a curvature radius of
+𝜌min = 40 m and pitch angle constraints 𝛾 ∈ [−𝜋/12, 𝜋/9]. The random-face algorithm was used with 𝑛pts = 8 samples
+per visibility volume, 𝑛𝜓 = 4 heading angles per sample, and 𝑛𝛾 = 1 pitch angle per sample-heading angle pair. The
+visibility volumes for targets that had no occlusions had a common dome shape, whereas targets located closer to objects
+had more arbitrary shapes. Another example Fig. 8b illustrates the solution for five targets in a model of New York City,
+New York. This example compares the three-dimensional random-face algorithm with 𝑛pts = 32, 𝑛𝜓 = 8, and 𝑛𝛾 = 3
+pitch angles, to the two-dimensional optimized altitude entry pose sampling algorithm with 𝑛pts = 32, 𝑛𝜓 = 8. The
+3D path can change altitude which allowed the algorithm to find a lower cost path of 3920m while the 2D algorithm
+maintained constant altitude and found a path of cost 4285m, a 10.9% reduction in path cost.
+12
+
+(a) 3D DTSP with neighborhoods (DTSPN) in Charlotte,
+North Carolina
+(b) 3D DTSP with neighborhoods (DTSPN) in New York
+City, New York
+Fig. 8
+Solutions to the 3D Dubins traveling salesperson problem with neighborhoods. Panel (a) was computed
+using the random face algorithm in light blue with 8 samples per target visibility volume, four heading angles
+per sample, and one pitch angle per sample-heading angle pair. Panel (b) was computed using the random face
+sampling algorithm in dark blue with 𝑛pts = 32 samples per target visibility volume, 𝑛𝜓 = 8 heading angles per
+sample, and 𝑛𝛾 = 3 pitch angles per sample-heading pair; the two-dimensional entry pose sampling from [6] in
+magenta with 𝑛pts = 32 samples per target visibility volume and 𝑛𝜓 = 4 heading angles per sample. The target
+visibility volume is translucent white with black edges and the targets are red spheres. The green spheres are
+the vehicle configurations for the solution to the DTSPN. The environment shown is a section of New York City,
+New York obtained from the OpenStreetMap database. Building heights are indicated by the varying color
+scale from yellow to purple.
+V. Numerical Performance Study
+The 2D algorithms from Sec. III were compared to the 3D algorithms from Sec. IV through a Monte-Carlo
+experiment that randomized target locations and a number of targets located in an urban environment. This section
+describes the implementation of the algorithms, the design of the Monte-Carlo study, and discusses the results.
+A. Implementation
+The algorithms in this work were written in python 3.9 ∗[21] using a number of packages, including Shapely [22]
+for polygonal operations and NumPy [23] for working with matrices. The GLKH traveling salesperson solver [24] was
+used to solve the generalized traveling salesperson problems that arise from DTSPs. The target visibility volumes were
+created with data from OpenStreetMap [25], inverse depth calculations from the target location using OpenGL [15], and
+Blender [26] was used for intersecting the triangular meshes within the feasible airspace 𝐹 as well as decimating the
+meshes (i.e., reducing the number of triangular faces). This work uses [27] to compute 2D Dubins paths for the 2D
+algorithms. The algorithm simulations were performed on an AMD Threadripper 3990X running Ubuntu 20.04 with
+one thread allocated to the algorithm.
+B. Monte-Carlo Experiment
+A Monte-Carlo experiment was designed using the environments described in Table 1. The environments were
+created by capturing all of the buildings in a rectangular area in New York City with the OpenStreetMap database and
+limiting the building heights to 300 m. Target locations were randomized for each trial and determined by sampling
+the environment and placing targets on the ground, the wall of buildings, or the roofs of buildings according to a
+user-defined distribution. The radius of the target visibility volumes was 300 m with each target being at least 600
+m apart. The proposed sampling methods and heuristics are independent and studied here in different combinations.
+The algorithms parameters were varied as follows: the number of samples per visibility volume was varied between
+𝑛pts = {2, 4, 8, 16, 32}, the number of heading angles per sample was 𝑛𝜓 = {2, 4, 8}. To reduce the number of trials,
+only one pitch angle (𝑛𝛾 = 1) of 0◦ was used by passing 0◦ for 𝛾min and 𝛾max to the random face sampling (Sec. IV.A.1),
+3D edge sampling (Sec. IV.A.2), and global weighted face sampling (Sec. IV.A.3) algorithms. The Dubins airplane had
+∗The implementation of this study can be found at https://github.com/robotics-uncc/VisualTour3DDubins.
+13
+
+Table 1
+Description of environments obtained from an OpenStreetMap database for New York City, USA, and
+used for the Monte-Carlo experiment.
+Number of Targets
+Number of objects
+Width
+Depth
+5
+5624
+1986 m
+2090 m
+10
+9202
+2809 m
+2857 m
+15
+11584
+3440 m
+3621 m
+20
+12119
+3972 m
+4181 m
+a minimum curvature radius of 𝜌min = 40 m and a pitch angle constrained between -𝜋/12 and 𝜋/9, similar to [10]. A
+total of 80 configurations of targets were generated, divided evenly among groups of 5, 10, 15, and 20 targets. Every
+combination of algorithm parameters was evaluated with the 80 configurations. The normalized tour cost (total length
+of the tour divided by the turn radius) and the computation time were recorded. The algorithms are denoted by acronyms
+wherein the prefix is either 2D-DTSP, 2D-DTSPN, 3D-DTSPN, or 3D-METSPN corresponding to the algorithms of
+Sections III.B, III.C, IV.A, and IV.B.1, respectively. The 2D-DTSP is followed by a dash and an integer representing the
+number of heading angles. The remaining two algorithms are described by a sampling method acronym: entry pose
+sampling (ETRY) from Sec. III.C, random face sampling (RFAC) from Sec. IV.A.1, 3D edge sampling (E3D) from
+Sec. IV.A.2, or global weighted face sampling (GWF) from Sec. IV.A.3 followed by a dash and an integer representing
+the number of heading angles and another dash and an integer representing the number of samples per target visibility
+volume (i.e., 2D-DTSPN-ETRY-4-16 corresponds to a 2D DSTPN using entry pose sampling with 4 heading angles and
+16 sample points per target visibility region).
+1. Analysis of Monte-Carlo Study
+In general, two-dimensional methods at a fixed altitude performed better if the targets are all located at similar
+heights; whereas, 3D methods trended towards better tour cost when targets occupy a wide range of altitudes. The
+median path length, normalized by dividing the cost by the minimum curvature radius, of each view-planning tour (i.e.,
+cost) of the Monte-Carlo runs for an increasing number of targets, the number of heading angles is held at 𝑛𝜓 = 8, and
+the number of pitch angles is held at 𝑛𝛾 = 1 is plotted in Fig. 9. The DTSP algorithms that only visit a single point
+(gray) have one location sample per visibility volume but the lines were extended along the abscissa for comparison.
+The DTSP is inefficient in our problem because shorter paths can be obtained between targets by flying through the
+boundary of their corresponding visibility volumes rather than requiring the paths to pass through the visibility volume
+centers. As the number of heading angles increases the mean cost of the solution decreases, as expected. The METSPN
+algorithms have a similar cost to the eight sampled heading angle solutions. Most of the medians for different algorithms
+approach an asymptote, suggesting that they are converging towards a fixed median tour cost (i.e., further increasing
+the number of samples has diminishing returns). For a large number of samples, the proposed random face sampling
+algorithm yields a lower tour cost than the optimized altitude 2D algorithm. However, the median of the 3D edge
+sampling algorithm is less than the optimal altitude 2D algorithm for all numbers of samples greater than 2. This may
+be due to the 3D algorithms spreading their samples across another dimension (altitude). The 3D algorithms that spread
+the samples along the vertical dimension of each visibility volume perform worse than the algorithm that only samples
+one altitude slice. This suggests that distributing the points in the horizontal plane is more important than distributing
+them in the vertical direction for this particular environment and visibility volume. The sensor model creates visibility
+volumes with the most horizontal variation at the bottom of the shape as seen in Fig. 8; therefore, sampling the visibility
+volumes at the bottom is the best way to produce samples with the greatest horizontal variation.
+To isolate the effects of the different sampling methods, the results are examined for the case where the number
+of samples is held at 𝑛pts = 32, the number of heading angles is held at 𝑛𝜓 = 8, and the number of pitch angles is
+𝑛𝛾 = 1. Box plots of those trials can be seen in Fig. 10. The medians of the 3D methods (black bar in the middle
+of the colored box) are lower than the medians of the 2D methods suggesting that the 3D methods are able to more
+consistently find lower-cost solutions. The difference between medians of 2D and 3D methods grows as the number of
+target visibility volumes increases. The range of solutions for the different methods, denoted by the vertical black bars,
+is large and suggests that the difference between the solutions produced by the 2D and 3D cases is variable and sensitive
+to the environment. The time for each algorithm to execute on a single thread is shown in Fig. 10. It can be seen that
+14
+
+2 4
+8
+16
+32
+Samples per Target
+170
+180
+190
+200
+Tour Cost (nondim.)
+5 Targets
+2 4
+8
+16
+32
+Samples per Target
+225
+250
+275
+Tour Cost (nondim.)
+10 Targets
+2 4
+8
+16
+32
+Samples per Target
+270
+300
+330
+360
+Tour Cost (nondim.)
+15 Targets
+2 4
+8
+16
+32
+Samples per Target
+325
+350
+375
+400
+425
+450
+475
+Tour Cost (nondim.)
+20 Targets
+Algorithm
+2D-DTSP
+2D-DTSPN-ETRY
+3D-DTSPN-E3D
+3D-DTSPN-GWF
+3D-DTSPN-RFAC
+3D-METSPN-E3D
+3D-METSPN-GWF
+3D-METSPN-RFAC
+Fig. 9
+The line plots show the median non-dimensional tour cost of the different algorithms as the number of
+samples per target visibility volume increases.
+the algorithms that only consider one point per region have lower execution times than the algorithms that consider
+neighborhoods. The 2D ETRY method has a similar execution time to the 3D DTSPN methods. However, the heuristic
+METSPN algorithm has a lower execution time compared to the other 3D methods because the graph that it creates
+is smaller and less computationally expensive. The results suggest that for a large number of samples the METSPN
+algorithm outperforms the 3D DTSPN algorithms since it produces tours of similar cost but with a computation time
+that is approximately two orders of magnitude lower.
+VI. Conclusion
+This paper studied the view planning problem of using a 3D Dubins airplane model to inspect points of interest in
+an urban environment in minimum time. Triangular meshes were used to compute approximate visibility volumes that
+correspond to locations where an unobstructed view of the target can be obtained while satisfying imaging and altitude
+constraints. The mesh-based approach for computing visibility volumes is flexible and can represent more complex
+geometries than have previously been considered. A range-based sensor model was assumed here, however mesh-based
+view planning can potentially support other sensor models, sensing modalities, and encode sensing performance
+15
+
+100
+150
+200
+250
+Tour Cost (nondim.)
+5 Targets
+250
+300
+350
+Tour Cost (nondim.)
+10 Targets
+200
+225
+250
+Tour Cost (nondim.)
+15 Targets
+325
+350
+375
+400
+Tour Cost (nondim.)
+20 Targets
+5
+10
+15
+20
+Number of Targets
+100
+101
+102
+103
+104
+Time (s)
+Algorithm
+2D-DTSP
+2D-DTSPN-ETRY
+3D-DTSPN-E3D
+3D-DTSPN-GWF
+3D-DTSPN-RFAC
+3D-METSPN-E3D
+3D-METSPN-GWF
+3D-METSPN-RFAC
+Fig. 10
+The box plots show the range of cost across all sets of target visibility volumes when the number of
+samples per target visibility is held at 𝑛pts = 32, the number of heading angles is held at 𝑛𝜓 = 8 and the number
+of pitch angles is held at 𝑛𝛾 = 1. The vertical black bars show the upper and lower quartiles of the data while
+the colored sections show the middle quartiles. The black bar in the middle of the box plots is the median of
+the data set. The black diamonds are outliers. The line graph shows the increase in computation time as the
+number of target visibility volumes increases on a log10 scale. The shaded region around each line shows the
+range of computation time.
+characteristics. The 3D Dubins airplane model used in this work can, in some circumstances, produce more efficient
+inspection tours by exploiting altitude changes that are otherwise not possible with constant-altitude Dubins path tours.
+In cases where visibility volumes occupy disjoint altitude segments, the 3D algorithms provide a feasible solution where
+the 2D algorithms are not feasible. However, the pitch angle constraints of a Dubins airplane limit the change in altitude
+over a tour. Altitude changes are accompanied by an increase in path length and thus are only efficient when they greatly
+improve access to the visibility volume.
+This work introduced a heuristic that computes edge costs by replacing the 3D Dubins path computation with a
+simpler lower bound and assigning heading and pitch angles based on the geometric relation of successive points in a
+tour. This strategy provides a similar tour cost to other 3D algorithms that use the exact 3D Dubins path planner for
+edge cost computation but with computation time reduced by two orders of magnitude. Future work may consider the
+view planning problem in the presence of obstacles that must be avoided, with target visibility volumes that overlap,
+and/or with uncertain moving targets to be inspected.
+Acknowledgments
+This work was supported by the William States Lee College of Engineering at the University of North Carolina at
+Charlotte through the Multidisciplinary Team Initiation (MTI) Grant.
+16
+
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+18
+

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+Creative beyond TikToks: Investigating Adolescents’ Social
+Privacy Management on TikTok
+Nico Ebert
+[email protected]
+Zurich University of Applied Sciences,
+School of Management and Law
+Winterthur, Zurich, Switzerland
+Tim Geppert
+Zurich University of Applied Sciences,
+School of Management and Law
+Winterthur, Zurich, Switzerland
+Joanna Strycharz
+University of Amsterdam, Faculty of
+Social and Behavioural Sciences
+Amsterdam, North Holland
+Netherlands
+Melanie Knieps
+University of Zurich, Digital Society
+Initiative
+Zurich, Zurich, Switzerland
+Michael Hönig
+Zurich University of Applied Sciences,
+School of Management and Law
+Winterthur, Zurich, Switzerland
+Elke Brucker-Kley
+Zurich University of Applied Sciences,
+School of Management and Law
+Winterthur, Zurich, Switzerland
+ABSTRACT
+TikTok has been criticized for its low privacy standards, but lit-
+tle is known about how its adolescent users protect their privacy.
+Based on interviews with 54 adolescents in Switzerland, this study
+provides a comprehensive understanding of young TikTok users’
+privacy management practices related to the creation of videos.
+The data were explored using the COM-B model, an established
+behavioral analysis framework adapted for sociotechnical privacy
+research. Our overall findings are in line with previous research
+on other social networks: adolescents are aware of privacy related
+to their online social connections (social privacy) and perform
+conscious privacy management. However, we also identified new
+patterns related to the central role of algorithmic recommenda-
+tions potentially relevant for other social networks. Adolescents
+are aware that TikTok’s special algorithm, combined with the app’s
+high prevalence among their peers, could easily put them in the spot-
+light. Some adolescents also reduce TikTok, which was originally
+conceived as a social network, to its extensive audio-visual capabil-
+ities and share TikToks via more private channels (e.g., Snapchat)
+to manage audiences and avoid identification by peers. Young users
+also find other creative ways to protect their privacy such as identi-
+fying stalkers or maintaining multiple user accounts with different
+privacy settings to establish granular audience management. Based
+on our findings, we propose various concrete measures to develop
+interventions that protect the privacy of adolescents on TikTok.
+KEYWORDS
+TikTok, adolescent, video, privacy management, social privacy,
+COM-B, Behavior Change Wheel
+1
+INTRODUCTION
+The global popularity and rapid growth of TikTok are accompanied
+by problems that are the subject of public debate. The platform has
+been criticized for several privacy issues such collecting personal
+This work is licensed under the Creative Commons Attribu-
+tion 4.0 International License. To view a copy of this license
+visit https://creativecommons.org/licenses/by/4.0/ or send a
+letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
+Proceedings on Privacy Enhancing Technologies 2023(2), 1–15
+© 2023 Copyright held by the owner/author(s).
+https://doi.org/XXXXXXX.XXXXXXX
+data from minors under the age of 13 [29, 33] or transferring U.S.
+minor’s data to China [70]. These issues are especially alarming as
+a large number of platform users are underage [74]. As a result, the
+private space of children such as the bedrooms from which they
+create their videos becomes visible to the world [43]. TikTok has
+reacted to public criticism by introducing several features to better
+protect the privacy of adolescents [75].
+Implicit to the current debate is the apparent consensus that
+adolescents lack the awareness or skill set to consider the possible
+privacy implications of their platform use. In fact, however, little
+evidence is publicly available on how TikTok is used by adolescents
+[63] and how they manage their privacy on the platform [40]. As
+short videos are the platform’s key purpose, this immediately raises
+the question of how younger users protect their privacy when they
+create videos. In this paper, we aim to answer the following re-
+search question: "How and why do adolescents manage their privacy
+when creating videos on TikTok?" We build on the COM-B model for
+behavioral analysis, an established conceptual framework for be-
+havior change widely applied in health communication and beyond
+[62]. This model allows us to explore not only privacy behavior, but
+also related users’ motivations, skills and desires. To explore adoles-
+cents’ privacy management in video creation from the perspective
+of their capabilities, motivations, and opportunities, we conducted
+interviews with 54 adolescent TikTok users in Switzerland, where
+TikTok has gained popularity among young people [6].
+This study contributes to the growing body of research that ad-
+dresses how adolescents manage their privacy on social networks.
+This topic, which is often viewed (and judged) through the moral
+lens of adults, is usually met with a sense of alarm. Empirical evi-
+dence that could serve as a better fact base about adolescents’ online
+privacy behavior is largely based on platforms that cater to a more
+general population (e.g., Facebook, Twitter). However, TikTok is
+not only explicitly geared towards a younger audience [47], but also
+strongly encourages the sharing of short personal videos. Although
+adding text is possible on TikTok, it takes a back seat in favor of
+video content. Given the particularly sensitive nature of one’s image
+and its link to an individual’s personal development [28], TikTok
+strikes us as a particularly relevant but under-researched new use
+case with the potential to enrich the ongoing debate about how –
+if at all – teenagers perceive and manage their privacy [40].
+1
+arXiv:2301.11600v1  [cs.SI]  27 Jan 2023
+
+Proceedings on Privacy Enhancing Technologies 2023(2)
+Ebert et al.
+This study is the first to examine how adolescents between the
+ages of 12 and 18 manage their privacy on TikTok when it comes
+to personal videos. Our findings are based on original data from
+personal interviews and offer unique insights into how privacy
+concerns influence young people’s online behavior. The qualitative
+nature of our study helped us to understand the components that
+shape sharing behavior on TikTok. Ultimately, this allowed us to
+make concrete suggestions on how to effectively promote privacy-
+protective behavior among adolescents on TikTok (e.g., specific
+training, improved app features, and policy enforcement).
+2
+RELATED WORK
+2.1
+TikTok and Privacy Issues
+As with many social media platforms, TikTok has come under
+scrutiny for its handling of personal data. TikTok is a video-focused
+social network originally started as U.S.-based musical.ly but later
+bought by Beijing ByteDance Technology Ltd. The TikTok app
+(available for Android and iOS) allows users to create short videos
+(which may only be a few seconds long) and live streams [42]. Like
+YouTube, TikTok is a manifestation of user-generated media where
+content is not primarily created by a limited number of producers
+but by a myriad of users [44]. Compared to other social networks
+such as Facebook or Instagram, users on TikTok do not need to
+communicate with each other to find a community. They can simply
+visit the “For You” default page to find like-minded users [16, 42, 43,
+87]. Via the primary button at the center of the home screen, users
+can easily record and edit short videos, apply various effects and
+sounds, and reuse content produced by other users. Videos can be
+saved as drafts or published immediately to be viewed by different
+audiences (myself, followers, everybody) [56]. As of September
+2021, 1 billion monthly active users were reported [79], and 740
+million first-time installs were estimated in 2021 [73]. Cloudflare, a
+provider of content delivery networks, ranked TikTok as the most
+popular website of 2021, before Google [68]. TikTok is currently
+also gaining in popularity among users below the official age limit
+of 13 years [19]. In Switzerland, three-quarters of all adolescents
+had a TikTok account in 2020 (behind Instagram and Snapchat with
+both over 90%) [6]. Younger adolescents (12-15 years) were even
+more likely to have a TikTok account than older adolescents (16-19
+years). Slightly more girls (78%) used it than boys (68%). 51% of all
+adolescents stated to use it at least multiple times per week, and 38%
+daily [6]. However, little is known about how young users think
+about the data they share on the platform.
+The app has raised numerous severe security and privacy con-
+cerns (e.g., [5, 23, 24, 46, 84]) and caught the attention of the inter-
+national authorities in the U.S. and EU [29, 69, 70]. For example, an
+analysis of the app revealed extensive aggressive user tracking (e.g.,
+including techniques such as fingerprinting) and data sharing with
+other websites (e.g., sharing searches with Facebook) [24]. The app
+could also potentially collect other personal data from the user’s
+smartphone (e.g., data from the clipboard [23]). Since young people
+have always been an important user group of TikTok, concerns
+have been raised about ByteDance’s handling of their personal
+data. For example, in February 2019 ByteDance was fined USD
+5.7 million by the U.S. Federal Trade Commission (FTC) because
+musical.ly had collected information from minors under the age
+of 13 in violation of the Children’s Online Privacy Protection Act
+[33]. Due to the death of a 10-year-old TikTok user, the Italian data
+protection authority has banned TikTok from processing the data
+of users whose age could not be determined with full certainty [29].
+Also, the transfer of minors’ data to China after the acquisition
+of U.S.-based musical.ly had caused a serious backlash in the US
+and EU [69, 70]. As recent as June 2022, evidence surfaced that
+ByteDance has repeatedly accessed U.S. user data from China –
+a practice that they had denied three years earlier when similar
+criticism was raised [26].
+TikTok has reacted to public criticism with several privacy-
+related updates to the original app. As part of its settlement with
+the FTC, the platform introduced an age-verification process for
+its users based on self-declaration, meaning users can provide a
+false age [48]. Further changes included extended parental control
+features [41] and privacy settings contingent on the app users’ age
+statement [75]. While children below 13 cannot use the app, ado-
+lescents between the ages of 13 and 15 are automatically switched
+to a “private account” as a default option, limiting those who can
+view their videos to approved followers. When 16- and 17-year-old
+users imitate an existing video in the form of a “duet” (split-screen
+video) or “stitch” (video incorporating a short clip of someone else’s
+content), these are automatically restricted to “friends only”. Only
+users who are 18 and older can buy and send virtual gifts. However,
+it is unclear if and how TikTok’s efforts have affected users’ privacy
+management.
+2.2
+Adolescents’ Privacy Management on
+Social Media
+From the moment adolescents started to use online social network-
+ing sites, “online privacy” has been a major topic of discussion [31].
+Informational privacy can be defined as “the claim of individuals,
+groups or institutions to determine for themselves when, how, and
+to what extent information about them is communicated to others”
+[81]. Research on online privacy and adolescents can be divided
+into two categories: “institutional privacy” and "social privacy"
+[67]. Institutional privacy refers to the data collection practices by
+organizations (e.g., for commercial purposes) [67, 85]. The focus of
+this paper is social privacy, i.e., issues related to sharing personal
+information with others (e.g., friends and family). According to the
+theory of "networked privacy," individuals do not have complete
+control over the sharing of their personal information within social
+connections (e.g., on social media) because privacy is not managed
+by individuals alone, but by networks of individuals collectively
+[58].
+Young people are often seen as particularly vulnerable social
+media users with limited capacities to protect their privacy [15, 58].
+At the same time, they are also portrayed as individuals who put
+themselves and others at risk with their naive and reckless social
+media behavior [18]. Following this logic, numerous guides for
+parents emphasize the importance of modifying privacy settings
+and monitoring their children’s behavior (e.g., [37]). However, there
+has also been a pushback to this alarmist perspective by scholars
+who suggest that adolescents’ online privacy should be addressed
+based on empirical research rather than paternal instinct [83].
+2
+
+Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok
+Proceedings on Privacy Enhancing Technologies 2023(2)
+Empirical evidence from social networks other than TikTok (e.g.,
+Facebook) suggests that adolescents are aware of their social privacy
+and actively manage their privacy on social media. As described
+by boyd [9], adolescents want to avoid surveillance from parents,
+teachers, friends and other meaningful persons in their lives (that
+is what “online privacy” means to them). Adolescents’ social media
+use seems to generally prompt increased disclosure of personal
+information [72]. However, frequent sharing of content does not
+imply that adolescents share indiscriminately, nor that the content
+is intended for a wider audience [58]. Indeed, adolescents are con-
+cerned about their privacy and capable of protecting it [1, 8, 17, 53].
+Contrary to conventional wisdom, young people are, in fact, more
+likely to protect their privacy on social media than older people
+[8]. Madden et. al found several strategies adolescents use on social
+media to manage their identity and protect sensitive information
+[57]. These strategies include deleting friends, faking names, delet-
+ing content, withholding/faking information, and changing privacy
+settings [20, 35, 64]. They also employed different “zones of privacy”
+by using different channels for disclosing personal information to
+maintain intimacy with friends while protecting their privacy from
+their parents and strangers [50]. Privacy management can also
+mean modifying social media content to shield it from audiences
+[57, 64]. This practice is referred to as “social steganography” or en-
+coding a message for a defined audience [58]. Adolescents’ privacy
+management is influenced by various factors such as their social
+environment (e.g., friends, parents), prior (negative) experiences as
+well as the saliency of privacy settings [53, 86].
+Despite the existing evidence on adolescents’ social media use
+on other social networks, researchers argue that existing findings
+might not be directly applicable to TikTok [63]. Compared to other
+networks such as Facebook or Instagram, TikTok mainly thrives
+on content exploration and (re)-creation [87]. The focus is not on
+the interaction between users and their social network but the
+interaction with users’ videos proposed by an algorithm [7]. The
+main feature, the “For You” page, presents an endless stream of
+personalized, publicly available videos. Seeing them will motivate
+users to react and create similar content (e.g., through features such
+as “duet” or “stitch”). TikTok might therefore pose a particular threat
+to adolescents’ privacy because a space previously conceptualized
+as private and safe can easily become a space of public visibility,
+surveillance, and judgment (such as in the case of a teenager being
+seen to perform a dance routine in their bedroom) [43].
+Only a few studies have investigated adolescents’ privacy man-
+agement on TikTok. There is some evidence that privacy manage-
+ment on TikTok is considered as crucial by adolescents [16] and
+becomes more stringent at higher perceived risks [40]. However,
+it is unclear how and why adolescents manage their privacy on
+TikTok.
+2.3
+COM-B Model
+As we were interested in the components that shape privacy be-
+havior, we chose the COM-B model, which has been used in ex-
+ploratory studies (e.g., [32]) and a series of contexts to change
+behavior (e.g., [3]), as the conceptual framework for our analysis.
+Many behavioral theories have been developed, often with overlap-
+ping but differently named constructs [60] and limited guidance on
+Capability
+Motivation
+Opportunity
+Behavior
+Reflective
+Automatic
+Psychological
+Physical
+Social
+Physical
+Environment
+Individual
+Figure 1: The COM-B model [62]. The three components capa-
+bility (C), opportunity (O) and motivation (M) must be present for a
+behavior (B) to occur. They interact over time and form a dynamic
+system with positive and negative feedback loops [80].
+choosing an appropriate theory for a particular, real-world context
+[62]. As a consequence, theories are often under-used to under-
+stand real-world contexts and to design real-world solutions, which
+makes replication, implementation, evaluation, and improvements
+difficult [25, 62]. Researchers have argued that a comprehensive
+meta-model or “supra-theory” model of behavior – like the COM-B
+model – is needed that is applicable across contexts [25, 62]. As a
+meta-model of behavior, the COM-B model does not come with a
+pre-determined set of context-specific predictions that are common
+for many behavioral theories. COM-B is based on several exist-
+ing social cognition models and has a broader understanding of
+behavior, having "also [...] automatic processing at its heart [like
+emotions and habits], broadening the understanding of behaviour
+beyond the more reflective, systematic cognitive processes that
+have been the focus of much behavioural research [...] (for example,
+social cognition models such as the Theory of Planned Behaviour)"
+[62]. Its comprehensive nature and flexibility made it a good fit
+for the exploratory nature of our study that was not constrained
+by the conceptual boundaries of a single theoretical framework.
+Furthermore, the model comes with hands-on actionable advice on
+appropriate interventions in a given context in form of a holistic
+behavior change framework (“Behavior Change Wheel”) (see [62]).
+As illustrated in Figure 1, the COM-B model is based on three
+components – capability (C), opportunity (O), and motivation (M)
+– that shape a person’s behavior (B) [62]. Firstly, capability is a
+subject’s psychological ability (including necessary comprehension,
+knowledge, and skills) as well as the physical ability (e.g., control
+of the body) to engage in a behavior. Secondly, motivation can
+be defined as the subject’s mental processes that energize and di-
+rect behavior. It includes the reflective motivation that involves
+conscious processes (e.g., goals, plans, and evaluations) as well as
+automatic processes (i.e., habitual, instinctive, drive-related, and
+affective processes). Finally, opportunity is defined as an attribute
+of the environmental system (unlike capability and motivation)
+3
+
+Proceedings on Privacy Enhancing Technologies 2023(2)
+Ebert et al.
+that enables or facilitates a behavior. Opportunities can be physical
+(e.g., technical features of an app, material, financial, and time) and
+social (e.g., norms and culture). In this study, we analyzed the par-
+ticipants’ capabilities, opportunities, and motivation to engage in
+privacy behaviors.
+Firstly, capability is a subject’s psychological ability (including
+necessary comprehension, knowledge, and skills) as well as the
+physical ability (e.g., control of the body) to engage in a behavior.
+Secondly, motivation can be defined as the subject’s mental pro-
+cesses that energize and direct behavior. It includes the reflective
+motivation that involves conscious processes (e.g., goals, plans, and
+evaluations) as well as automatic processes (i.e., habitual, instinctive,
+drive-related, and affective processes). Finally, opportunity is de-
+fined as an attribute of the environmental system (unlike capability
+and motivation) that enables or facilitates a behavior. Opportunities
+can be physical (e.g., technical features of an app, material, financial,
+and time) and social (e.g., norms and culture).
+In our exploratory study, we did not focus on identifying inter-
+actions between COM-B components that explain a specific target
+behavior. Rather, our scope was first to learn about the full range
+of behaviors and explanatory factors associated with adolescents’
+privacy management.
+3
+METHODOLOGY
+3.1
+Research Ethics
+This paper is based on semi-structured, one-to-one interviews with
+adolescents in the Canton of Zurich, Switzerland, conducted in
+November 2021. In total, we visited two secondary schools (one
+in the city of Zurich and one in the greater Zurich area) and three
+youth centers (all in the city of Zurich). All interviews were audio-
+recorded and transcribed verbatim. Ethical approval was obtained
+from our university’s institutional review board. Study participants
+provided written informed consent. For subjects below the age
+of 16, additional consent was sought from the parents. The in-
+terviews were voluntary and conducted at the institutions from
+which the subjects had been recruited. Digital, personalized shop-
+ping vouchers with a value of CHF 20 (~USD 19) were offered to
+study participants as compensation. The amount and type of the
+vouchers was determined beforehand together with the adolescents’
+supervisors (i.e., teachers, social workers) in order to not create an
+inappropriate but still sufficient incentive. After the interviews, in
+agreement with the participants, WhatsApp was used to deliver
+the individualized vouchers to the participants and to allow them
+to review their personal interview transcripts.
+Several steps were taken to protect the participants identity with-
+out compromising the transparency of our research process. To
+begin with, all personally identifiable information was removed
+(e.g., references to persons, locations) and participants’ names were
+replaced with pseudonyms. Furthermore, the study data was stored
+in line with our university’s storage policy and only the involved re-
+searchers had access to the files. Finally, the original audio files were
+deleted from all devices half a year after recording together with
+other remaining personal data (e.g., phone numbers, WhatsApp
+chats, digital vouchers).
+3.2
+Sample and Procedure
+Due to the lack of research on this topic, we chose a highly ex-
+ploratory approach. To identify information-rich cases and make
+optimal use of available resources, we drew a purposive sample
+[27]. We used social media and search engines to find institutions
+in the Canton of Zurich (e.g., secondary schools, youth work, youth
+associations, museums) with contact with adolescents between 12
+and 18 years of age. Afterward, principals of participating insti-
+tutions recruited interested teachers and social workers. They, in
+turn, contacted interested TikTok users in the required age group.
+To extend the participant base, we applied snowballing among the
+interested TikTok users. Based on our primary aim (i.e., to explore
+how adolescent TikTok users manage their privacy), we chose to
+sample based on study participants’ age and gender (equally dis-
+tributed). We decided to ignore other demographic information such
+as ethnic identity. Following a pragmatic definition of theoretical
+saturation [55], no new information emerged after approximately
+40 interviews, and we ended data collection after the 54th interview.
+We chose to employ semi-structured interviews for our study
+because it encourages two-way communication and provides the
+interviewer with the opportunity to learn the reasons behind an
+answer. Some of the questions were part of the interviewer’s guide
+(see Appendix), others were addressed at the moment. The inter-
+view guide was developed based on a previous study that applied
+the COM-B model in a qualitative setting [14] and adopted to the
+context of the current study. After asking for demographic informa-
+tion, we first explored general TikTok usage and motivation. The
+other questions followed the COM-B structure and were related to
+privacy-related behaviors as well as the explanatory components
+related to the target behavior “video creation”. We finished the
+interviews with questions about commercial privacy aspects (i.e.,
+targeted advertising and user tracking). After the interview pro-
+cess was completed, study participants received a copy of their
+interview transcript via WhatsApp and were invited to add infor-
+mation or make amendments. Minimal revisions were made by one
+participant.
+To analyze the content of the interviews, we used a two-step pro-
+cedure that first divided each statement into one of the four COM-B
+components (behavior, capability, opportunity, motivation) before
+further subdividing them into privacy-specific content. For phase
+one, we used a directed content analysis approach [38] to analyze
+the statements. To counter the subjectivity inherent to qualitative
+data analysis, three researchers read and coded all statements into
+the four COM-B domains (behavior, capability, opportunity, motiva-
+tion). On the grounds of economy in both cost and effort, we decided
+against using "intercoder reliability" (ICR). As full replication of
+results was deemed unnecessary due to the exploratory and quali-
+tative nature of data collection and analysis, we instead followed
+guidelines suggesting the use of "multiple coding" which allows
+independent researchers to cross check their coding strategies and
+interpretation of data [2]. The authors engaged in researcher tri-
+angulation [21] by discussing the emerging codes during the open
+coding process of the first three interviews and developed cod-
+ing guidelines. Disagreements were discussed and resolved. Using
+the MAXQDA 2020 software, all responses were coded consistent
+4
+
+Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok
+Proceedings on Privacy Enhancing Technologies 2023(2)
+with six COM-B labels1 (behavior, psychological capability, auto-
+matic/reflective motivation, social/physical opportunity). To ensure
+continued adherence to the agreed coding guidelines, the three
+researchers regularly communicated to ensure coding consistency.
+In phase two, all statements – previously labeled as one of the
+COM-B components – were further analyzed for their privacy-
+specific content. Therefore, an inductive thematic analysis [11]
+to identify themes within similarly coded statements was con-
+ducted (see Appendix for coding scheme). One researcher identified
+themes across identically coded statements and discussed them
+with the other researchers. A theme reflects a collection of similar
+responses from at least two different study participants. For exam-
+ple, responses that were coded under the COM-B label “reflective
+motivation” such as “I would be afraid of stupid remarks.”, “I have
+no desire to be bullied.”, and “I can do without being ridiculed in
+my class’s WhatsApp group.” were allocated to the privacy-specific
+theme “negative reaction avoidance”. This step resulted in a list
+of themes within each of the six COM-B labels. Ultimately, the
+researchers reviewed and discussed the emerging themes, merged
+similar themes, and re-labeled others. By playing the “devil’s ad-
+vocate” – a common way to scrutinize identified themes [2] – we
+sought to exploit the full potential of multiple coding to furnish
+alternative interpretations of our findings. The anonymized, coded
+interview transcripts are publicly available at osf.io/z8d3w.
+4
+RESULTS
+A total of 54 adolescents aged between 12-18 years (15 ± 1.82 years)
+were interviewed, of which half (27) were female (see Table 1).
+Interviews ranged from 5 to 21 minutes in length, with a mean of
+12.6 min per interview (SD = 3.91). Most users attended secondary
+school, and 80% had used the app for more than one year. Half
+of the study participants admitted using TikTok between one and
+three hours per day.
+Table 1: Characteristics of one-to-one interview participants
+(n = 54)
+Variables
+% (n)
+Gender (% of females)
+50% (27)
+Age
+15 ± 1.82
+Educational level
+Primary level
+2% (1)
+Lower secondary level
+54% (29)
+Upper secondary level
+44% (24)
+User since
+One year or less
+20% (11)
+Between one and two years
+33% (18)
+More than two years
+46% (25)
+Current app usage
+Daily >= 3h
+17% (9)
+Daily >= 1 and <3h
+50% (27)
+Daily < 1h
+28% (15)
+Less than daily
+6% (3)
+1We did not need to code “physical capability” as the participants did not have physical
+impairments.
+Building on the conceptional framework of the COM-B model,
+we identified 13 themes from the data analysis that described how
+and why adolescents protect their privacy on TikTok (see Table 2).
+These are described in more detail in the following. No weighting
+was associated with the themes in terms of their overall contribu-
+tion.
+4.1
+Behavior
+4.1.1
+Proactive privacy. The participants in our study mentioned
+various ways to control the content of their TikToks2 and their
+audience. Publishing content to audiences was described as reflec-
+tive and non-automatic (as opposed to a habitual, non-reflective
+publication of TikToks). This behavior is also referred to as the “ap-
+proach” privacy strategy [58]. For example, regarding the content,
+study participants described what they consider to be too sensitive
+for publication on TikTok and would not publish (e.g., TikToks that
+reveal too much about them). Lima (F, 14) creates public videos
+and has 50 different accounts. She has clear privacy boundaries
+regarding the video content: “I would not post TikToks where you
+can see a lot of myself. I wouldn’t post videos in which I’m drunk.”.
+Another form of restriction is to define who can see which type of
+content on the platform. This includes TikTok users making drafts
+only visible to themselves or blocking selected users from watching
+videos. Bärbel (F, 13) actively tries to keep her parents from seeing
+her videos: “To prevent my parents from seeing my videos, I can
+simply block them.”.
+We identified two subthemes within the proactive privacy theme:
+private creators (19 persons, 35% of the sample) and public creators
+(11 persons, 20%). Private creators create videos only for themselves
+or close friends but do not publish them for a broad audience. A few
+users described the practice of posting videos that are just visible to
+themselves, only to be able to then repost them on “more private”
+social media such as Snapchat or WhatsApp for a selected group
+of people: “I don’t post my videos. I download them, save them
+under photos, then send them on WhatsApp, for example. I only
+use TikTok for editing.” (Amy, F, 17).
+Public creators regularly create videos for their followers or the
+general public. An extreme case is Joy (F, 13), who has used TikTok
+since she was nine years old (when the app was still musical.ly). She
+maintains 50 thematic user accounts with different age settings and
+distinct followings (e.g., some accounts for gaming-related videos
+and others for YouTube reposts). In addition to managing multiple
+accounts, public creator Lima (F, 14) also uses the live feature. It
+is available to users with at least 1,000 followers and allows them
+to create personal live streams and interact with users in real time.
+Lima had to set her age to 16 years to enable the live feature.
+4.1.2
+Avoidance. Some study participants reported that they do
+not publish videos on TikTok at all to protect their privacy. In the
+literature, this is referred to as the avoidance privacy strategy [58].
+Peter (M, 14), one of 24 study participants (44%) we classified as a
+pure consumer, stated: “I’ve never created a TikTok. I don’t even
+know how to do it.”. Tim (M, 12) published once but decided to
+only watch TikToks afterward: “To try it out, I uploaded something
+2The term “TikToks” is used synonymously with videos.
+5
+
+Proceedings on Privacy Enhancing Technologies 2023(2)
+Ebert et al.
+Table 2: Identified themes for adolescents’ video privacy management on TikTok based on the COM-B model. Frequency is
+calculated across 54 interviews.
+Theme
+Description
+Frequency
+Behavior
+Proactive privacy
+Publishing videos with control over the content and the audience
+30
+Avoidance
+Publishing no videos on the platform
+24
+Capability (Psychological)
+Past privacy incidents
+Previous negative experiences related to privacy on the platform (e.g., lost
+account, accidental publication)
+15
+Privacy literacy
+Knowledge and skills related to privacy management in the app (e.g., audience
+understanding and configuration)
+53
+Opportunity (Social)
+Negative feedback
+Negative behavior of others affects privacy management (e.g., observation of
+cyber-bullying)
+16
+Linkability experience
+Observing that online personas can be linked to the personal sphere affects
+privacy management (e.g., my teacher is on the platform)
+39
+Restrictive influence
+Restrictive behavior of others affects privacy management (e.g., restrictive
+parental mediation)
+34
+Opportunity (Physical)
+Platform features
+Privacy-related features of the platform (e.g., audience settings, sharing via
+other social networks)
+46
+Device features
+Privacy-related features of the device (e.g., screen time limits, deleting videos
+on the smartphone)
+17
+Motivation (Automatic)
+Negative emotion avoidance
+Avoidance of negative emotions expected as a result of publication (e.g., shame,
+fear)
+15
+Motivation (Reflective)
+Negative reaction avoidance
+Goal to avoid expected negative consequences of publication
+10
+Privacy identity
+Privacy as a general value (e.g., also on other platforms)
+5
+Publicity avoidance
+Goal to avoid expected publicity of publication
+29
+once, but nothing from me. I thought that was funny. But I prefer
+to watch videos.”
+4.2
+Capability (Psychological)
+4.2.1
+Past privacy incidents. This theme refers to a specific form
+of privacy-related knowledge (cp. [60]) gained after experiencing
+potential or actual privacy incidents. Potential privacy incidents
+are perceived as minor threats but may lead to increased privacy
+awareness. “I posted my very first video by accident. It was only
+seen by three people,” reported Yasmina (F, 15). Lima (F, 14), a public
+creator, remembered: “I was half asleep and accidentally posted a
+TikTok. The next morning, I saw that someone had commented
+on the video. But I thought it was funny and not bad at all.” When
+TikTok updated its app and increased the size of the “publish” button
+to lower the threshold for publication, Lima decided to block app
+updates.
+Users have also realized that some of TikTok’s privacy features
+can be easily bypassed. Their awareness of the platform’s weak-
+nesses has contributed to a greater privacy awareness. An example
+is a feature that allows blocking certain users from viewing videos,
+which can be easily bypassed: “If I block people but they still want
+to see my TikToks, they immediately make an extra fake account
+and continue seeing them.” Roswitha (F, 15). However, she found
+a way to manage her privacy: “Since these users have too few fol-
+lowers, I simply block them again or ignore them depending on the
+video.”.
+A more serious subtheme are actual privacy incidents. Bärbel (F,
+13) had to realize that she was not anonymizing herself sufficiently:
+“I wore a mask on my face in the video, anonymously, so to speak.
+But the people who deal with me every day recognized me by my
+outfit, my room, and my hairstyle and posted the video in the class
+WhatsApp chat.”. Anna (F, 14) reported losing her account and not
+being able to reclaim it through TikTok’s customer support. At the
+same time, other users were still able to watch her videos: “I made
+videos of myself when I was 9 and then lost the account. Now the
+videos are still public, but I can no longer access them.”.
+4.2.2
+Privacy literacy. Privacy literacy can be defined as a com-
+bination of factual or declarative (’knowing that’) and procedural
+(’knowing how’) knowledge about online privacy [76]. Concern-
+ing the publication of videos on the platform, adolescents need to
+have the knowledge and skills to assess and manage audiences and
+content as needed.
+Respondents mentioned, for example, that the algorithm might
+present a video on TikTok’s center stage: “It depends on how pop-
+ular a video is and only then does it appear on the For You Page.”
+(Bärbel (F, 13)) or that public videos can also be watched without
+6
+
+Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok
+Proceedings on Privacy Enhancing Technologies 2023(2)
+having a TikTok account: “From Google or Safari you can type in
+TikTok and view the videos.” (Aron (M, 13)). They also described
+how to find out which of their peers used TikTok: “When you post
+a video, it spreads immediately and then you know who has TikTok
+and who does not. Because so many people have TikTok now, it has
+become weird for me to post TikToks.” (Elsa (F, 14)). Respondents
+also described their audience and content management skills. The
+private creator Bea (F,14) only publishes for a strictly curated list
+of followers and therefore has established an approval process that
+allows her to maintain the desired level of privacy: “I get to know
+new classmates first and only then give them my TikTok account.
+Afterward, they tell me they sent a request and I accept them as
+followers in the app.” (Bea (F, 14)). Furthermore, the adolescents
+interviewed were also able to assess different levels of sensitivity of
+content in terms of their privacy and select an adequate audience
+accordingly: “My buddy and I made 10 TikToks in which we share
+our weekend activities with people. Some have 60,000 views. But
+we think carefully what to make public.” (Alex (M, 18)).
+The adolescents also talked about various app settings needed to
+manage the audience, such as the activation of the private account
+“Switching to the private account takes only two minutes. This is
+not difficult.” (Alexandra (F, 12)) or knowing the publication status
+of a video: “A draft is rendered greyish and blurry. When published,
+it is bright and jumps right out at you.” (Alexander (M, 15)). Some
+adolescents also perform “digital housekeeping” activities by re-
+moving content related to a specific event or as a habit: “As I became
+older, I started to delete old videos.” (Ariane (F, 15)).
+4.3
+Opportunity (Social)
+4.3.1
+Negative feedback. Negative feedback refers to expected or
+observed negative feedback from others (such as harsh comments
+to videos). Study participants reported negative reactions on the
+platform (e.g., from strangers or people from the same school) as
+an explanation for their privacy protection behavior. Alexander (M,
+15) mentioned a general culture of mutual criticism: “Many of the
+famous TikTokers sometimes make mistakes. Afterwards, everyone
+makes fun of them in videos.”. Other respondents mentioned nega-
+tive reactions from their peers that had influenced their behavior:
+“A friend went viral with a video. Then she got yelled at on the
+street. It would annoy me.” (Katja (F, 17)).
+4.3.2
+Linkability experience. Similar to the perception of negative
+feedback, the realization of how easily online personas can be linked
+to the personal sphere can also lead to more restrictive publication
+behavior. Study participants perceived the platform as a public space
+shared by acquaintances and strangers. However, by recognizing
+people from their school on their “For You” page, study participants
+realized that they, too, could be easily recognized. As Georg (M,
+15) put it: “There are maybe ten or twenty people in the school
+building who do [public] TikToks regularly. You suddenly realize: I
+know that guy from TikTok. That’s the reason why I don’t publish.”.
+In addition to peers, respondents also described experiences that
+made them understand that acquainted adults in authority positions
+would be able to see their TikTok as well. Sibylle (F, 15) realized
+this: “My music teacher was on TikTok singing a song.”. Therefore,
+Sibylle also does not publish so as not to be recognized by everyone
+on the platform.
+4.3.3
+Restrictive influence. Restrictive influence refers to others
+(e.g., close friends or parents) perceived to be restrictive or restrict-
+ing study participants’ video creation behavior. Some interviewees
+reported that their friends did not publish on TikTok, which in part
+motivated why they did not publish, either. In mentioning his peers,
+Felix (M, 12) stated: “Most of the people I know don’t upload any-
+thing of themselves where they show their face.”. Another example
+is restrictive mediation by parents or relatives: “My eight-year-old
+cousin accidentally posted a video with my smartphone. His uncle
+saw it on his For You page, so I deleted it.” (Sibylle (F, 15)).
+4.4
+Opportunity (Physical)
+4.4.1
+Platform features. Age verification is a key platform feature
+intended to protect the privacy of young users (not limited to cre-
+ating videos) and the subject of much public discussion. In the
+semi-structured interviews, 29 of the interviewed participants were
+also asked what age they provided. Two-thirds admitted that they
+had given a false age when they registered (indicating, e.g., the age
+of their parents). The main motivation for this behavior was to be
+able to use TikTok in general (for those below the age of 13) or all
+its features. Some study participants, like Martin (M, 14), also had
+misconceptions about possible age restrictions: “Because otherwise,
+TikTok won’t let me watch videos.”.
+However, study participants also described how they used TikTok’s
+features for privacy purposes in general. This includes using a nick-
+name instead of their real name, limiting the use of personal infor-
+mation on their profile page, and not linking their TikTok account
+with other social media accounts (e.g., Instagram). While some in-
+terviewees do not use a name at all: “Why should people know my
+name? I have replaced my name and individual letters with an X.”
+(Ali (M, 12)), others actively involve their parents to make use of
+the in-app parental controls that restrict their app access.
+Study participants also reported using various features related to
+audience configuration, such as creating personal drafts, activating
+a private account, deleting videos, or blocking users. Public creators
+sometimes create multiple “privacy-tailored” user accounts with
+specific follower groups for content of special sensitivity. Where the
+features offered by the platform are perceived as too limited or in-
+effective, the adolescents used creative workarounds not originally
+anticipated by the platform provider. For example, it is not easily
+possible to download and share drafts of videos that are not yet
+published. Amy (F, 17), however, described a popular workaround:
+“I post videos on TikTok, but only for me. Afterward, I’m able to
+download them to share them with my friends on WhatsApp.”
+4.4.2
+Device features. As part of the greater sociotechnical system,
+some devices (e.g., smartphones) offer features that affect user pri-
+vacy. For example, study participants make use of the “digital well-
+being” functionality of their smartphone to limit their screentime:
+“I used TikTok three hours a day because I didn’t know anything
+better to do with myself. Now I’m trying to get a handle on this
+with a screen time limit.” stated Matthias (M, 17). Sandra (F, 14)
+was one of the study participants who used smartphone features to
+share videos more selectively: "You can take a screenshot of drafts
+with an iPhone and then send them via WhatsApp or Snapchat.".
+As mentioned earlier, Lima (F, 14) noticed that the size of the red
+“publish” button grew with each new app update compared to the
+7
+
+Proceedings on Privacy Enhancing Technologies 2023(2)
+Ebert et al.
+grey “save as draft” button. Fearing accidental publication, she by-
+passed this potentially manipulative design pattern (“dark pattern”)
+by using an old version of the app, which her operating system
+allowed her to do: “Therefore, I have blocked the updates for TikTok
+on my cell phone.”.
+4.5
+Motivation (Automatic)
+4.5.1
+Negative emotion avoidance. The interviewees describe vari-
+ous negative emotions if they appeared in a video on TikTok. For
+example, they mentioned feelings of discomfort, shame, awkward-
+ness, and annoyance. Milo (M, 12), who does not publish any videos,
+said: “I would be embarrassed to be seen in a video.” Elsa (F, 14)
+reported that her desire to avoid negative emotions had evolved.
+While she had posted videos on musical.ly, she didn’t publish on
+TikTok anymore: “Posting TikToks has become weird for me.”.
+4.6
+Motivation (Reflective)
+4.6.1
+Negative reaction avoidance. Another reason for not pub-
+lishing personal content was negative reactions by others to their
+videos such as being bullied in class (e.g., in the WhatsApp class
+chat). Alexander (M, 15), who does not publish any videos, com-
+mented: “You make a mistake, people from school see it, it gets sent
+on, and you get bullied.”. Avoidance can also relate to the negative
+long-term consequences of sharing personal content. As they get
+older, adolescents who are getting ready to join the job market real-
+ize that their activity on TikTok could harm their career prospects.
+“The Internet never forgets and if I eventually look for an appren-
+ticeship, it may be that my future employer sees that. That’s very
+bad for my reputation.” (Lima (F, 14)).
+4.6.2
+Privacy identity. With privacy identity, we refer to a coher-
+ent set of privacy-related behaviors and personal qualities of an
+individual in a social setting [60]. Some teenagers consider privacy
+as a value in itself and part of their identity. For example, for Yara
+(F, 14), the publication of videos on TikTok is no different from
+any social network activity: “It’s just not my thing. I don’t post in
+general either, not even on Instagram or anything.”. Lena (F, 17)
+explicitly stated that she considers privacy a significant personal
+value: “Privacy is important to me. I keep everything private that
+can be kept private.”.
+4.6.3
+Publicity avoidance. Another motivation for restricting the
+publication of personal videos on the platform is closely related to
+the linkability experience theme: the desire to not attract public
+attention. Study participants explained that publishing on TikTok
+means being in the public eye: “It’s a big platform, and I don’t
+want people around me to see that I make videos.” (Anna (F,14)).
+While in musical.ly, the public was described as a community of
+people with similar interests and ages, on TikTok, it is perceived
+as a heterogenous, superficial place with different people of all
+ages (including strangers, peers from the same school, teachers,
+extended family members, and parents). Lina (F, 17) described how
+the change in the audience had an impact on her behavior: “At
+musical.ly, there were also strangers, but more my age. But TikTok
+is now worldwide and there are adults everywhere. I don’t have
+to post anything there.”. Her comment shows that the platform is
+now perceived as completely public, whereas it used to be a more
+private community.
+5
+DISCUSSION
+Our general observation of adolescents’ on TikTok is in line with
+previous research on other social networks [1, 8, 17, 53]: Contrary
+to public perception which portrays the publication of TikToks
+by young people as automatic and unreflective, the adolescents in
+our sample actively engaged in privacy management. They demon-
+strated a strong awareness of the need to manage their online
+identity and social privacy on the platform. However, the interview
+participants were more concerned with protecting their privacy
+from their immediate social environment than with institutional or
+commercial privacy issues. That is, while they were generally aware
+that TikTok used algorithms to tailor video content to their partic-
+ular online behavior, they were more worried about the tangible
+aspects of the algorithm: that a published video could immediately
+appear on a classmate’s account.
+Next, we will discuss the results in more detail following the
+structure of the COM-B model. While many of our findings are
+consistent with themes found in previous research on other social
+media platforms (e.g., Facebook), a few themes and aspects are
+indeed unique and – best to our knowledge – have not yet been
+studied by researchers on TikTok or other platforms. The qualitative
+nature of our data inform the design of very concrete interventions
+on TikTok (Section 5.7).
+5.1
+Behavior
+In addition to previous research on other social networks [59],
+we were able to identify two very different types of proactive pri-
+vacy behavior: public and private creation. While public creators
+perform privacy management to share videos directly on TikTok,
+private creators merely use the platform to create and edit videos
+to share them on other social networks that they see more appro-
+priate for such content (e.g., Snapchat, WhatsApp). It indicates that
+adolescents have different "imagined audiences" (mental conceptu-
+alization of the people with whom the user is communicating, [49])
+on each social network and curate who sees what by switching
+between networks. A unique finding of our study is that private
+creators essentially reduce TikTok, which was originally conceived
+as a social network, to its extensive audio-visual capabilities and
+share their personal content where social connections already exist
+and a higher degree of perceived control and intimacy exists (e.g.,
+WhatsApp). It is possible that such a practice might also be found
+elsewhere (e.g., Instagram, YouTube). At a time when adolescents’
+increasingly use multiple social media platforms at once, privacy
+perceptions of and management between different platforms has to
+be addressed more comprehensively. That is, privacy management
+can no longer be seen as a single-platform-phenomenon – an obser-
+vation with important research implications. Rather than focusing
+on isolated social networks with their own privacy standards, re-
+searchers should expand their analysis to include a cross-network
+view of privacy management.
+8
+
+Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok
+Proceedings on Privacy Enhancing Technologies 2023(2)
+5.2
+Psychological Capabilities
+Similar to previous studies on other social media platforms [1, 52,
+53], we found that adolescents possess knowledge and skills on how
+to manage their privacy on TikTok (see "privacy literacy" theme).
+That is, adolescents were not only able to assess the audience of
+videos but also to actively manage the audience and content of
+their TikToks. As previously noted [58], privacy management can
+be very creative. This finding also holds true for TikTok: some
+of our respondents reported using various accounts for different
+audiences, blocking app updates to avoid receiving less privacy-
+friendly versions of the app, and making an effort to detect fake
+users trying to follow them. An interesting observation that can
+potentially inform other research on social privacy management
+in social networks is that adolescents on TikTok do not only use
+the technical features provided by the social network itself. Instead,
+some are also capable of using physical opportunities provided the
+device (e.g., blocking app updates, screen time management). This
+example illustrates how the existence of these generic physical
+opportunities provided by the operating system can influence the
+privacy management capability of young TikTok users to learn
+about additional ways to protect their privacy.
+In line with previous research we found that negative past expe-
+riences affect future privacy management behaviors [53]. Incidents
+can even serve as a learning opportunity [82]. In our sample, partici-
+pants experienced near or actual privacy incidents (e.g., accidentally
+publishing videos, loss of account with personal videos) that led
+them to adapt their privacy management (e.g., immediately deleting
+accidentally published videos, paying more attention to a publi-
+cation in the future). While our data support the hypothesis that
+incidents serve as learning opportunities, it must be said that cer-
+tain very extreme violations of privacy (e.g., persistent bullying or
+stalking) have not been reported in our study. It is unclear how such
+experiences affect privacy behavior in the long run. Nonetheless,
+our findings inform future research by showing that even minor pri-
+vacy incidents without severe consequences can lead to improved
+capabilities.
+5.3
+Physical Opportunities
+Adolescents in our sample used various features of TikTok and
+the operating system to manage their privacy (themes platform
+features and device features). At the same time, they were aware
+of TikTok’s privacy management limitations (e.g., the ineffective-
+ness of blocking users). Some of the measures TikTok has taken to
+protect the privacy of younger users in response to public criticism
+may not be very effective. Out of 29 study participants with whom
+we discussed the topic, two-thirds used a false age. Many teenagers
+we interviewed have been publishing on TikTok much before the
+legally allowed age of 13. Regardless of the normative standpoint,
+this calls into question TikTok’s fine-grained, age-based privacy
+features. Despite legislative measures such as the Children’s Online
+Privacy Protection Act of 1998, this problem has been described on
+other social networks in the past [51, 54]. Sometimes also parents
+help their underage children to access social networks [10]. Rea-
+sons for using social networks below the specified minimum age
+are diverse (e.g., wanting to stay in touch with classmates, want-
+ing unrestricted access to TikTok’s features) [10]. Consequently,
+technical measures to protect children such as non-public accounts
+or content restrictions are failing [65]. boyd et. al [10] called for
+abandoning ineffective age-based mechanisms. Instead, she advo-
+cates for an honest discussion about children’s use of social media
+and a rethinking of the industry to better incorporate the needs of
+children and parents when developing apps.
+Another issue on social networking sites is account loss [66].
+This issue was also highlighted by several of our respondents who
+reported that they were unable to reclaim a video they had posted
+after losing an account. As a consequence, they were unable to
+revoke their consent from publishing a childhood experiment that
+would now remain online forever. This is particularly problematic
+against the background of increasingly better algorithms for rec-
+ognizing people in images and videos and the resulting linkability
+risk (e.g., Clearview AI [36]). To exercise the "right to be forgotten"
+as embodied in the EU GDPR, for example, the ability to reclaim
+accounts and delete old videos is essential. It is unclear whether ac-
+count loss among adolescents is a broader phenomenon or whether
+other social networks are affected as well.
+5.4
+Social Opportunities
+Our findings on TikTok support previous research demonstrating
+that the social environment of teenagers shapes their privacy be-
+haviors [53]. Other social network users as well as the parents are
+major agents of socialization [31]. Social norms, which emerge as
+a response to observed behavior or expected attitudes of friends
+and parents, influence children’s intention to share personal infor-
+mation [77]. If friends and parents disapprove of such behavior,
+children tend to share less. A recent study on TikTok described, that
+restrictive mediation by parents can also lead to more restrictive
+disclosure behavior in children [40].
+In our study, we identified similar social influences on TikTok.
+Observing strangers being publicly criticized for videos (theme
+negative feedback) resulted in restrictive publication behavior by
+the adolescents we interviewed. In line with previous research [77],
+the restrictive norms and behavior of relatives, parents, and friends
+were also found to have the potential to affect behavior on TikTok
+(e.g., not publishing or blocking parents from videos).
+What makes TikTok stand out from other social networks, is its
+specific content algorithm based on a granular observation of user
+preferences [45]. The results of our study indicate that prevalent
+TikTok usage among peers in combination with the platform’s spe-
+cific algorithm that immediately displays the published content to
+cohorts with similar attributes – i.e., peers – may increase the social
+influence of others on adolescents’ privacy behavior (“linkability
+experience”). Unlike posting a video under a nickname on YouTube
+that may never be discovered by peers, adolescents were aware that
+posting on TikTok was potentially more privacy-invasive. They
+recognized that their videos could become visible to their personal
+environment (e.g., in the schoolyard). This experience led to re-
+stricted publication behavior.
+5.5
+Automatic and Reflective Motivations
+Adolescents’ motivations for protecting their privacy on TikTok
+were based on either wanting to avoid publicity, to avoid nega-
+tive reactions/emotions, or to actively achieve privacy (themes
+9
+
+Proceedings on Privacy Enhancing Technologies 2023(2)
+Ebert et al.
+negative reaction avoidance, publicity avoidance). The adolescents
+interviewed reported wanting to evade the public eye and feared
+negative feedback (e.g., public criticism). These are themes previ-
+ously described on other social networks [53]. To avoid a negative
+emotional outcome (e.g., shame), they refrain from having a too
+public profile (theme negative emotion avoidance) (see [13] for a
+similar finding).
+For some adolescents, privacy was a personal matter beyond
+TikTok (theme privacy identity). That is, these teenagers were in-
+trinsically motivated to keep their information private - a finding
+that stands in contrast with previous research on other social net-
+works. Research suggests that, on average, adolescents have fewer
+privacy concerns than young adults [4, 22]. However, our findings
+indicate that these concerns can vary greatly across adolescents,
+and some may place great value on their privacy on social media.
+Even though the theme was mentioned by only a few participants,
+it underscores that adolescents are not a homogeneous group when
+it comes to motives for managing privacy on social media. For some
+participants’ being private is a personal value and their goal is to
+achieve a coherent privacy behavior on TikTok and beyond.
+5.6
+Methodological Consideration
+For our study, the COM-B model helped to holistically understand
+adolescents’ privacy management on TikTok related to the creation
+of videos. It has a solid theoretical foundation and – according to its
+authors – can be applied across various contexts. However, much of
+the research to date has applied the COM-B model to health-related
+behaviors such as smoking cessation and lowering cardiovascular
+disease risk [60]. Our study, which showed that the COM-B model is
+also a suitable analytical framework for studying privacy behavior,
+provides yet another use case. By demonstrating its relevance to
+the privacy management of adolescents, we strengthen the model’s
+extrinsic validity.
+5.7
+Possible Approaches for Privacy
+Interventions
+Several of the themes we identified can be used as starting points
+for the development of privacy interventions. The COM-B model is
+part of a theory-driven intervention development framework called
+behavior change wheel (BCW), a synthesis of behavior change
+frameworks [62]. In the logic of the BCW, interventions are di-
+rected at desired “target behaviors” (e.g., enabling privacy settings).
+Building on the interview findings and our observations, Figure 2
+shows different parties and ideas for potential target behaviors af-
+fecting adolescents’ video privacy management. It focuses on which
+behaviors to address and does not answer the question of how to
+design interventions that address these behaviors (e.g., adequate
+behavior change techniques [61]).
+Any intervention schemes to improve the privacy of adoles-
+cent TikTok users should focus on the behavior of the adolescents
+themselves. The interviews provide concrete suggestions for be-
+haviors that adolescents already report that improve their privacy
+protection. This includes encouraging young users to remove in-
+appropriate videos from the platform and the use of alternative
+social media apps (e.g., WhatsApp) to share content (theme: proac-
+tive privacy). Some of our participants reported regular checks if a
+video with the status “published” should be set to private. They also
+removed their old TikToks from the app and their smartphone. Our
+private creators did seldom publish on TikTok but used alternative
+apps such as Snapchat or WhatsApp with a perceived higher level
+of privacy and the ability to automatically delete shared TikToks
+after being watched by their friends. Another possible target behav-
+ior derived from our observations is “backing up user credentials”
+(theme: privacy literacy). Some adolescents in our sample who had
+already created accounts in musical.ly could not delete published
+videos because they had forgotten their credentials, and were not
+able to prove their identity to the TikTok support to retrieve their
+account. An intervention could mitigate account loss, especially
+in cases where children have multiple accounts. Finally, teenagers
+should be made aware of the privacy settings (e.g., the private ac-
+count) and the potential risks of not correctly setting these (theme:
+reflective motivation). For example, in our interviews, participants
+accidentally published a TikTok upon their first usage of the app
+because they were not aware others would immediately see it.
+The platform must also play an important role in safeguarding
+the privacy of children and adolescents. Improving features directly
+related to privacy such as improved age verification, more effec-
+tive blocking of users, and facilitating access to lost user accounts
+are promising approaches (theme: platform features). As described
+earlier, many adolescents in our sample did not use their real age
+due to various reasons. For example, they were often unaware that
+the private account would have been activated by default if they
+had provided their real age. As a result of providing false infor-
+mation, the privacy settings were much more lenient and TikTok
+videos would not only be published to followers but to everybody.
+Following boyd et. al’s [10] philosophy, one possibility would be
+to abandon TikTok’s age-based mechanism and incorporate the
+needs of children and parents when developing the app. For TikTok,
+this could mean taking a certain level of responsibility for its con-
+tent and giving kids and parents ways to control what videos are
+shown (e.g., via a content configuration or a separate app similar
+to YouTube Kids). Even if the app adhered to the age-based privacy
+concept, describing the consequences of providing real age (e.g.,
+better privacy protection) might encourage some youth to provide
+their real age. Another approach has been lately launched by the
+twin app Douyin [30]. Douyin introduced an age verification that
+is not based on self-declaration only but requires – unlike the in-
+ternational counterpart TikTok - user authentication and imposes
+restrictions on the permitted daily use for users under 14.
+Some participants also criticized that they could not effectively
+block users who they wanted to prevent from seeing their videos.
+The problem persists because blocked users can immediately "respawn"
+under a different username. TikTok could prevent this issue with
+a feature that block all accounts of the same user (similar to Insta-
+gram [39]). Some study participants also reported feeling “nudged”
+by the user interface design towards publishing TikTok video for a
+broad audience. Others described publishing personal TikToks acci-
+dentally. While nudging teenagers towards better privacy behavior
+is also controversial [78], presenting them with simple alternatives
+(such as publishing a TikTok vs saving a local draft) could provide a
+welcome middle ground. Furthermore, TikTok might also do more
+to educate its users on how to protect their privacy. This suggestion
+10
+
+Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok
+Proceedings on Privacy Enhancing Technologies 2023(2)
+TikTok 
+Users
+Family & 
+Friends
+Schools
+& Youth 
+Work
+Policy-
+Makers & 
+Privacy 
+Advocates
+Other 
+Platform 
+Users
+OS 
+Vendors & 
+Other Apps
+ByteDance 
+(TikTok 
+Creator)
+Enable 
+privacy 
+settings
+Share/remove 
+personal content 
+consciously
+Backup 
+account
+credentials 
+Share TikToks
+using other apps
+(e.g., Whatsapp)
+Inform each other 
+about privacy 
+possiblities
+Do not nudge 
+users towards 
+publication
+Educate children about long-
+term privacy risks
+Improve privacy 
+features (e.g., 
+reclaiming ‘lost’ 
+accounts, age 
+verification)
+Use TikTok yourself to 
+understand privacy 
+issues
+Educate students early 
+about longterm 
+privacy risks
+Support childrens’
+privacy efforts
+Provide privacy 
+tutorials
+Create & enforce 
+privacy laws 
+(e.g., transparency 
+about PII usage,
+age verification)
+Housekeeping
+functionality for 
+TikTok
+Enforce app 
+privacy in OS
+Explain business 
+model of TikTok
+Use TikTok yourself to 
+understand privacy 
+issues
+Explain business 
+model of TikTok
+Respect the privacy
+of others
+(e.g., norms)
+Figure 2: Different parties and their potential target behaviors relevant for adolescents’ video privacy management on TikTok
+is based on our observation that capabilities varied between adoles-
+cents and TikTok users had begun to create such privacy tutorials.
+The latter indicates a demand for more support (e.g., via privacy
+tutorials provided by TikTok).
+Family, friends, schools, and youth workers can also positively in-
+fluence the privacy management of adolescents (social opportunity
+themes). In addition to supporting adolescents’ privacy efforts, their
+social network could use TikTok themselves to better understand
+specific privacy issues. In our sample, an uncle of an eight-year-old
+boy used TikTok himself and warned him about the possibility on
+TikTok of publishing a video by accident. The social environment
+can also advise about long-term privacy risks to the children and
+adolescents of which they might not yet be aware. Among a group
+of adolescents of the same class, we repeatedly heard the narrative
+of a classmate being recognized on TikTok despite her wearing
+a mask. Due to this “risk narrative” the whole class was aware
+of the potential risks of insufficient anonymization on TikTok. A
+collection of such tales could be used by teachers in the classroom
+to illustrate the privacy risk associated with the platform.
+As users do not only interact with each other when they share
+videos but also with the platform and its owner company, teenagers
+should also be made aware of commercial privacy issues. Our data
+confirmed that adolescents’ primary privacy focus was indeed so-
+cial. To this end, adolescents would need to understand TikTok’s
+business model, which heavily relies on their personal data, and
+the organization behind TikTok.
+Policymakers and privacy advocates are also relevant actors. Not
+only do they seek to create privacy laws to protect users but also to
+enforce these laws through, for example, insisting on effective age
+verification (theme: platform features). Ideally, these actions are
+guided by evidence in collaboration with researchers, adolescent
+users, and parents. For example, our findings indicate that ado-
+lescents did not know that TikTok had taken additional measures
+to protect them in 2021 [75]. While privacy legislation demands
+transparency for data subjects – especially for children – this ex-
+ample shows that there is room for improvement in terms of the
+implementation of laws.
+It should also be mentioned that other TikTok users can influence
+an adolescent’s privacy behavior (social opportunity themes). Older
+and more experienced teenagers may have capabilities (e.g., based
+on their negative experiences) that can benefit younger and less
+experienced users. One of our participants reported having learned
+about privacy settings from a video on TikTok. Indeed, some more
+experienced users have already begun to acts as mentors. This
+includes the user @seansvv with 1.1 million followers, who stated
+in his biography “I Read ToS [Terms of Service] So That You Don’t
+Have To” and regularly posts TikTok videos related to privacy topics
+[71].
+Finally, our interviews showed that OS vendors and the vendors
+of other apps contribute to teenagers’ privacy on TikTok (theme:
+device features). OS vendors have implemented more and more
+privacy control mechanisms for their end-users (e.g., granular rights
+management, location sharing notifications). These methods all
+work on low-level personal data (e.g., IP address, location, and
+email address). However, videos shared by adolescents on TikTok
+that possibly contain more sensitive personal data with higher risks
+involved are not yet covered by these mechanisms. At times when a
+user publishes a video accidentally, the OS could warn them in the
+same way that they are warned when sharing their location with
+the app. In our sample, participants reported also manually cleaning
+11
+
+Proceedings on Privacy Enhancing Technologies 2023(2)
+Ebert et al.
+up their TikToks in the app and on their phones. OS vendors could
+provide housekeeping functionalities that would simplify removing
+personal content across different social networks and on the phone.
+5.8
+Limitations and Future Research
+As with most qualitative research, our sample is small and was not
+drawn randomly. Therefore, we cannot claim that the results are
+representative of all young people in the region under consideration,
+and certainly not of Switzerland as a whole. Further validation
+with different samples is needed to strengthen the findings (e.g.,
+including subjects’ socioeconomic status).
+Choosing interviews as our data collection methodology was use-
+ful to learn more about the perspectives of adolescents in Switzer-
+land. Nevertheless, we are aware of the limitations associated with
+this method. Primarily, we relied on self-reporting rather than be-
+havioral observations. Self-reports can be biased due to various
+influences, such as subjects’ desire to portray themselves in a posi-
+tive light. Future studies might want to gather data from a wider
+range of sources, such as direct observations of privacy manage-
+ment behavior (e.g., through TikTok data donations).
+Based on our findings, future research could develop and system-
+atically test privacy interventions based on the BCW methodology.
+A necessary first step would be to identify appropriate target be-
+haviors with the greatest potential to improve privacy management
+among adolescents. Our research could be a starting point for select
+a “promising” target behavior reported by the adolescents (e.g., ac-
+tivating the private account) to address in a target population (e.g.,
+pupils of a local school). To identify a baseline for each of the poten-
+tial behaviors and to select a target behavior among them, further
+research would be necessary (e.g., in form of a survey among pupils).
+Furthermore, additional research is required to select appropriate
+behavior change techniques (e.g., increasing awareness for privacy
+settings) and evaluate their effectiveness (e.g., with an experiment).
+Importantly, such research could also control for factors such as
+socioeconomic status might also be relevant to explain privacy-
+related behaviors on TikTok [34]. Given that teenagers may have
+very heterogeneous privacy management capabilities, motivations,
+and opportunities, depending on their age and experience regarding
+the platform, interventions need to be tailored to the specific target
+group. Large-scale intervention studies using the BCW can help to
+identify effective and evidence-based policies to improve privacy
+management among young people on social media platforms like
+TikTok.
+Our interviews focused on social aspects of adolescents’ privacy
+management. That is, our interviewees were more concerned with
+protecting their privacy from their social environment than from
+the corporations dealing with their data for commercial purposes;
+see [53]. Yet, TikTok videos are not only shared with other users
+but also with ByteDance. Even the users we identified as pure
+consumers who only view but not create content may have privacy
+issues. As the video and ad algorithms are known for their high
+level of customization, they make the platform heavily reliant on
+personal data including detailed user behavior [45]. Both users’
+active and passive behavior on the app has consequences: The
+TikTok pixel allows companies to engage in detailed web tracking
+of TikTok users on websites (e.g., a user who sees the ad on TikTok
+might buy the product in the online shop) [12]. Further research
+could investigate if adolescent users are aware of these commercial
+privacy aspects and how they manage them.
+ACKNOWLEDGMENTS
+The research reported in this article was funded by the Digital
+Future Fund (DFF), which is part of the Digitalization Initiative of
+the Zurich Higher Education Institutions (DIZH), Switzerland. We
+would like to thank all adolescents, teachers, and social workers
+we contacted in conducting our study. We also thank Frank Wieber,
+Katja Kurz and Manuel Günther for their helpful comments.
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+They Use It. Retrieved 2022-11-02 from https://www.today.com/parents/tiktok-
+changes-privacy-settings-kids-under-18-t205733
+[76] Sabine Trepte, Doris Teutsch, Philipp K. Masur, Carolin Eicher, Mona Fischer,
+Alisa Hennhöfer, and Fabienne Lind. 2015. Do People Know About Privacy
+and Data Protection Strategies? Towards the “Online Privacy Literacy Scale”
+(OPLIS). In Reforming European Data Protection Law, Serge Gutwirth, Ronald
+Leenes, and Paul de Hert (Eds.). Springer Netherlands, Dordrecht, 333–365. https:
+//doi.org/10.1007/978-94-017-9385-8{_}14
+[77] Ellen Van Gool, Joris Van Ouytsel, Koen Ponnet, and Michel Walrave. 2015. To
+Share or Not to Share? Adolescents’ Self-Disclosure about Peer Relationships on
+Facebook: An Application of the Prototype Willingness Model. Computers in
+Human Behavior 44 (2015), 230–239. https://doi.org/10.1016/j.chb.2014.11.036
+[78] Mariana Veretilnykova and Leyla Dogruel. 2021. Nudging Children and Adoles-
+cents toward Online Privacy: An Ethical Perspective. Journal of Media Ethics 36,
+3 (2021), 128–140. https://doi.org/10.1080/23736992.2021.1939031
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+pany.
+https://www.reuters.com/technology/tiktok-hits-1-billion-monthly-
+active-users-globally-company-2021-09-27/
+[80] Robert West and Susan Michie. 2020. A Brief Introduction to the COM-B Model
+of Behaviour and the PRIME Theory of Motivation [V1]. Retrieved 2022-11-02
+from https://www.qeios.com/read/WW04E6
+[81] Alan F Westin. 1967. Privacy and Freedom. Vol. 7. Athenum, New York.
+[82] Pamela Wisniewski. 2018-03. The Privacy Paradox of Adolescent Online Safety:
+A Matter of Risk Prevention or Risk Resilience? IEEE Security Privacy 16, 2
+(2018-03), 86–90. https://doi.org/10.1109/MSP.2018.1870874
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+Adolescence. In Modern Socio-Technical Perspectives on Privacy, Bart P. Knij-
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+Proferes, and Jennifer Romano (Eds.). Springer International Publishing, Cham,
+315–336. https://doi.org/10.1007/978-3-030-82786-1{_}14
+[84] Queenie Wong. 2019. TikTok Accused of Secretly Gathering User Data and Send-
+ing It to China. Retrieved 2022-11-02 from https://www.cnet.com/news/privacy/
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+[85] Alyson Leigh Young and Anabel Quan-Haase. 2013. Privacy Protection Strategies
+on Facebook. Information, Communication & Society 16, 4 (2013), 479–500. https:
+//doi.org/10.1080/1369118X.2013.777757
+[86] Brahim Zarouali, Karolien Poels, Koen Ponnet, and Michel Walrave. 2018. “Ev-
+erything under Control?”: Privacy Control Salience Influences Both Critical
+Processing and Perceived Persuasiveness of Targeted Advertising among Ado-
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+(2018). https://doi.org/10.5817/CP2018-1-5
+[87] Diana Zulli and David James Zulli. 2020. Extending the Internet Meme: Concep-
+tualizing Technological Mimesis and Imitation Publics on the TikTok Platform.
+New Media & Society 24, 8 (2020). https://doi.org/10.1177/1461444820983603
+A
+APPENDIX
+A.1
+Interview Guide (translated from German)
+(1) What’s your first name? How old are you? In what grade
+are you?
+(2) How often do you use TikTok? How long have you been
+using TikTok? When did you start to use TikTok?
+(3) Do you remember how old you were when you started using
+the app?
+(4) How many people do you follow? How many followers do
+you have?
+(5) Do you share videos? How many? What types of videos?
+(Behavior)
+(a) Why do you/don’t you share videos? (Motivation)
+(b) If yes: How do you share videos on TikTok? (Psychological
+capability)
+(6) Who can see your videos when they are shared? (Psycholog-
+ical capability)
+(7) How can you influence who can see your videos? (Psycho-
+logical capability)
+(8) Do you restrict who can see your videos? (Behavior)
+(a) If yes: Why / When do you restrict your videos? (Motiva-
+tion)
+(b) If no: Why? Have you ever considered restricting your
+videos? (Motivation)
+(9) Do your friends or others restrict their/your videos? (Social
+Opportunity)
+(10) Have you ever accidentally posted a video? If yes: What did
+you do? (Psychological capability)
+(11) What do you think about TikTok’s features to share/restrict
+videos? (Physical opportunity)
+A.2
+Coding Scheme
+Table 3 shows the hierarchical coding scheme together with the
+frequency of each code calculated across the 54 interviews.
+14
+
+Creative beyond TikToks: Investigating Adolescents’ Social Privacy Management on TikTok
+Proceedings on Privacy Enhancing Technologies 2023(2)
+Table 3: Coding Scheme (translated from German). Frequency is calculated across 54 interviews.
+Code
+Description
+Frequency
+Usage_since
+Start of TikTok usage
+54
+Usage_frequency
+Frequency of TikTok usage
+53
+App_age
+Age entered into the app at first use
+29
+Video_Behavior
+Avoidance
+I normally do not create/publish TikToks.
+24
+Proactive
+PersonalCreator
+I regularly create/publish TikToks for myself and close friends.
+19
+PublicCreator
+I regularly create/publish TikToks for my followers/the public.
+11
+Video_PsyCapability
+PastPrivacyIncidents
+Minor
+I have perceived a potential/minor privacy incident.
+9
+Severe
+I have perceived a severe privacy incident.
+8
+PrivacyLiteracy
+AudienceContentLiteracy
+I’m aware of different audience/content types and have the ability to manage
+them.
+52
+TechnicalLiteracy
+I have the technical knowledge and skills to manage my audience.
+50
+Video_SocOpportunity
+NegativeFeedback
+Others show negative reactions to TikToks, that’s why I’m not active.
+16
+LinkabilityExperience
+Users can be easily recognized in real life.
+39
+RestrictiveInfluence
+I’m not active because others are also restrictive or enforce my privacy.
+34
+Video_PhyOpportunity
+PlatformFeatures
+TikTok helps to ensure my privacy.
+46
+DeviceFeatures
+The device helps to ensure my privacy.
+17
+Video_AutMotivation
+NegativeEmotionAvoidance
+I don’t publish content to avoid negative emotions.
+15
+Video_RefMotivation
+NegativeReactionAvoidance
+I don’t publish content to avoid negative reactions.
+10
+PrivacyIdentity
+I don’t publish content because privacy is important to me.
+5
+PublicityAvoidance
+I don’t publish content because I don’t want to be in the public eye.
+29
+15
+

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+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+JOHN R. DOYLE AND DAVID KRUMM
+Abstract. Among all the dynamical modular curves associated to quadratic polynomial
+maps, we determine which curves have infinitely many quadratic points.
+This yields a
+classification statement on preperiodic points for quadratic polynomials over quadratic fields,
+extending previous work of Poonen, Faber, and the authors.
+1. Introduction
+Let K be a field with algebraic closure K, and let f be a rational function in one variable
+over K. Corresponding to f there are a morphism of algebraic varieties P1
+K → P1
+K and a
+map on point sets P1(K) → P1(K), both of which we also denote by f. A point P ∈ P1(K)
+is called periodic for f if there exists a positive integer n such that f n(P) = P, where f n
+denotes the n-fold composition of f with itself; in that case, the smallest such n is called
+the (exact) period of P. More generally, the point P is preperiodic for f if there exists
+m ≥ 0 such that f m(P) is periodic; the smallest such m is then called the preperiod of P,
+and we call the period of f m(P) the eventual period of P. Here, f 0 is interpreted as the
+identity map, so that periodic points are considered preperiodic.
+For any intermediate field K ⊆ L ⊆ K we define a directed graph G(f, L), called the
+preperiodic portrait of f over L, whose vertices are the points P ∈ P1(L) that are
+preperiodic for f, and whose directed edges are the ordered pairs (P, f(P)) for all vertices
+P. In the terminology of graph theory, G(f, L) is a functional graph, i.e., a directed graph in
+which every vertex has out-degree 1. Throughout this paper we will use the term portrait
+instead of functional graph in order to emphasize our dynamical perspective.
+1.1. Portraits for quadratic maps. Assume henceforth that K is a number field. We will
+primarily, though not exclusively, be interested in the case where K is a quadratic extension
+of Q; we refer to such fields simply as quadratic fields. A type of problem that has received
+much attention in the field of arithmetic dynamics is that of classifying the portraits G(f, K)
+up to graph isomorphism as f is allowed to vary in an infinite family of rational functions.
+An early example of this classification problem is Poonen’s study [43] of the portraits G(f, Q)
+as f varies over the family of all quadratic polynomials with rational coefficients.
+Theorem 1.1 (Poonen [43]). Assume that there is no quadratic polynomial over Q having
+a rational periodic point of period greater than 3.
+Then, for every quadratic polynomial
+f ∈ Q[z], the portrait G(f, Q) is isomorphic to one of the following twelve graphs (using the
+labels from Appendix B):
+∅, 2(1), 3(1, 1), 3(2), 4(1, 1), 4(2), 5(1, 1)a, 6(1, 1), 6(2), 6(3), 8(2, 1, 1), 8(3).
+Date: January 3, 2023.
+2020 Mathematics Subject Classification. Primary 37P05, 37P35; Secondary 37P15, 11G30, 14G05.
+Key words and phrases. Arithmetic dynamics, dynatomic curve, preperiodic portrait, uniform bounded-
+ness conjecture.
+1
+arXiv:2301.00510v1  [math.NT]  2 Jan 2023
+
+2
+JOHN R. DOYLE AND DAVID KRUMM
+Regarding the assumption in Theorem 1.1, it is known that a quadratic polynomial over
+Q cannot have rational periodic points of period 4 (Morton [38]), period 5 (Flynn–Poonen–
+Schaefer [18]), or, assuming that the conclusions of the Birch and Swinnerton-Dyer conjecture
+hold for a certain Jacobian variety, period 6 (Stoll [47]). In addition, a substantial amount
+of empirical evidence supporting the assumption in Poonen’s theorem has been provided by
+Hutz and Ingram [23] and Benedetto et al. [1]. However, it remains an open problem to
+prove that this assumption is valid. Portraits for other families of quadratic maps over Q are
+studied in the articles [3,6,32,33]. The present paper concerns the preperiodic portraits of
+quadratic polynomials defined over quadratic fields, a topic previously explored in [10,12,13].
+1.2. Analogy with torsion points. A guiding principle that has proved fruitful in arith-
+metic dynamics is to regard the set of preperiodic points of a map as being analogous to
+the set of torsion points on an abelian variety. Thus, for instance, the well-known fact that
+the set of K-rational torsion points on an abelian variety is finite is viewed as analogous to
+a theorem of Northcott [41] stating that, for every rational function f over K of degree at
+least 2, the set of K-rational preperiodic points of f is finite.
+Motivated by this analogy, Morton and Silverman formulated the following dynamical
+analogue of a standard uniform boundedness conjecture for abelian varieties. We state the
+dynamical conjecture only in the case of endomorphisms of the projective line, although a
+similar statement applies to arbitrary projective spaces1.
+Conjecture 1.2 (Morton–Silverman [39]). For a number field K and morphism f : P1
+K → P1
+K
+of degree greater than 1, the number of K-rational preperiodic points of f is bounded above
+by a constant depending only on the degree of f and the absolute degree of K.
+This dynamical uniform boundedness conjecture would imply, in particular, that there are
+only finitely many isomorphism classes of portraits G(f, Q) as f ranges over all quadratic
+polynomials with rational coefficients, since the number of vertices in such a portrait is
+uniformly bounded. Theorem 1.1 can thus be seen as a refinement of the conjecture in this
+case, as it provides a (conditionally) complete list of all possible portraits for the family of
+quadratic polynomial maps. In the analogy with torsion points, Poonen’s list of portraits
+corresponds to the list of abelian groups that can be realized as the torsion subgroup of an
+elliptic curve over Q, the latter list being provided by a well-known theorem of Mazur [34].
+Similarly, the Morton–Silverman conjecture would imply that the portraits G(f, K), where
+K is a quadratic field and f is a quadratic polynomial over K, fall into finitely many iso-
+morphism classes. A conjecturally complete list of classes was first proposed in [13], and is
+included here in Appendix B. The list is comprised of 46 portraits, and can be viewed as anal-
+ogous to the list of 26 abelian groups, known by work of Kamienny [25] and Kenku–Momose
+[26], that can arise as torsion subgroups of elliptic curves over quadratic fields.
+1.3. Infinitely occurring portraits. Our primary objective in this paper is to determine,
+under a suitable notion of equivalence of maps, which of the 46 graphs in [13] arise as the
+preperiodic portrait of infinitely many inequivalent quadratic polynomials over quadratic
+fields. In the context of elliptic curves, both over Q and over quadratic fields, the correspond-
+ing question is well understood: every abelian group that arises as the torsion subgroup of
+an elliptic curve can be realized as such by infinitely many non-isomorphic curves (see [24]).
+1Interestingly, work of Fakhruddin [17] shows that the more general Morton–Silverman conjecture in fact
+implies its analogue for abelian varieties. See also [45, §3.3], where Merel’s theorem for elliptic curves is
+shown to follow from Conjecture 1.2
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+3
+In order to state our questions more precisely, we begin by defining the appropriate equiv-
+alence relation on maps. Two morphisms f, h : P1
+K → P1
+K are called linearly conjugate
+over K if there exists an automorphism σ ∈ PGL2(K) such that
+h = σ−1 ◦ f ◦ σ.
+(Similarly, one can define linear conjugacy over any extension of K.) In that case, a simple
+argument shows that the portraits G(f, K) and G(h, K) are isomorphic as directed graphs;
+hence, the isomorphism class of G(f, K) is determined by the linear conjugacy class of f.
+In the case of quadratic polynomials, it is well known that every such map f ∈ K[z] is
+linearly conjugate to a unique map of the form
+fc(z) := z2 + c,
+where c ∈ K. Thus, in studying the portraits of quadratic polynomials we may restrict
+attention to the one-parameter family of maps fc.
+Returning to the question of portraits arising infinitely often, the case of quadratic poly-
+nomials over Q was answered by Faber, who showed in addition that Poonen’s list in [43]
+does not omit any such portrait.
+Theorem 1.3 (Faber [16]). For a portrait P, the following are equivalent:
+(i) There exist infinitely many c ∈ Q such that G(fc, Q) ∼= P.
+(ii) P is isomorphic to one of the following graphs (using the labels from Appendix B):
+∅, 4(1, 1), 4(2), 6(1, 1), 6(2), 6(3), 8(2, 1, 1).
+Motivated by Faber’s theorem, we now state the main questions to be addressed here.
+Question 1.4. Among the 46 known isomorphism classes of portraits arising as G(fc, K),
+with K a quadratic field and c ∈ K, which ones can be realized as such by infinitely many
+algebraic numbers c? In addition, must every infinitely occurring portrait belong to one of
+the 46 known isomorphism classes?
+1.4. Main results. We define the following sets of portraits using labels as in Appendix B:
+Γ0 := {∅, 4(1, 1), 4(2), 6(1, 1), 6(2), 6(3), 8(2, 1, 1)};
+Γrat := {8(1, 1)a, 8(2)a, 8(4), 10(3, 1, 1), 10(3, 2)};
+Γquad := {8(1, 1)b, 8(2)b, 8(3), 10(2, 1, 1)a/b};
+Γ := Γ0 ∪ Γrat ∪ Γquad.
+We provide two answers to Question 1.4 which differ in their level of specificity. The
+simplest is Theorem 1.5, with Theorems 1.6 and 1.7 providing additional information in
+terms of the above subsets of Γ. For an integer n ≥ 1, we define
+Q(n) := {α ∈ Q : [Q(α) : Q] ≤ n}.
+Theorem 1.5. For a portrait P, the following are equivalent:
+(i) There exist infinitely many c ∈ Q(2) such that G(fc, K) ∼= P for some quadratic field
+K containing c.
+(ii) P ∈ Γ.
+
+4
+JOHN R. DOYLE AND DAVID KRUMM
+Theorem 1.5 can be refined in order to take into account certain subtleties illustrated by
+the following example: We see from Theorem 1.3 that the portrait P = 4(2) is realized as
+G(fc, Q) for infinitely many c ∈ Q. For each such c, the set of preperiodic points for fc is a
+set of bounded height, and therefore fc has only finitely many preperiodic points of algebraic
+degree 2 over Q. Hence, for each of the infinitely many c ∈ Q with G(fc, Q) ∼= P, we must
+also have G(fc, K) ∼= P for all but finitely many quadratic fields K.
+We therefore show that each of the portraits P ∈ Γ is realized infinitely often—even if
+one excludes the infinitely many “trivial” realizations in the sense of the previous paragraph.
+This is done in the next two theorems, which are stated separately in order to distinguish
+between polynomials with rational coefficients and those with quadratic algebraic coefficients.
+Theorem 1.6. For a portrait P, the following are equivalent:
+(i) There exist infinitely many c ∈ Q such that G(fc, Q) ⊊ G(fc, K) ∼= P for some
+quadratic field K.
+(ii) P ∈ Γ ∖ {∅, 6(3)}.
+Theorem 1.7. For a portrait P, the following are equivalent:
+(i) There exist infinitely many c ∈ Q(2) ∖ Q such that G(fc, Q(c)) ∼= P.
+(ii) P ∈ Γ0 ∪ Γquad.
+Note that every portrait in Γ is covered by at least one of Theorems 1.6 and 1.7, since ∅
+and 6(3) are elements of Γ0. In particular, these two theorems together imply Theorem 1.5.
+1.5. Quadratic points on dynamical modular curves. The proofs of our main results
+rely heavily on the concept of a dynamical modular curve. To each of the 46 portraits from
+[13], and more generally to any portrait that could potentially be realized as the preperi-
+odic portrait of a quadratic polynomial over a number field, we associate an algebraic curve
+parametrizing instances of the portrait as a preperiodic portrait G(fc, K). The curve cor-
+responding to a portrait P will be denoted X1(P) by analogy with the classical modular
+curves X1(N) parametrizing elliptic curves with a torsion point of order N. To avoid confu-
+sion, the latter curve will henceforth be denoted Xell
+1 (N). The details of the construction as
+well as basic properties of dynamical modular curves are discussed in [11]. A more general
+construction of dynamical moduli spaces appears in [15].
+The core of our analysis in this paper is a study of basic geometric invariants, such as
+genus and gonality, of the curves X1(P). We then turn this geometric data into arithmetic
+data using Faltings’ theorem on rational points on subvarieties of abelian varieties, via the
+following result of Harris and Silverman:
+Theorem 1.8 (Harris–Silverman [21, Cor. 3]). Let X be a smooth, irreducible, projective
+curve of genus g ≥ 2 defined over a number field K. If X is neither hyperelliptic or bielliptic
+over K, then X has only finitely many points that are quadratic over K.
+In addition, we consider arithmetic questions regarding the fields of definition of quadratic
+points on X1(P). In particular, if X1(P) has a point defined over a quadratic number field
+K, what can be said about basic arithmetic invariants of K, such as discriminant and class
+number? For the curves Xell
+1 (N), arithmetic questions of this kind have been discussed by
+several authors: Momose [35] shows that if K is the field of definition of a quadratic point
+on Xell
+1 (13), then the prime 2 splits in K, and 3 is unramified in K; Bosman et al. [5] show
+that K must be a real quadratic field, an observation also made in [13]. In the case of the
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+5
+modular curves Xell
+0 (N), Najman and Trbovi´c [40] prove arithmetic results of this type for
+several values of N.
+For the dynamical modular curves X1(P) we prove the following two theorems. Though
+our methods can be applied to several portraits in the set Γ (namely, those for which the
+corresponding modular curve is hyperelliptic), the portraits 8(4) and 10(3,1,1) are highlighted
+here due to their significance in the context of elliptic curves, explained below.
+By a quadratic point on an algebraic curve over a field k, we mean a point whose field
+of definition is a quadratic extension of k.
+Theorem 1.9. Let P denote the portrait 8(4).
+(a) For every prime p and every residue class c ∈ Z/pZ, there exist infinitely many
+squarefree integers d ∈ c such that X1(P) has a quadratic point defined over Q(
+√
+d).
+(b) There exist infinitely many imaginary quadratic fields K with class number divisible
+by 10 such that X1(P) has a quadratic point defined over K.
+As noted in [13], the above curve X1(P) is isomorphic to Xell
+1 (16). Thus, Theorem 1.9
+provides new information about the collection of quadratic fields K such that there exists
+an elliptic curve E/K with a K-rational torsion point of order 16.
+Similarly, taking P = 10(3, 1, 1), the curve X1(P) is known to be isomorphic to Xell
+1 (18).
+The next theorem strengthens earlier results by Kenku–Momose [26] regarding the splitting
+of rational primes in the fields of definition of quadratic points on this curve.
+Theorem 1.10. Let P denote the portrait 10(3, 1, 1) and let K be the field of definition of
+a quadratic point on X1(P).
+(a) The prime 2 splits in K, and 3 is not inert in K.
+(b) There exists an infinite and computable set of primes, denoted π, that is independent
+of K, and such that every prime in π is unramified in K.
+1.6. Points of higher degree. Though our primary focus here is on quadratic fields, we
+make one observation concerning arbitrary number fields. The next result is a straightforward
+consequence of a theorem of Frey [19] together with the main theorem of [14].
+Theorem 1.11. Fix a positive integer n. For any portrait P, let γ(P) denote the set of
+algebraic numbers c ∈ Q(n) such that P ∼= G(fc, K) for some number field K satisfying
+c ∈ K ⊂ Q(n). There are only finitely many portraits P such that γ(P) is infinite.
+Note that Theorem 1.5 is a more refined version of Theorem 1.11 in the case n = 2.
+1.7. Outline of the paper. In Section 2 we define the notion of a generic quadratic portrait
+and discuss basic facts concerning dynamical modular curves associated to such portraits,
+followed by general properties of algebraic curves in Section 3.
+In Section 4, we apply geometric arguments to determine all generic quadratic portraits P
+for which the curve X1(P) has infinitely many quadratic points. This proves, in particular,
+the implication (i) ⇒ (ii) in Theorem 1.5; see Theorem 4.1 and the immediately preceding
+discussion.
+Section 5 addresses the issue that K-rational points on X1(P) correspond to instances
+where G(fc, K) simply contains the portrait P; that is, we need not have an isomorphism
+G(fc, K) ∼= P, and in fact, in many cases we do not. The section culminates with the proofs
+of Theorems 1.6 and 1.7 in §5.3.
+
+6
+JOHN R. DOYLE AND DAVID KRUMM
+Finally, Section 6 is devoted to arithmetic questions concerning the fields of definition of
+quadratic points on the curves X1(P), and in particular to proving Theorems 1.9 and 1.10.
+Acknowledgements. We thank Joe Silverman for helpful comments, and especially for a
+suggestion that led to the more refined statements in Theorems 1.6 and 1.7. The first author
+was partially supported by NSF grant DMS-2112697.
+2. Dynamical modular curves
+2.1. Dynatomic polynomials. If f is a polynomial with coefficients in a field K and α ∈ K
+is a point of exact period n for f, then α is a root of the polynomial f n(z)−z. However, the
+roots of f n(z) − z may have period strictly dividing n, and indeed there is a factorization
+f n(z) − z =
+�
+d|n
+Φd,f(z),
+where (generically) the roots of Φd,f have exact period d for f. M¨obius inversion yields
+Φn,f(z) =
+�
+d|n
+(f d(z) − z)µ(n/d),
+where µ denotes the M¨obius function. We call Φn,f the nth dynatomic polynomial of f.
+More generally, for m, n ≥ 1 we define
+Φm,n,f(z) :=
+Φn,f(f m(z))
+Φn,f(f m−1(z)).
+Then Φm,n,f is a polynomial whose roots are (again, generically) points of preperiod m and
+eventual period n for f. (That Φn,f and Φm,n,f are indeed polynomials is proven in [22,45].)
+Since we are specifically interested in the family fc(z) = z2 + c, we write
+Φn(c, z) := Φn,fc(z)
+and
+Φm,n(c, z) := Φm,n,fc(z).
+Then Φn (resp., Φm,n) is a polynomial in Z[c, z], and the vanishing locus defines an affine
+curve Y1(n) (resp., Y1(m, n)), which we refer to as a dynatomic curve. Thus, for example,
+if α has period n for fc, then (c, α) is a point on the dynatomic curve Y1(n). We denote by
+X1(·) the normalization of the projective closure of the affine curve Y1(·), and we also refer
+to X1(·) as a dynatomic curve.
+2.2. Dynamical modular curves associated to portraits. The dynamical properties
+of quadratic polynomial maps impose certain restrictions on those portraits that may be
+realized as G(f, K) for some number field K and a quadratic polynomial f ∈ K[z]. First,
+no point may have more than two preimages under f. Also, for each positive n ∈ Z, the nth
+dynatomic polynomial for a quadratic polynomial f has degree
+(2.1)
+D(n) := deg Φn,f(z) =
+�
+d|n
+µ(n/d)2d.
+Thus, a quadratic polynomial has at most D(n) points of period n, partitioned into at most
+R(n) := D(n)/n cycles of length n. With these restrictions in mind, we make the following
+definition:
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+7
+•
+•
+•
+•
+Figure 1. A generic quadratic portrait
+Definition 2.1. A quadratic portrait is a portrait P satisfying the following properties:
+(a) Every vertex of P has in-degree at most 2.
+(b) For each n ≥ 1, the number of n-cycles in P is at most
+R(n) := 1
+n
+�
+d|n
+µ(n/d)2d.
+For any number field K and quadratic polynomial f ∈ K[z], the portrait G(f, K) is
+quadratic. However, for most quadratic polynomials (in a sense that can be made precise),
+we can say more about the structure of the set of K-rational preperiodic points. For the
+model fc(z) = z2 + c, if α is a preperiodic point for fc, then −α is also preperiodic, since
+both are preimages of f(α). Thus, a preperiodic point typically has either no K-rational
+preimages or exactly two K-rational preimages. The exception to this rule occurs when
+α = 0 is a preperiodic point, in which case exactly one preperiodic point (namely, c = fc(0))
+has a single K-rational preimage.
+Along the same lines, if a polynomial fc has a K-rational fixed point β, then β is a root of
+the quadratic polynomial fc(z)−z = z2−z+c; thus, unless we have disc(fc(z)−z) = 1−4c = 0
+(i.e., c = 1/4), there is a second fixed point β′, necessarily defined over K.
+With these observations in mind, we make the following definition.
+Definition 2.2. A generic quadratic portrait is a quadratic portrait P with the following
+additional properties:
+(a) The in-degree of any vertex of P is equal to 0 or 2.
+(b) If P has a fixed point, then P has exactly two fixed points.
+Remark 2.3. We will sometimes refer to the results of [11], in which the term “strongly
+admissible” is used instead of “generic quadratic.”
+Given a quadratic portrait P, there is a dynamical modular curve Y1(P), defined over
+Q, whose K-points—for any extension K/Q—correspond to tuples (c, z1, . . . , zn) such that
+z1, . . . , zn are preperiodic points forming a subportrait of G(fc, K) isomorphic to P. If Q
+is the point on Y1(P) corresponding to such a tuple, then the field of definition of Q is
+Q(c, z1, . . . , zn). We denote by X1(P) the smooth projective curve birational to Y1(P).
+A formal treatment of dynamical modular curves appears in [11], where the curves are de-
+fined only for generic quadratic portraits. We lose no generality in making such a restriction:
+Given any quadratic portrait P, there is a unique portrait P′ that is minimal among generic
+quadratic portraits containing P as a subportrait. In the language of [11], P′ is the generic
+quadratic portrait generated by the vertices of P. It follows from the results of [11, §2] that
+X1(P) ∼= X1(P′), so we may as well assume that P is generic quadratic.
+Rather than formally defining X1(P) (we refer the interested reader to [11] or, for a
+different approach in a more general setting, [15]), we give an example.
+Example 2.4. Consider the generic quadratic portrait P appearing in Figure 1. One could
+construct a curve birational to X1(P) simply by giving one equation for each relation coming
+
+8
+JOHN R. DOYLE AND DAVID KRUMM
+from an edge in P: if we label the vertices 1, 2, 3, 4 from left to right, we have
+z2
+1 + c = z2,
+z2
+2 + c = z3,
+z2
+3 + c = z2,
+z2
+4 + c = z3.
+(2.2)
+Note that we must also impose certain Zariski open conditions of the form zi ̸= zj to remove
+extraneous components; for example, there is a full component of the curve defined by (2.2)
+on which z1, z2, z3, and z4 are all equal. It is this approach that is taken in [15].
+An alternative approach, which is particular to quadratic polynomials, is to note that any
+generic quadratic portrait containing a point of period 2 must necessarily contain P. Thus,
+another (affine) model for X1(P) is the plane curve defined by the vanishing of
+Φ2(c, z) = (z2 + c)2 + c − z
+z2 + c − z
+= z2 + z + c + 1.
+In other words, X1(P) is isomorphic to the dynatomic curve X1(2). This second model has
+the advantage of being defined in a lower-dimensional affine space (A2, rather than A5), and
+it is this second approach which is described in detail in [11].
+We conclude this example by pointing out that X1(P) ∼= X1(2) also has “degenerate”
+points where two or more of the vertices of the portrait P collapse.
+For example, the
+equation Φ2(c, z) = 0 has the solution (c, z) =
+�
+− 3
+4, − 1
+2
+�
+despite the fact that − 1
+2 is a fixed
+point for f−3/4. However, for a given portrait P, there are only finitely many such degenerate
+points on X1(P).
+Before summarizing the required properties of dynamical modular curves, we recall the
+following terminology:
+Definition 2.5. Let X be a smooth, irreducible projective curve defined over a field k.
+The k-gonality of X, denoted gonk(X), is the minimal degree of a nonconstant morphism
+X → P1 defined over k.
+Proposition 2.6. Let P be a generic quadratic portrait, and let k be any field of character-
+istic 0.
+(a) The curve X1(P) is irreducible over k.
+(b) If P′ is a generic quadratic portrait properly contained in P, then there is a finite
+morphism πP,P′ : X1(P) → X1(P′) of degree at least 2 defined over k.
+(c) Given any ordering P1, P2, . . . of all generic quadratic portraits, the k-gonalities of
+the curves X1(Pi) tend to ∞.
+Proof. Parts (a) and (b) are proven in [11, Thm. 1.7] and [11, Prop. 3.3], respectively. Note
+that the morphism πP,P′ is obtained simply by forgetting the preperiodic points correspond-
+ing to vertices of P ∖ P′, hence is defined over the base field k.
+Statement (c) is a slight generalization of, but follows directly from, [14, Thm. 1.1(b)],
+which says that as m + n → ∞, the gonalities of the curves X1(m, n) tend to ∞. Given a
+bound B, there are only finitely many quadratic portraits P such that every vertex v of P
+has preperiod m and eventual period n satisfying m + n ≤ B. In other words, if for every
+generic quadratic portrait P we choose a vertex vP with preperiod mP and eventual period
+nP maximizing the sum mP + nP, we must have mP + nP → ∞ as P ranges over all generic
+quadratic portraits in any order. Since there is a nonconstant morphism from X1(P) to
+X1(mP, nP) (e.g., by part (b)), we have gonk(X1(P)) ≥ gonk(X1(mP, nP)), and the latter
+expression tends to ∞.
+□
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+9
+Proof of Theorem 1.11. Fix n ≥ 1 and a portrait P, and suppose there are infinitely many
+c ∈ Q(n) such that G(fc, K) ∼= P for some degree-n number field K containing c. Then the
+dynamical modular curve X1(P) has infinitely many points of degree at most n. It follows
+from [19, Prop. 2] (cf. [8, Thm. 5]) that X1(P) must have gonality at most 2n, hence there
+are only finitely many such portraits P by part (c) of Proposition 2.6.
+□
+3. Some useful properties of algebraic curves
+In this section, we collect a few facts about algebraic curves that will be used throughout
+the rest of the paper.
+First, we provide a statement that follows from Hilbert’s irreducibility theorem; see [44,
+§3.4] and [30, §9.2] for details.
+Proposition 3.1. Let K be a number field, let X be a curve defined over K, and let ϕ :
+X → P1 be a dominant morphism of degree d ≥ 2 defined over K. Then the set
+T :=
+�
+P ∈ P1(K) : [K(Q) : K] < d for some Q ∈ ϕ−1(P)
+�
+is a thin subset of P1(K).
+Remark 3.2. Thin subsets T ⊂ P1(K) have density 0, in the sense that
+lim
+N→∞
+���{P ∈ T : h(P) ≤ N}
+���
+���{P ∈ P1(K) : h(P) ≤ N}
+���
+= 0,
+where h is the na¨ıve Weil height on P1(Q). In particular, for any maximal ideal p ∈ Spec OK
+and any mod-p residue class c in P1(K), the set c \ T is infinite.
+Given an elliptic curve E with Weierstrass equation y2 = f(x), where f ∈ K[x] is square-
+free of degree 3, it is easy to construct infinitely many quadratic points on E: For “most”
+x ∈ K, the point (x, y) = (x,
+�
+f(x)) is quadratic over K. More precisely, since f is not a
+square in K[x], it follows from Hilbert irreducibility that f(x0) is a nonsquare in K for all
+x0 outside a thin subset of K. The following result, proven in [13, Lem. 2.2], gives a useful
+characterization of quadratic points (x, y) with x /∈ Q:
+Lemma 3.3. Let E/K be an elliptic curve defined by an equation of the form
+y2 = ax3 + bx2 + cx + d,
+where a, b, c, d ∈ K and a ̸= 0. Suppose (x, y) ∈ E(K) is a quadratic point with x /∈ K.
+Then there exist (x0, y0) ∈ E(K) and t ∈ k such that y = y0 + t(x − x0) and
+x2 + ax0 − t2 + b
+a
+x + ax2
+0 + t2x0 + bx0 − 2y0t + c
+a
+= 0.
+By Theorem 1.8, a curve with infinitely many quadratic points must admit a degree-2
+morphism to either P1 or an elliptic curve, hence must have gonality at most 4. Thus, to
+prove that a curve has finitely many quadratic points, it suffices to show that the gonality
+of the curve is greater than 4. The following inequality is a standard tool for finding lower
+bounds for gonalities.
+
+10
+JOHN R. DOYLE AND DAVID KRUMM
+Proposition 3.4 (Castelnuovo–Severi inequality [46, Thm. 3.11.3]). Let Y , Y1, and Y2 be
+curves of genera gY , g1, and g2, respectively. Suppose we have maps ϕ1 : Y → Y1 and
+ϕ2 : Y → Y2 of degrees d1 and d2, and suppose further that there is not an intermediate
+curve Z and a map ψ : Y → Z of degree greater than 2 such that both ϕ1 and ϕ2 factor
+through ψ. Then
+(3.1)
+gY ≤ d1g1 + d2g2 + (d1 − 1)(d2 − 1).
+4. Dynamical modular curves with infinitely many quadratic points
+The purpose of this section is to prove one direction of Theorem 1.5, namely that if there
+are infinitely many c ∈ Q(2) such that G(fc, K) ∼= P for some quadratic field K containing
+c, then P ∈ Γ. Since any such realization of P as G(fc, K) yields a quadratic point on the
+dynamical modular curve Y1(P), it suffices to prove the following:
+Theorem 4.1. Let P be a generic quadratic portrait.
+Then X1(P) has infinitely many
+quadratic points if and only if P ∈ Γ.
+Remark 4.2. If we just assume that P is a quadratic portrait (i.e., not necessarily generic),
+then X1(P) has infinitely many quadratic points if and only if P is a subportrait of some
+portrait in Γ. This follows from Theorem 4.1 as well as the fact that for any quadratic
+portrait P, if we let P′ be the minimal generic quadratic portrait containing P, then X1(P)
+and X1(P′) are isomorphic over Q. (See the discussion preceding Example 2.4.)
+One direction of Theorem 4.1 is straightforward: For every portrait P ∈ Γ, the curve
+X1(P) is described in at least one of the articles [38,43,48]. All the curves in those articles
+have genus at most 2 and at least one rational point, hence have infinitely many quadratic
+points. Thus, we must show that if P is generic quadratic but not in Γ, then X1(P) has only
+finitely many quadratic points.
+To help organize the arguments in the rest of this section, we introduce some terminology:
+Definition 4.3. The cycle structure of a portrait P is the nonincreasing sequence of cycle
+lengths appearing in P. Note that the empty portrait has cycle structure ( ).
+If K is a quadratic field and c ∈ K, then the cycle structure of G(fc, K) may contain the
+integer 1 at most twice and each of the integers 2, 3, and 4 at most once; for periods 1 and
+2 this follows from the fact that a quadratic polynomial can have at most two fixed points
+and at most one 2-cycle, and for periods 3 and 4 this comes from [10, Cor. 4.16]. More
+precisely, the results of [10] imply that the “period at most 4” portion of the cycle structure
+of G(fc, K) must be (4,1,1), (4,2), or one of the following:
+(4.1)
+( ), (1,1), (2), (3), (4), (2,1,1), (3,1,1), (3,2).
+Moreover, it follows from [13, Cor. 3.48] (resp., [10, Thm. 4.21]) that no portrait with
+both a 4-cycle and a 1-cycle (resp., 4-cycle and a 2-cycle) may be realized infinitely often as
+G(fc, K) for K a quadratic field and c ∈ K. For our purposes, therefore, we may exclude
+the cycle structures (4,1,1) and (4,2) from consideration.
+By enumerating generic quadratic portraits with few vertices, one can verify that if P is a
+generic quadratic portrait which is not in Γ, then P has a cycle of length n ≥ 5 or P properly
+contains a portrait in Γ1 or Γ2. We handle these two possibilities separately, showing in each
+case that the dynamical modular curve X1(P) has only finitely many quadratic points.
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+11
+4.1. Points of period n ≥ 5. If P is a generic portrait with a cycle of length n, then there
+is a dominant morphism X1(P) → X1(n) defined over Q. In particular, every quadratic
+point on X1(P) maps to a rational or quadratic point on X1(n), so we need only show that
+if n ≥ 5, then X1(n) has only finitely many points defined over quadratic fields; this is the
+content of Proposition 4.6.
+For n ≥ 1, the cyclic group Cn acts on X1(n) as follows: Given a point (c, z) ∈ X1(n), we
+also have σn(c, z) := (c, fc(z)) ∈ X1(n), so σ defines an order-n automorphism of X1(n). We
+denote by πn : X1(n) → X0(n) the quotient of X1(n) by this cyclic group action. (The curve
+X0(n) parametrizes maps fc together with a marked cycle of length n.) Let c1,n and c0,n
+denote the maps from X1(n) and X0(n), respectively, to the c-line; note that c1,n = c0,n ◦ πn.
+For a curve X, we will denote by gX its genus; for simplicity, for each n ≥ 1 we will write
+g1,n and g0,n for the genera of X1(n) and X0(n), respectively. Finally, recall that we denote
+by D(n) the degree (in z) of the polynomial Φn(c, z); a formula for D(n) is given in (2.1),
+and using that formula one can show that
+(4.2)
+2n−1 ≤ D(n) ≤ 2n,
+with equality on the right if and only if n = 1 and on the left if and only if n = 2.
+Lemma 4.4. For all n ≥ 5, we have g0,n ≥ 2.
+Proof. In [37, Thm. 13], Morton gives an explicit formula for g0,n as well as the lower bound
+g0,n ≥ 3
+2 +
+�1
+4 − 1
+n
+�
+2n − (n + 1)2n/2−1.
+Rewriting the right-hand expression and using the assumption that n ≥ 5, we have
+g0,n ≥ 3
+2 + 2n/2−1
+��1
+4 − 1
+5
+�
+2n/2+1 − (n + 1)
+�
+= 3
+2 + 2n/2−1
+� 1
+102n/2 − (n + 1)
+�
+.
+The expression
+� 1
+102n/2 − (n + 1)
+�
+is positive for all n ≥ 15, so for all such n we have
+g0,n > 3/2, hence g0,n ≥ 2. Finally, using the explicit formula for g0,n given by Morton,
+we can exactly compute g0,n for all 1 ≤ n ≤ 14, and we find that g0,n < 2 if and only if
+1 ≤ n ≤ 4, in which case g0,n = 0.
+□
+Lemma 4.5. Let n ≥ 6, and let Rn be the ramification divisor of πn. Then deg Rn > 4n.
+Proof. It suffices to replace Rn with R0
+n, the restriction of the ramification divisor to points
+that do not map to ∞ under c1,n. The ramification divisors of the maps c1,n and c0,n are
+explicitly computed by Morton in [37, Thms. 11, 13], from which it follows that
+(4.3)
+deg R0
+n = 1
+2
+�
+d|n
+d<n
+D(d)ϕ(n/d)(n − d).
+If n is prime, then
+deg R0
+n = 1
+2D(1)ϕ(n)(n − 1) = (n − 1)2,
+
+12
+JOHN R. DOYLE AND DAVID KRUMM
+which is greater than 4n when n ≥ 6. If n is composite, then let m be the largest proper
+divisor of n. Since √n ≤ m ≤ n/2, we have
+deg R0
+n ≥ 1
+2D(m)ϕ(n/m)(n − m)
+> 2m−2ϕ(n/m)n
+2
+(by (4.2))
+≥ 2
+√n−3ϕ(n/m)n.
+For all n ≥ 25, we have 2
+√n−3 ≥ 4, thus deg R0
+n > 4n. An explicit computation of deg R0
+n
+for all 6 ≤ n ≤ 25 (using (4.3)) completes the proof.
+□
+Proposition 4.6. Let n ≥ 5. Then X1(n) has only finitely many quadratic points.
+Remark 4.7. The fact that X1(n) has only finitely many quadratic points for sufficiently
+large n follows from Theorem 1.8, together with [14, Thm.
+1.1], which states that the
+gonality of X1(n) tends to infinity with n. For our purposes, however, we need the more
+precise statement of Proposition 4.6.
+Proof of Proposition 4.6. By Theorem 1.8, it suffices to show that for n ≥ 5, the curve X1(n)
+does not admit a degree-2 map to a curve of genus at most 1.
+Let ϕ : X1(n) → C be a dominant morphism, where C is a curve of genus gC ≤ 1. We
+claim that d := deg ϕ > 2.
+First, suppose n ≥ 6.
+In order to apply the Castelnuovo–Severi inequality (Proposi-
+tion 3.4), we consider two cases.
+Case 1: Suppose there is a curve Y such that both πn : X1(n) → X0(n) and ϕ : X1(n) →
+C factor through a map ψ : X1(n) → Y of degree at least 2. Since Y covers X0(n), we have
+gY ≥ 2 > gC by Lemma 4.4, hence the map Y → C has degree at least 2. Therefore, d ≥ 4.
+Case 2: Now suppose that there is no such curve Y , so that we can apply the Castelnuovo–
+Severi inequality to the maps ϕ and πn to get
+g1,n ≤ ng0,n + dgC + (n − 1)(d − 1).
+Rewriting this, we have
+g1,n − ng0,n + n − 1 ≤ d(gC + n − 1).
+By the Riemann–Hurwitz formula, the left hand side is equal to half the degree of the
+ramification divisor Rn, so by Lemma 4.5 we have
+d(gC + n − 1) > 2n.
+Since we assumed gC ≤ 1, this implies that d > 2.
+Finally, we consider n = 5. A calculation in Magma [4] shows that the map X1(5) → P1
+given by Φ2(c, z) = z2 + z + c + 1 has degree 7. If ϕ factors through Φ2, then d ≥ 7. If
+not, then ϕ and Φ2 do not simultaneously factor through a nontrivial intermediate map
+X1(5) → Y , since Φ2 has prime degree. By the Castelnuovo–Severi inequality, we have
+g1,5 ≤ dgC + 6(d − 1).
+The curve X1(5) has genus g1,5 = 14, and we assumed gC ≤ 1, so it follows that d > 2.
+□
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+13
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+•
+Figure 2. The portrait 10(4)
+Remark 4.8. The fact that Φ2 has low degree on X1(5) seems related to the fact that, as
+polynomials in Q[c, z], the dynatomic polynomials Φ2 and Φ5 have no common zeros (c, z).
+In particular, all zeros and poles of Φ2 on X1(5) lie above ∞, which restricts the possible
+number of such points.
+4.2. Generic quadratic portraits properly containing the portraits in Γ1 and Γ2.
+By enumerating portraits with few vertices, one finds that any generic quadratic portrait P
+that has its cycle structure listed in (4.1), but which is not contained in Γ, must properly
+contain a portrait from Γ1 or Γ2, and moreover, P must have a subportrait isomorphic to
+one of the following portraits:
+(4.4)
+10(1,1)a/b, 10(2), 10(3)a/b, 10(4), 12(2,1,1)a/b, or Gn for some 1 ≤ n ≤ 10.
+All portraits listed above appear in Appendix B except 10(4), which is the label we give to
+the subportrait of 12(4) shown in Figure 2.
+Proposition 4.9. For each of the portraits P appearing in (4.4), the curve X1(P) has only
+finitely many quadratic points.
+The cases P = 10(1, 1)b and P = 10(2) form the majority of the proof of Proposition 4.9.
+We include only the proof for 10(1, 1)b, as the argument for 10(2) is very similar.
+Lemma 4.10. Let C ⊂ Spec Q[x, z] be the curve of genus 5 defined by the equation
+�
+z2 − 2(x2 + 1)
+�2 = 2(x2 − 1)2(x3 + x2 − x + 1).
+Let (c, p) ∈ A2(K) be such that p has preperiod 4 and eventual period 1 for fc. Then there
+exists a point (x, z) ∈ C(K) such that c = −2(x2 + 1)/(x2 − 1)2.
+Proof. Let q = fc(p) = p2 + c, so that q has preperiod 3 (and still eventual period 1). A
+calculation in [43, p. 22] shows that there is an element x ∈ K ∖ {±1} such that
+c = −2(x2 + 1)
+(x2 − 1)2
+and q2 = 2(x3 + x2 − x + 1)
+(x2 − 1)2
+.
+Hence we have
+2(x3 + x2 − x + 1) = q2(x2 − 1)2 = (p2 + c)2(x2 − 1)2 =
+�p2(x2 − 1)2 − 2(x2 + 1)
+x2 − 1
+�2
+.
+Letting z = p(x2 − 1) we obtain
+�
+z2 − 2(x2 + 1)
+�2 = (x2 − 1)2 · 2(x3 + x2 − x + 1)
+with x, z ∈ K. Thus (x, z) ∈ C(K).
+□
+
+14
+JOHN R. DOYLE AND DAVID KRUMM
+Proposition 4.11. For the portraits P = 10(1, 1)b and P = 10(2), the set of quadratic
+points on X1(P) is finite.
+Proof. As mentioned above, we only give a proof for P = 10(1, 1)b. By Lemma 4.10, it
+suffices to show that the curve C has only finitely many quadratic points. The latter curve
+admits a dominant map to the elliptic curve with Weierstrass equation w2 = 2(x3+x2−x+1),
+which is the modular curve Xell
+1 (11). Explicitly, a natural map ϕ : C → Xell
+1 (11) is given by
+ϕ(x, z) =
+�
+x, z2 − 2(x2 + 1)
+x2 − 1
+�
+.
+The curve Xell
+1 (11) has exactly four affine rational points, namely, (±1, ±2). Suppose
+that (x, z) is a quadratic point on C with field of definition K. Then x /∈ {±1}, so the
+point ϕ(x, z) cannot be a rational point on X1(11). Thus ϕ(x, z) is a quadratic point, and
+K = Q(ϕ(x, z)). Letting
+(4.5)
+w = z2 − 2(x2 + 1)
+x2 − 1
+,
+we therefore have w2 = 2(x3 + x2 − x + 1) and K = Q(x, w). We now consider two cases.
+Suppose first that x ∈ Q. Then K = Q(w), and by (4.5) we have z2 = 2(x2+1)+(x2−1)w.
+Applying the norm map NK/Q to this equation we obtain
+y2 = 4(x2 + 1)2 − 2(x2 − 1)2(x3 + x2 − x + 1),
+where y = NK/Q(z).
+The above equation defines a hyperelliptic curve of genus 3, and
+therefore has only finitely many rational solutions. We conclude that C has only finitely
+many quadratic points with rational x-coordinate.
+Now suppose that x /∈ Q, so that K = Q(x). By Lemma 3.3 applied to the equation
+w2 = 2(x3 + x2 − x + 1), there is a rational number t and a point (x0, w0) ∈ {(±1, ±2)} such
+that w = w0 + t(x − x0) and
+(4.6)
+x2 + 2x0 − t2 + 2
+2
+x + 2x2
+0 + t2x0 + 2x0 − 2w0t − 2
+2
+= 0.
+For each point (x0, w0) ∈ {(±1, ±2)} we consider the relation
+(4.7)
+z2 = 2(x2 + 1) + (x2 − 1)(w0 + t(x − x0)).
+Using (4.6) we express the right-hand side of (4.7) as a linear combination of 1 and x.
+Applying the norm map NK/Q and letting u = 2 · NK/Q(z), we obtain a relation of the
+form u2 = g(t), where g is a polynomial of degree 7 with integral coefficients and nonzero
+discriminant. Each of the resulting four equations u2 = g(t) defines a hyperelliptic curve of
+genus 3, and therefore has only finitely many rational solutions. Since t has only finitely
+many possible values, (4.6) implies the same for x. Therefore C has finitely quadratic points
+with quadratic x-coordinate.
+□
+Proof of Proposition 4.9. The proposition has already been proven in [13] and [10] for all of
+the portraits except 10(1, 1)b, 10(2), and 10(3)a/b. Moreover, Proposition 4.11 shows that
+the statement is true for the portraits 10(1,1)b and 10(2), so all that remains is to show that
+X1(P) has only finitely many quadratic points when P = 10(3)a or P = 10(3)b.
+Each of the curves X1(P) with P = 10(3)a/b has genus 9, and each admits a degree-2
+map ϕ to the genus-2 curve X1(P′), where P′ = 8(3). Now suppose we have a degree-d map
+ψ : X1(P) → C, where C is a curve of genus gC ≤ 1. Then, by Proposition 3.4, either ψ
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+15
+factors through ϕ, in which case deg ψ > deg ϕ = 2, or the Castelnuovo–Severi inequality
+(3.1) applies to ϕ and ψ, in which case we have
+4 + dgC + (d − 1) ≥ 9, hence d ≥
+6
+gC + 1 ≥ 3.
+It follows that X1(P) is not hyperelliptic or bielliptic, hence X1(P) has only finitely many
+quadratic points by Theorem 1.8.
+□
+4.3. Proof of Theorem 4.1. We now combine Propositions 4.6 and 4.9 to complete the
+proof of Theorem 4.1, which in turn proves one direction of Theorem 1.5.
+Proof of Theorem 4.1. As mentioned previously, the fact that X1(P) has infinitely many
+quadratic points for each P ∈ Γ follows from the work of Walde–Russo [48] and Poonen
+[43]. Now suppose P is a generic quadratic portrait such that X1(P) has infinitely many
+quadratic points.
+Proposition 4.6 asserts that P cannot have a cycle of length n ≥ 5;
+combining this with the paragraph following Definition 4.3, the cycle structure of P must be
+one of those appearing in (4.1). By simply enumerating all small generic quadratic portraits
+with the allowable cycle structures, one finds that if P is not contained in Γ, then P has
+a subportrait isomorphic to one of the portraits listed in (4.4), hence there is a dominant
+morphism X1(P) → X1(P′) for some P′ in that list. Proposition 4.9 shows that each such
+X1(P′) has only finitely many quadratic points, hence X1(P) does as well.
+□
+5. Preperiodic portraits realized infinitely often over quadratic fields
+In this section, we show that if P ∈ Γ, then there are infinitely many c ∈ Q such that
+G(fc, K) ∼= P for some quadratic field K. We also determine for which portraits P the same
+is true for infinitely many c ∈ Q(2) ∖ Q.
+It follows from Theorem 4.1 that Γ is precisely the set of generic quadratic portraits that
+can be realized infinitely often as a subportrait of G(fc, K); the difficulty is in proving that
+we infinitely often have equality. This step requires two main tools: The first is Hilbert
+irreducibility, and the second is a dynamical result giving an upper bound for the lengths of
+periodic cycles of maps over number fields that depends only on the primes of bad reduction
+of those maps; see, for example, [42,45,49].
+Proposition 5.1. Let K be a number field, and let p ∈ Spec OK be a prime ideal of norm q.
+There exists a bound B := B(q) such that if c ∈ K and vp(c) ≥ 0, then fc has no K-rational
+points of period larger than B.
+We will also repeatedly use the observation that given any portrait P and any bound C,
+there are only finitely many generic portraits P′ that are minimal (relative to inclusion)
+among those generic portraits containing P and have no cycles of length larger than C. This
+is more or less due to the fact that there are only finitely many generic quadratic portraits
+with a given number of vertices.
+5.1. Rational c-values. For any c ∈ Q, there are infinitely many quadratic fields K for
+which G(fc, K) ∼= G(fc, Q).
+This is a consequence of Northcott’s theorem: The set of
+preperiodic points for fc has bounded height, hence there are only finitely many preperiodic
+points which are quadratic over Q, and therefore only finitely many quadratic fields over
+which fc gains new preperiodic points.
+
+16
+JOHN R. DOYLE AND DAVID KRUMM
+Table
+1. For
+each
+pair
+(P, P′),
+there
+is
+a
+degree-2
+morphism
+X1(P) → X1(P′) defined over Q.
+P
+4(1,1)
+4(2)
+6(1,1)
+6(2)
+8(2,1,1)
+P′
+∅
+∅
+4(1,1)
+4(2)
+4(1,1) or 4(2)
+In particular, if a portrait P is realized as G(fc, Q) for infinitely many c ∈ Q, then P must
+also be realized as G(fc, K) for infinitely many c ∈ Q and, for each such c, infinitely many
+quadratic fields K. The portraits realized infinitely often over Q are precisely the portraits
+in Γ0; this is the main result of [16].
+A more interesting problem, then, is to determine the set of portraits P for which there
+are infinitely many c ∈ Q with G(fc, Q) ⊊ P but G(fc, K) ∼= P for some quadratic field K.
+Proposition 5.2. Let P ∈ Γ. Then there exist infinitely many c ∈ Q such that G(fc, K) ∼= P
+for some quadratic field K. Moreover, if P ∈ Γ ∖ {∅, 6(3)}, the infinitely many c ∈ Q may
+be chosen so that
+G(fc, Q) ⊊ G(fc, K) ∼= P.
+Remark 5.3. The portraits ∅ and 6(3) are genuine exceptions to the second statement, as
+asserted in Theorem 1.6 and proven in §5.3.
+Proof of Proposition 5.2. It follows from the discussion preceding the statement of Proposi-
+tion 5.2 that for both P = ∅ and P = 6(3), which are elements of Γ0, there are infinitely
+many c ∈ Q such that G(fc, K) ∼= P for some quadratic field K. We henceforth assume
+P ∈ Γ ∖ {∅, 6(3)} and prove the stronger statement that there are infinitely many c ∈ Q
+such that G(fc, Q) ⊊ G(fc, K) ∼= P for some quadratic field K.
+There is a model for X1(P) of the form y2 = F(x), with F(x) ∈ Q[x] nonconstant and
+squarefree, such that the morphism c : X1(P) → P1 factors through x : X1(P) → P1. If
+X1(P) has genus 0, this is because there is a proper (generic quadratic) subportrait P′ ⊊ P
+for which the natural morphism
+πP,P′ : X1(P) −→ X1(P′)
+described in Proposition 2.6(b) has degree exactly 2; see Table 1 for the list of such pairs
+(P, P′). For the curves of genus 1 or 2, explicit models are given in Appendix A.
+Now fix a portrait P ∈ Γ ∖ {∅, 6(3)}, and let y2 = F(x) be the model for X1(P) described
+in the previous paragraph. Choose any value of x0 ∈ Q, and choose a prime p ∈ Spec Z
+of good reduction for c : X1(P) → P1 such that vp(c(x0)) ≥ 0. Let B = B(p2) be the
+bound from Proposition 5.1, let P1, . . . , Pn be the generic quadratic portraits that properly
+contain P and that have no cycles of length larger than B, and for each i = 1, . . . , n let
+πi := πPi,P : X1(Pi) → X1(P) be the natural projection morphism from Proposition 2.6(b).
+By Hilbert’s Irreducibility Theorem, the sets
+{x ∈ Q :
+�
+F(x) ∈ Q}
+and, for each i = 1, . . . , n,
+�
+x ∈ Q :
+�
+Q
+�
+π−1
+i
+�
+x,
+�
+F(x)
+��
+: Q
+�
+≤ 2
+�
+,
+are thin subsets of P1(Q). Since the residue class
+[x0]p := {x ∈ P1(Q) : x ≡ x0 (mod p)}
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+17
+is not thin, there are infinitely many x ∈ [x0]p such that K := Q
+��
+F(x)
+�
+is a quadratic
+field and
+�
+x,
+�
+F(x)
+�
+∈ X1(P)(K) does not lift to a K-rational point on X1(Pi) for any
+i = 1, . . . , n.
+Now let c = c(x) for any of the infinitely many x from the previous paragraph. Excluding
+at most finitely many x ∈ [x0]p, we may assume that G(fc, K) is a generic quadratic portrait.
+Since c lifts to a quadratic point Q on X1(P), we have
+G(fc, Q) ⊊ P ⊆ G(fc, K).
+On the other hand, since c does not lift to a quadratic point on X1(Pi) for any i = 1, . . . , n,
+the portrait G(fc, K) is either isomorphic to P or contains a cycle of length greater than B.
+Finally, since vp(c) ≥ 0, G(fc, K) has no cycles of length greater than B, and thus we have
+G(fc, Q) ⊊ G(fc, K) ∼= P.
+□
+5.2. Quadratic c-values. The purpose of this section is to determine which portraits P
+may be realized infinitely often as G(fc, Q(c)) for some quadratic algebraic number c. We
+begin with the simplest case, namely, where X1(P) has genus 0.
+Proposition 5.4. Suppose P ∈ Γ0, and let K/Q be a quadratic field.
+Then there are
+infinitely many c ∈ K ∖ Q such that G(fc, K) ∼= P.
+Proof. Let X := X1(P) ∼=Q P1. Choose an inert prime p ∈ Spec Z such that c : X → P1
+has good reduction at p and such that p > D := deg(c : X → P1). Let p ∈ Spec OK be the
+unique prime lying above p. The good reduction condition implies that the mod-p reduction
+�c of the map c : X → P1 has degree D, and the condition that D < p implies that �c cannot
+map all of P1(Fp2) to P1(Fp). In other words, we may choose a point P0 ∈ X(K) such that
+�
+c(P0) = �c(�
+P0) ∈ P1(Fp2) ∖ P1(Fp). Note that since ∞ ∈ P1(Fp) and �
+c(P0) /∈ P1(Fp), we must
+have vp(c(P0)) ≥ 0. Then, for any P in the residue class [P0]p, c(P) must be in K ∖ Q,
+and vp(c(P)) ≥ 0. In particular, for all P ∈ [P0]p, fc(P) has no K-rational points of period
+greater than B = B(p2), the bound from Proposition 5.1.
+Let P1, . . . , Pn be the complete list of generic quadratic portraits that minimally contain
+P and which have no cycles of length larger than B, and for each i = 1, . . . , n let
+πi := πPi,P : X1(Pi) −→ P1
+be the morphism from Proposition 2.6(b). If P ∈ [P0]P and G(fc(P), K) properly contains
+P, then G(fc(P), K) must contain one of the portraits Pi. By Hilbert irreducibility, the set
+[P0]p ∖
+n�
+i=1
+πi
+�
+X1(Pi)(K)
+�
+is infinite. Thus, there are infinitely many c ∈ K ∖ Q such that G(fc, K) ∼= P.
+□
+We now consider the portraits P ∈ Γ ∖ Γ0; that is, the portraits for which X1(P) has
+genus 1 or 2. We begin with a useful consequence of Theorem 4.1.
+Corollary 5.5. Let P be a generic quadratic portrait such that X1(P) has positive genus.
+If P′ is any generic quadratic portrait properly containing P, then X1(P′) has finitely many
+quadratic points.
+
+18
+JOHN R. DOYLE AND DAVID KRUMM
+Proof. If X1(P′) has infinitely many quadratic points, then so does X1(P), and therefore
+both P and P′ are elements of Γ. By simply inspecting the portraits in Γ, the only way we
+can have P, P′ ∈ Γ and P ⊊ P′ is if P ∈ Γ0; that is, if X1(P) has genus 0.
+□
+Combining Corollary 5.5 and Proposition 5.1 gives us the following sufficient condition for
+P to be realized infinitely often as G(fc, K) over quadratic fields K. For an algebraic curve
+X defined over a field k, we denote by X(k, 2) the set of all points on X of degree at most
+2 over k. For a prime p ∈ Spec Z and an element α ∈ Q, the phrase “vp(α) ≥ 0” should be
+read to mean “there exists some extension p ∈ Spec OQ(α) of p such that vp(α) ≥ 0.”
+Lemma 5.6. Let P be a generic quadratic portrait such that X1(P) has genus 1 or 2. Fix
+a prime p ∈ Spec Z, and consider the set
+Sp,P := {β ∈ X1(P)(Q, 2) : vp(c(β)) ≥ 0}.
+For all but finitely many β ∈ Sp,P, we have G(fc(β), Q(β)) ∼= P.
+Proof. Suppose β ∈ Sp,P, set c := c(β), and let K := Q(β). Removing at most finitely many
+points β ∈ Sp,P, we may assume that G(fc, K) is generic quadratic. Since β ∈ X1(P)(K)
+and G(fc, K) is generic quadratic, G(fc, K) has a subportrait isomorphic to P.
+Let p ∈ Spec OK be a prime lying above p. Since p has norm at most p2, the map fc
+has no K-rational points of period larger than B := B(p2). Thus, as explained following
+Proposition 5.1, if G(fc, K) ̸∼= P, then G(fc, K) must contain one of finitely many portraits
+P1, . . . , Pn properly containing P. But each of the curves X1(Pi) has only finitely many
+quadratic points by Corollary 5.5; this completes the proof.
+□
+Proposition 5.7. Let P ∈ Γrat. If K is a quadratic field and c ∈ K satisfies G(fc, K) ∼= P,
+then c ∈ Q.
+Proof. For P ∈ {8(4), 10(3, 1, 1), 10(3, 2)}, this follows immediately from Theorems 3.16,
+3.25, and 3.28 of [13]. For the remaining portraits P—namely, 8(1,1)a and 8(2)a—the proofs
+are similar, so we only provide the details for 8(1,1)a.
+Let P = 8(1, 1)a. As shown in Appendix A, X1(P) is isomorphic to the elliptic curve E
+with affine model y2 = x3 − x2 + x, which is the curve labeled 24A4 in [9] and 24.a5 in [31].
+We claim that if P = (x, y) is a quadratic point on E, then c(P) = − (x2+1)2
+4x(x−1)2 ∈ Q. This is
+certainly the case if x ∈ Q, so assume that x is quadratic.
+By Lemma 3.3, there exist P0 = (x0, y0) ∈ E(Q) ∖ ∞ and t ∈ Q such that
+(5.1)
+x2 + (x0 − t2 − 1)x + (x2
+0 + t2x0 − x0 − 2y0t + 1) = 0.
+The only affine rational points (x0, y0) on E are (0, 0) and (1, ±1), and for each of these
+points we can use (5.1) to rewrite c(P) as a function of t and x which has degree at most 1
+in x. For the points (0, 0), (1, 1), and (1, −1), respectively, we get
+c = −
+(t2 + 1)2
+4(t − 1)(t + 1),
+c = −t2 (t2 − 2t + 2)
+4(t − 1)2
+, and
+c = −t2 (t2 + 2t + 2)
+4(t + 1)2
+.
+In any case, since t ∈ Q, we must also have c ∈ Q.
+□
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+19
+Proposition 5.8. For all P ∈ Γ0 ∪ Γquad there exist infinitely many c ∈ Q(2) such that
+G(fc, Q(c)) ∼= P.
+Proof. For the portraits in Γ0, this follows from Proposition 5.4. The proofs for 8(1,1)b and
+8(2)b (resp., 10(2,1,1)a and 10(2,1,1)b) are very similar, so we only provide the details for
+8(1,1)b, 10(2,1,1)a, and 8(3).
+First, let P be the portrait 8(1,1)b. Appendix A shows that X1(P) is isomorphic to the
+curve X with affine model y2 = 2(x3 + x2 − x + 1), with c : X → P1 given by
+c = −
+2(x2 + 1)
+(x + 1)2(x − 1)2.
+Set P0 = (x0, y0) := (1, 2) ∈ X(Q). For t ∈ Q, the line y − 2 = t(x − 1) intersects the curve
+X at P0 and two additional points Pt and P t. Since t and P0 are rational, either Pt and P t
+are rational as well, or Pt and P t are quadratic conjugates. Since X(Q) is finite, we may,
+at the expense of excluding finitely many t, assume that Pt and P t are quadratic Galois
+conjugates. Let K := Q(Pt), and let τ be the nontrivial element of Gal(K/Q). With this
+setup, we have Pt + P t = −P0 ̸= O, so y(Pt) ̸= −y(P t) = −y(Pt)τ, and therefore x(Pt) /∈ Q.
+By calculating the intersection of y − 2 = t(x − 1) with the affine curve X, we find that the
+minimal polynomial of x(Pt) must be
+x2 − t2 − 4
+2
+x + t2 − 4t + 2
+2
+.
+Thus, we may rewrite c(Pt) as
+c(Pt) =
+t + 2
+8(t − 2)x(Pt) − t5 − 10t3 + 8t2 + 8t + 32
+16(t − 2)2t
+.
+Finally, for any t ∈ Q with t ≡ 1 (mod 3), we see that x(Pt) and c(Pt) are integral at p = 3,
+so Pt ∈ S3,P. By Lemma 5.6, we conclude that there are infinitely many quadratic c with
+G(fc, Q(c)) ∼= P.
+Next, we let P be the portrait 10(2,1,1)a, and we take X to be the curve defined by
+y2 + xy + y = x3 − x2 − x with map c : X → P1 given by
+c =
+x − 2
+4x(x − 1)y − x4 − x3 + 3x − 1
+4x2(x − 1)
+.
+By Hilbert irreducibility, there are infinitely many points on X with x ∈ Q, x ≡ 2 (mod 3),
+and y /∈ Q. For all such points P = (x, y), c(P) must be quadratic, and we have P ∈ S3,P.
+The desired conclusion again follows from Lemma 5.6.
+Finally, we let P be the portrait 8(3). Let X be the curve y2 = x6−2x4+2x3+5x2+2x+1
+with c : X → P1 given by
+c = −x6 + 2x5 + 4x4 + 8x3 + 9x2 + 4x + 1
+4x2(x + 1)2
+.
+The curve X has two rational points at infinity; we denote these by ∞+ and ∞−. These two
+points are transposed by the hyperelliptic involution ι : X → X given by ι(x, y) = (x, −y).
+Let J be the Jacobian of the genus-2 curve X. For a thorough treatment of the arithmetic
+of genus-2 curves, we recommend [7]; here, we just summarize the necessary properties.
+
+20
+JOHN R. DOYLE AND DAVID KRUMM
+Points on J correspond to degree-0 divisor classes on X. Moreover, by the Riemann–Roch
+theorem, every nontrivial divisor class can be written uniquely as
+{P, Q} := [P + Q − ∞+ − ∞−]
+with P, Q ∈ X and Q ̸= ι(P), up to swapping P and Q. (The trivial class O is equal to
+{P, ι(P)} for all P ∈ X.) A point {P, Q} ̸= O is rational if and only if either P and Q are
+both rational themselves, or P and Q are Galois-conjugate quadratic points.
+Fix the point
+P0 :=
+�
+−1
+4
+�
+1 +
+√
+−15
+�
+, − 1
+16
+�
+17 + 9
+√
+−15
+��
+∈ X(Q, 2),
+for which we have c(P0) =
+1
+48
+�
+7 + 8√−15
+�
+/∈ Q. If we let
+P 0 :=
+�
+−1
+4
+�
+1 −
+√
+−15
+�
+, − 1
+16
+�
+17 − 9
+√
+−15
+��
+be the Galois conjugate of P0, then D0 := {P0, P 0} ∈ J(Q). Note that P 0 ̸= ι(P0), so
+D0 ̸= O. Poonen showed in [43, Prop. 1] that J(Q) ∼= Z, so D0 has infinite order.
+The curve X (hence also its Jacobian J) has good reduction at the prime p = 7. A straight-
+forward computation (e.g., in Magma) shows that the reduction �D0 ∈ J(F7) has order 21,
+so Dn := (1 + 21n)D0 ≡ D0 (mod 7) for all n ∈ Z. For each n we write Dn = {Pn, P n}
+with Pn ≡ P0 (mod 7) and P n ≡ P 0 (mod 7). We claim that Pn ∈ S7,P for all n ∈ Z, from
+which the result follows by Lemma 5.6.
+Since −15 ≡ −1 (mod 7) is not a square in F7, �
+P0 is quadratic over F7, thus the same is
+true for �
+Pn for all n ∈ Z. This implies that Pn is quadratic over Q for all n.
+The map c : X → P1 has good reduction at p = 7, so Pn ≡ P0 (mod 7) implies that
+c(Pn) ≡ c(P0) (mod 7). Arguing as in the previous paragraph, we conclude that c(Pn)
+is quadratic over Q. Finally, we note that since c(Pn) ≡ c(P0) ̸≡ ∞ (mod 7), we have
+v7(c(Pn)) ≥ 0, so Pn ∈ S7,P.
+□
+5.3. Proofs of Theorems 1.6 and 1.7.
+Proof of Theorem 1.6. That (ii) implies (i) is precisely the second statement in Proposi-
+tion 5.2, so it remains only to show that (i) implies (ii).
+Assume there are infinitely many c ∈ Q such that G(fc, Q) ⊊ G(fc, K) ∼= P for some
+quadratic field K.
+Every such occurrence of P as G(fc, K) yields a quadratic point on
+Y1(P), so there are infinitely many quadratic points on X1(P). Thus P ∈ Γ by Theorem 4.1.
+All that remains for us to show is that P cannot be isomorphic to ∅ or 6(3). Certainly
+one cannot have G(fc, Q) ⊊ G(fc, K) ∼= ∅, so we need only show that if c ∈ Q and K is a
+quadratic field with G(fc, K) ∼= 6(3), then in fact G(fc, Q) ∼= 6(3).
+Supposing that G(fc, K) ∼= 6(3), the map fc then has a period-3 point α ∈ K, and the six
+K-rational preperiodic points prescribed by the portrait 6(3) are ±α, ±fc(α), and ±f 2
+c (α).
+It therefore suffices to show that α ∈ Q.
+Let τ be the nontrivial element of the Galois group Gal(K/Q). Since fc is defined over Q,
+ατ is also a K-rational point of period 3, hence lies in the cycle {α, fc(α), f 2
+c (α)} (because
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+21
+the portrait 6(3) has only one 3-cycle2). Write ατ = f k
+c (α) for some k ∈ {0, 1, 2}. Then
+α = (ατ)τ = f k
+c (ατ) = f 2k
+c (α).
+Since α has exact period 3, this implies that k = 0; that is, ατ = α. Thus α ∈ Q, and
+therefore G(fc, Q) ∼= 6(3).
+□
+Proof of Theorem 1.7. First suppose that (i) holds; i.e., there are infinitely many c ∈ Q(2) ∖ Q
+such that G(fc, Q(c)) ∼= P. The curve X1(P) then has infinitely many quadratic points, and
+therefore P ∈ Γ by Theorem 4.1. By Proposition 5.7, we must have
+P ∈ Γ ∖ Γrat = Γ0 ∪ Γquad.
+Thus, (i) implies (ii). The converse is precisely Proposition 5.8.
+□
+6. Fields of definition of quadratic points
+In this section we address the question of whether the existence of a given preperiodic
+portrait over a given quadratic field has implications regarding standard arithmetic invariants
+of that field. In particular, we focus on the portraits that occur infinitely often over quadratic
+fields, namely those in the set Γ.
+To be precise, we are interested in arithmetic invariants of the fields of definition of qua-
+dratic points on dynamical modular curves X1(P). To partially justify the transition from
+realizations of portraits to simply points on dynamical modular curves, we begin by proving
+the following result:
+Proposition 6.1. For a number field K and a generic quadratic portrait P, the following
+are equivalent:
+(i) There exist infinitely many c ∈ K such that G(fc, K) ∼= P.
+(ii) The curve X1(P) has infinitely many K-rational points.
+Proof. That (i) implies (ii) is immediate from the definition of X1(P), so suppose that
+X1(P)(K) is infinite. Since X1(P) is irreducible (see Proposition 2.6(a)), this implies that
+X1(P) is isomorphic over K either to P1 or to an elliptic curve.
+In the former case, the result follows from essentially the same proof as for Proposition 5.4.
+In fact, the appropriate modification of that proof shows that there are infinitely many c ∈ K
+such that Q(c) = K and such that G(fc, K) ∼= P.
+In the latter case, choose a non-torsion point Q0 ∈ X1(P)(K), and choose a prime p ∈
+Spec OK of good reduction for the morphism c : X1(P) → P1 such that vp(Q0) ≥ 0. Since Q0
+is non-torsion, there are infinitely many points Q ∈ X1(P)(K) such that Q ≡ Q0 (mod p),
+and since the morphism c has good reduction at p, we have c(Q) ≡ c(Q0) (mod p) for every
+such Q. In particular, we have infinitely many Q ∈ X1(P)(K) such that vp(c(Q)) ≥ 0, so
+the desired result follows from Lemma 5.6.
+□
+We now move on to a result concerning the splitting of rational primes in the quadratic
+fields over which the portrait 10(3, 1, 1) is realized as a preperiodic portrait. This example
+is included in order to illustrate our methods, but similar reasoning can be applied to any
+portrait in Γ for which the corresponding modular curve is hyperelliptic.
+2In fact, if K is any quadratic field and c ∈ K, then fc has at most one 3-cycle defined pointwise over K.
+This follows from [36, Thm. 3] when c ∈ Q and [10, Thm. 4.5] in general.
+
+22
+JOHN R. DOYLE AND DAVID KRUMM
+As noted in [43], the dynamical modular curve corresponding to the portrait 10(3, 1, 1)
+is isomorphic to Xell
+1 (18). If K is the field of definition of a quadratic point on this curve,
+Kenku and Momose [26, Prop. 2.4] show that
+• either 2 splits or 3 does not split in K;
+• 3 is not inert in K; and
+• 5 and 7 are unramified in K.
+In what follows, for every polynomial f ∈ Z[x], we denote by πf the set of all integer primes
+p such that f does not have a root modulo p. Extending the results of Kenku and Momose,
+we prove the following. (Note that this proves Theorem 1.10.)
+Theorem 6.2. Let K be a quadratic field such that G(fc, K) ∼= 10(3, 1, 1) for some c ∈ K.
+Then the prime 2 splits in K, 3 is not inert in K, and letting
+f(x) = x6 + 2x5 + 5x4 + 10x3 + 10x2 + 4x + 1,
+every prime in the set πf (which includes 5 and 7) is unramified in K. Moreover, πf has
+Dirichlet density 13/18.
+For every nonzero rational number r, we let sqf(r) denote the squarefree part of r, i.e.,
+the unique squarefree integer d such that r/d is the square of a rational number.
+Lemma 6.3. Let f ∈ Z[x] be a monic polynomial of even degree, and let p be an odd prime.
+If p ∈ πf, then p is unramified in every quadratic field of the form Q(
+�
+f(r)) with r ∈ Q.
+Proof. Given a quadratic field K = Q(
+�
+f(r)), we must show that p does not divide the
+discriminant of K. Let D = sqf(f(r)), so that K = Q(
+√
+D). Since p is odd, it suffices to
+show that p does not divide D. Set g(x, y) = y2kf(x/y) ∈ Z[x, y], where deg(f) = 2k, and
+write r = n/d with gcd(n, d) = 1. Then D = sqf(g(n, d)), so that
+g(n, d) = Ds2,
+s ∈ Z.
+We now consider two cases. If d ≡ 0 mod p, then the above equation can be reduced
+modulo p to obtain n2k ≡ Ds2 mod p. Since p cannot divide n (given that n and d are
+coprime), we conclude that p does not divide D, as required.
+Suppose now that d ̸≡ 0 mod p. We can then consider the equation d2kf(n/d) = Ds2 as
+taking place in the ring Zp. If p | D, then reducing modulo p we obtain f(n/d) ≡ 0 mod p,
+contradicting the hypothesis that f has no root modulo p. Therefore p cannot divide D.
+□
+Proof of Theorem 6.2. By [13, Thm. 3.25], we have K = Q(
+�
+f(r)) for some r ∈ Q∖{0, −1}.
+Writing r = n/d in lowest terms, it follows that K = Q(
+�
+g(n, d)), where
+g(n, d) := d6f(n/d) = n6 + 2n5d + 5n4d2 + 10n3d3 + 10n2d4 + 4nd5 + d6.
+We claim that g(n, d) ≡ 1 mod 8. If n, d are both odd, then
+g(n, d) ≡ 1 + 2nd + 5 + 10nd + 10 + 4nd + 1 = 17 + 16nd ≡ 1 mod 8.
+If n is even and d is odd, then g(n, d) ≡ d6 ≡ 1 mod 8. If n is odd and d is even, then
+g(n, d) ≡ 1 + 2nd + 5d2 mod 8. Writing n = 2k + 1 for some integer k we see that
+g(n, d) ≡ 5d2 + 2d + 1 ≡ (d + 1)2 ≡ 1 mod 8,
+which proves the claim. Letting D = sqf(g(n, d)), the fact that g(n, d) ≡ 1 (mod 8) implies
+that D ≡ 1 mod 8 and therefore 2 splits in Q(
+√
+D) = K.
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+23
+Similar reasoning shows that g(n, d) is congruent to either 0 or 1 modulo 3. Considering
+all possible values of n and d modulo 9, we find that if g(n, d) is divisible by 9, then n and
+d are both divisible by 3, which is a contradiction; hence 9 ∤ g(n, d). Writing g(n, d) = Ds2
+for some integer s, this implies that s is not divisible by 3, and therefore g(n, d) ≡ D mod 3.
+Hence, D is congruent to 0 or 1 modulo 3, and therefore 3 is not inert in K.
+A computation in Magma based on [27, Thm. 2.1] shows that πf has Dirichlet density
+13/18. Finally, Lemma 6.3 implies that every odd prime in πf is unramified in K, and we
+have already shown that 2 is unramified in K.
+□
+An argument very similar to the proof of Theorem 6.2 yields the following result, in which
+the relevant dynamical modular curve is known to be isomorphic to Xell
+1 (13).
+Theorem 6.4. Let K be a quadratic field such that G(fc, K) ∼= 10(3, 2) for some c ∈ K.
+Then the prime 2 splits in K, and every prime in πf is unramified in K, where
+f(x) = x6 + 2x5 + x4 + 2x3 + 6x2 + 4x + 1.
+Moreover, the set πf has Dirichlet density 13/18.
+Our next result concerns the curve Xell
+1 (16), which is isomorphic to the modular curve for
+the portrait 8(4); see Section 3.7 of [13]. In contrast to Theorems 6.2 and 6.4, we show that
+the discriminants of quadratic fields defined by points on Xell
+1 (16) are not restricted to any
+residue class. (Note that this proves Theorem 1.9(a).)
+Theorem 6.5. Let P denote the portrait 8(4). For every prime integer p and every residue
+class c ∈ Z/pZ, there exist infinitely many squarefree integers d ∈ c such that the curve
+X1(P) has a quadratic point defined over the field Q(
+√
+d).
+For the proof of the theorem we use the methods of [28] and [29]; the following lemma,
+which follows from Proposition 14 in [29], collects the main tools to be used.
+Lemma 6.6. Let f ∈ Z[x] be a squarefree polynomial of degree at least 3, and such that
+every irreducible factor of f has degree at most 6. Let
+S(f) = {sqf(f(x)) : x ∈ Q and f(x) ̸= 0}.
+Let D be the largest integer dividing all integer values of f. Fix a prime p such that f has
+an irreducible factor whose discriminant is not divisible by p, and let ε = εp ∈ {0, 1} be
+the parity of ordp(D). Finally, for c ∈ Z/pZ and v ∈ Z, let σ(p, c, v) denote the following
+statement.
+(6.1)
+σ(p, c, v) :
+�
+�
+�
+�
+�
+There exist h ∈ c and x0, y0 ∈ Z satisfying
+• hy2
+0 ≡ f(x0) (mod p2(v+ε)+1) and
+• ordp(y0) = v + ε.
+Suppose c is nonzero and σ(p, c, v) holds for some v ≥ 0. Then the set S(f) ∩ c is infinite.
+Proof of Theorem 6.5. By [38, p. 93], the curve X1(P) is hyperelliptic and has an affine
+model given by y2 = f(x), where f(x) = −x(x2 + 1)(x2 − 2x − 1).
+In order to prove the theorem it suffices to show that, for every prime p and residue class
+c ∈ Z/pZ, the set S(f)∩c is infinite. Indeed, if d belongs to this set, we may write dy2
+0 = f(x0)
+for some rational numbers x0, y0 with y0 ̸= 0. The pair (x0, y0
+√
+d) then represents a quadratic
+point on X1(P) whose field of definition is Q(
+√
+d). Hence, the theorem follows.
+
+24
+JOHN R. DOYLE AND DAVID KRUMM
+In the notation of Lemma 6.6, for the above polynomial f(x) we have D = 2; moreover,
+the irreducible factor x of f(x) has discriminant 1, which is coprime to every prime p. It
+follows in particular that
+εp =
+�
+0
+if p is odd,
+1
+if p = 2.
+Fix a prime p. For the class c = 0 ∈ Z/pZ, Theorem 2.1 in [28] implies that the set
+S(f) ∩ c is infinite, as desired. (When p = 2, the hypotheses of the cited theorem are not all
+satisfied, but the proof still applies.) Next, we claim that
+for every nonzero c ∈ Z/pZ, either σ(p, c, 0) or σ(p, c, 1) must hold.
+Assuming this claim for the moment, Lemma 6.6 implies that the set S(f) ∩ c is infinite,
+completing the proof of the theorem.
+To prove the claim we consider first the case p = 2: taking c = 1, the statement σ(p, c, 1)
+can be shown to hold by setting (h, x0, y0) = (1, 16, 4) in (6.1). The remainder of the proof
+is divided into three cases.
+Case p ≤ 5: Taking p = 3, we check that σ(p, c, 1) holds for c = 1, 2 by using the tuples
+(h, x0, y0) = (1, 9, 3)
+and
+(h, x0, y0) = (2, 18, 3).
+Similarly, taking p = 5, we check that σ(p, c, 1) holds for c = 1, 2, 3, 4, respectively, by using
+the following tuples (h, x0, y0):
+(1, 25, 5), (2, 18, 5), (3, 75, 5), (4, 7, 5).
+In the remaining two cases we show that σ(p, c, 0) holds. For r ∈ c, let Xr be the hy-
+perelliptic curve over Fp defined by the equation ry2 = f(x). Since εp = 0, the statement
+σ(p, c, 0) is equivalent to the requirement that Xr have an affine point (x0, y0) ∈ Xr(Fp) with
+y0 ̸= 0; we refer to such points as nontrivial points on Xr. Thus, it remains to show that Xr
+has at least one nontrivial point.
+Case 7 ≤ p ≤ 23: A straightforward search for points verifies that #Xr(Fp) ≥ 7 for every
+nonzero r ∈ Fp. (Note that it suffices to check this for just two values of r, one in each
+square class modulo p.) The number of affine points (x0, y0) ∈ Xr(Fp) having y0 = 0 is at
+most 5, so there must exist at least one nontrivial point in Xr(Fp), as required.
+Case p ≥ 29: For r ∈ Fp ∖ 0, the curve Xr has genus 2, so the Hasse–Weil bound yields
+#Xr(Fp) ≥ ⌊p + 1 − 4√p⌋ ≥ 7.
+The same reasoning as in the previous case implies that Xr has a nontrivial Fp-point.
+□
+We end the paper by proving Theorem 1.9(b).
+Proposition 6.7. Let P = 8(4). There exist infinitely many imaginary quadratic fields K
+with class number divisible by 10, such that X1(P) has a quadratic point defined over K.
+Proof. As noted earlier, the curve X1(P) is isomorphic to Xell
+1 (16). The result follows from
+[20, Cor. 3.2], since the Jacobian J1(16) has a rational torsion point of order 10.
+□
+Remark 6.8. Experimental evidence supports a statement stronger than Proposition 6.7:
+for every imaginary quadratic field K ̸= Q(√−15) that is the field of definition of a point on
+X1(P), the class number of K is divisible by 10. One approach to proving this is suggested
+by the methods of [2, 20]; however, the required computational tools (in particular, for
+computing quotients of abelian varieties) do not seem to be presently available.
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+25
+Appendix A. Dynamical modular curves of genera 1 and 2
+We provide here models for all dynamical modular curves X1(P) of genus 1 or 2, together
+with an explicit description of the morphism c : X1(P) → P1. Each of these models appears
+in [43]. Note that in some cases we provide two models—one of the form y2 = F(x), and
+another that turns out to be more useful for certain aspects of our proofs.
+Portrait P
+Model(s) for X1(P)
+Morphism c : X1(P) → P1
+8(1,1)a
+y2 = x3 − x2 + x
+− (x2 + 1)2
+4x(x − 1)2
+8(1,1)b
+y2 = 2(x3 + x2 − x + 1)
+−
+2(x2 + 1)
+(x + 1)2(x − 1)2
+8(2)a
+y2 = x3 − 2x + 1
+−(x2 − 2x + 2)(x2 + 2x − 2)
+4x2(x − 1)
+8(2)b
+y2 = 2(x3 + x2 − x + 1)
+−x4 + 2x3 + 2x2 − 2x + 1
+(x + 1)2(x − 1)2
+10(2,1,1)a
+y2 = 5x4 − 8x3 + 6x2 + 8x + 5
+−(3x2 + 1)(x2 + 3)
+4(x + 1)2(x − 1)2
+y2 + xy + y = x3 − x2 − x
+x − 2
+4x(x − 1)y − x4 − x3 + 3x − 1
+4x2(x − 1)
+10(2,1,1)b
+y2 = (5x2 − 1)(x2 + 3)
+−(3x2 + 1)(x2 + 3)
+4(x + 1)2(x − 1)2
+y2 + xy + y = x3 + x2
+−
+x + 2
+4x(x + 1)y − x4 + 4x3 + 6x2 + 3x + 1
+4x2(x + 1)
+8(3)
+y2 = x6 − 2x4 + 2x3 + 5x2 + 2x + 1
+−x6 + 2x5 + 4x4 + 8x3 + 9x2 + 4x + 1
+4x2(x + 1)2
+8(4)
+y2 = −x(x2 + 1)(x2 − 2x − 1)
+(x2 − 4x − 1)(x4 + x3 + 2x2 − x + 1)
+4x(x + 1)2(x − 1)2
+10(3,1,1)
+y2 = x6 + 2x5 + 5x4 + 10x3 + 10x2 + 4x + 1
+−x6 + 2x5 + 4x4 + 8x3 + 9x2 + 4x + 1
+4x2(x + 1)2
+10(3,2)
+y2 = x6 + 2x5 + x4 + 2x3 + 6x2 + 4x + 1
+−x6 + 2x5 + 4x4 + 8x3 + 9x2 + 4x + 1
+4x2(x + 1)2
+
+26
+JOHN R. DOYLE AND DAVID KRUMM
+Appendix B. Tables of preperiodic portraits
+B.1. Preperiodic portraits realized over quadratic fields. We list here the 46 portraits
+known to be realized as G(fc, K) for some quadratic field K and c ∈ K. These were found in
+the search described in [13]. The label of each portrait is in the form N(ℓ1, ℓ2, . . .), where N
+is the number of vertices in the portrait and ℓ1, ℓ2, . . . are the lengths of the directed cycles
+in the portrait in nonincreasing order. If more than one isomorphism class with this data
+was observed, we add a lowercase Roman letter to distinguish them. For example, the labels
+5(1,1)a and 5(1,1)b correspond to the two isomorphism classes of portraits observed that
+have five vertices and two fixed points. In all figures, we omit the vertex corresponding to
+the fixed point at infinity.
+0
+2(1)
+3(1,1)
+3(2)
+4(1)
+4(1,1)
+4(2)
+5(1,1)a
+5(1,1)b
+5(2)a
+5(2)b
+6(1,1)
+6(2)
+6(2,1)
+6(3)
+7(1,1)a
+7(1,1)b
+7(2,1,1)a
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+27
+7(2,1,1)b
+8(1,1)a
+8(1,1)b
+8(2)a
+8(2)b
+8(2,1,1)
+8(3)
+8(4)
+9(2,1,1)
+10(1,1)a
+10(1,1)b
+10(2)
+10(2,1,1)a
+10(2,1,1)b
+10(3)a
+10(3)b
+10(3,1,1)
+
+28
+JOHN R. DOYLE AND DAVID KRUMM
+10(3,2)
+12(2)
+12(2,1,1)a
+12(2,1,1)b
+12(3)
+12(4)
+12(4,2)
+12(6)
+14(2,1,1)
+14(3,1,1)
+14(3,2)
+
+QUADRATIC POINTS ON DYNAMICAL MODULAR CURVES
+29
+B.2. Additional portraits. The following portraits are not known to be realized over qua-
+dratic fields (and, in some cases, have been shown not to be); however, they make an
+appearance in the discussion in Section 4.2, so we include them here. The labels G1, . . . , G10
+are taken from [10].
+G1
+G2
+G3
+G4
+G5
+G6
+G7
+G8
+G9
+G10
+
+30
+JOHN R. DOYLE AND DAVID KRUMM
+References
+[1] Robert L. Benedetto, Ruqian Chen, Trevor Hyde, Yordanka Kovacheva, and Colin White, Small dy-
+namical heights for quadratic polynomials and rational functions, Exp. Math. 23 (2014), no. 4, 433–447.
+MR 3277939
+[2] Yuri Bilu and Jean Gillibert, Chevalley-Weil theorem and subgroups of class groups, Israel J. Math. 226
+(2018), no. 2, 927–956. MR 3819714
+[3] J´er´emy Blanc, Jung Kyu Canci, and Noam D. Elkies, Moduli spaces of quadratic rational maps with
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+MR 3431627
+[4] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language,
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+URL: http://maths.dk
+Email address: [email protected]
+

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+Thermodynamic Correlation Inequality
+Yoshihiko Hasegawa∗
+Department of Information and Communication Engineering,
+Graduate School of Information Science and Technology,
+The University of Tokyo, Tokyo 113-8656, Japan
+(Dated: January 10, 2023)
+Uncertainty relations place fundamental limits on the operations that physical systems can per-
+form. In this Letter, we obtain uncertainty relations that give bounds for the correlation function,
+which measures the relationship between a system’s current state and its future state, in both
+classical and quantum Markov processes. The obtained bounds, referred to as thermodynamic cor-
+relation inequality, state that the change in the correlation function has an upper bound comprising
+the dynamical activity, a measure of the activity of a Markov process. Moreover, applying the ob-
+tained relation to the linear response function, we show that the effect of perturbation has a bound
+comprising the dynamical activity.
+Introduction.—Uncertainty relations imply that there
+are ultimate impossibilities in the physical world that
+cannot be overcome by any technological advances. The
+most well-known example is the Heisenberg uncertainty
+relation [1, 2], which establishes a limit on the precision of
+position-momentum measurement. The quantum speed
+limit is interpreted as the energy-time uncertainty rela-
+tion and places a limit on the speed at which the quan-
+tum state can be changed [3–10] (see [11] for a review).
+It has many applications in quantum computation [12],
+quantum communication [13, 14], and quantum thermo-
+dynamics [5]. Recently, the concept of speed limit has
+also been considered in classical systems [15–17]. In par-
+ticular, the Wasserstein distance can be used to obtain
+the minimum entropy production required for a stochas-
+tic process to transform one probability distribution into
+another [18–22]. Moreover, the speed limit has been gen-
+eralized to the time evolution of the observables [23–27],
+where the speed of the observables is the quantity of in-
+terest. A closely related principle was recently found in
+stochastic thermodynamics, which is known as the ther-
+modynamic uncertainty relation [28–50] (see [51] for a re-
+view), stating that, for thermodynamic systems, higher
+accuracy can be achieved at the expense of larger ther-
+modynamic cost. Nowadays, the thermodynamic uncer-
+tainty relations become a central topic in nonequilibrium
+thermodynamics, and their importance is also recognized
+from a practical standpoint because thermodynamic un-
+certainty relations can be used to infer entropy produc-
+tion without detailed knowledge on the system [52–55].
+In the present Letter, we obtain uncertainty relations
+that confer bounds for the correlation function in classical
+and quantum Markov processes. The correlation function
+is a statistical measure that quantifies the correlation be-
+tween the current state of a system and its future or past
+states. In a Markov process, the correlation function can
+be used to analyze the dependence of the current state on
+past states, and to identify any patterns in the system’s
+∗ [email protected]
+behavior over time. We derive the thermodynamic corre-
+lation inequality stating that the amount of the correla-
+tion change has an upper bound that comprises the dy-
+namical activity, which quantifies the activity of a system
+of interest. Our derivation is based on the continuous ma-
+trix product state representation [56, 57], which is a real-
+ization of the bulk/boundary correspondence in Markov
+processes. It allows us to represent a classical or quan-
+tum Markov process by the corresponding quantum field
+state, where jump events in the Markov process are rep-
+resented by particle creations in the field state. Since the
+dynamics of the continuous matrix product state is as-
+sumed to obey that of quantum mechanics, we can apply
+the techniques developed in quantum information [58].
+The obtained bound exhibits unexpected generality; it
+holds for classical as well as quantum Markov processes.
+Moreover, it can be generalized to multi-point correlation
+functions and multivariate Markov processes. The cor-
+relation function gives the spectral information via the
+Wiener-Khinchin theorem and plays a fundamental role
+in the linear response theory [59]. The linear response
+function can be represented by a time derivative of the
+corresponding correlation function, which is the state-
+ment of the fluctuation-dissipation theorem.
+Applying
+the obtained correlation bound to the linear response
+function, we derive an upper bound to the perturbation.
+Results.—We derive the thermodynamic correlation in-
+equality for a classical Markov process. A quantum gen-
+eralization will be discussed later. Consider a classical
+Markov process with N states B ≡ {B1, B2, · · · , BN}.
+Let {X(t)|t ≥ 0} be a collection of discrete random vari-
+ables that take values in B (that is X(t) ∈ B). Let P(ν; t)
+be the probability that X(t) is Bν at time t and Wνµ be
+the transition rate of X(t) from Bµ to Bν. The time evo-
+lution of P(t) ≡ [P(1; t), . . . , P(N; t)]⊤ is governed by
+the following master equation:
+dP(t)
+dt
+= WP(t),
+(1)
+where W ≡ {Wνµ}. Next, we define the scoring function
+S(·) that takes a state Bi (i ∈ {1, 2, . . . , N}) and returns
+a real value of (−∞, ∞). When it is clear from the con-
+arXiv:2301.03060v1  [quant-ph]  8 Jan 2023
+
+2
+State
+Time
+State
+Time
++1
+-1
++1
+-1
+(a)
+(b)
+FIG. 1.
+Illustration of Markov processes.
+(a) Classical
+Markov process (dichotomous process) two states {B1, B2}.
+The score function is specified by S(B1) = −1 and S(B2) = 1.
+(b) Quantum Markov process (two level atom driven by a
+classical laser field).
+The time evolution of the quantum
+Markov process consists of continuous evolution induced by
+the effective Hamiltonian Heff and discontinous evolution due
+to the jump operator L.
+The score function is given by
+S(ρ) = 2Tr[ρ |e⟩ ⟨e|] − 1, in which the ground and excited
+states give S(|g⟩ ⟨g|) = −1 and S(|e⟩ ⟨e|) = 1.
+text, we use the notation S(t) ≡ S(X(t)) for simplicity.
+Moreover, we define
+Smax ≡ max
+Bi∈B |S(Bi)|,
+(2)
+which is the maximum absolute value of the score func-
+tion within B. We are interested in the correlation func-
+tion C(t) ≡ ⟨S(0)S(t)⟩, where
+⟨S(0)S(t)⟩ =
+�
+µ,ν
+S(Bν)S(Bµ)P(µ; 0)P(ν; t|µ; 0)
+= 1SeWtSP(0).
+(3)
+Here, P(ν; t|µ; 0) is the conditional probability that
+X(t) = Bν given X(0) = Bµ, 1 ≡ [1, 1, . . . , 1] is the
+trace state, and S ≡ diag[S(B1), . . . , S(BN)]. The cor-
+relation function C(t) is widely explored in the field of
+stochastic process [60, 61]. Recently, the correlation func-
+tion is considered in the context of quantum speed limit
+[26, 62], which is obtained as particular cases of speed
+limit on observables.
+As an example of the classical
+system, Fig. 1(a) shows the dichotomous process, which
+comprises two states {B1, B2}. X(t) in this process ex-
+hibits random switching between B1 and B2. For the di-
+chotomous process, the score function is typically given
+by S(B1) = −1 and S(B2) = 1. To quantify the Markov
+process, we define the dynamical activity A(t) as follows
+[63]:
+A(t) ≡
+� t
+0
+dt′
+�
+ν,µ,ν̸=µ
+P(µ; t′)Wνµ.
+(4)
+A(t) represents the average number of jumps during the
+interval [0, t] and it quantifies the activity of the stochas-
+tic process. The dynamical activity plays a fundamental
+role in classical speed limits [15] and thermodynamic un-
+certainty relations [30, 32].
+In the classical Markov process, we obtain an upper
+bound on the correlation function C(t). For 0 ≤ t1 < t2,
+we obtain the following bound:
+|C(t1) − C(t2)| ≤ 2S2
+max sin
+�
+1
+2
+� t2
+t1
+�
+A(t)
+t
+dt
+�
+,
+(5)
+which holds for 0 ≤ 1
+2
+� t2
+t1
+√
+A(t)
+t
+dt ≤ π
+2 . For t1 and t2
+outside this range, the upper bound is |C(t1) − C(t2)| ≤
+2S2
+max, which holds trivially. Equation (5) is the main
+result of this Letter. It should be emphasized that all
+the quantities appeared in Eq. (5) are physically inter-
+pretable. The proof of Eq. (5) is based on the continuous
+matrix product state and inequalities in quantum infor-
+mation, which is shown in Appendix D. Equation (5)
+holds for an arbitrary time-independent Markov process
+starting from an arbitrary initial probability distribution
+with an arbitrary score function S(Bi).
+Equation (5)
+states that higher dynamical activity allows the system
+to forget its current state more quickly, which agrees with
+our intuition. For a simple consistency check, consider
+the null dynamics (i.e., Wνµ = 0 for all ν and µ), in
+which there is no jump at all. In this case, the dynamical
+activity becomes A(t) = 0 and thus the right-hand side
+of Eq. (5) vanishes to yield ⟨S(0)S(t)⟩ = ⟨S(0)2⟩, which
+is trivially true. For the steady state case, C(t2) − C(t1)
+for t1 < t2 is negative, and hence it seems that we do not
+have to consider absolute operation in Eq. (5). However,
+when the system is not in steady state, this is not the
+case. Note that a weaker bound can be obtained via a
+thermodynamic uncertainty relation derived in Ref. [64].
+Let us consider particular cases of Eq. (5). Simply taking
+t1 = 0 and t2 = t with t > 0, Eq. (5) provides an upper
+bound for |C(0) − C(t)|:
+|C(0) − C(t)| ≤ 2S2
+max sin
+�
+1
+2
+� t
+0
+�
+A(t′)
+t′
+dt′
+�
+,
+(6)
+where 0 ≤
+1
+2
+� t
+0
+√
+A(t′)
+t′
+dt′ ≤
+π
+2 . Moreover, let ϵ be an
+infinitesimally small positive value. Substituting t1 = t−
+ϵ and t2 = t into Eq. (5) and using the Taylor expansion
+to the sinusoidal function, we obtain
+����
+dC(t)
+dt
+���� ≤ S2
+max
+�
+A(t)
+t
+.
+(7)
+Equation (7) states that the absolute change of the corre-
+lation function is determined by the dynamical activity.
+Equation (6) holds for 0 ≤
+1
+2
+� t
+0
+√
+A(t′)
+t′
+dt′ ≤
+π
+2 and
+thus the predictive power of the bound is lost at a finite
+time. An alternative bound to Eq. (5) is given by
+|C(0) − C(t)| ≤ 2S2
+max
+�
+1 − η(t),
+(8)
+where η(t) is the Loschmidt echo [65] between time
+evolved state and the initial state in the continuous
+
+3
+(a)
+(b)
+FIG. 2.
+Results of numerical simulations.
+(a) The ratio
+|∂tC(t)|/(S2
+max
+�
+A(t)/t) for the dichotomous process.
+The
+result obtained with W12 = 1, W22 = −1, P(0) = [0, 1] is plot-
+ted by the dashed line. The results obtained by random pa-
+rameters are plotted by the solid lines. The parameter ranges
+for the random realizations are W12 ∈ [0, 1], W21 ∈ [0, 1],
+S(B1) ∈ [−1, 0], and S(B2) ∈ [0, 1].
+The initial distribu-
+tion is first sampled from P1(0) ∈ [0, 1] and P2(0) ∈ [0, 1]
+and then normalize the sampled distribution.
+(b) The ra-
+tio |∂tC(t)|/(S2
+max
+�
+B(t)/t) for the driven two level atom
+model. The results obtained by random parameters are plot-
+ted by the solid lines. The parameter ranges are Ω ∈ [0.1, 1],
+∆ ∈ [0.1, 1], and κ ∈ [0.1, 1]. The initial density is sampled
+from ⟨g|ρ(0)|g⟩ ∈ [0, 1] and ⟨e|ρ(0)|e⟩ ∈ [1, 2] and normalized
+the sampled density (non-diagonal elements are zero).
+matrix product state representation (see Appendix C).
+Equation (8) is the second result of this paper, whose
+proof is provided in Appendix E. Following Ref. [66], we
+can compute η(t) for the classical Markov process as fol-
+lows:
+η(t) ≡
+��
+µ
+P(µ; 0)
+�
+e−t �
+ν(̸=µ) Wνµ
+�2
+,
+(9)
+which can be represented by quantities of the Markov
+process. Note that the Loschmidt echo η(t) constitutes a
+lower bound in a quantum and classical thermodynamic
+uncertainty relation [66].
+The term within the square
+root in η(t) represents the survival probability that there
+is no jump starting from the state Bµ. Therefore, when
+the activity of dynamics is lower, the survival probabil-
+ity becomes higher and in turn η(t) yields a higher value.
+Although the Loschmidt echo η(t) has fewer physical in-
+terpretations than dynamical activity, it has the advan-
+tage over Eq. (5) that the bound of Eq. (8) holds for any
+value of t.
+We can extend Eq. (5) to a quantum Markov pro-
+cess.
+Let ρ(t) be a density operator of a quantum
+Markov process at time t. We assume that the dynamics
+of ρ(t) is governed by the following Lindblad equation
+˙ρ(t) = L(ρ(t)), where L is the Lindblad superoperator
+[67, 68]:
+L(ρ(t)) ≡ −i [H, ρ(t)] +
+�
+m
+D (ρ(t), Lm) .
+(10)
+Here H is a Hamiltonian, D(ρ, L) ≡ LρL† − {L†L, ρ}/2
+is the dissipator, Lm is the mth jump operator.
+We
+can unravel Eq. (10) to obtain a quantum trajectory,
+which is a measurement record when observing the en-
+vironment. The dynamics of the quantum trajectory is
+represented by a stochastic Schr¨odinger equation. Simi-
+lar to the classical case, we assign the score function to
+the quantum state ρ(t) via S(ρ(t)) = Tr[ρ(t)O], where
+O is an Hermitian operator.
+Figure 1(b) illustrates
+an example of a quantum trajectory, which consists of
+continuous state change by the effective Hamiltonian
+Heff ≡ H − (i/2) �
+m L†
+mLm and discontinuous jumps
+by Lm. Let ρ(0) be the initial density operator. Then
+the correlation function C(t) is calculated by
+C(t) = S(ρ(0))S(ρ(t)).
+(11)
+For 0 ≤ t1 < t2, the following relation holds:
+|C(t1) − C(t2)| ≤ 2S2
+max sin
+�
+1
+2
+� t2
+t1
+�
+B(t)
+t
+dt
+�
+,
+(12)
+which holds for 0 ≤ 1
+2
+� t2
+t1
+√
+B(t)
+t
+dt ≤ π
+2 . Here B(t) is the
+quantum dynamical activity defined in Ref. [64], which
+is the quantum generalization of Eq. (4) (see Eq. (F2) in
+Appendix F). B(t) is defined through the quantum Fisher
+information. The quantum dynamical activity plays an
+important role in a speed limit and a thermodynamic
+uncertainty relation [64]. The proof of Eq. (12) is shown
+in Appendix E. Equation (12) is the same as Eq. (5)
+except that A(t) in Eq. (5) is replaced by its quantum
+counter part B(t). Following the same procedure as in
+Eq. (7), we obtain
+����
+dC(t)
+dt
+���� ≤ S2
+max
+�
+B(t)
+t
+.
+(13)
+Moreover, the bound of Eq. (8) also holds for the quan-
+tum Markov process, where the Loschmidt echo for the
+quantum case becomes [66] (Appendix C):
+η(t) =
+��Tr
+�
+e−iHefftρ(0)
+���2 .
+(14)
+The Loschmidt echo shown in Eq. (14) also constitutes
+the lower bound in a quantum thermodynamic uncer-
+tainty relation [66].
+Numerical simulations.—We perform numerical sim-
+ulations to verify the correlation bounds [Eqs. (7) and
+(13)].
+We first demonstrate Eq. (7) with the classical
+dichotomous process [69], which takes only two states
+B = {B1, B2}. The dichotomous process has a number of
+applications in communication engineering and physics.
+We are interested in the ratio between the left and right
+hand sides of Eq. (7), i.e., |∂tC(t)|/(S2
+max
+�
+A(t)/t) which
+must be no larger than 1 according to Eq. (7). We set the
+score function to S(B1) = −1 and S(B2) = 1, in which
+Smax = 1. The transition rate is set to W12 = 1 and
+W22 = −1, where the other elements are set to 0. The
+
+4
+initial distribution is P(0) = [0, 1].
+We plot the ratio
+as a function of t in Fig. 2(a) with the dashed line. We
+also randomly determine the score function S(Bi), the
+transition rate Wnm, and the initial distribution P(0)
+and calculate the ratio. The ratio as a function of t for
+the random realizations is plotted by the solid line in
+Fig. 2(a) (the parameter ranges are shown in the cap-
+tion of Fig. 2(a)). We see that all the results are below
+1 (the dotted line), which numerically verifies the bound
+of Eq. (7).
+Next, we consider a simple two-level atom driven by
+a classical laser field to check the correlation bound of
+Eq. (13), whose dynamics is represented by a Lindblad
+equation: H = ∆ |e⟩ ⟨e| + Ω
+2 (|e⟩ ⟨g| + |g⟩ ⟨e|) and L =
+√κ |g⟩ ⟨e|, where ∆, Ω, and κ are model parameters, and
+|e⟩ and |g⟩ are the excited and ground states, respectively.
+For the score function, we choose S(ρ) = 2Tr[ρ |e⟩ ⟨e|]−1,
+which ranges within [−1, 1] and thus Smax = 1. We calcu-
+late |∂tC(t)|/(S2
+max
+�
+B(t)/t), which is the ratio between
+the left and right-hand side of Eq. (13). We randomly de-
+termine the model parameters and the initial state (the
+parameter ranges are shown in the caption of Fig. 2(b)).
+The random realizations are shown by the solid lines in
+Fig. 2(b).
+Different from the classical case, the corre-
+lation oscillates due to the contribution of the effective
+Hamiltonian Heff. Since all the random realizations are
+below 1 (the dotted line), we numerically verify Eq. (13).
+Linear response.—The correlation function C(t) is
+closely related to the linear response theory [59]. Here,
+we apply the correlation bound of Eqs. (5) and (7) to
+the linear response theory (see Appendix G for details).
+Suppose that the Markov process is in the steady state
+Pst = [Pst(1), . . . , Pst(N)], that satisfies WPst = 0. We
+apply a weak perturbation χFf(t) to the master equa-
+tion of Eq. (1), that is W → W + χFf(t) in Eq. (1),
+where 0 < χ ≪ 1 and F is an N × N matrix, and f(t) is
+arbitrary real function of time t. We expand the proba-
+bility distribution as P(t) = Pst +χP1(t), where P1(t) is
+the first-order correction to the probability distribution.
+Collecting the first-order contribution O(χ) in Eq. (1),
+P1(t) is given by
+P1(t) =
+� t
+−∞
+eW(t−t′)FPstf(t′)dt′.
+(15)
+Let us consider a scoring function G(Bn), which may
+be different from S(Bn) at the moment, and define the
+expectation of G(Bn) by
+⟨G⟩ =
+�
+n
+G(Bn)Pst(n) = 1GPst,
+(16)
+where G ≡ diag[G(B1), . . . , G(BN)]. The change in ⟨G⟩
+due to the perturbation, represented by ∆G ≡ 1GP(t)−
+1GPst, is
+∆G(t) = χ
+� ∞
+−∞
+RG(t − t′)f(t′)dt′,
+(17)
+where RG(t) is the linear response function:
+RG(t) =
+�
+1GeWtFPst
+t ≥ 0
+0
+t < 0 .
+(18)
+In the linear response regime, any input-output relation
+can be expressed through RG(t). From Eq. (3), the time
+derivative of C(t) reads ∂tC(t) = 1SeWtWSPst. Com-
+paring Eq (18) and ∂tC(t), when G = S and F = WS,
+∂tC(t) gives the linear response function of Eq. (18),
+which is the statement of the fluctuation-dissipation the-
+orem.
+As a particular case, let us consider the pulse pertur-
+bation, f(t) = δ(t), where δ(t) is the Dirac delta func-
+tion. This perturbation corresponds to the application of
+a sharp pulsatile perturbation at t = 0. Then the change
+of the expectation of S(Bn) under the perturbation F =
+WS, represented by ∆S(p), is ∆S(p)(t) = χ∂tC(t) (the
+superscript (p) represents that it is the pulse response).
+The correlation bound of Eq. (7) gives
+���∆S(p)(t)
+��� ≤ χS2
+max
+�
+a
+t ,
+(19)
+where
+a
+is
+the
+rate
+of
+dynamical
+activity
+a
+≡
+�
+ν,µ,ν̸=µ Pst(µ)Wνµ (note that A(t) = at for a steady
+state). Equation (19) relates the dynamical activity with
+the effect of the pulse perturbation on the Markov pro-
+cess. A step response can be calculated in a similar way.
+We apply a constant perturbation switched on at t = 0,
+which can be modeled by f(t) = Θ(t) with Θ(t) being
+the Heaviside step function. From Eq. (17), we obtain
+∆S(s)(t) = χ
+� t
+0
+RS(t′)dt′.
+(20)
+Equation (20) leads to the following bound:
+|∆S(s)(t)| ≤ 2χS2
+max sin
+�√
+at
+�
+,
+(21)
+which holds for 0 ≤
+√
+at ≤ π/2. For t outside this range,
+the trivial inequality |∆S(s)(t)| ≤ 2χS2
+max holds.
+Generalizations.—So far we have been concerned with
+the two-point correlation function. It is straightforward
+to extend the bounds to the multi-point correlation func-
+tions. Let us consider a J-point correlation function:
+⟨S(t1)S(t2) · · · S(tJ)⟩
+≡
+�
+S(Bn1)S(Bn2) · · · S(BnJ)P(n1; t1)
+× P(n2; t2|n1; t1) · · · P(nJ; tJ|nJ−1; tJ−1),
+(22)
+where 0 ≤ t1 < t2 < · · · < tJ. We can obtain analogous
+relations of Eqs. (6) and (8) for Eq. (22).
+Markov processes are often represented by multiple
+variables.
+For example, in stochastic thermodynam-
+ics, a multipartite process can reveal the relation be-
+tween dissipated heat and information flow [70, 71]. For
+
+5
+simplicity, here we consider a bivariate Markov pro-
+cess defined in (X(t), Y (t)), {(X(t), Y (t))|t ≥ 0} that
+satisfies in X(t) ∈ BX and Y (t) ∈ BY .
+Moreover,
+we define different score functions for X(t) and Y (t),
+which are expressed by SX(·) and SY (·), respectively,
+and define SX,max ≡ maxB∈BX SX(B) and SY,max ≡
+maxB∈BY SY (B). We are often interested in the correla-
+tion CXY (t) ≡ ⟨SX(t)SY (0)⟩. Then, |CXY (0) − CXY (t)|
+obeys the same upper bounds of Eqs. (5) and (8) except
+that S2
+max is replaced by SX,maxSY,max, which gives a
+bound that is tighter than or equal to Eqs. (5) and (8).
+Conclusion.—In this Letter, we present a relation be-
+tween the correlation function and dynamical activity in
+classical and quantum Markov processes. The obtained
+bounds hold for arbitrary time-independent transition
+rate starting from an arbitrary initial distribution. By
+applying the obtained bounds to the linear response the-
+ory, we demonstrate that the effect of perturbations on
+a steady state system is bounded by the dynamical ac-
+tivity. We expect that our findings have the potential to
+enhance our understanding of nonequilibrium dynamics,
+as the correlation function plays a fundamental role in
+thermodynamics.
+Appendix A: Continuous matrix product state
+The derivation of the correlation bounds employ the
+continuous matrix product state [56, 57], which bridges
+the quantum field and the stochastic process. The con-
+tinuous matrix product state is a type of tensor net-
+work representation that is used to describe many-body
+quantum systems. In one direction, quantum field states
+are analyzed via the corresponding continuous measure-
+ment problem. In the opposite direction, the continuous
+matrix product state can map a classical or quantum
+Markov process into a quantum field so that we can an-
+alyze trajectory information from the view point of the
+quantum field.
+We consider a Lindblad equation [Eq. (10)]. The clas-
+sical Markov process given by Eq. (1) can be covered by
+the Lindblad equation by setting H = 0 and the jump
+operator to be of the form Lm = Lνµ =
+�
+Wνµ |Bν⟩ ⟨Bµ|,
+where {|Bν⟩}ν constitutes the orthonormal basis, corre-
+sponding to the classical states B = {Bν}ν, and Wνµ is
+the transition rate from |Bµ⟩ to |Bµ⟩.
+Applying the continuous measurement on the Lindblad
+equation [Eq. (10)], we obtain a trajectory Γ, which is a
+record of the measurement, as follows:
+Γ ≡ [(t1, m1), (t2, m2), . . . , (tK, mK)],
+(A1)
+where K is the number of total jumps, tk and mk are
+time and type of the kth jump event, respectively. The
+evolution of ρ(t) in a given trajectory Γ is governed
+by a stochastic Schr¨odinger equation.
+By taking the
+average of all possible measurements in the stochastic
+Schr¨odinger equation, we can recover the original Lind-
+blad equation of Eq. (10).
+Applying continuous measurement, we obtain a par-
+ticular trajectory Γ. In the continuous matrix product
+state, such a trajectory is recorded in the following state:
+|Γ⟩ ≡ φ†
+mK(tK) · · · φ†
+m2(t2)φ†
+m1(t1) |vac⟩ ,
+(A2)
+where φ(t) is the field operator that satisfies the commu-
+tation relation [φm(t), φ†
+m′(t′)] = δmm′δ(t − t′), and |vac⟩
+is the vacuum state of φm(t), where φ†
+m(t) creates a mth
+particle at t. The time evolution of the system and field
+state |Γ⟩ is given by
+|Φ(t)⟩ = U(t; H, {Lm}) |Φ(0)⟩ ,
+(A3)
+where U(t; H, {Lm}) is given by
+U(t; H, {Lm}) ≡ T exp
+�
+−i
+� t
+0
+ds (Heff ⊗ IF
++
+�
+m
+iLm ⊗ φ†
+m(s))
+�
+,
+(A4)
+In Eq. (A4), the initial state is represented by |Φ(0)⟩ =
+|ψ(0)⟩ ⊗ |vac⟩, with |ψ(0)⟩ being the initial state of the
+system; T is the time ordering operator, and IF is the
+identity operator in the field. |Φ(t)⟩ records the jump
+events occurring within the interval [0, t]. The continuous
+matrix product state |Φ(t)⟩ comprises the system, which
+corresponds to the state of the Markov process, and the
+field, which records jump events. The time of the original
+Lindblad equation is expressed by t while that of the
+continuous matrix product state is by t. All information
+about measurement is recorded by creating particles in
+the quantum field through the application of an operator
+φ†
+m(t) to the vacuum state |vac⟩.
+For a small time increment ∆t, considering the time
+evolution Eq. (A3) and tracing over the field, the time
+evolution of the system is given by the Kraus represen-
+tation:
+ρ(t + ∆t) =
+�
+m
+Vmρ(t)V †
+m,
+(A5)
+where Vm are Kraus operators:
+V0 ≡ I − i∆tH,
+(A6)
+Vm ≡
+√
+∆tLm
+(1 ≤ m).
+(A7)
+Dividing the interval [0, t] into Z ≫ 1 equipartitioned
+intervals, the time evolution from t = 0 to t can be rep-
+resented by successive applications of Eq. (A5):
+ρ(t) =
+�
+mZ
+· · ·
+�
+m1
+VmZ · · · Vm1 |ψ(0)⟩ ⟨ψ(0)| V †
+m1 · · · V †
+mZ.
+(A8)
+Using the continuous matrix product state, we can obtain
+all the information about the Markov processes. Given
+the initial state |ψ(0)⟩, the trajectory probability within
+[0, t] can be obtained via
+P(Γ, t) = ⟨Φ(t)|IS ⊗ |Γ⟩ ⟨Γ| |Φ(t)⟩ .
+(A9)
+
+6
+The system state ρ(t) can be computed as follows:
+ρ(t) = TrF [|Φ(t)⟩ ⟨Φ(t)|] ,
+(A10)
+where TrF denotes the trace operation with respect to
+the field state.
+Next, we explain a scaled continuous matrix product
+state, which was recently introduced in Ref. [64].
+We
+want to study the time evolution of the continuous ma-
+trix product state.
+Initially, we might consider using
+the unitary operator defined in Eq. (A4) as the time-
+evolution operator. However, this approach has a prob-
+lem when we try to calculate the fidelity between two
+continuous matrix product states at different times, be-
+cause the integration ranges for |Φ(t1)⟩ and |Φ(t2)⟩ are
+different. Therefore, we instead use the scaled represen-
+tation. Let us define τ > 0, which is the final time of
+the evolution. For 0 ≤ t ≤ τ, the scaled matrix product
+state representation is given by
+|Ψ(t)⟩ = U
+�
+τ; t
+τ H,
+��
+t
+τ Lm
+��
+|Ψ(0)⟩ ,
+(A11)
+where |Ψ(0)⟩ = |ψ(0)⟩ ⊗ |vac⟩. Here, |Φ(t)⟩ and |Ψ(t)⟩
+represent the states of the genuine and scaled continuous
+matrix product states, respectively. In the scaled con-
+tinuous matrix product state [Eq. (A11)], H and Lm are
+scaled as (t/τ)H and
+�
+t/τLm, respectively, which corre-
+sponds to the Lindblad equation that generates dynamics
+that are t/τ times as fast as that of the original dynam-
+ics. The scaling allows us to have the same integration
+range for all values of t, making it possible to evaluate
+the fidelity at different times, that is ⟨Ψ(t2)|Ψ(t1)⟩. As
+mentioned above, since the scaled matrix product state
+is the same as the original one except for their time scale,
+both states provide us with the same information except
+for the time scale. At the final time τ, both the origi-
+nal and the scaled representations give the same state,
+|Φ(τ)⟩ = |Φ(τ)⟩.
+Moreover, |Ψ(0)⟩ corresponds to the
+null dynamics, that is, the dynamics without any state
+change. For instance, the system state can be obtained
+by
+ρ(t) = TrF [|Ψ(t)⟩ ⟨Ψ(t)|] = TrF [|Φ(t)⟩ ⟨Φ(t)|] .
+(A12)
+When deriving the correlation bounds, we employ the
+scaled representation.
+Appendix B: Initially mixed state case
+The continuous matrix product state given by Eq. (A3)
+only considers initially pure state |ψ(0)⟩. Let us consider
+the initially mixed state case. Let ρ(0) be the initial den-
+sity operator, which is mixed in general. Let us consider
+the ancilla A that purifies ρ(0), that is
+ρ(0) = TrA[| ˜ψ(0)⟩ ⟨ ˜ψ(0)|],
+(B1)
+where TrA is the trace operation with respect to the an-
+cilla A. Let us introduce the continuous matrix product
+state operator corresponding to Eq. (A4), that is applied
+to the purified state:
+˜U(t; H, {Lm}) ≡T exp
+�
+−i
+� t
+0
+ds(Heff ⊗ IA ⊗ IF
++
+�
+m
+iLm ⊗ IA ⊗ φ†
+m(s))
+�
+,
+(B2)
+The Kraus operators corresponding to Eq. (B2) is given
+by
+˜Vm = Vm ⊗ IA,
+(B3)
+where IA is the identity operation in the ancilla and Vm
+are defined in Eqs. (A6) and (A7). Using Eq. (B3), it
+can be confirmed that the one-step evolution yields
+TrA
+��
+m
+˜Vm |˜Ψ(0)⟩ ⟨˜Ψ(0)| ˜V †
+m
+�
+=
+�
+m
+VmTrA
+�
+|˜Ψ(0)⟩ ⟨˜Ψ(0)|
+�
+V †
+m
+=
+�
+m
+Vmρ(0)V †
+m,
+(B4)
+which actually yields the consistent time evolution.
+Appendix C: Fidelity calculation of continuous
+matrix product states
+The bounds considered in this Letter relate to the cal-
+culation of the quantum Fisher information. Specifically,
+we need to calculate the following fidelity:
+⟨Ψ(t2)|Ψ(t1)⟩ = TrSF [|Ψ(t1)⟩ ⟨Ψ(t2)|]
+= TrS [ζ(τ; t1, t2)] ,
+(C1)
+where ζ(τ; t1, t2) ≡ TrF [|Ψ(t1)⟩ ⟨Ψ(t2)|]. ζ(τ; t1, t2) sat-
+isfies the two-sided Lindblad equation [72, 73]:
+d
+dtζ(t; t1, t2) = −iH1ζ + iζH2 +
+�
+m
+L1,mζL†
+2,m
+− 1
+2
+�
+m
+�
+L†
+1,mL1,mζ + ζL†
+2,mL2,m
+�
+,
+(C2)
+where H1 ≡ (t1/τ)H and L1,m ≡
+�
+t1/τLm (H2 and
+L2,m are defined in a similar manner).
+Note that
+ζ(τ; t1, t2) is not a density operator, since its trace is not
+necessarily equal to unity. To calculate the fidelity using
+Eq. (C2), we solve Eq. (C2) from t = 0 to t = τ with the
+initial value ζ(0; t1, t2) = ρ(0).
+Using Eq. (C2), we can compute the fidelity between
+two scaled continuous matrix product states:
+η(τ) ≡ |⟨Ψ(τ)|Ψ(0)⟩|2
+(C3)
+
+7
+From Eq. (C2), |ζ(t; τ, 0)|2 = η(τ) obeys the following
+equation [66]:
+˙ζ = −iHeffζ = −iHζ − 1
+2
+�
+m
+L†
+mLmζ.
+(C4)
+Then the fidelity is obtained as follows:
+η(τ) =
+��TrS
+�
+e−iHeffτρ(0)
+���2 .
+(C5)
+The classical case can be calculated by setting H = 0
+[66].
+Appendix D: Derivation of Eq. (5)
+Here we provide the derivation of Eq. (5).
+Using
+the scaled continuous matrix product state, a classical
+Markov process can be analyzed via quantum mechan-
+ics, and thus we can take advantage of inequalities in
+quantum information. Let O be an arbitrary Hermitian
+operator, and ⟨O⟩t ≡ Tr[ρ(t)O].
+In the field of quan-
+tum speed limit, the following relation was recently used
+[25, 26]:
+��⟨O⟩t2 − ⟨O⟩t1
+�� = Tr [O(ρ(t2) − ρ(t1))]
+≤ ∥O∥op ∥ρ(t2) − ρ(t1)∥tr
+= 2 ∥O∥op TD (ρ(t2), ρ(t1)) .
+(D1)
+The second line of Eq. (D1) is due to the H¨older inequal-
+ity (see Eq. (H6)). We will use Eq. (D1) for the deriva-
+tion. The sketch of the proof for Eq. (5) is as follows:
+• Consider the scaled continuous matrix product
+state for ρ(t)
+• Assign the Hermitian operator that calculates the
+correlation function for O
+• Obtain an upper bound for the trace distance
+TD (ρ(t2), ρ(t1)) using the dynamical activity
+When considering classical probability and quantum
+spaces in Eq. (D1), Eq. (D1) leads to the classical and
+quantum bounds, respectively.
+We consider Eq. (D1) for the classical probability
+space. Let us assume that two density operators ρ and σ
+only have diagonal elements:
+ρ =
+�
+x
+p(x) |x⟩ ⟨x| ,
+(D2)
+σ =
+�
+x
+q(x) |x⟩ ⟨x| ,
+(D3)
+where p(x) and q(x) are arbitrary probability distribu-
+tions and {|x⟩}x constitutes the orthonormal basis. By
+calculating the trace distance [Eq. (H7)] for Eqs. (D2)
+and (D3), TD(ρ, σ) reduces to the total variation dis-
+tance [Eq. (H12)]:
+TD(ρ, σ) = TVD(p, q).
+(D4)
+Now we consider a particular probability distribution.
+The probability of measuring a trajectory Γ and Bν at
+the end time is
+P(Γ, ν, t) ≡ ⟨Ψ(t)|(|Bν⟩ ⟨Bν| ⊗ |Γ⟩ ⟨Γ|)|Ψ(t)⟩ ,
+(D5)
+where |Ψ(t)⟩ is the scaled continuous matrix product
+state. When considering initially mixed state, we may
+use |˜Ψ(t)⟩.
+Because arccos Bhat(·, ·) constitutes the
+geodesic distance under the Fisher information metric
+[74], the following relation holds [64]:
+1
+2
+� t2
+t1
+�
+A(t)
+t
+dt ≥ arccos [Bhat (P(Γ, ν, t1), P(Γ, ν, t2))] ,
+(D6)
+which yields
+cos
+�
+1
+2
+� t2
+t1
+�
+A(t)
+t
+dt
+�
+≤ Bhat (P(Γ, ν, t1), P(Γ, ν, t2)) ,
+(D7)
+for 0 ≤ 1
+2
+� t2
+t1
+√
+A(t)
+t
+dt ≤ π
+2 . Substituting Eq. (D7) into
+Eq. (H17) to obtain
+TVD(P(Γ, ν, t1), P(Γ, ν, t2))
+≤
+�
+1 − Bhat (P(Γ, ν, t1), P(Γ, ν, t2))2
+≤
+�
+�
+�
+�1 − cos
+�
+1
+2
+� t2
+t1
+�
+A(t)
+t
+dt
+�2
+= sin
+�
+1
+2
+� t2
+t1
+�
+A(t)
+t
+dt
+�
+.
+(D8)
+From Eqs. (D1), (D4), and (D8), we obtain
+��⟨O⟩t2 − ⟨O⟩t1
+��
+≤ 2 ∥O∥op sin
+�
+1
+2
+� t2
+t1
+�
+A(t)
+t
+dt
+�
+,
+(D9)
+which holds for 0 ≤ 1
+2
+� t2
+t1
+√
+A(t)
+t
+dt ≤ π
+2 . Equation (D9)
+is the central inequality for deriving the thermodynamic
+correlation inequality.
+We now implement the correlation calculation C(τ) =
+⟨S(0)S(τ)⟩ with an Hermitian operator acting on the
+scaled continuous matrix product state. Given a trajec-
+tory Γ and the final state Bν, we can calculate the cor-
+relation S(0)S(τ) using |Ψ(τ)⟩. We assume that a real
+function M(Γ, ν) calculates the correlation given such in-
+formation:
+M(Γ, ν) ≡ S(X(0))S(X(τ)) = S(0)S(τ).
+(D10)
+Now we introduce an Hermitian operator M, whose
+eigendecomposition reads
+M =
+�
+Γ,ν
+M(Γ, ν) |Γ, ν⟩ ⟨Γ, ν| .
+(D11)
+
+8
+Since Eq. (D11) is the eigendecomposition of M, from
+Eq. (H2), the operator norm of M is
+∥M∥op = max
+Γ,ν M(Γ, ν)
+=
+max
+Bi,Bj∈B [S(X(0) = Bi)S(X(τ) = Bj)]
+= S2
+max,
+(D12)
+where Smax is the maximum absolute value of S(Bi) for
+Bi ∈ B defined in Eq. (2).
+When we evaluate M in
+|Ψ(τ)⟩, it gives
+⟨Ψ(τ)|M|Ψ(τ)⟩ = ⟨Ψ(τ)|
+�
+Γ,ν
+M(Γ, ν) |Γ, ν⟩ ⟨Γ, ν| |Ψ(τ)⟩
+=
+�
+Γ,ν
+M(Γ, ν)P(Γ, ν, τ)
+= ⟨S(0)S(τ)⟩ .
+(D13)
+Because |Ψ(0)⟩ corresponds to the null dynamics (the
+state does not evolve at all), ⟨Ψ(0)|M|Ψ(0)⟩ = ⟨S(0)2⟩.
+In a similar way, when we consider |Ψ(t)⟩ for 0 < t <
+τ, we have ⟨Ψ(t)|M|Ψ(t)⟩ = ⟨S(0)S(t)⟩.
+Substituting
+Eqs. (D12) and (D13) into Eq. (D9), we finally obtain
+Eq. (5) in the main text.
+Appendix E: Derivation of Eqs. (8) and (12)
+In this section, we derive Eqs. (8) and (12). We evalu-
+ate TD(ρ(τ), ρ(0)) in Eq. (D1). Since continuous matrix
+product states are pure, we have [Eq. (H10)]
+TD(|Ψ(t1)⟩ , |Ψ(t2)⟩) =
+�
+1 − | ⟨Ψ(t2)|Ψ(t1)⟩ |2.
+(E1)
+As explained in Appendix C, the fidelity can be com-
+puted, which leads to Eq. (8) in the main text.
+The quantum case can be derived in a similar manner.
+As explained in Eqs. (D10) and (D11), the correlation
+function can be computed given a trajectory Γ for the
+quantum case as well.
+Then, the quantum version of
+Eq. (8) is obtained in the same way as the classical bound.
+We next derive Eq. (12). Since the Bures angle con-
+stitutes the geodesic length under the quantum Fisher
+information metric [6, 75], similar to Eq. (D6), the fol-
+lowing inequality holds [64]:
+arccos |⟨Ψ(t2)|Ψ(t1)⟩| ≤ 1
+2
+� t2
+t1
+�
+B(t)
+t
+dt,
+(E2)
+where B(t) is the quantum dynamical activity [64] (Ap-
+pendix F). For 0 ≤ 1
+2
+� t2
+t1
+√
+B(t)
+t
+dt ≤ π
+2 , we have
+cos
+�
+1
+2
+� t2
+t1
+�
+B(t)
+t
+dt
+�
+≤ |⟨Ψ(t2)|Ψ(t1)⟩| .
+(E3)
+Substituting Eq. (E3) into Eq. (E1), we obtain
+TD (|Ψ(t1)⟩ , |Ψ(t2)⟩) ≤ sin
+�
+1
+2
+� t2
+t1
+�
+B(t)
+t
+dt
+�
+.
+(E4)
+From Eqs. (D1) and (E4), we obtain Eq. (12) in the main
+text.
+Appendix F: Quantum dynamical activity
+The quantum dynamical activity B(t) is defined
+through the quantum Fisher information [64]. The quan-
+tum Fisher information for the scaled continuous matrix
+product state is calculated as follows:
+J (t) =
+8
+∆t2 [1 − | ⟨Ψ(t) | Ψ(t + ∆t)⟩ |],
+(F1)
+where ∆t is a sufficiently small increment. The fidelity
+| ⟨Ψ(t) | Ψ(t + ∆t)⟩ | can be computed by the two-sided
+Lindblad equation [Eq. (C2)]. The quantum dynamical
+activity is defined by [64]
+B(t) ≡ J (t)
+t2 .
+(F2)
+Appendix G: Linear response
+Here, we show detailed calculations of the linear re-
+sponse theory. Let us consider applying a weak pertur-
+bation χFf(t) to the master equation (1). Considering
+the perturbation expansion with respect to χ, upto the
+first order, the probability distribution is expanded as
+P(t) = Pst + χP1(t),
+(G1)
+where P1(t) is the first-order term. Substituting Eq. (G1)
+into Eq. (1), we have
+d
+dt (Pst + χP1(t)) = (W + χFf(t)) (Pst + χP1(t)) ,
+(G2)
+in which collecting the terms with respect to the order of
+χ yields
+O(χ0)
+d
+dtPst = WPst,
+(G3)
+O(χ1)
+d
+dtP1(t) = WP1(t) + FPstf(t).
+(G4)
+The zeroth order equation vanishes in definition since Pst
+is assumed to be the stationary solution of Eq. (1). P1(t)
+is given by Eq. (15) in the main text. In the main text,
+we consider a scoring function G(Bn) and its expecta-
+tion given by Eq. (16). The change of ⟨G⟩ due to the
+perturbation can be represented by
+∆G(t) ≡ χ1GP1(t)
+= χ
+� t
+−∞
+1GeW(t−t′)FPstf(t′)dt′
+= χ
+� ∞
+−∞
+RG(t − t′)f(t′)dt′,
+(G5)
+
+9
+where RG(t) is the linear response function [Eq. (18)].
+From Eq. (3), the time derivative of C(t) reads
+d
+dtC(t) = 1SeWtWSPst.
+(G6)
+In the main text, we consider the case G = S and
+F = WS, which will be assumed in the following. The
+perturbation WS can be expressed by
+WS =
+�
+����
+S11W11
+S22W12
+· · ·
+SNNW1N
+S11W21
+S22W22
+SNNW2N
+...
+...
+...
+S11WN1 S22WN2 · · · SNNWNN
+�
+���� .
+(G7)
+We immediately obtain
+RS(t) = d
+dtC(t).
+(G8)
+For the pulse perturbation, f(t) = δ(t), where δ(t) is
+the Dirac delta function, we obtain
+∆S(p)(t) = χ
+� ∞
+−∞
+RS(t − t′)δ(t′)dt′
+= χRS(t)
+= χ d
+dtC(t).
+(G9)
+Using Eq. (7), we obtain Eq. (19).
+Next, we consider the step perturbation, i.e., f(t) =
+Θ(t), where Θ(t) is the Heaviside step function:
+Θ(t) =
+�
+0
+(t < 0)
+1
+(t ≥ 0) .
+(G10)
+Then we have
+∆S(p)(t) = χ
+� ∞
+−∞
+RS(t − t′)Θ(t′)dt′
+= χ
+� t
+0
+RS(t − t′)dt′
+= χ
+� t
+0
+RS(t′)dt′
+= χ
+� t
+0
+dC(t′)
+dt′
+dt′
+= χ (C(t) − C(0)) ,
+(G11)
+which yields Eq. (21) in the main text.
+Appendix H: Norm and distance measures
+For readers’ convenience, we here review the norm and
+distance measures for quantum and classical systems. Let
+A and B be arbitrary Hermitian operators. The Shattan
+p-norm is defined by
+∥A∥p ≡
+�
+Tr
+��√
+A2
+�p�� 1
+p =
+�
+�
+�
+λ∈evals(A)
+|λ|p
+�
+�
+1
+p
+.
+(H1)
+For particular p, we have
+∥A∥op = ∥A∥∞ =
+max
+λ∈evals(A) |λ|,
+(H2)
+∥A∥tr = ∥A∥1 = Tr
+�√
+A2
+�
+,
+(H3)
+∥A∥hs = ∥A∥2 =
+�
+Tr [A2],
+(H4)
+where evals(A) gives a set of eigenvalues of A.
+Equa-
+tions (H2), (H3), and (H4) are referred to as the opera-
+tor norm, the trace norm, and the Hilbert-Schmidt norm,
+respectively. The H¨older inequality states
+|Tr [AB]| ≤ ∥A∥p ∥B∥q .
+(H5)
+where p and q should satisfy 1/p+1/q = 1. When p = q =
+2, Eq. (H5) reduces to the Cauchy-Schwarz inequality. In
+particular, we use p = ∞ and q = 1 case:
+|Tr [AB]| ≤ ∥A∥op ∥B∥tr .
+(H6)
+Let us define the trace distance and quantum fidelity:
+TD(ρ, σ) ≡ 1
+2 ∥ρ − σ∥1 ,
+(H7)
+Fid(ρ, σ) ≡
+�
+Tr
+�√ρσ√ρ
+�2
+.
+(H8)
+When considering pure states |ψ⟩ and |φ⟩, the fidelity
+reduces to the inner product:
+Fid (|ψ⟩ , |φ⟩) = |⟨ψ|φ⟩|2 ,
+(H9)
+TD(|ψ⟩ , |φ⟩) =
+�
+1 − | ⟨ψ|φ⟩ |2
+(H10)
+These two distances are related via
+TD(ρ, σ) ≤
+�
+1 − Fid(ρ, σ).
+(H11)
+The equality of Eq. (H11) holds when both ρ and σ are
+pure [58].
+Let us introduce related classical probability distance
+measures. Let p(x) and q(x) be probability distributions.
+The total variation distance and the Hellinger distance
+are given, respectively, by
+TVD(p, q) ≡ 1
+2
+�
+x
+|p(x) − q(x)|,
+(H12)
+Hel2(p, q) ≡ 1
+2
+�
+x
+��
+p(x) −
+�
+q(x)
+�2
+(H13)
+= 1 − Bhat(p, q)
+(H14)
+
+10
+where Bhat(p, q) is the Bhattacharyya coefficient:
+Bhat (p, q) ≡
+�
+x
+�
+p(x)q(x).
+(H15)
+Between the total variantion and the Hellinger distances,
+the following relations are known to hold [76]:
+Hel2(p, q) ≤ TVD(p, q)
+(H16)
+≤
+�
+Hel2(p, q)(2 − Hel2(p, q))
+(H17)
+≤
+�
+2Hel2(p, q).
+(H18)
+ACKNOWLEDGMENTS
+This work was supported by JSPS KAKENHI Grant
+Number JP22H03659.
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+ 
+ 
+TopoBERT: Plug and Play Toponym Recognition Module 
+Harnessing Fine-tuned BERT∗ 
+Bing Zhou 
+ Department of Geography 
+ Texas A&M University 
+ College Station, Texas, US 
+ [email protected] 
+Yingjie Hu  
+Department of Geography 
+ University at Buffalo 
+ Buffalo, New York, US 
+ [email protected] 
+Lei Zou  
+Department of Geography 
+ Texas A&M University 
+ College Station, Texas, US 
+ [email protected] 
+Yi Qiang  
+School of Geoscience 
+ University of South Florida 
+Tampa, Florida, US 
+[email protected] 
+ABSTRACT 
+Extracting precise geographical information from textual contents 
+is crucial in a plethora of applications. For example, during 
+hazardous events, a robust and unbiased toponym extraction 
+framework can provide an avenue to tie the location concerned to 
+the topic discussed by news media posts and pinpoint humanitarian 
+help requests or damage reports from social media. Early studies 
+have leveraged rule-based, gazetteer-based, deep learning, and 
+hybrid approaches to address this problem. However, the 
+performance of existing tools is deficient in supporting operations 
+like emergency rescue, which relies on fine-grained, accurate 
+geographic information. The emerging pretrained language models 
+can better capture the underlying characteristics of text information, 
+including place names, offering a promising pathway to optimize 
+toponym recognition to underpin practical applications. In this 
+paper, TopoBERT, a toponym recognition module based on a one-
+dimensional Convolutional Neural Network (CNN1D) and 
+Bidirectional Encoder Representation from Transformers (BERT), 
+is proposed and fine-tuned. Three datasets (CoNLL2003-Train, 
+Wikipedia3000, WNUT2017) are leveraged to tune the 
+hyperparameters, discover the best training strategy, and train the 
+model. Another two datasets (CoNLL2003-Test and Harvey2017) 
+are used to evaluate the performance. Three distinguished 
+classifiers, linear, multi-layer perceptron, and CNN1D, are 
+benchmarked to determine the optimal model architecture. 
+TopoBERT achieves state-of-the-art performance (f1-score=0.865) 
+compared to the other five baseline models and can be applied to 
+diverse toponym recognition tasks without additional training. 
+KEYWORDS 
+Natural Language Processing; Geoparser; Convolutional Neural 
+Network; Toponym Recognition; BERT 
+ 
+1 Introduction 
+Since the emergence of social sensing, scholars have been 
+endeavoring to sense the pulse of society with the help of satellite 
+images, sensor networks from IoT and various forms of textual 
+information from the Internet. Extra attention has been paid to 
+mining knowledge from social media because people nowadays are 
+consciously or unconsciously sharing their views towards ongoing 
+events online, which propels social media to become one of the few 
+agents that reflects the real-time societal awareness, reactions and 
+impacts of particular events. This trait is a rare feature seldom 
+shared by other forms of data sources.  
+In the light of this feature, Avvenuti et al. presented an early 
+earthquake detecting and warning system using Twitter data, which 
+offers prompt detection of events [1]. Several case studies 
+processed social media data with geocoding and sentiment analysis 
+tools to analyze the spatial patterns of changing public awareness 
+and emotions toward hurricanes in different phases of the disaster 
+management cycle [2,3]. Huang et al. scrutinized the human 
+mobility patterns during the COVID-19 pandemic at multiple 
+scales based on geotagged Twitter data [4]. Zhou et al. proposed 
+VictimFinder which is capable of harvesting social media help 
+requests during hurricanes [5].  
+Let alone the fact that geographical information being one of the 
+key elements of knowledge generation, the aforementioned studies 
+and other similar spatial analysis and modeling are highly 
+dependent on the location information of the social media data. 
+However, social media users start to pay more attention to user 
+privacy, which results in a significant drop of the number of 
+geotagged tweets. Simultaneously, Twitter published policies 
+forbidding users to attach precise longitudes and latitudes to tweets. 
+Moreover, the geographical information bound up with the social 
+media posts might not necessarily be equivalent to the place names 
+described in the textual content of the post. Thus, extracting 
+location information from the textual content of social media data 
+has inevitably become an issue that needs to be addressed. This 
+breeds the process of geoparsing, a two-step approach which 
+includes toponym recognition (identifying place names from texts) 
+and toponym resolution (transforming location names to 
+geographical coordinates). This paper focuses on the first 
+component of geoparsing.  
+
+ 
+ 
+ 
+ 
+Existing studies on toponym recognition can be categorized into 
+four parties based on the character of the solutions, namely rule-
+based, gazetteer-based, statistical learning-based, and hybrid 
+approaches. In general, statistical learning and hybrid methods that 
+incorporate deep learning techniques render better performance 
+than methods that solely rely on rules or gazetteers [6,7,8,9]. Based 
+on Bidirectional Long Short-Term Memory (BiLSTM), Wang et al. 
+introduced NeuroTPR to extract place names [6]. Qi et al. extended 
+CoreNLP and brought about an open-sourced named entity 
+recognition python toolkit called Stanza, which is able to detect 
+place names and support multiple languages [7]. SAVITR is a 
+system that combines both NLP techniques and gazetteers for real-
+time location extraction [8]. Hu et al. addressed the incompleteness 
+of gazetteers and fused gazetteers, rules, and deep learning to 
+render a reliable place name extractor, GazPNE [9].  
+However, those studies suffer from several limitations. First, some 
+models do not focus only on place names, so their prediction of 
+location name extraction might be disturbed. Second, recurrent 
+neural network based deep learning models might suffer from 
+information vanishing problems when the input sequence gets 
+larger and network deeper. Third, complicated deep neural 
+networks frequently require large, annotated datasets and are time-
+consuming to train to achieve promising results.  
+To address the aforementioned latent flaws, this paper proposes 
+TopoBERT, a toponym recognition module based on a one-
+dimensional 
+Convolutional 
+Neural 
+Network 
+(CNN) 
+and 
+Bidirectional Encoder Representation from Transformers (BERT).  
+It contributes in the following directions. First, several classifiers 
+were tested and one feasible model and classifier combination 
+based on the evaluation result of a standard dataset is determined. 
+Second, TopoBERT was tested by an unseen dataset together with 
+some other existing tools to verify its generalizability. Third, the 
+tool is ready-to-use and the dataset we generated in this study can 
+be used by other scholars to train, test, and compare different 
+toponym recognition models and tools. 
+The remainder of this paper is structured as follows. The datasets 
+involved in fine-tuning and testing the framework, a concise 
+introduction of the holistic design of the framework, the 
+implementation of the framework, and the parameters used in fine-
+tuning the framework are detailed in section 2. The results of the 
+experiments conducted are documented in section 3. Section 4 
+illustrates the potential limitations of this work and lists several 
+future research directions. Section 5 epitomizes the findings of this 
+paper and presents the implications of this study.  
+2 Methodology  
+2.1 Datasets 
+Totally four different datasets were utilized to train the module and 
+evaluate the performance. CoNLL2003 is a shared task that 
+concerns named entity recognition, which has been widely applied 
+to training deep learning models [10]. The data contains entities of 
+five types: persons (PER), organizations (ORG), locations (LOC) 
+and miscellaneous names (MISC) and other words that are 
+irrelevant to named entities of the aforementioned four groups (O). 
+The prefix “B-” and “I-” are used to tag the beginning of a named 
+entity and words that fall inside a named entity [10]. The dataset is 
+originally divided into training, validation, and test data which are 
+noted 
+as 
+CoNLL2003-Train, 
+CoNLL2003-Validation 
+and 
+CoNLL2003-Test. Training data is used to train a deep learning 
+model, validation data is used to tune the hyperparameters of the 
+model, and the test data is used to evaluate the performance of the 
+trained model. The data distribution of each label type in the three 
+datasets is depicted in Figures 1(a), 1(b), and 1(c), respectively. The 
+dataset is later modified to suit the purpose of this study by labeling 
+all the named entities as “O” except for the location entities.  
+Around 4.1% of the tags are location entities in these datasets. 
+  
+ 
+  
+(a)                              (b)                           (c) 
+Figure 1: Data Distribution of CoNLL2003 Dataset 
+WNUT2017 is a relatively smaller dataset collected from Twitter 
+and manually annotated, the objective of which is to tackle the 
+issues caused by novel, emerging, singleton named entities in noisy 
+text [11]. It aims to offer support to sustainable named entity 
+recognition systems. This dataset contains seven different groups: 
+person, location, corporation, product, creative work, group and 
+none of the above. Considering the main focus of this paper and 
+different tags used to label the dataset, this dataset is preprocessed 
+to retain only the location entities tag and to unify the tag symbols 
+used based on CoNLL2003 (location entities are tagged with “B-
+LOC” or “I-LOC” while the rest are tagged with “O”). The 
+distribution of data under each label type in the modified dataset is 
+shown in Figure 2(a). The total number of location names in this 
+dataset is 1140. 
+  
+  
+  
+ 
+(a)          
+         (b)                             (c) 
+Figure 2: Data Distribution of WNUT2017, Wiki300 and 
+Harvey2017 Dataset 
+Wiki3000 is an automatically generated dataset from Wikipedia 
+articles by a data producing workflow proposed by Wang et al. [6]. 
+The proposed auto-annotation approach utilizes the first paragraph 
+of Wikipedia articles which usually encompass various entities 
+presented with hyperlinks. These hyperlinks are later checked if 
+they are associated with a geographical location. If so, the 
+
+CoNLL2003-TrainDataset
+200000
+169578
+150000
+Count
+100000
+50000
+82971002511128
+4593
+0
+LOCORGPER
+RMISO
+ClassNamesCoNLL2003-ValidationDataset
+50000
+42759
+40000
+Count
+30000
+20000
+10000
+3149
+20942092
+1268
+0
+LOCORG
+PER
+MISC
+ClassNamesCoNLL2003-TestDataset
+4000038323
+30000
+Count
+20000
+10000
+192524962773
+918
+0
+LOCORG
+PER
+MISC
+ClassNamesWNUT2017Dataset
+10673
+10000
+Count
+5000
+1140
+0
+0
+LOC
+ClassNamesWiki3000Dataset
+40466
+40000
+30000
+Count
+20000
+16000
+10000
+0
+0
+LOC
+ClassNamesHarvey2017Dataset
+15295
+15000
+Count
+10000
+5000
+3973
+0
+0
+LOC
+ClassNames 
+ 
+ 
+hyperlinked word will be labeled as a toponym. Then the Wikipedia 
+article is divided into multiple short sentences within 280 
+characters with additional strategies such as random flipping to 
+mimic the general patterns of Twitter posts [6]. The distribution of 
+data under each label type is shown in Figure 2(b). 
+Harvey2017 is a dataset originally collected from the North Texas 
+University repository (https://digital.library.unt.edu/ark:/67531 
+/metadc993940/), which contains 7,041,866 tweets collected based 
+on hashtag query. It was pruned, randomly subsampled and 
+manually annotated by Wang et al. to form a new dataset with 1000 
+tweets aiming to evaluate NeuroTPR [6]. This dataset is adopted by 
+this paper to test the performance of TopoBERT. The distribution 
+of data under each label type is shown in Figure 2(c). 
+2.2 Framework Design and Implementation 
+As mentioned in section 1, there is an acute conflict between robust 
+spatial analysis on social media or news media and the diminishing 
+availability of geolocated textual context. Additionally, the location 
+mentioned in the textual content of the tweets might differ from the 
+geotags attached. A reliable and ready-to-use geoparser can be the 
+mediator of such conflicts. Therefore, we present a general location 
+extractor that can be used upon social media and news media. The 
+workflow is shown in Figure 3.  
+The existing geotags of the data will be retained, and the textual 
+contents will go through a rule-based data preprocessing module 
+before they are fed to a zip code extractor and place name extractor. 
+Once the place names are pulled out, a geocoding service will be 
+applied to transform the place names into precise coordinates. The 
+place name extractor is marked with an orange dashed rectangle in 
+Figure 3 and serves as the crucial backbone of the entire workflow.  
+ 
+  
+Figure 3: Holistic Design of Location Extraction Framework 
+for Textual Contents 
+ 
+Figure 4: Demonstration of token classification workflow. 
+Identifying location names from input sentences is a token 
+classification task (Figure 4), which contains two parts. A language 
+model and a classifier. It behaves similar to how human beings 
+analyze whether the given words are place names or not. First the 
+language model attempts to understand the language by 
+transforming the tokenized input data into higher dimensional 
+space which captures the meaning of words in a given sentence, 
+then the classifier makes predictions based on the transformed 
+vectors and determines whether the input word belongs to location 
+entity. 
+The heart of the proposed toponym recognition module, 
+TopoBERT, is the Bidirectional Encoder Representation from 
+Transformers (BERT). It is structured by stacking the encoder 
+components of the Transformer architecture and is designed to be 
+pretrained in an unsupervised manner. BERT takes advantage of 
+the Attention [25] mechanism, which resolves the information 
+vanishing issue that often upsets recurrent neural networks such as 
+Long Short-Term Memory [26] and Gated Recurrent Neural 
+Network [27] when the input sequence gets longer. Moreover, 
+distinguished from many other bidirectional language models, such 
+as ELMo designed by Peters et al. [28], in which the contextual 
+representation of every word is the concatenation or summation of 
+the forward and backward representations, BERT reads the entire 
+sequence of words at once and is trained using a Masked Language 
+Model (MLM) approach and a Next Sentence Prediction (NSP) 
+approach which genuinely implemented the bidirectional concept 
+or unidirectional concept. These two features combined facilitate 
+better language understanding and bring the trophy to BERT 
+
+Geotag
+Geotagged
+Coordinates
+Bounding
+Box
+Center of Bounding
+Box
+Place
+Social Media
+Coordinatesfrom
+Name
+Data
+PlaceName
+No
+4
+ZipCode
+Rule-based
+Extractor
+Data
+Google
+ZipCodes
+NewsMedia
+Geocoding
+Preprocess
+Data
+Toponym
+IdentifiedLocation
+Recognition
+Names
+Coordinatesfrom
+GeocodingTokenClassification
+B-LOC
+I-LOC
+Outputpredictionforeach
+token
+Classifier
+Language Model
+［101,1030,17870,...102,...
+0,
+0,
+0]
+Tokenizer
+#HarveyRescueHoustonTX77074waitingforwater
+rescueintheattic.Pleasehelp!" 
+ 
+ 
+ 
+throughout a number of NLP tasks under the General Language 
+Understanding Evaluation (GLUE) benchmark [12]. 
+Off-the-shelf pretrained BERT model weights can be separated into 
+several categories based on the size of the model, whether upper 
+and lower cases are taken into consideration, the targeted language, 
+and 
+unique 
+training 
+strategies 
+(https://huggingface.co/transformers/v3.3.1/pretrained_models.ht
+ml). Since place names are highly case sensitive and only the 
+English language is involved in this study, ‘bert-base-cased’ and 
+‘bert-large-cased’ are selected as the candidate pretrained models 
+to be evaluated. The ‘bert-base-cased’ model comprises 12 layers, 
+and each hidden layer has 768 nodes, with 12 self-attention heads 
+and a total number of 110 million parameters. The ‘bert-large-cased’ 
+model consists of 24 layers, and each hidden layer has 1024 nodes, 
+with 16 self-attention heads and 340 million parameters. The 
+parameters are pretrained with English text from BooksCorpus 
+(800 million words) and English Wikipedia (2,500 million words). 
+By stacking a classifier on top of BERT, the combo can be fine-
+tuned to accomplish this downstream. Recent study showed that 
+model performance can be enhanced by applying classifiers more 
+complex than simple linear classifier or Conditional Random Field 
+(Zhou et al. 2022). Therefore, three classifiers were examined in 
+this study, namely linear classifier, multi-layer perceptron (MLP, 
+Figure 5) and one-dimensional CNN (CNN1D, Figure 6). The 
+simple linear classifier connects the output of the language model 
+to the final prediction results with the softmax activation function. 
+MLP applied in this study contains three fully connected layers and 
+links the language model output with a layer with the input size 
+equivalent to the output vector size. The number of hidden layer 
+nodes is 256 and the output layer size equals the number of distinct 
+labels from the training dataset. The CNN models are competent in 
+detecting underlying features [29] and one-dimensional CNN has 
+been successfully applied to process natural language [30, 31]. 
+Realizing 
+location 
+names 
+might 
+share 
+some 
+common 
+characteristics, the idea of CNN1D is adopted. The vector output of 
+the language model can be considered as a one-dimensional signal 
+and a CNN1D with kernel size 3 is applied. The output channel of 
+the convolution is 16. Followed by a max pooling layer of size 2, 
+which further generalizes the features and reduces model 
+complexity. All channels of the max pooling layer output are 
+concatenated into a single vector and is fed to a fully connected 
+MLP with hidden layer size equals to 128. 
+All model combinations were implemented using Python language 
+and pertinent packages. The dataset splitting took advantage of the 
+ScikitLearn library and the BERT models were implemented based 
+on 
+the 
+huggingface 
+Transformer 
+library 
+(https://huggingface.co/transformers/). The model finetuning 
+pipeline was built using PyTorch functions. 
+ 
+ 
+ 
+  
+Figure 
+5: 
+TopoBERT 
+Architecture 
+with 
+Multi-layer 
+Perceptron as Classifier 
+ 
+Figure 6: TopoBERT Architecture with One-Dimensional 
+Convolutional Neural Network as Classifier 
+2.3 Training and Evaluation 
+TopoBERT is envisioned to be a ready-to-use module that renders 
+optimal performance in toponym recognition. Models with 
+different architectures were trained and evaluated with six datasets 
+specified in Section 2.1 to determine the best model architecture 
+and training strategy. The training process utilized CoNLL2003-
+Train as the training dataset by default and compared to another 
+larger dataset fusing CoNLL2003, Wiki3000, and WNUT2017. 
+The original dataset is labelled at word-level which cannot be input 
+to BERT directly due to BERT’s word-piece encoding, otherwise 
+it will lead to large numbers of out of vocabulary words. To tackle 
+
+BERT
+E(cLs
+En
+ESEP
+StackedTransformerEncoders
+食
+价
+EICLS)
+E
+Em
+ER
+EISER
+[CLS]
+TOKEN 1
+TOKEN.m
+TOKENn
+[SEP]
+Word
+Embeddings
+Multi-layer
+Perceptron
+"TheEuropeanComissionsaidonTuesday...tohumanbeings."Max
+Pooling
+Word
+Embeddings
+Multi-layer
+Convolutional
+Concatenated
+Perceptron
+Layers
+Layers
+BERT
+ECLs
+Ei
+En
+ESEP)]
+StackedTransformerEncoders
+食
+E(cLs)
+E
+E
+En
+E(SEP)
+[CLS]
+TOKEN1
+TOKEN
+TOKENT
+[SEP]
+"The European Comission saidon Tuesday...tohuman beings." 
+ 
+ 
+with this issue, we first split the input data at word-level, and 
+applied BERT word-piece tokenizer to each word. The same label 
+was assigned to each word-piece of a single word. The labeled 
+word-pieces are then merged to form the new input data which 
+could be processed by BERT. This experiment aimed at measuring 
+the performance fluctuations caused by training data size and 
+heterogeneity. CoNLL2003-Validation was used during the 
+training process to tune several fundamental hyperparameters such 
+as training epochs and learning rate. CoNLL2003-Test and 
+Harvey2017 datasets were used to evaluate the model performance. 
+The Harvey2017 dataset was also used to benchmark TopoBERT 
+with five prevailing toponym recognition models, namely Stanford 
+NLP [32], spaCy (https://spacy.io/), Bidirectional LSTM-CRF [33], 
+DM_NLP [34], and NeuroTPR [6]. 
+The parameters of the classifier component of the module were 
+initialized with random non-zero numbers and the BERT 
+component was initialized with pre-trained parameters. The entire 
+module was trained with the fine-tuning approach [12], and the 
+parameters were updated using a mini-batch gradient descent 
+approach with early stopping. The maximum length of the input 
+sequence was limited to 128 in this paper. The maximum number 
+of training epochs was set to 50. As recommended by the original 
+BERT paper, the initial learning rate and the training batch size 
+were set to 2e-5 and 32 respectively [12]. Most commonly used loss 
+function for multi-class classification task, the cross-entropy loss 
+was employed. AdamW was selected as the optimizer during 
+training which adjusts the learning rate dynamically to accelerate 
+parameter convergence and implements weight decay to lower the 
+chance of overfitting. Warm up steps, which is using a very low 
+learning rate for the first several weight updating iterations, were 
+also introduced during training to reduce the impact of deviating 
+the model drastically from sudden exposure to unseen datasets. 
+Three commonly used evaluation metrics, precision, recall, and F1-
+score (Equation 1-3), were applied to gauge the performance and 
+bias of the models. Precision calculates the percentage of correctly 
+identified location names (noted as True Positives, TP) among all 
+the location names predicted by the model, which combines both 
+TP and False Positives (FP). Recall measures the percentage of 
+correctly identified ones amongst all ground truth, which is the 
+combination of TP and False Negatives (FN). F1-score is the 
+harmonic mean of precision and recall, providing a comprehensive 
+metric to evaluate model performance.    
+   𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 =  
+��
+�����                             (Equation 1) 
+  𝑅𝑒𝑐𝑎𝑙𝑙 = 
+��
+�����             
+             (Equation 2) 
+𝐹1–𝑠𝑐𝑜𝑟𝑒 = 2 ∗ 
+���������∗ ������
+���������� ������ 
+ (Equation 3) 
+ 
+The outputs of BERT models are at word-piece level and they are 
+concatenated using the special prefix ‘##’ and the word-level labels 
+are assigned base on the starting word-piece of the word. The 
+evaluation metrics are based on ‘per-token’ scores. Additionally, 
+location name entity consists of two types of labels (B-LOC and I-
+LOC). In order to gauge the comprehensive performance of the 
+model on toponym recognition, the evaluation metrics were 
+calculated using a micro average approach, which computes a 
+global average of precision, recall, and F1-score. It calculates the 
+TP, FP and FN by counting the total number of TP, FP and FN 
+under each class, namely, “B-LOC” and “I-LOC”. 
+ 
+3 Results and Analysis 
+The first step of the experiment targeted at determining the optimal 
+pretrained parameters for BERT model. We hypothesize that larger 
+models outperform smaller models. To verify this hypothesis, the 
+performance of the models initialized with ‘bert-base-cased’ and 
+‘bert-large-cased’ with a linear classifier stacked on top were tested. 
+The results are displayed in Table 1.  
+Table 1: Evaluation results for testing on different pretrained 
+parameters. 
+BERT Model 
+Classifie
+r 
+Precisio
+n 
+Recal
+l 
+F1-
+score 
+bert-base-cased 
+Linear 
+0.900  
+0.904  
+0.902  
+bert-large-
+cased 
+Linear 
+0.934  
+0.901  
+0.917  
+ 
+These two models were trained with CoNLL2003-Train and 
+evaluated with CoNLL2003-Test. Compared to ‘bert-base-cased’, 
+the precision of the prediction increased from 0.900 to 0.934 by 
+using ‘bert-large-cased’ while the recall almost remained static. 
+The F1-scores showed that ‘bert-large-cased’ rendered better 
+results which is in conformity with the original BERT paper [12] 
+and validated our initial hypothesis. Therefore, ‘bert-large-cased’ 
+was harnessed in all the follow-up experiments.  
+The second step of the experiments aimed to measure the influence 
+of the training data and determine the optimal classifier. The model 
+performances were evaluated using two different datasets, 
+CoNLL2003-Test and Harvey2017. We hypothesize that (a) the 
+model with CNN1D classifier yield better results and (b) models 
+trained with larger datasets perform better in placename recognition. 
+Table 2 and Table 3 list the evaluation metrics of all the tests.  
+Table 2: Evaluation results with CoNLL2003-Test dataset for 
+testing on training data variation and classifier types. 
+Training Data 
+Classifier 
+Precision 
+Recall 
+F1-score 
+CoNLL2003 
+Linear 
+0.934  
+0.901  
+0.917  
+CoNLL2003 
+MLP 
+0.904  
+0.910  
+0.907  
+CoNLL2003 
+CNN1D 
+0.923  
+0.920  
+0.921  
+Combined 
+Linear 
+0.889  
+0.844  
+0.866  
+Combined 
+MLP 
+0.941  
+0.884  
+0.912  
+Combined 
+CNN1D 
+0.942  
+0.916  
+0.929  
+Table 3: Evaluation results with Harvey2017 dataset for 
+testing on training data variation and classifier types. 
+
+ 
+ 
+ 
+ 
+Training Data 
+Classifier 
+Precision 
+Recall 
+F1-score 
+CoNLL2003 
+Linear 
+0.895 
+0.804 
+0.847 
+CoNLL2003 
+MLP 
+0.885 
+0.811 
+0.846 
+CoNLL2003 
+CNN1D 
+0.898 
+0.835 
+0.865 
+Combined 
+Linear 
+0.872 
+0.589 
+0.703 
+Combined 
+MLP 
+0.932 
+0.541 
+0.685 
+Combined 
+CNN1D 
+0.941 
+0.668 
+0.781 
+The “CoNLL2003” under the Training Data column means 
+CoNLL2003-Train dataset and the “Combined” represents the 
+dataset merging CoNLL2003-Test, Wiki3000 and WNUT2017.  
+In Table 2, when models were trained with CoNLL2003-Train, the 
+one with a simple linear classifier produced the best precision 
+(0.934), and the one with CNN1D produced the best recall (0.920) 
+and F1-score (0.921). MLP performed the worst among the three 
+classifiers. When models were trained with a combined dataset, the 
+model with CNN1D outperformed the rest in all three metrics with 
+precision equal to 0.942, recall of 0.916, and F1-score of 0.929. The 
+one with a linear classifier produced the worst results with an F1-
+score of 0.866. In Table 3, when models were trained with 
+CoNLL2003-Train, the one with the CNN1D classifier 
+outperformed the rest with precision equal to 0.898, recall of 0.835, 
+and F1-score of 0.865. When models were trained with a combined 
+dataset, the model with CNN1D successfully defended its trophy 
+by rendering precision of 0.941, recall of 0.668, and F1-score of 
+0.781. The models with MLP worked slightly worse than the ones 
+with linear classifiers. 
+ The above elucidation certifies the hypothesis that models with 
+CNN1D generate the optimal performance. It also shows that more 
+complicated classifiers like multi-layer perceptron do not 
+necessarily render better results.  
+However, when viewing Tables 2 and 3 contemporaneously, the 
+results from training with different datasets, the metrics indicated 
+that the model trained with the combined dataset generally 
+performed worse than the ones trained with merely CoNLL2003-
+Train. This phenomenon contradicts the hypothesis that models 
+trained with larger datasets perform better. After scrutinizing the 
+dataset used for training, we noticed some inconsistencies in the 
+labeling criteria of the datasets. Some examples are listed in Table 
+4 and the unexpected phenomenon can be interpreted by the 
+heterogeneity of the datasets. 
+Table 4: Examples of different labels across the datasets used 
+for training the model. 
+Example Entity 
+Dataset 
+CoNLL200
+3 
+Wiki300
+0 
+WNUT201
+7 
+"Canadian" 
+B-MISC 
+O 
+B-LOC 
+"Planet" 
+O 
+O 
+B-LOC 
+"east" 
+O 
+O 
+B-LOC 
+"orchard" 
+"academy" 
+B-ORG/ 
+I-ORG 
+O 
+B-LOC/ 
+I-LOC 
+"earth" 
+O 
+N/A 
+B-LOC 
+It can be seen from Table 4 that the word “Canadian,” which is 
+labeled as “B-MISC” (beginning of a miscellaneous name), is 
+identified as “B-LOC” (beginning of a location) in the WNUT2017 
+dataset. The words “Planet”, “east,” and “earth” are misclassified 
+as locations in the WNUT2017 dataset. The phrase “orchard 
+academy,” regarded as an organization under the CoNLL2003 
+criteria, is also labeled as a location entity. In this case, combining 
+several heterogeneous datasets can be considered adding some 
+helpful unseen samples to the original training data while 
+introducing a substantial amount of noise.  
+Rolnick et al. [13] experimented on several deep learning models 
+when trained with noisy data and claimed that the CNN model is 
+more resilient to noise than MLP and linear models. The trend of 
+performance change shown in Tables 2 and 3 when trained with 
+different datasets is in accordance with this statement. It is 
+noticeable that the models experience an increase in precision and 
+a drastic decrease in recall when trained with a combined dataset. 
+This incident can as well be triggered by noisy data. Since deep 
+learning models attempt to learn the underlying patterns of the 
+training data, the existing noise will confuse the model, resulting in 
+a fewer number of positive predictions. This might result in an 
+increase in precision and a decrease in recall.  
+Based on the observation and interpretation above, the BERT 
+model initialized with ‘bert-large-cased’, stacked with a CNN1D 
+classifier and fine-tuned with CoNLL2003-Train was selected as 
+the finalized TopoBERT module. Table 5 shows a comparison 
+between TopoBERT and five other models and tools based on the 
+Harvey2017 dataset.  
+Table 5: Evaluation results with Harvey2017 dataset for 
+comparing TopoBERT with other existing models. 
+Model 
+Precisio
+n 
+Recal
+l 
+F1-
+score 
+Stanford NER (broad 
+location) 
+0.729  
+0.440  
+0.548  
+SpaCy NER (broad location) 
+0.461  
+0.304  
+0.366  
+BiLSTM-CRF 
+0.703  
+0.600  
+0.649  
+DM_NLP 
+0.729  
+0.680  
+0.703  
+NeuroTPR 
+0.787  
+0.678  
+0.728  
+TopoBERT 
+0.898  
+0.835  
+0.865  
+The SpaCy version v3.0 is used with model “en_core_web_sm” 
+loaded. Broad location indicates that we include entities in both 
+LOCATION and ORGANIZATION for Stanford NER, and we 
+include entities in the types of LOC, ORG, FACILITY, and GPE 
+for spaCy NER. Evaluation results show that TopoBERT prevailed 
+in the competition with precision equals to 0.898, recall 0.835 and 
+F1-score 0.865. This result outperformed other baseline models by 
+at least 18%. 
+TopoBERT has been developed as a ready-to-use module. The 
+output data of TopoBERT includes word labels and confidence of 
+the prediction. It complies with JSON file format for ease of use. 
+
+ 
+ 
+ 
+The source code has been uploaded to GitHub and can be accessed 
+with the link: https://github.com/SPGBarrett/gearlab_topobert. 
+ 
+4 Discussion 
+This paper presents a geoparsing framework and breeds a plug and 
+play toponym recognition module which can facilitate spatial 
+analysis based on social media or news media data. Figure 7 shows 
+a practical application of this framework in locating Twitter posts 
+under fine-grained topics during hazardous events. The study area 
+is the State of Florida, and the dots in multiple colors displayed on 
+the map are tweets posted during Hurricane Irma harvested by 
+Twitter developer API. The locations of those tweets without 
+geotags are retrieved by running TopoBERT and google geocoding 
+service. The module also enjoys the potential of being used for 
+location name detection for news media to pinpoint the discussed 
+topics [14,15] and help to identify fake news [16]. 
+ 
+ 
+Figure 7: Toponym recognition applied to locate Twitter posts 
+during disasters.  
+This paper concentrates mainly on designing a novel architecture 
+of a reliable and versatile module for toponym recognition. 
+However, the performance enhancement can continue by 
+addressing the following issues.  
+First, the models are trained and evaluated based on well prepared 
+datasets. This can be regarded as a best-case scenario compared to 
+real life situations. Place name usage can be highly ambiguous and 
+random, especially within social media platforms. Typos are 
+extremely common which might cause out-of-vocabulary words in 
+language models. Place name abbreviations such as “Boulevard” 
+and “blvd”, “Drive” and “Dr.”, “Street” and “St.” and so forth are 
+frequently utilized interchangeably. People might unconsciously 
+ignore the correct upper-case and lower-case usage, such as 
+“college station” and “College Station”, “mexico” and “MEXICO”. 
+Meticulous data preprocessing methods can be incorporated to 
+tackle this problem in order to achieve better overall performance.  
+Second, several rule-base approaches can be leveraged to further 
+boost the performance. Enlightened by the success of hybrid 
+models [9], sets of grammar rules based on the composition of 
+nouns, determiners, adjectives, conjunctions, numbers and 
+possessive ending can be designed [17]. Additionally, commonly 
+used gazetteers such as OpenStreetMap and GeoNames can be used 
+as extra named entity matching criteria which will enhance the True 
+Positives of the model. Regional criteria can be appended to the 
+model while identifying place names by making country name, 
+state names, county names, or bounding boxes as input variables of 
+the model. This will allow the model to add constraints during 
+processing. The top-N words from word embedding models [9,35], 
+which are not place names, can be applied to filter words during 
+data preprocessing. This will to some extent eliminate the False 
+Positives of the prediction.  
+Third, due to the data-hungry nature of deep learning, data 
+availability and quality are topics being inevitably discussed when 
+large complicated deep learning models are involved. It is common 
+knowledge in the deep learning world that larger datasets lead to 
+better generalizability and performance. However, this statement 
+fails to hold true in this paper due to the fact that the larger datasets 
+are derived from several distinguished smaller datasets labeled 
+under their own unique regime. Therefore, there is an urgent need 
+to define criteria and build unified datasets for toponym recognition 
+model training, evaluating and benchmarking. The dataset can be 
+manually modified based on existing datasets and augmented using 
+rule-based methods, gazetteers or Generative Adversarial Network 
+[18,19,20].  
+Fourth, fine-tuned language models can be few-shot or zero-shot 
+learners, which means that the models can be applied directly to 
+certain downstream tasks with very little or even no further training 
+[21,22,23]. This is because advanced language models can better 
+capture the meaning of the text. This claim is also underpinned by 
+the result of this paper which leverages BERT to boost the module 
+capability. Therefore, incorporating gigantic models such as GPT-
+3 [24] might lead to another round of performance enhancement. 
+5 Conclusion 
+To further enhance the performance of toponym recognition by 
+better understanding natural language, TopoBERT, which 
+incorporate pretrained language model, BERT, is introduced. 
+Experiments on the pretrained parameters, training dataset 
+combinations, and model architecture reveal the following findings. 
+First, the toponym recognition model performance is sensitive to 
+the architecture of pre-trained language models and classifiers. The 
+models initialized with a larger-structured BERT model (“bert-
+large-cased”) show an advantage over the models initialized with a 
+basic BERT model (“bert-base-cased”). More complicated 
+classifiers like MLP do not necessarily win over simple linear 
+classifiers. Second, increasing training data size produces worse 
+results, especially for the recall, due to data heterogeneity. The 
+model trained with single dataset, CoNLL2003-Train, and stacked 
+on top with a CNN1D classifier renders the optimum results both 
+on CoNLL2003-Test and Harvey2017 datasets. Finally, the 
+developed TopoBERT module outperforms existing models in 
+
+Hurricane Category
+IrmaRouteLine
+Florida_Census_ Tract_2019
+0
+Human_Help_Florida
+Animal HelpFlorida
+3
+Infrastructure Florida
+ShelterFlorida 
+ 
+ 
+ 
+recognizing place names in texts. The clinched TopoBERT with the 
+optimal model architecture and training strategy produces reliable 
+toponym prediction and achieves F1-score of 0.865 on Harvey2017 
+dataset, which surpasses other prevailing models or tools by at least 
+18%. 
+In nutshell, the discoveries of this paper contribute in determining 
+the optimal model structure on toponym recognition tasks and 
+urges a large standardized dataset labeled with unified regime to 
+support model training and benchmarking. A plug and play module 
+is implemented and open sourced to support pertinent applications 
+and similar research.  
+ACKNOWLEDGMENTS 
+The research is supported by a project funded by the U.S. National 
+Science Foundation: Reducing the Human Impacts of Flash Floods 
+- Development of Microdata and Causal Model to Inform 
+Mitigation and Preparedness (Award No. 1931301).  
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+MITP/21-047
+Refactorisation in subleading ¯B → Xsγ
+Tobias Hurtha and Robert Szafron,b
+aPRISMA+ Cluster of Excellence and Institute of Physics (THEP),
+Johannes Gutenberg University, D-55099 Mainz, Germany
+bDepartment of Physics, Brookhaven National Laboratory, Upton, N.Y., 11973, U.S.A.
+Abstract
+We establish refactorisation conditions between the subleading O8-O8 contributions to
+the inclusive ¯B → Xsγ decay suffering from endpoint divergences and prove a factorisa-
+tion theorem for these contributions to all orders in the strong coupling constant.
+This
+allows for higher-order calculations of the resolved contributions and consistent summation
+of large logarithms, consequently reducing the recently found large-scale dependence in these
+contributions. We implement the concept of refactorisation in a heavy flavour application
+of SCET, which includes nonperturbative functions as additional subtlety not present in
+collider applications.
+1
+arXiv:2301.01739v1  [hep-ph]  4 Jan 2023
+
+Contents
+1
+Introduction
+3
+2
+General setup
+4
+2.1
+Hard matching
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+3
+Bare factorisation theorem
+7
+3.1
+B-type current (direct) contribution . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.2
+A-type (resolved) contribution
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4
+Refactorisation of the endpoint contribution
+12
+4.1
+Refactorisation at leading order . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+4.2
+Bare refactorised factorisation theorem . . . . . . . . . . . . . . . . . . . . . . . .
+17
+4.3
+Refactorised factorisation theorem after renormalisation . . . . . . . . . . . . . . .
+17
+5
+Summary and Outlook
+20
+2
+
+1
+Introduction
+There has been a general belief that soft-collinear factorisation at subleading power in ΛQCD/mb
+expansion is well established for inclusive B−decay modes such as ¯B → Xsγ, ¯B → Xsℓℓ, or
+¯B → Xuℓ¯ν [1] – in contrast to exclusive B decays where factorisation theorems do not exist at
+the subleading power in general. There are two types of subleading contributions to the inclusive
+¯B → Xsγ decay, direct and resolved ones. In the latter, the photon does not directly couple to an
+effective electroweak vertex, but they contain subprocesses in which the photons couple to light
+partons instead. These subleading corrections are nonlocal in the endpoint region, and they stay
+nonlocal even in the region where the local heavy mass expansion is applicable. In this sense,
+they represent an irreducible uncertainty of this decay mode. Analogous subleading contributions
+exist in the inclusive ¯B → Xsℓℓ decay but not in the inclusive ¯B → Xuℓ¯ν decay because, in this
+case, the leptons can couple to light partons via the W vector boson only.
+The first systematic analysis of resolved contributions to the inclusive ¯B → Xsγ decay [2,3] was
+worked out in Refs. [4,5], the corresponding 1/mb contributions to the inclusive ¯B → Xsℓℓ decay
+were discussed in Refs. [6,7], using soft collinear effective theory (SCET). Recently, the uncertainty
+due to the resolved contribution was reduced with the help of a new hadronic input [8, 9]. But
+these resolved contributions still represent the largest uncertainty in the inclusive ¯B → Xsγ decay.
+Moreover, a large scale dependence and also a large charm mass dependence were identified in
+the lowest order result of the resolved contribution, which calls for a systematic calculation of αs
+corrections and renormalisation group (RG) summation [9]. A mandatory prerequisite for this
+task is an all-order in the strong coupling constant αs factorisation formula for the subleading
+power corrections.
+The factorisation of resolved contributions introduces a new ingredient, namely an anti-
+hardcollinear jet function [4], typically referred to as a radiative or amplitude-level jet function
+in collider and flavour applications [10–18]. They are not represented by cut propagators as the
+usual jet functions but as full propagator functions (both dressed by Wilson lines). But as al-
+ready noticed in Ref. [4], the specific resolved O8 − O8 contribution does not factorise because
+the convolution integral is UV divergent. The authors of Ref. [4] emphasised that there is an
+essential difference between divergent convolution integrals in power-suppressed contributions of
+exclusive B decays and the divergent convolution integrals in the present case, while the former
+were of IR origin, the latter divergence of UV nature. However, a solution at the lowest order
+was established by considering the sum of direct and resolved O8 − O8 contributions, which was
+shown to be scale and scheme dependent by using a hard cut-off in the resolved contribution. But
+the failure of factorisation does not allow for a consistent resummation of large logarithms.
+In this paper, we identify the divergences in the resolved and in the direct contributions
+as endpoint divergences by showing that also the divergence in the direct contribution can be
+traced back to a divergent convolution integral.
+Recently, new techniques were presented in
+specific collider applications of SCET [19–27]. The so-called refactorisation conditions or endpoint
+factorisation [22,23,25–27] allow for an operator-level reshuffling of terms within the factorisation
+formula so that all endpoint divergences cancel out. In this work, we now implement this idea
+in a flavour application of SCETI, which includes nonperturbative soft functions not present in
+collider applications – often referred to as subleading shape functions [28,29].
+As a first step, we derive the matching of the hard function for the two operators involved
+in the O8 − O8 subleading contributions. In the second step, we establish the bare factorisation
+theorem for the direct and the resolved contribution at an operational level. Then we derive
+3
+
+the refactorisation conditions to all orders, leading us finally to the renormalised factorisation
+theorem. We present all steps for the inclusive ¯B → Xsγ, but all the details can also be taken
+over for the corresponding ¯B → Xsℓℓ case.
+2
+General setup
+The starting point for all calculations concerning the ¯B → Xsγ decay is the weak effective
+Lagrangian defined at a scale µb parametrically equal to the b-quark mass µb ∼ mb. The weak
+effective Lagrangian is obtained from the SM Lagrangian after integrating out the heavy particles
+like the heavy gauge bosons and the top quark. We use the convention of Ref. [30]. Assuming
+Standard Model CKM unitarity, with λq = VqbV ∗
+qs and λu + λc + λt = 0, the effective Hamiltonian
+may be written as
+Heff = GF
+√
+2
+�
+q=u,c
+λq
+�
+C1 Oq
+1 + C2 Oq
+2 + C7γ O7γ + C8g O8g +
+�
+i=3,...,6
+Ci Oi
+�
+.
+(1)
+Here we concentrate on the O8 operator:
+O8g = − gs
+8π2 mb ¯sσµν(1 + γ5)Gµνb .
+(2)
+Our sign convention is that iDµ = i∂µ+gs taAa
+µ+e qfAµ, where ta are the SU(3) colour generators,
+and Qf is the electric charge of the fermion in units of e. We consider the CP-averaged ¯B → Xsγ
+photon energy spectrum in the endpoint region where Mb − 2Eγ = O(ΛQCD). Soft-collinear-
+effective theory (SCET) offers the appropriate framework for this multi-scale problem.
+The
+kinematics of the decay is given as follows: the initial meson carries momentum pB, and it decays
+into a photon with momentum q and a jet whose total momentum is PX. From pB − q = PX in
+the B meson rest frame, we have 2MBEγ = M 2
+B − M 2
+X. Thus, the jet invariant mass MX is much
+smaller than the photon energy Eγ and jet energy EX. We set PX⊥ = 0 and choose reference
+vectors n2 = n2 = 0, v2 = 1, such that n + ¯n = 2v and nn = 2. Choosing
+qµ = Eγ¯nµ
+and
+pµ
+B = MBvµ ,
+(3)
+we find MB = ¯nPX and MB = nPX + 2Eγ or equivalently
+P µ
+X = (MB − 2Eγ)nµ + MB¯nµ .
+(4)
+Thus, there is only one independent kinematical variable in the ¯B → Xsγ decay. One may choose
+the photon energy Eγ or nPX = MB − 2Eγ.
+Three dynamical scales describe the endpoint region, a hard scale of O(MB), an intermedi-
+ate (anti-)hardcollinear scale of O(
+�
+MBΛQCD), and a soft scale of O(ΛQCD). The expansion
+parameter in our present analysis is defined as λ2 = ΛQCD/MB1. The photon can be treated as
+anti-hardcollinear. The hadronic final state factorises into a hardcollinear jet and soft wide-angle
+radiation. Since the soft modes have parametrically smaller virtuality than the hardcollinear
+1Alternative convention often used in the literature is to define λ = ΛQCD/MB.
+4
+
+modes, the problem at hand is described by the SCETI setup [31–34]. Using a shorthand nota-
+tion a ∼ (na, ¯na, a⊥) to indicate the scaling of the momentum components in powers of λ, we
+have: hard momentum scales like phard ∼ (1, 1, 1)mb, a hardcollinear one phc ∼ (λ2, 1, λ)mb, an
+anti-hardcollinear region phc ∼ (1, λ2, λ)mb and a soft momentum psoft ∼ (λ2, λ2, λ2)mb.
+The first step in the derivation of a factorisation theorem is hard matching. We have to match
+the electroweak operator onto SCET. We will see that the direct contribution is represented by a
+next-to-leading power (NLP) B-type current in SCET, i.e. power-suppressed current composed of
+two collinear building blocks. The resolved contribution is represented by a time-ordered product
+of a leading-power (LP) A-type current with a subleading L(1)
+ξq Lagrangian (see Ref. [19, 35–37]
+for a precise definition of the different types of currents).
+In the second step, we integrate out the hardcollinear fields, which lead to the appearance of
+jet functions. The latter are, technically speaking, matching coefficients of SCET on the pure
+soft effective field theory. Kinematics forbids the emission of anti-hardcollinear partons. Thus
+anti-hardcollinear fields have to be integrated out at the amplitude level. The physics in the
+anti-hardcollinear direction is similar to the threshold Drell-Yan expansion, where kinematical
+constraints forbid hardcollinear emissions into the final state. For more details on SCET NLP
+factorisation and resummation in the threshold Drell-Yan, see Refs. [13,14,20,38]. In the hard-
+collinear sector, the formalism resembles, for example, the thrust factorisation, where NLP jet
+functions are defined at the cross-section level [15,39–42].
+2.1
+Hard matching
+The electroweak operator O8 matches onto two possible SCET operators. First, we consider the
+A-type current, which enters the resolved contribution. Within the present section, the SCET
+operators are always given before the soft decoupling transformation [32] is performed.
+The
+A-type SCET operator is given by
+OA0
+8g (0) = χhc (0) /n
+2γµ⊥Aµ
+hc⊥ (0) (1 + γ5) h (0) ,
+(5)
+where h is the heavy quark field, and the SCET building blocks are hardcollinear gauge-invariant
+due to the introduction of hardcollinear Wilson lines W. The fermionic building block is χhc =
+W †
+hcξ and gluon field is
+Aµ
+hc⊥ = W †
+hc [Dµ
+hc⊥Whc] = Aaµ
+hc⊥ta.
+(6)
+Note that the colour and Dirac structure for the A-type operator is uniquely fixed.
+For the
+matching, we can use the partonic QCD amplitude b (pb) → s (ps) g (r), where the momenta pb is
+hard, ps anti-hardcollinear and r hardcollinear. The matching condition is given by
+GF
+√
+2λt C8g ⟨Q8g⟩ = CA0 (mb)
+�
+OA0
+8g
+�
+.
+(7)
+The brackets ⟨ ⟩ indicate that the matrix element of the operators is considered. At leading order
+in αs we find
+CA0
+LO (mb) = m2
+b
+4π2
+GF
+√
+2λqC8g .
+5
+
+The B-type current which enters the direct contribution is the following SCET operator:2
+OB1
+8g (u) =
+�
+dt
+2πe−iumbtχhc (t¯n) γν⊥ Qs Bν
+hc⊥ (0) γµ⊥ Aµ
+hc⊥ (0) (1 + γ5) h (0) ,
+(8)
+with Qs as the electric charge of the strange quark in units of e and the electromagnetic gauge-
+invariant transverse photon field
+Bν
+hc⊥ = e
+�
+Aν
+⊥ − ∂ν
+⊥
+n∂ nA
+�
+.
+(9)
+We note that beyond the LO, a second operator with an alternative Dirac structure γµγν appears.
+These operators do not mix under renormalisation [43]. We checked by explicit computation at
+the one-loop order that the matching coefficient for the operator with γµγν structure does not
+develop any endpoint divergence.
+We use the partonic QCD amplitude b (pb) → g (r) s (ps) γ (q) to fix the matching coefficient.
+Here the momenta are pb hard, r and ps hardcollinear and q anti-hardcollinear. On the QCD
+side, a time-ordered product of the operator O8 and of the QED current,
+LQED,q (x) = eq Aµ (x) q (x) γµq (x) ,
+(10)
+is needed. The matching condition at the leading order in QED is given by
+GF
+√
+2λtC8g i
+�
+d4x ⟨T [O8g (0) , LQED,b (x) + LQED,s (x)] ⟩ =
+� 1
+0
+duCB1 (mb, u)
+�
+OB1
+8g (u)
+�
+.
+(11)
+We do not consider QED corrections. At leading order in αs, we find that only the QED current
+with an s quark contributes, and we arrive at
+CB1
+LO (mb, u) = (−1)u
+u
+m2
+b
+4π2
+GF
+√
+2λt C8g = (−1)u
+uCA0
+LO (mb) ,
+(12)
+where we use hardcollinear momentum conservation ¯ns + ¯nr = mb and introduce hardcollinear
+momentum fraction u = ¯nps
+mb and u = 1 − u.
+Finally, we compare the different kinematics of the A- and B-type currents in Figure 1. The
+external s-quark carries hardcollinear momentum. Therefore the intermediate propagator is hard.
+This situation is represented in SCET by the B-type current. When the momentum of the external
+s-quark tends to zero, the propagator becomes anti-hardcollinear and cannot be integrated out –
+it must be reproduced by a dynamical field in the low energy EFT. This situation is represented
+in SCET by the time-order product of subleading Lagrangian and the A-type current.
+The
+degeneracy in the EFT description is the reason why the SCET develops divergencies in the
+convolution integrals.
+2Note that this operator is equal to J2 (τ) in Ref. [43] (eq. 16), with J2 (τ) = 2mbOB1
+8g (τ).
+6
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+h/hc
+b
+�hc
+O8
+ghc
+shc
+Figure 1: The full theory LO diagram which induces an endpoint divergence in SCET, see text.
+3
+Bare factorisation theorem
+The derivation of the factorisation theorem follows the standard approach. [44, 45].
+We first
+perform the soft decoupling transformation [32], but we do not use a new notation for the hard-
+collinear fields after decoupling.
+The decay rate is obtained from the imaginary part of the
+current-current correlator. The states factorise and thus allow taking matrix elements separately
+in hardcollinear, anti-hardcollinear and soft sectors.
+The common to both A- and B-type contributions is the anticollinear matrix element of the
+photon. It is given by a discontinuity of the photon propagator
+−gµν
+⊥ e2 Jγ
+�
+q2�
+=
+1
+2πi Disc[ i
+�
+d4xeiqx ⟨0| T [Bµ
+hc⊥ (x) , Bν
+hc⊥ (0)] |0⟩ ]
+(13)
+= −gµν
+⊥ e2 δ
+�
+q2�
+θ
+�
+q0�
+= −gµν
+⊥ e2 δ+ �
+p2�
+(14)
+Since we are only interested in the photon-final state, the above expression is exact to all orders
+in perturbation theory.
+3.1
+B-type current (direct) contribution
+There are several functions entering the factorisation formula of the direct contribution with the
+B-type current. The soft function – the leading power shape function – is defined as
+S (ω) =
+1
+2mB
+�
+dt
+2πe−iωt ⟨B| h (tn) Sn (tn) S†
+n (0) h (0) |B⟩ ,
+(15)
+or with open indices [46,47]3
+1
+2mB
+�
+dt
+2πe−iωt ⟨B| [hα (tn) Sn (tn)]i
+�
+S†
+n (0) hβ (0)
+�
+j |B⟩ = δij
+2 Nc
+�1 + /v
+2
+�
+βα
+S (ω) .
+(16)
+3We use Greek for spinor indices and Latin for colour indices.
+7
+
+The hardcollinear jet function is a genuine NLP object. In analogy to the LP jet function, we
+define it as a vacuum matrix element of a product of hardcollinear fields
+J
+�
+p2, u, u′�
+= (−1)
+2Nc
+1
+2π
+�
+dtdt′
+(2π)2 d4x e−imb(ut−u′t′)+ipx
+(17)
+Disc
+�
+⟨0| tr
+�1 + /v
+2
+(1 − γ5) /Ahc⊥ (x) γν
+⊥χhc (t′¯n + x) χhc (t¯n) γν⊥ /Ahc⊥ (0) (1 + γ5)
+�
+|0]⟩
+�
+.
+The field operators are time- or anti-time-ordered according to the Keldysh formalism.4
+The
+trace is taken both with respect to colour and spinor spaces. Using projection properties of the
+hardcollinear fields and 2/v = /n+ + /n− the Dirac algebra in eq.(17) can be simplified to
+J
+�
+p2, u, u′�
+= (−1)
+2Nc
+1
+2π
+�
+dtdt′
+(2π)2 d4x e−imb(ut−u′t′)+ipx (d − 2)2
+(18)
+Disc
+�
+⟨0| tr
+�/¯n
+4(1 − γ5)Aµ
+hc⊥ (x) χhc (t′¯n + x) χhc (t¯n) Ahc⊥
+µ
+(0) (1 + γ5)
+�
+|0⟩
+�
+where, as mentioned before, the trace is also applied in the colour space.
+With these definitions, we find the bare factorisation theorem for the direct contribution
+dΓ
+dEγ
+= NB
+� 1
+0
+duCB1 (mb, u)
+� 1
+0
+du′CB1∗ (mb, u′)
+� Λ
+−p+
+dωJ (MB (p+ + ω) , u, u′) S (ω)
+(19)
+with the prefactor
+NB = [(2π)] [e2Q2
+s] [
+1
+(2π)3 2 Eγ
+E2
+γ 4π] = e2Q2
+s
+Eγ
+2π
+(20)
+The three pieces of the prefactor correspond to the phase space factors of the photon, to its
+charges and to the redefinition of the jet function with a 2π factor.
+Finally, we prove to all orders in αs that the jet-function is symmetric in u and u′ up to
+complex conjugation:
+J(p2, u, u′) = J∗(p2, u′, u).
+(21)
+This can be read off from the factorisation theorem of the direct contribution. The photon energy
+spectrum is real. The leading power shape function is also real to all orders. This can be shown
+by complex conjugation of Eq.(15) and by using translation invariance [4]. Then the jet function
+inherits the symmetry property given in Eq.(21), from the product of the Wilson coefficients,
+CB1 (mb, u) CB1∗ (mb, u′), in the convolution integral. An anti-symmetric part of the jet function
+would cancel out in the convolution integral. We emphasise that this property is also valid when
+the other B-type operator with the reversed Dirac structure is present. In particular, the sum of
+the two mixed terms has this property. In the latter terms, the reduction of the Dirac structure
+leads to (4 − d) (d − 2), and hence these terms vanish for d = 4.
+The symmetry property is crucial for the refactorisation because it implies that no double
+subtraction regarding the variables u and u′ is needed in the B-type (direct) current contribution.
+This can be seen in the following way. We showed above that the integrand of the convolution
+integral of the Wilson coefficients and the jet function in the two variables u and ¯u is real, so the
+4For a brief summary, see appendix of Ref. [48].
+8
+
+complete integrand is symmetric in u and u′. Then the subsequent rearrangement is possible (we
+here only write the convolution variables u and u′):
+� 1
+0
+duCB1 (u)
+� 1
+0
+du′CB1∗ (u′) J (u, u′) = 2
+� 1
+0
+duCB1 (u)
+� 1
+u
+du′CB1∗ (u′) J (u, u′) .
+(22)
+As the endpoint divergence manifests for small u and u′, we need to ensure that only the last
+integral over u is rendered finite by an appropriate subtraction.
+At the leading order, the jet function is real, and we find that the jet function is symmetric
+in u and u′. Explicitly, we find using the dimensional MS regulator (µ2ϵ → µ2ϵ exp(γEϵ)/(4π)ϵ):
+J
+�
+p2, u, u′�
+= CF
+αs
+4π mb
+θ(p2) A(ϵ) δ(u − u′)u1−ϵ(1 − u)−ϵ
+�p2
+µ2
+�−ϵ
+,
+(23)
+with
+A(ϵ) = (2 − 2ϵ)2 (1 − 1/2 ϵ) Γ(1 − ϵ)−1 exp(γEϵ) = 4 − 10ϵ + O(ϵ2) .
+(24)
+We compute the convolution integrals explicitly5 using this leading order result for the jet
+function and also the hard function at leading order, Eq.(12),
+dΓ
+dEγ
+|B = 2NB
+��CA0
+LO (mb)
+��2 � 1
+0
+du ¯u
+u
+� 1
+u
+du′ ¯u′
+u′
+(25)
+CFA(ϵ)
+αs
+(4π) mb
+� Λ
+−p+
+dω S (ω)
+�mb(p+ + ω)
+µ2
+�−ϵ
+u1−ϵ(1 − u)−ϵδ(u − u′)
+= NB
+��CA0
+LO (mb)
+��2 CF
+αs
+(4π) mb
+� Λ
+−p+
+dω S(ω) A(ϵ)B(3 − ϵ, −ϵ)
+�mb(ω + p+)
+µ2
+�−ϵ
+,
+where B(x, y) denotes the Beta function. We see that the divergence in the direct contribution
+is now identified as an endpoint point divergence in the convolution integral of the hard and the
+jet function in the u integration for u ≪ 1.
+We emphasise that this endpoint divergence can be regularised within the dimensional regu-
+larisation scheme6. This leads to additional poles after performing the convolution. Consequently,
+due to endpoint divergences, the bare factorisation formula is already invalid for the d → 4 limit
+at the leading order.
+5Symmetry of the original integral implies that
+� 1
+u du′δ(u − u′) = θ(0) with θ(0) = 1/2, for u ∈ [0, 1].
+6We note that we do not confirm the leading order result of the direct contribution of Ref. [4] in the dimensional
+regularisation scheme. In the notation of Ref. [4] we get
+F (a)
+88 (Eγ, µ) = CF αs(µ)
+4π
+� mb
+2Eγ
+�2 � ¯Λ
+−p+
+dω
+�2
+9 ln mb(ω + p+)
+µ2
++ 2
+9
+�
+S(ω, µ) .
+9
+
+3.2
+A-type (resolved) contribution
+For the resolved contribution with the A-type current, we start with the time-ordered product
+OTξq = i
+�
+ddxT
+�
+Lξq (x) , OA0
+8g (0)
+�
+= i
+�
+ddxT
+�
+qs (x+) Sn(x+)
+�
+Qs /Bhc⊥ + /Ahc⊥
+�
+(x) χhc (x) ,
+χhc (0) S†
+n(0)Sn(0)/n
+2 /Ahc⊥(0) (1 + γ5) S†
+n(0)h (0)
+�
+.
+(26)
+The operator in the hardcollinear sector contains only gluon fields. Hence the standard leading
+power gluon jet function appears
+−g2
+sδabgµν
+⊥ Jg
+�
+p2�
+=
+1
+2πi Disc
+�
+i
+�
+d4xeipx ⟨0| T
+�
+Aaµ
+hc⊥ (x) , Abν
+hc⊥ (0)
+�
+|0⟩
+�
+.
+(27)
+At leading order we find the standard result Jg (p2) = δ+ (p2).
+Besides photons, there are no energetic particles emitted in the anti-hardcollinear directions.
+Thus, the anti-hardcollinear jet function is defined at the amplitude level:
+OTξq =
+�
+dω
+�
+dt
+2πe−itω [qs]α (tn)
+�
+J (ω)
+�a νµ
+αβ
+Qs Bν
+hc⊥ (0) Aµ
+hc⊥ (0) [h (0)]β .
+(28)
+The anti-hardcollinear jet function can be decomposed as
+�
+J (ω)
+�a νµ
+αβ = J (ω) ta
+�
+γν
+⊥γµ
+⊥
+/¯n/n
+4
+�
+αβ
+,
+(29)
+to all orders. The other structure γµ
+⊥γν
+⊥ does not appear as one can read off from the structure of
+the T product in eq. (26) and the fact that the gluon and heavy quark fields are only spectators.
+The Dirac structure can then be simplified at the level of the cross-section with the help of the
+following relation:
+�
+γν
+⊥γµ
+⊥
+/¯n/n
+4
+�
+αβ
+�
+γµ
+⊥γν
+⊥
+/n/¯n
+4
+�
+α′β′
+= (d − 2)2
+�/¯n/n
+4
+�
+αβ
+�/n/¯n
+4
+�
+α′β′
+.
+(30)
+At leading order, the anti-hardcollinear jet function is given by
+�
+J (ω)
+�a νµ
+αβ =
+ta
+(ω + i ϵ)
+�
+γν
+⊥γµ
+⊥
+/¯n/n
+4
+�
+αβ
+.
+(31)
+Having defined hardcollinear and anti-hardcollinear functions, we now focus on the soft sector.
+The operatorial definition of the soft function in position space with open Dirac indices is
+Sαβ,α′β′ (u, t, t′) =
+= g2
+s ⟨B|
+�
+h (un) (1 − γ5)
+�
+α′
+�
+Sn (un) taS†
+n (un)
+�
+S¯n (un)
+�
+S†
+¯n (t′¯n + un) qs (t′¯n + un)
+�
+β′
+× [qs (t¯n) S¯n (t¯n)]α S†
+¯n (0)
+�
+Sn (0) taS†
+n (0)
+�
+[(1 + γ5)h (0)]�� |B⟩ / (2mB) .
+(32)
+10
+
+We can now plug in all the objects into the matrix element squared, and we find the resolved
+contribution
+dΓ
+dEγ
+= NA
+��CA0 (mb)
+��2 � Λ
+−p+
+dωJg (mb (p+ + ω))
+�
+dω1
+�
+dω2J (ω1) J
+∗ (ω2) S (ω, ω1, ω2) ,
+(33)
+with the prefactor
+NA = NB ≡ N ,
+(34)
+and the scalar soft function obtained after contracting spinor indices according to
+S (u, t, t′) = (d − 2)2g2
+s ⟨B| h (un) (1 − γ5)
+�
+Sn (un) taS†
+n (un)
+�
+S¯n (un) S†
+¯n (t′¯n + un)
+(35)
+/n/¯n
+4 qs (t′¯n + un) qs (t¯n) /¯n/n
+4 S¯n (t¯n) S†
+¯n (0)
+�
+Sn (0) taS†
+n (0)
+�
+(1 + γ5) h (0) |B⟩ / (2mB) .
+The soft function in momentum space, which appears in eq. (33), is obtained through the Fourier
+transform of the position space expression according to
+S (ω, ω1, ω2) =
+� du
+2πe−iuω
+�
+dt
+2πe−itω1
+� dt′
+2πeit′ω2S (u, t, t′) .
+(36)
+As the NLP jet function in u and u′ variables, the soft function S (ω, ω1, ω2) is symmetric in
+ω1 and ω2 up to complex conjugation:
+S (ω, ω1, ω2) = S∗ (ω, ω2, ω2) .
+(37)
+This property stems from the fact that the gluon jet function is real to all orders. Thus, the
+soft function inherits the symmetry property from the product of the anti-hardcollinear jet func-
+tions, J (ω1) J
+∗ (ω2) in the factorisation formula of the resolved contribution. Any anti-symmetric
+part would cancel in the convolution integral. This symmetry property implies that within the
+refactorisation, a double-subtraction regarding the variables ω1 and ω2 in the A-type current
+contribution is not needed either. The symmetry implies that the integrand in the convolution
+integral between anti-hardcollinear jet functions and soft function in the two variables ω1 and ω2
+is real and symmetric and allows for the following rearrangement of the convolution integral
+� ∞
+−∞
+dω1
+� ∞
+−∞
+dω2J (ω1) J
+∗ (ω2) S (ω1, ω2) = 2
+� ∞
+−∞
+dω1
+� ω1
+−∞
+dω2J (ω1) J
+∗ (ω2) S (ω1, ω2) ,
+(38)
+which is motivated by the fact in the resolved contribution. As we will see explicitly in section 4.1,
+the convolution integral of the jet and shape function is logarithmically divergent for ω1,2 → ∞.
+At leading order, we find the factorisation formula in case of the A-type current (resolved)
+contribution:7
+dΓ
+dEγ
+= 2N
+��CA0
+LO (mb)
+��2 � Λ
+−p+
+dωδ (mb (p+ + ω))
+� ∞
+−∞
+dω1
+� ω1
+−∞
+dω2
+1
+(ω1 − iϵ)
+1
+(ω2 + iϵ) S (ω, ω1, ω2) .
+(39)
+We keep the soft function unevaluated at this point since this is a nonperturbative object. For
+ω1,2 ≫ ω, the soft function can be shown to be asymptotically constant, which leads to endpoint
+divergence in the convolution integrals for large ω1,2 (see section 4.1).
+7This confirms the leading order result of the resolved contribution of Ref. [4] when the asymptotic limit of the
+soft function is not yet considered.
+11
+
+Figure 2: Scales relevant to refactorisation of the endpoint divergent contribution. The left part of
+the diagram represents the standard hierarchy of three scales for SCETI. Near the endpoint, when
+the momentum fraction u is no longer u ∼ O(1), i.e. u ≪ 1, we introduce additional, unphysical
+scales which make it possible to factorise further objects appearing in the bare factorisation
+theorem.
+4
+Refactorisation of the endpoint contribution
+We here state the three refactorisation relations, which are based on the fact that in the limits
+u ∼ u′ ≪ 1 and ω1 ∼ ω2 ≫ ω the two terms of the subleading O8−O8 contribution have the same
+structure. The refactorisation relations are operatorial relations that guarantee the cancellation
+of endpoint divergences between the two terms to all orders in αs.
+The refactorisation conditions result from the overlap between soft and hardcollinear modes.
+The hierarchy of scales near the endpoint is shown in Fig. 2.
+We will refer to these overlap
+modes as softcollinear modes.
+They play a similar role as the z-SCET modes introduced in
+Ref. [24] to prove the refactorisation of the B1-type matching coefficients.
+The parameter z
+corresponds to the momentum fraction u in the present analysis. On the one hand, we can think
+of the softcollinear mode as a limit of hardcollinear mode when the large momentum fraction
+tends to zero8.
+On the other hand, the softcollinear modes can be understood as a limit of
+the soft modes when the n+k momentum component becomes much larger than the remaining
+components, mb ≫ n+k ≫ λ2mb. We want to emphasise that softcollinear and u-hardcollinear
+modes are not physical but help introduce refactorisation. The softcollinear fields obey the same
+projection properties and have the same transformation properties regarding gauge invariance as
+their hardcollinear counterparts.
+• Following Refs. [22–24], we find that in the limit u → 0, the matching coefficient can be
+8Thus, they do not appear in the leading power problems, where only operators with a single hardcollinear
+field in each direction occur.
+12
+
+2
+hard
+B1
+CAO 7
+umb
+u-hardcollinear
+hardcollinear
+J&S→J,3
+softcollinear
+S↑
+softfurther factorised
+�
+CB1 (mb, u)
+�
+= (−1)CA0 (mb) mbJ (umb) ,
+(40)
+where �g(u)� only denotes the leading term of a function g(u) in the limit u → 0 and
+without any higher power corrections in u ≪ 1. The function J (umb), which appears here,
+is exactly the same radiative jet function (29) we introduced before in the context of A-type
+contribution.
+This refactorisation condition stems from the fact that in the limit u → 0, the amplitude
+used in the matching of the B-type current can be represented by a time-ordered product
+[24],
+CB1 (mb, u)
+�
+OB1
+8g (u)
+� ���
+u→0 = CA0 (mb) i
+�
+ddxe−i (nx/2) umb
+�
+T
+�
+L(1)
+ξqsc (x) , OA0−u
+8g
+(0)
+��
+,
+(41)
+of the leading power current OA0−u
+8g
+(0) = χu−hc(0) S†
+n(0) Sn(0) /n
+2 /Au−hc⊥(0) (1 + γ5) S†
+n(0) h(0) ,
+equal to (5) up to a replacement of the hardcollinear fields by the u-hardcollinear fields, and
+subleading Lagrangian
+L(1)
+ξqsc (x) = qsc(x+)S†
+n(0) Sn(0)
+�
+Qs /Bu−hc⊥ + /Au−hc⊥
+�
+χu−hc(x) + h.c.
+(42)
+The jet-function J (umb) appears after integrating out the u-anti-hardcollinear quark fields.
+We note a close resemblance to the structure of the resolved contribution, where a similar
+time-ordered product appears (see Eq. (26)).
+• We find the new soft function �S (ω, ω1, ω2) which corresponds to the function S (ω, ω1, ω2) in
+the limit ω1 ∼ ω2 ≫ ω. In this limit, we can consider the light soft quarks to be softcollinear
+qs → qsc. In this function �S (ω, ω1, ω2) higher power corrections in ω/ω1,2 are neglected.
+• In the limit, where the momentum fractions u → 0 and u′ → 0, the jet function
+J (mb (p+ + ω) , u, u′) fulfills the following relation
+� Λ
+−p+
+dω �J (mb (p+ + ω) , u, u′) S(ω)� =
+� Λ
+−p+
+dωJg(mb(p+ + ω)) �S(ω, mbu, mbu′) ,
+(43)
+where the brackets indicate that the u → 0 and u′ → 0 limits have to be taken and that the
+hardcollinear quark fields in J are regarded as softcollinear fields, χhc → qsc in accordance
+with (41).
+It is crucial that the soft function �S (ω, ω1, ω2) appears both in the A-type contribution in
+the limit ω1 ∼ ω2 ≫ ω and in the B-current term if one expands for small u and u′.
+Before we proceed, let us comment on the structure of �S. In the asymptotic regime, where
+ω1,2 ≫ ω, we can match the �S on the leading power shape function
+�S (ω, ω1, ω2) =
+�
+dω′K(ω, ω′, ω1, ω2)S(ω′) .
+(44)
+13
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+hc
+b
+�hc
+L(1)
+⇠q
+A0
+ghc
+ssoft
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+vFDZfNFZube/A3n4wQ+</latexit>
+b
+�hc
+B1
+ghc
+shc
+Figure 3:
+The SCET representations of the full theory diagram in Fig.1, see text.
+The matching kernel K(ω, ω′, ω1, ω2) introduced in (44) can be computed perturbatively, i.e. to
+extract K, we can replace B-meson state by a b-quark in the definition of the soft function and
+calculate both sides of the equation on the partonic level. The LP soft function here appears
+since the limit ω1,2 → ∞ is equivalent to the treatment of t and t′ as infinitesimal variables in
+(32) and, consequently, the soft Wilson lines obtained from decoupling in the anti-hardcollinear
+direction Sn cancel. At the same time, the softcollinear quark field produces an additional soft
+Wilson line associated with the hardcollinear direction Sn because we require the softcollinear
+quark to have the same gauge transformation as a hardcollinear field. Finally, the structure of
+the soft function corresponds to LP shape function S(ω). Consistency of the second and third
+refactorisation conditions, which approach the softcollinear limit from two different directions as
+shown in Figure 2, leads to
+� Λ
+−p+
+dω �J (mb (p+ + ω) , u, u′) S(ω)� =
+� Λ
+−p+
+dωJg(mb(p+ + ω))
+�
+dω′K(ω, ω′, ω1, ω2)S(ω′).
+(45)
+This relation implies that the kernel K can be obtained from the quark-gluon jet function in the
+limit when momentum fraction of the quark tends to zero. Furthermore, it confirms that the
+kernel K is a perturbative object and that the softcollinear scale can be treated perturbatively.
+Finally, we note that softcollinear quarks must appear on both sides of the cut. Fermion num-
+ber conservation implies that only in this case we get a non-vanishing decay rate. Consequently,
+the endpoint divergences only appear in the limit when both u and u′ are small or when ω1 and
+ω2 are large.
+Figure 3 shows that the A- and B-type current have the same structure in the refactorisation
+limit. On the left, the s-quark is soft and emitted through the insertion of the subleading power
+Lagrangian. On the right, the s-quark is hardcollinear and emitted directly from the hard B-type
+vertex. When the fraction of the hardcollinear momentum of the s-quark tends to zero, the B-type
+current refactorises into the time-ordered product represented on the left, and both diagrams rep-
+resent the same full theory configuration. This duality in the description leads to the appearance
+of the endpoint divergences. A similar problem has already been identified in Refs. [49,50], in the
+context of QED corrections in Bs → µ+µ− due to O7 operator at the amplitude level.
+14
+
+4.1
+Refactorisation at leading order
+Based on the refactorisation conditions, we first discuss the procedure of refactorisation at the
+leading order. We explicitly verify the conditions using the leading order results. Starting with the
+last refactorisation condition, we consider the factorisation theorem of the A-type contribution
+when the soft function is considered in the limit ω1 ∼ ω2 ≫ ω. This asymptotic limit of the
+soft function can be analysed by means of semi-perturbative methods [51], where the energetic
+softcollinear quarks are treated perturbatively, while ordinary soft modes are assumed to be
+nonperturbative. In the leading order, this corresponds to the replacement of the softcollinear
+quarks by a cut propagator. We anticipate the endpoint divergence in the convolution of the
+soft and the anti-hardcollinear jet functions and use dimensional MS regularisation within the
+calculation. We find the following expression of the asymptotic soft function at leading order [51],:
+�S (ω, ω1, ω2) = CFA(ϵ) αs
+(4π) ω1−ϵ
+1
+δ(ω1 − ω2)
+� Λ
+ω
+dω′ S(ω′)
+�(ω′ − ω)
+µ2
+�−ϵ
+,
+(46)
+which includes the leading power shape function S(ω). Note that this expression, in principle,
+receives corrections of higher order in αs and ΛQCD/ω1,2, which we do not take into account in
+the leading order analysis within this section. A(ϵ) was defined in eq. (24) 9
+We convolute the asymptotic soft function with the anti-hardcollinear jet functions for large
+ω1 and ω2 only by restricting the limits of the ω1 integral to mb and +∞. These integration
+limits will become clear once we consider the B-type current contribution. Starting with the
+factorisation formula of the A type current given in eq. (39), the asymptotic contribution of the
+A type current reads at leading order:
+dΓ
+dEγ
+|asy
+A
+= 2N |CA0
+LO(mb)|2
+� Λ
+−p+
+dωJLO
+g
+(mb(p+ + ω))
+� ∞
+mb
+dω1JLO(ω1)
+� ω1
+0
+dω2J
+∗
+LO(ω2) �S(ω, ω1, ω2)
+= N|CA0
+LO (mb) |2 αsCF
+(4π) mb
+1
+ϵA(ϵ)
+� Λ
+−p+
+dω SLO(ω′)
+�mb(ω + p+)
+µ2
+�−ϵ
+.
+(47)
+The 1
+ϵ pole is the manifestation of the endpoint divergence in the resolved contribution in the
+limit ω1 ∼ ω2 ≫ ω. In the next step, we will see that the specific choice mb as a lower limit of
+the ω1 integration is induced by the refactorisation conditions. The lower limit in the ω2 integral
+can be chosen to be non-negative due to the delta function δ(ω1 − ω2).
+Now we take the limit u → 0 in the factorisation theorem of the B-type current at leading
+order, which we derived in eq. (25) before performing the integrals over u and u′. This leads to
+dΓ
+dEγ
+|u,u′→0
+B
+= −N
+��CA0
+LO (mb)
+��2
+αsCF
+(4π) mb
+1
+ϵ A(ϵ)
+� Λ
+−p+
+dω SLO(ω)
+�mb(ω + p+)
+µ2
+�−ϵ
+.
+(48)
+This result differs from eq. (47) only by an overall sign. The sum of these two terms is finite and
+equal to zero. This leading-order result is a special case of the all-order relation, which follows
+from refactorisation conditions. In the ω1 ∼ ω2 ≫ ω (asymptotic) limit of the A-type current
+(with integration limits over ω1 from mb to +∞), we exactly single out the same term as in the
+9We note that we do not confirm the leading order result of the asymptotic soft function of Ref. [4] in the
+dimensional regularisation scheme.
+15
+
+u → 0 of the B-type current up to a minus sign. This reflects the fact that in the limits u → 0
+and ω1 ∼ ω2 ≫ ω the two terms of the subleading O8 − O8 contribution have the same structure.
+Moreover, we see that with the relations mbu = ω1 and mbu′ = ω2, the u, u′ → 1 limit corresponds
+to the limit ω1, ω2 → mb, which fixes the integration limit in the subtraction term of the A-type
+current.
+We can summarise the relation we just verified at LO as
+dΓ
+dEγ
+|asy
+A
+= (−1) dΓ
+dEγ
+|u,u′→0
+B
+.
+(49)
+The refactorisation conditions guarantee that the eq. (49) holds to all orders in perturbation
+theory. To make this relation useful for the reshuffling of the factorisation theorem, let us consider
+an integral of �S(ω1, ω2, ω)J(ω1)J
+∗(ω2) over the ω1,2 ∈ [0, ∞]. Since �S(ω1, ω2, ω) is expanded for
+ω1,2 ≫ ω, this integral is scaleless and equal to zero in dimensional regularisation. We then
+perform the following manipulations of the integration limits
+0 =
+� ∞
+0
+dω1
+� ∞
+0
+dω2 = 2
+� ∞
+0
+dω1
+� ω1
+0
+dω2 = 2
+�� mb
+0
+dω1 +
+� ∞
+mb
+dω1
+� � ω1
+0
+dω2.
+(50)
+In the second step, we made use of the fact that the integrand is symmetric in ω1 and ω2 as we
+derived to all orders in eq. (38). Finally, we split the integration region into two parts suitable
+for the subtraction.
+The term integrated over ω1 from mb to ∞ is already in the form suitable for the subtraction
+of the A-type term and equal to (47). To bring the second term into the form of eq. (48), we
+perform substitutions ω1 = mb u and ω2 = mb u′, use (40) to replace J(ω1) by the singular part
+of the CB1 matching coefficient and then use the second refactorisation condition to derive
+2N
+��CA0
+LO(mb)
+��2 � mb
+0
+dω1JLO(ω1)
+� ω1
+0
+dω2J
+∗
+LO(ω2)
+� Λ
+−p+
+dωJLO
+g
+(mb(p+ + ω)) �S(ω, ω1, ω2)
+= 2N
+� 1
+0
+du
+�
+CB1
+LO (mb, u)
+� � 1
+u
+du′ �
+CB1
+LO (mb, u′)
+� � Λ
+−p+
+dω �JLO (mb (p+ + ω) , u, u′) SLO(ω)� (51)
+We rewrite Eq. (49) using the functions which enter the factorisation theorem:
+2N
+��CA0
+LO(mb)
+��2 � ∞
+mb
+dω1JLO(ω1)
+� ω1
+0
+dω2J
+∗
+LO(ω2)
+� Λ
+−p+
+dωJLO
+g
+(mb(p+ + ω)) �S(ω, ω1, ω2)
+= − 2N
+� 1
+0
+du
+�
+CB1
+LO (mb, u)
+� � 1
+u
+du′ �
+CB1
+LO (mb, u′)
+� � Λ
+−p+
+dω �JLO (mb (p+ + ω) , u, u′) SLO(ω)� .
+(52)
+We see with the help of (51) that the fact that the sum of asymptotic contributions is equal to
+zero is a consequence of our refactorisation conditions. It is now clear that these two subtraction
+terms, which add up to zero, make it possible to reshuffle the factorisation theorem and cancel
+the endpoint divergences at the leading order.
+16
+
+4.2
+Bare refactorised factorisation theorem
+The generalisation of the LO order result to all orders is straightforward.
+Since we are still
+working in d-dimensons with bare objects, we can insert a scaleless expression into the factorisation
+theorem using the integral manipulations we performed at LO, see eq. (50)
+Using the all-orders refactorisation conditions discussed at the beginning of this section, we
+then can cast the subtraction term into the following form with the help of the same manipulations
+as in the LO case and generalise eq. (52) to all orders:
+0 = 2N
+��CA0 (mb)
+��2 � Λ
+−p+
+dωJg (mb (p+ + ω))
+� ∞
+mb
+dω1J (ω1)
+� ω1
+0
+dω2J
+∗ (ω2) �S (ω, ω1, ω2)
++ 2N
+� 1
+0
+du
+�
+CB1 (mb, u′)
+� � 1
+u
+du′ �
+CB1∗ (mb, u′)
+� � Λ
+−p+
+dω �J (mb (p+ + ω) , u, u′) S(ω)� . (53)
+Starting from the all-order bare factorisation theorem
+dΓ
+dEγ
+= 2N
+��CA0 (mb)
+��2 � ∞
+−∞
+dω1J (ω1)
+� ω1
+−∞
+dω2J
+∗ (ω2)
+� Λ
+−p+
+dωJg (mb (p+ + ω)) S (ω, ω1, ω2)
++ 2N
+� 1
+0
+duCB1 (mb, u)
+� 1
+u
+du′CB1∗ (mb, u′)
+� Λ
+−p+
+dωJ (mb (p+ + ω) , u, u′) S(ω)
+(54)
+and subtracting eq. (53) we arrive at
+dΓ
+dEγ
+|A+B = 2N
+� Λ
+−p+
+dω
+�
+Jg(mb(p+ + ω))
+��CA0 (mb)
+��2
+(55)
+×
+� ∞
+−∞
+dω1
+� ω1
+−∞
+dω2J(ω1) J
+∗(ω2)
+�
+S (ω, ω1, ω2) − θ(ω1 − mb)θ(ω2) �S(ω, ω1, ω2)
+�
++
+� 1
+0
+du
+� 1
+u
+du′ �
+CB1
+LO (mb, u) CB1∗ (mb, u′) J (mb (p+ + ω) , u, u′) S (ω)
+−
+�
+CB1 (mb, u)
+� �
+CB1∗ (mb, u′)
+�
+�J (mb (p+ + ω) , u, u′) S(ω)�
+��
+,
+where �J (mb (p+ + ω) , u, u′) S(ω)� = Jg(mb(p+ + ω)) �S(ω, mbu, mbu′) and
+�
+CB1 (mb, u′)
+�
+= (−1)CA0 (mb) mbJ (umb). We note here that the second term effectively restricts the integration
+range over ω1 to a finite range in the first line and consequently removes endpoint divergence.
+Thus these terms need to be added together before the ω1 integral is performed. Similarly, the
+last term removes the endpoint divergence of the third term, and therefore u integration has to be
+performed after these two terms are added up. In addition, we note that the integrals in the first
+term are finite for large negative values of ω1 and ω2 due to nonperturbative dynamics [4]. At this
+point, the convolutions integrals in the A- and B-type contributions are no longer divergent, and
+we can renormalise the functions entering the factorisation theorem and take the limit d → 4.
+4.3
+Refactorised factorisation theorem after renormalisation
+We achieved refactorisation at the level of the bare factorisation theorem. It has been pointed
+out that refactorisation and renormalisation do not commute in general [23, 52]. Therefore, for
+17
+
+the result to be helpful for the resummation of the large logarithms, we must prove that we
+can express the factorisation theorem in terms of renormalised objects. To this end, we have to
+replace bare quantities with renormalised ones. The renormalisation of hard matching coefficients
+is well-established
+CA0
+bare(mb) = ZA0(µ) CA0
+ren(µ, mb) ,
+(56)
+CB1
+bare(u) =
+� 1
+0
+du′ ZB1(µ, u, u′) CB1
+ren(µ.u′) ,
+(57)
+where the one-loop renormalisation factors can be found in Ref. [43]. The LP jet function is
+renormalised according to
+Jbare
+g
+(p2) =
+� p2
+o
+dp′2 ZJg(µ, p2 − p′2) Jren
+g (µ, p′2) ,
+(58)
+with the ZJg factor given in Refs. [53, 54] up to the three-loop order. Similarly, the LP shape
+function
+Sbare(ω) =
+�
+dω′ ZS(µ, ω − ω′) Sren(µ, ω′)
+(59)
+is well-known [55]
+Much less is known about NLP objects. The radiative jet function is a notable example which
+appeared before in the context of B → γℓν [12]. It has recently been computed at the two-loop
+order in Ref. [17]. The most important detail is that the time-like (ω > 0) and space-like (ω < 0)
+radiative jet functions do not mix under renormalisation
+J
++
+bare/ren(ω) = θ(ω)Jbare/ren(ω) ,
+(60)
+J
+−
+bare/ren(ω) = θ(−ω)Jbare/ren(ω) ,
+(61)
+and
+J
++
+bare(ω) =
+� ∞
+0
+dω′ Z+
+J (µ, ω, ω′) , J
++
+ren(µ, ω′) ,
+(62)
+J
+−
+bare(ω) =
+� 0
+−∞
+dω′ Z−
+J (µ, ω, ω′) J
+−
+ren(µ, ω′) .
+(63)
+This separation into time-like and spec-like jet functions is necessary since we choose to integrate
+the subtraction term only over non-negative values of ω1,2. Finally, we define the renormalisation
+of the NLP soft and jet functions
+Sbare(ω, ω1, ω2) =
+�
+dω′dω′
+1dω′
+2 ZS(µ, ω, ω′, ω1, ω′
+1, ω2, ω′
+2) Sren(µ, ω′, ω′
+1, ω′
+2) ,
+(64)
+Jbare(p2, u1, u2) =
+�
+dp′2
+� 1
+0
+du′
+1
+� 1
+0
+du′
+2 ZJ(µ, p2 − p′2, u1, u′
+1, u2, u′
+2) Jren(p′2, u′
+1, u′
+2) .
+(65)
+These renormalisation kernels are currently unknown.
+18
+
+We require that A- and B-type contributions are separately RG invariant (see Ref. [56] for
+analogous treatment). This leads to the following conditions on the renormalisation kernels
+|ZA0|2
+�
+dω
+�
+dω1
+�
+dω2 ZJg(ω − ω′)ZJ(ω1, ω′
+1) Z†
+J(ω2, ω′
+2) ZS(ω, ω′′, ω1, ω′′
+1, ω2, ω′′
+2)
+=δ(ω′ − ω′′) δ(ω′
+1 − ω′′
+1) δ(ω′
+2 − ω′′
+2) ,
+(66)
+and
+� 1
+0
+du1
+� 1
+0
+du2
+�
+dω ZB1(u1, u′
+1) ZB1†(u2, u′
+2) ZJ(ω − ω′, u1, u′
+1, u2, u′
+2) ZS(ω − ω′′)
+= δ(ω′ − ω′′) δ(u′
+1 − u′′
+1) δ(u′
+2 − u′′
+2) ;
+(67)
+and further, RG invariance of the subtraction term leads to
+|ZA0|2
+� ∞
+0
+dω1
+� ∞
+0
+dω2
+�
+dω Z+
+J (ω1, ω′
+1) Z+†
+J (ω2, ω′
+2) ZJg(ω − ω′) Z�S(ω − ω′′, ω1, ω′′
+1, ω2, ω′′
+2)
+= δ(ω′ − ω′′) δ(ω′
+1 − ω′′
+1) δ(ω′
+2 − ω′′
+2) .
+(68)
+These conditions are sufficient to prove that renormalisation and refactorisation commute and
+there is no leftover term.
+We can now insert the above definitions into eq. (55),
+dΓ
+dEγ
+|A+B = 2N
+� Λ
+−p+
+dω
+�
+Jren
+g (mb(p+ + ω))
+��CA0
+ren (mb)
+��2
+(69)
+×
+� ∞
+−∞
+dω1
+� ω1
+−∞
+dω2J
++
+ren(ω1) J
++∗
+ren(ω2)
+�
+Sren (ω, ω1, ω2) − θ(ω1 − mb)θ(ω2) �Sren(ω, ω1, ω2)
+�
++
+� 1
+0
+du
+� 1
+u
+du′ �
+CB1
+ren (mb, u) CB1∗
+ren (mb, u′) Jren (mb (p+ + ω) , u, u′) Sren (ω)
+−
+�
+CB1
+ren (mb, u)
+� �
+CB1∗
+ren (mb, u′)
+�
+�Jren (mb (p+ + ω) , u, u′) Sren(ω)�
+��
+.
+This is our final result. Endpoint divergences are manifestly absent, assuming one performs
+the integrals over ω1 after adding the first and second terms together. Similarly, the integrals
+over u should be performed after adding the last two lines.
+This renormalised factorisation
+theorem allows for a consistent resummation of large logarithms within the resolved O8 − O8,
+using standard RG methods owing to the fact that each object appearing in the above equation
+is a single scale object. However, a judicious choice of scale might be necessary.
+19
+
+5
+Summary and Outlook
+In the present paper, we identified the divergences in the resolved, but also in the direct subleading
+O8 − O8 as endpoint divergences which lead to a breakdown of the factorisation theorem already
+at leading order in four space-time dimensions. The failure of naive factorisation does not allow
+for consistent separation of scales and, consequently, resummation of large logarithms.
+However, it was recently shown [9] that the resolved contributions still represent the most
+significant uncertainty in the inclusive ¯B → Xsγ decay. Large scale dependence and also a large
+charm mass dependence were identified in the lowest order result of the resolved contribution,
+which calls for a systematic calculation of αs corrections and RG summation of all resolved
+contributions [9]. A mandatory input for this task is a well-defined factorisation formula for these
+subleading corrections. This critical step was established in this paper. The next step consists of
+computing renormalisation kernels for the NLP soft and jet functions, extracting the anomalous
+dimensions and solving the RG equations to resum large logarithms.
+Recent intensive studies of the power corrections in collider applications of SCET [19,22–24,
+42,52] lead to the development of new techniques that allow for a reshuffling of terms within the
+factorisation formula so that all endpoint divergences cancel out. We used these new techniques in
+our flavour application which includes nonperturbative functions typically not present in collider
+applications of SCET. Unlike in the h → γγ decay [22], in the considered SCETI problem, there
+are no leftover terms present after renormalisation.
+To derive a consistent factorisation theorem, we first established the bare factorisation theorem
+for the resolved and direct contributions on the operatorial level. Then we derived the all-orders
+refactorisation conditions applicable to our process. This idea is based on the fact that in certain
+limits, the two terms of the subleading O8 − O8 contribution have the same structure, which
+guarantees that the endpoint divergences cancel between the two terms to all orders. Finally, we
+proved that we could express the factorisation theorem in terms of renormalised objects so that the
+result can be used for the resummation of the large logarithms within the resolved contributions.
+Acknowledgements
+We thank Martin Beneke and Matthias Neubert for their valuable discussions. RS would also like
+to thank Mathias Garny and Jian Wang for many discussions on power corrections in SCET and
+Mikolaj Misiak for a discussion on the theoretical predictions for B → Xsγ. TH is grateful to
+Michael Benzke for uncounted discussions on the resolved contributions. RS is supported by the
+United States Department of Energy under Grant Contract DESC0012704. TH is supported by
+the Cluster of Excellence “Precision Physics, Fundamental Interactions, and Structure of Matter"
+(PRISMA+ EXC 2118/1) funded by the German Research Foundation (DFG) within the German
+Excellence Strategy (Project ID 39083149), as well as by the BMBF Verbundprojekt 05H2018 -
+Belle II. TH also thanks the CERN theory group for its hospitality during his regular visits to
+CERN where part of the work was done.
+20
+
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+24
+

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+arXiv:2301.03131v1  [math.AT]  9 Jan 2023
+INTRINSIC CONVERGENCE OF THE HOMOLOGICAL TAYLOR TOWER FOR
+r-IMMERSIONS IN Rn
+GREGORY ARONE AND FRANJO ˇSARˇCEVI´C
+Abstract. For an integer r ≥ 2, the space of r-immersions of M in Rn is defined to be the
+space of immersions of M in Rn such that at most r − 1 points of M are mapped to the same
+point in Rn. The space of r-immersions lies “between” the embeddings and the immersions.
+We calculate the connectivity of the layers in the homological Taylor tower for the space of r-
+immersions in Rn (modulo immersions), and give conditions that guarantee that the connectivity
+of the maps in the tower approaches infinity as one goes up the tower. We also compare the
+homological tower with the homotopical tower, and show that up to degree 2r − 1 there is a
+“Hurewicz isomorphism” between the first non-trivial homotopy groups of the layers of the two
+towers.
+Contents
+1.
+Introduction
+1
+2.
+Prerequisites
+5
+2.1.
+Cubical diagrams
+5
+2.2.
+Manifold calculus of functors
+6
+2.3.
+Spectra
+9
+3.
+The homological Taylor tower for reduced r-immersions in Rn
+12
+4.
+r-configuration spaces in Rn as complements of subspace arrangements
+13
+5.
+Total fiber of a retractive cubical diagram
+16
+6.
+The cube of r-configuration spaces is retractive
+18
+7.
+Connectivity of the cube of (co)homologies of r-configuration spaces
+19
+8.
+Convergence result
+22
+9.
+Comparing with the unstable tower
+23
+10.
+Further questions
+26
+References
+27
+1. Introduction
+Let M be a smooth manifold of dimension m, and fix an integer r ≥ 2. An r-immersion of M
+in Rn is an immersion of M in Rn such that the preimage of every point in Rn contains at most
+r − 1 points of M. The space of r-immersions of M in Rn is denoted by rImm(M, Rn). For
+2020 Mathematics Subject Classification. Primary: 57R42; Secondary: 55R80, 57R40, 55P42.
+Key words and phrases. calculus of functors, manifold calculus, Taylor tower, embeddings, immersions, r-
+immersions, homotopy of spectra, homological convergence, partial configuration space.
+Acknowledgements. F. ˇSarˇcevi´c was partially supported by the grant P20 01109 (JUNTA/FEDER, UE).
+1
+
+r = 2, 2-immersions are the same thing as injective immersions, which are essentially the same
+as embeddings in nice cases. In any case, we have inclusions of subspaces
+Emb(M, Rn) ⊆ 2 Imm(M, Rn) ⊂ 3 Imm(M, Rn) ⊂ · · · ⊂ rImm(M, Rn) ⊂ · · · ⊂ Imm(M, Rn).
+In this paper we study the homological Taylor tower of the r-immersions functor. The “Taylor
+tower” is meant in the sense of manifold calculus (also known as embedding calculus) developed
+by Weiss [Wei99] and Goodwillie-Weiss [GW99].
+The basic idea of manifold calculus is the following. In order to study the homotopy type of a space
+such as rImm(M, Rn), one views it as a particular value of the presheaf rImm(−, Rn) defined on
+M (one can also consider more general target manifolds than Rn, but we will content ourselves
+with maps into Rn). A presheaf is a contravariant functor on the poset O(M) of open subsets
+of M. Inside O(M) there is a sequence of subposets O1(M) ⊂ · · ·Ok(M) ⊂ · · · ⊂ O∞(M),
+where Ok(M) is the poset of open subsets of M that are diffeomorphic to the disjoint union of
+at most k copies of Rm. By restricting a presheaf F to Ok(M) and then extrapolating back to
+O(M) one obtains a tower of approximations to F, which is usually denoted as follows
+F → (T∞F → · · · → TkF → Tk−1F → · · · T0F).
+This is called the “Taylor tower” of F. Manifold calculus, and the Taylor tower in particular, has
+had many consequences and applications [Mun05], [Vol06], [ALV07], [Mun11], [DH12], [ST16],
+[BdBW18].
+In this paper we investigate the Taylor tower that calculates the homology of the space rImm(M, Rn).
+In practice, this means the following. First of all, it is convenient to replace the space of r-
+immersions with r-immersions modulo immersions. Let us suppose that we fix a basepoint in
+Imm(M, Rn), and let rImm(M, Rn) be the homotopy fiber of the inclusion map rImm(M, Rn) →
+Imm(M, Rn). Let HZ denote the Eilenberg-MacLane spectrum. We are interested in the Taylor
+tower of the presheaf of Spectra, defined by the formula
+U �→ HZ ∧ rImm(U, Rn).
+(more precise definitions are given in Section 2).
+Our main result concerns the rate of convergence of the Taylor tower of this functor.
+The
+question of convergence is a fundamental one.
+We will distinguish between two aspects of
+convergence: how strongly the tower converges to its limit, and what it converges to. We will
+say that the Taylor tower of a functor F converges intrinsically at M if the connectivity of the
+map TkF(M) → Tk−1F(M) approaches ∞ as k approaches ∞. We say that the Taylor tower
+of F converges strongly to F(M) if the connectivity of the map F(M) → TkF(M) approaches
+∞ as k approaches ∞. Strong convergence implies intrinsic convergence, but the converse does
+not have to be true. In practice it seems that for “natural” functors that we know, whenever the
+Taylor tower of F converges intrinsically, it converges strongly to F. But intrinsic convergence
+is usually much easier to prove than strong convergence.
+Before we state our main result, let us recall, for context, that one of the deepest results in
+functor calculus is the Goodwillie-Klein-Weiss convergence theorem [GW99], [GK08], [GK15].
+Theorem 1.1 (Convergence of the Taylor tower for spaces of embeddings). If M is a smooth
+closed manifold of dimension m, and N is a smooth manifold of dimension n, then the map
+Emb(M, N) → Tk Emb(M, N)
+2
+
+is
+k(n − m − 2) + 1 − m-connected.
+In particular, if n−m−2 > 0, then the connectivities grow with k and the Taylor tower therefore
+converges strongly to Emb(M, N).
+There is an easier, but also important convergence result for the homological version of the tower,
+which is more directly relevant to this paper. Define Emb(M, Rn) to be the homotopy fiber of
+the inclusion Emb(M, Rn) → Imm(M, Rn). Consider the contravariant functor from O(M) to
+Spectra that sends U to HZ ∧ Emb(U, Rn). This functor represents the homology of the space
+of embeddings modulo immersions. The Taylor tower of this functor is known to converge when
+n > 2m + 1 [Wei04].
+Now let us state our main result
+Theorem 1.2. Let M be m-dimensional. Assume that n ≥ 2. If r ≤ n + 1, the Taylor tower
+for HZ ∧ rImm(M, Rn) converges intrinsically when
+n > rm + 1
+r − 1 .
+If r ≥ n + 1 then the Taylor tower converges intrinsically when n > m + 1.
+Remarks 1.3.
+(1) When r = n + 1 the two statements are equivalent.
+Indeed, the function f(n) =
+n2 − nm − m − 1, n ∈ N, is positive only for n > m + 1.
+(2) When r = 2 we get the condition n > 2m + 1, which is the known condition for the
+convergence of the Taylor tower of HZ ∧ Emb(M, Rn).
+(3) The condition n > rm+1
+r−1 is equivalent to rm−(r−1)n < −1. The number rm−(r−1)n
+equals, at least when it is positive, to the dimension of the intersection of r copies of Rm
+embedded in Rn in a general position.
+Next let us discuss the proof. Let F be a presheaf defined on a suitable category of m-dimensional
+manifolds and codimension zero embeddings. The basic building blocks in the construction of
+the Taylor tower of F are spaces of the form F(�
+i Rm), for i = 0, 1, 2, . . .. The homotopy fiber
+of the map TkF → Tk−1F depends on the total homotopy fiber of the following cubical diagram,
+indexed by the poset of subsets of k = {1, . . . , k}:
+(1)
+S �→ F
+
+�
+k\S
+Rm
+
+
+This homotopy fiber is sometimes called the k-th derivative (or the k-th cross-effect) of F at
+∅. The following fact is particularly important for analysing intrinsic convergence. Recall that a
+cubical diagram is called c-cartesian if the map from the initial object to the homotopy limit of the
+rest of the cubical diagram is c-connected. Suppose the cubical diagram (1) is ck-cartesian. Then
+the map TkF(M) → Tk−1F(M) is ck − mk-connected. Thus the Taylor tower of F converges
+intrinsically at M if the number ck − mk approaches ∞ as k approaches ∞.
+When F(M) = Emb(M, Rn), there is a well-known equivalence Emb(�
+k Rm, Rn) ≃ Conf(k, Rn),
+where Conf(k, Rn) is the configuration space of ordered k-tuples of pairwise distinct points in Rn.
+3
+
+Similarly, there is an equivalence between rImm(�
+k Rm, Rn) and the so-called r-configuration
+space, also called no r-equal configuration space, defined by
+rConf(k, Rn) := rImm(k, Rn).
+This is the space of ordered k-tuples of points in Rn where at most r −1 are allowed to be equal.
+A proof of the equivalence
+rImm(
+�
+k
+Rm, Rn)
+≃−→ rConf(k, Rn)
+is given in [AˇS22]. Thus r-configuration spaces are basic building blocks in the Taylor tower of
+rImm(M, Rn).
+To analyse the intrinsic convergence of the Taylor tower of the functor HZ ∧ rImm(−, Rn), one
+needs to calculate how cartesian the following k-dimensional cubical diagram is
+(2)
+S �→ HZ ∧ rConf(k \ S, Rn).
+The space rConf(i, Rn) is the complement of a subspace arrangement in Rni. It follows that the
+homology of r-configuration spaces is accessible by means of the Goresky-MacPherson formula
+and other such tools. The homology of r-configuration spaces was studied by a number of people,
+starting with Bj¨orner and Welker [BW95].
+Using the Goresky-MacPherson formula and the results in [BW95] we prove the following result
+(it is combining Proposition 7.7 and Theorem 8.1)
+Theorem 1.4. When r ≤ n + 1, the cube (2) is k(n − 1) +
+� k
+r
+�
+(r − n − 1)-cartesian, and the
+map
+pk : TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn)
+is
+k
+�
+nr − 1
+r
+− m − 1
+r
+�
+− (k mod r)
+r
+(r − n − 1)-connected.
+Here (k mod r) := k − r
+� k
+r
+�
+.
+When r ≥ n + 1, the cube (2) is k(n − 1) + r − n − 1-cartesian, and the map pk is
+k(n − m − 1) + r − n − 1-connected.
+Theorem 1.2 follows easily from Theorem 1.4.
+In Section 9 we compare the tower of the homological functor HZ ∧ rImm(M, Rn) with that of
+the tower of the homotopical functor rImm(M, Rn). Let us suppose that we chose a basepoint in
+the space rImm(M, Rn). In this case the presheaf rImm(−, Rn) takes values in pointed spaces,
+and we have the following diagram of presheaves:
+(3)
+rImm(−, Rn)
+i←− rImm(−, Rn)
+h−→ Ω∞HZ ∧ rImm(−, Rn).
+It is well-known that the map i induces an equivalence of all layers except the first one. Indeed,
+the map i is the homotopy fiber of the map from rImm(−, Rn) to its linear approximation. Thus
+we can view the map h as a map from the higher layers/derivatives of rImm(−, Rn) to the
+corresponding layers/derivatives of Ω∞HZ∧rImm(−, Rn), which are essentially the same as the
+layers/derivatives of HZ ∧ rImm(−, Rn), since Ω∞ commutes with Taylor approximations.
+4
+
+When r = 2, the second derivative of rImm(−, Rn) is equivalent to Sn−1, and the second
+derivative of HZ ∧ rImm(−, Rn) is HZ ∧ Sn−1. It follows that in the case r = 2, the map h
+in (3) induces the Hurewicz homomorphism from the second derivatives of rImm(−, Rn) to the
+second derivative of HZ ∧ rImm(−, Rn). In particular, it follows that the connectivity of the
+quadratic layers of the Taylor towers of rImm(−, Rn) and of HZ ∧ rImm(−, Rn) is the same,
+and their first non-trivial homotopy groups are isomorphic.
+By contrast, at degrees higher than 2, the layers of the homotopical tower rImm(−, Rn) and of
+the homological tower of the functor HZ ∧ rImm(−, Rn) have different connectivities, and there
+is no Hurewicz type isomorphism between them.
+And again by contrast, in Section 9 we show that for r > 2 the map h in diagram (3) induces
+a Hurewicz type isomorphism between first non-trivial homotopy groups of layers roughly up to
+degree 2r − 1. See Theorem 9.1 for precise statement.
+Organization of the paper. In Section 2 we review some background material on cubical diagrams,
+manifold calculus and spectra. In Section 3 we introduce the homological Taylor tower that is
+the main subject of this paper.
+In Section 4 we make an excursion into the subspace arrangements. We describe r-configuration
+spaces via subspace arrangements and compute their cohomology using the Goresky-MacPherson
+theorem.
+In Section 5 we define the notion of a retractive cubical diagram. This is a diagram where the
+maps have sections that satisfy a certain hypothesis. We prove that the homotopy groups of
+the total homotopy fiber of a retractive cube are isomorphic to the total kernel of the cube of
+homotopy groups.
+In Section 6 we prove that the cube of r-configuration spaces that controls the layers in the Taylor
+tower is retractive. In Section 7 we prove our main result about the homological connectivity
+of the cube of r-configuration spaces. In Section 8 we prove the main result about the intrinsic
+convergence of the Taylor tower of HZ ∧ rImm(M, Rn).
+In Section 9 we compare the tower of HZ ∧ rImm(M, Rn) with the tower of rImm(M, Rn) in
+low degrees. We prove that the layers in the two towers have the same connectivity up to degree
+2r − 1 (with some exceptions in the cases r = 2, 3).
+In Section 10 we discuss some possible directions for further exploration.
+2. Prerequisites
+2.1. Cubical diagrams. Cubical diagrams play an important role in functor calculus, and in this
+paper in particular, so we will recall a few elementary facts about them. All the results in this
+subsection, and much more, can be found in [Goo92].
+Let k denote the standard set with k elements {1, . . . , k}. Let P(k), or just P(k), denote the
+poset of subsets of k. A k-dimensional cubical diagram in a category C is a functor χ: P → C.
+It is easy to see that P(k) is equivalent to P(k)op, so a contravariant functor from P(k) to C is
+called a cubical diagram as well. We will mostly consider cubical diagrams in (pointed) spaces
+and spectra, and also in abelian groups.
+5
+
+Given a cubical diagram χ in topological spaces or spectra, there is a natural map
+iχ : χ(∅) → holim
+∅̸=S⊂k χ(S).
+We say that χ is c-cartesian, if this map is c-connected. The homotopy fiber of this map is called
+the total homotopy fiber of χ. The total homotopy fiber of χ is denoted by tfiber(χ). Clearly if
+χ is c-cartesian then tfiber(χ) is c−1-connected. The converse always holds for cubical diagrams
+of spectra, and it holds for spaces under the additional assumption that iχ is surjective on path
+components.
+One can identify a k-dimensional cubical diagram with a map of two k − 1-dimensional cubical
+diagrams. Given a k-dimensional cubical diagram χ, let us define two k − 1-dimensional cubical
+diagrams χ1 and χ2 as follows: χ1(U) = χ(U), and χ2(U) = χ(U ∪ {k}). Then χ can be
+identified with the map of cubes χ1 → χ2. Furthermore, there is a homotopy fibration sequence
+whose meaning is that total homotopy fiber can be calculated as an iterated homotopy fiber
+tfiber(χ) ≃ hofiber(tfiber(χ1) → tfiber(χ2)).
+When χ is a cubical diagram of abelian groups, we de��ne the total kernel of χ to be
+tkernel(χ) := ker(χ(∅) →
+k
+�
+i=1
+χ({i})).
+Just as with total fibers, the total kernel can be calculated as an iterated kernel. There is a
+natural isomorphism
+tkernel(χ) ∼= ker(tkernel(χ1) → tkernel(χ2)).
+When χ is a cubical diagram of spaces or spectra, there is a natural homomorphism of graded
+groups
+π∗(tfiber χ) → tkernel(π∗χ).
+This homomorphism is not an isomorphism in general. In Section 5 we will investigate a condition
+on a cubical diagram that guarantees for it to be an isomorphism.
+2.2. Manifold calculus of functors. Let M be a smooth manifold of dimension m. Define
+O(M) to be the poset category of open subsets of M. Objects of O(M) are open sets U ⊆ M,
+and morphisms U → V are the inclusions U ⊆ V .
+Manifold calculus of functors, developed by Weiss [Wei99] and Goodwillie-Weiss [GW99], studies
+contravariant functors from O(M) to a category that supports a reasonable notion of homo-
+topy. In their foundational papers, Goodwillie and Weiss only considered functors with values in
+topological spaces, and maybe spectra. Nowadays it is natural to let the target category to be
+an ∞-category. We will content ourselves with functors with values in (pointed) spaces and in
+spectra.
+Technically speaking, manifold calculus applies to functors that are good, in the sense that they
+satisfy the following two conditions:
+(i) they are isotopy functors, and
+(ii) they are finitary.
+A functor is an isotopy functor if it takes isotopy equivalences to weak homotopy equivalences
+(for the definition of isotopy equivalence see [MV15, Definition 10.2.2]). It is finitary if for every
+6
+
+monotone union �
+i Ui (where Ui ⊂ Ui+1 for i = 1, 2, ...) the canonical map from F(�
+i Ui) to
+holimi F(Ui) is a weak homotopy equivalence.
+If F is a ”half-good” contravariant functor (cofunctor), i.e. an isotopy functor which is not a
+finitary functor, then we need to tame this functor. We call V ∈ O(M) tame if V is the interior
+of a compact smooth codimension zero submanifold of M. As mentioned in [GKW01], property
+(ii) ensures that a good cofunctor F on O(M) is essentially determined by its behavior on tame
+open subsets of M.
+In particular, suppose F is a cofunctor from O(M) to Top having property (i). Then the functor
+defined by
+F #(V ) := holimtame U⊂V F(U)
+for V ∈ O(M) has also property (ii), i.e. F # is a good cofunctor on O(M). We call F # the
+taming of F.
+There exists a natural transformation F → F #. The map F(V ) → F #(V ) is an equivalence
+whenever either F or V is tame.
+The motivating example for the development of the manifold calculus of functors is the embedding
+functor.
+Definition 2.1. (Space of embeddings) Let M and N be smooth manifolds.
+• A smooth embedding of M in N is a smooth map f : M → N such that
+1. the map of tangent spaces
+Dxf : TxM → Tf(x)N
+is an injection for all x ∈ M, i.e. the derivative of f is a fiberwise injection, and
+2. f : M → f(M) is a homeomorphism.
+• The space of embeddings, Emb(M, N), is the subspace of the space of smooth maps
+from M to N consisting of smooth embeddings of M in N. The space Emb(M, N) is
+topologized using Whitney C∞-topology; for an explanation see [MV15, Appendix A.2.2]).
+An important example of a space of embeddings with very rich theory is the space of classical
+knots defined to be the space Emb(S1, R3).
+Definition 2.2. (Embedding functor)
+For a smooth n-dimensional manifold N, the embedding functor Emb(−, N) : O(M) → Top is
+a contravariant functor given by U �→ Emb(U, N).
+The contravariance follows from the fact that an inclusion of open subsets of a manifold M gives
+a restriction map of embedding spaces of manifolds.
+A related notion is the space of immersions Imm(M, N), which is a space of smooth maps
+f : M → N such that just the derivative of f is a fiberwise injection, (property 1.
+from
+Definition 2.1). If M is a compact manifold and f is an injective immersion M → N, then f is
+an embedding.
+The corresponding functor is the immersion functor Imm(−, N) : O(M) → Top given by
+U �→ Imm(U, N).
+Functors Emb(−, N) and Imm(−, N) are examples of good functors (see [Wei99] and [GKW01]).
+7
+
+The idea of the manifold calculus of functors is to approximate a good functor with simpler,
+polynomial functors.
+Definition 2.3. (Polynomial functor)
+A good contravariant functor F : O(M) → Top is called polynomial of degree ≤ k if for all
+U ∈ O(M) and for all pairwise disjoint closed subsets A0, ..., Ak ⊂ U, the (k + 1)-cube
+P(k + 1) → Top
+S �→ F(U −
+�
+i∈S
+Ai)
+is homotopy cartesian; equivalently, the map F(U) → holimS̸=∅ F(U − �
+i∈S Ai) is a homotopy
+equivalence. Here P(k + 1) is the poset category of all subsets of the set k + 1 = {1, ..., k + 1}
+with ⊂ as the relation of partial order. Its shape is an (k + 1)-dimensional cubical diagram.
+It is well known that a polynomial f : R → R of degree k such that f(0) = 0 is uniquely
+determined by its values on k distinct points. In analogy, a polynomial functor is completely
+determined by its values on the category of at most k open discs.
+[Mun10] provides more
+analogies between the ordinary calculus of functions and the manifold calculus of functors.
+More precisely, let Ok(M) be the full subcategory of M consisting of open subsets of M diffeo-
+morphic to ≤ k disjoint discs. We have the following theorem due to Weiss ([Wei99, Theorem
+5.1]).
+Theorem 2.4. Suppose F, G : O(M) −→ Top are good functors that are polynomials of degree
+≤ k. If T : F → G is a natural transformation that is an equivalence for all U ∈ Ok(M), then
+T is an equivalence for all U ∈ O(M).
+Example 2.5.
+• The functor U �→ Imm(U, N) is a polynomial of degree ≤ 1.
+• The functor U �→ Emb(U, N) is not a polynomial of degree ≤ k for any k.
+For the details, see [MV15, Example 10.2.10], [Wei99, Example 2.3], [Mun10, Examples 4.7 and
+4.8].
+Definition 2.6. (Polynomial approximations)
+For a good functor F, define for each U ∈ O(M) the kth polynomial approximation of F to be
+TkF(U) = holimV ∈Ok(U) F(V ).
+As Weiss proved in [Wei99, Theorems 3.9. and 6.1], such defined TkF is polynomial of degree
+≤ k. Also, higher derivatives of such defined polynomial functors vanish and derivatives of a
+functor and derivatives of its kth polynomial approximation agree up to kth degree, where the
+derivatives of functors are defined as follows:
+Definition 2.7. (Derivative of a functor)
+Let Dm
+1 , ..., Dm
+k be pairwise disjoint open discs in M. Define a k-cube of spaces by the rule
+S �→ F(�
+i/∈S Dm
+i ). We define the kth derivative of F at the empty set, denoted F (k)(∅), to be
+the total homotopy fiber of the cube S �→ F(�
+i/∈S Dm
+i ).
+8
+
+For example, the 1st derivative of embeddings are immersions. Also, the linearization of the space
+of embeddings is the space of immersions, namely there exists an equivalence T1 Emb(−, N) ≃
+Imm(−, N) ([Wei99]).
+For more details and intuition behind this, see Munson’s survey [Mun10]. For other relevant
+results, see [MV15, Theorem 10.2.16] and [Wei99].
+The inclusion Ok−1(U) → Ok(U) induces a map TkF(U) → Tk−1F(U) and so we obtain a tower
+of functors, called the manifold calculus Taylor tower of F:
+(4)
+F(−)
+�❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥
+�
+�▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+▼
+�❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+❳
+T0F(−)
+· · ·
+�
+Tk−1F(−)
+�
+TkF(−)
+�
+· · ·
+�
+T∞F(−)
+�
+Here T∞F denotes the homotopy inverse limit of this tower. TkF is also called the kth stage of
+the Tower.
+By evaluating diagram (4) on U ∈ O(M), we get a diagram of spaces with maps between the
+stages that are fibrations. In particular, we can set U = M.
+Definition 2.8. (Layer)
+Define the kth layer of the manifold calculus Taylor tower of F to be the homotopy fiber of the
+map between two successive stages of the tower, that is,
+LkF = hofiber(TkF → Tk−1F).
+We need to work here with a based Taylor tower. It can be accomplished by choosing a basepoint
+in the space F(M) which then also bases the spaces TkF(U) for all k and U.
+One of the fundamental results, which is a consequence of the Classification of homogeneous
+functors theorem ([Wei99, Theorem 8.5], see also [MV15, Theorem 10.2.23 and Proposition
+10.2.26]) is the following
+Proposition 2.9. For a good functor F defined on m-dimensional manifolds, if the cube
+S �→ F
+
+�
+k\S
+Dm
+
+
+is ck-cartesian, then the map TkF(M) → Tk−1F(M) is ck − km-connected. More generally, if
+U has handle dimension j, then the map TkF(U) → Tk−1F(U) is (ck − kj)-connected.
+For the definition of handle dimension, see [MV15, Appendix A.2.1].
+It follows that the Taylor tower of F converges intrinsically at M if the number ck−mk approaches
+∞ as k approaches ∞.
+2.3. Spectra. The subject of this paper is a functor that represents homology. We had a choice
+between working with chain complexes and the singular chains functor, or working with spectra
+and using smash product with the Eilenberg-MacLane spectrum to represent homology. We chose
+the latter.
+9
+
+We adopt a naive, old-fashioned view of spectra as sequences of spaces equipped with structure
+maps between them.
+Definition 2.10. (Spectrum)
+A spectrum E is a sequence of based spaces {En}n∈N0 together with basepoint-preserving maps
+(called structure maps)
+(5)
+ΣEn → En+1,
+or, equivalently, the maps
+(6)
+En → ΩEn+1,
+where Σ and Ω denote suspension and loop space, respectively.
+If the maps (6) are weak equivalences, then E is called an Ω-spectrum.
+Each En from an
+Ω-spectrum is called an infinite loop space.
+Example 2.11. (Eilenberg-MacLane spectrum)
+Let n be an arbitrary positive integer and G be an arbitrary group, abelian for n > 1. Then there
+exists a CW complex X such that
+(7)
+πn(X) ∼= G and πk(X) is trivial for k ̸= n.
+A topological space X with property (7) is called an Eilenberg-MacLane space K(G, n). For
+example, K(Z, 1) ≃ S1.
+For an abelian group G, the Eilenberg-MacLane spectrum, denoted by HG, is defined to be the
+spectrum {En}n∈N0 with En = K(G, n + 1) and maps
+(8)
+K(G, n + 1) → ΩK(G, n + 2).
+The maps (8) are weak equivalences, hence HG is an Ω-spectrum.
+Since for a spectrum E there exist maps
+πi+n(En) → πi+n+1(En+1)
+(for details, see [Hat02, Section 4.F]), it makes sense to define the ith homotopy group of the
+spectrum E as
+πi(E) = colimn πi+n(En).
+Definition 2.12. A map of spectra f : E → F is a collection of maps
+fn : En → Fn, n ≥ 0
+that commute with the structure maps in E = {En} and F = {Fn}.
+Taking spectra as objects and maps of spectra as morphisms we can define the category of
+spectra. It is denoted by Spectra.
+A spectrum can be smashed with a pointed space.
+Definition 2.13. Let E = {En} be a spectrum and X be a based space. The spectrum E ∧ X
+is defined by
+(E ∧ X)n = En ∧ X.
+10
+
+Since Σ(En ∧ X) ∼= (ΣEn) ∧ X, the structure maps in the spectrum E ∧ X are the products of
+structure maps in E and the identity map. For a spectrum E∧X the homotopy groups πi(E∧X)
+are the groups colimn πi+n(En∧X). These groups define a generalized reduced homology theory,
+determined by E.
+The following result is a consequence of Proposition 4F.2 in [Hat02]. See also [Whi62] for more
+details on representing generalized homology theories with spectra.
+Proposition 2.14. For the Eilenberg-MacLane spectrum HZ there exists an isomorphism
+πi(X ∧ HZ) ∼= �Hi(X; Z).
+If a spectrum E = {En}n≥0 is an Ω-spectrum, then πn(E) is
+πn(E) =
+�
+πn(E0),
+for n ≥ 0
+π0(E−n),
+for n ≤ 0
+Let us note that smash product with a spectrum can be extended from pointed to unpointed
+spaces.
+Definition 2.15. Let E be a spectrum and X an unpointed space. Define the smash product
+of E and X to be the homotopy fiber of the map
+E ∧ X+ → E
+induced by the canonical map X+ → S0.
+For any choice of basepoint in X, there is a canonical equivalence between the new and the old
+definition E ∧ X. But the new definition does not depend on a choice of basepoint. This is a
+variant of the fact that reduced homology can be defined as relative homology to a basepoint,
+but also can be defined independently of basepoint, using the augmented chain complex.
+However, it is also important to note that without a choice of basepoint in X, there is no natural
+map X → Ω∞HZ ∧ X representing the Hurewicz homomorphism. Such a map is defined only
+with a choice of basepoint.
+We can assume that each spectrum is an Ω-spectrum up to weak equivalence. Precisely, the
+following result holds.
+Proposition 2.16. Every spectrum is weakly equivalent to an Ω-spectrum.
+If two spectra E and F are weak equivalent, we write E ≃ F.
+Operation Σ∞ which assigns to a based space X its suspension spectrum Σ∞X, defined by
+En = ΣnX with identities as structure maps, is a functor
+Σ∞ : Top∗ → Spectra.
+Its adjoint functor
+Ω∞ : Spectra → Top∗
+is defined to be the functor which takes a spectrum E = {En}n≥0, then replaces it by an
+equivalent Ω-spectrum F = {Fn}n≥0 (which exists using proposition 2.16) and finally picks off
+the first place F0. In short, Ω∞(E) = F0 where F = {Fn}n≥0 ≃ E. This F0 is an infinite loop
+space, which explains the notation.
+11
+
+It follows from the results and comments above that nth homotopy group of a spectrum E equals
+the nth homotopy group of the space Ω∞(E).
+Finally, let us mention that in addition to the smash product of a spectrum with a space, there is
+a very important notion of smash product of spectra. For our purposes, the most naive version
+of the construction suffices. Given two spectra E = {En} and F = {Fn}, we define their smash
+product E ∧ F by the formulas (E ∧ F)2n = En ∧ Fn, and (E ∧ F)2n+1 = En+1 ∧ Fn, with
+the structure maps being induced from the structure maps in E and F in the obvious way. The
+sphere spectrum is the unit (up to homotopy) for this smash product.
+One feature of smash product of spectra that plays a role in this paper is that unlike smash
+product of spaces, smash product with a fixed spectrum commutes with finite homotopy limits
+of spectra. More generally, it commutes with homotopy limits over a category whose classifying
+space is compact. This is discussed in some detail in [LRV03]. The significance for us is that
+if F is a good presheaf of spectra on M, and E is a fixed spectrum, then there are natural
+equivalences
+E ∧ TkF ≃ TkE ∧ F
+and
+E ∧ LkF ≃ LkE ∧ F.
+3. The homological Taylor tower for reduced r-immersions in Rn
+The main goal of this paper is to give a convergence result about the homological Taylor tower
+for the space of r-immersions of a smooth manifold M in Rn. As is often the case, when studying
+the homological tower, it is convenient to replace the functor of r-immersions by r-immersions
+��modulo immersions”. This enables us to express the layers in the Taylor tower in terms of
+r-configuration spaces.
+Let M be a smooth manifold. Assume that a basepoint in the space Imm(M, Rn) is chosen, and
+therefore the functor U �→ Imm(U, Rn) is a presheaf of pointed spaces on M. Recall that for
+U ⊂ M, rImm(U, Rn) denotes the homotopy fiber of the map rImm(U, Rn) → Imm(U, Rn).
+Let HZ denote the Eilenberg-Mac Lane spectrum. The functor
+X �→ HZ ∧ X
+represents reduced homology, in the sense that there is a natural isomorphism
+(9)
+π∗(HZ ∧ X) ∼= �H∗(X; Z).
+Furthermore, recall that the functor can be extended to unpointed spaces, by defining HZ ∧ X
+for unpointed X to be the homotopy fiber of the map HZ ∧ X+ → HZ. In this paper we study
+the following functor
+HZ ∧ rImm(−, Rn):
+O(M)
+→
+Spectra
+U
+�→
+HZ ∧ rImm(U, Rn)
+This functor is representing the homology of rImm(−, Rn).
+12
+
+Remark 3.1. Instead of using spectra and the functor HZ ∧ − to represent homology, we could
+have used chain complexes and the singular chains functor. One reason for choosing spectra is
+their topological nature. The category of spectra, and of HZ-module spectra, is tensored and
+cotensored over topological spaces, while the category of chain complexes is not. Of course,
+this is a minor technical issue that can be overcome, but anyway it was one reason for us to
+work with HZ-modules rather than chain complexes. Another reason is that working with HZ-
+modules readily points to generalizations. In particular, most of our results about the functor
+HZ ∧ rImm(−, Rn) can be extended to the functor Σ∞rImm(−, Rn), which in turn can be used
+to obtain information about the unstable Taylor tower of rImm(−, Rn).
+Remark 3.2. In [GKW01] and [Wei04], Goodwillie, Weiss and Klein point out that for a con-
+travariant functor F : O(M) → Top, the cofunctor λJF given by
+U �→ F(U)+ ∧ J
+for a fixed spectrum J is only ”half-good”, even if F is good. Namely, it is an isotopy functor
+but it fails to be finitary. As mentioned in Section 2.2, to fix this they suggest to use the taming
+of λJF. We will denote the taming of a functor such as λJF by λJF #. The functor λJF # is
+a good cofunctor, and there is a natural transformation λJF → λJF #, which is an equivalence
+when evaluated on a tame subset of M, where by a tame subset we mean an open subset which
+is diffeomorphic to the interior of a compact manifold with boundary. From now on, whenever
+we write HZ ∧ rImm(−, Rn) we really mean the taming of this functor. In practice it makes no
+difference since we only are interested in evaluating our functors on tame manifolds.
+So we need to figure out the connectivity of the kth layer of the Taylor tower for the space
+HZ∧rImm(M, Rn). By Proposition 2.9, this is determined by the homotopy fiber of the cubical
+diagram, indexed by subsets of {1, . . . , k},
+S �→ HZ ∧ rImm
+
+�
+k\S
+Dm, Rn
+
+ .
+There is a natural map rImm(�
+k\S Dm, Rn) → rConf(k \ S, Rn), which is the composition of
+the natural map into rImm(�
+k\S Dm, Rn), followed by evaluation at the centers of the discs.
+By the main result of [AˇS22], this map is an equivalence. It follows that the connectivity of the
+layers of HZ ∧ rImm(M, Rn) is determined by the connectivity of the total fiber of the cubical
+diagram
+S �→ HZ ∧ rConf(k \ S, Rn).
+To analyze the total fiber of this cube, we need to review some facts about the homology of
+r-configuration spaces. This will be done in the next section.
+4. r-configuration spaces in Rn as complements of subspace arrangements
+We saw in the previous section that the convergence of the Taylor tower of the functor HZ ∧
+rImm(−, Rn) is determined by the homology of r-configuration spaces rConf(k, Rn). These con-
+figuration spaces can be interpreted as the complement of an arrangement of subspaces of (Rn)k.
+The combinatorics and topology (in particular, homology and cohomology) of subspace arrange-
+ments and their complements are well studied. Some of main references are [OS80], [GM80],
+[GM83a], [GM83b], [BEZ90]. In particular, the (co)homology of r-configuration was studied from
+13
+
+this perspective first by Bj¨orner and Welker in [BW95], and by a number of people after that.
+In this section we review a qualitative description of the cohomology of r-configuration spaces,
+based on the Goresky-MacPherson formula. We will also describe the effect on cohomology of
+restriction maps between configuration spaces.
+Recall that an r-configuration space of k points in Rn is defined to be the space
+rConf(k, Rn) = {(v1, ..., vk) ∈ (Rn)k : ∄1 ≤ i1 < · · · < ir ≤ k such that vi1 = ... = vir}.
+The space rConf(k, Rn) is an example of the complement of a subspace arrangement. Let us
+now recall some formal definitions.
+Definition 4.1. Suppose I is an r-tuple of integers I = (i1, . . . , ir), where 1 ≤ i1 < · · · < ir ≤ k.
+Let us denote the set of all such r-tuples by
+�k
+r
+�
+. De���ne
+AI = {(v1, . . . , vk) ∈ (Rn)k | vi1 = · · · = vir}.
+Let A =
+�
+AI | I ∈
+�k
+r
+��
+. When we need to make the set k explicit, we write Ak. More generally,
+for any set T define AT to be the set of “r-equal” diagonals in (Rn)T.
+Note that one can identify rConf(k, Rn) with the complement of the union of the AIs:
+rConf(k, Rn) = (Rn)k \
+�
+I∈(k
+r)
+AI.
+Example 4.2.
+• If k < r, rConf(k, Rn) ∼= (Rn)k ≃ ∗
+• If k = r, rConf(k, Rn) ∼= (Rn)r − ∆ ≃ S(r−1)n−1, where ∆ is the thin diagonal in (Rn)r
+and S(r−1)n−1 is the sphere of dimension (r − 1)n − 1.
+The collection A of linear subspaces of Rnk is an example of a subspace arrangement. Recall
+that the intersection lattice of A is the poset LA consisting of all the intersections AI1 ∩· · ·∩AIt
+of elements of A, ordered by reverse inclusion. We include in LA the “empty intersection” of
+AIs, which is Rnk. The space Rnk is the minimal element of LA. It will be denoted by ˆ0. The
+maximal element of LA is the intersection of all the AI, which, assuming k ≥ r, is the diagonal
+copy of Rn in Rnk. We denote the maximal elements of LA by ˆ1.
+The poset LA is isomorphic to the poset Πk,r of partitions of {1, . . . , k} whose every block is
+either a singleton or contains at least r elements. We call elements of Πk,r r-equal partitions
+of {1, . . . , k}. The partitions are ordered from finer to coarser. The isomorphism Πk,r → LA
+sends a partition λ of {1, . . . , k} to the space of k-tuples of vectors (v1, . . . , vk) ∈ (Rn)k with
+the property that vi = vj whenever i and j are in the same block of λ. Equivalently, one can
+say that λ is sent to the space of functions from k to Rn that are constant on each block of λ.
+From now on we will identify the posets LA and Πk,r.
+Because LA is a partially ordered set, we can define the open interval (x, y) in LA to be the set
+(x, y) = {z ∈ LA | x < z < y}.
+Definition 4.3. The order complex ∆(x, y) of an open interval (x, y) in LA, is the abstract
+simplicial complex whose vertices are the elements of (x, y) and whose p-simplices are the chains
+x0 < ... < xp in (x, y).
+14
+
+Let �Hi(x, y) denote the ith reduced simplicial homology group of ∆(x, y) with integer coefficients.
+Similarly, �H
+i(x, y) denotes the ith reduced cohomology group of ∆(x, y).
+The (reduced) cohomology groups of the space rConf(k, Rn) = Rnk\�
+I∈(k
+r) AI can be described
+in terms of (reduced) homology groups of the order complex of intervals in the intersection lattice
+of A. This is known as the Goresky-MacPherson formula. For the original proof of the Goresky-
+MacPherson formula by means of stratified Morse theory see [GM88, Part III]. An elementary
+proof was given by Ziegler and ˇZivaljevi´c in [ZˇZ93]. For the original calculation of the cohomology
+rConf(k, Rn) using the Goresky-MacPherson formula see [BW95]. Here is the statement, in the
+case relevant to us.
+Theorem 4.4 (Special case of Goresky-MacPherson formula). There is an isomorphism
+(10)
+�H
+i(rConf(k, Rn)) ∼=
+�
+x∈L>ˆ0
+A
+�Hcodim(x)−2−i(ˆ0, x)
+Here, the direct sum is indexed by all x ̸= ˆ0 in LA, and codim(x) is the codimension of the space
+x as the subspace of Rnk.
+For each diagonal x ∈ LA, let c(x) denote the number of components of the partition of k which
+determines the diagonal x. Obviously, dimension of x in (Rn)k is dim(x) = n · c(x), so
+(11)
+codim(x) = n(k − c(x)).
+The following easy example of 3-configuration spaces of 4 points illustrates the application of
+formula (10).
+Example 4.5. Let A is the set of all (at least 3)-diagonals in (Rn)4. Then 3 Conf(4, Rn) =
+(Rn)4 − A. The intersection lattice LA of A is pictured in Figure 1. Using Theorem 4.4, we find
+that for every n > 1,
+H0(3 Conf(4, Rn)) ∼= Z,
+H2n−1(3 Conf(4, Rn)) ∼= Z4,
+H3n−2(3 Conf(4, Rn)) ∼= Z3,
+and other cohomology groups are trivial. For n = 1, the formula is still valid, except that in this
+case 2n−1 = 3n−2 = 1, so the two cohomology groups add together. So H0(3 Conf(4, R)) ∼= Z
+and H1(3 Conf(4, R)) ∼= Z7. For n = 1, 2, the cohomology of 3 Conf(4, Rn) can be read off the
+tables at the end of [BW95].
+For the purpose of analysing the layers in the homological Taylor tower for r-immersions it also is
+desirable to know the effect of restriction maps between r-configuration spaces on cohomology.
+Suppose we have a subset T ⊂ {1, . . . , k}. Then we have a restriction map rConf(k, Rn) →
+rConf(T, Rn). We want to describe the induced homomorphism on cohomology, in terms of
+formula (10). The inclusion T ֒→ {1, . . . , k} induces an inclusion of the poset of r-equal partitions
+of T into the poset of r-equal partitions of {1, . . . , k}, by making each element of {1, . . . , k} \ T
+into a singleton. Notice that for every r-equal partition of T, the codimension of the corresponding
+diagonal is the same whether it is considered a diagonal in (Rn)T or in (Rn)k. This is so because
+the codimension of a diagonal determined by a partition is determined by the difference between
+15
+
+(1)(2)(3)(4)
+(1)(2, 3, 4)
+(1, 2, 3, 4)
+(2)(1, 2, 3)
+(3)(1, 2, 4)
+(4)(1, 2, 3)
+Figure 1. Intersection lattice for 3 Conf(4, Rn), also known as Π4,3
+the cardinality of the set and the number of blocks of the partition, by formula (11). This number
+remains unchanged if one adds some singletons to a partition. Thus we have a homomorphism
+(12)
+�
+x∈L>ˆ0
+AT
+�Hcodim(x)−2−i(ˆ0, x) →
+�
+x∈L>ˆ0
+Ak
+�Hcodim(x)−2−i(ˆ0, x)
+which is defined by the inclusion L>ˆ0
+AT ֒→ L>ˆ0
+Ak, and uses the fact that for every x ∈ L>ˆ0
+AT , the
+number codim(x) is the same whether x is considered an element of L>ˆ0
+AT or of L>ˆ0
+Ak.
+Lemma 4.6. The homomorphism �H
+i(rConf(T, Rn)) → �H
+i(rConf(k, Rn)) corresponds, under
+the isomorphism (10), to the homomorphism (12) that we just described.
+Proof. This follows easily from the fact that the Goresky-MacPherson formula is natural with
+respect to inclusions of subarrangements [Hu94, Corollary 2.1]
+□
+5. Total fiber of a retractive cubical diagram
+In general homotopy groups do not commute with total homotopy fibers of cubical diagrams.
+In this section we will show that for a class of cubes that we call retractive they do commute.
+More precisely, we show that for retractive cubes, the homotopy groups of the total fiber are
+canonically isomorphic to the total kernel of the cube of homotopy groups.
+Suppose we have a two-dimensional cubical diagram of spaces or spectra
+(13)
+E∅
+i∅,1 �
+i∅,2
+�
+E1
+i1,12
+�
+E2
+i2,12
+� E12
+16
+
+Suppose that all the maps in the square (13) have homotopy sections, so that the square of
+sections
+E12
+s12,1 �
+s12,2
+�
+E1
+s1,∅
+�
+E2
+s2,∅
+� E0
+commutes up to homotopy, and so that the following mixed square
+E2
+i2,12 �
+s2,∅
+�
+E12
+s12,1
+�
+E0
+i∅,1
+� E1
+also commutes up to homotopy. Note that the vertical maps in the mixed square are sections,
+while the horizontal maps are from the original square.
+Let us call a square (13) with such sections a retractive square.
+More generally, let us define a retractive cubical diagram as follows.
+Definition 5.1. Let χ be a k-dimensional cubical diagram. We say that χ is retractive if for
+every U ⊂ {1, . . . , k} and every i /∈ U, the map χ(U) → χ(U ∪ {i}) has a homotopy section,
+the cube of sections commutes up to homotopy, and furthermore whenever U ⊂ {1, . . . , k}, and
+i, j ∈ {1, . . . , k} \ U, with i < j, the following mixed square commutes up to homotopy
+χ(U ∪ {j})
+�
+�
+χ(U ∪ {i, j})
+�
+χ(U)
+� χ(U ∪ {i})
+.
+Lemma 5.2. Let χ be a retractive k-dimensional cubical diagram of spectra. Let E∗ be any
+homology theory, and let E∗ be a cohomology theory. Then E∗(tfiber χ) (resp. E∗(tfiber χ)) is a
+direct summand of E∗(χ(∅)) (resp. of E∗(χ(∅))). Moreover, the following natural homomorphism
+is an isomorphism:
+E∗(tfiber χ)
+∼
+=−→ tkernel (E∗χ) .
+Similarly, there is a natural isomorphism
+tcokernel(E∗χ)
+∼
+=−→ E∗(tfiber χ).
+Proof. We will prove the claim for homology. The proof of the cohomological statement is the
+same, reversing all arrows. The proof is by induction on k, starting with with the case k = 1,
+which is elementary and well-known. Let us review it anyway. A retractive 1-dimensional cube
+is a map χ(∅) → χ(1), together with a homotopy section χ(1) → χ(∅). The total fiber of
+the cube is the homotopy fiber of the map χ(∅) → χ(1). By homotopy section we mean that
+the composition χ(1) → χ(∅) → χ(1) is a weak equivalence. It follows that the composition
+E∗χ(1) → E∗χ(∅) → E∗χ(1) is an isomorphism. From here it readily follows that the long
+exact sequence in E∗ associated with the fibration sequence tfiber χ → χ(∅) → χ(1) splits as a
+17
+
+direct sum of split short exact sequences in each degree. Furthermore it readily follows that the
+following homomorphisms are isomorphisms
+E∗ tfiber χ
+∼
+=−→ ker (E∗χ(∅) → E∗χ(1))
+∼
+=−→ coker (E∗χ(1) → E∗(χ(∅))) .
+Now suppose the lemma holds for cubes of dimension less than k and let χ be a retractive
+cube of dimension k. Let χ1 and χ2 be k − 1-dimensional cubes defined by χ1(U) = χ(U)
+and χ2(U) = χ(U ∪ {k}). Then χ can be identified with the natural map of cubes χ1 → χ2.
+The cubes χ1 and χ2 are retractive, so by induction hypothesis, the lemma holds for them. The
+retractions do not quite define a map of cubes χ2 → χ1, because we only assumed that the mixed
+squares commute up to homotopy. But they do define a homomorphism of cubes E∗χ2 → E∗χ1,
+which is a section of the homomorphism of cubes E∗χ1 → E∗χ2. We have the following diagram
+E∗ tfiber χ
+E∗ tfiber χ1
+E∗ tfiber χ2
+tkernel E∗χ2
+tkernel E∗χ1
+tkernel E∗χ2
+∼
+=
+∼
+=
+∼
+=
+The top row is induced by applying E∗ to a fibration sequence of spectra. The vertical homomor-
+phisms are isomorphisms by induction hypothesis. It follows that the upper right homomorphism
+is a split surjection, and the top row is a split short exact sequence in each dimension. Fur-
+thermore, E∗ tfiber χ maps isomorphically onto the kernel of the bottom right map, which is
+tkernel E∗χ.
+□
+6. The cube of r-configuration spaces is retractive
+Lemma 6.1. The k-cube of spaces
+S �→ rConf(k \ S, Rn)
+is retractive for n ≥ 2.
+Proof. Let T be a finite set and suppose x /∈ T. Our first step it to construct a section to the
+restriction map
+rT∪{x},T : rConf(T ∪ {x}, Rn) → rConf(T, Rn).
+Let p1: Rn → R be projection onto the first coordinate. Define a map
+sT,T∪{x}: rConf(T, Rn) → rConf(T ∪ {x}, Rn)
+as follows. An element of rConf(T, Rn) is a function f : T → Rn with the property that no r
+points of T go to the same point. Extend f to a function from T ∪ {x} by sending x to
+(max{p1f(t) | t ∈ T} + 1, 0, . . . , 0).
+In words, x is sent to the point of Rn whose first coordinate is one more than the maximal
+first coordinate of the existing points, and all other coordinates are zero. It is clear that the
+image of x is different from all the other points in the configuration.
+Thus if f was an r-
+immersion, then the resulting map T ∪ {x} → Rn is still an r-immersion. We have defined a
+18
+
+map sT,T∪{x}: rConf(T, Rn) → rConf(T ∪ {x}, Rn). It is clear that the following composition
+is the identity (not even just homotopic to the identity but is the actual identity map)
+rConf(T, Rn)
+sT,T ∪{x}
+−−−−−→ rConf(T ∪ {x}, Rn)
+rT ∪{x},T
+−−−−−→ rConf(T, Rn).
+It follows that sT,T∪{x} is a section of rT∪{x},T. Next, we need to show that whenever x, y /∈ T,
+the following diagram commutes up to homotopy
+rConf(T, Rn)
+�
+�
+rConf(T ∪ {x}, Rn)
+�
+rConf(T ∪ {y}, Rn)
+� rConf(T ∪ {x, y}, Rn)
+It is for this step that we need to assume n ≥ 2.
+Let f : T → Rn represent an element
+of rConf(T, Rn). The images of f in rConf(T ∪ {x, y}, Rn) under the two ways around the
+diagram are two extensions of f from T to T ∪ {x, y}.
+One of the extensions sends x to
+(max{p1f(t) | t ∈ T} + 1, 0, . . . , 0), and sends y to (max{p1f(t) | t ∈ T} + 2, 0, . . . , 0). The
+other extension does the same thing, with x and y switched. It is clear that one can write a
+homotopy between the two maps, by swapping the images of x and y along a circle in the plane
+spanned by the first two coordinates of Rn.
+Finally we need to check that the following mixed square commutes up to homotopy
+rConf(T ∪ {x}, Rn)
+�
+�
+rConf(T, Rn)
+�
+rConf(T ∪ {x, y}, Rn)
+� rConf(T ∪ {y}, Rn)
+.
+This, too, is clear. In fact, it is easy to check that there is a well-defined straight line homotopy
+between the two maps around the square.
+We have shown that the section maps that we have defined make the cube of r-configuration
+spaces and restriction maps between them into a retractive cube.
+□
+7. Connectivity of the cube of (co)homologies of r-configuration spaces
+We have seen that the cube of spaces S �→ rConf(k \ S, Rn), where S ranges over the subsets
+of {1, . . . , k} is retractive (Lemma 6.1). It follows that the cube of spectra obtained by applying
+the suspension spectrum functor to it, i.e., the cube
+(14)
+S �→ Σ∞ rConf(k \ S, Rn),
+is also retractive.
+Our goal is to analyse how cartesian is the cube S �→ HZ∧Σ∞ rConf(k\S, Rn). Smash product
+commutes with total fibers of cubical diagrams of spectra. Therefore, the answer is the same
+as for the cubical diagram (14). However, we want to use the description of the cohomology
+of r-configuration spaces given by the Goresky-MacPherson formula. The following lemma says
+that the homology and cohomology groups of the relevant spectrum are isomorphic.
+Lemma 7.1. The homology and cohomology groups of the total fiber of (14) are (non-canonically)
+isomorphic.
+19
+
+Proof. It is known, for example by the results of [BW95], that the homology groups of the space
+rConf(k, Rn), and therefore also of the suspension spectrum of this space, are finitely generated
+free abelian groups. Since the cube Σ∞ rConf(k \ S, Rn) is retractive, it follows by Lemma 5.2
+that the homology of the total fiber of the cube Σ∞ rConf(k \ S, Rn) is a direct summand of
+the homology of Σ∞ rConf(k, Rn). Therefore, the homology groups of the total fiber are also
+finitely generated free abelian groups. Therefore they are isomorphic to the cohomology groups
+of the total fiber, by the universal coefficients theorem.
+□
+It follows that the homological connectivity of the total fiber of (14) is equivalent to the coho-
+mological connectivity. Next, we give a qualitative description of the cohomology of the total
+fiber, in the style of Theorem 4.4.
+Let Π≥r(k) denote the set partitions of k with the property that each component has at least r
+elements (i.e., elements of Πk,r without singletons).
+Lemma 7.2. The i-th cohomology group of the total fiber of the cube (14) is isomorphic to the
+following direct sum:
+(15)
+�
+x∈Π≥r(k)
+�Hcodim (x)−2−i(ˆ0, x)
+Proof. The cube (14) is retractive. Using the cohomological part of Lemma 5.2, we conclude
+that the i-th cohomology of the total fiber is isomorphic to the cokernel of the homomorphism
+k
+�
+i=1
+�H
+i rConf(k \ {i}, Rn) → �H
+i rConf(k, Rn).
+By Lemma 4.6, this homomorphism can be identified with the following homomorphism
+(16)
+k
+�
+i=1
+�
+x∈L>ˆ0
+Ak\{i}
+�Hcodim(x)−2−i(ˆ0, x) →
+�
+x∈L>ˆ0
+Ak
+�Hcodim(x)−2−i(ˆ0, x)
+The homomorphism maps each summand in the source isomorphically onto a summand in the tar-
+get (some summands in the source go to the same summand in the target, so the homomorphism
+is not injective). The image of the homomorphism is the sum of terms corresponding to r-equal
+partitions with at least one singleton. The cokernel is the direct sum of terms corresponding to
+r-equal partitions that do not have a singleton.
+□
+It follows from Lemma 7.2 that to find how cartesian the cube (14) is, we need to find the
+smallest i for which the homology group
+(17)
+�Hcodim(x)−2−i(ˆ0, x)
+is non-trivial for some x ∈ Π≥r(k).
+Throughout this section, let x be a partition of {1, . . . , k} where each block has at least r
+elements. Recall that c(x) denotes the number of blocks of x. Note that if k1, . . . , kc(x) are the
+sizes of the blocks of x, then k1 + · · · + kc(x) = k. Let [ˆ0, x] be the closed interval in Πk,r.
+Lemma 7.3. Let x be as above. Suppose x has c(x) blocks, of sizes k1, . . . , kc(x). Then there
+is an isomorphism of posets
+[ˆ0, x] ∼= Πk1,r × · · · × Πkc(x),r.
+20
+
+Proof. The interval [ˆ0, x] consists of r-equal partitions of {1, . . . , k} that are refinements of x.
+This is the same data as an r-equal partition of each block of x, which is the same as an element
+of Πk1,r × · · · × Πkc(x),r.
+□
+Given a poset P with a minimum and maximum element, let P0 be the poset P with the minimum
+and maximum removed.
+Corollary 7.4. Let x be as in the previous lemma. Then there is a homotopy equivalence (∗
+denotes joint)
+|∆(ˆ0, x)| ≃ Σc(x)−1|Π0
+k1,r| ∗ · · · ∗ |Π0
+kc(x),r|.
+Proof. This follows from the lemma, and the well-known fact that given two posets P and Q
+with minimum and maximum objects, there is a homotopy equivalence [Wal88, Theorem 5.1 (d)]
+|(P × Q)0| ≃ Σ|P0| ∗ |Q0|.
+□
+Lemma 7.5. Let x be as in the previous lemma and corollary. Then |∆(ˆ0, x)| is homotopy
+equivalent to a complex of dimension k−c(x)(r−1)−2. Furthermore, the homology of |∆(ˆ0, x)|
+in dimension k − c(x)(r − 1) − 2 is non zero.
+Proof. By the corollary, the space |∆(ˆ0, x)| is homotopy equivalent to Σc(x)−1|Π0
+k1,r| ∗ · · · ∗
+|Π0
+kc(x),r|. By the results of [BW95], |Π0
+k,r| is homotopy equivalent to a wedge of spheres, not all
+of the same dimension, and the top homology of this space occurs in dimension k − r − 1. It
+follows that the space Σc(x)−1|Π0
+k1,r|∗· · ·∗|Π0
+kc(x),r| is a wedge of spheres, with the top homology
+occurring in dimension
+c(x) − 1 + (k1 − r − 1) + · · · + (kc(x) − r − 1) + c(x) − 1 = k − c(x)(r − 1) − 2.
+□
+Example 7.6. If r ≤ k < 2r, there is only one summand x in (15) - this is the partition {k}, or
+in other words the thin diagonal. For this x, dim ∆(ˆ0, x) = k − r − 1.
+Now we can state and prove the main result of this section
+Proposition 7.7. When r ≤ n + 1, the cube (14) is k(n − 1) +
+� k
+r
+�
+(r − n − 1)-cartesian.
+When r ≥ n + 1, the cube (14) is k(n − 1) + r − n − 1-cartesian.
+Remark 7.8. Note that when r = n+1 both formulas say that the cube (14) is k(n−1)-cartesian.
+Proof. Given x, the smallest i for which the homology (17) might be non-trivial is one that
+satisfies codim(x)−2−i = dim ∆(ˆ0, x). Using Lemma 7.5 we have that the smallest i for which
+the total cokernel (15) might be non-trivial is one that satisfies
+codim(x) − 2 − i = k − c(x)(r − 1) − 2.
+Because codim(x) = n(k − c(x)) for x ∈ Π≥r(k), it follows that
+(18)
+i = k(n − 1) + c(x)(r − n − 1).
+21
+
+We have to see for which x this number i is the smallest possible. We distinguish between two
+overlapping cases, depending on the sign of r − n − 1.
+1) When r − n − 1 ≤ 0, i.e. when r ≤ n + 1, finding i as small as possible is the same as finding
+x ∈ Π≥r(k) with the biggest number c(x) of components. Since all components have to be of
+the size at least r, the largest number of them is attained when there is a maximum number of
+them of the size r. In that case, c(x) =
+� k
+r
+�
+, so the smallest i is
+i = k(n − 1) +
+�k
+r
+�
+(r − n − 1).
+So in this case, the cubical diagram (14) is k(n − 1) +
+� k
+r
+�
+(r − n − 1)-cartesian.
+2) When r − n − 1 ≥ 0, i.e. r ≥ n + 1, finding i as small as possible is the same as finding
+x ∈ Π≥r(k) with the smallest number c(x) of components. Thus we need c(x) to be equal to 1.
+This x is actually the thin diagonal in the space (Rn)k that corresponds to the partition {k} of
+k. In that case,
+i = k(n − 1) + r − n − 1,
+hence (14) is k(n − 1) + r − n − 1-cartesian.
+□
+8. Convergence result
+Let M be a smooth manifold of dimension m. Now we finally can calculate the connectivity of
+the map
+(19)
+TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn).
+Knowing that ck-connectivity of the total fiber of the cube (14) implies (ck−km+1)-connectivity
+of the map (19), we can find the conditions under which the Taylor tower converges, using results
+from Section 7. There are three different cases.
+1) For r − n − 1 < 0, the connectivity of the map (19) is
+(20)
+k(n − 1) +
+�k
+r
+�
+(r − n − 1) − 1 − mk + 1 = k(n − m − 1) +
+�k
+r
+�
+(r − n − 1)
+= k(n − m − 1) +
+�k
+r − k mod r
+r
+�
+(r − n − 1)
+= k
+�
+n − m − n
+r − 1
+r
+�
+− k mod r
+r
+(r − n − 1)
+= k
+�
+nr − 1
+r
+− m − 1
+r
+�
+− k mod r
+r
+(r − n − 1)
+where we noted that
+� k
+r
+�
+= k/r − (k mod r)/r. Note now that
+−k mod r
+r
+(r − n − 1)
+22
+
+is nonnegative since r − n − 1 < 0. This means that, as long as
+nr − 1
+r
+− m − 1
+r > 0,
+the connectivities increase with k.
+2) For r − n − 1 = 0, the connectivity of the map (19) is
+(21)
+k(n − 1) − 1 − mk + 1 = k(n − m − 1),
+which goes to +∞ as k −→ +∞ if n − m − 1 > 0.
+3) For r − n − 1 > 0, the connectivity of the map (19) is
+(22)
+k(n − 1) + r − n − 2 − mk + 1 = k(n − m − 1) + r − n − 1,
+which goes to +∞ as k −→ +∞ if n − m − 1 > 0.
+Thus we proved the following theorem.
+Theorem 8.1. (Homological convergence of the Taylor tower for r-immersions in Rn)
+Let M be an m-dimensional smooth manifold and Rn the n-dimensional Euclidean space. Assume
+n > 1. Let rImm(M, Rn) be the space of r-immersions of M in Rn. Consider the map
+pk : TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn).
+a) For r ≤ n + 1 the map pk is
+k
+�
+nr − 1
+r
+− m − 1
+r
+�
+− k mod r
+r
+(r − n − 1)
+-connected. The tower converges intrinsically if n > rm+1
+r−1 .
+b) For r ≥ n + 1 the map pk is k(n − m − 1) + r − n − 1-connected. The tower converges
+intrinsically if n > m + 1.
+Proof. Only the assertions regarding intrinsic convergence remain to be checked.
+The tower
+converges intrinsically if the connectivity of pk approaches ∞ with k. In the case r ≤ n+1, since
+(k mod r) is a bounded function of k, this is equivalent to the condition nr−1
+r
+− m − 1
+r > 0,
+which is the same as n > rm+1
+r−1 . In the case r ≥ n + 1, the formula for the connectivity of pk
+clearly tells us that the connectivity goes to ∞ if n > m + 1.
+□
+9. Comparing with the unstable tower
+In this section we will compare the layers, and the connectivities of the maps in the Taylor tower
+of HZ ∧ rImm(M, Rn) with those in the Taylor tower of the unstabilized functor rImm(M, Rn).
+We will show that roughly up to degree 2r − 1 the connectivities of the maps in the two towers
+are the same, and the first non-trivial homotopy groups of the layers are isomorphic.
+23
+
+In this section, let us assume that we chose a basepoint in rImm(M, Rn) rather than just in
+Imm(M, Rn), so that the presheaf rImm(−, Rn) takes values in pointed spaces. We have a
+diagram of presheaves
+(23)
+rImm(−, Rn)
+i←− rImm(−, Rn)
+s−→ Ω∞Σ∞rImm(−, Rn)
+h−→ Ω∞HZ ∧ rImm(−, Rn).
+The map i induces an equivalence of derivatives and layers beyond the first one. The map h
+induces the Hurewicz homomorphism. In particular, it induces a Hurewicz isomorphism on the
+first non-trivial homotopy group of each layer. We focus on the question for which k the map s,
+and therefore also h ◦ s, induces an isomorphism on the first nontrivial homotopy group of the
+k-th layer. When r = 2, the answer is known to be: only for k = 2. We show that for r > 2 the
+answer is: for all k ≤ 2r − 1, with a small caveat for r = 3.
+Theorem 9.1. Assume 0 < dim(M) < n, r > 2.
+For 1 < k < r, the following maps are equivalences:
+Tk rImm(M, Rn) → T1 rImm(M, Rn) ≃ Imm(M, Rn)
+and
+TkHZ ∧ rImm(M, Rn) → T1HZ ∧ rImm(M, Rn) ≃ ∗.
+For r ≤ k ≤ 2r − 1, the connectivity of the map Tk rImm(M, Rn) → Tk−1 rImm(M, Rn) is the
+same as the connectivity of the map TkHZ ∧ rImm(M, Rn) → Tk−1HZ ∧ rImm(M, Rn).
+When r = 3, k = 2r − 1 = 5, the map s, and therefore also h ◦ s in diagram (23), induces an
+epimorphism on the first non-trivial homotopy group of the k-th layer. In all other cases when
+r > 2, r ≤ k ≤ 2r − 1, the maps s and h ◦ s induce an isomorphism on the first non-trivial
+homotopy group of the k-th layer.
+Remark 9.2. The case k = r, r + 1 of the last assertion of the theorem can be obtained by
+comparing our Theorem 8.1 with the calculations done in [SˇSV20].
+Proof. The assertion that T1 rImm(M, Rn) ≃ Imm(M, Rn) follows from the fact that when
+M = Dm, the following maps are equivalences [AˇS22]
+Emb(Dm, Rn)
+���−→ rImm(Dm, Rn)
+≃−→ Imm(Dm, Rn),
+together with the fact that the functor Imm(−, Rn) is linear, at least on manifolds whose handle
+dimension is less than n.
+The assertion that both towers are constant for k < r follows from the fact that the derivatives of
+both functors vanish below degree r. Indeed, the k-th layer in the Taylor tower of rImm(M, Rn)
+is determined by the following k-dimensional cubical diagram
+S �→ rImm(
+�
+k\S
+Dm, Rm).
+By the result of [AˇS22], this cubical diagram is equivalent to the diagram
+S �→ L(Rm, Rn)k\S × rConf(k \ S, Rn).
+24
+
+Here L(Rm, Rn) is the space of injective linear maps from Rm to Rn. This is the “tangential
+data” of an immersion. When k > 1 the tangential data cancels out, and the last cube is as
+cartesian as the following cube
+(24)
+S �→ rConf(k \ S, Rn).
+On the other hand, the k-th layer in the Taylor tower of Ω∞Σ∞ rImm(M, Rn) is determined by
+the following k-dimensional cubical diagram
+(25)
+S �→ Ω∞Σ∞ rConf(k \ S, Rn).
+To prove the assertion about the connectivities of the maps in the two towers, we need to
+show that the cubical diagram (24) is as cartesian as the diagram (25) in the indicated cases.
+Furthermore, we want to prove that the map s in (23) induces an isomorphism/epimorphism on
+the first non-trivial homotopy groups of the total homotopy fibers in the appropriate cases.
+The map s induces the following map of cubical diagrams, indexed by the poset of subsets S ⊂ k,
+(26)
+rConf(k \ S, Rn)
+s−→ Ω∞Σ∞ rConf(k \ S, Rn).
+The spaces rConf(k \ S, Rn) are (r − 1)n − 2-connected. By Freudenthal suspension theorem,
+the maps (26) are 2(r−1)n−3-connected. On the other hand, both cubes are retractive cubes by
+Lemma 6.1. It follows that the homotopy groups of the total homotopy fibers of both cubes are
+isomorphic to the total kernels of the corresponding cubes of homotopy groups. Proposition 7.7
+tells us the connectivity of the total homotopy fiber of (25), and therefore also the connectivity
+of the total kernel of the corresponding cube of homotopy groups. If this connectivity is smaller
+than (resp. equals to) the connectivity of the maps in (26), then (26) induces an isomorphism
+(resp: an epimorphism) between the first non-trivial homotopy groups of the total homotopy
+fibers. So we have to check that the range provided by Proposition 7.7 is smaller than (or equals
+to) 2(r − 1)n − 3 in the cases indicated in the statement that we are trying to prove.
+Suppose first that r > n+ 1. In this case, Proposition 7.7 says that (25) is k(n−1) + r −n−1-
+cartesian. So we have to check that the inequality
+k(n − 1) + r − n − 1 < 2(r − 1)n − 3
+holds whenever k < 2r. Simplifying, we obtain the inequality
+k < (2n − 1)r − 3
+n − 1
+− 1.
+So it is enough to check the inequality
+2r ≤ (2n − 1)r − 3
+n − 1
+− 1.
+Multiplying by n − 1 we obtain the inequality
+2r(n − 1) ≤ (2n − 1)r − 3 − n + 1,
+which is equivalent to r ≥ n + 2, which is what we assumed.
+Now suppose that r ≤ n + 1. Then Proposition 7.7 says that (25) is k(n − 1) +
+� k
+r
+�
+(r − n − 1)-
+cartesian. So we have to check that the inequality
+k(n − 1) +
+�k
+r
+�
+(r − n − 1) < 2(r − 1)n − 3
+25
+
+holds when r ≤ k < 2r, with the exception that when r = 3, k = 5 it is in fact an equality. The
+reader can check that in this case we do indeed obtain the equality
+5(n − 1) +
+�5
+3
+�
+(3 − n − 1) = 4n − 3.
+In other cases, the assumption r ≤ k < 2r implies
+� k
+r
+�
+= 1. So we have to check the inequality
+k(n − 1) + r − n − 1 < 2(r − 1)n − 3.
+We can rewrite the inequality as follows
+k(n − 1) < (2r − 1)(n − 1) + r − 3,
+or equivalently
+k < 2r − 1 + r − 3
+n − 1.
+For r = 3 this inequality is equivalent to k < 5. For 3 < r ≤ n + 1, this holds for all k ≤ 2r − 1,
+as stated.
+□
+10. Further questions
+1. We gave conditions on the m and n that guarantee intrinsic convergence of the Taylor tower
+of HZ ∧ rImm(M, Rn). The next question is, what does the Taylor tower from Theorem 8.1
+converge to? It is natural to guess that whenever the Taylor tower converges intrinsically, it
+actually converges to HZ ∧ rImm(M, Rn).
+2.
+What can one say about the convergence of the Taylor tower for the unstable functor
+rImm(M, Rn)? The question of intrinsic convergence of the unstable tower might be tractable,
+and is a good place to start. One can use the methods of this paper to describe the layers
+of the functor Σ∞rImm(M, Rn). Given this, one can try to analyse the layers of the functor
+rImm(M, Rn) via the cobar construction
+cobar(Ω∞, Σ∞Ω∞, Σ∞rImm(M, Rn)),
+in the style of [AC11]. It is conceivable that one can use these methods to obtain conditions on
+m, n, and r that guarantee that the tower converges intrinsically.
+Then there is a question of what the tower actually converges to. Once again, it seems reasonable
+to guess that whenever the Taylor tower of a “natural” functor converges intrinsically, then it
+actually converges to the functor.
+3. What can one say about r-immersions into a general manifold N? In order to understand the
+layers of the tower of the functor Σ∞rImm(M, N) one needs to understand (the stable homotopy
+type of) the homotopy fiber of the map rConf(k, N) → Nk. For r = 2 this homotopy fiber was
+analysed in [Aro09], and it seems likely that a similar analysis can be done for general r.
+4. Construct interesting invariants/obstructions to existence of r-immersions, using the Taylor
+tower. In this paper we focused on situations where the connectivity of the k-th layer in the
+tower goes to infinity as k goes to infinity. But situations when the connectivity does not go to
+infinity also can be interesting. Of particular potential interest are situations where the layers are
+all either −1-connected or −2-connected. In the former case, the bottom homotopy groups of
+the layers give invariants, in the latter case they give obstructions to existence.
+26
+
+For example, it follows from Theorem 8.1 that when n = m+1 and r = n+1, then all the layers
+of HZ∧(n+ 1) Imm(M, Rn) are −1-connected. The 0-th homotopy groups of the layers should
+give invariants of r-immersions. In the case n = 2, and say M = S1, 3 Imm(S1, R2) is the space
+of smooth curves in R2 that do not have triple intersections. Spaces of such curves were studied
+quite intensely, starting with Arnol’d [Arn94, Tab96, Shu95]. In particular, Arnol’d developed the
+theory of finite type invariants for such curves. We expect these invariants to show up in the
+Taylor tower of HZ ∧ 3 Imm(S1, R2). In particular we speculate that the first non-trivial layer of
+the tower, which by Theorem 9.1 is the third layer, detects the “Strangeness” invariant, defined
+in [Arn94] and studied further in [Tab96] and [Shu95].
+References
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+furcations, volume 21 of Adv. Soviet Math., pages 33–91. Amer. Math. Soc., Providence, RI. 1994.
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+Department of Mathematics, Stockholm University
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+URL: pmf.unsa.ba/franjos
+28
+

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+arXiv:2301.13474v1  [math.NT]  31 Jan 2023
+GENERALIZED FRUIT DIOPHANTINE EQUATION AND
+HYPERELLIPTIC CURVES
+OM PRAKASH AND KALYAN CHAKRABORTY
+Abstract. We show the insolvability of the Diophantine equation axd −y2 −z2 +xyz −
+b = 0 in Z for fixed a and b such that a ≡ 1 (mod 12) and b = 2da − 3, where d is an
+odd integer and is a multiple of 3. Further, we investigate the more general family with
+b = 2da − 3r, where r is a positive odd integer. As a consequence, we found an infinite
+family of hyperelliptic curves with trivial torsion over Q. We conclude by providing some
+numerical evidence corroborating the main results.
+1. Introduction
+One of the earliest topics in number theory is the study of Diophantine equations. In
+the third century, Greek mathematician Diophantus of ‘Alexandria’ began this study. A
+polynomial equation of the form
+P(x1, x2, · · · , xn) = 0
+is known as a Diophantine equation. Finding all of its integer solutions, or all of the
+n−tuples (x1, x2, · · · , xn) ∈ Z that satisfy the above equation, is of prime interest. The
+main task is to investigate whether solutions exist for a given Diophantine equation. If
+they do, it would be the aim to know how many are there and how to find all. There
+are certain Diophantine equations which has no non zero integer solutions, for example,
+Fermat’s equation xn + yn = zn for n ≥ 3. The tenth of Hilbert’s 23 problems, which
+he presented in 1900, dealt with Diophantine equations. Hilbert asked, is there an al-
+gorithm to determine weather a given Diophantine equation has a solution or not? and
+Matiyasevich in 1970 answered it negatively.
+We investigate a class of Diophantine equations of the form axd−y2−z2+xyz−b = 0 for
+fixed a and b. Due to its emergence when attempting to solve an equation involving fruits,
+this type of Diophantine equations were given the name “Fruit Diophantine equation” by
+B. Sury and D. Majumdar [5] and they proved the following:
+2010 Mathematics Subject Classification. Primary: 11D41, 11D72. Secondary: 11G30.
+Key words and phrases. Diophantine equation, Quadratic residue, Elliptic curves, Hyperelliptic curves.
+1
+
+2
+OM PRAKASH AND KALYAN CHAKRABORTY
+Theorem 1.1. [5] The equation
+y2 − xyz + z2 = x3 − 5
+has no integer solution in x, y and z.
+Similar type of equations were previously studied by F. Luca and A. Togb´e. In particu-
+lar, Luca and Togb´e [4] studied the solution of the Diophantine equation x3+by+1−xyz =
+0 and later, Togb´e [7] independently studied the equation x3 + by + 4 − xyz = 0.
+As a consequence of Theorem 1.1 Majumdar and Sury proved the following:
+Theorem 1.2. [5] For any integer m, the elliptic curve
+Em : y2 − mxy = x3 + m2 + 5
+has no integral point.
+L. Vaishya and R. Sharma expanded on Majumdar and Sury’s work in [8]. A class of
+fruit Diophantine equations without an integer solution was found by them. In particular
+Vaishya and Sharma showed,
+Theorem 1.3. [8] For fixed integers a and b with a ≡ 1 (mod 12) and b = 8a − 3. The
+Diophantine equation
+ax3 − y2 − z2 + xyz − b = 0
+has no integer solution.
+Using Nagell-Lutz theorem [6] and Theorem 1.3 they got hold of an infinite family of
+elliptic curves with torsion-free Mordell-Weil group over Q.
+Theorem 1.4. [8] Let a and b be as in Theorem 1.3.
+• For any even integer m the elliptic curve
+Ee
+m,a,b : y2 = x3 + 1
+4m2x2 − a2 �
+m2 + b
+�
+has torsion-free Mordell-Weil group.
+• For any odd integer m the elliptic curve
+Eo
+m,a,b : y2 = x3 + m2x2 − 64a2 �
+m2 + b
+�
+has torsion-free Mordell-Weil group.
+
+GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES
+3
+We extend Vaishya and Sharma’s results [8] for higher exponents. We obtain a family
+of hyperelliptic curves, by carrying out some appropriate transformations. In 2013, D.
+Grant gave an analogue of Nagell-Lutz theorem for hyperelliptic curves [3], using which
+we conclude that the Mordell-Weil group of each member of the corresponding family of
+hyperelliptic curves is torsion-free.
+2. Insolvability
+Here we state and prove the main theorem and derive a couple of interesting corollaries.
+We end this section by looking into a couple of examples.
+Theorem 2.1. The equation
+axd − y2 − z2 + xyz − b = 0
+has no integer solutions for fixed a and b such that a ≡ 1 (mod 12) and b = 2da − 3,
+where d is an odd integer and divisible by 3.
+Proof. Consider
+axd − y2 − z2 + xyz − b = 0.
+(2.1)
+If possible, let (x, y, z) be an integer solution of (2.1). Let us fix x = α. Then (2.1) can
+be re-written as,
+y2 + z2 + b = aαd + αyz.
+(2.2)
+We consider the cases of α being even or odd separately.
+Case 1. If α is even. Then, we write (2.2) as:
+�
+y − αz
+2
+�2
+−
+�α2
+4 − 1
+�
+z2 = aαd − b
+(2.3)
+and set Y = y − αz
+2 , β = α
+2 and z = Z. Thus (2.3) becomes,
+Y 2 −
+�
+β2 − 1
+�
+Z2 = aαd − b = 2dβda − b.
+(2.4)
+• If β is even, say β = 2n for some integer n, then reducing (2.4) modulo 4 gives,
+Y 2 + Z2 ≡ 3
+(mod 4),
+(2.5)
+which is not possible in Z/4Z.
+
+4
+OM PRAKASH AND KALYAN CHAKRABORTY
+• If β is odd, then β = 2n + 1 for some integer n. Reduction of (2.4) modulo 4
+entails,
+Y 2 ≡ 3
+(mod 4)
+(2.6)
+which is impossible.
+Case 2. If α is odd, say, α = 2n + 1 for some integer n. Then,
+y2 + z2 + b
+=
+aαd + αyz
+y2 + z2 + a2d − 3
+=
+a (2n + 1)d + αyz
+y2 + z2 − (2n + 1) yz
+=
+a (2n + 1)d − a2d + 3.
+Now
+y2 + z2 + yz
+≡
+a + 3
+(mod 2),
+⇒ y2 + z2 + yz
+≡
+0
+(mod 2).
+Note that y2 + z2 + yz ≡ a + 3 (mod 2) has only solution y ≡ 0 ≡ z in Z/2Z, that is, y
+and z are even. Thus (2.3) becomes
+aαd − b ≡ 0
+(mod 4).
+If we write a = 12l + 1 for some integer l, then,
+αd −
+�
+a2d − 3
+�
+≡
+0
+(mod 4),
+⇒ αd + 3
+≡
+0
+(mod 4),
+⇒ αd
+≡
+1
+(mod 4),
+⇒ α
+≡
+1
+(mod 4).
+Let us consider
+�
+y − αz
+2
+�2
+−
+�α2
+4 − 1
+�
+z2
+=
+aαd − b,
+i.e.
+�
+y − αz
+2
+�2
+−
+�
+α2 − 4
+� �z
+2
+�2
+=
+aαd − b.
+Further, we set Y = y − αz
+2 and Z = z
+2. Then,
+Y 2 −
+�
+α2 − 4
+�
+Z2 = aαd − b
+(2.7)
+where α ≡ 1 (mod 4), a ≡ 1 (mod 12) and b = a2d − 3. Three sub cases need to be
+considered.
+
+GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES
+5
+Sub-case 1. If α ≡ 1 (mod 12), write α = 12l + 1 for some integer l. Then,
+α ≡ 1
+(mod 3)
+⇒ α + 2 ≡ 0
+(mod 3).
+Substituting α = 12l + 1 in 2.7, we get
+Y 2 −
+�
+(12l + 1)2 − 4
+�
+Z2
+=
+aαd − b,
+⇒ Y 2 ≡ aαd − b
+(mod 3),
+⇒ Y 2 ≡ a (12l + 1)d − a2d + 3
+(mod 3),
+⇒ Y ≡ 1 − 2d
+(mod 3),
+⇒ Y 2 ≡ 2
+(mod 3).
+A contradiction as 2 is not square modulo 3.
+Sub-case 2. If α ≡ 9 (mod 12). Then, there is a prime factor p ≡ 5 or 7 (mod 12) of
+(α − 2). Let p ≡ 5 or 7 (mod 12) be a prime factor of (α − 2). Thus,
+Y 2 ≡ aαd − b
+(mod p).
+Let α = pl + 2 for some integer l. Then,
+Y 2
+≡
+a (pl + 2)d − b
+(mod p),
+⇒ Y 2
+≡
+3
+(mod p).
+This leads to a contradiction as 3 is not a quadratic residue modulo p.
+Sub-case 3. When α ≡ 5 (mod 12), we substitute α = 3k + 2 for some integer k and
+get,
+Y 2 −
+�
+(3l + 2)2 − 4
+�
+Z2
+=
+(12l + 1) (3k + 2) − 2d (12l + 1) + 3,
+⇒ Y 2
+≡
+2 − 2d ≡ 0
+(mod 3),
+⇒ Y
+≡
+0
+(mod 3).
+
+6
+OM PRAKASH AND KALYAN CHAKRABORTY
+Further, we substitute Y = 3m and α = 12n + 5 for some integers n and m in 2.7 and
+arrive onto,
+9m2 − (12n + 3) (12n + 7) Z2
+=
+a (12n + 5)d − b,
+⇒ − (n + 1) Z2
+≡
+d−1
+�
+i=0
+(12n + 5)d−1−i 2i
+(mod 3),
+⇒ − (n + 1) Z2
+≡
+1
+(mod 3),
+⇒ n
+≡
+1
+(mod 3).
+Hence, α ≡ 17 (mod 36).
+Note that 3 divides (α − 2). Thus there is a prime factor p ≡ 5 or 7 (mod 12) of (α−2)
+3
+,
+otherwise it would mean that α−2
+3
+is congruent to ±1, which is not the case. Therefore,
+α − 2 ≡ 0
+(mod p).
+Thus,
+Y 2 ≡ aαd − b
+(mod p).
+Substituting α = pl + 2 for some integer l, we have
+Y 2 ≡ 3
+(mod p),
+which contradicts the fact that 3 is quadratic residue modulo p if p ≡ ±1 (mod 12).
+□
+Remark 1. The result of Sury and Majumdar [5] follows by substituting a = 1 and d = 3
+in Theorem 2.1. The particular case d = 3 in the same theorem deduces the results of
+Vaishya and Sharma [8].
+By increasing the exponents in the expression for b to 3, we will now examine the
+Diophantine equation with a little more generality. The potential of a solution in this
+scenario is described by the following two corollaries, along with a few examples.
+Corollary 2.1. The equation
+axd − y2 − z2 + xyz − b = 0
+has no integer solution (x, y, z) with x even for fixed integers a and b such that a ≡ 1
+(mod 12) and b = 2da − 3r with positive odd integers r and d as in Theorem 2.1.
+
+GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES
+7
+Proof. We follow exactly the same steps as in Case 1 of Theorem 2.1. Suppose there is a
+solution with x = α even, then we write (2.2) as:
+�
+y − αz
+2
+�2
+−
+�α2
+4 − 1
+�
+z2 = aαd − b.
+(2.8)
+Let Y = y − αz
+2 , β = α
+2 and z = Z. Then (2.8) can be written as,
+Y 2 −
+�
+β2 − 1
+�
+Z2 = aαd − b = 2dβda − b.
+(2.9)
+• If β is even, say β = 2n for some integer n, then the reduction modulo 4 of (2.9)
+will give,
+Y 2 + Z2 ≡ 3r ≡ 3
+(mod 4),
+(2.10)
+which is not feasible in Z/4Z.
+• If β is odd, say β = 2n + 1 for some integer n. Then, the reduction modulo 4 of
+(2.9) provides,
+Y 2 ≡ 3r ≡ 3
+(mod 4),
+(2.11)
+which again is not possible.
+□
+The following corollary deals with solutions having x, an odd integer:
+Corollary 2.2. The equation
+axd − y2 − z2 + xyz − b = 0
+has no integer solution in x, y and z with x ≡ 1 or 9 (mod 12), for fixed integers a, b
+such that a ≡ 1 (mod 12) and b = 2da − 3r, for r and d as in Corollary 2.1.
+Proof. Analogous steps as in Sub-case 2 and 3 of Theorem 2.1 will give the proof.
+□
+Remark 2. Corollary 2.2 says that, if there is a solution of axd − y2 − z2 + xyz − b = 0
+with a and b as described in the Corollary 2.2, then x must be 5 modulo 12.
+We will see some examples.
+Example 1. For a = 25, d = 3 and r = 3. The equation
+25x3 − y2 − z2 + xyz − 173 = 0
+(2.12)
+has no integer solution.
+
+8
+OM PRAKASH AND KALYAN CHAKRABORTY
+Example 2 shows that the equation may not have solution even with x ≡ 5 (mod 12).
+However, the next examples tell us the other possibility as well.
+Example 2. If a = 13, d = 3 and r = 3, then
+13x3 − y2 − z2 + xyz − 77 = 0
+(2.13)
+has an integer solution (5, = 18, −102).
+Remark 3. The condition that r should be odd is rigid.
+Example 3. For a = 13, d = 3 and r = 2, the equation
+13x3 − y2 − z2 + xyz − 95 = 0
+(2.14)
+has an integer solution (2, −10, −7).
+3. Hyperelliptic curves
+A hyperelliptic curve H over Q is a smooth projective curve associated to an affine plane
+curve given by the equation y2 = f (x), where f is a square-free polynomial of degree at
+least 5. If the degree of f is 2g + 1 or 2g + 2, then the curve has genus g. We write H (Q)
+for the set of Q-points on H. Determining rational points on hyperelliptic curve is one
+of the major problems in mathematics. The following is the general result regarding the
+size of H (Q), which was conjectured by Mordell and was proved by Faltings:
+Theorem 3.1. [2] If C is a smooth, projective and absolutely irreducible curve over Q of
+genus at least 2, then C (Q) is finite.
+We may thus, at least theoretically, write down the finite set C (Q). It is still a signifi-
+cant unresolved problem to perform this practically for a given curve.
+Given a hyperelliptic curve H, we can define the height (classical) function to be the
+maximum of absolute values of the coefficients. The Northcott property tells us that there
+are finitely many equations with bounded height. Thus, one may talk about the density
+and averages. In this regard, Bhargava [1] has proved that most of the hyperelliptic curve
+over Q has no rational point. So, most of the times calculating H (Q) means proving
+H (Q) = φ.
+In this section, we construct hyperelliptic curves corresponding to the equation axd −
+y2 − z2 + xyz − b = 0 with a and b as mentioned in Theorem 2.1.
+Then, we prove
+that H (Q) = φ (corroborating Bhargava [1]). The main ingredient to prove this is the
+following Nagell-Lutz type theorem (Theorem 3, [3]) proved by D. Grant.
+
+GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES
+9
+Theorem 3.2. [3] Let C be a nonsingular projective curve of genus g ≥ 1 given by
+y2 = x2g+1 + b1x2g + · · · + b2gx + b2g+1, where bi ∈ Z. Suppose
+ψ : C (Q) → J (Q)
+be the Abel-Jacobi map, defined by ψ (p) = [p − ∞], where J (Q) is the Jacobian variety.
+If p = (x, y) ∈ C (Q) \ {∞} and ψ (p) ∈ J (Q)tors, then, x, y ∈ Z and either y = 0 or y2
+divides discriminant of the polynomial x2g+1 + b1x2g + · · · + b2gx + b2g+1.
+For fixed m we define hyperelliptic curves,
+Hm,a,b : y2 − mxy = axd − m2 − b.
+• Suppose m is even. Then write (2.1) as:
+�
+y − mx
+2
+�2
+− m2x2
+4
+= axd − m2 − b.
+(3.1)
+Multiplying (3.1) by ad−1 throughout, and using the fact that d is odd and divisible
+by 3, we have,
+��
+y − mx
+2
+�
+a
+d−1
+2
+�2
+− ad−1m2x2
+4
+= (ax)d − m2ad−1 − bad−1.
+(3.2)
+We get the following hyperelliptic curve by substituting
+��
+y − mx
+2
+�
+a
+d−1
+2
+�
+= Y and
+ax = X,
+He
+m,a,b : Y 2 − ad−3m2X2
+4
+= Xd − m2ad−1 − bad−1.
+(3.3)
+• Now if m is odd, multiply (3.2) by 4d throughout to get
+��
+y − mx
+2
+�
+a
+d−1
+2 2d�2
+− (4a)d−1 m2x2 = (4ax)d − m2ad−14d − bad−14d.
+Finally substitute
+��
+y − mx
+2
+�
+a
+d−1
+2 2d�
+= Y and 4ax = X, to get
+Ho
+m,a,b : Y 2 − (4a)d−3 m2X2 = Xd − m2ad−14d − bad−14d.
+(3.4)
+Let,
+Hm,a,b =
+
+
+
+He
+m,a,b
+if m is even
+Ho
+m,a,b
+if m is odd,
+(3.5)
+be the hyperelliptic curves.
+Theorem 3.3. Let a and b be as defined in Theorem 2.1. For any m ∈ N, the hyperelliptic
+curve Hm,a,b has torsion-free Mordell-Weil group over Q.
+
+10
+OM PRAKASH AND KALYAN CHAKRABORTY
+Proof. Let a and b be fixed positive integers with a ≡ 1 (mod 12) and b = 2da − 3.
+• For any even integer m, consider the hyperelliptic curve
+He
+m,a,b : Y 2 − ad−3m2X2
+4
+= Xd − m2ad−1 − bad−1.
+(3.6)
+By Theorem 3 of [3], if (3.6) has an integer solution (X0, Y0),
+then
+�
+aX0,
+��
+Y0 − mX0
+2
+�
+a
+d−1
+2
+�
+, m
+�
+is a solution of (2.1). However, in Theorem
+2.1 we have proved that it has no integer solutions.
+• For an odd integer m, consider the hyperelliptic curve
+Ho
+m,a,b : Y 2 − (4a)d−3 m2X2 = Xd − m2ad−14d − bad−14d.
+(3.7)
+Suppose (3.7) has a solution (X0, Y0), then
+�
+4aX0,
+��
+Y0 − mX0
+2
+�
+a
+d−1
+2 2d�
+, m
+�
+is a
+solution of (2.1), which is a contradiction.
+□
+4. Numerical examples
+In this section we give some numerical examples corroborating our results in Corollary
+2.2 and Remark 2.
+a
+d
+r
+Equation
+Solution
+1
+3
+3
+x3 − y2 − z2 + xyz + 19 = 0
+(5, 0, −12)
+1
+3
+5
+x3 − y2 − z2 + xyz + 235 = 0
+(29, 12, −60)
+1
+3
+7
+x3 − y2 − z2 + xyz + 2179 = 0
+(5, 0, −48)
+1
+3
+9
+x3 − y2 − z2 + xyz + 19675 = 0
+(−31, 12, −30)
+13
+3
+3
+13x3 − y2 − z2 + xyz − 77 = 0
+(5, −18, −102)
+13
+3
+5
+13x3 − y2 − z2 + xyz + 139 = 0
+(5, 0, −42)
+13
+3
+7
+13x3 − y2 − z2 + xyz + 2083 = 0
+?
+25
+3
+3
+25x3 − y2 − z2 + xyz − 173 = 0
+(5, 0, −42)
+Acknowledgement
+This work is done during the first author’s visit to Institute of Mathematical Sci-
+ences (IMSc), Chennai, and he is grateful to the Institute for the hospitality and the
+wonderful working ambience. Both the authors are grateful to Kerala School of Mathe-
+matics(KSoM), Kozhikode, for it’s support and wonderful ambience.
+
+GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES
+11
+References
+[1] M. Bhargava, Most hyperelliptic curve over Q have no rational point, arXiv:1308.0395.
+[2] G. Faltings, “Finiteness theorems for abelian varieties over number fields”, Invent. Math., 73 (1983),
+349–366.
+[3] D. Grant, On an analogue of the Lutz-Nagell theorem for hyperelliptic curves, J. Number Theory,
+133 (2013), 963–969.
+[4] F. Luca and A. Togb´e, On the positive integral solution of the Diophantine equation x3+by+1−xyz,
+Bull. Malays. Math. Sci. Soc., 31 (2008), 129–134.
+[5] D. Majumdar and B. Sury, Fruit Diophantine Equation,https://arxiv.org/abs/2108.02640.
+[6] J.H. Silverman, , J.T. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics.
+Springer-Verlag, New York (1992).
+[7] A. Togb´e, On the positive integral solution of the Diophantine equation x3 + by + 4 − xyz, Afr.
+Diaspora J. Math., 8 (2009), 81–89.
+[8] L. Vaishya and R. Sharma, A class of fruit Diophantine equations, Monatshefte f¨ur Mathematik, 199
+(2022), 899–907.
+Kerala School of Mathematics, Kozhikode - 673571, Kerala, India.
+Email address: [email protected]
+Kerala School of Mathematics, Kozhikode - 673571, Kerala, India.
+Email address: [email protected]
+

KtFRT4oBgHgl3EQfDzfl/content/tmp_files/load_file.txt

@@ -0,0 +1,291 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf,len=290
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='13474v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='NT]  31 Jan 2023  GENERALIZED FRUIT DIOPHANTINE EQUATION AND  HYPERELLIPTIC CURVES  OM PRAKASH AND KALYAN CHAKRABORTY  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' We show the insolvability of the Diophantine equation axd −y2 −z2 +xyz −  b = 0 in Z for fixed a and b such that a ≡ 1 (mod 12) and b = 2da − 3, where d is an  odd integer and is a multiple of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Further, we investigate the more general family with  b = 2da − 3r, where r is a positive odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' As a consequence, we found an infinite  family of hyperelliptic curves with trivial torsion over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' We conclude by providing some  numerical evidence corroborating the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Introduction  One of the earliest topics in number theory is the study of Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' In  the third century, Greek mathematician Diophantus of ‘Alexandria’ began this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' A  polynomial equation of the form  P(x1, x2, · · · , xn) = 0  is known as a Diophantine equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Finding all of its integer solutions, or all of the  n−tuples (x1, x2, · · · , xn) ∈ Z that satisfy the above equation, is of prime interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The  main task is to investigate whether solutions exist for a given Diophantine equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' If  they do, it would be the aim to know how many are there and how to find all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' There  are certain Diophantine equations which has no non zero integer solutions, for example,  Fermat’s equation xn + yn = zn for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The tenth of Hilbert’s 23 problems, which  he presented in 1900, dealt with Diophantine equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Hilbert asked, is there an al-  gorithm to determine weather a given Diophantine equation has a solution or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' and  Matiyasevich in 1970 answered it negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  We investigate a class of Diophantine equations of the form axd−y2−z2+xyz−b = 0 for  fixed a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Due to its emergence when attempting to solve an equation involving fruits,  this type of Diophantine equations were given the name “Fruit Diophantine equation” by  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Sury and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Majumdar [5] and they proved the following:  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Primary: 11D41, 11D72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Secondary: 11G30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Diophantine equation, Quadratic residue, Elliptic curves, Hyperelliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  1  2  OM PRAKASH AND KALYAN CHAKRABORTY  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' [5] The equation  y2 − xyz + z2 = x3 − 5  has no integer solution in x, y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Similar type of equations were previously studied by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Luca and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Togb´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' In particu-  lar, Luca and Togb´e [4] studied the solution of the Diophantine equation x3+by+1−xyz =  0 and later, Togb´e [7] independently studied the equation x3 + by + 4 − xyz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  As a consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1 Majumdar and Sury proved the following:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' [5] For any integer m, the elliptic curve  Em : y2 − mxy = x3 + m2 + 5  has no integral point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Vaishya and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Sharma expanded on Majumdar and Sury’s work in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' A class of  fruit Diophantine equations without an integer solution was found by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' In particular  Vaishya and Sharma showed,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' [8] For fixed integers a and b with a ≡ 1 (mod 12) and b = 8a − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The  Diophantine equation  ax3 − y2 − z2 + xyz − b = 0  has no integer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Using Nagell-Lutz theorem [6] and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='3 they got hold of an infinite family of  elliptic curves with torsion-free Mordell-Weil group over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' [8] Let a and b be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  For any even integer m the elliptic curve  Ee  m,a,b : y2 = x3 + 1  4m2x2 − a2 �  m2 + b  �  has torsion-free Mordell-Weil group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  For any odd integer m the elliptic curve  Eo  m,a,b : y2 = x3 + m2x2 − 64a2 �  m2 + b  �  has torsion-free Mordell-Weil group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES  3  We extend Vaishya and Sharma’s results [8] for higher exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' We obtain a family  of hyperelliptic curves, by carrying out some appropriate transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' In 2013, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Grant gave an analogue of Nagell-Lutz theorem for hyperelliptic curves [3], using which  we conclude that the Mordell-Weil group of each member of the corresponding family of  hyperelliptic curves is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Insolvability  Here we state and prove the main theorem and derive a couple of interesting corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  We end this section by looking into a couple of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The equation  axd − y2 − z2 + xyz − b = 0  has no integer solutions for fixed a and b such that a ≡ 1 (mod 12) and b = 2da − 3,  where d is an odd integer and divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Consider  axd − y2 − z2 + xyz − b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1)  If possible, let (x, y, z) be an integer solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Let us fix x = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1) can  be re-written as,  y2 + z2 + b = aαd + αyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2)  We consider the cases of α being even or odd separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' If α is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then, we write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2) as:  �  y − αz  2  �2  −  �α2  4 − 1  �  z2 = aαd − b  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='3)  and set Y = y − αz  2 , β = α  2 and z = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='3) becomes,  Y 2 −  �  β2 − 1  �  Z2 = aαd − b = 2dβda − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='4)  If β is even, say β = 2n for some integer n, then reducing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='4) modulo 4 gives,  Y 2 + Z2 ≡ 3  (mod 4),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='5)  which is not possible in Z/4Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  4  OM PRAKASH AND KALYAN CHAKRABORTY  If β is odd, then β = 2n + 1 for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Reduction of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='4) modulo 4  entails,  Y 2 ≡ 3  (mod 4)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='6)  which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' If α is odd, say, α = 2n + 1 for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then,  y2 + z2 + b  =  aαd + αyz  y2 + z2 + a2d − 3  =  a (2n + 1)d + αyz  y2 + z2 − (2n + 1) yz  =  a (2n + 1)d − a2d + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Now  y2 + z2 + yz  ≡  a + 3  (mod 2),  ⇒ y2 + z2 + yz  ≡  0  (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Note that y2 + z2 + yz ≡ a + 3 (mod 2) has only solution y ≡ 0 ≡ z in Z/2Z, that is, y  and z are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='3) becomes  aαd − b ≡ 0  (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  If we write a = 12l + 1 for some integer l, then,  αd −  �  a2d − 3  �  ≡  0  (mod 4),  ⇒ αd + 3  ≡  0  (mod 4),  ⇒ αd  ≡  1  (mod 4),  ⇒ α  ≡  1  (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Let us consider  �  y − αz  2  �2  −  �α2  4 − 1  �  z2  =  aαd − b,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  �  y − αz  2  �2  −  �  α2 − 4  � �z  2  �2  =  aαd − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Further, we set Y = y − αz  2 and Z = z  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then,  Y 2 −  �  α2 − 4  �  Z2 = aαd − b  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='7)  where α ≡ 1 (mod 4), a ≡ 1 (mod 12) and b = a2d − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Three sub cases need to be  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES  5  Sub-case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' If α ≡ 1 (mod 12), write α = 12l + 1 for some integer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then,  α ≡ 1  (mod 3)  ⇒ α + 2 ≡ 0  (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Substituting α = 12l + 1 in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='7, we get  Y 2 −  �  (12l + 1)2 − 4  �  Z2  =  aαd − b,  ⇒ Y 2 ≡ aαd − b  (mod 3),  ⇒ Y 2 ≡ a (12l + 1)d − a2d + 3  (mod 3),  ⇒ Y ≡ 1 − 2d  (mod 3),  ⇒ Y 2 ≡ 2  (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  A contradiction as 2 is not square modulo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Sub-case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' If α ≡ 9 (mod 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then, there is a prime factor p ≡ 5 or 7 (mod 12) of  (α − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Let p ≡ 5 or 7 (mod 12) be a prime factor of (α − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Thus,  Y 2 ≡ aαd − b  (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Let α = pl + 2 for some integer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then,  Y 2  ≡  a (pl + 2)d − b  (mod p),  ⇒ Y 2  ≡  3  (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  This leads to a contradiction as 3 is not a quadratic residue modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Sub-case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' When α ≡ 5 (mod 12), we substitute α = 3k + 2 for some integer k and  get,  Y 2 −  �  (3l + 2)2 − 4  �  Z2  =  (12l + 1) (3k + 2) − 2d (12l + 1) + 3,  ⇒ Y 2  ≡  2 − 2d ≡ 0  (mod 3),  ⇒ Y  ≡  0  (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  6  OM PRAKASH AND KALYAN CHAKRABORTY  Further, we substitute Y = 3m and α = 12n + 5 for some integers n and m in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='7 and  arrive onto,  9m2 − (12n + 3) (12n + 7) Z2  =  a (12n + 5)d − b,  ⇒ − (n + 1) Z2  ≡  d−1  �  i=0  (12n + 5)d−1−i 2i  (mod 3),  ⇒ − (n + 1) Z2  ≡  1  (mod 3),  ⇒ n  ≡  1  (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Hence, α ≡ 17 (mod 36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Note that 3 divides (α − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Thus there is a prime factor p ≡ 5 or 7 (mod 12) of (α−2)  3  ,  otherwise it would mean that α−2  3  is congruent to ±1, which is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Therefore,  α − 2 ≡ 0  (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Thus,  Y 2 ≡ aαd − b  (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Substituting α = pl + 2 for some integer l, we have  Y 2 ≡ 3  (mod p),  which contradicts the fact that 3 is quadratic residue modulo p if p ≡ ±1 (mod 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  □  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The result of Sury and Majumdar [5] follows by substituting a = 1 and d = 3  in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The particular case d = 3 in the same theorem deduces the results of  Vaishya and Sharma [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  By increasing the exponents in the expression for b to 3, we will now examine the  Diophantine equation with a little more generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The potential of a solution in this  scenario is described by the following two corollaries, along with a few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The equation  axd − y2 − z2 + xyz − b = 0  has no integer solution (x, y, z) with x even for fixed integers a and b such that a ≡ 1  (mod 12) and b = 2da − 3r with positive odd integers r and d as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES  7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' We follow exactly the same steps as in Case 1 of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Suppose there is a  solution with x = α even, then we write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2) as:  �  y − αz  2  �2  −  �α2  4 − 1  �  z2 = aαd − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='8)  Let Y = y − αz  2 , β = α  2 and z = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='8) can be written as,  Y 2 −  �  β2 − 1  �  Z2 = aαd − b = 2dβda − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='9)  If β is even, say β = 2n for some integer n, then the reduction modulo 4 of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='9)  will give,  Y 2 + Z2 ≡ 3r ≡ 3  (mod 4),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='10)  which is not feasible in Z/4Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  If β is odd, say β = 2n + 1 for some integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then, the reduction modulo 4 of  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='9) provides,  Y 2 ≡ 3r ≡ 3  (mod 4),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='11)  which again is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  □  The following corollary deals with solutions having x, an odd integer:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The equation  axd − y2 − z2 + xyz − b = 0  has no integer solution in x, y and z with x ≡ 1 or 9 (mod 12), for fixed integers a, b  such that a ≡ 1 (mod 12) and b = 2da − 3r, for r and d as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Analogous steps as in Sub-case 2 and 3 of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1 will give the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2 says that, if there is a solution of axd − y2 − z2 + xyz − b = 0  with a and b as described in the Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2, then x must be 5 modulo 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  We will see some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' For a = 25, d = 3 and r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The equation  25x3 − y2 − z2 + xyz − 173 = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='12)  has no integer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  8  OM PRAKASH AND KALYAN CHAKRABORTY  Example 2 shows that the equation may not have solution even with x ≡ 5 (mod 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  However, the next examples tell us the other possibility as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' If a = 13, d = 3 and r = 3, then  13x3 − y2 − z2 + xyz − 77 = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='13)  has an integer solution (5, = 18, −102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The condition that r should be odd is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' For a = 13, d = 3 and r = 2, the equation  13x3 − y2 − z2 + xyz − 95 = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='14)  has an integer solution (2, −10, −7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Hyperelliptic curves  A hyperelliptic curve H over Q is a smooth projective curve associated to an affine plane  curve given by the equation y2 = f (x), where f is a square-free polynomial of degree at  least 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' If the degree of f is 2g + 1 or 2g + 2, then the curve has genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' We write H (Q)  for the set of Q-points on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Determining rational points on hyperelliptic curve is one  of the major problems in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The following is the general result regarding the  size of H (Q), which was conjectured by Mordell and was proved by Faltings:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' [2] If C is a smooth, projective and absolutely irreducible curve over Q of  genus at least 2, then C (Q) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  We may thus, at least theoretically, write down the finite set C (Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' It is still a signifi-  cant unresolved problem to perform this practically for a given curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Given a hyperelliptic curve H, we can define the height (classical) function to be the  maximum of absolute values of the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The Northcott property tells us that there  are finitely many equations with bounded height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Thus, one may talk about the density  and averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' In this regard, Bhargava [1] has proved that most of the hyperelliptic curve  over Q has no rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' So, most of the times calculating H (Q) means proving  H (Q) = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  In this section, we construct hyperelliptic curves corresponding to the equation axd −  y2 − z2 + xyz − b = 0 with a and b as mentioned in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Then, we prove  that H (Q) = φ (corroborating Bhargava [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' The main ingredient to prove this is the  following Nagell-Lutz type theorem (Theorem 3, [3]) proved by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES  9  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' [3] Let C be a nonsingular projective curve of genus g ≥ 1 given by  y2 = x2g+1 + b1x2g + · · · + b2gx + b2g+1, where bi ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Suppose  ψ : C (Q) → J (Q)  be the Abel-Jacobi map, defined by ψ (p) = [p − ∞], where J (Q) is the Jacobian variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  If p = (x, y) ∈ C (Q) \\ {∞} and ψ (p) ∈ J (Q)tors, then, x, y ∈ Z and either y = 0 or y2  divides discriminant of the polynomial x2g+1 + b1x2g + · · · + b2gx + b2g+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  For fixed m we define hyperelliptic curves,  Hm,a,b : y2 − mxy = axd − m2 − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Suppose m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Then write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1) as:  �  y − mx  2  �2  − m2x2  4  = axd − m2 − b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1)  Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1) by ad−1 throughout, and using the fact that d is odd and divisible  by 3, we have,  ��  y − mx  2  �  a  d−1  2  �2  − ad−1m2x2  4  = (ax)d − m2ad−1 − bad−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2)  We get the following hyperelliptic curve by substituting  ��  y − mx  2  �  a  d−1  2  �  = Y and  ax = X,  He  m,a,b : Y 2 − ad−3m2X2  4  = Xd − m2ad−1 − bad−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='3)  Now if m is odd, multiply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2) by 4d throughout to get  ��  y − mx  2  �  a  d−1  2 2d�2  − (4a)d−1 m2x2 = (4ax)d − m2ad−14d − bad−14d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Finally substitute  ��  y − mx  2  �  a  d−1  2 2d�  = Y and 4ax = X, to get  Ho  m,a,b : Y 2 − (4a)d−3 m2X2 = Xd − m2ad−14d − bad−14d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='4)  Let,  Hm,a,b =  \uf8f1  \uf8f2  \uf8f3  He  m,a,b  if m is even  Ho  m,a,b  if m is odd,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='5)  be the hyperelliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Let a and b be as defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' For any m ∈ N, the hyperelliptic  curve Hm,a,b has torsion-free Mordell-Weil group over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  10  OM PRAKASH AND KALYAN CHAKRABORTY  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Let a and b be fixed positive integers with a ≡ 1 (mod 12) and b = 2da − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  For any even integer m, consider the hyperelliptic curve  He  m,a,b : Y 2 − ad−3m2X2  4  = Xd − m2ad−1 − bad−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='6)  By Theorem 3 of [3], if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='6) has an integer solution (X0, Y0),  then  �  aX0,  ��  Y0 − mX0  2  �  a  d−1  2  �  , m  �  is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' However, in Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1 we have proved that it has no integer solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  For an odd integer m, consider the hyperelliptic curve  Ho  m,a,b : Y 2 − (4a)d−3 m2X2 = Xd − m2ad−14d − bad−14d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='7)  Suppose (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='7) has a solution (X0, Y0), then  �  4aX0,  ��  Y0 − mX0  2  �  a  d−1  2 2d�  , m  �  is a  solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='1), which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Numerical examples  In this section we give some numerical examples corroborating our results in Corollary  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='2 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  a  d  r  Equation  Solution  1  3  3  x3 − y2 − z2 + xyz + 19 = 0  (5, 0, −12)  1  3  5  x3 − y2 − z2 + xyz + 235 = 0  (29, 12, −60)  1  3  7  x3 − y2 − z2 + xyz + 2179 = 0  (5, 0, −48)  1  3  9  x3 − y2 − z2 + xyz + 19675 = 0  (−31, 12, −30)  13  3  3  13x3 − y2 − z2 + xyz − 77 = 0  (5, −18, −102)  13  3  5  13x3 − y2 − z2 + xyz + 139 = 0  (5, 0, −42)  13  3  7  13x3 − y2 − z2 + xyz + 2083 = 0  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  25  3  3  25x3 − y2 − z2 + xyz − 173 = 0  (5, 0, −42)  Acknowledgement  This work is done during the first author’s visit to Institute of Mathematical Sci-  ences (IMSc), Chennai, and he is grateful to the Institute for the hospitality and the  wonderful working ambience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Both the authors are grateful to Kerala School of Mathe-  matics(KSoM), Kozhikode, for it’s support and wonderful ambience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  GENERALIZED FRUIT DIOPHANTINE EQUATION AND HYPERELLIPTIC CURVES  11  References  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Bhargava, Most hyperelliptic curve over Q have no rational point, arXiv:1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='0395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Faltings, “Finiteness theorems for abelian varieties over number fields”, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=', 73 (1983),  349–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Grant, On an analogue of the Lutz-Nagell theorem for hyperelliptic curves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Number Theory,  133 (2013), 963–969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content='  [4] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Luca and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
+page_content=' Togb´e, On the positive integral solution of the Diophantine equation x3+by+1−xyz,  Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFRT4oBgHgl3EQfDzfl/content/2301.13474v1.pdf'}
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+Exploring Current Constraints on Antineutrino Production by 241Pu and Paths Towards the
+Precision Reactor Flux Era
+Yoshi Fujikake, Bryce Littlejohn,* and Ohana B. Rodrigues†
+Physics Department, Illinois Institute of Technology, Chicago, IL 60616, USA
+Pranava Teja Surukuchi‡
+Wright Laboratory, Yale University, New Haven, CT 06520, USA
+By performing global fits to inverse beta decay (IBD) yield measurements from existing neutrino experi-
+ments based at highly 235U enriched reactor cores and conventional low-enriched cores, we explore current
+direct bounds on neutrino production by the sub-dominant fission isotope 241Pu. For this nuclide, we determine
+an IBD yield of σ241 = 8.16 ± 3.47 cm2/fission, a value (135 ± 58)% that of current beta conversion models.
+This constraint is shown to derive from the non-linear relationship between burn-in of 241Pu and 239Pu in con-
+ventional reactor fuel. By considering new hypothetical neutrino measurements at high-enriched, low-enriched,
+mixed-oxide, and fast reactor facilities, we investigate the feasible limits of future knowledge of IBD yields
+for 235U, 238U, 239Pu, 241Pu, and 240Pu. We find that first direct measurement of the 240Pu IBD yield can be
+performed at plutonium-burning fast reactors, while a suite of correlated measurements at multiple reactor types
+can achieve a precision in direct 238U, 239Pu, and 241Pu yield knowledge that meets or exceeds that of current
+theoretical predictions.
+I.
+INTRODUCTION
+A nuclear reactor primarily generates thermal energy as
+product nuclei inherit (as kinetic energy) and deposit (through
+repeated elastic collisions) excess rest mass energy from the
+fission of heavy nuclides in the reactor’s fuel, such as 235U,
+238U, 239Pu, 241Pu, and more. Successive decays of these
+neutron-rich product nuclei release additional energy in the
+form of beta particles, gamma-rays, and antineutrinos. While
+the two former product types are additional, sub-dominant
+contributors to heat generation in a reactor, the antineutri-
+nos (νe ) and their associated kinetic energy entirely escape
+the reactor core, offering an attractive avenue for studying
+the properties of neutrinos [1–3], interrogating state-of-the-
+art nuclear data [4], and non-intrusively monitoring nuclear
+reactor cores [5]. Reactor-based νe detectors have demon-
+strated that neutrinos have mass [6–9], and have searched for
+the existence of new heavy neutrino states [10–15] and other
+new physics phenomena [16–23]. By observing discrepan-
+cies with respect to existing theoretical νe flux and energy
+spectrum predictions, they have also highlighted limitations of
+and/or inaccuracies in community fission yield and beta decay
+databases [7, 9, 24–32]. Antineutrino monitoring case studies
+have explored a variety of potential use case scenarios, such
+as thermal power load-following and determination of reactor
+fissile inventory [33–37], and existing νe detectors have con-
+firmed the feasibility of some of these activities [25, 38, 39].
+The average number of antineutrinos released or detected
+per nuclear fission depends on the fission isotope in question:
+different fission isotopes have different fission product yields,
+with each product varying in its distance from the line of sta-
+bility and having its own unique nuclear structure and decay
+* [email protected]
+† [email protected]
+‡ [email protected]
+scheme. Thus, reactor cores with differing fuel compositions
+are expected to differ in their rate of νe output. These ex-
+pected differences have been explicitly demonstrated in recent
+νe experiments using the inverse beta decay (IBD) interaction
+process, p + νe → e++ n, which has a 1.8 MeV interaction
+threshold and a precisely-predicted cross-section, σIBD(Eν),
+versus νe energy Eν [40]. For these experiments, measured
+νe fluxes have been expressed in terms of an IBD yield per
+fission σf [1]:
+σf(t) =
+�
+i
+Fi(t)σi.
+(1)
+In this expression, Fi(t) is the fraction of fissions contributed
+by isotope i in the sampled reactor core(s) during the experi-
+ment’s measurement period and σi its IBD yield per fission,
+σi =
+�
+Si(Eν)σIBD(Eν)dEν.
+(2)
+Here, Si(Eν) is the true produced νe energy (Eν) spectrum
+per fission for isotope i, and σIBD is the inverse beta decay
+interaction cross-section.
+In a straightforward demonstration of variations in νe emis-
+sion between fission isotopes, reported IBD yields σf are
+clearly offset [41] between measurements at 235U-burning
+highly enriched reactor cores [42–48] and measurements per-
+formed at commercial cores burning a mixture of 235U, 238U,
+239Pu, and 241Pu [7, 49–56]. In a separate demonstration, the
+Daya Bay and RENO experiments have compared IBD yields
+measured in the same detectors at differing points in their sam-
+pled commercial reactors’ fuel cycles, observing higher yields
+during periods with higher (lower) 235U (239Pu) fission frac-
+tions [25, 55].
+By performing fits to a set of σf measurements at reactors
+of well-known fission fraction Fi, one can use Equations 1
+and 2 to directly determine the isotopic IBD yield σi of one
+or more fission isotopes. With a single HEU-based experi-
+ment, the IBD yield for 235U, σ235, can be trivially deter-
+mined as σ235 = σf, since F235 approaches unity for these
+arXiv:2301.13123v1  [hep-ph]  30 Jan 2023
+
+2
+cores. On their own, HEU-based σ235 measurements exhibit
+deficits [57] with respect to commonly-used beta-conversion
+predictions [58, 59], indicating issues in modeling either the
+core’s νe emissions or νe behavior during propagation [60].
+Daya Bay and RENO σf measurements, which encompass
+multiple data points with differing LEU fuel composition
+F235, 238, 239, 241, when combined with modest theoretical
+constraints on σ238, 241, yields from the sub-dominant iso-
+topes 238U and 241Pu, enable determination of isotopic yields
+for both 235U and 239Pu [25, 55]. These measurements show
+a deficit with respect to 235U conversion predictions, but no
+such deficit for 239Pu, providing further credence to the νe
+emission mis-modelling hypothesis. Going further, global fits
+of both LEU and HEU datasets can be used to simultaneously
+determine σ235, 238, 239 [61]: the measured σ238 shows a sig-
+nificant (33±14)% deficit [62] with respect to summation pre-
+dictions based on community-standard nuclear databases [59],
+suggesting potential issues in current 238U fission yield mea-
+surements or evaluations. Future direct determination of iso-
+topic IBD yields for a wider array of fission isotopes beyond
+235U, 239Pu, and 238U, as well as improved precision for these
+three isotopes, can lead to further understanding or improve-
+ment of existing nuclear data, reactor νe models, and reactor-
+based fundamental physics studies.
+Improved isotopic IBD yield measurements also hold po-
+tential benefits for future νe -based applications. Some ad-
+vanced reactor technologies present unique safeguards chal-
+lenges that may be satisfied by near-field νe -based monitor-
+ing capabilities [63]. However, neutrino emissions have never
+been measured at advanced reactor cores, some of which dif-
+fer substantially from measured HEU and LEU reactor types
+in both fuel composition and core neutronics [35, 64, 65].
+For example, mixed-oxide reactor fuels, which, unlike con-
+ventional low-enriched fuel, are produced from a mixture of
+uranium and plutonium isotopes, may be deployed in future
+reactors to realize a closed nuclear fuel cycle or as a means of
+disposing of existing plutonium stockpiles. Fast fission reac-
+tor technologies, which, unlike conventional thermal reactors,
+rely on fast neutron induced fission to maintain criticality, may
+offer safety and sustainability advantages with respect to con-
+ventional reactor types. For these reactors, better direct deter-
+minations of true underlying σi can enable more robust and
+reliable future monitoring capabilities than would be possible
+using existing demonstrably imperfect models of νe produc-
+tion per fission.
+In this paper, we study how existing and potential future
+IBD measurements can provide first direct glimpses at νe
+production by previously unexplored fission isotopes and im-
+prove our precision in understanding of the more-studied iso-
+topes 235U, 239Pu, and 238U. By performing loosely con-
+strained fits of isotopic IBD yields to existing LEU and HEU
+datasets, we demonstrate the feasibility of achieving non-
+trivial future bounds on νe production by 241Pu. By applying
+the same fit techniques to hypothetical future high-precision
+IBD yield measurements at HEU, LEU, MOX, and fast fis-
+sion reactors, we show that direct IBD yield determinations
+for all four primary fission isotopes (235U, 238U, 239Pu, and
+241Pu) can meet or exceed the claimed precision of exist-
+ing conversion-based predictions while also placing the first
+meaningful bounds on 240Pu νe production.
+We begin in Section II with a description of the global fit
+and existing and hypothetical future IBD yield datasets. Re-
+sults of the fit to existing datasets and studies of 241Pu limits
+are presented and discussed in Sections III. In Section IV, we
+describe the set of considered future hypothetical experiments
+and the result of applying global fits to the hypothetical re-
+sults of these experiments. Main results are then summarized
+in Section V.
+II.
+GLOBAL DATASETS AND FIT TECHNIQUE
+In this analysis we perform fits to a set of IBD rate measure-
+ments with varying degrees of systematic correlation between
+each measurement set. For an individual measurement, the
+number of IBD interactions N detected per time interval t can
+be described as:
+N = NpεP(L)
+4πL2
+� Wth(t)σf(t)
+¯E(t)
+dt,
+(3)
+where Np is the number of target protons, ε is the efficiency
+of detecting IBDs, P(L) is the survival probability due to
+neutrino oscillations, and L is the core-detector distance. Of
+the time-dependent quantities, Wth(t) is the reactor’s thermal
+power, ¯E(t) = �
+i Fi(t)ei is the core’s average energy re-
+leased per fission, ei is the average energy released per fission
+of isotope i, and Fi(t) and σf(t), as in Equations 1 and 2, are
+the fission yields and IBD yields of isotope i. In order to per-
+form one or multiple measurements of σf, a reactor νe flux
+experiment must measure N while characterizing the other
+reactor and detector inputs in Equation 3.
+A.
+Existing Datasets
+Many experiments have successfully measured σf values
+and associated statistical and systematic uncertainties. As in-
+put for this study, we include time-integrated IBD yield mea-
+surements and uncertainties reported by the Goesgen, Bugey-
+3, Bugey-4, Rovno, Palo Verde, CHOOZ, and Double Chooz
+LEU-based experiments and the ILL, Savannah River, Kras-
+noyarsk, Nucifer and STEREO HEU-based experiments, as
+well as the highly-correlated datasets at varying Fi from the
+Daya Bay and RENO experiments. Calculated fission frac-
+tions and measured yields for these experiments, as well as as-
+sociated uncertainties and cross-measurement systematic cor-
+relations, have been summarized in Ref. [66], and are used
+for portions of this paper’s analysis. Input data tables are pro-
+vided in the public GitHub repository [67] provided by the
+authors as an accompaniment to this analysis. Since we do
+not consider short-baseline oscillations as part of this analysis,
+reactor-detector baselines are not used in analysis of existing
+datasets, but are nonetheless provided in these tables.
+
+3
+B.
+Hypothetical Future Datasets
+For this study, we also generate hypothetical future IBD
+yield datasets and uncertainty budgets matching the expected
+capabilities of experimental deployments at HEU, LEU, MOX
+and fast reactor types. These are also provided in the GitHub
+supplementary materials, along with assumed uncertainty co-
+variance matrices for all considered hypothetical measure-
+ments.
+Hypothetical IBD yield measurements are Asimov
+datasets free of statistical and systematic fluctuations that are
+generated according to Equation 3. As input to this equa-
+tion, fission fractions are required for each host reactor and
+are described below.
+To match general indications from
+recent summations [32] and fission beta [68], and νe flux
+evolution [25] measurements, and matching the approach in
+Ref. [61], we choose input ‘true’ IBD yield values match-
+ing a scenario where Huber-Mueller modelled yields [69]
+are only incorrect for 235U: (σ235, σ238, σ239, σ241) =
+(6.05,10.10,4.40,6.03) ×10−43 cm2/fission.
+The yield for
+240Pu has not been predicted in the literature to our knowl-
+edge, so we estimate it by applying a 3Z-A scaling suggested
+in Ref. [70] to the four previously-mentioned isotopes; the
+determined central value is σ240 = 4.96 ×10−43 cm2/fission.
+Other experimental assumptions regarding detector, reactor,
+and experimental layout parameters are then required to de-
+fine the statistical and systematic uncertainties associated with
+each hypothetical IBD yield measurement.
+The HEU-based measurement is modeled after the HFIR
+facility at Oak Ridge National Laboratory, sporting 85 MW
+of thermal power, a 100% 235U fission fraction, and a 7 m
+reactor-detector center-to-center distance. LEU-based mea-
+surements are assumed to occur at a 20 m center-to-center dis-
+tance from a core following the attributes of a 2.9 GWth Daya
+Bay core with an 18 month fuel cycle. Assumed fission frac-
+tions are chosen to fall roughly in the middle of the range
+reported for Daya Bay’s cores in Fig. 1 of Ref. [75], and cor-
+respond to a fully-loaded core with roughly 1/3 of its rods con-
+taining fresh (pure uranium oxide at start-up) fuel; this level
+of partial reloading is customary when operating cores of this
+type.
+MOX-based measurements are modelled after the MOX
+reactor studies of Ref. [72], and is assumed to occur 20 m
+from a core with a 3.2 GWth thermal power and 18 month cy-
+cle length, and fission fractions matching those of the sim-
+ulated 50% weapons-grade MOX-burning core.
+Weapons-
+grade (WG) plutonium is characterized by low 240Pu and
+241Pu isotopic fractions, and thus a low F241 fission fraction
+at reactor start-up. These WG-MOX core parameters corre-
+spond to a realizable operational scenario implemented for
+the goal of plutonium stockpile disposition in a commercial
+reactor core. We will also reference a similar case where 50%
+reactor-grade (RG) MOX fuel is used in the same reactor type;
+these parameters correspond to an operational scenario for a
+commercial complex operated as part of a closed nuclear fuel
+cycle program. Following recommendations of the authors
+of Ref. [72], fission fractions for the WG-MOX core exam-
+ple are assumed to match the reported fission fractions for the
+first third of pictured 50% WG-MOX running in Ref. [72],
+while fractions from the RG-MOX case are assumed to match
+the those of 50% WG-MOX running between days 800 and
+1350 [76]; fission fractions were extracted by interpolating
+fission rates from this reference and normalizing such that the
+sum for the four primary fission isotopes is equal to unity.
+It should be stressed that modeled fuel content evolution for
+LEU and MOX cores is highly dependent on the initial condi-
+tions of the fuel, on the neutronics of the involved core type,
+and on reactor operations. In this study, we include one spe-
+cific fission fraction set for each fuel type – LEU, WG-MOX,
+and RG-MOX; the impact or potential benefits of further vari-
+ations between LEU or MOX core types is not considered.
+Finally, two experiments are assumed to occur at the base-
+lines of 20 m and 7 m distances from the primarily plutonium-
+burning 1.25 GWth PFBR fast breeder reactor in India [73]
+and the 300 GWth Versatile Test Reactor fast reactor [74] re-
+spectively. The former reactor plays a central role in plans for
+realization of an independent, sustainable nuclear fuel cycle
+in India, while the latter has been developed as a US-based
+reactor materials and irradiation R&D facility based at Idaho
+National Laboratory [64]. Assumed reactor and site parame-
+ters for all measurements are summarized in Table I; fission
+fraction values for all hypothetical measurement data points
+used in this study are illustrated in Figure 1.
+0
+5
+10
+15
+20
+25
+30
+Data Points
+0
+0.2
+0.4
+0.6
+0.8
+1
+Fission Fraction
+Fission fractions in PROS HEU+LEU+WGMOX+RGMOX+VTRRx+PFBR
+U235
+U238
+Pu239
+Pu240
+Pu241
+HEU
+LEU
+WG MOX
+RG MOX
+VTR
+PFBR
+FIG. 1. Fission fractions used for hypothetical future measurement
+data points described in this section. See text for details.
+For all experiments, an IBD detector matching qualities of
+the 4 ton PROSPECT reactor νe detector are used [77]; rel-
+evant parameters are also listed in Table I. In some cases, a
+1 ton detector with otherwise similar experimental parame-
+ters is also considered; this case enables investigation of the
+value of using a near-future compact νe monitoring detector,
+such as the Mobile Antineutrino Demonstrator [78] (MAD),
+to perform IBD yield benchmarking measurements at multi-
+ple reactor locations. In all cases, the statistical uncertain-
+ties associated with each datapoint for each reactor-detector
+combination are estimated using the associated detector and
+reactor parameters quoted in Table. I and lie between 0.15
+% and 0.2 %. For simplicity, we do not consider statistical
+
+4
+Parameter
+HEU LEU MOX Fast (PFBR) Fast (VTR)
+Reactor
+Thermal Power (MWth)
+85
+2900 3200
+1250
+300
+Burnup Profile
+-
+[71]
+[72]
+[73]
+[74]
+Reactor Cycle Length
+24 d
+1.5 y 1.5 y
+1.5 y
+100 d
+Experimental
+Core-Detector Distance (m)
+7 m
+20 m 20 m
+20 m
+20 m
+Data-Taking Length
+3 y
+1.5 y 1.5 y
+1 y
+100 d
+Detector
+Active Mass
+4 ton (1 ton)
+Target Protons
+2×1029 (0.5×1029)
+IBD Detection Efficiency
+40%
+Uncertainty, Reactor
+Thermal Power
+1.0% 0.5% 0.5%
+1.0%
+1.0%
+Fission Fractions
+-
+0.6% 0.6%
+0.6%
+0.6%
+Energy per Fission
+0.1% 0.2% 0.2%
+0.2%
+0.2%
+Uncertainty, Detector
+Target Protons
+1.0%
+Detection Efficiency
+0.75%
+IBD Cross Section
+0.1%
+Total Reactor Systematic
+0.5% 0.8% 0.8%
+1.2%
+1.2%
+Total Detector Systematic
+1.3%
+TABLE I. Assumed reactor and site parameters for the hypothetical future short-baseline reactor experiments described in the text.
+and systematic uncertainty contributions from IBD-like back-
+grounds; for a PROSPECT-like detector expecting signal-to-
+background ratios of better than 4 (10) deployed on-surface at
+an HEU (LEU) reactor [79], IBD counts would be expected to
+dominate measurement statistical uncertainties.
+Hypothetical measurements should also be accompanied by
+predicted reactor- and detector-related systematic uncertain-
+ties, which are also summarized in Table I. Systematics for
+most cores are dominated by the uncertainty in the thermal
+power produced by the operating core. Commercial reactor
+companies have provided sub-percent precision in reported
+thermal powers for existing IBD yield measurements at nu-
+merous reactor sites [80, 81]; for this reason, we choose 0.5%
+uncertainty for LEU and MOX cores. While similar thermal
+power measurement devices and strategies could be applied
+to HEU facilities, in practice, legacy systems used in exist-
+ing HEU facilities have recently provided thermal power un-
+certainties closer to 2% [48]; for this analysis, we optimisti-
+cally assume implementation of upgraded measurement sys-
+tems or techniques capable of providing 1% precision at an
+HEU core. Advanced technologies for time-stable and high-
+precision thermal power monitoring in sodium-cooled fast re-
+actors like PFBR and VTR are under active development, due
+to the difficulties associated with the coolant’s high temper-
+ature and chemical corrosiveness; given the lack of available
+quantitatively demonstrated capabilities, we also assume a 1%
+thermal power uncertainty for this core type. While thermal
+power uncertainties for different reactors are assumed to be
+uncorrelated, this uncertainty is correlated between multiple
+measurements at the same core. Measured IBD yields for an
+experiment will also be uncertain due to the limits in knowl-
+edge of fission fractions in the core, which is defined via de-
+tailed reactor core simulations. In the absence of these cal-
+culations for all core types, we will assume an uncertianty
+of 0% for the HEU experiment and 0.6% for all other cores,
+following the value quoted by Daya Bay and others for LEU
+cores [71]. This uncertainty is also assumed to be uncorre-
+lated between cores, but correlated between measurements at
+the same core. Isotopic energy release per fission ei – re-
+quired for calculating expected experiment statistics – have
+minor IBD yield uncertainty contributions of 0.1% to 0.2%
+depending on core fuel content [82]; the ei central value and
+uncertainty for 240Pu is assumed to match that of 241Pu.
+On the detector side, uncertainties are dominated by the
+limited knowledge of IBD detection efficiency, assumed to be
+known with 0.75% precision, as well as knowledge of the to-
+tal number of protons within the detector’s target region, as-
+sumed to be known to 1%; these chosen values reflect those
+achieved in a range of recent large- and small-detector IBD
+experiments [48, 55, 83, 84]. In this analysis, we consider
+the possibility of moving a single reactor neutrino detector to
+multiple reactor core types to perform systematically corre-
+lated IBD yield measurements; for this reason, unless other-
+wise mentioned, we treat detector systematic uncertainties as
+correlated between all measurements.
+
+5
+C.
+Global Fit Approach
+To obtain isotopic IBD yields in this analysis, we use a
+least-squares test statistic:
+χ2 =
+�
+a,b
+�
+σf,a − r
+�
+i
+Fi,aσi
+�
+V−1
+ab
+�
+σf,b − r
+�
+i
+Fi,bσi
+�
++
+�
+j,k
+(σth
+j − σj)V−1
+ext,jk(σth
+k − σk).
+(4)
+In this fit, experimental inputs Fi and σf are as described
+above, and the sum i is run over five fission isotopes, 235U,
+238U, 239Pu, 241Pu, and 240Pu, with five attendant IBD yield
+fit parameters.
+The experimental covariance matrix V de-
+fines the uncertainties for each experiment and their cross-
+correlations, as described in the previous-subsection.
+The
+final term is used to constrain fitted σi values to theoreti-
+cal predictions by adding a penalty that increases as the two
+quantities diverge.
+In contrast to most recent global IBD
+yield fits [25, 57, 85], we are interested in examining weakly-
+constrained or un-constrained simultaneous fits of all relevant
+fission isotopes’ IBD yields. For this reason, j and k sum only
+over the three sub-dominant isotopes, 238U, 241Pu, and 240Pu,
+and the 3×3 V−1
+ext is diagonal (no assumed uncertainty correla-
+tion between isotopes), with elements set to achieve wide 1σ
+theoretical constraints of 75% of the predicted yield. To com-
+pare to previous IBD yield fits [61, 86], we occasionally con-
+sider the much tighter (2.6%) bounds on σ241 quoted by the
+Huber model [58]. For fits not involving fast reactor datasets,
+σ240 is pegged to the theoretically-predicted value, and has no
+effect on the subsequent 4-parameter fit.
+III.
+FITS TO EXISTING DATASETS AND 241PU IBD
+YIELD CONSTRAINTS
+We first consider IBD yield fits applied to the existing
+global yield datasets described briefly in Section II A. By first
+applying tight 2.6% constraint on 241Pu, we largely reproduce
+unconstrained 235U, 238U, and 239Pu yield best-fit values re-
+ported for the oscillation-free fit in Ref. [86]. Test statistic
+values with respect to the best-fit (∆χ2) versus input value
+are shown for each isotope in Figure 2, while minimizing over
+the three other isotopic yield parameters. We observe a best-
+fit 235U yield more than 3σ (5%) below the Huber-predicted
+value, and a best-fit 238U yield that deviates from the pre-
+dicted central value by (36±20), slightly more than in previ-
+ous fits [86]. As in previous fits, the 239Pu yield is found to be
+consistent with Huber-predicted values within a 5%, ∼ 1σ un-
+certainty band. This similarity in results indicates that the rel-
+atively new STEREO data point [48], while qualitatively bol-
+stering confidence in historical observations of a ∼5% yield
+deficit at HEU cores [87], has fairly modest quantitative im-
+pact on the primary issues surrounding data-model agreement
+for conversion-predicted uranium IBD yields.
+With consistency established with respect to previous anal-
+yses, we proceed with loosening of yield constraints for all
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1 1.2
+i
+R
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+2
+χ
+ 
+∆
+ 0.013
+±
+ = 0.953 
+235
+σ
+ 0.201
+±
+ = 0.642 
+238
+σ
+ 0.044
+±
+ = 1.047 
+239
+σ
+ 0.026
+±
+ = 1.001 
+241
+σ
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+i
+R
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+2
+χ
+ 
+∆
+ 0.013
+±
+ = 0.953 
+235
+σ
+ 0.264
+±
+ = 0.730 
+238
+σ
+ 0.252
+±
+ = 0.910 
+239
+σ
+ 0.426
+±
+ = 1.353 
+241
+σ
+FIG. 2. Isotopic IBD yield fit results for the existing global dataset
+with tight (top, 2.6%) and loose (bottom, 75%) external constraints
+on the 241Pu yield. Test statistic values with respect to the best-fit
+(∆χ2) are shown versus input value for each of the four primary
+fission isotopes. For each isotope’s curve, the fit is marginalized over
+the other isotopes.
+fission isotopes. Figure 2 shows reported isotopic ∆χ2 test
+statistic values versus input σ value for each isotope while
+applying a looser constraint on 241Pu of 75%. Best-fit param-
+eters and 1σ ranges are found to be:
+σ235 = 6.37 ± 0.08;
+σ238 = 7.37 ± 1.95;
+σ239 = 4.00 ± 1.01;
+σ241 = 8.16 ± 3.47.
+(5)
+The best-fit χ2
+min is found to be 26.2 for 38 degrees of free-
+dom (41 data points, 3 fit parameters), indicating an accept-
+
+6
+able goodness-of-fit.
+However, this value is only slightly
+lower than that provided by the more-constrained fit (χ2
+min
+= 26.6), indicating that this enhanced freedom has not sub-
+stantially improved data-model agreement.
+Central values
+of 235U, 238U, and 239Pu fit parameters are relatively sta-
+ble, remaining within 15% of those provided by the more-
+constrained fit. Meanwhile, the newly freed 241Pu yield in-
+creases by 35%, although σ241 nonetheless remains consistent
+with its model-predicted value within its large 43% relative
+uncertainty band. Thus it appears that the current global IBD
+yield dataset does not have the statistical power to provide
+meaningful tests of underlying modelling issues for 241Pu.
+The disappointing lack of new insight should not be too sur-
+prising, given the small (O(5%) or less) fractional contribu-
+tion of 241Pu fissions in all existing measured reactor cores.
+However, it is interesting to note that σ241 1σ error bands
+are found to be tighter than the externally-applied constraint.
+This indicates that there are features in the existing global
+dataset that provide the power to specifically constrain 241Pu.
+To attempt to identify these features, we examined correla-
+tions between fitted isotopic yields, which are depicted in Fig-
+ure 3 as best-fit parameter space regions in two dimensions be-
+tween 241Pu and the other three isotopes. Substantial 241Pu-
+239Pu and 241Pu-238U degeneracies can be observed, with the
+former reflected in a more than five-fold increase in uncertain-
+ties on σ239 between the more-constrained (4% uncertainty)
+and less-constrained (25% uncertainty) fits. Degeneracies can
+also be expressed by calculating correlation coefficients be-
+tween the fitted yield parameters, which are also given in the
+legends of Fig 3:
+ρσi,σj = (σi − σi)(σj − σi)
+σσiσσj
+(6)
+The extreme 241Pu-239Pu correlation can be understood by
+observing the fission fraction evolution trends experienced by
+LEU reactors, as depicted in Figure 1. In these cores, F239
+and F241 rise in tandem with reactor fuel burn-up, making it
+hard for unconstrained fits to simultaneously determine both
+σ239 and σ241. It can also be understood as a simple reality
+of underlying nuclear physics in the core: 241Pu is produced
+by via multi-neutron capture on 239Pu, and thus its build-up in
+the core is dependent on the build-up of the latter. In aspects
+of previous multi-datapoint LEU analyses, such as those of
+Daya Bay [25, 88] and RENO [55], 241Pu and 239Pu fission
+fractions are treated as explicitly linearly correlated.
+We examine the limits of this linear correlation by gener-
+ating hypothetical LEU reactor IBD yield datasets following
+the method described in Section II B and fission yields from
+Figure 1: one dataset assumes isotopic yields matching the
+best-fit for the existing global dataset, and the other assumes
+true 241Pu and 239Pu yields close to the axis of anti-correlation
+between the two datasets, but beyond the 1σ bounds allowed
+by the data. Chosen true yields for this test are illustrated
+in the right panel of Figure 3; the 238U yield for this case,
+8.8 cm2/fission, was chosen to vertically align the two yield
+datasets for easier comparison of trends. Hypothetical yields
+for these two cases are pictured in Figure 4. The test cases
+clearly differ in the change in slope, or curvature, present in
+the LEU data points, providing an indication of the primary
+source of unique 241Pu yield information in current and fu-
+ture experimental data. The extreme 239Pu-241Pu yield offset
+in this example amplifies the impact of a modest non-constant
+relationship between F239 and F241 in LEU-based datasets,
+which is also illustrated in Figure 4. To test the validity of
+this hypothesis with existing datasets, we perform a fit to only
+the RENO and Daya Bay LEU datasets while applying loose
+75% external constraints on all four isotopes. While large un-
+certainty increases are seen in σ235 and σ238, σ239 and σ241
+fractional uncertainties are altered by <30%, and fractional
+bounds on σ241 (43%) remain tighter than the 75% external
+constraint. Thus, in the existing global dataset, it does ap-
+pear that that the Daya Bay and RENO LEU data points are
+responsible for the modest breaking of degeneracy between
+239Pu and 241Pu yields.
+Adding this to previously-established trends, it is straight-
+forward to recount the independent features of the global IBD
+yield dataset that enable determination of all four isotopes’
+IBD yields:
+• HEU-based experiments’ σf measurements directly
+constrain σ235 [57].
+• The measured relative linear σf slope versus fuel
+burn-up at LEU-based experiments directly constrains
+σ239 [25].
+• The time-integrated offset in σf between HEU and LEU
+cores constrains σ238 [61].
+• The curvature of σf slope versus fuel burn-up at LEU
+experiments constrains σ241.
+As we move on to consider possible future IBD yield mea-
+surement scenarios, these high-level principles serve to guide
+attention toward those with particular promise for improving
+global knowledge of isotopic yields. In particular, we will
+look to explore new multi-dataset measurements that can pro-
+vide an enhanced view of σf curvature with host reactor fuel
+evolution.
+We end this Section by noting that within the current global
+dataset, Daya Bay contains currently-unexploited potential.
+Ref. [75] indicates O(5%) F241/F239 variations between re-
+actor cycles that are averaged out in its current fuel content
+binning scheme. To estimate the achievable gains in the fis-
+sion yields, we generate an Asimov IBD yield dataset with fis-
+sion fractions taken from a combination of rates, RENO and
+Daya Bay-like experiment is divided into two halves; one with
+the default fission fractions while the other having F241/F239
+relatively reduced by 2.5%. The systematic and statistical un-
+certainties are assumed to match the existing global dataset
+and the yields are generated using best-fit results from the
+global dataset. Such a joint fit provides a modest improvement
+in the precision of fission yield of (σ235, σ238, σ239, σ241) =
+(1.3%, 24.8%, 19.7%, 39.2%) compared to the precision of
+(1.3%, 26.4%, 25.2%, 42.6%) for the existing global dataset.
+If we further double the statistics of the Daya Bay Asimov
+data–as expected from the full Daya Bay dataset—in conjunc-
+tion with the modified binning in fission fractions, we find
+
+7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+/fission]
+2
+cm
+-43
+ [10
+239
+σ
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+/fission]
+2
+cm
+-43
+ [10
+241
+σ
+Correlation: -0.990
+6
+6.2
+6.4
+6.6
+6.8
+/fission]
+2
+cm
+-43
+ [10
+235
+σ
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+/fission]
+2
+cm
+-43
+ [10
+241
+σ
+Correlation: 0.013
+0
+2
+4
+6
+8
+10
+12
+14
+/fission]
+2
+cm
+-43
+ [10
+238
+σ
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+/fission]
+2
+cm
+-43
+ [10
+241
+σ
+Correlation: 0.757
+FIG. 3. Isotopic IBD yield fits for the existing global dataset with loose (75%) external constraints on the 241Pu IBD yield, σ241. Contours
+are pictured for σ241 relative to the other isotopic yields, with the fit marginalized over the non-pictured isotopes. Correlation coefficients
+between fitted σ241 and the other yields are given in the plot legends. Also shown in dashed lines are the theoretical IBD yields predicted by
+the Huber-Mueller model. Stars indicate IBD yields chosen for illustration in Fig. 4.
+5.65
+5.7
+5.75
+5.8
+5.85
+5.9
+5.95
+6
+6.05
+6.1
+/fission]
+2
+ cm
+-43
+IBD yield [10
+Nominal IBD yields
+Modified IBD yields
+0.45
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+235
+F
+0.14
+0.16
+0.18
+0.2
+0.22
+0.24
+239
+/F
+241
+F
+Daya Bay
+RENO
+FIG. 4. Top: IBD yield sets for two hypothetical LEU measure-
+ments: one assuming measurements align with isotopic IBD yields
+matching the best-fit for the existing global dataset, and another as-
+suming alignment with σ239 and σ241 values matching those indi-
+cated in Figure 3. The latter scenario’s values lie outside of the 1
+σ region preferred by the global IBD yield dataset; for this scenario,
+σ238 is reduced to enable better vertical alignment of the two datasets
+and easier comparison of slopes. Bottom: Ratio (F241/F239) of the
+fission yields of 241Pu and239Pu for the hypothetical LEU dataset.
+Realized F235 ranges for RENO and Daya Bay datasets are also pic-
+tured.
+a further improvement in precision to (1.3%, 21.7%, 16.4%,
+30.8%). Thus, we conclude that it may be worthwhile for
+Daya Bay to consider a more diversified fuel content binning
+scheme in a future analysis of its final full-statistics IBD yield
+dataset. This observation may also be applicable to other high-
+statistics datasets spanning many LEU reactor cycles, such as
+those recorded by RENO and DANSS [12].
+IV.
+FUTURE IMPROVEMENTS FROM NEW
+MEASUREMENTS AT MULTIPLE CORE TYPES
+We now turn to consideration of future improvements in
+global knowledge of isotopic IBD yields by performing new
+measurements at a range of different reactor core types. We
+will begin by considering the most imminently-achievable
+next steps: short baseline measurements of a single LEU
+core over a full fuel cycle, and a subsequent systematically-
+correlated measurement at an HEU using the same νe de-
+tector.
+We will then proceed to study possible improve-
+ments gained by making measurements at mixed-oxide and
+plutonium-burning fast reactor core types.
+A.
+Benefits of New HEU and LEU Measurements
+Some benefits of new measurements of IBD yields at short
+distances from a full LEU reactor core cycle have already been
+discussed in the literature [61], and have served as part of the
+physics motivation for the NEOS-II experiment [89]. In par-
+ticular, this configuration enables access to a wider range of
+F239 and F235 values beyond those achieved at θ13 exper-
+iments sampling multiple cores, which should result in im-
+proved σ239 constraints. When coupled with a systematically-
+correlated HEU-based measurement, which could be achieved
+
+8
+via two site deployments of the same detector system, di-
+rect constraints on σ238 may exceed the claimed precision
+of the summation prediction of Mueller et al. [59]. Multi-
+ple current or near-future efforts, such as PROSPECT-II [79]
+or MAD [78], are well-suited to realize part or all of this com-
+bined LEU-HEU measurement program.
+Such a setup would also broaden access to LEU fuel content
+regimes with less linear relationships between F239 and F241,
+allowing for improved constraint of σ241. This improvement
+was demonstrated above for the hypothetical LEU measure-
+ments in Figure 4. Realized effective F239 ranges for Daya
+Bay and RENO are also highlighted with shaded bands; we
+note that offsets in median F235 (and, while not pictured, also
+F241/F239) between hypothetical LEU and Daya Bay/RENO
+cases is due to the specifics of the single cycle core loading
+simulated in Ref. [71]. A new short-baseline LEU measure-
+ment set can capture periods earlier and later in the fuel cycle
+of a conventional LEU core with respect to RENO and Daya
+Bay, when relative contributions of 239Pu and
+241Pu fissions
+deviate most strongly from their cycle-integrated mean. For
+the hypothetical short-baseline LEU measurement, F239/F241
+varies roughly 6%, from 17% to 23%, over a cycle. Daya
+Bay’s and RENO’s F241/F239 ratios, meanwhile vary by only
+3% or less, with maximums and minimums of 20% and 17%,
+respectively [25, 55].
+The extent to which these HEU and LEU measurements can
+improve constraints on σ241 has so far not been investigated in
+the literature. To do so, we apply the four-parameter yield fit
+of Eq. 5 to the hypothetical HEU and LEU datasets described
+in Section II B, Table I, and Figure 1. Table II gives the result-
+ing precision in measurements of the four isotopic IBD yields
+probed by this new HEU+LEU dataset. The most striking dif-
+ference with respect to the current global dataset is the sub-
+stantial improvement in knowledge of 239Pu and 241Pu yields.
+Uncertainties in σ239 and σ241 are improved from 25.2% and
+42.6% in the existing dataset to 4.6% and 10.5%, respectively,
+greater than four-fold improvement in both values. As illus-
+trated in Figure 5, this improvement can be partially attributed
+to the reduction in degeneracy between these two isotopes’ fis-
+sion fraction variations over a full LEU fuel cycle. If all mea-
+surements are instead performed with a 1 ton detector, more
+closely approximating the expected size of the MAD detec-
+tor, uncertainties are similar in size, with σ235,238,239,241 shift-
+ing from (1.6%, 11.2%, 4.6%, 10.5%) for the PROSPECT-II
+sized detector case to (1.62%, 11.7%, 6.1%, 14.6%) for the
+MAD detector case. Thus, the HEU+LEU deployment sce-
+nario may yield major benefits for both physics-oriented or
+smaller applications-oriented future detectors.
+As noted in Ref. [61], σ238 constraints are also significantly
+improved, primarily due to the correlated nature of the detec-
+tor systematics assumed between the HEU and LEU measure-
+ments. If this correlation is removed, or if the chosen opti-
+mistic 1% HEU thermal power uncertainties are increased to
+the currently-achievable 2% level, precision in knowledge of
+the 238U yield is substantially reduced – to 18.1% and 17.2%
+for these two cases, respectively – while precision in knowl-
+edge of the 241Pu yield is virtually unchanged. Thus, follow-
+ing the next generation of short-baseline HEU and LEU mea-
+surements, the precision of knowledge of the 241Pu yield may
+rival that of its sub-dominant 238U counterpart, and will be
+less dependent on a detailed understanding of host reactors’
+thermal powers and on movement-induced changes in detec-
+tor response. At this point, direct νe -based measurements of
+241Pu fission attributes may begin to have useful application
+in testing the general accuracy of nuclear data knowledge for
+this isotope – similar to the value provided by νe -based con-
+straints of 238U from the current global dataset.
+B.
+Benefits from MOX Reactor Measurements
+Reactors burning mixed-oxide (MOX) fuels are another
+promising venue for performing IBD yield measurements
+with unique Fi combinations.
+In particular, the RG-MOX
+measurement case may be an imminently realizable one, given
+the presence and operation of RG-MOX commercial cores in
+Europe and Japan. The 50% reactor-grade mixed-oxide (RG-
+MOX) core described in Section II B features F239 far higher
+than an LEU core and broad variations in F241 from nearly
+15% at reactor start-up to roughly 25% after one cycle. Ra-
+tios F239/F241 vary much more widely from cycle beginning
+(27%) to end (45%) compared to the LEU reactor case above.
+Amidst these substantial fission fraction variations, 238U frac-
+tions remain relatively consistent between LEU and RG-MOX
+cases, offering further opportunity for reduction in degeneracy
+between 238U and the other isotopes.
+Addition of a hypothetical ten-datapoint IBD yield dataset
+from this RG-MOX reactor core provides substantial enhance-
+ments in IBD yield precision when added to those of the short-
+baseline HEU and LEU datasets, which are also summarized
+in Table II. Expected precision of yields σ239 and σ241 are im-
+proved by another factor of ∼ 2 and ∼ 3 respectively when the
+hypothetical RG-MOX is added to the fit alongside the hypo-
+thetical HEU and LEU datasets. Meanwhile, σ238 yield preci-
+sion is also tightened to 9.7% expected relative uncertainty.
+Correlations between yield fit parameters for this case are
+also pictured in Figure 5, and appear further reduced between
+239Pu and 241Pu with respect to the hypothetical HEU+LEU
+case. As with the HEU+LEU case, if measurements are per-
+formed instead with a MAD-sized 1 ton detector target, only
+modest degradation in precision is seen: σ235,238,239,241 un-
+certainties shift from (1.6%, 9.7%, 2.2%, 3.4%) for a 4 ton
+target to (1.6%, 10.3%, 2.5%, 3.9%) for a 1 ton target. un-
+certainty. On the other hand, if the correlation between the
+reactor measurements are removed, or if the chosen opti-
+mistic 1% HEU thermal power uncertainties are increased to
+the currently-achievable 2% level, precision in knowledge of
+the 238U and 241Pu yields are reduced—to 14.9%, 15.4% and
+4.3%, 5.0% respectively— and are moderately worse than the
+theoretical yields. Comparing this with the HEU+LEU case
+where the precision achievable on 238U yield is 11.1%, the
+improvement provided by the addition of RG-MOX reactor
+data doesn’t fully compensate for the loss in precision due to
+the lack of correlation or a reduction in thermal power uncer-
+tainty.
+With measurements at three reactor types – HEU, LEU, and
+
+9
+Case
+Description
+Precision on σi (%)
+235U 238U 239Pu 240Pu 241Pu
+-
+Existing Global Data
+1.3
+26.4
+25.2
+-
+42.6
+1
+HEU + LEU
+1.6
+11.1
+4.6
+-
+10.5
+3
+HEU + LEU + RG-MOX
+1.6
+9.7
+2.2
+-
+3.4
+2
+HEU + LEU + WG-MOX
+1.6
+9.9
+2.5
+-
+3.6
+4
+HEU + LEU + Fast
+1.6
+10.9
+4.6
+27.2
+10.3
+5
+All
+1.6
+9.5
+2.1
+23.6
+3.3
+6
+All, Uncorrelated
+1.5
+14.3
+2.1
+36.2
+4.2
+-
+Model Uncertainty [66]
+2.1
+8.2
+2.5
+-
+2.2
+TABLE II. Constraints on IBD yields of 235U, 238U, 239Pu, 240Pu, and 241Pu, from future hypothetical datasets from LEU and HEU reactors,
+given as a percentage of the best fit yield. For all cases unless noted, detector systematic uncertainties are assumed to be correlated between
+measurements, and a 75% external constraint is used for 241Pu and for 240Pu when applicable. The ‘All’ case considers inclusion of HEU, LEU,
+RG-MOX, VTR and PFBR yield measurements employing the same detector. Model prediction uncertainties from [66] are also provided.
+5.8
+6
+6.2
+6.4
+/fission]
+2
+cm
+-43
+ [10
+235
+σ
+7
+8
+9
+10
+11
+12
+13
+14
+/fission]
+2
+cm
+-43
+ [10
+238
+σ
+Correlation: -0.383
+5.8
+6
+6.2
+6.4
+/fission]
+2
+cm
+-43
+ [10
+235
+σ
+4
+4.1
+4.2
+4.3
+4.4
+4.5
+4.6
+4.7
+4.8
+/fission]
+2
+cm
+-43
+ [10
+239
+σ
+Correlation: 0.772
+5.8
+6
+6.2
+6.4
+/fission]
+2
+cm
+-43
+ [10
+235
+σ
+5.2
+5.4
+5.6
+5.8
+6
+6.2
+6.4
+6.6
+6.8
+/fission]
+2
+cm
+-43
+ [10
+241
+σ
+Correlation: 0.727
+7
+8
+9
+10
+11
+12
+13
+14
+/fission]
+2
+cm
+-43
+ [10
+238
+σ
+4
+4.1
+4.2
+4.3
+4.4
+4.5
+4.6
+4.7
+4.8
+/fission]
+2
+cm
+-43
+ [10
+239
+σ
+Correlation: -0.511
+7
+8
+9
+10
+11
+12
+13
+14
+/fission]
+2
+cm
+-43
+ [10
+238
+σ
+5.2
+5.4
+5.6
+5.8
+6
+6.2
+6.4
+6.6
+6.8
+/fission]
+2
+cm
+-43
+ [10
+241
+σ
+Correlation: -0.681
+4
+4.1 4.2
+4.3
+4.4
+4.5
+4.6
+4.7
+4.8
+/fission]
+2
+cm
+-43
+ [10
+239
+σ
+5.2
+5.4
+5.6
+5.8
+6
+6.2
+6.4
+6.6
+6.8
+/fission]
+2
+cm
+-43
+ [10
+241
+σ
+Correlation: 0.490
+FIG. 5. Isotopic IBD yield contours for a combined fit of hypothetical HEU, LEU, and RG-MOX datasets. In each panel, fits are marginalized
+over the undepicted isotopes. Correlation coefficients between each pair of isotopes are provided in the legend.
+MOX – with a common detector, direct IBD-based constraints
+on νe production by the four primary fission isotopes may
+be expected to rival or exceed the precision of conversion-
+based predictions. Most of these direct isotopic yield uncer-
+tainties are also smaller and more well-defined in origin than
+the O(5%) uncertainty attributed to summation predictions for
+these isotopes. Thus, with a global HEU+LEU+MOX dataset,
+one could generate IBD-based reactor νe flux predictions for
+many existing or future reactor types free from biases known
+to be present in conversion-predicted models without sacrific-
+ing relative model precision.
+Expected isotopic IBD yield measurement precision de-
+livered by instead combining a ten datapoint weapons-grade
+mixed-oxide (WG-MOX) measurement with the hypothetical
+HEU and LEU datasets has also been considered. IBD yield
+uncertainties for a HEU+LEU+WG-MOX measurement set
+are slightly worse than a HEU+LEU+RG-MOX set for σ238,
+σ239, and σ241 as shown in Table II. Similarity in results be-
+tween MOX fuel types should not be too surprising, since both
+WG-MOX and RG-MOX cycles roughly span a ∼ 16−17%
+
+10
+range in F239/F241 fission fraction ratios.
+It is worth noting that wide variations in F239/F241 should
+also expected to be provided by conventional LEU cores burn-
+ing entirely fresh fuel, such as would occur upon first oper-
+ation of a new commercial power plant [90]. In this case,
+F239/F241 fission fraction ratios should be expected to vary
+by well over 10% over course of a fuel cycle [76]. Thus, in
+lieu of MOX-based options, IBD yield measurement regimes
+including newly started commercial cores likely serve as an-
+other promising avenue for producing precise constraints on
+all main fission isotopes.
+C.
+Benefits from Fast Reactor Measurements
+Since fast fission cross-sections of many minor actinides
+– particularly 240Pu – are substantially higher than ther-
+mal fission cross-sections, fission fractions in the VTR and
+PFBR fast reactors are substantially different than those of
+the high-MOX-fraction conventional core configurations de-
+scribed in [72]. In particular, 240Pu fissions now compose a
+non-negligible fraction of the total, and, as a result, 241Pu fis-
+sion fractions are substantially lower. The addition of the two
+fast reactor dataset to the hypothetical HEU and LEU datasets
+is also summarized in Table II. The most striking product of
+introducing these datasets to the fit is the potential for set-
+ting the first-ever meaningful constraints on νe production
+by 240Pu. We find roughly comparable 240Pu yield measure-
+ments when either VTR or PFBR are fitted separately with the
+other datasets. Such a measurement could prompt new and
+deeper study of fission yields and decay data for this minor
+actinide, which plays a major role in the operation of next-
+generation fast reactor systems. The level of achievable preci-
+sion in the σ240 measurement is primarily driven by the preci-
+sion in understanding the thermal output of these fast reactor
+cores – an instrumentation challenge under active investiga-
+tion in the nuclear engineering community.
+Inclusion of fast reactor datasets generates only minor im-
+provements in the knowledge of σi for the other primary
+fission isotopes beyond that achievable with the HEU+LEU
+measurement scenario. While this results primarily from the
+general lack of knowledge of the value of σ240, it also high-
+lights the value delivered by multiple highly systematically
+correlated measurements at differing fuel composition, like
+that provided by the MOX reactor cases, in contrast to the sin-
+gle measurement provided by the relatively static composition
+of these fast reactor cores. Were F240 to evolve in a meaning-
+ful way for either core, it is likely that the isotopic IBD yield
+knowledge delivered by this core would be substantially im-
+proved.
+V.
+DISCUSSION AND SUMMARY
+After observing that the current global IBD yield dataset
+exhibits some capability to constrain antineutrino production
+by 235U, 238U, 239Pu, and 241Pu, we have investigated how
+suites of future systematically-correlated measurements at di-
+verse reactor core types can improve knowledge for these
+and other fission isotopes. We have observed that with the
+simplest combination of correlated HEU and LEU measure-
+ments using a PROSPECT-sized or MAD-sized IBD detec-
+tor, an IBD yield measurement precision of 12% or better can
+be achieved for all four fission isotopes. With a combina-
+tion of HEU, LEU, and RG-MOX datasets, all isotopic yields
+can be directly measured with a precision rivaling or exceed-
+ing the precision claimed by conversion-predicted models. If
+measurements of fast reactors are also included in the global
+dataset, first constraints of order 25% precision can be placed
+on antineutrino production by 240Pu. Beyond future measure-
+ments, we also noted other avenues for improving knowledge
+of isotopic IBD yields with current data: in particular, mea-
+surements performed over multiple LEU fuel cycles, such as
+Daya Bay and DANSS, can benefit from exploiting known
+variations in 241Pu between cycles.
+With a combined global dataset in hand from multiple
+reactor types, one can generate IBD-based reactor νe flux
+predictions for many existing or future reactor types free
+from biases known to be present in conversion-predicted
+models without sacrificing in relative model precision.
+If
+one considers the full suite of correlated HEU, LEU, RG-
+MOX and fast reactor measurements (the “All” scenario
+in Table II), the resultant data-based model would include
+(σ235, σ238, σ239, σ240, σ241,) uncertainties of (1.6, 9.5, 2.1,
+23.6, 3.3)%.
+The correlation between these achievable
+directly-constrained uncertainties has also been calculated,
+and can be seen in Figure 6, alongside those of the Huber-
+Mueller model [91]. Besides representing the similar mag-
+nitudes in uncertainty, Figure 6 shows direct measurements’
+reduced correlations between 235U, 239Pu, and 241Puwith re-
+spect to conversion predictions, which are primarily caused
+by the common experimental apparatus used at ILL for input
+fission beta measurements [92, 93].
+This kind of direct and precise understanding of all of the
+major fission isotopes’ contributions to reactor antineutrino
+emissions would represent movement into an era of ‘preci-
+sion flux physics’ offering many potential pure and applied
+physics benefits. On the applications side, it would enable
+unbiased, high-fidelity monitoring, and performing of robust
+case studies for, a broad array of current and future reactor
+types. Well-measured isotopic antineutrino fluxes could be
+compared to summation-predicted ones to provide enhanced
+benchmarking and improvement of nuclear data associated
+with the main fission isotopes and their daughters, as well
+as the first meaningful integral datasets for validating the nu-
+clear data of 240Pu.
+These models and correlated datasets
+would allow for precise independent tests of each of the four
+IBD yield predictions provided by the Huber-Mueller model,
+enabling thorough investigation of the hypothesis that mis-
+modelling of one or more isotopes’ yields is responsible for
+the reactor antineutrino anomaly. Precise and reliable IBD-
+based flux constraints would also improve the reach of be-
+yond standard model searches with signal-dominated coherent
+neutrino-nucleus scattering detectors [3]. Finally, by probing
+for persistent residual IBD yield deficits common to all iso-
+
+11
+1.6
+-2.4
+1.6
+3.6
+2.0
+-2.4
+9.5
+-3.2
+-12.8
+-4.6
+1.6
+-3.2
+2.1
+3.6
+1.9
+3.6
+-12.8
+3.6
+23.6
+7.0
+2.0
+-4.6
+1.9
+7.0
+3.3
+ 
+235
+U
+ 
+238
+U
+ 
+239
+Pu
+ 
+240
+Pu
+ 
+241
+Pu
+ 
+ 
+235
+U
+ 
+238
+U
+ 
+239
+Pu
+ 
+240
+Pu
+ 
+241
+Pu
+ 
+15
+−
+10
+−
+5
+−
+0
+5
+10
+15
+20
+25
+Uncertainty [%]
+2.4
+2.6
+2.5
+8.2
+2.6
+2.9
+2.7
+100.0
+2.5
+2.7
+2.6
+ 
+235
+U
+ 
+238
+U
+ 
+239
+Pu
+ 
+240
+Pu
+ 
+241
+Pu
+ 
+ 
+235
+U
+ 
+238
+U
+ 
+239
+Pu
+ 
+240
+Pu
+ 
+241
+Pu
+ 
+15
+−
+10
+−
+5
+−
+0
+5
+10
+15
+20
+25
+Uncertainty [%]
+FIG. 6. Left: Uncertainties in isotopic IBD yield measurements based on a hypothetical global dataset including HEU, LEU, RG-MOX, and
+fast reactor IBD yield measurements. Diagonal elements correspond to the uncertainty in isotopic yields given for the “All” case in Table II,
+while off-diagonal elements describe the correlations between them. The values are extracted by taking the square root of the corresponding
+elements of the correlation matrix and are assigned a negative value where the correlations are negative. Full covariance matrices are provided
+in the supplementary materials accompanying this paper. Right: Uncertainties in IBD yields predicted by the Huber-Mueller model [66]. Since
+there are no theoretical models predicting σ240, we assign 100% uncertainty on it.
+topes with respect to conversion or summation models, the
+community can search for enduring hints of sterile neutrino
+oscillations, even in the presence of other confounding effects,
+such as neutrino decay or wave packet de-coherence [94]. We
+encourage the use of the forecasted flux uncertainty matrix
+provided above and in the supplementary materials as input
+for future physics sensitivity and use case studies; these ex-
+ercises would help to directly demonstrate the value of this
+achievable advance reactor neutrino flux knowledge.
+VI.
+ACKNOWLEDGEMENTS
+This work was supported by DOE Office of Science, un-
+der award No. DE-SC0008347, as well as by the IIT College
+of Science. We thank Anna Erickson, Jon Link, and Patrick
+Huber for useful comments and discussion, and Nathaniel
+Bowden and Carlo Giunti for comments on early manuscript
+drafts.
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+[91] Both hypothetical future and Huber-Mueller model IBD yield
+covariance matrices are included in supplementary materials
+accompanying this paper.
+[92] F. Von Feilitzsch, A. A. Hahn,
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+[94] C. A. Arg¨uelles, I. Esteban, M. Hostert, K. J. Kelly, J. Kopp,
+P. A. N. Machado, I. Martinez-Soler, and Y. F. Perez-Gonzalez,
+(2021), arXiv:2111.10359 [hep-ph].
+

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+Superconductivity of anomalous pseudospin
+Han Gyeol Suh1, Yue Yu1,2, Tatsuya Shishidou1, Michael Weinert1, P. M. R. Brydon3, and Daniel F. Agterberg1
+1Department of Physics, University of Wisconsin, Milwaukee, Wisconsin 53201, USA
+2Department of Physics, Stanford University, 476 Lomita Mall, Stanford, CA 94305, USA and
+3Department of Physics and MacDiarmid Institute for Advanced Materials and Nanotechnology,
+University of Otago, P.O. Box 56, Dunedin 9054, New Zealand
+In materials with both time-reversal (T) and inversion symmetry (I), superconductivity is formed
+by pairing fermion pseudospin partners at momenta k and −k. Typically, pseudospin shares the
+same symmetry properties as usual spin-1/2. Here we consider non-symmorphic materials with mo-
+mentum space spin-textures that exhibit an anomalous pseudospin with different symmetry prop-
+erties than usual spin-1/2. We provide a comprehensive list of space groups for which anomalous
+pseudospin occurs on planes in momentum space and carry out a complete categorization and anal-
+ysis of superconductivity for Fermi surfaces centered on all possible T, I invariant momenta (TRIM)
+in these planes. We show that superconductivity from this anomalous pseudospin leads to a vari-
+ety of unusual consequences for superconductivity including: extremely large Pauli limiting fields
+and residual Knight shifts for pseudospin singlet superconductors; field induced pair density wave
+states; field induced pseudospin singlet to pseudospin triplet transitions; fully gapped ‘nodal’ super-
+conductors; and additional insight into the breakdown of Blount’s theorem for pseudospin triplet
+superconductors. We apply our results to UPt3, BiS2-based superconductors, Fe-based supercon-
+ductors, and paramagnetic UCoGe.
+I.
+INTRODUCTION
+Momentum space spin-textures of electronic bands are known to underlie spintronic and superconducting properties of
+quantum materials [1–3]. In the spintronics context, Rashba-like spin textures allow control of electronic spin through
+applied electric fields [1, 3].
+In superconductors, these same spin textures lead to unusual and counter-intuitive
+magnetic response, such as the robustness of spin-singlet superconductivity to applied magnetic fields, pair density
+wave states, and singlet-triplet mixing [2]. While such spin-textures are common when inversion symmetry is broken,
+it has been realized that these can occur when inversion symmetry is present. This has lead to the notion of hidden
+spin-textures [4] and locally non-centrosymmetric superconductivity [5], where inversion related sectors each allow a
+Rashba-like spin-texture due to the local inversion symmetry breaking. These spin-textures are of opposite sign on
+the two sectors, so that global inversion symmetry is restored. These hidden spin-textures allows the novel physics
+associated with spin-orbit coupling (SOC) to emerge even when inversion symmetry is not broken. It further allows
+for new physics to emerge. One notable example is a field induced transition from an even-parity (pseudospin singlet)
+to odd-parity (pseudospin triplet) observed in CeRh2As2 [6–9].
+Key to observing novel physics associated with these spin-textures in inversion symmetric materials, is that the
+inversion related sectors are weakly coupled [5, 9–11]. Theoretical proposals for how to achieve this fall under two
+approaches: the first is to tailor weak coupling between the inversion related sectors, for example by separating two
+inversion symmetry related layers so that the interlayer coupling is weak [6]; the second is to exploit symmetries that
+ensure that this inter-sector coupling vanishes. The symmetry based approach has been applied to points and lines
+in momentum space. Examples include two-dimensional (2D) transition metal dichalcogenides near the K-point [12]
+and non-symmorphic symmetries near the X −M line in BaNiS2 with space group 129 (P4/nmm) [10]. In these cases,
+the only energy splitting between the inversion-related sectors is due to SOC - a situation conceptually similar to
+materials with broken inversion symmetry, where the usual two-fold pseudopsin degeneracy is broken solely by SOC.
+Here we generalize this symmetry based approach by identifying electronic band degeneracies that are split solely
+by SOC in materials with both inversion, I, and time-reversal, T, symmetries. This requires bands that are at least
+four-fold degenerate when SOC is ignored. Such band degeneracies are not generic and require symmetries beyond the
+usual two-fold pseudospin (or Kramers) degeneracy that arises from TI symmetry. Here we focus on 2D momentum
+planes, nodal planes, where this occurs. This is the largest region in momentum space for which the required four-
+fold electronic degeneracies can appear when SOC is ignored. As discussed in a variety of contexts [13–16], such
+nodal planes arise in non-symmorphic crystal structures. Here we provide a complete list of space groups for which
+this occurs and provide symmetry based kp theories for all time-reversal-invariant momenta (TRIM) on these nodal
+planes. As discussed later, many relevant superconductors exhibit Fermi surfaces near these TRIM. We find that the
+SOC-split electronic states on these nodal planes generically exhibit a pseudospin that has a different symmetry than
+that of usual spin-1/2 fermions (this generalizes a result we found for space group P4/nmm [9]). Here we name this
+anomalous pseudospin and examine the consequences of this anomalous pseudospin on superconductivity. We find
+that this anomalous pseudospin plays a central role on the superconducting magnetic response and on the properties
+arXiv:2301.11458v1  [cond-mat.supr-con]  26 Jan 2023
+
+2
+of spin-triplet superconductivity. Our results complement and provide further insight on earlier nodal and topological
+classifications of superconductivity in non-symmorphic materials [17–21].
+In this paper we begin by defining anomalous pseudospin on nodal momenta planes, we then characterize all possible
+symmetry based kp theories near TRIM points on these nodal planes. Using these kp theories, we analyse the magnetic
+response and nodal excitations of superconducting states formed from anomalous pseudospin. We apply this analysis
+to a series of materials that exhibit Fermi surfaces that lie on or near these nodal planes. More specifically we reveal
+how anomalous pseudospin: explains critical fields that far exceed the Pauli field in BiS2-based materials [22] and the
+observed magnetic response 3D Fe-based superconductors [23]; identifies which space groups and TRIM are ideal to
+find a field induced even parity to odd parity transition akin to that observed in CeRh2As2 [7]; provides insight into
+the gap symmetry of UPt3 [24]; and shines new light on re-entrant superconductivity in UCoGe [25].
+II.
+ANOMALOUS PSEUDOSPIN: SYMMETRY ORIGIN
+Our aim is to exploit symmetry to find nodal plane band degeneracies that are lifted solely by SOC. As discussed
+below, once these band degeneracies are lifted, a two-fold pseudospin degeneracy will remain. We find that generically,
+the pseudospin that results from this procedure does not share the same symmetry properties as usual spin 1/2 and
+hence we name this anomalous pseudospin.
+Pseudospin describes the two-fold Kramers degeneracy that arises at each momentum point k when the product of
+time-reversal T and inversion I symmetries, TI, is present. The product TI is anti-unitary and for fermions satisfies
+(TI)2 = −1, ensuring at least a two-fold degeneracy. It is often the case that this pseudospin behaves as spin-1/2
+under rotations [26]. However, when symmetries beyond TI are present, it is possible that this is not the case. One
+example of this is the angular momentum jz = ±3/2 electronic states that arise when cubic symmetry or a three-fold
+rotation axis is present [2, 27, 28]. In the latter case, this gives rise to so-called type-II Ising superconductivity in 2D
+materials [28, 29] where large in-plane critical fields appear when the Fermi surface is sufficiently close to momentum
+points with this three-fold rotation symmetry. In our case, the anomalous pseudospin appears on momentum planes
+in the Brillouin zone, allowing a larger phase space for the physical properties of anomalous pseudospin to manifest.
+To ensure the requisite band degeneracy on a nodal plane, consider the symmetry elements that keep a momentum
+point on the plane invariant (here taken to be normal to the ˆn axis). These are {E, ˜
+Mˆn, TI, T ˜C2,ˆn}, where
+˜
+Mˆn
+is a translation mirror symmetry and ˜C2,ˆn is a translation two-fold rotation symmetry. Their point group rotation
+and translation component can be denoted using Seitz notation, for example ˜
+Mˆn = {Mˆn|t1, t2, t3} where Mˆn is a
+point group mirror symmetry along ˆn and (t1, t2, t3) is a fractional translation vector. Since we are searching for a
+degeneracy that appears without SOC, we consider orbital or sublattice degrees of freedom for which (TI)2 = 1. The
+only remaining symmetry that can enforce a two-fold degeneracy is T ˜C2,ˆn, since this is anti-unitary, it must satisfy
+(T ˜C2,ˆn)2 = −1 to do so. Since T commutes with rotations, this implies ˜C2
+2,ˆn = −1. When operating on orbital or
+sublattice degrees of freedom, ˜C2
+2,ˆn is typically 1, suggesting it is not possible to have the required degeneracy. However,
+in non-symmorphic groups, ˜C2,ˆn can be a screw axis, for which it is possible to satisfy ˜C2
+2,ˆn = −1. In particular, using
+Seitz notation ˜C2,ˆn = {C2ˆn|t1, t2, 1/2} (here t1 and t2 correspond to either a half in-plane translation vector or to no
+translation) we have ( ˜C2,ˆn)2 = {E|0, 0, 1}. When operating on a state carrying momentum k, ( ˜C2,ˆn)2 is represented
+by eik·ˆn. Hence if the nodal plane sits at momentum k · ˆn = π, then ˜C2
+2,ˆn = −1 and a two-fold orbital or sublattice
+degeneracy is ensured. When spin-degeneracy is also included, these states are then four-fold degenerate when SOC
+is ignored.
+When SOC is included, it is possible to show that the TI pseudospin partners have the same Mˆn mirror eigenvalue
+(this result is generalization of that given in Ref. [9] where t1 = 0 and t2 = 0 was used). That is, labeling the two
+Kramers degenerate states as |+⟩ and TI|+⟩, both belong to the same eigenstate of ˜
+Mˆn. As a consequence, all Pauli
+matrices ˜σi made from the two states |+⟩ TI|+⟩ must all be invariant under ˜
+Mˆn. It is this feature that differs from
+usual spin-1/2. Of the three Pauli matrices σi, constructed from usual spin-1/2 states, two will be odd under ˜
+Mˆn and
+one will be even under ˜
+Mˆn. It is this symmetry distinction between the anomalous pseudospin operators (˜σi) and
+usual spin 1/2 operators (σi) that underlie the unusual superconducting properties discussed below.
+The above argument can also be applied to nodal lines generated by the symmetry elements {E, ˜C2,ˆn, TI, T ˜
+Mˆn}
+with (T ˜
+Mˆn)2 = −1 when applied to orbital or sublattice degrees of freedom.
+In this case, repeating the same
+arguments above show that SOC will also split the band degeneracy and lead to anomalous pseudospin. Here, due
+to the larger available momentum phase space, we restrict our analysis and classification to nodal planes and leave
+an analysis of nodal lines to a later work. For all space groups that host nodal planes, we develop symmetry-based
+kp theories valid near all TRIM on theses nodal planes. We emphasis these TRIM since Cooper pairs are formed by
+pairing states at momenta k and −k with the momentum origin given by a TRIM. We then consider Fermi surfaces
+
+3
+Z
+E
+B
+D
+Y
+C
+A
+Γ
+FIG. 1.
+Example from space group 14 where the green shading reveals the planes and lines in momentum space on which
+anomalous pseudospin exists. A Fermi surface located near the momentum plane kz = π (as depicted by the dark Fermi surface
+near the Z point) will have its superconducting properties governed by pairing of anomalous pseudospin. However, Fermi
+surfaces far from these planes (such as that depicted near the Γ point) will exhibit more usual superconducting properties.
+near these TRIM and discuss the resultant superconducting properties. Figure 1 illustrates our approach. Here, in
+green, we show the nodal planes and lines that exhibit anomalous pseudospin. Here we examine the properties of
+superconductivity for a Fermi surface near the Z point, which is a TRIM on the nodal plane. The properties of
+superconductivity for a Fermi surface near the Γ point, for which pseudospin is typically not anomalous, are described
+in earlier review articles [30, 31]. We note that many superconducting materials, including the examples discussed in
+this paper, exhibit Fermi surfaces near nodal planes.
+III.
+NODAL PLANE SPACE GROUPS AND SINGLE-PARTICLE kp HAMILTONIANS
+Here we identify all space groups that allow anomalous pseudospin on nodal planes and construct the corresponding
+symmetry-based kp-like Hamiltonians for all TRIM on these planes.
+A.
+Space groups with nodal planes
+To identify these nodal planes, all space groups containing inversion symmetry I = {I|0, 0, 0} and the screw axis
+˜C2,ˆn = {C2ˆn|t1, t2, 1/2} (where t1 = 0, 1/2 and t2 = 0, 1/2) were identified. For these space groups, the nodal planes
+lie on the Brillouin zone boundary. Table 1 lists the resultant space groups, point groups, nodal planes, and types
+of kp theories allowed for these space groups. As discussed in the previous section, the degeneracies of these nodal
+planes is generically lifted by SOC, yielding anomalous pseudospin.
+B.
+Symmetry based kp theories near TRIM
+To understand the consequences of anomalous pseudospin on superconductivity requires a theory for the normal
+state. Cooper pairs rely on the degeneracy between states of momenta k and −k and this degeneracy is ensured by
+both T and I symmetries. For this reason, we develop symmetry-based kp theories expanded around TRIM. To derive
+these kp-like Hamiltonians, we have used the real representations for the TRIM given in the Bilbao Crystallographic
+
+4
+Crystal Type
+Number
+Name
+Nodal planes
+kp theory classes
+Monoclinic (C2h)
+11
+P21/m
+(u, 1/2, w)
+Ctype1
+2h,1
+14
+P21/c
+(u, 1/2, w)
+Ctype1
+2h,1 , Ctype2
+2h,2
+Orthorhombic (D2h)
+51
+Pmma
+(1/2, v, w)
+Dtype1
+2h,3
+52
+Pnna
+(u, 1/2, w)
+Dtype1
+2h,3 , Dtype2
+2h,4 , 8-fold
+53
+Pmna
+(u, v, 1/2)
+Dtype1
+2h,3 , Dtype2
+2h,4
+54
+Pcca
+(1/2, v, w)
+Dtype1
+2h,3 , 8-fold
+55
+Pbam
+(1/2, v, w), (u, 1/2, w)
+Dtype2
+2h,2 , Dtype1
+2h,3
+56
+Pccn
+(1/2, v, w), (u, 1/2, w)
+Dtype1
+2h,1 , Dtype2
+2h,2 , Dtype1
+2h,3 , 8-fold
+57
+Pbcm
+(u, v, 1/2), (u, 1/2, w)
+Dtype1
+2h,3 , 8-fold
+58
+Pnnm
+(1/2, v, w), (u, 1/2, w)
+Dtype1
+2h,1 , Dtype2
+2h,2 , Dtype1
+2h,3 , Dtype2
+2h,4
+59
+Pmmn
+(1/2, v, w), (u, 1/2, w)
+Dtype1
+2h,1 , Dtype1
+2h,3
+60
+Pbcn
+(1/2, v, w), (u, v, 1/2)
+Dtype1
+2h,3 , Dtype2
+2h,4 , 8-fold
+61
+Pbca
+(1/2, v, w), (u, v, 1/2), (u, 1/2, w)
+Dtype1
+2h,3 , 8-fold
+62
+Pnma
+(1/2, v, w), (u, v, 1/2), (u, 1/2, w)
+Dtype1
+2h,1 , Dtype1
+2h,3 , 8-fold
+63
+Cmcm
+(u, v, 1/2)
+Ctype1
+2h,1 , Dtype1
+2h,3
+64
+Cmce
+(u, v, 1/2)
+Ctype2
+2h,2 , Dtype1
+2h,3
+Tetragonal (D4h)
+127
+P4/mbm
+(u, 1/2, w)
+Dtype1
+2h,3 , Dtype2
+4h,2 , Dtype2
+4h,4
+128
+P4/mnc
+(u, 1/2, w)
+Dtype1
+2h,3 , Dtype2
+2h,4 , Dtype2
+4h,2 , Dtype2
+4h,4 , Dtype1
+4h,5 , 8-fold
+129
+P4/nmm
+(u, 1/2, w)
+Dtype1
+2h,3 , Dtype1
+4h,1 , Dtype1
+4h,3
+130
+P4/ncc
+(u, 1/2, w)
+Dtype1
+2h,3 , Dtype1
+4h,1 , Dtype1
+4h,3 , 8-fold
+135
+P42/mbc
+(u, 1/2, w)
+Dtype1
+2h,3 , Dtype2
+4h,2 , Dtype2
+4h,4 , 8-fold
+136
+P42/mnm
+(u, 1/2, w)
+Dtype1
+2h,3 , Dtype2
+2h,4 , Dtype1
+4h,1 , Dtype2
+4h,2 , Dtype1
+4h,3 , Dtype2
+4h,4
+137
+P42/nmc
+(u, 1/2, w)
+Dtype1
+2h,3 , Dtype1
+4h,1 , Dtype1
+4h,3 , Dtype1
+4h,5 , 8-fold
+138
+P42/ncm
+(u, 1/2, w)
+Dtype1
+2h,3 , Dtype1
+4h,1 , Dtype2
+4h,2 , Dtype1
+4h,3 , Dtype2
+4h,4 , 8-fold
+Hexagonal (C6h)
+176
+P63/m
+(u, v, 1/2)
+Ctype1
+2h,1 , Ctype1
+6h
+, 8-fold
+Hexagonal (D6h)
+193
+P63/mcm
+(u, v, 1/2)
+Dtype1
+2h,3 , Dtype1
+6h
+, 8-fold
+194
+P63/mmc
+(u, v, 1/2)
+Dtype1
+2h,3 , Dtype1
+6h
+, 8-fold
+Cubic (Th)
+205
+Pa3
+(u, 1/2, w)
+Dtype1
+2h,3 , 8-fold
+TABLE I. Space groups with nodal planes
+server [32–34]. For these TRIM, we initially consider space group irreducible representations that do not include spin,
+which, for simplicity, we name orbital representations. These representations are either 2-fold or 4-fold degenerate
+(when spin is added, these becomes 4-fold and 8-fold degenerate respectively). The full kp-like Hamiltonians are only
+listed for the 2-fold degenerate representations. We present a partial classification of the 4-fold degenerate orbital
+representations near the end of this paper.
+In constructing the kp theories for the 2-fold orbital degenerate TRIM points, we choose τi to be Pauli matrices that
+encode the orbital degrees of freedom, and σi to be spin Pauli matrices. We take T = τ0(iσy)K where K is the complex
+conjugation operator, hence the τ2 operator is odd under time-reversal. For a given doubly degenerate space group
+representation on a TRIM, constructing its direct product leads to four irreducible point group representations. These
+four representations each correspond to an orbital operator τi, and this partially dictates the momentum dependencies
+of symmetry allowed terms in the kp Hamiltonian. We present our results for the kp Hamiltonians in Table 2. The
+first row of each box gives the type of the kp theory class and the point group representations of the orbital operators
+that are given by Pauli matrices τi. In this decomposition, the square brackets correspond to the antisymmetric τ2
+operator and remaining terms correspond to τ0, τ1, and τ3. The second row of a box gives the kp Hamiltonian, and
+the last part of a box lists the space groups and TRIM points representations that belong to the kp Hamiltonian class.
+We have tabulated the kp Hamiltonians for 122 TRIM points and we find that only 13 different kp theories appear.
+These are of two types, which we call type 1 and type 2. Type 1 kp theories have degenerate even and odd parity
+orbital basis functions. Type 2 kp theories has two degenerate orbital basis functions with the same parity symmetry.
+The generic form of these kp theories are
+H(k) = ε0,k + t1,kτ1 + tα,kτα + τβ(λk · σ) = ε0,k + Hδ(k) ,
+(1)
+
+5
+(I, τα, τβ) =
+�
+(τ1, τ2, τ3)
+for type 1 ,
+(τ0, τ3, τ2)
+for type 2 ,
+(2)
+where Hδ(k) = H(k) − ε0,k and α and β are type indices will be used the remaining context. For parity mixed, type
+1, kp theories, the degeneracy at TRIM points is not broken by SOC. This is because the non-symmorphic symmetries
+combined with topological arguments imply these TRIM must have an odd number of Dirac lines passing through
+them [35]. These Dirac lines lie in the nodal plane. Elsewhere in the nodal plane, SOC lifts the 4-fold degeneracy.
+We will discuss some consequences of these Dirac lines later. The non trivial inversion symmetry for type 1, I = τ1,
+implies the parity of the momentum functions that ε0,k = ε0,−k, t1,k = t1,−k, t2,k = −t2,−k, and λk = −λ−k.
+This form of Hamiltonian has often been used to understand locally non-centrosymmetric superconductors [2] and
+hidden spin polarization in inversion symmetric materials [11].
+In these contexts, the orbital degrees of freedom
+reside on different sectors that are related by inversion symmetry and there is typically no symmetry requirement
+that ensures the SOC dominates. The τ3 matrix is odd under inversion symmetry, allowing the odd-parity SOC
+λk to appear. Many superconductors of interest have Fermi surfaces near type 1 TRIM points, examples include:
+Fe-based superconductors, which often have electron pockets near the M point in space group 129 (classes Dtype1
+4h,1
+or
+Dtype1
+4h,3 ) [23], in this context the high Tc superconductor monolayer FeSe is of interest, since it only has Fermi surfaces
+near the M point [36]; CeRh2As2 which exhibits a field induced transition from an even parity to an odd-parity
+superconducting state [7, 8] and has Fermi surfaces near the M point in space group 129 (classes Dtype1
+4h,1
+or Dtype1
+4h,3 );
+BiS2-based superconductors [22] which has superconductivity that survives to very high fields and which has electron
+pockets near the X point in space group 129 (class Dtype1
+2h,3 ); the odd-parity heavy fermion superconductor UPt3
+[24] which has a pancake-like Fermi surface at kz = π/c in space group 193 (class Dtype1
+6h
+); and the ferromagnetic
+superconductor UCoGe [25] with space group 62 and a Fermi surface near the T point (class Dtype1
+2h,1 ).
+For type 2 kp theories, the 4-fold degeneracy is sometimes already split into 2 at the TRIM point when SOC is
+added, unlike what occurs for type 1 kp theories. This happens in classes Ctype2
+2h,2
+and Dtype2
+2h,1 . For the other type 2
+classes, this degeneracy at the TRIM point is not split. In these cases, an even number of Dirac lines pass through
+the TRIM point. These Dirac lines lie in the nodal plane. Since I = τ0 for type 2, all terms in the Hamiltonian are
+even parity, that is, unchanged under k → −k. One example where type 2 kp theories apply is in strain induced
+superconductivity in RuO2[37, 38]. Without strain, RuO2 is thought to be a non-superconducting altermagnet [39].
+When strain is applied, bands near the X-M-R-A Brillouin zone face are most strongly affected [37]. RuO2 has space
+group 136 with the R and M points belonging to classes Dtype2
+2h,4 , Dtype2
+4h,2 , or Dtype2
+4h,4 . Later we discuss the ferromagnetic
+superconductor UCoGe with space group 62 [25]. In this example we highlight the role of 8-fold degenerate points
+which exhibit some properties similar to that found for type 2 TRIM points.
+Type 1 and type 2 kp Hamiltonians share some common features that play an important role in understanding
+the properties of the superconducting states. The first is that the non-symmorphic symmetry dictates that these
+Hamiltonians are best described as two-band systems with eigenenergies given by
+E±(k) = ε0,k ±
+�
+t2
+1,k + t2
+α,k + |λk|2 = ε0,k ± εδ,k ,
+(3)
+where α is the type index in Eq. 2. The second feature is that both simplify dramatically on the nodal plane, where
+only the coefficient functions ε0,k and λk·ˆn are non-vanishing (that is t1,k = t2,k = t3,k = |λk׈n| = 0). This property
+is a direct consequence of the anomalous pseudopspin. The symmetry arguments discussed in the previous section
+enforce this condition. In particular, for momenta on the nodal plane, the mirror operator through the nodal plane,
+UM, takes the from UM = −iτβ(σ · ˆn). The requirement that these Hamiltonians obey time-reversal and inversion
+symmetries and commute with UM lead to this simple form of the kp theories in the nodal plane. The final important
+property of these kp Hamiltonians is that the SOC terms are often the leading order terms in the kp expansions, that
+is, they appear with the lowest powers of ki. This is the case for classes Ctype2
+2h,2 , Dtype1
+2h,1 , Dtype2
+2h,4 , Dtype1
+4h,2 , Dtype1
+4h,3 , and
+Dtype1
+4h,5 . This feature ensures that there exists a limit in which the SOC is the dominant single-particle interaction on
+the Fermi surface and hence the unusual magnetic superconducting response we later discuss must exist.
+IV.
+SUPERCONDUCTING STATES
+In the previous section, complete symmetry-dictated kp theories were found for anomalous pseudospin.
+These
+theories are complete in the sense that they include all operators of the form τiσj allowed by symmetry. For super-
+conductivity, the orbital degree of freedom enlarges the corresponding space of possible gap functions compared to
+the usual even-parity (pseudospin-singlet) ˜∆(k) = ψk(iσy) and odd-parity (pseudospin-triplet) ˜∆(k) = dk · σ(iσy)
+
+6
+Class
+Symmetry
+Hamiltonian
+Space Group Momenta
+Ctype1
+2h,1
+Ag + Bg + [Au] + Bu
+H = ϵ0 + (t1xkx + t1zkz)kyτ1 + t2kyτ2
++ τ3[λxkyσx + (λyxkx + λyzkz)σy + λzkyσz]
+11(C1, D1, E1, Z1), 14(Z1),
+63(R1(yz)), 176(L1(yz))
+Ctype2
+2h,2
+Ag + 2Bg + [Ag]
+H = ϵ0 + (t1xkx + t1zkz)kyτ1 + (t3xkx + t3zkz)kyτ3
++τ2[(λxxkx + λxzkz)kyσx + λyσy + (λzxkx + λzzkz)kyσz]
+14(D±
+1 D±
+2 ), 64(R±
+1 R±
+2 (yz))
+Dtype1
+2h,1
+Ag + B1g + [Au] + B1u
+H = ϵ0 + t1kxkyτ1 + t2kxkykzτ2
++ τ3[λxkyσx + λykxσy + λzkxkykzσz]
+56(S1,2), 58(R1,2)
+59(S1,2, R1,2), 62(T1,2(xz))
+Dtype2
+2h,2
+Ag + 2B1g + [Ag]
+H = ϵ0 + t1kxkyτ1 + t3kxkyτ3
++ τ2[λxkykzσx + λykxkzσy + λzkxkyσz]
+55(S±
+1 S±
+2 , S±
+3 S±
+4 , R±
+1 R±
+2 , R±
+3 R±
+4 )
+56(R±
+1 R±
+2 , R±
+3 R±
+4 ), 58(S±
+1 S±
+2 , S±
+3 S±
+4 )
+Dtype1
+2h,3
+Ag + B2g + [B1u] + B3u
+H = ϵ0 + t1kxkzτ1 + t2kzτ2
++ τ3[λxkxkykzσx + λykzσy + λzkyσz]
+51(X1,2, S1,2, U1,2, R1,2), 52(R1,2(xy), Y1,2(xyz))
+53(Z1,2(zyx), T1,2(zyx)), 54(X1,2, S1,2)
+55(U1,2(yz), X1,2(yz), Y1,2(xyz), T1,2(xyz))
+56(X1,2, Y1,2(xy))
+57(S1,2(xyz), Y1,2(xyz), Z1,2(zyx), U1,2(zyx))
+58(X1,2(yz), Y1,2(xyz))
+59(X1,2, U1,2, T1,2(xy), Y1,2(xy)), 60(X1,2, Z1,2(zyx))
+61(X1,2, Y1,2(xyz), Z1,2(zyx))
+62(X1,2, Z1,2(xz), Y1,2(xyz))
+63(T1,2(zyx), Z1,2(zyx)), 64(T1,2(zyx), Z1,2(zyx))
+127(X1,2(xyz), R1,2(xyz)), 128(X1,2(xyz))
+129(X1,2(xy), R1,2(xy)), 130(X1,2(xy))
+135(X1,2(xyz), R1,2(xyz)), 136(X1,2(xyz))
+137(R1,2(xy), X1,2(xy)), 138(X1,2(xy))
+193(L1,2), 194(L1,2(xy))
+205(X1,2(xyz))
+Dtype2
+2h,4
+Ag + B1g + B3g + [B2g]
+H = ϵ0 + t1kxkyτ1 + t3kykzτ3
++ τ2[λxkxkyσx + λyσy + λzkykzσz]
+52(T ±
+1 ), 53(U ±
+1 (yz), R±
+1 (yz))
+58(T ±
+1 , U ±
+1 (xy)), 60(S±
+1 (xy))
+128(R±
+1 ), 136(R±
+1 )
+Dtype1
+4h,1
+A1g + B2g + [A1u] + B2u
+H = ϵ0 + t1kxkyτ1 + t2kxkykz(k2
+x − k2
+y)τ2
++ τ3[λx(kxσy + kyσx) + λ3kxkykzσz]
+129(M1,2, A1,2), 130(M1,2)
+136(A3,4), 137(M1,2), 138(M1,2)
+Dtype2
+4h,2
+A1g + 2B2g + [A1g]
+H = ϵ0 + t1kxkyτ1 + t3kxkyτ3
++ τ2[λx(kykzσx + kxkzσy) + λzkxky(k2
+x − k2
+y)σz]
+127(M ±
+1 M ±
+4 , M ±
+2 M ±
+3 , A±
+1 A±
+4 , A±
+2 A±
+3 )
+128(M ±
+1 M ±
+4 , M ±
+2 M ±
+3 ), 135(M ±
+1 M ±
+4 , M ±
+2 M ±
+3 )
+136(M ±
+1 M ±
+4 , M ±
+2 M ±
+3 ), 138(A±
+1 A±
+4 , A±
+2 A±
+3 )
+Dtype1
+4h,3
+A1g + B2g + [B1u] + A2u
+H = ϵ0 + t1kxkyτ1 + t2kxkykzτ2
++ τ3[λx(kxσy − kyσx) + λzkxkykz(k2
+x − k2
+y)σz]
+129(M3,4, A3,4), 130(M3,4)
+136(A1,2), 137(M3,4), 138(M3,4)
+Dtype2
+4h,4
+A1g + A2g + B2g + [B1g]
+H = ϵ0 + t1kxky(k2
+x − k2
+y)τ1 + t3kxkyτ3
++ τ2[λx(kykzσx + kxkzσy) + λzkxkyσz]
+127(M ±
+5 , A±
+5 ), 128(M ±
+5 )
+135(M ±
+5 ), 136(M ±
+5 ), 138(A±
+5 )
+Dtype1
+4h,5
+A1g + A2g + [B1u] + B2u
+H = ϵ0 + t1kxky(k2
+x − k2
+y)τ1 + t2kxkykzτ2
++ τ3[λx(kxσy + kyσx) + λzkxkykzσz]
+128(A1,2), 137(A1,2)
+Ctype1
+6h
+Ag + Bg + [Au] + Bu
+H = ϵ0 + (t1xkx(k2
+x − 3k2
+y) + t1yky(3k2
+x − k2
+y))kzτ1
++ t2kzτ2 + τ3[λxkz(2kxkyσx + (k2
+x − k2
+y)σy)
++ (λzxkx(k2
+x − 3k2
+y) + λzyky(3k2
+x − k2
+y))σz]
+176(A1)
+Dtype1
+6h
+A1g + B2g + [A2u] + B1u
+H = ϵ0 + t1kxkz(k2
+x − 3k2
+y)τ1 + t2kzτ2
++τ3[λxkz(2kxkyσx + (k2
+x − k2
+y)σy) + λzky(3k2
+x − k2
+y)σz]
+193(A1,2), 194(A1,2(xy))
+TABLE II. Classification of kp theories. Subscript numbering of momenta represents different real representations on the same
+momentum point, and a permutation of the axes is denoted by the cyclic notation. For example, 128(X1,2(xyz)) represents
+that there are two representations X1 and X2 on X = (0, 1/2, 0) space group 128, and their local theory is obtained by
+Dtype1
+2h,3
+Hamiltonian under x → y → z → x relabelling. The representation convention is following Bilbao Crystallographic
+servera[32–34] except for the L point in 193 and 194.
+a https://www.cryst.ehu.es/ Representations and Applications → Point and Space Groups → - Representations → SG Physically
+irreducible representations given in a real basis
+
+7
+states that appear in single-band theories [30, 31]. Nevertheless, it is possible to understand some general properties
+of the allowed pairing states.
+To deduce the symmetry properties of possible pairing channels in this larger space of electronic states, it is useful
+to define gap function differently than usual [40, 41]. In particular, we take
+H =
+�
+i,j,k
+Hij(k)c†
+k,ick,j + 1
+2
+�
+i,j,k
+[∆ij(k)c†
+k,i˜c†
+k,j + h.c.].
+(4)
+where i, j are combined spin and orbital indices, h.c. means Hermitian conjugate, ck(c†
+k) is the Fermionic spin-half
+particle creation(annihilation) operator, and ˜ck(˜c†
+k) is the time reversed partner of ck(c†
+k). In the usual formulation ˜c†
+k,j
+is replaced c†
+−k,j which leads a different gap function ˜∆ij and to difficulties in interpreting the symmetry transformation
+properties of this gap function [40, 41]. For a single-band, these new gap functions become ∆(k) = ψkσ0 for even-
+parity and ∆(k) = dk · σ for odd-parity. The key use of Eq. 4 is that the ∆ij(k) transform under rotations in the
+same way as the Hij(k), allowing the symmetry properties of the gap functions to be deduced. The disadvantage of
+this approach is that the antisymmetry of the gap functions that follows from the Pauli exclusion principle is not as
+readily apparent compared to the usual formulation [40, 41].
+Enforcing the Pauli exclusion principle leads to eight types of gap functions that generalize the pseudospin-singlet
+and pseudospin-triplet of single-band gap functions. Six of these are simple generalizations of the single-band gap
+functions: τiψk and τi(dk · σ) for i = 0, 1, and 3 where ψ−k = ψk and d−k = −dk. Two are new gap functions:
+τ2(ψk · σ) and τ2dk with ψ−k = ψk and d−k = −dk. It is possible to determine whether these gaps functions are
+either even or odd-parity and this depends upon whether the kp Hamiltonian is type 1 or type 2. These gap functions
+and their parity symmetry are listed in Table III. Without further consideration of additional symmetries, the gap
+function will in general be a linear combination all the even (or odd) parity gap functions.
+To gain an understanding of the relative importance of these pairing states it is useful to project these gaps onto the
+band basis. Such a projection is meaningful if the energy separation between the two bands is much larger than the gap
+magnitude. For many of the kp Hamiltonians, due to the presence of Dirac lines, there will exist regions in momentum
+space for which this condition is not satisfied. However, these regions represent a small portion of the Fermi surface
+when the SOC energies are much larger than the gap energies, so that an examination of the projected gap is still
+qualitatively useful in this limit. Provided the superconducting state does not break time-reversal symmetry, the
+projected gap magnitude on band a can be found through [42]
+˜∆2
+± = Tr[|{Hδ, ∆}|2P±]
+Tr[|Hδ|2]
+.
+(5)
+where P±(k) = 1
+2(1 ± Hδ(k)/εδ,k) which is a projection operator onto ± band by the energy dispersion Eq. 3. This
+projected gap magnitude is related to superconducting fitness [43, 44]: if it vanishes, the corresponding gap function
+is called unfit and will have a Tc = 0 in the weak coupling limit. Table III gives the projected gap functions for
+the pairing states discussed above. The projection generally reduces the size of the gap, with the exception of the
+usual even-parity τ0ψk state (interestingly, the odd-parity τ0(dk ·σ) state has a gap that is generically reduced). This
+reduction strongly suppresses the Tc of the pairings state, where it enters exponentially in the weak-coupling limit.
+We later examine the different kp classes to identify fit gap functions since the Tc of these states will be the largest,
+given a fixed attractive interaction strength.
+On the nodal plane, the projected gap functions, shown in Table III, simplify considerably since only ε0 and λk · ˆn
+are non-zero. For both type 1 and type 2 Hamiltonians, this leads to two gap functions that are fully fit, that is,
+not reduced by the projection. For type 1 Hamiltonians, these fully fit states are τ0ψk and τ3ψk. The state τ0ψk is
+even-parity and the state τ3ψk is odd-parity and, as discussed later, these two states play an important role in the
+appearance of a field-induced transition from even to odd parity superconductivity as observed in CeRh2As2. For
+gap functions described by vectors, for example dk, the projected gaps on the nodal plane are of the form |dk · ˆn|2
+or |dk × ˆn|2. This is qualitatively different than the usual odd-parity single-band gap, where the gap magnitude is
+|dk|2. The latter requires that all three components of dk must vanish to have nodes. For the projected gaps on the
+nodal planes, this requirement less stringent: only one or two components of dk need to vanish to have nodes. This
+is closely related to the violation of Blount’s theorem on the nodal planes.
+A.
+Gap projection and the violation of Blount’s theorem
+Blount’s theorem states that time-reversal symmetric odd-parity superconductors cannot have line nodes when SOC
+is present [40]. Key to Blount’s theorem is the assumption that pseudsopsin shares the same symmetry properties
+
+8
+Type 1
+Type 2
+Gap function Inversion
+Gap projection
+Gap on nodal plane Inversion
+Gap projection
+Gap on nodal plane
+τ0ψ
++
+|ψ|2
+|ψ|2
++
+|ψ|2
+|ψ|2
+τ0(d · σ)
+−
+(t2
+1 + t2
+2)|d|2 + |d · λ|2
+t2
+1 + t2
+2 + |λ|2
+|d · ˆn|2
+−
+(t2
+1 + t2
+2)|d|2 + |d · λ|2
+t2
+1 + t2
+2 + |λ|2
+|d · ˆn|2
+τ3ψ
+−
+|λ|2|ψ|2
+t2
+1 + t2
+2 + |λ|2
+|ψ|2
++
+t2
+3|ψ|2
+t2
+1 + t2
+3 + |λ|2
+0
+τ3(d · σ)
++
+|d · λ|2
+t2
+1 + t2
+2 + |λ|2
+|d · ˆn|2
+−
+t2
+3|d|2 + |d × λ|2
+t2
+1 + t2
+3 + |λ|2
+|d × ˆn|2
+τ1ψ
++
+t2
+1|ψ|2
+t2
+1 + t2
+2 + |λ|2
+0
++
+t2
+1|ψ|2
+t2
+1 + t2
+3 + |λ|2
+0
+τ1(d · σ)
+−
+t2
+1|d|2 + |d × λ|2
+t2
+1 + t2
+2 + |λ|2
+|d × ˆn|2
+−
+t2
+1|d|2 + |d × λ|2
+t2
+1 + t2
+2 + |λ|2
+|d × ˆn|2
+τ2d
++
+t2
+2|d|2
+t2
+1 + t2
+2 + |λ|2
+0
+−
+|λ|2|d|2
+t2
+1 + t2
+3 + |λ|2
+|d|2
+τ2(ψ · σ)
+−
+t2
+2|ψ|2 + |ψ × λ|2
+t2
+1 + t2
+2 + |λ|2
+|ψ × ˆn|2
++
+|ψ · λ|2
+t2
+1 + t2
+3 + |λ|2
+|ψ · ˆn|2
+TABLE III. Classification of allowed pairing states for the kp theories. For both type I and II TRIMs we give the symmetry
+under inversion, the gap projection onto the Fermi surface, and the gap on the nodal plane. The momentum subscript indices
+k of the coefficient functions are omitted here.
+as usual spin [40]. While the violation of Blount’s theorem in non-symmorphic space groups has been demonstrated
+earlier [18, 20, 21], here we present a simple proof that closely links anomalous pseudopsin to the violation of Blount’s
+theorem.
+The existence of anomalous pseudospin requires the presence of the translation mirror symmetry ˜
+Mˆn. Consequently,
+the gap function can be classified as even or odd under this symmetry. Momenta on the nodal plane are invariant
+under ˜
+Mˆn. Hence, for these momenta, U †
+M∆(k)UM = ±∆(k) where the + (−) holds for a mirror-even (mirror-odd)
+gap function. For our basis choice UM = −iτβ(σ · ˆn). Importantly, for both types the kp theories on the nodal plane
+are given by H(k) = ε0,k +iUM(λk · ˆn). This defines the two bands E±(k) = ε0,k ±|λk · ˆn|. Written in the band basis,
+we can divide the pairing potential into intraband and interband components. On the nodal plane the intraband gap
+functions are explicitly given by
+P±∆P± = 1
+4(−UM ± i sgn(λk · ˆn)){UM, ∆} ,
+(6)
+while the interband components are
+P±∆P∓ = 1
+4(−UM ± i sgn(λk · ˆn))[UM, ∆]
+(7)
+We observe that since a mirror-even gap function satisfies [UM, ∆] = 0, the interband gap components must vanish
+on the nodal plane, i.e. the pairing only involves particles from the same band. The general form of the BdG energy
+dispersion relation is then
+±′ �
+(ε0,k ± |λk · ˆn|)2 + |∆±±|2 ,
+(8)
+where intraband gap magnitude |∆±±|2 =
+1
+4Tr[|P±∆P±|2] and ±′ is the particle-hole symmetry index which is
+independent of band index ±. Since there is no requirement that |∆±±|2 = 0, line nodes are therefore not expected
+on the nodal plane, but rather we should generically find two-gap behavior with different size gaps on the two bands.
+In contrast, for the mirror-odd gap functions we have {UM, ∆} = 0, so there is no intraband pairing on the nodal
+plane. The eigenenergies for this interband pairing state are then
+±′ �
+±|λk · ˆn| +
+�
+ϵ2
+0,k + |∆±∓|2
+�
+,
+(9)
+where intraband gap magnitude |∆±∓|2 = 1
+4Tr[|P±∆P∓|2]. The gap has line nodes provided |λk · ˆn|2 > |∆±∓|2. This
+result depends only on the mirror-odd symmetry of the gap, and not on the parity symmetry. Since gaps which are
+odd under both mirror and parity symmetry are allowed, this result shows that odd-parity gaps can have line nodes,
+thus demonstrating a violation of Blount’s theorem.
+
+9
+The origin of these nodes due to purely interband pairing implies that the nodes are shifted off the Fermi surface
+[45]. If the spin-orbit coupling is too weak, i.e. |λk · ˆn|2 < |∆±∓|2, the nodes can annihilate with each other and are
+absent. This possibility has been discussed in the context of monolayer FeSe [46] and UPt3 [47]. The analysis above
+is valid even when Dirac lines pass through the TRIM points, as is the case in most of the derived kp theories. On
+the Dirac lines, the condition |λk · ˆn|2 < |∆±∓|2 must occur and the spectrum is therefore gapped. In the Appendix
+A we present exact expressions for the energy eigenstates on the nodal plane for all possible combinations of mirror
+and parity gap symmetries.
+B.
+Unconventional pairing states from electron-phonon interactions
+To highlight how pairing of anomalous pseudospin can differ from the single-band superconductivity, it is instructive
+to consider an attractive U Hubbard model. Such a model is often used to capture the physics of electron-phonon
+driven s-wave superconductivity in single-band models. Here we show that this coupling also allows unconventional
+pairings states. In particular, odd-parity states in type 1 kp Hamiltonians. Such a state has recently likley been
+observed in CeRh2As2.
+Here we consider a local Hubbard-U attraction on each site of the lattice and do not consider any longer range
+Coulomb interactions. These sites are defined by their Wyckoff positions. Importantly, for the non-symmorphic groups
+we have considered here, each Wyckoff position has a multiplicity greater than one. Here we limit our discussion to
+Wyckoff positions with multiplicity two, which implies that there are two inequivalent atoms per unit cell.
+An
+attractive U on these sites stabilizes a local spin-singlet Cooper pair. Since there are two sites per unit cell this
+implies that there are two stable superconducting degrees of freedom per unit cell. These two superconducting states
+can be constructed by setting the phase of Cooper pair wavefunction on each site to be the same or opposite. Since
+only local interactions are included, both these two states will have the same pairing interaction. The in-phase state
+is a usual s-wave τ0ψk state. Identifying the other, out of phase, superconducting state requires an understanding of
+the relationship between the basis states for the kp Hamiltonians and orbitals located at the Wyckoff positions. In
+general, this will depend on the specific orbitals included in the theory. However, the condition that the resultant
+pairing states must be spin-singlet and local in space (hence momentum independent) allows only two possibilities
+for this additional pairing state: it is either a τ1ψk or a τ3ψk pairing state. Of these states, for two reasons, the τ3ψk
+state for type 1 Hamiltonains is of particular interest. The first reason is that this state is odd-parity and therefore
+offers a route towards topological superconductivity [48, 49]. The second reason is that of the four possible states
+(τ1ψk or τ3ψk for type 1 or type 2 Hamiltonians), this is the only state that is fully fit on the nodal plane (as can be
+seen in Table III, the other three states have zero gap projection on the nodal plane). This implies that for type 1
+Hamiltonians, the odd-parity τ3ψk and the s-wave τ0ψk states can have comparable Tc since they both have the same
+pairing interaction. In practice, the τ3ψk state will have a lower Tc than the τ0ψk state since it will not be fully fit away
+from the nodal plane. Table III reveals that this projection is given by the ratio |λk|2/(t2
+1,k +t2
+2,k +|λk|2). For classes
+Dtype1
+2h,1 , Dtype1
+4h,1 , Dtype1
+4h,3 , and Dtype1
+4h,5 , this ratio is nearly one since the SOC terms are the largest in the kp Hamiltonian.
+This suggests that these classes offer a promising route towards stabilizing odd-parity superconductivity. We stress
+that because |λk|2/(t2
+1,k + t2
+2,k + |λk|2) is slightly less than one, the Tc of the odd-parity τ3ψk will be comparable but
+less than that of the usual s-wave state. However, as we discuss later, the τ3ψk state can be stabilized over the usual
+s-wave τ0ψk state in an applied field. The identification of classes Dtype1
+2h,1 , Dtype1
+4h,1 , Dtype1
+4h,3 , and Dtype1
+4h,5
+that maximize
+the Tc of odd-parity pairing from electron-phonon interactions allows the earlier theory for a field induced even to
+odd parity transition CeRh2As2 [9] (with space group 129) to be generalized to many other space groups.
+While the above odd-parity state is only relevant for type 1 Hamiltonians, for type 2 Hamiltonians, the usual s-wave
+interaction can develop a novel structure. In particular, for the classes Ctype2
+2h,2
+and Dtype2
+2h,4 , Table II shows that the
+state τ2σy is maximally fit and has s-wave symmetry. Consequently, this state will admix with the usual s-wave τ0ψ
+state. The theory describing this admixture formally resembles that of a Hund pairing mechanism proposed to explain
+the appearance of nodes in the likely s-wave superconductor KFe2As2 [50]. The results of this analysis and a follow
+up analysis [51] allows some of the properties of this state to be understood. An important conclusion of these works
+is that an s-wave superconducting state can emerge even when pairing for the usual s-wave state is repulsive (that is
+for the Hubbard U > 0). This holds if two conditions are met: the effective interaction for the τ2σy state is attractive
+(to first approximation, this effective interaction does not depend upon U [50, 51]) and the two bands that emerge in
+the kp theory both cross the chemical potential. This s-wave pairing state naturally lead to nodes.
+
+10
+V.
+ROLE OF MAGNETIC FIELDS
+The role of anomalous pseudopsin is perhaps most unusual in response to magnetic fields. In many superconductors,
+there has been a push to drive up the magnetic field at which these are operational. Ising superconductors are one
+class of materials for which this has been successful, the in-plane critical field far surpasses the Pauli field, opening
+the door to applications [52]. Another relevant example is the field induced transition from an even parity to an
+odd-parity state observed in CeRh2As2 [7, 8].
+Recently, a powerful method to examine the response of superconductors to time-reversal symmetry-breaking fields
+has been developed by the projection onto the band-basis[42]. The form of the kp theories we have developed allows
+for the direct application of this projection method. The response of superconductivity to time-reversal symmetry-
+breaking is described by a time-reversal symmetry-breaking interaction Hh(k). A common form of TRSB Hamiltonian,
+and the one we emphasize here, is the Zeeman field interaction term, which is represented by
+Hh(k) = τ0(h · σ) ,
+(10)
+where h is a magnetic field parameter in the system. We note that our qualitative results apply to a broader range of
+TRSB Hamiltonians. In particular, this is true if the TRSB field shares the same symmetry properties as a Zeeman
+field (for example if Hh(k) describes the coupling between orbital angular momentum and an applied field).
+The theory introduces two parameters that quantify the response of superconductivity to time-reversal symmetry-
+breaking. The first parameter is an effective g-factor given by
+˜g2
+±,k,h = 2Tr[|{Hδ, Hh}|2P±]
+Tr[|Hδ|2]Tr[|Hh|2] .
+(11)
+The second parameter is the field-fitness, given by
+˜F±,k,h =
+Tr[|{{Hδ, ˜∆}, {Hδ, Hh}}|2P±]
+2Tr[|{Hδ, Hh}|2P±]Tr[|{Hδ, ˜∆}|2P±]
+.
+(12)
+This field-fitness function ranges in value from zero to one. When the field-fitness is zero, the superconducting state
+is not suppressed by the time-reversal symmetry breaking perturbation. With these two parameters, the response of
+superconductivity to applied fields and the temperature dependence of magnetic susceptibility in the superconducting
+state can be determined. With the choice of the time-reversal symmetry-breaking field as the Zeeman field, Eq. 10,
+one finds
+˜g2
+±,k,h =
+t2
+1,k + t2
+α,k + (λk · ˆh)2
+t2
+1,k + t2
+α,k + λ2
+k
+(13)
+where α is a type index that is 2 for type 1 and 3 for type 2. This agrees with results in [53] derived for Hamiltonians
+that resemble type 1 Hamiltonians. We note that the band index ± and the magnitude of field h in the field-fitness
+and the g-factor do not change the outcome, thus they will be omitted in the subsequent sections and they will be
+denoted by ˜F 2
+k,ˆh and ˜g2
+k,ˆh.
+A.
+Even parity superconductors
+It can be shown that the field-fitness parameter in Eq. 12 is 1 for all even parity states. Consequently, the magnetic
+response is governed solely by the generalized g-factor given in Eq. 13. For momenta on the nodal plane, where
+t1,k = t2,k = t3,k = λk × ˆn = 0, the g-factor vanishes for magnetic fields orthogonal to ˆn. This is a direct consequence
+of the anomalous pseudospin, since the symmetries of the Pauli matrices formed from anomalous pseudospin do
+not allow any coupling to a Zeeman field perpendicular to ˆn. An immediate consequence is that superconductivity
+survives to much stronger fields than expected for these field orientations. However, momenta that do not sit on
+the nodal plane also contribute to the superconducting state and their contribution needs to be included as well. To
+quantify this, we solve for the Pauli limiting field within weak coupling theory at T = 0. For an isotropic s-wave
+superconductor, we find
+ln
+hP,ˆh
+h0
+= −⟨ln |˜gk,ˆh|⟩k
+(14)
+
+11
+-1.0
+-0.5
+ 0.0
+ 0.5
+ 1.0
+Γ
+X
+M
+(a)
+Energy (eV)
+-1.0
+-0.5
+ 0.0
+ 0.5
+ 1.0
+Γ
+X
+M
+(b)
+FIG. 2.
+DFT bands of BiS2 near the X point (a) without and (b) with the SOC. The bands highlighted in the box are our
+focus.
+for field along direction ˆh, where h0 is the usual Pauli limiting field (found when the SOC is ignored), and ⟨·⟩k means
+an average over the Fermi surface weighted by the density of states. Below, we apply this formula to BiS2-based
+superconductors. We note that the spin susceptibility in the superconducting state can also be expressed using ˜gk,ˆh
+as well [42], and this shows that a non-zero spin susceptibility is predicted at zero temperature whenever the critical
+field surpasses h0.
+1.
+Enhanced in plane field Pauli for BiS2-based superconductors
+Here we turn to recent experimental results on BiS2-based superconductors [22, 54]. This material has the tetragonal
+space group 129 (P4/nmm) and it exhibits two electron pockets about the two equivalent X points [55]. When S is
+replaced with Se, it has been observed that the in-plane upper critical field surpasses the usual Pauli limiting field by
+a factor of 7 [54]. While it has been suggested that the local non-centrosymmetric structure is the source of this large
+critical field [54], there has been no quantitative calculation for this. Here we apply Eq. 14 to the kp theory at the
+X-point to see if it is possible to account for this large critical field. The X point in space group 129 belongs to class
+Dtype1
+2h,3 .For BiS2, the dispersion is known to be strongly two-dimensional (2D) [22, 55] so we consider the kp theory in
+the 2D limit. This kp theory is
+HBiS2 = ¯h2
+2m
+�
+k2
+x + γ2k2
+y
+�
+− µ + t2kyτ2 + λxkyτ3σx + λykxτ3σy.
+(15)
+Assuming s-wave superconductivity and accounting for the two equivalent pockets yields
+hP,ˆx = h0
+�
+t2
+2 + λ2x + |γλy|
+�
+|t2| + |γλy|(t2
+2 + λ2x)1/4
+(16)
+where h0 is the usual Pauli limiting field. For simplicity we consider γ = 1 in the following. Eq. 16 reveals that a large
+enhancement of the limiting field is possible and requires two conditions. The first is that t2 << λx, λy and second is
+that these is substantial anisotropy in λx and λy. To understand if these conditions are reasonable, we have carried
+out density-functional theory (DFT) calculations on LaO1/2F1/2BiS2 with and without SOC. DFT calculations for
+LaO1/2F1/2BiS2 were carried out by the full-potential linearized augmented plane wave method [56]. The Perdew-
+Burke-Ernzerhof form of the exchange correlation functional [57], wave function and potential energy cutoffs of 14 and
+200 Ry, respectively, muffin-tin sphere radii of 1.15, 1.2, 1.3, 1.0 ˚A for Bi, S, La, O atoms, respectively, the experimental
+lattice parameters [58], and an 15 × 15 × 5 k-point mesh were employed for the self-consistent field calculation. The
+virtual crystal approximation was used by setting the nuclear charge Z = 8.5 at O(F) sites. The resultant bands are
+shown in Fig. 2. Without SOC, the band splitting along Γ to X yields an estimate for t2. When SOC is present, the
+band splitting along the X to M yields λy and the band splitting along Γ to X yields
+�
+λ2x + t2
+2. The DFT calculated
+splittings suggest that λx is the largest parameter by a factor of 3-4, while t2 and λy are comparable. This suggests
+that the conditions to achieve a large critical field are realistic in BiS2-based superconductors. Note that the largest
+
+12
+observed Pauli fields are found when the S is substituted by Se [54]. Se has a larger SOC than S, suggesting that the
+λi parameters will be increased from what we estimate here. This is currently under exploration.
+It is worthwhile contrasting the above theory with that for Fe-based materials in which electron pockets exist near
+the M point of space group 129. The M-point is described by class Dtype1
+4h,1 . In this case, an analysis similar to to
+BiS2 gives an enhancement of only
+√
+2 of the Pauli field for in-plane fields. For c-axis fields, this class implies a
+significantly enhanced Pauli limiting field. These results are consistent with experimental fits to upper critical fields
+in Fe-based superconductors that reveal that the upper critical field for in-plane fields are Pauli suppressed while those
+for field along the c-axis are not [59]. The contrast bewteen Fe-based materials and BiS2-based materials highlights the
+importance of the different classes. In particular, the lower orthorhombic symmetry of the X point allows protection
+to in-plane fields not afforded to the M point, where the theory is strongly constrained by tetragonal symmetry.
+2.
+Pair density wave states
+In BCS theory, a spin-singlet superconductor is suppressed by the Zeeman effect.
+Under a sufficiently strong
+magnetic field, the pairing susceptibility can be peaked at non-zero Cooper pair momenta, leading to a pair density
+wave or FFLO state [60–62]. A schematic phase diagram for a centrosymmetric system is shown in the left panel
+of Fig.3. The typically first order phase transition (double solid line) between the uniform and FFLO state ends at
+a bicritical point (Tb, Hb), i.e. FFLO state only exists for T < Tb. A weak-coupling calculation reveals that for the
+usual FFLO phase, Tb/Tc = 0.56
+It is known that for locally non-centrosymmetric superconductors, FFLO-like phases can appear at lower fields Hb
+and higher temperatures Tb than the usual FFLO-like instability [5]. This is closely linked to the symmetry required
+instability to a pair density wave state for non-centrosymmetric superconductors when a field is applied [2]. For a
+non-centrosymmetric system under magnetic field, both inversion and time-reversal symmetry are broken. As a result,
+the pairing susceptibility is generically peaked at non-zero momentum and Tb = Tc. For locally non-centrosymmtric
+superconductors, inversion symmetry is locally broken on each sublattice. In an extreme case, if the two sublattices
+are decoupled, then the system effectively becomes non-centrosymmetric, and under a small magnetic field, an FFLO
+state can exists right below the zero-field superconducting Tc. However, these sublattices are generically coupled so
+that Tb = Tc is not realized in practice. Here we show that for type 1 Hamiltonians, FFLO-like states can in principle
+exist up to Tb = Tc.
+FIG. 3. Schematic phase diagram for a spin-singlet superconductor under Zeeman effect. Single solid lines denote continuous
+phase transitions while double solid lines denote first-order phase transitions.
+To show this, we consider the 2D version of class Dtype1
+4h,1
+and use the pairing susceptibility to calculate Tb and Hb.
+In 2D, class Dtype1
+4h,1
+has the following normal state Hamiltonian:
+HD4h,1 = ¯h2
+2m(k2
+x + k2
+y) − µ + t1kxkyτ1 + λxτ3(kyσx + kxσy) + Hxσx
+(17)
+λx denotes the strength of the local inversion symmetry breaking (local Rashba SOC), while t1 is the inter-sublattice
+
+13
+coupling. The pairing susceptibility for an s-wave state with gap function τ0ψk is
+χpairing(Q) = − 1
+β
+�
+ωn
+�
+(p,p+Q)∈FS
+Tr [G0(Q + p, ωn)G0(p, ωn)] ,
+(18)
+where G0 is the normal state Green’s function written in Nambu space. The FFLO state is favored, if the pairing
+susceptibility is peaked at non-zero Q. We examine the position of the bicritical point (Tb, Hb), as a function of
+λx/(t1kF ). We use the following two equations to locate the bicritical point: (1) The bicritical point lies on the BCS
+transition for the uniform superconductivity. (2) The bicritical point is a continuous phase transition between uniform
+and FFLO superconductivity, where ∇2
+Qχpairing(Q) = 0. The result is in Fig. 4. 1000 × 1000 points are sampled in
+the 2D Brillouin zone. Other parameters are t1 = 0.2, t = µ = 1. An energy cutoff of Ec = 0.1 is applied to determine
+the position of the Fermi surface.
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+x/t 1/kF
+0
+0.2
+0.4
+0.6
+0.8
+1
+Tb/Tc
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+x/t 1/kF
+0
+0.2
+0.4
+0.6
+0.8
+1
+Hb/Hb( x=0)
+FIG. 4. The position of the bicritical point (Tb, Hb), as a function of λx/kF t1.
+These results show that for zero λx/kF t1, a usual FFLO phase is found (that is Tb/Tc ≈ 0.56). As the SOC λx
+increases or equivalently, as kF decreases, Tb increases and approaches the zero-field critical temperature. In the
+meantime, Hb monotonically decreases.
+We have shown that the FFLO phase can exist up to Tb = Tc for a 2D version of class Dtype1
+4h,1 . Key is that SOC is
+the leading order term in the kp theory and this is also the case for other type 1 Hamiltonians. Hence the optimal
+conditions for an enhanced FFLO phase to occur are when fields are applied in-plane (perpendicular to the c-axis)
+for classes Dtype1
+2h,1 , Dtype1
+4h,1 , Dtype1
+4h,3 , and Dtype1
+4h,5 .
+B.
+Odd-parity superconductors
+For odd parity superconductors, the field fitness parameter ˜Fk,ˆh can become less than 1 [42]. Of particular interest
+is when ˜Fk,ˆh = 0 since this implies that Tc is unchanged by the time-reversal symmetry breaking field (this is
+independent of the effective g-factor) [42]. For anomalous pseudospin this possibility leads to two consequences not
+expected for spin-triplet states made from usual spin-1/2 fermions. The first is a field induced transition from an
+even to an odd parity state. The second is that, in spite of the presence of strong SOC, the superconducting state is
+immune to magnetic fields for all field orientations. We discuss these each in turn.
+1.
+Field induced even to odd parity transitions
+In CeRh2As2, a field induced even to odd parity transition has been observed for the field oriented along the c-axis
+in this tetragonal material [7, 8]. Earlier, we argued that this was due the anomalous pseudospin that arises on the
+Brillouin zone faces in the non-symmorphic space group P4/nmm [9]. Here we show how this can be generalized
+to other space groups that admit type 1 kp theories and determine which classes are optimal for observing such a
+transition. As discussed in Section IV C, an attractive electron-phonon like interaction gives rise to both both a usual
+
+14
+s-wave τ0ψk state and an odd-parity τ3ψk state. These two states have the same pairing interaction, but the gap
+projected onto the band basis is generally smaller for the τ3ψk state than for the τ0ψk state, implying that τ0ψk state
+has the higher Tc. For the type 1 classes Dtype1
+2h,1 , Dtype1
+4h,1 , Dtype1
+4h,3 , and Dtype1
+4h,5 , anomalous pseudospin leads to Tc’s that
+are nearly the same for the even τ0ψ and odd-parity τ3ψ states. These classes are therefore promising for observing
+a field induced transition from an even-parity to an odd-parity state.
+To determine if a such a field induced transition occurs we compute ˜Fk,ˆh for a pairing state ˜∆ = τ3. We find for
+type 1 kp theories
+˜Fk,ˆh =
+(ˆh · λk)2(t2
+1,k + t2
+2,k + |λk|2)
+|λk|2[ˆh2(t2
+1,k + t2
+2,k) + (ˆh · λk)2]
+.
+(19)
+Notice if ˆh · λk = 0, then ˜Fk,ˆh = 0 which maximizes Tc. To determine the field orientations for which ˜Fk,ˆh = 0, we
+examine the form of λk in the type 1 classes discussed above. In all these classes, the λz,k component appears with a
+higher power of momenta than the other components. Consequently, the field should be applied along the ˆz direction.
+As an example, consider the class Dtype1
+4h,3 . Here λz,k ∝ kxkykz(k2
+x − k2
+y) while λx,k ∝ ky and λy,k ∝ ky. In this case
+λk will be in-plane to an excellent approximation, and an even to odd-parity transition can be expected for the field
+along the c-axis. Consequently, classes Dtype1
+2h,1 , Dtype1
+4h,1 , Dtype1
+4h,3 , and Dtype1
+4h,5
+and, hence, space groups 56, 58, 59, 62,
+128, 129, 130, 136, 137, and 138 are promising for realizing a field-induced even to odd parity transition.
+2.
+Field immune odd-parity superconductivity
+For a conventional spin-triplet superconductor (with ∆ = dk · σ) formed from usual spin-1/2 pseudospin, SOC
+typically pins the direction of the vector dk. If the applied field is perpendicular to dk, that is if dk · ˆh = 0, then the
+Tc for this field orientation is unchanged [63–65]. Since there exists at least one field direction for which dk · ˆh ̸= 0, it
+is not expected that usual spin-triplet superconductors are immune to fields applied in all directions. For anomalous
+pseudopsin, this is not the case, it is possible for an odd-parity state to be robust against suppression for arbitrarily
+oriented magnetic fields. To show how this is possible, we calculate ˜Fk,ˆh for ∆ = τ0(dk · σ) for type 1 kp theories,
+this yields
+˜Fk,ˆh =
+[(t2
+1,k + t2
+2,k)dk · ˆh + (dk · λk)(λk · ˆh)]2
+[(t2
+1,k + t2
+2,k)ˆh2 + (λk · ˆh)2][(t2
+1,k + t2
+2,k)|dk|2 + (dk · λk)2]
+.
+(20)
+We first note that near the nodal plane, the effective g-factor is small for in-plane fields ˆn·⃗h = 0, so that for these field
+orientations superconductivity is not strongly suppressed (this is true for both even and odd-parity superconducting
+states). Hence, to show that an odd-parity state survives for all field orientations, we need to show that ˜Fk,ˆh ≈ 0
+for a field applied along the nodal plane normal where λk · ˆh becomes maximal. Near the plane we expect that
+λk · ˆh ≫
+�
+t2
+1,k + t2
+2,k. Also, (t2
+1,k + t2
+2,k) is small compared to λ2
+k, so ˜Fk,ˆh is dominated by the dk · λk term in the
+numerator. Hence if the denominator |t1,2dk| is much bigger than dk ·λk, then ˜Fk,ˆh ≈ 0. Given that λˆn is the largest
+SOC component, this requirement is equivalent to λ⊥ ≪ t1,2 and dk ⊥ ˆn (where λ⊥ is the magnitude of the SOC
+perpendicular to ˆn).
+As a relevant example of the above mechanism we consider UPt3 [24]. The superconducting state in UPt3 is believed
+to be an E2u state, with order parameter ∆ = ηp(σxky + σykx) + ηfσzkzkxky (we only include one component of
+this two-component order parameter since similar arguments hold for the second component). In general, since the
+p-wave and f-wave components have the same symmetry, both ηp and ηf are non-zero. However, theories based on
+the usual pseudospin typically require ηp = 0 due to the experimental observations discussed below [66–68]. Below we
+further show that ηp = 0 is not required for these experimental observations when anomalous pseudospin is considered.
+Indeed, these experiments are consistent with ηf = 0 and ηp ̸= 0 if pairing occurs predominantly near the nodal plane
+kz = π/c.
+Thermal conductivity experiments suggest the existence of line nodes [24]. For usual pseudospin, the state σxky +
+σykx is either fully gapped or has only point nodes. This is one reason to expect that ηp = 0. However, as illustrated
+in Table II, line nodes are expected for this state on the kz = π/c plane (note this conclusion also follows from
+Refs [18, 19, 21]). This is relevant for UPt3 since it is known to have the ‘starfish’ Fermi surface near this nodal plane
+[24] which belongs to class Dtype1
+6h
+In terms of paramagnetic suppression, the superconducting state is known to be more robust under B ⊥ z compared
+to B ∥ z [68]. For the usual pseudospin, this requires dk ∥ z, and thus ηp = 0. However, on the ‘starfish’ Fermi
+
+15
+surface, the small g-factor for B ⊥ z can serve to protect the p-wave state against paramagnetic suppression. As
+discussed above, the suppression from B ∥ z depends on the ratio λx,y/t1,2, while the g-factor for B ⊥ z depends
+on the ratio (t1,2, λx,y)/λz. The requirement λx,y/t1,2 > (t1,2, λx,y)/λz is thus sufficient to match the observations
+on the upper critical fields. If both ratios are much smaller than one, the p-wave state is immune to paramagnetic
+suppression for field along arbitrary directions. This could be relevant to the approximately unchanged Knight shift
+in the superconducting state [69]. We note that the use of ˜Fk,ˆh to determine the magnetic response relies on the
+validity of projection to a single band. However, for class Dtype1
+6h
+band degeneracies exist along three Dirac lines for
+which this projection is not valid. In Appendix B we include a detailed numerical calculation that includes interband
+effects.
+VI.
+8-FOLD DEGENERATE POINTS: APPLICATION TO UCOGE
+The arguments presented above relied on the 4-fold degeneracy at TRIM points when SOC is not present. However,
+some of these TRIM points have an 8-fold degeneracy without SOC. It is reasonable to ask if the conclusions found for
+kp theories of 4-fold degenerate points discussed above survive to 8-fold degenerate points. To address this, we have
+determined the symmetries of all orbital operators in Appendix C. We find that in most cases, the 8-fold degeneracy
+at these TRIM is split by a single SOC term of the form Oσi where O is a momentum independent 4 by 4 orbital
+matrix. In Table.IV, we give the direction of the spin component σi that appears in this SOC term at the TRIM point.
+The existence of this single SOC term ensures small effective g-factors for fields perpendicular to the spin-component
+direction. Consequently, the conclusions associated with the effective g-factor anisotropy discussed in Section V still
+hold for these 8-fold degenerate points. We note that the 8-fold degeneracy at the A point of space groups 130 and
+135 are not split by SOC and these points provide examples of double Dirac points examined in [70, 71].
+Spin Alignment
+Space Group Momenta
+σx
+54(U1U2),54(R1R2),56(U1U2),60(R1R2),61(S1S2),62(S1S2),205(M1M2)
+σy
+52(S1S2),56(T1T2),57(T1T2),57(R1R2),61(T1T2),130(R1R2),138(R1R2)
+σz
+60(T1T2),60(U1U2),61(U1U2),62(R1R2),128(A3A4),137(A3A4),176(A2A3),193(A3),194(A3)
+TABLE IV. Spin alignment of 8-fold degenerate TRIM.
+One material for which these 8-fold degenerate points are likely to be relevant is the ferromagetic superconductor
+UCoGe, which crystalizes in space group 62 (Pnma) [25]. UCoGe is believed to be a possibly topological odd-parity
+superconductor [17, 25]. Our Fermi surface (given in Figure 3) reveals that all Fermi surface sheets lie near nodal
+planes with anomalous pseudospin and further reveal tube-shaped pockets that enclose the zone-boundary S point
+and stretch along the S-R axis. Here we focus on these Fermi surfaces. This feature reasonably agrees with previous
+works [72–74] using local density approximation and the existence of these tube shaped Fermi surfaces is consistent
+with quantum oscillation measurements [75]. Here density-functional theory calculations for UCoGe were carried
+out by the full-potential linearized augmented plane wave method [56]. Perdew-Burke-Ernzerhof form of exchange
+correlation functional [57], wave function and potential energy cutoffs of 16 and 200 Ry, respectively, muffin-tin sphere
+radii of 1.4 ˚A for U and 1.2 ˚A for Co and Ge, respectively, the experimental lattice parameters [76], and an 8 × 12 × 8
+k-point mesh were employed for the self-consistent field calculation. Spin-orbit was fully taken into account in the
+assumed nonmagnetic state. Fermi surface was determined on a dense 30 × 50 × 30 k-point mesh and visualized by
+using FermiSurfer [77].
+Both the R and S points are 8-fold degenerate TRIM when SOC is not included for space group 62. Interestingly,
+from Table IV, the effective g-factors for fields along ˆy and ˆz directions are zero at the S-point and are zero for fields
+along ˆx and ˆy directions at the R-point. This indicates that superconductivity (both even and odd-parity) on the
+tube-shaped Fermi surfaces will be robust against magnetic fields applied along the ˆy direction. This is the field
+direction for which the upper critical field is observed to be the highest and for which an unusual S-shaped critical
+field curve appears [25]. We leave a detailed examination of the consequences of anomalous pseudospin in space group
+62 on superconductivity to a later work.
+VII.
+CONCLUSIONS
+Non-symmorphic symmetries allow the existence of nodal planes at Brillouin zone edges when no SOC is present.
+When SOC is added, the pseudospin on these nodal planes has different symmetry properties than usual pseudospin-
+
+16
+X
+U
+Z
+Y
+S
+R
+!
+T
+FIG. 5.
+DFT Fermi surface of UCoGe.
+1/2. Here we have classified all space groups and effective single-particle theories near TRIM points on these nodal
+planes and examined the consequences of this anomalous pseudospin on the superconducting state. We have shown
+how this enhances the Tc for odd-parity superconducting states due to attractive interactions, leads to unexpected
+superconducting nodal properties, allows large Pauli limiting fields and pair density wave states for spin-singlet
+superconductors, gives rise to field immune odd-parity superconductivity, and to field driven even to odd-parity
+superconducting transitions. While we have emphasized nodal planes on which anomalous pseudospin exists, there
+are also materials for which anomalous pseudospin develops on nodal lines and not on nodal planes. Some such
+materials also exhibit unusual response to magnetic fields [78–80], suggesting a broader range of applicability for
+anomalous pseudospin superconductivity.
+VIII.
+ACKNOWLEDGEMENTS
+DFA, HGS, and YY were supported by the US Department of Energy, Office of Basic Energy Sciences, Division of
+Materials Sciences and Engineering under Award DE-SC0021971 and by a UWM Discovery and Innovation Grant.
+MW and TS were supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials
+Sciences and Engineering under Award DE-SC0017632. PMRB was supported by the Marsden Fund Council from
+Government funding, managed by Royal Society Te Aparangi. We acknowledge useful discussions with Mark Fischer,
+Elena Hassinger, Seunghyun Khim, Igor Mazin, and Manfred Sigrist.
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+20
+Appendix A: Full excitation spectrum on the nodal plane
+On the nodal plane, the Bogoliubov de-Gennes Hamiltonian takes the form
+H =
+�
+k
+Ψ†
+k
+�
+ε0,k + τ3(λk · ˆn)(σ · ˆn)
+∆k
+Ơ
+k
+−ε0,k − τ3(λk · ˆn)(σ · ˆn)
+�
+Ψk,
+(A1)
+It is possible to classify the gap symmetry as even or odd under both inversion and mirror symmetries. For momenta
+on the nodal surface we have,
+U †
+P ∆kUP = ± ∆−k
+U †
+M∆kUM = ± ∆k
+(A2)
+where for type 1 TRIM UP = τ1 and UM = −iτ3σz and for type 2 TRIM UP = τ0 and UM = −iτ2σz. We label the
+gaps as ∆1(2),i,j where i = ± labels the parity symmetry and j = ± labels the mirror symmetry. Here, for clarity,
+we drop the k labels (note that k is unchanged by the mirror symmetry). For the type 1 TRIM, we write the gap
+functions in terms of the complete set of gap functions with the correct symmetries given in Table III as
+∆1,++ =
+ψ0τ0 + (dz · ˆn)(σ · ˆn)τ3
+∆1,+− =
+ψxτ1 + (dz × ˆn) · (σ × ˆn)τ3 + dyτ2
+∆1,−+ =(d0 · ˆn)(σ · ˆn)τ0 + (dx × ˆn) · (σ × ˆn)τ1 + ψzτ3 + (ψ × ˆn) · (σ × ˆn)τ2
+∆1,−− =
+(d0 × ˆn) · (σ × ˆn)τ0 + (dx · ˆn)(σ · ˆn)τ1 + (ψ · ˆn)(σ · ˆn)τ2
+(A3)
+where di are odd functions of k and ψi are even functions of k.
+Using Eq. A1, the corresponding quasiparticle
+excitation energies can be found to be
+E1,++ =
+±′ �
+(ϵ0 ± λ · ˆn)2 + (ψ0 ± dz · ˆn)2
+E1,+− =
+±′ ��
+ϵ2
+0 + ψ2x + (dz × ˆn)2 + d2y ± λ · ˆn
+�
+E1,−+ = ±′ �
+(ϵ0 ± λ · ˆn)2 + (ψz ± d0 · ˆn)2 + (dx × ˆn)2 + (ψ × ˆn)2 ± 2(dx × ψ) · ˆn
+E1,−− =
+±′ ��
+ϵ2
+0 + (d0 × ˆn)2 + (dx · ˆn)2 + (ψ · ˆn)2 ± λ · ˆn
+�
+(A4)
+where the prime denotes independent choices of the sign. For type 2 TRIM we similarly have
+∆2,++ =
+ψ0τ0 + (ψ · ˆn)(σ · ˆn)τ2
+∆2,+− =
+ψxτ1 + ψzτ3 + (ψ × ˆn) · (σ × ˆn)τ2
+∆2,−+ =(d0 · ˆn)(σ · ˆn)τ0 + (dx × ˆn) · (σ × ˆn)τ1 + (dz × ˆn) · (σ × ˆn)τ3 + dyτ2
+∆2,−− =
+(d0 × ˆn) · (σ × ˆn)τ0 + (dx · ˆn)(σ · ˆn)τ1 + (dz · ˆn)(σ · ˆn)τ3
+(A5)
+The quasiparticle excitation spectra for these states are
+E2,++ =
+±′ �
+(ϵ0 ± λ · ˆn)2 + (ψ0 ± ψ · ˆn)2
+E2,+− =
+±′ ��
+ϵ2
+0 + ψ2x + ψ2z + (ψ × ˆn)2 ± λ · ˆn
+�
+E2,−+ = ±′ �
+(ϵ0 ± λ · ˆn)2 + (dy ± d0 · ˆn)2 + (dx × ˆn)2 + (dz × ˆn)2 ± 2(dx × dz) · ˆn
+E2,−− =
+±′ ��
+ϵ2
+0 + (d0 × ˆn)2 + (dx · ˆn)2 + (dz · ˆn)2 ± λ · ˆn
+�
+(A6)
+
+21
+Appendix B: Magnetic susceptibility UPt3
+In the main text, we illustrated how the p-wave state in UPt3 is immune to the magnetic field along arbitrary
+directions.
+An important step is to consider the small g-factor for field B ⊥ z.
+However, the discussion is not
+complete. In the normal state, there exist 4-fold degenerate Dirac lines on the plane kz = π/c, where the g-factor is
+not small. In terms of the field fitness, Eq.20 in the main text only considered doubly degenerate bands. In principle,
+extra terms in the field fitness are needed for to describe these Dirac lines. However, the Fermi surface is not right
+on the nodal plane. This can make the Dirac lines unimportant. In this section, we will explicitly check the field
+response in the superconducting state through a numerical calculation on a tight-binding model for UPt3.
+In the following calculations, we will focus on the Knight shift (spin-susceptibility). Knight shift measures spin
+polarization at atom sites. By extracting spin susceptibility χs, one can determine pairing functions of an uncon-
+ventional superconductor. For a single-band spin-triplet superconductor, the change of Knight shift depends on the
+orientation of magnetic field with respect to the d-vector of the superconducting state. If the magnetic field is per-
+pendicular to the d-vector, the Knight shift should be a constant across superconducting Tc. If the magnetic field is
+parallel to the d-vector, the Knight shift will decrease to zero as temperature approaches zero. For the multi-band
+non-symmorphic superconductor UPt3, Knight shift is almost unchanged for all field orientations, suggesting the
+importance of spin-orbit coupling in this heavy fermion material.
+One of the Fermi surfaces (‘starfish’) of UPt3 is flat and located near the high symmetry plane kz = π/c. Zeeman
+terms Bxσx and Byσy then becomes inter-band. From non-generate perturbation theory, spin susceptibilities are
+inversely proportional to the band gap. This is different from the intra-band Zeeman effect, where susceptibilities are
+proportional to the density of states on Fermi surface, according to degenerate perturbation theory.
+Since the superconducting gap is much smaller than the band gap, inter-band susceptibilities will be unchanged
+across Tc. If the superconductivity is mainly developed on the above flat Fermi surface, then Knight shift is expected
+to be unchanged for in-plane magnetic fields, regardless of the superconducting pairing symmetry. If the d-vector is
+in-plane, then Knight shift will also be unchanged for a perpendicular magnetic field. In this section, we will explicitly
+illustrate this idea to understand the experimental results on UPt3.
+FIG. 6. Crystal structure of UPt3 with the unit vector e1 = (1, 0, 0).
+The 4 × 4 normal state Hamiltonian reads [67]:
+H = ε(k) + gz(k)σzτ3 + a1(k)τ1 + a2(k)τ2 + [gx(k)σx + gy(k)σy] τ3
+εk = 2t
+�
+i=1,2,3
+cos k∥ · ei + 2t3 cos kz − µ,
+gz(k) = gz0
+�
+i
+sin k∥ · ei
+a1(k) = 2t′ sin kz
+2
+�
+i=1,2,3
+sin k∥ · ri,
+a2(k) = 2t′ sin kz
+2
+�
+i=1,2,3
+cos k∥ · ri
+gx(k) = gx0fxfy sin kz,
+gy(k) = gy0(f 2
+x − f 2
+y ) sin kz
+fx ≡ sin k∥ · e1 − sin k∥ · e2 + sin k∥ · e3
+2
+,
+fy ≡
+√
+3 sin k∥ · e2 − sin k∥ · e3,
+(B1)
+here (kx, ky, kz) are relative to the high symmetry point (0, 0, π). Relevant vectors ei and ri can be found in Fig.6.
+τi matrices live in the sublattice space. On the high-symmetry plane kz = 0, the inter-sublattice hopping a1,2 and
+the spin-flip SOC gx,y vanish. |k, m = 1, ↑⟩ and |k, m = 2, ↓⟩ states form a pseudospin band, while |k, m = 2, ↑⟩ and
+|k, m = 1, ↓⟩ states form another band.
+
+22
+We now study spin susceptibilities. We will focus on a p-wave state in the E2u channel. Its d-vector is in-plane:
+d = ∆(T)(fx, −fy, 0). fx and fy are introduced in Eq.B1, and they transform as kx and ky. The gap magnitude is
+taken to be ∆(T) = ∆0
+�
+1 − T/Tc. t = 1, t3 = −4, gz0 = 2, µ = 12 and ∆0 = Tc = 0.001 is taken in the calculation.
+0
+0.2
+0.4
+0.6
+0.8
+1
+T/Tc
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+t'=0, gx0=gy0=0
+x
+z
+0
+0.2
+0.4
+0.6
+0.8
+1
+T/Tc
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+t'=0, gx0=gy0=0.1
+x
+z
+x,inter
+z,inter
+0
+0.2
+0.4
+0.6
+0.8
+1
+T/Tc
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+t'=0.1, gx0=gy0=0
+x
+z
+x,inter
+z,inter
+FIG. 7. Spin susceptibilities as a function of temperature, for (left) a1 = a2 = gx = gy = 0, which would be the case if the
+Fermi surface exactly lied on the high-symmetry plane. (middle) non-zero spin-flip SOC but zero inter-sublattice hopping.
+(right) non-zero inter-sublattice hopping but zero spin-flip SOC.
+To illustrate the effect of the anomalous pseudospin, we start with a toy model with zero inter-sublattice hopping
+and spin-flip SOC: t′ = gx0 = gy0. The corresponding four terms vanish in the normal state Hamiltonian: a1 = a2 =
+gx = gy = 0. In this extreme case, the spin susceptibilities are unchanged across Tc, as shown in the left panel of
+Fig.7.
+We now turn on the spin-flip SOC (gx0 and gy0), while keeping the inter-sublattice hopping t′ to be zero. hxσx
+develops an intra-band component, which will be suppressed in the superconducting state. As a result, the total
+χx deep in the superconducting state starts to decrease as function of temperature. For χz, spin-flip SOC induces
+higher-order terms in the E2u channel. The d-vector develops non-zero z-component in the band basis. This causes
+a decrease in χz. The result for gx0 = gy0 can be found in the middle panel of Fig.7. The inter-band susceptibilities
+in the normal state are included in dashed lines.
+We now turn on the inter-sublattice hopping t′, while keeping the spin-flip SOC (gx0 and gy0) to be zero. A similar
+effect is expected for χx due to the intra-band contribution. For χz, since σz is a good quantum number, χz will be
+unchanged. The result can be found in the right panel of Fig.7.
+Experimentally, the superconducting state is known to be more robust under B ∥ x compared to B ∥ z. In other
+words, the decrease in χx needs to be smaller than χz. This scenario is closer to the second limit.
+
+23
+Appendix C: 8-fold Representations
+Here, we list the symmetries of all orbital operators near the 8-fold degenerate points. The point group that keeps
+the TRIM point invariant can be found in the title. The bracket notation [·] is also used for antisymmetric operators
+which was τ2 in the main context, but in 8-fold cases, the antisymmtric component is not unique due to the higher
+degrees of freedom.
+Space group momenta
+Point group D2h
+54(U1U2)
+Ag + 2B1g + 2B2g + B3g + 2Au + B1u + B2u + [Ag] + [B3g] + [B1u] + [B2u] + 2[B3u]
+54(R1R2)
+Ag + 2B1g + 2B2g + B3g + 2Au + B1u + B2u + [Ag] + [B3g] + [B1u] + [B2u] + 2[B3u]
+56(U1U2)
+Ag + 2B1g + 2B2g + B3g + 2Au + B1u + B2u + [Ag] + [B3g] + [B1u] + [B2u] + 2[B3u]
+60(R1R2)
+Ag + 2B1g + 2B2g + B3g + Au + 2B2u + B3u + [Ag] + [B3g] + [Au] + 2[B1u] + [B3u]
+61(S1S2)
+Ag + 2B1g + 2B2g + B3g + Au + 2B1u + B3u + [Ag] + [B3g] + [Au] + 2[B2u] + [B3u]
+62(S1S2)
+Ag + 2B1g + 2B2g + B3g + Au + 2B1u + B3u + [Ag] + [B3g] + [Au] + 2[B2u] + [B3u]
+205(M1M2)
+Ag + 2B1g + 2B2g + B3g + Au + 2B1u + B3u + [Ag] + [B3g] + [Au] + 2[B2u] + [B3u]
+52(S1S2)
+Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u]
+56(T1T2)
+Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u]
+57(T1T2)
+Ag + 2B1g + B2g + 2B3g + Au + B2u + 2B3u + [Ag] + [B2g] + [Au] + 2[B1u] + [B2u]
+57(R1R2)
+Ag + 2B1g + B2g + 2B3g + Au + B2u + 2B3u + [Ag] + [B2g] + [Au] + 2[B1u] + [B2u]
+61(T1T2)
+Ag + 2B1g + B2g + 2B3g + Au + B2u + 2B3u + [Ag] + [B2g] + [Au] + 2[B1u] + [B2u]
+130(R1R2)
+Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u]
+138(R1R2)
+Ag + 2B1g + B2g + 2B3g + 2Au + B1u + B3u + [Ag] + [B2g] + [B1u] + 2[B2u] + [B3u]
+60(T1T2)
+Ag + B1g + 2B2g + 2B3g + 2Au + B2u + B3u + [Ag] + [B1g] + 2[B1u] + [B2u] + [B3u]
+60(U1U2)
+Ag + B1g + 2B2g + 2B3g + Au + B1u + 2B2u + [Ag] + [B1g] + [Au] + [B1u] + 2[B3u]
+61(U1U2)
+Ag + B1g + 2B2g + 2B3g + Au + B1u + 2B2u + [Ag] + [B1g] + [Au] + [B1u] + 2[B3u]
+62(R1R2)
+Ag + B1g + 2B2g + 2B3g + 2B1u + B2u + B3u + [Ag] + [B1g] + 2[Au] + [B2u] + [B3u]
+Space group momenta
+Point group D4h
+128(A3A4)
+A1g + A2g + 2B1g + 2B2g + A1u + A2u + 2B2u + [A1g] + [A2g] + [A1u] + [A2u] + 2[B1u]
+137(A3A4)
+A1g + A2g + 2B1g + 2B2g + A1u + A2u + 2B2u + [A1g] + [A2g] + [A1u] + [A2u] + 2[B1u]
+Space group momenta
+Point group C6h
+176(A2A3)
+Ag + Bg + E1g + E2g + Au + Bu + E1u + [Ag] + [Bg] + [Au] + [Bu] + [E2u]
+Space group momenta
+Point group D6h
+193(A3)
+A1g + B2g + E1g + E2g + A1u + B1u + E1u + [A2g] + [B1g] + [A2u] + [B2u] + [E2u]
+194(A3)
+A1g + B1g + E1g + E2g + A1u + B2u + E1u + [A2g] + [B2g] + [A2u] + [B1u] + [E2u]
+TABLE V. Symmetries of orbital operators at the 8-fold degenerate points.
+