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+arXiv:2301.11976v1  [stat.ME]  27 Jan 2023
+A. Philip Dawid* and Stephen Senn
+Personalised Decision-Making without
+Counterfactuals
+Keywords: decision theory, counterfactual, potential response, intention to treat
+This article is a response to recent proposals by Pearl and others for a new approach to personalised
+treatment decisions, in contrast to the traditional one based on statistical decision theory. We argue that
+this approach is dangerously misguided and should not be used in practice.
+1 Introduction
+In recent works [1–4], Judea Pearl and collaborators have set out an approach to personalised treatment
+that is radically different from that based on traditional statistical decision theory. It is based on the
+conception that we should care, not only about the outcome that actually materialises, but also about
+the (necessarily unobserved, counterfactual) outcome that, it is supposed, would have occurred under the
+treatment that was not applied. A similar conception forms the basis of other recent work [5–7].
+We consider this approach to be dangerously misguided, and believe that real harm will ensue if it
+is applied in practice. We argue our case from a number of different viewpoints, and explain why this
+approach should not be regarded as a viable alternative to standard statistical decision theory.
+1.1 Basic set-up
+The context is that of a “target” patient suffering from a disease, for which a treatment is available. The
+treatment is far from perfect, so that not all treated patients recover, while some untreated patients may
+recover anyway. There is information available on recovery rates for treated and untreated patients; both
+these rates may depend on measured individual patient characteristics. The basic problem is to decide, on
+the basis of the target patient’s own characteristics, whether or not to treat him. A variation is how to
+prioritise patients for treatment when there are limited doses available.
+We introduce notation as follows:
+Treatment decision Binary decision variable X, coded 1 for treat, 0 for don’t treat
+Response Binary stochastic variable Y , coded 1 for recovery, 0 for no recovery
+Individual background characteristics Stochastic variable L, potentially multivariate, unaffected by
+the treatment decision
+We suppose that there are available substantial data on (L, X, Y ), from either experimental or uncon-
+founded observational studies on patients we can regard as similar to the target1, from which we can
+estimate, essentially perfectly2, the distribution of Y , conditional on L, under either treatment interven-
+1 See § 7 for further discussion of this point.
+2 In a Bayesian setting it is straightforward to relax this condition, using predictive distributions based on finite
+samples. However the main issues are most clearly expressed in the case of essentially known probabilities.
+*Corresponding Author: A. Philip Dawid: University of Cambridge: [email protected]
+Stephen Senn: Statistical Consultant, Edinburgh: [email protected]
+
+2
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+tion. That is, we know the probability Pr(Y = 1 | L = l, X ← x), for any value l of L and x = 0 or 1.
+(Here X ← x denotes an external intervention to set X to x.)
+1.2 Outline
+In § 2 we recall the straightforward decision-theoretic analysis of this problem. Then in § 3 we briefly outline
+the approach proposed by Pearl et al., followed by some critical comments in § 4. Section 5 describes this
+approach in more detail, following a logical path that relates it to other problems, in particular the use
+of general covariate information to strengthen conclusions, and the specific case of an “intention to treat”
+covariate, whose properties can be identified by combining experimental and observational data. In § 6 we
+give critical consideration to some examples from Mueller and Pearl [2]. Section 7 notes some important
+assumptions that are implicitly made in the analysis, and points out that they are unlikely to hold in
+practice. Section 8 summarises our analysis and conclusions.
+2 Decision-theoretic approach
+We first describe the standard decision-theoretic (DT) approach to treatment selection.
+2.1 Single patient decision problem
+Consider first the case of the single target patient. We have to decide whether to offer this patient treatment,
+or not.
+Having access only to the target patient’s value L = l, our objective is to choose the treatment that will
+maximise the probability of recovery. We should thus treat this patient if p := Pr(Y = 1 | L = l, X ← 1) >
+q := Pr(Y = 1 | L = l, X ← 0). That is, we should treat just when CATE(l) > 0, where CATE(l) = p − q
+is the “conditional average treatment effect”.
+If the outcome Y is not necessarily binary, for example a survival time, we need to associate a utility
+U(y) with the outcome y, and treat the patient just when E{U(Y ) | L = l, X ← 1)} > E{U(Y ) | L =
+l, X ← 0)}. Applied to the binary case this reduces to the prescription above (so long as U(1) > U(0)).
+In [8], a companion paper to this one which treats utilities explicitly, the above is termed the inter-
+ventionist utility and approach, and contrasted with the counterfactual utility and approach implicit in [2]
+and explicit in [6, 7]. Here we restrict to the binary case and do not use utilities.
+Faced with a large collection G of patients to treat, and unlimited supplies of the treatment, managing
+each patient (each with their own value l of L) according to the above rule will maximise the number of
+recoveries. That is, any other (deterministic or randomised) decision rule that uses only the information
+on L would lead to fewer recoveries.
+Example 1. Consider a case where
+Pr(Y = 1 | L = l, X ← 1)
+=
+0.49
+Pr(Y = 1 | L = l, X ← 0)
+=
+0.21.
+The conditional average treatment effect is CATE(l) = 0.49−0.21 = 0.28. Since CATE(l) > 0, the optimal
+action is to treat this patient. If we have a large collection of similar patients, with the same value L = l,
+they should all be treated—in which case the overall proportion of recovered patients will be 49%. This is
+the best outcome that can be achieved by any treatment strategy.
+
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+3
+2.1.1 Missing information
+It may happen that, while we have full information on (L, X, Y ) for the study individuals, the value l of L
+for the target patient is not available. In that case we can not condition on L = l, and we have no option but
+to base the management of the patient on the unconditional probabilities Pr(Y = 1 | X ← x) (x = 1, 0).
+Nothing is gained by, for example, trying to impute the unknown value of L. If this is not obvious (as it
+should be), suppose we tried to do so. The recovery probabilities, conditional on a hypothesised value l for
+L, are Pr(Y = 1 | L = l, X ← x) (x = 1, 0). But as we do not know l, we need to take the expectation
+of Pr(Y = 1 | L, X ← x) over the distribution of L, when setting X ← x (which is the known marginal
+distribution of L, unaffected by the intervention). And this is just the unconditional recovery probability
+Pr(Y = 1 | X ← x). 3
+2.2 Unit selection
+Again consider a large collection G of patients i = 1, . . . , N, with individual recovery probabilities pi =
+Pr(Yi = 1 | Li = li, Xi ← 1), qi = Pr(Yi = 1 | Li = li, Xi ← 0). If we treat just those in a subset S, the
+expected number of recoveries will be �
+i∈S pi + �
+i∈G\S qi = �
+i∈G qi + �
+i∈S CATEi. Consequently, to
+maximise this expected number, we should choose S, subject to any constraints, to maximise �
+i∈S CATEi.
+If we have limited treatments available, we should thus prioritise individuals in decreasing order of their
+CATE (while of course not treating any one for whom CATE < 0.) Again, any other policy (subject to the
+same constraints) will have a smaller number of recoveries.
+2.3 Potential outcomes?
+The “potential outcome” approach to causal inference [9] conceives of the existence, even prior to treatment
+choice, of the pair of variables Y = (Y (1), Y (0)), where Y (x) denotes the value that, it is supposed, Y will
+take if intervention X ← x is applied. The pretreatment variables (Y (1), Y (0), L) are supposed to have a
+joint distribution, unaffected by treatment. With this interpretation, we have
+Pr(Y = y, L = l | X ← x) = Pr(Y (x) = y, L = l),
+(1)
+and p = E{Y (1) | L = l}, q = E{Y (0) | L = l},
+In this approach, inference is ideally desired for the “individual treatment effect”, ITE := Y (1) − Y (0),
+which can take values +1 (treatment benefits the patient), −1 (treatment harms the patient) or 0 (treatment
+has no effect). Then CATE = E(ITE | L = l). However, typically ITE is unobservable, since it is impossible
+simultaneously both to treat and not to treat the same patient. In particular, no information can be gained
+about the dependence between Y (1) and Y (0), nor about the distribution (marginal, or conditional on L)
+of ITE. All that can be inferred is the (conditional) expectation, as above, of ITE, depending as this does
+only on the individual distributions of Y (1) and Y (0), which can be identified from experimental data.
+In certain very special and atypical cases, essentially those where we have a fully deterministic and
+completely understood mechanistic system, it may be that the background knowledge L is detailed enough
+to support perfect prediction of the eventual response, under either intervention. Then we will know, in
+advance of treatment choice, both potential outcome variables. In this case p = Y (1), q = Y (0), and
+CATE = ITE. Clearly we should treat as many of those who will (we know for sure) benefit from the
+treatment as we can.
+3 With finite data, on taking account of known structure in the interventional distributions of (L, Y ) it may be possible
+to improve the estimation of Pr(Y = 1 | X ← x). But this still remains what we need to focus on.
+
+4
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+However, in typical cases perfect prediction is impossible, and then it is arguable whether the potential
+responses even have any meaningful existence. In any case, there is nothing to be gained by trying to impute
+potential responses: as in § 2.1.1 above (taking L = Y), we should again simply focus on CATE = p − q.
+In summary, consideration of potential responses (even if regarded as meaningful) does not add any
+value to the decision-theoretic approach.
+3 The approach of Mueller and Pearl [2]
+In contrast to the above decision-theoretic approach, Mueller amd Pearl [2] (henceforth MP) opt to take
+potential outcomes seriously, and focus attention on ITE = Y (1)−Y (0). They argue that we should ideally
+aim to treat those patients having ITE = 1, for whom the treatment made a difference: they would not
+have recovered without it. It would be wasteful to treat a patient with ITE = 0, for whome the treatment
+made no difference, and positively harmful to treat a patient with ITE = −1, who would have recovered if
+untreated, but not if treated.
+However, this ideal is unattainable, as we will not know a patient’s ITE before treatment. Concern is
+therefore transferred to the “probability of benefit”, PB = Pr(Y (1) = 1, Y (0) = 0) = Pr(ITE = 1), and
+the “probability of harm”, PH = Pr(Y (1) = 0, Y (0) = 1) = Pr(ITE = −1), which are now regarded as the
+criteria by which to assess any treatment strategy.
+But not only can we not know a patient’s ITE before the treatment decision is made, we can not even
+know it later, when the outcome Y is observed. For if we treat the patient we will observe Y = Y (1),
+but can not then observe the counterfactual outcome Y (0) relevant when we don’t treat; similarly, for an
+untreated patient we can observe Y (0), but not Y (1). So ITE is always unobservable. This means that,
+even with extensive data on other patients, it will not be possible fully to identify PB and PH. Such data
+can, however, be used to set interval bounds on these quantities. MP [2] further show how combining
+experimental and observational data can narrow these bounds. In certain very special cases the bounds
+narrow to a single point, leading to full identification of PB and PH.
+4 Comments on the approach
+Our comments on the MP programme are arranged along several dimensions.
+4.1 Philosophy
+Potential responses such as Y (0) and Y (1), first introduced by Neyman [10], have been considered as fun-
+damental to the conduct of causal inference ever since reintroduced by Rubin [9]. However this conception
+was challenged by Dawid [11, 12], who pointed out that, so far from being fundamental, they are entirely
+unnecessary, and that a fully satisfactory theory can be based on standard decision-theoretic elements.
+Indeed, there are serious philosophical objections to regarding potential responses as having real existence.
+Only if we take a fully Laplacean view of the universe, in which the future of the universe is entirely
+determined by its present state and the laws of Physics, does this make any sense at all—and even then, it
+is difficult to incorporate the whims of an unconstrained external agent who decides whether or not to give
+treatment, or to account for the effect of external conditions arising after treatment. Even under Laplacean
+determinism, our ignorance of the information needed to predict the future means that we are unable to
+make use of it. Whether or not we believe in a deep-down deterministic universe, our predictions of the
+future can only be based on the limited information we do have at our disposal, and must necessarily be
+
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+5
+probabilistic.4 Imagining what we could know or do, if only we had more information than we actually do
+have, is just pointless.
+4.2 Applicability
+Another important dimension of criticism is that the strong conditions needed for application of the MP
+theory will almost never obtain in practice. See § 7 below for details.
+4.3 Helpfulness
+The output of an MP analysis will, at very best, be point estimates of the probabilities of benefit and of
+harm—in most cases, we won’t even get these, but can only bound these quantities within an interval. But
+even when we have these quantities, it is far from clear how they help to inform treatment decisions.
+4.4 Ethics
+Our final criticism is the simplest, but most incisive. The treatment decisions made using the DT approach
+are guaranteed to be better than those made by any other decision rule, in the sense that they will maximise
+the number of recoveries in the population. So whenever the MP approach leads to different decisions, it
+will produce a decrease in the number of recoveries. We find it hard to construe this as ethical.
+5 Analysis
+Here we provide a deconstruction of the analysis of MP [2]—which should not, however, be taken as
+agreement with their arguments and interpretations. There are a number of crucial assumptions required,
+but to avoid cluttering the argument we leave these implicit, postponing specification and discussion of
+them to § 7.
+We develop the story-line in a number of stages.
+In § 5.1 we consider the case where we have access to experimental data on treatment X and response
+Y , and show how this can be used to bound the probabilities of benefit and of harm. We also discuss the
+special circumstances in which these interval bounds shrink to a point.
+In § 5.2 we further suppose that we can measure additional covariate information L on individuals. If
+we have these values in the experimental data, this additional information can lead to a narrowing of the
+bounds for PB and PH for the target case, even when L for that case is unobserved.
+Section 5.3 introduces a particular, potentially useful, covariate, “intention to treat”, X∗—the treat-
+ment that a patient (or their doctor) would like to choose, if unconstrained. This may well be informative
+about their state of health, and thus their outcome. In some experiments it may be possible to obtain
+information about X∗, and this can then be used as L in § 5.2. However it will often not be possible to
+observe X∗ in the experiment. Section 7.2 considers how this problem can be overcome by the incorpora-
+tion of observational data, if we can assume that, in such data, the desired treatment was the one actually
+applied, so that X∗ = X becomes observable. The combination of experimental and observational data
+allows us to identify the distribution of X∗ (together with the other variables), and so once again allows
+us to apply the theory of § 5.2 to obtain improved bounds for PB and PH, which are detailed in § 5.5.
+4 See Dawid [13] for an approach to understanding non-extreme probabilities based on imperfect information about a
+deterministic world.
+
+6
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+5.1 Simplest case
+We start by presenting the basis of the approach in the simplest case, where the data are experimental, and
+there is no additional covariate information. We thus have access to the interventional response probabilities
+Pr(Y = y | X ← x) = Pr(Y (x) = y), (x, y = 0, 1). What can be inferred, from these, about the probabilities
+of benefit and of harm?
+As described by Dawid and Musio [14], it is helpful to express the interventional probabilities in terms
+of parameters τ and ρ, where
+τ
+:=
+Pr(Y = 1 | X ← 1) − Pr(Y = 1 | X ← 0)
+(2)
+ρ
+:=
+Pr(Y = 1 | X ← 1) − Pr(Y = 0 | X ← 0).
+(3)
+Then τ is the average treatment effect, ATE, of X on Y , while ρ = Pr(Y = 1 | X ← 1) + Pr(Y = 1 | X ←
+0) − 1 is a measure of how common the outcome is.
+The transition matrix (Pr(Y = y | X ← x)) from X to Y is
+P = P(τ, ρ) :=
+�
+1
+2(1 + τ + ρ)
+1
+2(1 − τ − ρ)
+1
+2(1 − τ + ρ)
+1
+2(1 + τ − ρ)
+�
+,
+(4)
+where the row and column labels are implicitly 1 and 0 in that order. The necessary and sufficient condition
+for all the transition probabilities to be non-negative is
+|τ| + |ρ| ≤ 1.
+(5)
+We have equality in (5) only in the degenerate case that one of the entries of P is 0.
+We can express the joint distribution for Y = (Y (1), Y (0)) as in Table 1. The margins are determined
+Y (0) = 1
+Y (0) = 0
+Y (1) = 1
+1
+2 (1 + ρ − ξ)
+1
+2 (ξ + τ)
+1
+2 (1 + τ + ρ)
+Y (1) = 0
+1
+2(ξ − τ)
+1
+2 (1 − ρ − ξ)
+1
+2 (1 − τ − ρ)
+1
+2(1 − τ + ρ)
+1
+2(1 + τ − ρ)
+1
+Table 1. Joint probability distribution of (Y (1), Y (0)
+by (1) (with L absent) and (4); but the internal entries are indeterminate, having one degree of freedom
+crystallised in the unspecified “slack variable” ξ, which is not identified by the experimental data. The only
+constraint on ξ is the logical one that all internal entries of Table 1 be non-negative. This holds if and only
+if
+|τ| ≤ ξ ≤ 1 − |ρ|.
+(6)
+This interval information is all that can be concluded about the joint distribution for Y when we have data
+on the behaviour of Y under intervention on X, and no additional information.
+Remark 1. The interval (6) shrinks to a point, so that the joint distribution of Y is fully determined by
+the experimental data, if and only if we have equality in (5), i.e., just when P is degenerate, so that, for
+some x, y = 0, 1, Pr(Y = y | X ← x) = 0. That is to say, for at least one of the interventions, the resulting
+outcome Y can be predicted with certainty—a most unusual state of affairs. In this case Pr(Y (x) = y) = 0,
+so that both joint events (Y (x) = y, Y (x) = 0) and (Y (x) = y, Y (x) = 1) (where x = 1−x) have probability
+0.
+5.1.1 Benefit and harm
+The probability of benefit PB is the upper right entry of Table 1, PB = Pr(Y (1) = 1, Y (0) = 0) = 1
+2(ξ +τ),
+which by (6) is bounded between PB− := 1
+2(|τ|+τ) = max{τ, 0} and PB+ := 1
+2(1−|ρ|+τ) = min{Pr(Y =
+
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+7
+1 | X ← 1), Pr(Y = 0 | X ← 0)}. The probability of harm is the lower left entry of Table 1, PH =
+Pr(Y (1) = 0, Y (0) = 1) = 1
+2(ξ − τ) = PB − τ.
+For the case of Example 1, we have τ = 0.28, ρ = −0.3. Without any further information, we can only
+infer 0.28 ≤ PB ≤ 0.49, and correspondingly 0 ≤ PH ≤ 0.21.
+5.2 Covariate information
+Now suppose that, again with experimental data, we can obtain additional information on some pre-
+treatment covariate information L (for simplicity assumed discrete), unaffected by intervention (so Pr(L =
+l | X ← x) = Pr(L = l), assumed known and positive). We thus have access to the conditional interventional
+probabilities Pr(Y = y | L = l, X ← x).
+Let τ(l), ρ(l) be defined as in (2) and (3), but with probabilities further conditioned on L = l. If,
+for the target case, we observe L = l, then we simply apply the above analysis, conditional on L = l. In
+particular, the joint distribution for Y, given L = l, will be as in Table 1, with ρ, τ, ξ replaced, respectively,
+by ρ(l), τ(l), ξ(l), where ξ(l) is subject only to
+|τ(l)| ≤ ξ(l) ≤ 1 − |ρ(l)|.
+(7)
+Finally, suppose that, while having access, from the experimental data, to the probabilities Pr(Y =
+y | L = l, X ← x), we do not observe L for the target patient. In this case (and unlike the situation
+for decision theory) the additional background knowledge can make a difference. In Table 1 we now have
+ξ = �
+s ξ(l) × Pr(L = l), and we get the new interval bound
+L :=
+�
+s
+|τ(l)| × Pr(L = l) ≤ ξ ≤ 1 −
+�
+s
+|ρ(l)| × Pr(L = l) =: U.
+(8)
+Since τ = �
+s τ(l) × Pr(L = l), ρ = 1 − �
+s ρ(l) × Pr(L = l), this interval will be strictly contained in
+that of (6) so long as not all the (τ(l)), or not all the (ρ(l)), have the same sign.
+The probability of benefit is now bounded below by �
+l PB−(l) Pr(L = l) and above by �
+l PB+(l) Pr(L =
+l), where PB−(l) and PB+(l) can be computed as in § 5.1.1 with τ and ρ replaced by τ(l) and ρ(l), re-
+spectively.
+Remark 2. Applying Remark 1, and noting |τ(l)| ≤ 1 − |ρ(l)|, all l, we see that the interval (8) will reduce
+to a point, yielding full identification of the joint distribution of Y, if and only if |τ(l)| = 1 − |ρ(l)|, all l,
+so that, for each l, at least one of Pr(Y = y | L = l, X ← x), for x, y = 0, 1, is zero. In this case, both
+Pr(Y (x) = y, Y (x) = 0 | L = l) and Pr(Y (x) = y, Y (x) = 1 | L = l) will be 0. Knowing the value of L
+will then always allow us to predict at least one of the interventional outcomes with certainty. However,
+the relevant x and y may vary with l, in which case such certainty will not be possible in the absence of
+knowledge of L.
+5.2.1 Observational data
+Consider now the case that our data are observational, rather than experimental. Suppose we can observe
+a “sufficient covariate”: a covariate L such that, conditional on L, we can assume there is no residual
+confounding. That is to say, the observational probability Pr(Y = y | L = l, X = x) can be equated with
+the interventional probability Pr(Y = y | L = l, X ← x). To ensure meaningful conditioning, we further
+need the positivity condition: in the observational setting,
+Pr(L = l, X = x) > 0
+all l, and x = 0 or 1.
+(9)
+We can then proceed exactly as in § 5.2 above.
+
+8
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+5.3 Intention to treat
+Allocation of treatment to patients can be usefully decomposed into two steps:
+Intention The patient, or their doctor, decides on which treatment they would ideally want. This decision
+will typically be related to their health status and other background information that could be predic-
+tive of recovery, so that we cannot regard those who desire, and those who reject, active treatment as
+comparing like with like. This is the genesis of confounding.
+We introduce a binary stochastic “intention to treat” (ITT) variable X∗ to denote the treatment
+desired.
+Application A treatment X is imposed on the patient.
+It is important to distinguish X and X∗.5 The ITT variable X∗ exists prior to application of treatment,
+and can thus be regarded as independent of it:
+Pr(X∗ = x∗ | X ← x) = Pr(X∗ = x∗).
+(10)
+This expresses the covariate nature of X∗.
+We assume that, in an observational setting, the desired treatment is the one that is actually admin-
+istered (there being no reason to do otherwise). Thus the received treatment X will be the same as the
+desired treatment X∗. In particular, since we observe X, we can infer the value of X∗.
+In an experiment, however, the treatment X will be imposed (e.g., by randomization), in a way that
+will typically take no account of X∗. Even though we can still conceive of the ITT variable X∗ as existing,
+it may or—more usually—may not be possible to observe it. When X∗ is observable, it can be used, just
+like any other covariate, to improve decision-making, as in § 2 (when X∗ is observed for the target patient),
+or, in the approach of MP, to narrow the bounds on PB and PH, as in § 5.2.
+5.3.1 ITT as a sufficient covariate
+In an observational setting, where X∗ = X is observed, it is natural to assume “distributional consistency”
+[12]: the distribution of Y given intended treatment X∗ = x—and so, also, given received treatment
+X = x—is the same as that of Y , given X∗ = x, under an imposed intervention X ← x that happens to
+coincide with the treatment that would have been chosen anyway:
+Pr(Y = y | X∗ = x, X = x) = Pr(Y = y | X∗ = x, X ← x).
+(11)
+For x∗ ̸= x, the event (X∗ = x∗, X = x) does not occur in the observational regime, so we can interpret
+Pr(Y = y | X∗ = x∗, X = x) however we want, in particular as
+Pr(Y = y | X∗ = x∗, X = x) = Pr(Y = y | X∗ = x∗, X ← x),
+(12)
+and then (11) implies that (12) holds for all x, x∗.
+Properties (10) and (12) imply that X∗, which is observed in the observational setting, behaves as a
+sufficient covariate.
+5.4 Combination of data
+It would be nice if, with observational data, we could profit from the fact that X∗ is a sufficient covariate,
+as in § 5.2.1. However, this is not straightforward, since the positivity condition (9) fails: for x∗ ̸= x, even
+5 We should further distinguish between imposed treatment and received treatment, as in Dawid [12]. Here we notate
+both as X, hoping this will cause no confusion. We write X ← x when X refers to the imposed treatment, and X = x
+when X refers to the received treatment.
+
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+9
+though we may assume Pr(Y = y | X∗ = x∗, X ← x) = Pr(Y = y | X∗ = x∗, X = x), we have no
+data to estimate the latter term. Again, when our data are experimental but we can not directly observe
+X∗, we can not identify Pr(Y = y | X∗ = x∗, X ← x). However, it turns out that we can do so if we
+can also obtain observational data: the combination of both types of data allows us, after all, to identify
+Pr(Y = y | X∗ = x∗, X ← x), even for x ̸= x∗. This we show in the following theorem.
+Theorem 1. Suppose we can identify the joint distribution of X and Y in the observational context, where
+0 < Pr(X = 1) < 1, and can also identify the distribution of Y under either intervention X ← x (x = 0, 1).
+Then, under conditions (10) and (11), all the probabilities Pr(Y = y | X∗ = x∗, X ← x) (x, x∗ = 0, 1) are
+identified. Specifically,
+Pr(Y = y | X∗ = 1, X ← 1)
+=
+Pr(Y = y | X = 1)
+(13)
+Pr(Y = y | X∗ = 0, X ← 0)
+=
+Pr(Y = y | X = 0)
+(14)
+Pr(Y = y | X∗ = 1, X ← 0)
+=
+Pr(Y = y | X ← 0) − Pr(Y = y, X = 0)
+Pr(X = 1)
+(15)
+Pr(Y = y | X∗ = 0, X ← 1)
+=
+Pr(Y = y | X ← 1) − Pr(Y = y, X = 1)
+Pr(X = 0)
+.
+(16)
+Proof. (13) and (14) follow from (11).
+To identify Pr(Y = y | X∗ = 0, X ← 1), we argue as follows. We have
+Pr(Y = y | X ← 0)
+=
+Pr(Y = y | X∗ = 0, X ← 0) × Pr(X∗ = 0 | X ← 0)
++ Pr(Y = y | X∗ = 1, X ← 0) × Pr(X∗ = 1 | X ← 0)
+=
+Pr(Y = y | X = 0) × Pr(X = 0)
++ Pr(Y = y | X∗ = 1, X ← 0) × Pr(X = 1),
+(17)
+on using (10) and (11), and the fact that X∗ = X in the observational setting. Since all the other terms in
+(17) are identifiable in either the observational or the experimental context, and Pr(X = 1) ̸= 0, we can
+solve for Pr(Y = y | X∗ = 1, X ← 0), obtaining (15). Then (16) follows similarly.
+The above proof relies on X (and so X∗) being binary, but Y need not be. Versions of this argument have
+appeared in [14–17].
+Corollary 1. The joint distribution of (X∗, Y ) under the intervention X ← x is then identified.
+Proof. Follows since, by (10), Pr(X∗ = x∗ | X ← x) = Pr(X = x∗) is identified in the observational
+context.
+Remark 3. Since Pr(Y = y | X∗ = 1, X ← 0) ≥ 0, etc., we deduce from (15) and (16) the consistency
+constraint Pr(Y = y | X ← x) ≥ Pr(Y = y, X = x), all x, y. When this fails, and that failure can not
+be ascribed to sampling variation or bias, that is evidence of violation of the conditions of § 7 below, that
+have, implicitly, been used to justify the above argument.
+Theorem 1 and Corollary 1 express just what the combination of observational and experimental data is
+doing for us: it allows us to identify distributions involving the ITT variable X∗.
+5.5 Benefit and harm
+Taking now X∗ as our sufficient covariate L, we can apply the formulae of (13)–(16) to compute the
+quantities τ(x∗), ρ(x∗) required for the analysis of § 5.2. Noting that X∗ = X in the observational regime,
+
+10
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+so that Pr(X = x) = Pr(X∗ = x), we obtain
+Pr(X∗ = 1) × τ(1)
+=
+Pr(Y = 1) − Pr(Y = 1 | X ← 0)
+Pr(X∗ = 0) × τ(0)
+=
+Pr(Y = 1 | X ← 1) − Pr(Y = 1)
+Pr(X∗ = 1) × ρ(1)
+=
+K − Pr(Y = 0 | X ← 0)
+Pr(X∗ = 0) × ρ(0)
+=
+Pr(Y = 1 | X ← 1) − K
+where K = Pr(Y = 1, X = 1) + Pr(Y = 0, X = 0). Then from (8) we bound ξ within (L, U), where
+L
+=
+| Pr(Y = 1) − Pr(Y = 1 | X ← 0)| + | Pr(Y = 1) − Pr(Y = 1 | X ← 1)|
+1 − U
+=
+| Pr(Y = 0 | X ← 0) − K| + | Pr(Y = 1 | X ← 1) − K|
+Then PB lies in ( 1
+2(L + τ), 1
+2(U + τ)), and PH = PB − τ lies in ( 1
+2(L − τ), 1
+2(U − τ)). Although expressed
+differently, these results agree with those of MP [2].
+By Remark 2, the joint distribution of Y, and in particular PB, PH, will be point identified just when,
+for both x∗ = 0 and x∗ = 1, there exist x, y such that Pr(Y = y | X∗ = x∗, X ← x) = 0. In non-trivial
+cases we will have Pr(Y = y | X = x) ̸= 0, in which case, by (13) and (14), this would need to happen
+with x ̸= x∗. For that, by (15) and (16), we require
+Pr(Y = y | X ← 0) = Pr(Y = y, X = 0)
+(18)
+for either y = 1 or y = 0; as well as
+Pr(Y = y | X ← 1) = Pr(Y = y, X = 1)
+(19)
+for either y = 1 or y = 0.
+6 Examples
+MP [2, Table 1] consider two cases, in both of which the interventional probabilities of recovery are as in
+our Example 1, having Pr(Y = 1 | X ← 1) = 0.49, Pr(Y = 1 | X ← 0) = 0.21, and so ATE = 0.28.
+However, they have different observational data. We now analyse these in detail.
+Example 2. This example relates to females, for whom the observational joint probabilities are as in
+Table 2.
+Y = 1
+Y = 0
+X = 1
+0.19
+0.51
+0.70
+X = 0
+0.21
+0.09
+0.30
+0.40
+0.60
+1
+Table 2. Joint observational distribution of (X, Y ) for females
+Applying the formulae of § 5.5 we find:
+0.7 × τ(1)
+=
+0.19
+0.3 × τ(0)
+=
+0.09
+0.7 × ρ(1)
+=
+−0.51
+0.3 × ρ(0)
+=
+0.11
+It follows that PB−(1) = τ(1) = 19/70. Also, PB+(1) = Pr(Y = 1 | X∗ = 1, X ← 1) = Pr(Y = 1 |
+X = 1) = 19/70. Hence, given X∗ = 1, we have exact identification: PB(1) = 19/70. This occurs because
+
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+11
+Pr(Y = 1, X = 0) = 0.21 = Pr(Y = 1 | X ← 0), implying the deterministic property Pr(Y = 1 | X∗ =
+1, X ← 0) = 0: a female who desires treatment will never recover if untreated. Consequently such a female
+should be treated.
+Also, PB−(0) = τ(0) = 0.3, while PB+(0) = Pr(Y = 0 | X∗ = 0, X ← 0) = Pr(Y = 0 | X = 0) = 0.3.
+Given X∗ = 0, we again have exact identification: PB−(0) = 0.3. This occurs because Pr(Y = 0, X =
+1) = 0.51 = Pr(Y = 0 | X ← 1), so that Pr(Y = 0 | X∗ = 0, X ← 1) = 0: a female who does not desire
+treatment will always recover if treated. Again, such a female should be treated.
+Finally we obtain exact identification marginally: PB = 0.28. Correspondingly, PH = PB − τ = 0. As
+there is no possibility of harm, any female should be treated.
+All the above conclusions agree with the DT prescription, based on the experimental data alone: since
+ATE > 0, a female should be treated.
+Example 3. For males, the observational joint probabilities are as in Table 3. Proceeding similarly to
+Y = 1
+Y = 0
+X = 1
+0.49
+0.21
+0.70
+X = 0
+0.21
+0.09
+0.30
+0.70
+0.30
+1
+Table 3. Joint observational distribution of (X, Y ) for males
+Example 2, we find that PB and PH are again identified exactly: PB = 0.49, PH = 0.21. Indeed, Pr(Y =
+1, X = 1) = 0.49 = Pr(Y = 1 | X ← 1), implying the deterministic property Pr(Y = 1 | X∗ = 0, X ← 1) =
+0: a male who does not desire treatment will never recover if treated. Consequently such a male should not
+be treated. Also, Pr(Y = 1, X = 0) = 0.21 − Pr(Y = 1 | X ← 0), so that Pr(Y = 1 | X∗ = 1, X ← 0) = 0:
+a male who desires treatment will never recover if untreated, so that such a male should be treated.
+However, if we do not observe X∗ for the target male patient, the above does not tell us how to proceed.
+We might try to balance PH (= 0.49) and PB (= 0.21) somehow: for example, treat just when PB > λPH
+for some chosen value of λ. In the light of the clinical maxim primum non nocere, a value λ = 3 might be
+chosen—in which case the target male would not be treated.6
+By contrast, in the absence of knowledge of X∗ for the target male, the DT approach would take no
+account of the observational data, again focusing simply on ATE = 0.28—and so decide to treat. In a large
+population of similar cases, this would lead to an overall recovery rate of 49%, the maximum possible;
+whereas the above strategy based on balancing PB and PH would only have a 21% recovery rate. It is
+difficult to see how this could be regarded as ethical.
+Example 4. Consider another case. Again, the interventional probabilities are Pr(Y = 1 | X ← 1) = 0.49,
+Pr(Y = 1 | X ← 0) = 0.21, with τ = 0.28. Now the observational joint probabilities are as in Table 4. We
+Y = 1
+Y = 0
+X = 1
+0.2
+0.5
+0.7
+X = 0
+0.1
+0.2
+0.3
+0.3
+0.7
+1
+Table 4. Another joint observational distribution of (X, Y )
+6 This argument parallels one in MP [1], having different numbers, and λ = 2.
+
+12
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+compute
+0.7 × τ(1)
+=
+0.09
+0.3 × τ(0)
+=
+0.19
+0.7 × ρ(1)
+=
+−0.39
+0.3 × ρ(0)
+=
+0.09.
+Using (8) we find 0.28 ≤ ξ ≤ 0.52, whence PB = 1
+2(ξ + τ) is bounded between 0.28 and 0.40—and so
+PH = PB − τ lies between 0 and 0.12. So, even with the aid of the additional observational data, we have
+not been able to identify these probabilities exactly. And even if we were to resolve the ambiguity somehow,
+for example by taking the midpoints of these intervals as suggested by Li and Pearl [3], we would be no
+better off than we were in Example 3, where trying to balance PB against PH could lead to a decision
+opposite to the rcommendation of the simple DT analysis, so leading to fewer recoveries.
+7 Assumptions and critical comments
+Here we identify and discuss some of the assumptions underlying the foregoing analyses.
+7.1 Representative data
+A fundamental assumption underlying both the decision-theoretic analysis of § 2 and the alternative ap-
+proach of § 3 is that the data available for estimating the interventional probabilities Pr(Y = y, L = l |
+X ← x) are on individuals who can be regarded as “similar to” (“exchangeable with”) the target case, so
+that these estimated probabilities are applicable to the target.7 In reality this is highly implausible. For
+example, a clinical trial will have entry criteria and processes that make its subjects quite untypical of
+the population from which they are drawn, or indeed of the individuals recruited into another such trial.
+In any case, despite the name, entry criteria govern who does not get into a trial: they cannot guarantee
+that those who enter are representative even of a target individual meeting the same criteria. A clinical
+trial gains its value, not from representativeness, but from the internal randomisation that ensures that a
+comparison between its treated and untreated groups is indeed a comparision of like with like, and that
+valid probability statements can be made about likely differences, so enforcing internal validity. Because
+of unrepresentativeness it would not be appropriate to regard Pr(Y = y, L = l | X ← x), estimated from
+the data, as being directly relevant to the target case—the problem of external validity. (One cheating way
+round this is to focus on a hypothetical target individual who can be regarded as exchangeable with those
+in the study.) Nevertheless, it may still be reasonable to regard the estimated ATE or CATE as applying
+to the target—if not in its exact numerical value, at least in its sign, which is what is required, for DT
+application, to solve the single patient treatment problem; or in its ordering of the CATEi, as required to
+solve the DT unit selection problem.
+To underline how unreasonable the representative assumption is, it should be noted that even when
+clinical trials with similar protocols are compared this assumption is not made. A striking example of its
+failure for nearly identical protocols is given by the TARGET study [18], in which osteoarthritis patients in
+some centres were randomised to receive either lumiracoxib or naproxen, and patients in other centres either
+lumiracoxib or ibuprofen. The degree of comparability in design of the two sub-studies thus defined was
+7 For application to the MP arguments of § 3, the representativeness assumption should apparently be extended to
+the (typically unidentifiable) bivariate distribution, along with the other variables, of the pair of potential responses
+(Y (1), Y (0)). For the interval-valued inferences made, however, this is not crucial, since these allow for arbitrary de-
+pendence in this bivariate distribution.
+
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+13
+greater than one would typically expect between two randomised controlled trials (RCTs), and a fortiori
+than between an RCT and an observational study, such as MP consider. Nevertheless, very important
+differences at baseline were seen between the two sub-studies, even though within-sub-study treatment
+arms were comparable. Furthermore, it was possible to demonstrate differences at outcome between the
+two studies using lumiracoxib data only, a striking illustration of a study effect. It is generally accepted by
+sponsors and regulators that as soon as concurrent control is abandoned the greatest of care must be taken
+in drawing inferences. Modern work on using data on historical controls to try and improve the efficiency
+of clinical trials takes such study-to-study variation as a given that must be allowed for [19].
+7.2 Combination of data
+An essential requirement for the application of Theorem 1 is that the observational and experimental
+datasets comprise similar individuals, so that the same probabilities for X, X∗, Y apply to both groups.
+This is even more implausible than the representativeness of either group. In particular, the assumption of
+a common distribution for the desired treatment X∗, in both datasets and in the target patient, is vital but
+highly questionable. Even if we were to accept the arguments of MP [2] based on combining observational
+and experimental data, without this property they are simply irrelevant.
+7.3 What do clinical trialists do in practice?
+The key to using RCTs is to identify reasonable assumptions, and use theory to transfer results from trial
+to practice. A striking example is given by bioequivalence studies. The subjects are usually young healthy
+volunteers, frequently male. However, the results will be used to decide on appropriate treatments for
+elderly frail patients, some female. There is no pretence of representativeness. Instead, tight control and
+sensible scales of analysis are used. The purpose of such studies is to compare two formulations in terms of
+bioavailability, and this is typically done using a cross-over trial in which each subject is their own control,
+the order of administration being randomised. On separate days, concentration of the test and reference
+pharmaceuticals are measured, and the ratio of the areas under the two concentration time curves (AUCs)
+are calculated for each subject, then analysed over all subjects, typically after log-transformation. What is
+relevant for treating an individual patient is their own AUC: too low and efficacy may be disappointing,
+too high and the drug may not be tolerated. However, no inference is made from a bioequivalence study
+in terms of AUCs alone, since they would be quite different in healthy volunteers and patients. Instead,
+the idea is that the ratio between test and reference ought to be the same in volunteers and patients, and
+this ratio can be used to make predictions as to how the test drug will behave in clinical practice. An
+interesting example of such a study is reported by Shumaker and Metzler [20]. They used a more elaborate
+design in which test and reference drugs were given in a double cross-over, thus permitting them to analyse
+the formulation-by-subject interaction. They were able to demonstrate that there was no evidence of an
+individual bioequivalence effect: although you could estimate the individual relative bioavailability, using
+the average over all subjects would be superior than any such naïve estimate. This raises a further issue
+with MP, who assume that individual causal effects are stable over time. Moreover, typical causal analysis
+assumes an infinite sample size, but no infinities are available for individual subjects, and estimating
+individual causal effects requires close attention to components of variance. Bioequivalence studies are an
+extreme example, but the general idea of transferring results using a suitable scale for analysis, and back-
+transforming to a scale suitable for decision analysis, is commonplace: see [21] for a general discussion and
+[22] in the specific context of vaccine efficacy. Of course, as the COVID-19 pandemic has reminded us, there
+are no guarantees. Things that work at one time may not do so at another. It behoves all those proposing
+solutions to be cautious and humble.
+
+14
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+8 Summary
+We have given careful accounts of the DT and MP approaches to individualised treatment choice. The DT
+approach is simple in the extreme, and selects the treatment strategy that maximises the number of recov-
+eries. In contrast, the MP approach fixates on philosophically questionable and unknowable counterfactual
+concepts, and when its recommendations differ from those of DT will lead to fewer recoveries. This has
+been illustrated in a number of examples.
+One feature of the MP approach is the combination of experimental and observational data. When
+some very strong, and practically implausible, conditions are satisfied, this permits identification of the
+distribution of a special covariate, the intention to treat (ITT). As with any other covariate whose distri-
+bution is known, this can then feed back to tighten the MP inferences. But it would be better to observe
+this—or any other—covariate in the target patient, which would then lead to better results from the DT
+point of view. In particular we have shown that, in just those very special cases that use of ITT leads to
+point identification of the MP probabilities of benefit and of harm, knowledge of the target patient’s ITT
+value allows perfect prediction of the outcome under at least one of the treatment interventions, and so to
+a trivial solution to the decision problem.
+The DT approach has a long history of fruitful application to an enormous variety of fields, from clinical
+trials to rocket science. Attempts to replace it with another approach, based on counterfactuals, are totally
+unnecessary and dangerously misguided. This approach should not be used in practice.
+Acknowledgments
+We have benefited greatly from discussions with Mats Stensrud and Aaron Sarvet.
+Conflict of interest: Prof. Philip Dawid is a member of the Editorial Board in the Journal of Causal
+Inference but was not involved in the review process of this article.
+References
+[1]
+Scott Mueller and Judea Pearl.
+Which patients are in greater need: A counterfactual analysis with reflections on
+COVID-19. Blog post, April 2020.
+[2]
+Scott Mueller and Judea Pearl. Personalized decision making – a conceptual introduction. Technical Report 513,
+Department of Computer Science, UCLA, 2022.
+[3]
+Ang Li and Judea Pearl. Unit selection based on counterfactual logic. In Proceedings of the Twenty-Eighth Inter-
+national Joint Conference on Artificial Intelligence, IJCAI-19, pages 1793–1799. International Joint Conferences on
+Artificial Intelligence Organization, 7 2019. DOI:10.24963/ijcai.2019/248.
+[4]
+Ang Li and Judea Pearl. Unit selection: Case study and comparison with A/B test heuristic. Preprint, UCLA, 2022.
+[5]
+Kosuke Imai, Zhichao Jiang, D. James Greiner, Ryan Halen, and Sooahn Shin. Experimental evaluation of algorithm-
+assisted human decision-making: Application to pretrial public safety assessment (with Discussion). Journal of the
+Royal Statistical Society, Series A, 2022. To appear.
+[6]
+Jonathan G. Richens, Rory Beard, and Daniel H. Thompson. Counterfactual harm. In Advances in Neural Information
+Processing Systems 35 (NeurIPS 2022), 2022.
+https://arxiv.org/abs/2204.12993.
+[7]
+Eli Ben-Michael, Kosuke Imai, and Zhichao Jiang. Policy learning with asymmetric utilities, 2022.
+[8]
+Aaron L. Sarvet and Mats J. Stensrud. Perspectives on harm in personalized medicine. Submitted to the American
+Journal of Epidemiology, 2022.
+[9]
+Donald B. Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of
+Educational Psychology, 66:688–701, 1974.
+
+A. Philip Dawid and Stephen Senn, Personalised Decision-Making
+15
+[10] Jerzy Neyman. On the application of probability theory to agricultural experiments. Essay on principles (in Polish).
+Roczniki Nauk Rolniczych, X:1–51, 1923. English translation of Section 9 (D. M. Dabrowska and T. P. Speed): Sta-
+tistical Science 9 (1990), 465–480.
+[11] A. Philip Dawid. Causal inference without counterfactuals (with Discussion). Journal of the American Statistical
+Association, 95:407–448, 2000.
+[12] A. Philip Dawid. Decision-theoretic foundations for statistical causality. Journal of Causal Inference, 9:39–77, 2021.
+DOI:10.1515/jci-2020-0008.
+[13] A. Philip Dawid. Probability, causality and the empirical world: A Bayes/de Finetti/Popper/Borel synthesis. Statistical
+Science, 19:44–57, 2004.
+[14] A. Philip Dawid and Monica Musio. What can group level data tell us about individual causality? In A. Carriquiry,
+J. Tanur, and W. Eddy, editors, Statistics in the Public Interest: In Memory of Stephen E. Fienberg, pages 235–256.
+Springer International Publishing, 2022.
+DOI: 10.1007/978-3-030-75460-0_13.
+[15] James M. Robins, Tyler J. Vanderweele, and Thomas S. Richardson. Comment on “Causal effects in the presence of
+non compliance: A latent variable interpretation” by Antonio Forcina. Metron, LXIV:288–298, 2007.
+[16] Sara G. Geneletti and A. Philip Dawid. Defining and identifying the effect of treatment on the treated. In Phyllis M.
+Illari, Federica Russo, and Jon Williamson, editors, Causality in the Sciences, pages 728–749. Oxford University Press,
+2011.
+[17] Mats J. Stensrud and Aaron L. Sarvet. Optimal regimes for algorithm-assisted human decision-making. arXiv preprint
+arXiv:2203.03020, 2022.
+[18] Stephen Senn. Lessons from TGN1412 and TARGET: Implications for observational studies and meta-analysis. Phar-
+maceutical Statistics, 7:294–301, 2008.
+[19] Heinz Schmidli, Sandro Gsteiger, Satrajit Roychoudhury, Anthony O’Hagan, David Spiegelhalter, and Beat Neuen-
+schwander. Robust meta-analytic-predictive priors in clinical trials with historical control information. Biometrics,
+70:1023–1032, 2014.
+[20] Robert C. Shumaker and Carol M. Metzler. The phenytoin trial is a case study of “individual bioequivalence”. Drug
+Information Journal, 32:1063–1072, 1998.
+[21] Jacobus Lubsen and Jan G. Tijssen. Large trials with simple protocols: Indications and contraindications. Controlled
+Clinical Trials, 10:151–160, 1989.
+[22] Stephen Senn. The design and analysis of vaccine trials for COVID-19 for the purpose of estimating efficacy. Pharma-
+ceutical Statistics, 21:790–807, 2022.
+

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+A Symbolic Emulator for Shuffle Synthesis
+on the NVIDIA PTX Code
+Kazuaki Matsumura
+Barcelona Supercomputing Center (BSC)
+[email protected]
+Simon Garcia De Gonzalo
+Sandia National Laboratories
+[email protected]
+Antonio J. Peña
+Barcelona Supercomputing Center (BSC)
+[email protected]
+Abstract
+Various kinds of applications take advantage of GPUs
+through automation tools that attempt to automatically
+exploit the available performance of the GPU’s parallel
+architecture. Directive-based programming models, such as
+OpenACC, are one such method that easily enables parallel
+computing by just adhering code annotations to code loops.
+Such abstract models, however, often prevent programmers
+from making additional low-level optimizations to take
+advantage of the advanced architectural features of GPUs
+because the actual generated computation is hidden from
+the application developer.
+This paper describes and implements a novel flexible
+optimization technique that operates by inserting a code
+emulator phase to the tail-end of the compilation pipeline.
+Our tool emulates the generated code using symbolic
+analysis by substituting dynamic information and thus
+allowing for further low-level code optimizations to be
+applied. We implement our tool to support both CUDA and
+OpenACC directives as the frontend of the compilation
+pipeline, thus enabling low-level GPU optimizations for
+OpenACC
+that
+were
+not
+previously
+possible.
+We
+demonstrate the capabilities of our tool by automating
+warp-level shuffle instructions that are difficult to use by
+even advanced GPU programmers. Lastly, evaluating our
+tool with a benchmark suite and complex application code,
+we provide a detailed study to assess the benefits of shuffle
+instructions across four generations of GPU architectures.
+CCS Concepts
+• Software and its engineering →
+Source code generation.
+Keywords
+Compiler, Symbolic Analysis, Code Generation,
+GPUs, NVIDIA PTX, Program Optimization
+Permission to make digital or hard copies of all or part of this work for
+personal or classroom use is granted without fee provided that copies are not
+made or distributed for profit or commercial advantage and that copies bear
+this notice and the full citation on the first page. Copyrights for components
+of this work owned by others than ACM must be honored. Abstracting with
+credit is permitted. To copy otherwise, or republish, to post on servers or to
+redistribute to lists, requires prior specific permission and/or a fee. Request
+permissions from [email protected].
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+© 2023 Association for Computing Machinery.
+ACM ISBN 979-8-4007-0088-0/23/02...$15.00
+https://doi.org/10.1145/3578360.3580253
+ACM Reference Format:
+Kazuaki Matsumura, Simon Garcia De Gonzalo, and Antonio J. Peña.
+2023. A Symbolic Emulator for Shuffle Synthesis on the NVIDIA
+PTX Code. In Proceedings of the 32nd ACM SIGPLAN International
+Conference on Compiler Construction (CC ’23), February 25–26, 2023,
+Montréal, QC, Canada. ACM, New York, NY, USA, 12 pages. https:
+//doi.org/10.1145/3578360.3580253
+1
+Introduction
+Effectively utilizing the vast amount of computational
+performance available in modern supercomputers remains a
+challenge to this day. Hardware, middleware, and parallel
+algorithms should be carefully orchestrated so that ideal
+efficiency may be obtained for solving large real-world
+problems in high-performance computing (HPC). Compiler
+technologies are developed with highly-automated program
+optimizations that use domain-specific knowledge and
+target architecture specialization to solve a part of this
+puzzle. With the end of Moore’s Law [19] approaching, the
+focus on supercomputing technology is shifting toward
+even more specialized accelerators, which in turn increases
+their complexity. This trend further signifies the importance
+of compiler technology to relieve programmers from the
+burden of understanding the complex architecture of
+modern accelerators to be able to efficiently optimize their
+applications.
+Currently, Graphics Processing Units (GPUs) are the most
+widely adopted accelerator technology, as these are present
+in seven out of the top 10 systems in the TOP500 list [29].
+GPUs work for accelerating application execution time
+through
+their
+highly
+parallelized
+yet
+cooperative
+architecture. To benefit the most from GPUs, however,
+programmers must be proficient in writing complex
+low-level GPU code, often a largely time-consuming task.
+To overcome the complexity of low-level GPU code
+development, pragma-based programming models such as
+OpenACC/OpenMP [3, 24] have been developed or adapted
+to be able to automatically retarget existing code for
+acceleration. Although these automation tools have
+improved the utilization of GPU acceleration by many
+different types of applications, they lack the ability to
+benefit from low-level architecture-specific optimizations.
+One such type of optimizations is the use of warp-level
+primitives, which have been available since NVIDIA Kepler
+GPUs. Warp-level primitives, such as shuffle operations,
+may be used to fill a gap between threads and thread-blocks
+arXiv:2301.11389v1  [cs.DC]  26 Jan 2023
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+K. Matsumura, S. G. De Gonzalo, A. J. Peña
+working as collaborative mechanisms, instead of relying on
+shared and global memory accesses.
+The main operation across the warp is the shuffle, which
+delivers computed elements to neighbor threads to suppress
+the redundancy of computation and memory accesses.
+However, as many existing efforts [5, 7, 13, 25] have
+demonstrated, those primitives often require non-trivial
+modification of algorithms in the fundamental part of their
+codes. Since the latency of the shuffle is similar to that of
+shared memory loads [7] (apart from storing and
+synchronization), it may serve as a cache system, holding
+data in registers [5]. However, the effectiveness of this
+technique is still unknown when disregarding domain-
+specific knowledge.
+Our work provides a middle-end environment to extend
+the code of the NVIDIA GPU assembly PTX and enables, for
+the first time in the literature, automatic shuffle synthesis to
+explore the opportunity of this operation. Our environment,
+PTXASW§ (Wrapper of PTX optimizing ASsembler),
+addresses the entire computational flow of PTX, leveraging
+a
+symbolic
+emulator
+that
+can
+symbolically
+extract
+memory-access patterns. We introduce a Satisfiability
+Modulo Theories (SMT) solver to prune avoidable control
+flows while tracking down the register update.
+Following the emulating results, PTXASW utilizes the
+solver and detects the global-memory loads that are
+possible to be covered by the shuffle operation. Around
+those loads, additional instructions are implanted, while
+supporting corner cases and circumventing overheads. We
+conduct the shuffle synthesis on an OpenACC benchmark
+suite, a directive-based programming model having no user
+exposure to warp-level instructions. Our implementation
+functions as a plugin of the compilation tool yielding
+moderate overhead.
+Applying our technique, we find various opportunities to
+enable the shuffle over the original code of the benchmarks.
+The performance improvement achieved is up to 132% with
+no user intervention on the NVIDIA Maxwell GPU.
+Additionally, based on the results of the experiments using
+several generations of GPUs, we analyze the latency caused
+for the shuffle operations to provide guidelines for shuffle
+usage on each GPU architecture. In summary, the
+contributions of our work are:
+1. We create a symbolic emulator to analyze and optimize
+GPU computing code, equipped with an SMT solver
+for the comparison of symbolic expressions, induction
+variable recognition for loops, and various optimizations
+to reduce overheads.
+2. Through symbolic analysis, we automatically find the
+possible cases to utilize the shuffle operation, which
+previously required in-depth domain knowledge to be
+§The artifact is available at https://github.com/khaki3/ptxas-wrapper.
+applied. Then, we synthesize those to the applications,
+while avoiding expensive computation.
+3. Using a directive-based programming model, we
+generate various shuffle codes on several generations
+of GPUs and show the cases that attain performance
+improvement with no manual effort.
+4. We show the latency breakdown of the optimization on
+each GPU architecture and provide general guidelines
+for the use of shuffle operations.
+Our work is the first attempt at general utilization of
+shuffles. Although manual warp-level operations often
+contributed to domain-specific optimizations, the metrics to
+be addressed by warp-level efforts have not been studied.
+Even when computation or memory accesses are reducible,
+the trade-offs have remained unknown to date, especially
+when thread divergence is involved.
+The rest of the paper is structured as follows. Section 2
+provides
+the
+necessary
+background
+on
+GPUs
+for
+general-purpose
+computing,
+PTX
+code,
+and
+shuffle
+operations. Section 3 provides a high-level overview of our
+work. Sections 4 and 5 describe our symbolic emulator and
+shuffle synthesis, while Section 6 details our overall
+methodology. Sections 7 and 8 provide the results of our
+experimental evaluation and in-depth analysis. Section 9
+discusses previous related work and Section 10 provides
+concluding remarks.
+2
+Background
+This section provides the necessary background on GPUs
+for general-purpose computing, low-level PTX code, and
+warp-level shuffle operations.
+2.1
+GPUs
+A Graphics Processing Unit (GPU), is a massively parallel
+accelerator architecture having with several computational
+and communication layers. The minimum execution unit is
+a thread. Each thread can collaborate with other threads
+bound to a certain thread-block and grid, through per-block
+shared memory and/or grid-wise global memory. The
+architecture is composed of many streaming multiprocessors
+(SMs), which execute distributed thread-blocks in groups of
+threads (usually 32), called warps. Using inner parallel
+processing
+units,
+the
+SM
+takes
+advantage
+of
+instruction-level parallelism (ILP), as well as parallelism
+among warps and thread-blocks. Since the memory-access
+latency increases through the levels of the memory
+hierarchy, the concept of locality is highly respected for
+performance, while locality optimizations bring additional
+synchronization and resource use to programs. Warp-level
+primitives, available since the NVIDIA Kepler generation of
+GPUs, allow for the communication among threads within
+the same warp [21], avoiding access to either shared or
+global memory.
+
+A Symbolic Emulator for Shuffle Synthesis on the NVIDIA PTX Code
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+All threads execute the same program code, known as
+GPU kernels, customarily written in CUDA [22] for NVIDIA
+GPUs, in a single-instruction multiple-data fashion. Threads
+operate
+on
+different
+data,
+specified
+in
+kernels
+by
+programmers, deriving from thread and thread-block
+identifiers. Kernels accept arguments, and the number of
+threads and thread-blocks is specified as variables.
+2.2
+NVIDIA PTX
+User-level code implemented manually in CUDA or
+OpenACC is brought to execution on GPUs through
+NVIDIA PTX [23], a virtual machine and ISA for
+general-purpose parallel thread execution. PTX programs
+feature the syntax and sequential execution flow of
+assembly language. Thread-specific variables are replicated
+to be run over SMs in parallel using the same program but
+different parameters. Since the actual machine code (SASS)
+cannot be modified from official tools [35], PTX is the
+nearest documented and standard GPU code layer that may
+be modified.
+PTX code consists of kernel and function declarations.
+Those have parameters and instruction statements along
+with variable declarations, labels, and predicates. Listing 2
+provides the CUDA-generated PTX kernel from Listing 1.
+Variable declarations from several data spaces and types
+correspond to the usage of on-chip resources, especially
+__global__ void add(
+float *c, float *a, float *b, int *f) {
+int i = threadIdx.x + blockIdx.x * blockDim.x;
+if (f[i]) c[i] = a[i] + b[i];
+}
+Listing 1. Addition kernel in CUDA
+.visible .entry add(. param .u64 c, .param .u64 a,
+.param .u64 b, .param .u64 f){
+/* Variable Declarations */ .reg .pred %p<2>;
+.reg .f32 %f<4>;.reg .b32 %r<6>;.reg .b64 %rd <15>;
+/* PTX Statements */
+ld.param.u64 %rd1 , [c];
+ld.param.u64 %rd2 , [a];
+ld.param.u64 %rd3 , [b];
+ld.param.u64 %rd4 , [f];
+cvta.to.global.u64 %rd5 , %rd4;
+mov.u32 %r2 , %ntid.x;
+mov.u32 %r3, %ctaid.x;
+mov.u32 %r4 , %tid.x; mad.lo.s32 %r1, %r3, %r2 ,%r4;
+mul.wide.s32 %rd6 , %r1, 4; add.s64 %rd7 ,%rd5 ,%rd6;
+// if (!f[i]) goto $LABEL_EXIT;
+ld.global.u32 %r5, [%rd7]; setp.eq.s32 %p1 ,%r5 ,0;
+@%p1 bra $LABEL_EXIT;
+// %f3 = a[i] + b[i]
+cvta.u64 %rd8 , %rd2; add.s64 %rd10 , %rd8 , %rd6;
+cvta.u64 %rd11 ,%rd3; add.s64 %rd12 , %rd11 ,%rd6;
+ld.global.f32 %f1, [%rd12];
+ld.global.f32 %f2, [%rd10]; add.f32 %f3, %f2 , %f1;
+// c[i] = %f3
+cvta.u64 %rd13 ,%rd1; add.s64 %rd14 , %rd13 ,%rd6;
+st.global.f32 [%rd14], %f3;
+$LABEL_EXIT: ret;
+}
+Listing 2. Addition kernel in PTX (simplified)
+registers. Accepting options and types (e.g. .eq, .s32), PTX
+instructions leverage defined registers and compute results,
+while some of these enable access to other resources (e.g.,
+ld.global.u32). Predicates (@%p1) limit the execution of
+the instructions stated under them, which may lead to
+branching based on the thread-specific values, such as
+thread and thread-block IDs (%tid.x, %ctaid.x). Labels
+(e.g., $LABEL_EXIT) are branch targets and allow backward
+jumps that may create loops.
+2.3
+Shuffle Operation
+In GPU architectures prior to NVIDIA Kepler, each
+sequential execution of a given thread was allowed to
+transfer data to another thread only through non-local
+memories, accompanied by a block-level or grid-level
+synchronization barrier. Modern GPU architectures now
+support additional data sharing within warps. Intra-warp
+communication
+is performed via
+shuffle operations.
+Listing 3 shows the shfl.sync instruction in PTX, in which
+data gets shifted unidirectionally (.up, .down) across the
+threads of the warp, swapped in a butterfly way (.bfly), or
+exchanged by precise indexing (.idx).
+In the unidirectional shuffle, the delta part, which has no
+source lane from the same warp, will be unchanged and
+obtain a false value in the resultant predicate (%p1); only the
+active threads (%mask) of the same control flow participate
+in the same shuffle. Inactive threads or threads from
+divergent flows produce neither valid results nor predicates
+to destination lanes. Each operation is accompanied by the
+warp-level synchronization, some of which are optimized
+away during compilation. While shuffle instructions allow
+for sub-warp granularity, our paper focuses on the
+unidirectional instruction with 32 threads using 32-bit data,
+as applying sub-warp granularity to applications tends to
+feature corner cases and suffers from exception handling for
+intricate patterns.
+activemask.b32 %mask;
+// val[warp_id] = %src; %dst = val[warp_id -%i]
+shfl.sync.up.b32
+%dst1|%p1 , %src , %i,
+0, %mask;
+// val[warp_id] = %src; %dst = val[warp_id +%i]
+shfl.sync.down.b32 %dst2|%p2 , %src , %i, 31, %mask;
+// val[warp_id] = %src; %dst = val[warp_id ^%i]
+shfl.sync.bfly.b32 %dst3|%p3 , %src , %i, 31, %mask;
+// val[warp_id] = %src; %dst = val[%i]
+shfl.sync.idx.b32
+%dst4|%p4 , %src , %i, 31, %mask;
+Listing 3. The use of shfl.sync in PTX
+Table 1 shows the latencies (clock cycles) of shared
+memory (SM; no-conflict) and L1 cache as reported by [16],
+besides that of shuffle, from a microbenchmark based
+on [33]. In the table, Kepler is NVIDIA Tesla K80, Maxwell
+is M60, Pascal is P100 and Volta is V100, while Tesla
+K40c/TITAN X are used for the shuffle of Kepler/Maxwell.
+This table reveals that shuffle brings benefits over shared
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+K. Matsumura, S. G. De Gonzalo, A. J. Peña
+NVHPC
+User Program
+OpenACC/OpenMP
+NVCC
+CUDA
+PTX Code
+PTXAS
+Execution Binary
+Compiling
+Assembling
+PTXAS
+PTXASW
+1
+Allocate symbolic
+registers
+2A
+Update registers through the PTX execution
+2B
+Gather branch conditions & memory accesses
+3
+Detect shuffle
+opportunities
+4
+Synthesize
+shuffles
+𝑁 = ?
+𝑁 =
+Figure 1. Overview of PTXASW
+memory as a communication mechanism when data
+movement is not redundantly performed, so storing and
+synchronization are avoidable. In particular, latencies of L1
+cache
+on
+Maxwell/Pascal
+are
+higher
+compared
+to
+Kepler/Volta, which integrate shared memory with L1 cache.
+Those allow the shuffle to be utilized as a register cache for
+performance improvement, but the engineering efforts in
+order to modify the fundamental parts of parallel
+computation are considerably high.
+name
+Shuffle (up)
+SM Read
+L1 Hit
+Kepler
+24
+26
+35
+Maxwell
+33
+23
+82
+Pascal
+33
+24
+82
+Volta
+22
+19
+28
+Table 1. Latencies (clock cycles) as reported by [16, 33]
+3
+Overview
+Our work PTXASW can substitute the original PTX
+assembler, which accepts input code from arbitrary sources.
+We do not rely on specific information of any certain
+language or any certain generation of GPU architecture.
+Figure 1 provides a high-level overview of PTXASW’s
+execution flow. PTXASW primarily aims at shuffle synthesis
+on PTX code. The input is produced by user-level code
+compilers, while directive-based programming models
+(OpenACC/OpenMP) do not expose control over warp-level
+operations, and CUDA prevents code extension due to its
+code complexities. Once PTXASW inserts shuffles, the
+resultant code is assembled to GPU binary by the original
+PTX assembler.
+PTXASW emulates the PTX execution based on the input.
+Since runtime information is not provided, we employ
+symbolic evaluation for each operation. First, 1 register
+declarations are processed to be mapped in a symbolic
+register environment (described in Section 4.1). Second, 2A
+for each statement of PTX instructions, a corresponding
+operation is performed to update registers (Section 4.1).
+While continuing the execution,
+2B PTXASW gathers
+branch
+conditions
+for
+avoiding
+unrealizable
+paths
+(Section 4.2) and creates memory traces (Section 4.3). When
+the entire emulation is finished, 3 we discover shuffle
+opportunities from memory traces (Section 5.1). Finally, 4
+we insert shuffle operations to the input code (Section 5.2);
+then, the generated code is consumed by the original PTX
+assembler.
+4
+Symbolic Emulator
+Analysis of high-level code has posed questions about its
+applicability to abstract program structures or other
+user-level languages. While high-level code analysis may
+process intact code information, enormous engineering
+efforts are required just for specific forms within one
+language [13, 32]. Therefore, virtual machines are utilized
+for providing a cushion between real architectures and user
+codes. In particular, analysis and optimization of the
+virtual-machine code tend to be reusable without the
+restriction of input types [12, 14, 34].
+Our work uses PTX as the virtual machine layer and
+performs general analysis through code emulation. We
+introduce symbolic emulation to encapsulate the runtime
+information in symbol expressions and compute concolic
+(concrete + symbolic) values for each register. Although a
+number of previous work have been conducted on symbolic
+emulation for the purpose of software testing [4], our work
+(PTXASW) especially aims at code optimization of memory
+access on GPUs, since it is often regarded as one of the
+bottlenecks of GPU computing [7]. Those computed values
+are utilized for code generation as described in Section 5.
+4.1
+Instruction Encoding
+Since the subsequent PTX assembler, while generating SASS
+code, will eliminate redundant operations and resources, we
+may abundantly use registers while not causing register
+pressure by unnecessary data movement outside of the
+static single assignment form (SSA). First, PTXASW
+recognizes variable declarations and prepares a symbolic
+bitvector of the corresponding size for each register. Since
+arithmetic calculation and bitwise operations are supported
+
+A Symbolic Emulator for Shuffle Synthesis on the NVIDIA PTX Code
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+on the combination of concrete and symbolic bitvectors, we
+encode each PTX instruction as the computation over
+vectors. For example, addition for 16-bit vectors is encoded
+as in the following pseudocode:
+a = [a_0 , a_1 , .., a_15]; //a_N is a 1-bit element
+b = [b_0 , b_1 , .., b_15];
+c = a + b
+= [a_0 + b_0 , a_1 + b_1 , .., a_15 + b_15];
+With the add instruction corresponding to the above
+calculation, we detect the instruction type and source
+registers (%a, %b) and compute the result:
+add.u16 %c, %a, %b; // dst: %c; src: %a, %b
+Then, having the binding with the name of the
+destination register (%c), we keep the computed value in the
+register environment. PTXASW defines each instruction to
+update the destination registers according to the instruction
+options and types, and those registers may be fully concrete
+with the movement or computation from constant values.
+Also, to support floating-point instructions, we insert the
+conversion by uninterpreted functions at loading and
+storing
+bitvectors
+to
+and
+from
+floating-point
+data.
+Regarding casting operands among integer types and binary
+types, truncating or extending is performed based on the
+PTX specification. The computational instructions under
+predicates issue conditional values in registers. Since
+registers are not used before initialization, these always
+have evaluated values, except for special registers, such as
+thread IDs and uninterpreted functions of loops and
+memory loads, which are described in following sections.
+!$acc
+kernels loop independent gang (65535)
+!$acc& present(w0(1:nx ,1:ny), w1(1:nx ,1:ny))
+do j = 2, ny -1
+!$acc loop independent vector (512)
+do i = 2, nx -1
+w1(i,j)=c0*w0(i,j) + c1*(w0(i-1,j)+w0(i,j -1)+&
+w0(i+1,j)+w0(i,j+1)) + c2*(w0(i-1,j-1)+&
+w0(i-1,j+1)+w0(i+1,j-1)+w0(i+1,j+1))
+enddo enddo
+Listing 4. Jacobi kernel in Fortran and OpenACC
+4.2
+Execution Branching
+Branching is caused by jumping to labels under binary
+predicates that are computed by preceding instructions.
+Since inputs and several parameters are unknown at
+compilation time, unsolvable values of predicates are often
+observed leading to undetermined execution flows where
+computation is boundless. Thus, we abstract the repeated
+instructions in the same execution flow. At the entry point
+to the iterative code block, we modify each iterator of the
+block to have uninterpreted functions with unique identities
+and perform operations only once upon those uninterpreted
+functions. Since those uninterpreted functions produce
+incomparable values, we clip the initial values out and add
+them to registers containing uninterpreted functions at the
+block entry, for better accuracy in the case of incremental
+iterators
+to
+be
+found
+by
+induction
+variable
+recognition [10, 11].
+We continue each branching while duplicating the
+register environment for succeeding flows. All the flows
+finish at re-entry to iterative blocks or at the end of
+instructions, completing their own results. The symbolic
+expressions in predicates used at the prior divergence are
+recorded as assumptions while updating those predicates, to
+have constant booleans in the register environment, based
+on whether it is assumed as true. Conflicting values in
+assumptions are removed according to an SMT solver
+(Z3 [9]) when new expressions are added. If the destination
+of a new branch can be determined providing assumptions
+to the solver, unrealizable paths are pruned for faster
+emulation. Also, we skip redundant code-block entry
+bringing the same register environment as other execution
+flows by memoization, to force new results at each entry.
+4.3
+Memory Analysis
+We collect memory loads forwardly through the emulation
+and express them by uninterpreted functions accepting
+addresses and returning data of corresponding sizes. The
+trace of memory loads is intervened by memory stores, and
+both loads and assumptions are invalidated by stores that
+possibly overwrite them, using the same mechanism for
+conflicting assumptions mentioned in Section 4.2.
+LD: 0xc + (load(param2) + ((((0x1 + %ctaid.x) * load(param6) // w0(i-1, j+1)
++ ((% tid.x + %ctaid.y << 0x9) + (- load(param5 )))) + loop(0, 14)) + loop(0, 53)) << 0x2)
+LD: 0xc + (load(param2) + ((( load(param6) * (0x3 + %ctaid.x) // w0(i+1, j+1)
++ ((% tid.x + %ctaid.y << 0x9) + (- load(param5 )))) + loop(0, 13)) + loop(0, 52)) << 0x2)
+LD: 0x4 + (load(param2) + ((((0x1 + %ctaid.x) * load(param6) // w0(i-1, j-1)
++ ((% tid.x + %ctaid.y << 0x9) + (- load(param5 )))) + loop(0, 14)) + loop(0, 53)) << 0x2)
+/* LD: w0(i+1, j-1), w0(i
+, j+1), w0(i+1, j
+), w0(i
+, j-1), w0(i-1, j
+), w0(i
+, j
+) */
+ST: 0x8 + (load(param3) + (((% tid.x + %ctaid.y << 0x9)
+// w1(i
+, j
+)
++ loop(0, 57)) + ((- load(param5 )) + load(param6) * ((0x2 + %ctaid.x) + loop(0, 21)))) << 0x2)
+Listing 5. Global-memory trace of Jacobi kernel through the symbolic emulation in order. Sign extensions are omitted.
+Numerical numbers, shown in hexadecimal, are originally in bitvectors. load/loop are uninterpreted functions for parameter
+loads having addresses and loop iterators having unique identities, respectively )
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+K. Matsumura, S. G. De Gonzalo, A. J. Peña
+Listing 4 shows a Jacobian kernel implemented in Fortran
+for GPUs using OpenACC. Its memory trace is obtained as
+in Listing 5 by PTXASW emulating the PTX code generated
+by NVHPC compiler 22.3. The address of each load is
+symbolically calculated as register values, thus containing
+uninterpreted functions and special registers. In the case of
+divergence, branched flows maintain such traces while
+sharing the common parts of the original flow.
+5
+Shuffle Synthesis
+Mapping programs over thread-level parallelism, while
+pursuing the performance of modern complex architectures
+and ensuring correctness, is a far–from–easy task. Most
+likely, existing GPU programs are already optimized in
+terms of resource use and scheduling, which does not
+smoothly allow for further optimization, especially at the
+low-level code. The shuffle operation performs at its best
+when the communication is fully utilized [25], but such
+cases are not common in compiler-optimized code or even
+manually-tuned code in HPC. The big trouble is corner
+cases. Not only halo, but fractional threads emerged from
+rounding up dynamic input sizes, demand exceptional cases
+to be operated on GPUs. While the generality and
+applicability of GPU shuffle instructions for all types of
+applications or computational patterns are yet unknown,
+the level of difficulty in manually applying shuffle
+instructions in different cases adds further hardness to the
+already complex task of understanding the true nature of
+the performance of shuffle operations.
+Hence, we implement automatic shuffle synthesis
+through PTXASW to drive the lower-latency operations
+seen in Section 2.3, while supporting corner cases and
+covering
+global-memory
+loads
+with
+warp-level
+communication. PTXASW is accordingly extended to seek
+shuffle candidates among loads, and embed shuffle
+instructions into code while alleviating register pressure.
+5.1
+Detection
+Warps are comprised of neighboring threads. We do not
+consider adjacent threads in non-leading dimensions, since
+those tend to generate non-sequential access patterns. Upon
+finding a global-memory load, PTXAS compares its load
+address to those of previous loads found through the same
+execution flow and not invalidated by any store. If for all
+threads in a warp the load is overlapped with existing loads,
+those instructions are recorded as possible shuffle sources.
+To utilize a load with an address represented as 𝐴(%tid.x)
+for another having the address 𝐵(%tid.x), there must exist
+an integer 𝑁 such that 𝐴(%tid.x + 𝑁) = 𝐵(%tid.x) and
+−31 ⩽ 𝑁 ⩽ 31. For example, when 𝑁 = 0, the load can be
+fully utilized in the same thread. When 𝑁 = 1, we can adapt
+the shfl.sync.down instruction to convey existing register
+values to next threads while issuing the original load for the
+edge case (%warp_id = 31). In the case of the memory trace
+in Listing 5, the load accesses of w0(i-1, j+1) and w0(i-1,
+j-1) are uniformly aligned with the close addresses to each
+other, so we can search the variable 𝑁, which satisfies the
+above condition, by supplying 𝑁 along with those addresses
+to the solver and find 𝑁 = −2.
+We make sure that each shuffle candidate has the same 𝑁
+as a shuffle delta in all the execution flows. This delta must
+be constant regardless of runtime parameters. Since the steps
+of loop iterators in PTX code could be any size (e.g. NVHPC
+Compiler uses the thread-block size), shuffles are detected
+only in straight-line flows, whereas live variable analysis is
+employed to exclude the case in which source values possibly
+reflect a different iteration from the destination. For faster
+analysis, we construct control-flow graphs before shuffle
+detection, while pruning unrelated instructions to memory
+operations and branches, and at the use of the SMT solver,
+uninterpreted functions are converted to unique variables.
+5.2
+Code Generation
+Warp divergence may be caused by various reasons,
+including the dynamic nature of the program execution,
+which
+is
+inconvenient
+to
+optimization,
+where
+the
+uniformity of threads matters for collaboration. Not only
+inactive threads, but an insufficient number of threads to
+constitute complete warps, raises corner cases in which
+original computation should be retained. Our shuffle
+synthesis handles both situations by adding dynamic
+checkers for uniformity.
+Listing 6 presents an example of the synthesis by
+PTXASW. Once all the emulation is finished, the results are
+collected and filtered to satisfy all the above-mentioned
+conditions. Then, PTXASW selects the possible shuffle for
+each load with the smallest shuffle delta (𝑁) and allows only
+the least corner cases. At the code generation, each source
+load instruction is extended to be accompanied by the mov
+instruction to prepare the source register (%source). The
+destination load is covered with the shuffle operation and a
+corner-case checker. First, we check if the thread has no
+source from the same warp (%out_of_range). Second, the
+ld.global.nc.f32 %f4 , [%rd31 +12]; // w0(i-1, j+1)
+/* ... */
+ld.global.nc.f32 %f7 , [%rd31 +4]; // w0(i-1, j-1)
+ld.global.nc.f32 %f4 , [%rd31 +12];
+mov.f32 %source , %f4; /* ... */
+mov.u32 %wid , %tid.x; rem.u32 %wid , %wid , 32;
+activemask.b32 %m; setp.ne.s32 %incomplete , %m, -1;
+setp.lt.u32 %out_of_range , %wid , 2;
+or.pred %pred , %incomplete , %out_of_range;
+shfl.sync.up.b32 %f7 , %source , 2, 0, %mask;
+@%pred ld.global.nc.f32 %f7 , [%rd31 +4];
+Listing 6. Shuffle synthesis on Jacobi kernel (Upper is
+original and lower is synthesized code; variable declarations
+are omitted and the naming is simplified)
+PTXASW
+
+A Symbolic Emulator for Shuffle Synthesis on the NVIDIA PTX Code
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+incompleteness of the warp (%incomplete) is confirmed
+with a warp-level querying instruction. In any case, the
+shuffle operation is performed at the position of the original
+load, shifting the value of the source register with the
+distance of the extracted shuffle delta. Finally, only the
+threads participating in an incomplete warp or assuming no
+source lane execute the original load under the predicate
+(%pred). When 𝑁 < 0, the shfl instruction takes the .up
+option and when 𝑁 > 0, the .down option is selected. If
+𝑁 = 0, just the mov instruction is inserted instead of all the
+synthesized code. In actual code, the calculation of
+%warp_id is shared among shuffles and set at the beginning
+of the execution to reduce the computational latency.
+To preserve the original program characteristics, such as
+the register use, uniformity, and ILP, following ways of
+generation are avoided. We can produce the correct results
+even if shfl is predicated by %incomplete, but it often
+imperils the basic efficiency with an additional branch,
+which limits ILP. On the other hand, our code introduces
+only one predicate to each shuffle and does not leave any
+new branch in the resultant SASS code. Also, we do not use
+a select instruction for merging the results between shuffles
+and corner cases, because it would aggravate register
+pressure. The output predicate by shuffle poses execution
+dependency and provides the invalid status of inactive
+threads; thus, it is ignored. Moreover, we only create
+shuffles from direct global-memory loads and do not
+implement shuffles over shuffled elements for better ILP.
+6
+Experimental Methodology
+We build PTXASW using Rosette [30], a symbolic-
+evaluation system upon the Racket language. PTXASW is
+equipped with a PTX parser and runs the emulation of the
+parsed code while expressing runtime parameters as
+symbolic bitvectors provided by Rosette. Our shuffle
+synthesis is caused at code generation, which prints the
+assembler-readable
+code.
+We
+evaluate
+our
+shuffle
+mechanism with the NVHPC compiler [20] by hooking the
+assembler invocation and overwriting the PTX code before
+it is assembled. The NVHPC compiler accepts the
+directive-based
+programming
+models
+OpenACC
+and
+OpenMP to generate GPU code, which have no control over
+warp-level instructions. The emulation is also tested for
+GCC with OpenACC/OpenMP code and LLVM with
+OpenMP code, but these use a master-worker model to
+distribute computation across thread-blocks [15] and do not
+directly refer to the thread ID in each thread, so mainly
+ineffective results are obtained. Our synthesis is not limited
+to global-memory loads and works on shared memory (such
+as Halide [26]), but the performance is not improved due to
+the similar latency of shared-memory loads and shuffles.
+The NVHPC compiler utilizes the same style to translate
+both
+OpenACC
+and
+OpenMP
+codes
+written
+in
+C/C++/Fortran to PTX, hence supporting any combinations.
+name
+Lang
+Shuffle/Load
+Delta
+Analysis
+divergence
+C
+1 / 6
+2.00
+4.281s
+gameoflife
+C
+6 / 9
+1.50
+3.470s
+gaussblur
+C
+20 / 25
+2.50
+7.938s
+gradient
+C
+1 / 6
+2.00
+4.668s
+jacobi
+F
+6 / 9
+1.50
+4.119s
+lapgsrb
+C
+12 / 25
+1.83
+14.296s
+laplacian
+C
+2 / 7
+1.50
+4.816s
+matmul
+F
+0 / 8
+-
+13.971s
+matvec
+C
+0 / 7
+-
+4.929s
+sincos
+F
+0 / 2
+-
+1m41.424s
+tricubic
+C
+48 / 67
+2.00
+1m39.476s
+tricubic2
+C
+48 / 67
+2.00
+1m41.855s
+uxx1
+C
+3 / 17
+2.00
+7.466s
+vecadd
+C
+0 / 2
+-
+3.281s
+wave13pt
+C
+4 / 14
+2.50
+6.967s
+whispering
+C
+6 / 19
+0.83
+6.288s
+Table 2. The KernelGen benchmark suite. Lang indicates
+the programming language used (C or Fortran). Shuffle/Load
+shows the number of shuffles generated among the total
+number of global-memory loads. Delta is the average shuffle
+delta. Analysis is the execution time of PTXASW on Intel
+Core i7-5930K
+For the evaluation, we use the KernelGen benchmark
+suite for OpenACC [18], shown in Table 2. Each benchmark
+applies the operator indicated in the benchmark name, to
+single or multiple arrays and updates different arrays. The
+benchmarks gameoflife, gaussblur, jacobi, matmul,
+matvec and whispering are two-dimensional, whereas
+others are three-dimensional, both having a parallel loop for
+each dimension, in which other loops might exist
+inside—except matvec, which features only one parallel
+loop. The thread-level parallelism is assigned to the
+innermost parallel loop and the thread-block level
+parallelism to the outermost. We show the total time of
+running the shuffle-synthesized kernel ten times on Kepler
+(NVIDIA Tesla K40c with Intel i7-5930K CPU), Maxwell
+(TITAN X with Intel i7-5930K), Pascal (Tesla P100 PCIE with
+Intel Xeon E5-2640 v3), and Volta (Tesla V100 SXM2 with
+IBM POWER9 8335-GTH). We use NVHPC compiler 22.3
+with CUDA 11.6 at compilation, but due to environmental
+restrictions,
+run
+the
+programs
+using
+CUDA
+driver
+11.4/11.4/10.0/10.2
+for
+Kepler/Maxwell/Pascal/Volta,
+respectively. The compiler options in NVHPC are "-O3
+-acc -ta=nvidia:cc(35|50|60|70),cuda11.6,loadcac
+he:L1". To fully utilize computation, 2D benchmarks select
+32768x32768 as their dynamic problem sizes and 3D
+compute
+512x1024x1024
+grids,
+except
+uxx1,
+which
+leverages 512x512x1024 datasets and whispering, where
+more buffers are allocated, computing over 8192x16384 data
+elements. To assess a performance breakdown, we prepare
+two other versions of PTXASW: NO LOAD and NO
+CORNER. The former eliminates loads that are covered by
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+K. Matsumura, S. G. De Gonzalo, A. J. Peña
+shuffles, whereas the latter only executes shuffles instead of
+original loads, without the support of corner cases.
+The shuffle synthesis fails on four benchmarks. In
+matmul and matvec, the innermost sequential loop
+contains loads, but these do not have neighboring accesses
+along the dimension of the thread ID. The benchmarks
+sincos and vecadd do not have several loads sharing the
+same input array.
+7
+Evaluation
+Figure 2 shows the speed-ups of benchmarks on each GPU
+with original code and PTXASW-generated code along with
+the NO LOAD and NO CORNER versions. The line plots
+provide the SM occupancy of each benchmark. Since there
+is no resource change other than the register use from the
+original execution, the occupancy rate is directly affected by
+the number of registers. The performance improvement on
+Kepler/Maxwell/Pascal/Volta is confirmed with 7/6/9/4
+benchmarks
+showing
+up
+to
+16.9%/132.3%/9.1%/14.7%
+performance
+improvement,
+respectively.
+We
+see
+performance degradation with Volta in the case where more
+than ten shuffles are generated. Other GPUs mostly gain
+better performance with such cases. With increased shuffle
+deltas, more corner cases are expected. Volta shows optimal
+efficiency when 𝑁 ⩽ 1.5, while other GPUs benefit from the
+case of 𝑁 = 2.5. For example, Maxwell attains the best
+performance with gaussblur (𝑁 = 2.5), although Volta’s
+performance drops by half for the same case. The average
+improvement across all GPU generations is -3.3%/10.9%/
+1.8%/-15.2% for Kepler/Maxwell/Pascal/Volta, respectively.
+Overall, the performance improvement by PTXASW is
+found when NO LOAD and NO CORNER have sufficiently
+better performance compared to the original and when the
+occupancy typically rises on Kepler/Maxwell and drops on
+Pascal/Volta. The average number of additional registers
+with NO LOAD/NO CORNER/PTXASW compared to the
+original is -6.4/-5.2/2.7 on Kepler, -6.6/-5.9/4.2 on Maxwell,
+-7.0/-5.9/3.8 on Pascal, and -6.4/6.8/9.2 on Volta.
+8
+Analysis
+This section provides the performance detail of our shuffle
+synthesis on each GPU. Figure 3 shows the ratio of stall
+reasons sampled by the profiler for all the benchmarks. Those
+characteristics of computation appear as the results of the
+program modification (e.g. register use, shuffle delta) and the
+architecture difference (e.g. computational efficiency, cache
+latency).
+8.1
+Kepler
+The Kepler GPU has long stalls on computational
+operations with each benchmark. The average execution
+dependency is 24.7% and pipeline busyness is 7.5% with the
+original. When we look at the memory-bound benchmarks
+such as gameoflife, gaussblur, and tricubic, NO LOAD
+significantly reduces the amounts of memory-related stalls.
+Especially, tricubic has 56.0 percentage points below
+memory throttles from the original to NO LOAD, yielding
+2.53x performance. From NO LOAD to NO CORNER, the
+execution dependency increases by 4.0 percentage points
+and the pipeline busyness decreases by 1.6 percentage
+points on average. The performance degradation at NO
+CORNER with the memory-bound benchmarks is observed
+with the latency of the pipelines and the wait for the SM
+scheduler. PTXASW suffers from memory throttling and
+additional computation for the corner cases, which limit the
+improvement up to 16.9%.
+The memory throttling and the additional computation
+bottlenecks suffered by PTXASW may be hidden if the
+shuffle operations reduce the original computation and
+communication into just one transfer among threads,
+functioning as a warp-level cache. Otherwise, there is a
+need to face a trade-off between the redundancy of
+operations and the efficiency on the architecture. On Kepler,
+both heavy computation and memory requests are imposed
+by the corner case. Therefore, in the general use of shuffles,
+the uniformity of calculation is crucial and it requires
+domain-specific knowledge.
+8.2
+Maxwell
+There are two obvious compute-bound benchmarks:
+gameoflife and tricubic. For these, no improvement is
+perceived with NO LOAD, and there are no particular
+changes in occupancy or stalls throughout the four different
+versions. In summary, gameoflife experiences -0.1%/5.7%/
+6.2% lower performance and tricubic shows -1.6%/7.7%/
+15.4% lower performance with NO LOAD/NO CORNER/
+PTXASW, respectively, compared to the original version. In
+other cases, memory dependency is dominant. However, the
+merit of NO LOAD is limited to gaussblur and lapgsrb,
+which
+experience
+large
+texture-memory
+latency
+of
+read-only cache loads, successfully replaced with shuffles by
+PTXASW. The texture stall was reduced from 47.5% to 5.3%
+in gaussblur and from 23.0% to 0.1% in lapgsrb from the
+original to PTXASW, attaining 132.2% and 36.9% higher
+throughput. Other benchmarks do not feature stalls that
+allow for clear performance improvement by NO LOAD. As
+it can be observed in Figure 3, the memory dependency
+stalls are maintained for most benchmarks, except for that
+of tricubic2, which shows 32.9 percentage points lower
+memory dependency and only 14.3% overall improvement
+with NO LOAD. Those values are mostly absorbed by the
+corner cases.
+On the Maxwell GPU, only the texture stalls are
+improvable for efficiency in the tested cases. Since we
+observe a moderate overhead of the corner cases, our
+synthesis tool may enhance the overall performance. The
+memory-dependency stalls work as a good indicator of the
+memory utilization. If, in addition, a high execution
+
+A Symbolic Emulator for Shuffle Synthesis on the NVIDIA PTX Code
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+0.0
+1.0
+2.0
+3.0
+4.0
+Speed-up
+Kepler
+0.00
+0.25
+0.50
+0.75
+1.00
+Occupancy
+0.0
+1.0
+2.0
+3.0
+4.0
+Maxwell
+0.00
+0.25
+0.50
+0.75
+1.00
+0.0
+0.5
+1.0
+1.5
+2.0
+Pascal
+0.00
+0.25
+0.50
+0.75
+1.00
+0.0
+0.5
+1.0
+1.5
+Volta
+0.00
+0.33
+0.66
+1.00
+divergence
+gameoflife
+gaussblur
+gradient
+jacobi
+lapgsrb
+laplacian
+tricubic
+tricubic2uxx1
+wave13pt
+whispering
+Kepler
+−0.05
+0.00
+0.05
+Original
+NO LOAD
+NO CORNER
+PTXASW
+Figure 2. Speed-up compared to Original. NO LOAD/NO
+CORNER produce invalid results
+dependency would exist, it would provide the warp-level
+shuffle optimization the opportunity to be beneficial to
+speed up the computation.
+8.3
+Pascal
+Even more than in Maxwell, texture stalls are found in most
+benchmarks and those produce higher throughput with NO
+LOAD.
+Especially,
+gameoflife
+and
+tricubic,
+the
+compute-bound kernels on Maxwell, become memory
+intensive on Pascal and the performance increases by 5.9%
+and 5.4% with PTXASW. The unspecific latency ("Other")
+fills many parts of computation on Pascal. Further
+investigation shows that this mainly consists of the latency
+from register bank conflicts and the instructions after
+branching. With the optimization adding a predicate to
+check the activeness of the warp (@!incomplete) before the
+shuffle and generating a uniform branch, the ratio of this
+latency improves from 34.4% to 8.6% with PTXASW at
+gameoflife, obtaining 150.8% efficiency compared to the
+original. However, as mentioned in Section 5.2, it decreases
+the average relative execution time to 0.88x slowdown.
+Since the latency of the L1 cache is higher than that of
+one shuffle operation, the computation may be hidden by
+data transfers. Once the memory-dependency stall ratio
+0
+25
+50
+75
+100
+Stall (%)
+Kepler
+0
+25
+50
+75
+100
+Maxwell
+0
+25
+50
+75
+100
+Pascal
+0
+25
+50
+75
+100
+Volta
+divergence
+gameoflife
+gaussblur
+gradient
+jacobi
+lapgsrb
+laplacian
+tricubic
+tricubic2
+uxx1
+wave13pt
+whispering
+0.00
+−0.05
+0.00
+0.05
+Mem Dep
+Inst Fetch
+Mem Throtle
+Not Selected
+Exec Dep
+Texture
+Pipe Busy
+Other
+Figure 3. Stall breakdown in the order of Original/NO
+LOAD/NO CORNER/PTXASW from left to right for each
+benchmark
+increases due to replacing the texture stalls, Pascal may
+maintain the efficiency with the corner cases, resulting in
+speed-up in nine benchmarks. For shuffle instructions to be
+beneficial, the execution should be less divergent and
+careful register allocation is recommended to maximize the
+thread utilization.
+8.4
+Volta
+On Volta, most benchmarks become memory-bound and
+memory-intensive applications become sensitive to memory
+throttles. Nevertheless, the speed-up by NO LOAD is
+limited to up to 1.35x (gameoflife), due to the highly
+efficient cache mechanism. As argued in Section 7, some of
+the benchmarks attain higher performance with NO
+CORNER than in the case of NO LOAD for the lower
+occupancy. Other than that, we observe performance
+degradation due to increased execution dependency for
+lapgsrb and tricubic with NO CORNER. Those further
+reduce the efficiency with PTXASW while featuring stalls
+for instruction fetching. Also, the memory dependency of
+tricubic develops a large latency for memory accesses with
+PTXASW even though the corner cases experience fewer
+loads. This leads to unstable speed-ups between 0.315x and
+1.15x.
+The calculation through shuffles is expected to be
+effective depending on the utilization of communication,
+and the nonentity of warp divergence. Especially, as Volta
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+K. Matsumura, S. G. De Gonzalo, A. J. Peña
+shows minimal latency at each operation, the penalty of
+non-aligned computation becomes apparent and must be
+avoided by the algorithm.
+8.5
+Application Example
+We also apply PTXASW for the compilation of CUDA
+benchmarks extracted from applications. We select three
+benchmarks that appeared as complex 3D stencil operations
+in [27]: hypterm, rhs4th3fort, and derivative, to run on
+the Pascal GPU. hypterm is a routine from a compressible
+Navier-Stokes mini-app [1]. rhs4th3fort and derivative
+are
+stencils
+from
+geodynamics
+seismic
+wave
+SW4
+application code [2]. Each thread in the benchmarks
+accesses
+152/179/166
+elements
+over
+13/7/10
+arrays,
+respectively. We modify the execution parameters to
+execute at least 32 threads along the leading thread-block
+dimension and use the float data type. Since we saw in the
+prior section the overhead of long-distance shuffles, which
+generate many corner cases, we limited the shuffle synthesis
+to be |𝑁 | ⩽ 1 and found shuffles only with |𝑁 | = 1.
+hypterms contains three kernels that work along
+different dimensions. In the kernel for the leading
+dimension, 12 shuffles are generated over 48 loads,
+producing
+0.48%
+improvement.
+rhs4th3fort
+and
+derivative feature a single kernel each. rhs4th3fort
+experiences 2.49% higher throughput by PTXASW while
+placing 44 shuffles among 179 loads. For derivative, having
+52 shuffles from 166 loads, PTXASW attains 3.79% speed-up
+compared to the original execution.
+9
+Related Work
+Ever since warp-shuffle instructions were introduced during
+the Kepler generation of GPUs, these have been the subject of
+various lines of research. Early work described their manual
+use for specific computational patterns such as reduction
+operations [17] and matrix transposition [6]. Other research
+described the use of warp-shuffle instructions in the context
+of domain-specific optimizations such as employing them
+as a register cache for stencil operations [5], or to replace
+memory access for Finite Binary Field applications [5].
+Research on the automatic generation of warp-shuffle
+instructions has been explored. Swizzle Inventor [25] helps
+programmers implement swizzle optimizations that map a
+high-level "program sketch" to low-level resources such as
+shuffle operations. The authors meticulously design the
+abstraction of shuffles, synthesize actual code roughly based
+on algorithms found in previous literature, and attain
+enhanced performance while reducing the amounts of
+computation. Tangram, a high-level kernel synthesis
+framework, has also shown the ability to automatically
+generate warp-level primitives [13]. Unlike the work
+presented in this paper, both of the above-mentioned efforts
+leverage domain-specific information to map computational
+patterns
+such
+as
+stencil,
+matrix
+transposition,
+and
+reductions to shuffle operations.
+Recent code-generation techniques allow for obtaining
+optimal SIMD code generation. Cowan et al. [8] generate
+program sketches for execution on ARM processors, by
+synthesizing additional instructions, as well as input/output
+registers, to implement the shortest possible SIMD code of
+reduction. Unlike PTXASW, which uses an SMT solver to
+find the optimal shuffle deltas,
+this work runs a
+comprehensive search of multiple possible code versions;
+thus, the search space is exponential to the number of
+instructions. VanHattum et al. [31] attain faster execution
+on digital signal processors while employing equality
+saturation [28], a modern way of optimization that
+generates possible code as much as possible from a basic
+program according to the rules of term rewriting. They
+derive shuffles along with vector I/O and computation from
+sequential C code. Their intermediate code contains
+instructions in one nested expression and the shuffle
+operation only works for memory loads that appear as
+arguments of the same vector operation. Therefore, the
+code rewriting for shuffles assumes a top-down style where
+outer expressions have to be vectorized first, in order to
+vectorize inner expressions containing shuffled loads. While
+their technique may provide a powerful method to the
+implementation of libraries, irregular patterns such as
+corner cases found in HPC applications are out of scope.
+10
+Conclusion
+This paper introduces symbolic emulation to compiling
+GPU code in order to discover hidden opportunities for
+optimization. We employ several languages, enabling
+OpenACC directives such as in C and Fortran, for the
+frontend to generate GPU assembly code. Then, our tool
+emulates the code upon symbols that substitute dynamic
+information. While pruning control flows to reduce the
+emulation time, we automatically find possible warp-level
+shuffles that may be synthesized to assembly code to bypass
+global-memory accesses. We apply this technique to a
+benchmark suite and complex application code showing
+results that improve multiple benchmarks on several
+generations of GPUs. We also provide the latency analysis
+across multiple GPUs to identify the use case of shuffles.
+Acknowledgement
+We are funded by the EPEEC project from the European
+Union’s Horizon 2020 research and innovation program
+under grant agreement No. 801051 and the Ministerio de
+Ciencia e Innovación—Agencia Estatal de Investigación
+(PID2019-107255GB-C21/AEI/10.13039/501100011033). This
+work has been partially carried out on the ACME cluster
+owned by CIEMAT and funded by the Spanish Ministry of
+Economy
+and
+Competitiveness
+project
+CODEC-OSE
+(RTI2018-096006-B-I00).
+
+A Symbolic Emulator for Shuffle Synthesis on the NVIDIA PTX Code
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+References
+[1] ExaCT 2013. 2013.
+ExaCT: Center for Exascale Simulation of
+Combustion in Turbulence: Proxy App Software.
+http://www.
+exactcodesign.org/proxy-app-software/
+[2] SW4 2014. 2014. Seismic Wave Modelling (SW4) - Computational
+Infrastructure for Geodynamics. https://geodynamics.org/resources/
+sw4
+[3] The OpenMP ARB. 1997. OpenMP. https://www.openmp.org/
+[4] Roberto Baldoni, Emilio Coppa, Daniele Cono D’elia, Camil
+Demetrescu, and Irene Finocchi. 2018. A survey of symbolic execution
+techniques. ACM Comput. Surv. 51, 3, Article 50 (may 2018), 39 pages.
+https://doi.org/10.1145/3182657
+[5] Eli Ben-Sasson, Matan Hamilis, Mark Silberstein, and Eran Tromer.
+2016. Fast multiplication in binary fields on GPUs via register cache.
+In Proceedings of the 2016 International Conference on Supercomputing
+(ICS ’16). Association for Computing Machinery, New York, NY, USA,
+Article 35, 12 pages. https://doi.org/10.1145/2925426.2926259
+[6] Bryan Catanzaro, Alexander Keller, and Michael Garland. 2014. A
+decomposition for in-place matrix transposition. In Proceedings of the
+19th ACM SIGPLAN Symposium on Principles and Practice of Parallel
+Programming (PPoPP ’14). Association for Computing Machinery, New
+York, NY, USA, 193–206. https://doi.org/10.1145/2555243.2555253
+[7] Peng Chen, Mohamed Wahib, Shinichiro Takizawa, Ryousei Takano,
+and Satoshi Matsuoka. 2019. A versatile software systolic execution
+model for GPU memory-bound kernels. In Proceedings of the
+International Conference for High Performance Computing, Networking,
+Storage and Analysis (SC ’19). Association for Computing Machinery,
+New York, NY, USA, Article 53, 81 pages.
+https://doi.org/10.1145/
+3295500.3356162
+[8] Meghan Cowan, Thierry Moreau, Tianqi Chen, James Bornholt, and
+Luis Ceze. 2020. Automatic generation of high-performance quantized
+machine learning kernels. Association for Computing Machinery, New
+York, NY, USA, 305–316. https://doi.org/10.1145/3368826.3377912
+[9] Leonardo de Moura and Nikolaj Bjørner. 2008. Z3: An efficient SMT
+solver. In Tools and Algorithms for the Construction and Analysis of
+Systems. Springer Berlin Heidelberg, Berlin, Heidelberg, 337–340.
+[10] Robert van Engelen. 2000. Symbolic evaluation of chains of recurrences
+for loop optimization. Technical Report TR-000102. Computer Science
+Dept., Florida State University.
+[11] Robert van Engelen. 2001. Efficients symbolic analysis for optimizing
+compilers. In Proceedings of the 10th International Conference on
+Compiler Construction (CC ’01). Springer-Verlag, Berlin, Heidelberg,
+118–132.
+[12] Philip Ginsbach, Toomas Remmelg, Michel Steuwer, Bruno Bodin,
+Christophe Dubach, and Michael F. P. O’Boyle. 2018.
+Automatic
+matching of legacy code to heterogeneous APIs: An idiomatic approach.
+Association for Computing Machinery, New York, NY, USA, 139–153.
+https://doi.org/10.1145/3173162.3173182
+[13] Simon Garcia De Gonzalo, Sitao Huang, Juan Gómez-Luna, Simon
+Hammond, Onur Mutlu, and Wen-mei Hwu. 2019.
+Automatic
+generation of warp-level primitives and atomic instructions for fast and
+portable parallel reduction on GPUs. In 2019 IEEE/ACM International
+Symposium on Code Generation and Optimization (CGO). 73–84. https:
+//doi.org/10.1109/CGO.2019.8661187
+[14] Tobias Grosser, Armin Groesslinger, and Christian Lengauer.
+2012.
+Polly — Performing polyhedral optimizations on a low-
+level intermediate representation.
+Parallel Processing Letters 22,
+04 (2012), 1250010.
+https://doi.org/10.1142/S0129626412500107
+arXiv:https://doi.org/10.1142/S0129626412500107
+[15] Arpith Chacko Jacob, Alexandre E Eichenberger, Hyojin Sung,
+Samuel F Antao, Gheorghe-Teodor Bercea, Carlo Bertolli, Alexey
+Bataev, Tian Jin, Tong Chen, Zehra Sura, Georgios Rokos, and Kevin
+O’Brien. 2017. Efficient fork-join on GPUs through warp specialization.
+In 2017 IEEE 24th International Conference on High Performance
+Computing (HiPC). 358–367. https://doi.org/10.1109/HiPC.2017.00048
+[16] Zhe Jia, Marco Maggioni, Benjamin Staiger, and Daniele P. Scarpazza.
+2018.
+Dissecting the NVIDIA Volta GPU architecture via
+microbenchmarking. https://doi.org/10.48550/ARXIV.1804.06826
+[17] Justin Luitjens. 2014. Faster parallel reductions on Kepler.
+https:
+//developer.nvidia.com/blog/faster-parallel-reductions-kepler/
+[18] Dmitry Mikushin, Nikolay Likhogrud, Eddy Z. Zhang, and Christopher
+Bergström. 2014. KernelGen – The design and implementation of a
+next generation compiler platform for accelerating numerical models
+on GPUs. In 2014 IEEE International Parallel Distributed Processing
+Symposium Workshops. 1011–1020. https://doi.org/10.1109/IPDPSW.
+2014.115
+[19] G. E. Moore. 1998. Cramming more components onto integrated
+circuits. Proc. IEEE 86, 1 (Jan 1998), 82–85. https://doi.org/10.1109/
+JPROC.1998.658762
+[20] NVIDIA Corporation. 2022. High Performance Computing (HPC) SDK
+| NVIDIA. https://developer.nvidia.com/hpc-sdk
+[21] NVIDIA Corporation. 2022. Kepler Tuning Guide :: CUDA Toolkit
+Documentation. https://docs.nvidia.com/cuda/kepler-tuning-guide/
+index.html
+[22] NVIDIA Corporation. 2022. Programming Guide :: CUDA Toolkit
+Documentation. https://docs.nvidia.com/cuda/cuda-c-programming-
+guide/index.html
+[23] NVIDIA Corporation. 2022. PTX ISA :: CUDA Toolkit Documentation.
+https://docs.nvidia.com/cuda/parallel-thread-execution/index.html
+[24] The OpenACC Organization. 2011. OpenACC. https://www.openacc.
+org/
+[25] Phitchaya Mangpo Phothilimthana, Archibald Samuel Elliott, An
+Wang, Abhinav Jangda, Bastian Hagedorn, Henrik Barthels, Samuel J.
+Kaufman, Vinod Grover, Emina Torlak, and Rastislav Bodik. 2019.
+Swizzle Inventor: Data movement synthesis for GPU kernels.
+In Proceedings of the Twenty-Fourth International Conference on
+Architectural Support for Programming Languages and Operating
+Systems (ASPLOS ’19). Association for Computing Machinery, New
+York, NY, USA, 65–78. https://doi.org/10.1145/3297858.3304059
+[26] Jonathan Ragan-Kelley, Connelly Barnes, Andrew Adams, Sylvain
+Paris, Frédo Durand, and Saman Amarasinghe. 2013.
+Halide: A
+language and compiler for optimizing parallelism, locality, and
+recomputation in image processing pipelines. In Proceedings of the
+34th ACM SIGPLAN Conference on Programming Language Design and
+Implementation (PLDI ’13). Association for Computing Machinery, New
+York, NY, USA, 519–530. https://doi.org/10.1145/2491956.2462176
+[27] Prashant Singh Rawat, Fabrice Rastello, Aravind Sukumaran-Rajam,
+Louis-Noël Pouchet, Atanas Rountev, and P. Sadayappan. 2018.
+Register Optimizations for Stencils on GPUs. In Proceedings of the
+23rd ACM SIGPLAN Symposium on Principles and Practice of Parallel
+Programming (PPoPP ’18). Association for Computing Machinery, New
+York, NY, USA, 168–182. https://doi.org/10.1145/3178487.3178500
+[28] Ross Tate, Michael Stepp, Zachary Tatlock, and Sorin Lerner. 2009.
+Equality Saturation: A new approach to optimization. In Proceedings
+of the 36th Annual ACM SIGPLAN-SIGACT Symposium on Principles
+of Programming Languages (POPL ’09). Association for Computing
+Machinery, New York, NY, USA, 264–276.
+https://doi.org/10.1145/
+1480881.1480915
+[29] TOP500.org. 2022. November 2022 | TOP500.
+https://www.top500.
+org/lists/top500/2021/11/
+[30] Emina Torlak and Rastislav Bodik. 2013.
+Growing solver-aided
+languages with Rosette. In Proceedings of the 2013 ACM International
+Symposium on New Ideas, New Paradigms, and Reflections on
+Programming & Software (Onward! 2013). Association for Computing
+Machinery, New York, NY, USA, 135–152.
+https://doi.org/10.1145/
+2509578.2509586
+[31] Alexa VanHattum, Rachit Nigam, Vincent T. Lee, James Bornholt, and
+Adrian Sampson. 2021. Vectorization for Digital Signal Processors
+via Equality Saturation (ASPLOS 2021). Association for Computing
+
+CC ’23, February 25–26, 2023, Montréal, QC, Canada
+K. Matsumura, S. G. De Gonzalo, A. J. Peña
+Machinery, New York, NY, USA, 874–886.
+https://doi.org/10.1145/
+3445814.3446707
+[32] Sven Verdoolaege and Tobias Grosser. 2012. Polyhedral Extraction Tool.
+In Second International Workshop on Polyhedral Compilation Techniques
+(IMPACT). Paris, France. http://impact.gforge.inria.fr/impact2012/
+[33] Henry Wong, Misel-Myrto Papadopoulou, Maryam Sadooghi-Alvandi,
+and Andreas Moshovos. 2010. Demystifying GPU microarchitecture
+through microbenchmarking. In 2010 IEEE International Symposium
+on Performance Analysis of Systems Software (ISPASS). 235–246. https:
+//doi.org/10.1109/ISPASS.2010.5452013
+[34] Xiaolong Xie, Yun Liang, Xiuhong Li, Yudong Wu, Guangyu Sun,
+Tao Wang, and Dongrui Fan. 2015. Enabling coordinated register
+allocation and thread-level parallelism optimization for GPUs. In 2015
+48th Annual IEEE/ACM International Symposium on Microarchitecture
+(MICRO). 395–406. https://doi.org/10.1145/2830772.2830813
+[35] Da Yan, Wei Wang, and Xiaowen Chu. 2020. Optimizing batched
+Winograd convolution on GPUs. Association for Computing Machinery,
+New York, NY, USA, 32–44. https://doi.org/10.1145/3332466.3374520
+Received 2022-11-10; accepted 2022-12-19
+

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+1 
+Deep leakage from gradients 
+Yaqiong Mu 
+College of Information Science and Technology, Donghua University, 201620, Shanghai China 
+[email protected] 
+Abstract. With the development of artificial intelligence technology, Federated Learning (FL) 
+model has been widely used in many industries for its high efficiency and confidentiality. Some 
+researchers have explored its confidentiality and designed some algorithms to attack training data 
+sets, but these algorithms all have their own limitations. Therefore, most people still believe that 
+local machine learning gradient information is safe and reliable. In this paper, an algorithm based on 
+gradient features is designed to attack the federated learning model in order to attract more attention 
+to the security of federated learning systems.  
+In federated learning system, gradient contains little information compared with the original train-
+ing data set, but this project intends to restore the original training image data through gradient in-
+formation. Convolutional Neural Network (CNN) has excellent performance in image processing. 
+Therefore, the federated learning model of this project is equipped with Convolutional Neural Net-
+work structure, and the model is trained by using image data sets. The algorithm calculates the virtual 
+gradient by generating virtual image labels. Then the virtual gradient is matched with the real gradi-
+ent to restore the original image.  
+This attack algorithm is written in Python language, uses cat and dog classification Kaggle data 
+sets, and gradually extends from the full connection layer to the convolution layer, thus improving 
+the universality. At present, the average squared error between the data recovered by this algorithm 
+and the original image information is approximately 5, and the vast majority of images can be com-
+pletely restored according to the gradient information given, indicating that the gradient of federated 
+learning system is not absolutely safe and reliable.  
+Keywords: Federated Learning, CNN, reconstruction attack, Gradient feature 
+1 
+Introduction 
+In modern Federated Learning (FL) systems [1-3], model updating by exchanging gra-
+dient information among multiple participants is a very common approach. The user 
+data of each participant is always stored locally, and only the gradient information is 
+propagated between different models. This type of algorithm does not need to establish 
+a dedicated central node for data processing, which protects the privacy of users and 
+the local model can be fully trained with the help of a federated learning system. For 
+example, medical systems can share the same data model while protecting the patient's 
+private information [4]. Therefore, it is not easy to extract the data information of local 
+models from the gradient, which has long been believed to be able to be propagated 
+among different models without worrying about privacy leakage, but in fact, stealing 
+local information from the gradient is still traceable. 
+With the rapid development of AI technology, federation learning models are in-
+creasingly used as a fundamental technique in AI technology. Federal learning keeps 
+the data of each participant locally, and the databases of each participant remain inde-
+pendent of each other during modeling, while the information interaction during joint 
+training is encrypted to ensure the confidentiality and efficiency of the system. In 
+
+2 
+addition, the federated learning system can guarantee that the training effect of the local 
+training model is almost the same as that of the original centralized training model. 
+Nowadays, the development of artificial intelligence and deep learning is rapidly 
+changing, and federated learning solves the problem that data from all parties in the 
+previous centralized model can only be used at the central node, and ensures the privacy 
+and confidentiality of users at each node. Federated learning is suitable for training 
+models with large volumes of data and can be applied in a variety of contexts. Nowa-
+days, the concept of smart cities has gained widespread attention, and federal learning 
+models have greatly contributed to the construction of smart cities. In terms of economy 
+and finance, it can combine data from various banks to build a model of economic 
+fluctuation, which can better predict the future economy, etc. In terms of politics and 
+people's livelihood, it can build a bridge between governments at all levels and the 
+masses, realize effective information sharing between governments and the masses, 
+build a good platform for communication between the masses and the government, and 
+help various governments to build a good system of people's city built by the people, 
+so that the authorities can do their work more efficiently and the masses can do their 
+work more conveniently, etc. efficient, more convenient for the masses, etc. 
+The high efficiency and confidentiality of the federal learning system make it 
+more and more widely used. However, the confidentiality of the federal model needs 
+to be further explored, and if the data involved in the training can be restored by some 
+means, it proves that the system still needs to be improved. With the continuous pro-
+gress of artificial intelligence, the protection of Internet privacy has gradually become 
+a hot topic of discussion. By studying the vulnerability of the system, the confidential-
+ity of the federation learning system is gradually improved, which can also provide 
+some new ideas for the protection of Internet privacy nowadays. 
+This thesis focuses on the gradient information leakage problem in convolutional 
+neural network-based federal learning systems, and explores how to restore the original 
+data image from the gradients containing very little information. After introducing the 
+basic principles, the effect of Deep Leakage from Gradients (DLG) algorithm to restore 
+the original image is studied, and certain improvements are made based on it, and fi-
+nally the corresponding conclusions are drawn by comparison. 
+The structure of the thesis is as follows: Chapter 1 briefly introduces the research 
+background, status and significance of this thesis, and briefly composes the content to 
+be studied in this thesis. Chapter 2 briefly introduces the federal learning system, the 
+structure, functions and common models of CNN, and some attack algorithms against 
+the federal learning system. Chapter 3 mainly introduces the general principle of local 
+information leakage, and the working principle and derivation process of DLG algo-
+rithm. Chapter 4 mainly shows the implementation of the depth gradient algorithm, 
+analyzes the shortcomings of the algorithm, proposes improvement methods and com-
+pares them. Chapter 5 mainly integrates and summarizes the research content of this 
+topic, presents the shortcomings and areas for improvement, and provides an outlook 
+for the gradient attack algorithm for FL. 
+
+3 
+2 
+Related Technologies 
+This section introduces the basic concepts and related techniques needed to under-
+stand the reconstruction attack based on gradient features, including the introduction of 
+the federal learning model, the convolutional neural network structure used to train the 
+model, the related models, the role of the functions involved in the network, and some 
+methods for gradient-based attacks. 
+2.1 
+Federal Learning Model 
+The system for federated learning [22] first utilizes an encryption-based user sample 
+alignment technique where data owners identify the common users of each party while 
+securing the data of their respective users in order to federate the features of these users 
+for modeling, and the modeling training process requires federated models to secure the 
+privacy of each local database. First, the federated model sends the public key to the 
+local database to ensure that the local place completes the local data encryption before 
+performing data exchange. After that, the local place transmits the data to the joint 
+model in encrypted form. The data has been initially calculated by the local place and 
+the gradient is calculated based on the tag value, and then the gradient is encrypted and 
+transmitted to the joint model. The joint model combines the gradients calculated by 
+each local model to find the total gradient value, decrypts it and sends it to each local 
+model, so the local model can update its own model parameters according to the new 
+gradient value and improve the optimized model. The above process is repeated until 
+the gradient is infinitely close to the set value, which completes the training of the whole 
+model. During the model training process, the data of each data owner is not exposed 
+to the federated model and other local models, and the data exchange during training 
+does not lead to data privacy threats. As a result, all parties are able to cooperate in 
+training the model with the help of the federated learning model. 
+2.2 
+Convolutional Neural Networks 
+Convolutional Neural Network (CNN) is a deep learning model inspired by biologi-
+cal neural networks [23], formed by interconnecting multiple layers of neurons, where 
+the number of input data in each layer is equal to the number of neurons in the previous 
+layer, and each neuron can receive multiple inputs but can only output one data. This 
+network is often applied in image processing, and the structure and role of each layer 
+will be described next [24]. 
+Input Layer. 
+Convolutional neural networks first need to convert image information into input 
+data. The color of a color picture pixel consists of three attributes: red, green and blue, 
+which are called RGB three channels, and the number of pixels in each row and column 
+of each picture is the resolution of the picture. However, for black and white pictures, 
+the color of the pixels is determined only by the attribute grayscale value. Assume that 
+the value of each channel is between 0 and 511. A color photo with a resolution of 
+
+4 
+100×100 can be converted to a tensor of (100,100,3), and a black and white photo of 
+the same size can be converted to a tensor of (100,100,1). 
+The main work of this layer is to perform a pre-processing of the original image, 
+which consists of three main categories: Centering, which subtracts the average of this 
+dimension from each dimension of the input data, so that the center of the data lies at 
+the zero point. Normalization, which makes the standard deviation of the data to be 1, 
+reduces the effect of different values taken by the data. PCA is used to reduce the cor-
+relation between the feature values and strives to eliminate the correlation between im-
+age bands; and whitening, which weakens the effect of the magnitude on the feature 
+axis of the data. 
+Convolutional Layer. 
+ 
+Fig. 1. Two-dimensional convolution example 
+The three hyperparameters of the convolution kernel are Stride, Zero Padding and 
+Depth. Stride is the number of frames that the data frame moves, which in Figure 2-3 
+is equal to 1. Zero padding protects the edge information of the image from being 
+blurred or lost during the network training process. Depth is the number of convolution 
+kernels, which should be the same as the number of neurons in the next layer. The 
+number of neurons in the convolutional layer is calculated by subtracting the number 
+of neurons from the size of the convolution plus twice the sum of the zero padding, 
+dividing by the step size, and finally adding one to the resulting result. 
+Without parameter sharing, 10×64×64×5×5×3=3072000 parameters are required, 
+and with parameter sharing, 10×5×5×3=750 parameters are required. It can be seen that 
+parameter sharing reduces the number of features obtained by the convolutional nuclei, 
+which leads to the loss of local features if the image size is large. An effective way to 
+solve this problem is to set multiple convolutional kernels in each convolutional layer. 
+
+1
+1
+1
+1
+1
+-1
+0
+-3
+0
+1
+1
+0
+0
+0
+-2
+-1
+2
+1
+1
+-1
+0
+0
+0
+0
+=
+2
+2
+4
+¥0
+-1
+1
+2
+1
+0
+0
+-1
+-1
+0
+0
+1
+2
+1
+1
+15 
+ 
+Fig. 2. Feature Mapping 
+Activating Layer 
+The role of this layer is, as the name suggests, both to take the output of the con-
+volutional layer and to process it nonlinearly. Commonly used nonlinear mapping func-
+tions will be introduced in the following. 
+Sigmoid function 
+Advantages: take the value range (0, 1), simple, easy to understand. 
+Disadvantages: too much data may paralyze the neuron, so that the gradient infor-
+mation cannot be transmitted; the function output data center point does not lie at the 
+zero point. 
+ 
+Fig. 3. Sigmoid function 
+Pooling Layer 
+The pooling layer, also called subsampling layer, is used for feature extraction, which 
+reduces the number of neurons to some extent and prevents the appearance of overfit-
+ting. This layer removes redundant information and retains only key features, which 
+can improve robustness. The pooling layer, also known as the downsampling layer, 
+causes the features of the input information to be lost, which in turn cuts the number of 
+
+3
+N
+0
+1
+2
+3
+2
+3
+0
+1
+2
+m
+16
+0
+1
+x
+0
+2
+m
+0
+15
+16
+3
+0
+1
+2
+m
+0
+2
+m
+2
+15
+16
+0
+1
+6
+15
+3
+01.0
+0.8
+0.6
+0.4
+0.2
+0.0
+-
+-
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+86 
+parameters, making the network less computationally burdensome; while keeping the 
+important features unchanged (cropping, stretching, scaling, etc.). 
+One is average pooling, which requires summing the feature points in the neighbor-
+hood and then dividing the total feature value equally among the feature points; the 
+other is maximum pooling, which, as the name implies, excludes all smaller feature 
+values in the domain and takes them out. The pooling often makes mistakes in obtaining 
+the feature values: first, the variance of the estimate increases; second, the shift of the 
+mean of the estimate. In terms of the prevailing theory, in image processing, the first 
+error handling method mostly uses the mean pooling operation to moderate the size 
+limitation of the domain to reduce the variance, thus making the image background 
+clearer; while the second error handling method mostly uses the maximum pooling op-
+eration, which basically ignores the parameter error of the convolutional layer and guar-
+antees the mean accuracy, thus preserving the texture of the image. Therefore, one of 
+these two methods is missing in convolutional neural networks. 
+Flatten layer and fully connected layer 
+The role of the flatten layer is to flatten multidimensional data into one-dimensional 
+data. The fully-connected layer limits the dimensionality of the data, and thus flattening 
+the data for re-input is essential. 
+The fully connected layer is often used as the closing layer in the convolutional neural 
+network structure, using different activation functions to match different classification 
+requirements. 
+Output Layer 
+The role of this layer is to output the final target result. 
+Structure of convolutional neural networks [26] 
+The layers introduced above are combined to become the complete convolutional 
+neural network structure [27]. Figure 4 shows the basic structure of a CNN, where each 
+convolutional layer applies an activation function for quadratic sampling and then two 
+fully connected layers to give predictions. 
+ 
+Fig. 4. Basic structure of CNN 
+
+C3:feature maps
+S4:feature maps
+16(@10*10
+S2:feature maps
+16@5*5
+C5:layer
+F6:layer OUTPUT
+C1:feature maps
+6(@14*14
+120
+O1
+6@28*28
+INPUT
+32*32
+Full connection
+Gaussian
+Convolutions
+Subsampling
+Convolutions
+Subsampling
+connections
+Full connection7 
+2.3 
+Common models of convolutional neural networks 
+Many models of convolutional neural networks exist, and several commonly used 
+models will be presented here. 
+LeNet 
+LeNet is mainly used to identify and classify non-printed fonts, and it has an accuracy 
+rate of 98%. As a result, the United States put this model into use in the financial in-
+dustry in the late 20th century. This model is used as the basis of convolutional neural 
+network, with a total of six layers of network, and the convolutional kernels are all 5×5 
+with a step size of 1,using average pooling: conv → pool → conv → pool → conv(fc) 
+→ fc. 
+AlexNet 
+This model uses the ReLU function as the activation function, and optimizes the 
+problem that the gradient of the sigmoid function is prone to be uncomputable in a 
+network with more layers. And some improvements are made in the final fully con-
+nected layer, where only some neurons are randomly selected to participate in the com-
+putation of the network, which can prevent overfitting.  
+Convolutional neural networks usually use average pooling and maximum pooling 
+alternately, but in this model, only maximum pooling is used, basically ignoring the 
+parameter error of the convolutional layer and the size limitation of the neighborhood. 
+This model reduces the step size to achieve a pooling kernel size larger than the step 
+size value, so the output of the pooling layer enhances the feature richness. 
+A local response normalization layer is created for the first time, so that the neuron 
+responses in this layer show bipolarity and improve generalization ability. 
+VGGNet. 
+The LRN layer used in AlexNet was not found to bring significant performance 
+improvement to the network in later practice, so the LRN layer in VGGNet has no 
+performance gain (A-LRN) and is not extended to other network models. 
+VGGNet increases the number of network layers compared with other previous 
+networks, and the number of layers in its network structure is twice or more than 
+AlexNet without counting the pooling and softmax layers here. The concept of convo-
+lutional block is proposed for the first time, and 2~3 convolutional layers form a con-
+volutional block, which can reduce the number of parameters and enhance the learning 
+ability by using ReLU activation function. 
+GoogLeNet. 
+Inception V1 increases the convolution module function compared to several pre-
+viously proposed network structures. The previous network structure improves the 
+training effect, but the effect benefits from its increased number of network layers also 
+deepens the network depth. However, the deeper depth also brings many problems, 
+such as overfitting, gradient cannot be found in the network and the computational ef-
+fort increases. 
+
+8 
+SqueezeNet. 
+SqueezeNet's model compression uses 3 strategies. 
+(1) replacing 3×3 convolution with 1×1 convolution: the number of parameters of 
+convolution is reduced to 1/9 of the original one, which helps to improve the speed of 
+network operation; (2) reducing the number of channels of 3×3 convolution: the com-
+putation of a 3×3 convolution is 3×3×a×b (where a, b are the number of channels of 
+input Feature Map and output Feature Map respectively), reducing the number of chan-
+nels to reduce the number of parameters The number of channels is reduced to reduce 
+the number of parameters, which helps to simplify the operation and improve the per-
+formance of the network; (3) the downsampling is set back: the larger Feature Map 
+contains more information, so the downsampling is moved to the classification layer. 
+Such an operation can improve the accuracy of the network, but it will increase the 
+burden of network computation. 
+ResNet. 
+Before introducing the model, it is necessary to understand the concept of residu-
+als, first of all, it is necessary to distinguish between residuals and errors. The error is 
+the measured value minus the reference value, and the residual is the difference between 
+the actual observed value and the predicted value, and the residual can detect whether 
+the prediction is accurate or not. The function of one layer in the residual network is set 
+as y=F(x), and the residual model can be expressed as H(x)=G(x) + x, that is, 
+G(x)=H(x)-x. In the unit mapping, y=x is the actual observed value, and H(x) is the 
+fitted value, so G(x) corresponds to the residual, so it is called the residual network. 
+ 
+Fig. 5. Residual network 
+Losing the residuals, as shown in the connection on the left side of the figure, the 
+error in training and the network depth show a negative correlation as the number of 
+networks increases. In contrast, theoretically, the increase of network depth and the 
+model training effect should show a positive correlation. Theoretical and practical de-
+viations often exist, and for an ordinary network without jump connections, the deeper 
+the depth will make the computation more complicated, and the improvement and 
+
+X
+weight
+G(x)
+X
+relu
+identity
+weight
+G(x)+x
+relu9 
+enhancement of the algorithm will be more difficult to achieve. Therefore, in reality, 
+there is a positive correlation between the depth of the network and the number of train-
+ing errors. 
+To solve this problem, the network needs to detect the existence of redundant lay-
+ers by itself, which makes the optimization algorithm complicated and does not achieve 
+constant mapping. The ResNet model is able to solve this problem in a very fitting way 
+by updating the parameters of the redundant layers with the residual G(x)=0 instead of 
+the fitted value H(x)=x, and by doing so, updating the parameters of the redundant lay-
+ers. That is, after the network spontaneously detects and infers which layers are redun-
+dant and useless, the residual function G(x)=0 makes the network of that layer, after 
+removing the redundant layer, match the input of the previous layer accurately. In this 
+way, the effect of errors caused by redundant layers is almost eliminated, effectively 
+solving the network degradation problem. 
+As an example to explore the cause of network degradation, when one designs the 
+network in the first place, one does not perform the actual operation to grasp the number 
+of layers needed for the network structure. To be on the safe side and to enable the 
+network to train well, people tend to set up more layers of network structure. When the 
+network is actually trained, it is found that only half the number of layers may be needed 
+to complete the task of this network, and then the extra layers are redundant. Therefore, 
+we hope that during the training process, the model can find out that the other half of 
+the layers are redundant and make a constant mapping for only half of the layers, so 
+that the input data will be identical to the output data after passing through the model. 
+But often the model is likely to learn this half of the constant mapping incorrectly, 
+so it may not work as well as a model with 2/3 of the original number of layers set. 
+Therefore, as the number of layers of the network increases, the effect of model training 
+may degrade, which is caused by the redundant layers learning the wrong constant map-
+pings. 
+DenseNet. 
+In a comprehensive view, DenseNet has the following advantages over the previ-
+ous models. 
+(i) the use of dense connectivity, which mainly improves the back propagation 
+speed of gradients to accelerate the training of convolutional neural networks. (2) The 
+parameters are reduced and the values are decreased to improve the efficiency of com-
+putation and to reduce the feature maps specific to each layer; (3) Feature reuse is used 
+to reuse the low-level features for the last layer to play the role of classification. 
+MobileNet. 
+①MobileNet-v1 
+In a nutshell, V1 replaces the usual convolutional layers in vgg with depth-sepa-
+rable convolution, and therefore can greatly reduce the number of parameters; and adds 
+the hyperparameters α and β on top of vgg. 
+②MobileNet-v2 
+
+10 
+MobileNetV2 is proposed by Google in 2018, with better accuracy and smaller 
+model compared to V1. The model highlights have Inverted Residuals structure (In-
+verted Residuals) and Linear bottlenecks. 
+Deep Residual Learning. 
+The core difference of this algorithm is that it proposes a new structure with a 
+topological spreading to form a new block structure, replacing the convolutional block 
+structure of the previous model, which can optimize the performance of the model pre-
+diction and improve the accuracy while adding almost no new parameters. The topo-
+logical spreading also reduces the number of hyperparameters and improves the gener-
+ality of the model. 
+ShuffelNet. 
+①ShuffelNet-v1 
+ShuffleNet is improved by two new operations: point-state group convolution and 
+channel scrubbing, similar to the previous model, which can ensure the accuracy of the 
+network structure output results and reduce the computational complexity. The basic 
+cell structure of the model is optimized and improved based on the residual model cells. 
+②ShuffelNet-v2 
+The number of neurons in this model is relatively small, and the number of 
+branches between layers is thus reduced to speed up the model convergence. The model 
+input speed depends on the number of input and output feature channels, but too many 
+grouping parameters can affect the model convergence speed. 
+EfficientNet. 
+Convolutional neural networks are usually built after resource evaluation, and the 
+more resources are available, the better the performance of the network model will be. 
+This model delves into how to scale the model up and down and finds that making the 
+depth and width of the network converge across the layers or reducing the gap in reso-
+lution can both improve the network's effectiveness. Therefore, a new method is pro-
+posed to balance the above three characteristics of the network with composite coeffi-
+cients, etc. 
+This model was born out of the desire to find a new balance between network 
+depth, width and resolution to measure the accuracy of the network. Previous models 
+have used only one of these aspects to evaluate the effectiveness of the network. This 
+model found that these three aspects together have an impact on the scaling of the net-
+work, and explored the evidence of the interaction between the three, based on which 
+the best combination of the three was found. 
+2.4 
+General Methods for Gradient-Based Attacks 
+Membership inference. 
+Membership inference [28] refers to speculating whether these data points have 
+been used in the process of training the model based on the known training model and 
+
+11 
+the delimited range of data points. In federation learning, the updated gradient infor-
+mation is fed back to the server every round, so the server is able to have certain local 
+model information. With this attack algorithm, the server is able to know whether the 
+delimited data points are used for model training or not. Sometimes, in certain situa-
+tions, this attack can directly lead to a privacy breach. For example, if the attack learns 
+that a patient's clinical records are used for training a model for a particular disease, the 
+fact that the patient has that disease is compromised. In practice, Melis et al. demon-
+strated that this attack approach is extremely accurate on the FourSquare location da-
+taset [29] and can almost determine whether a particular data point data point is used 
+for category classification training.  
+Attribute inference. 
+Attribute inference refers to inferring whether the corresponding training set con-
+tains the same labeled attributes as the known model based on the known training 
+model. Note that the attribute is not important in terms of its relevance to the main task. 
+When training a model on the LFW dataset [30] for identifying gender or race, attribute 
+inference can infer whether they wear a mask or not, in addition to the two known 
+labels. In practice, this also poses a potential risk of privacy compromise. If the patient's 
+age, gender, race, and whether they wear a mask or not are known, there is a high risk 
+that the patient's personal information will be compromised, even if the name and clin-
+ical records remain confidential. 
+Model inversion. 
+Model inversion is a greater threat to the privacy of the training dataset compared 
+to the first two aggressive ones. Since the learning process is always ongoing, this attack 
+exploits this property by having the adversary train a generative adversarial network 
+(GAN) [31] to generate samples that match the training dataset. The results of the attack 
+show that the images obtained are almost identical to the original images, since the 
+GAN is able to create matching samples that are nearly identical to the original training 
+dataset. Moreover, the higher the similarity of the training set, the better the perfor-
+mance of this attack. 
+The above three attack strategies reveal that the information in the gradient is at 
+risk of leakage to some extent, but each of these three attacks has its own limitations. 
+The membership inference attack relies on delimited data, and the attack will be much 
+more difficult when the input data is not textual information (e.g., images, voice). At-
+tribute inference relaxes the constraint that only a label is needed to perform the attack. 
+However, the attack result will narrow the scope and there is no guarantee to find the 
+specific data. For model inversion, although it can generate synthetic images directly 
+from the statistical distribution of the training data, the results are similar alternatives 
+(rather than the original data) and only work when all class members are similar. What 
+will be investigated and demonstrated in this paper is how to steal the training data 
+completely from the gradient information without prior training data. 
+
+12 
+2.5 
+Summary of this chapter 
+This chapter introduced the types of networks and their structures used in this at-
+tack. The first section starts with the federal learning system and outlines how it updates 
+the model by gradients; the second section describes the working principle of convolu-
+tional neural networks suitable for training classification images and the structure of 
+each level; the third section briefly describes the commonly used convolutional neural 
+network models and provides the basis for the next study on how to select and apply 
+such models for training; the fourth section introduces the The fourth subsection intro-
+duces some methods that can be used to perform gradient attacks with prior knowledge 
+of the training data. The theoretical foundation is laid for the subsequent research in 
+this paper to prove the attack algorithm based on gradient features only. 
+3 
+Design of reconstruction attack algorithm based on gradient features 
+The subject under study is a reconstruction attack based on gradient features, using 
+a convolutional neural network for the training of a federal learning system for image 
+classification. In this paper, we need to use the gradient derived from the image and its 
+label information trained by the convolutional neural network to restore the original 
+information. This chapter first introduces the principle of the attack that can obtain part 
+of the original data, and then delves into the analysis and study of the algorithm that 
+restores the complete original information based on the gradient. 
+3.1 
+Local leakage of specific layers 
+First, this chapter starts with a few special layers to study and optimize the attack 
+algorithm step by step. The first one is the fully-connected layer (FC). The fully con-
+nected layer is indispensable in both neural networks and convolutional neural net-
+works. For the biased fully connected layer, it is mathematically proven that the reduc-
+tion of the original input data from the gradient information is done without considering 
+the position of this layer and the class of layers before and after this layer. 
+Lemma 1: Suppose a fully connected layer of a neural network contains weights and 
+biases with input 𝑋 ∈ ℝ𝑛 and output 𝑌 ∈ ℝ𝑚, weight 𝑊 ∈ ℝ𝑚×𝑛and bias 𝐵 ∈ ℝ𝑚, 
+then it is obtained 
+ 
+𝑌 = 𝑊𝑋 + 𝐵  
+（3-1） 
+If there exists 
+𝑑𝐿
+𝑑(𝐵𝑖) ≠ 0, then the input data X can B be reconstructed from 
+𝑑𝐿
+𝑑𝑊 and 
+𝑑𝐿
+𝑑𝐵. The following proof is carried out: it is known that  
+𝑑𝐿
+𝑑(𝐵𝑖) =
+𝑑𝐿
+𝑑𝑌𝑖 and 
+𝑑(𝑌𝑖)
+𝑑(𝑊𝑖) = 𝑋𝑇, then 
+ 
+𝑑𝐿
+𝑑(𝑊𝑖) =
+𝑑𝐿
+𝑑(𝑌𝑖) ⋅
+𝑑(𝑌𝑖)
+𝑑(𝑊𝑖) =
+𝑑𝐿
+𝑑(𝐵𝑖) ⋅ 𝑋𝑇  
+（3-2） 
+where𝑌𝑖 𝑊𝑖 and𝐵𝑖 denote the ith row of output Y, weight W and bias B. Therefore, 
+the input X can be reconstructed from this formula as long as 
+𝑑𝐿
+𝑑(𝐵𝑖) ≠ 0 is satisfied. 
+The derivative as well as the bias 
+𝑑𝐿
+𝑑𝐵 are crucial for reconstructing the input layer. To 
+make the gradient attack more general, Geiping et al. delved deeper and found that if 
+the bias B is eliminated, the original input data can also be restored from a small amount 
+
+13 
+of gradient information as long as a suitable activation function (e.g., ReLU activation 
+function) is found. The proof process is similar, and the reconstruction of the input data 
+in the fully connected layer still works well. 
+If the function is not derived, the input data information is still implied in the gradi-
+ent. For example, in the language classification task, the federal learning system gener-
+ates corresponding gradients only for the words in the input model, and the attack tells 
+which words and phrases were used for model training in each local data set, respec-
+tively. The cross-entropy layer in the classification task, on the other hand, can only 
+generate negative gradients for the data with corrected completion labels. This property 
+gives away the true data labels to some extent. 
+However, there are many more factors to consider when extending from the fully 
+connected layer (FC) to the more complex convolutional layer (CONV), where the 
+number of features in the convolutional layer and the dimensionality of the input occu-
+pation are much larger than the size of the gradient values. A parsing reconstruction 
+method like the one in Lemma 1 will no longer be applicable. Modern convolutional 
+neural networks require a more general attack algorithm. 
+3.2 
+Complete leakage of the gradient 
+Zhu et al [33] proposed a new and improved algorithmic method that is able to 
+solve the above problem by using neural networks with the same structure and matching 
+gradients to restore the reconstructed original dataset. Thus it can ensure that the dataset 
+is private and non-interoperable, and the generality and attack capability of this method 
+are broader and more powerful than the methods in the previous subsection, and this 
+technique is called Deep Gradient Leakage algorithm (DLG). 
+DLG is a reconstruction attack based on gradient features. The attacker receives 
+the gradient update ∇𝑊𝑡,𝑘, k from the other participants k in round t, in order to obtain 
+the training set (𝑥𝑡,𝑘, 𝑦𝑡,𝑘) of participant k from the shared exchange information. Figure 
+3-1 shows how it works in stealing image information: normal participants input an 
+image from the original private data and derive a prediction by the F-model, then use 
+the difference between the prediction and the labeled value to calculate the gradient, 
+which is returned to the participants to update the model. The algorithm first generates 
+a virtual pixel point image with the same size as the real image, and then initializes a 
+virtual label indicating the probability, such as the cat and dog classification explored 
+in this topic, which sets the label value of 0 for the cat and 1 for the dog. then a softmax 
+layer is generated. the DLG iterates the matching of the image and the label on the 
+intermediate local model to compute the virtual gradient. Note that most FL models 
+share the privacy difference module 𝐹（𝑥, 𝑊） and the weights W by default. 
+The loss function is set to be the difference between the true gradient and the vir-
+tual gradient, and then the squared number is obtained to ensure that the loss function 
+is positive. The key point of this reconstruction attack is to narrow the gap between the 
+real gradient and the virtual gradient by continuously iterating, and then return to the 
+models of both parties, update their respective parameters, and retrain the attacker's 
+model so that the attacker's gradient value can continuously approximate the real 
+
+14 
+gradient value. When the target loss function is close to zero, the virtual data image will 
+also be infinitely close to the original data image. 
+ 
+Fig. 6. DLG algorithm 
+In Figure 6, the variables to be updated are marked in bold blue. While the local 
+training model is trained using its differential privacy module and calculates the corre-
+sponding 𝛻𝑊, the attacker uses its own randomly generated input images with label 
+values to derive the gradient 𝛻𝑊′and calculates the difference between the two gradi-
+ents, which the attacker uses as a basis to adjust the parameters and computationally 
+update its virtual input X and label Y so that the gradient loss function converges to a 
+minimum. When the optimization is complete, the attacker can restore the original data 
+information from the local model. 
+The flow of the algorithm is shown next in mathematical form. 
+ 𝐱′∗, 𝐲′∗ = arg 𝑚𝑖𝑛
+𝐱′,𝐲′  ∥∇𝑊′ − ∇𝑊∥2 = arg 𝑚𝑖𝑛
+𝐱′,𝐲′  ∥∥∥∂ℓ(𝐹(𝐱′,𝑊),𝐲′)
+∂𝑊
+− ∇𝑊∥∥∥
+2
+ （3-3） 
+This equation to show how the virtual input 𝐱′∗ and the label value 𝐲′∗如 are ob-
+tained from the gradient reduction. 
+Let the input be 𝐹（）: the microscopic machine learning model; W: the parame-
+ter weights; 𝛻𝑊: the gradient computed from the training data; 𝜂: the learning rate used 
+for DLG optimization. The outputs are the original private training data 𝑥 and the labels 
+𝑦. 
+① DLG algorithm（𝐹，𝑊，𝛻𝑊） 
+② 𝐱′
+1 ← 𝒩(0,1), 𝐲1
+′ ← 𝒩(0,1)                      Initialize virtual inputs and labels. 
+③  for 𝑖 ⟵ 1 to 𝑛 do 
+④  𝐋𝑖
+′ = softmax (𝐲𝑖
+′) 
+⑤  ∇𝑊𝑖
+′ ← ∂ℓ(𝐹(𝐱𝑖
+′, 𝑊), 𝐋𝑖
+′)/ ∂𝑊𝑡               Calculate the virtual gradient. 
+⑥  𝔻𝑖 ← ∥∥∇𝑊𝑖
+′ − ∇𝑊∥∥2 
+⑦  𝐱𝑖+1
+′
+⟵ 𝐱𝑖
+′ − 𝜂∇𝐱𝑖
+′𝔻𝑖
+。。。                              Update the input data according to 
+the gradient. 
+⑧  𝐲𝑖+1
+′
+⟵ 𝐲𝑖
+′ − 𝜂∇𝐲𝑖
+′𝔻𝑖
+。                              Update the labels according to the 
+gradient. 
+
+差分隐私模块
+F(x,W)
+Pred
+LOSS
+[0,1,0]
+VW
+Try to match
+VW'
+差分隐私模块
+Pred'
+Loss'
+[0.2,0.7,0.1]
+F(x,W)
+aD /ax
+aDa
+D=IVW.VWI215 
+It is important to note that the distance of the gradient, i.e., the loss function 
+∥∥∇𝑊𝑖
+′ − ∇𝑊∥∥2must be derivable, so that the virtual input data 𝑥 and label 𝑦 can be op-
+timized using a standard gradient-based approach. it follows that such optimization re-
+quires a second-order derivable function. Here it is assumed that F is a second-order 
+derivable function and this algorithm is applicable to most modern AI models, most 
+neural networks and related tasks. 
+3.3 
+Optimization of DLG algorithm 
+The DLG algorithm can restore the complete original data image in most of the scenes, 
+but in this topic, we found that there is a problem that some of the images cannot be 
+restored completely in practice, and we propose an improvement method based on this 
+problem. 
+Since the original gradient information is generated based on the pixel information 
+of the input image and the label through constant matching, then the richer and more 
+vivid the image color is, the more information the RGB three channels carry, the more 
+pixel information they contain, the more complex the generated gradient is, and the 
+more information can be obtained through the attack, and it is easier to restore the orig-
+inal image. Observe the part of the image that cannot be fully converged, there are 
+mostly large blank areas, which contain relatively less pixel information, so the com-
+plete image cannot be restored. 
+The uneven distribution of image pixel information and the small amount of infor-
+mation in local areas lead to difficulties in image restoration. Thus, the improved algo-
+rithm adds the calculation of the average value of the amount of information contained 
+in the image, based on which the hue of the whole image is inferred, and then the vari-
+ance of each pixel point from the average value is calculated and returned to calculate 
+the gradient and adjust the parameters. When most of the light-colored areas exist in 
+the image, the average value of the image is relatively small, and after other color-rich 
+areas are restored, after iteration, that is, it is possible to calculate the remaining areas 
+based on the average value of the pixel information as light-colored, and to reduce the 
+frequency of random pixel points and dark pixel points to some extent. 
+3.4 
+Summary of this chapter 
+Starting from the simplest fully connected layer, this chapter analyzes the principle 
+of reconstructing the input data from the gradient, but this method also has its limita-
+tions and is not applicable on CNN networks. Then, an optimization algorithm based 
+on this method is introduced, which not only breaks through the original limitations, 
+but also is better in restoring the original data, and can completely restore the original 
+image and labels based on the gradient. Finally, based on the shortcomings of the DLG 
+algorithm, an improvement method is proposed. 
+
+16 
+4 
+Performance evaluation of the reconstruction attack algorithm based on 
+gradient features 
+This chapter shows the implementation of the gradient feature-based reconstruction 
+attack algorithm and the performance evaluation of it and the improved algorithm. 
+4.1 
+System Environment 
+The implementation of the attack in this paper is based on the algorithm written in 
+python language, using the self-contained libraries in PyCharm to support the writing 
+of the program, and the libraries, versions, and configurations used are described in 
+Table 1 below. 
+Table 1. Software Configuration Description 
+Database 
+Versions 
+Description 
+opencv-python 
+4.5.5.62 
+Converts images into pixel 
+information. 
+Pillow 
+8.4.0 
+Image processing. 
+scikit-learn 
+1.0.2 
+Contains algorithms such 
+as classification, regression, 
+clustering 
+scipy 
+1.7.3 
+Differentiation, optimiza-
+tion, image processing 
+tensorboard 
+2.8.0 
+View training 
+torch 
+1.10.1 
+Convert data units 
+torchvision 
+0.11.2。 
+Process image data 
+The subject is trained on CPU, but the CPU is slow in training images, if conditions 
+allow, it is recommended to use GPU for model training to improve the training effi-
+ciency. 
+This section will compare the DLG algorithm and its improved algorithms, using the 
+two metrics of intuitive image presentation and image restoration as a measure. image 
+restoration This paper uses the mean square error between the restored image and the 
+original image data. 
+4.2 
+Implementation of reconstruction attacks based on gradient features 
+Dogs and cats classification dataset 
+The training set of this model uses the cat and dog dataset disclosed by Kaggle in 
+2013, which consists of 25,000 examples, including 12,500 examples of cats and 12,500 
+examples of dogs. Therefore, in this paper, 20,000 images are selected as the training 
+dataset and 2,500 as the test dataset. The data consists of RGB three-channel images of 
+various sizes, in which the types of cats and dogs vary in form and the environment 
+they are in, and the label values of cats and dogs are set to 0 and 1, respectively. 
+Implementation of DLG algorithm 
+
+17 
+The attack process is shown in the figure below. All DLG attacks start with a ran-
+domly generated pixel point (the first image) and try to infinitely approximate the gen-
+erated virtual gradient to the real gradient value. As shown in Table 4-2, the decrease 
+of the mean square error between the virtual image data and the original image data 
+indicates the degree of image convergence, reflecting that the virtual data image grad-
+ually approaches the original data image. 
+ 
+Fig. 7. Restore to get the cat picture 
+ 
+Fig. 8. Restore to get the dog picture 
+Table 2 Mean square error of the leaked image and the original image 
+Number of iterations  
+Image 
+Mean square error 
+20 
+ 
+105.68 
+
+iter=0
+iter=10
+iter=20
+iter=30
+iter=40
+iter=50
+iter=60
+iter=70
+iter=80
+iter=90
+iter=100
+iter=110
+iter=120
+iter=130
+iter=140
+iter=150
+iter=160
+iter=170
+iter=180
+iter=190
+iter=200
+iter=210
+iter=220
+iter=230
+iter=240
+iter=250
+iter=260
+iter=270
+iter=280
+iter=290iter=0
+iter=10
+iter=20
+iter=30
+iter=40
+iter=50
+iter=60
+iter=70
+iter=80
+iter=90
+iter=100
+iter=110
+iter=120
+iter=130
+iter=140
+iter=150
+iter=160
+iter=170
+iter=180
+iter=190
+iter=200
+iter=210
+iter=220
+iter=230
+iter=240
+iter=250
+iter=260
+iter=270
+iter=280
+iter=29018 
+40 
+ 
+99.63 
+50 
+ 
+89.63 
+80 
+ 
+54.25 
+200 
+ 
+3.22 
+Improved implementation of the algorithm 
+Table 3 Comparison of DLG algorithm and improved algorithm 
+Original image 
+DLG algorithm 
+DLG 
+Mean Square 
+Error 
+Improved al-
+gorithms 
+Improved algo-
+rithms Mean 
+Square Error 
+ 
+ 
+24.06 
+ 
+19.54 
+ 
+ 
+47.55 
+ 
+42.45 
+ 
+ 
+40.11 
+ 
+25.36 
+ 
+ 
+28.81 
+ 
+24.34 
+ 
+ 
+30.30 
+ 
+22.41 
+ 
+ 
+28.35 
+ 
+15.28 
+From the above table, it can be seen that the number of pixel points present in the 
+images is positively correlated with the mean square error between the images during 
+the restoration of the dog and cat images. It can be visually seen from the image 
+
+19 
+rendering effect that the improved algorithm has relatively fewer random pixel points 
+present and the mean squared error between the images and the original image is 
+smaller. 
+4.3 
+Experimental results and analysis 
+The DLG attack algorithm used in this paper can attack and restore the vast majority 
+of the original cat and dog pictures based on the gradient, as shown in Figure 7 and 
+Figure 8. Meanwhile, as shown in Table 2, the mean square error between the original 
+data and the original data also tends to the minimum value, which basically stays around 
+3. However, in the training of a large number of images, it was found that there existed 
+a part of images with poor convergence, which still left randomly generated pixel 
+points. Such images usually have some areas with lighter color nearly white, and after 
+improving the algorithm, as shown in Table 3, it can be observed that the improved 
+algorithm has better restoration of the lighter color areas and the mean square error 
+between the original pixel images is smaller. It illustrates that the reconstruction attack 
+based on gradient features is basically able to restore the local data images in the federal 
+learning system. 
+4.4 
+Summary of this chapter 
+This chapter is the implementation and improvement of the gradient feature-based 
+reconstruction attack. The first subsection introduces the programming language used 
+to implement the algorithm, the programming environment, and all the libraries used; 
+the second subsection describes the dataset used and shows the results of the imple-
+mentation of the attack algorithm in detail; the third subsection analyzes the results and 
+demonstrates that the gradient feature-based reconstruction attack can be a threat to the 
+local data of the federal learning system[34-55]. 
+5 
+Conclusion and Outlook 
+5.1 
+ Conclusion 
+In this paper, we study the reconstruction attack based on gradient features, mainly 
+using deep learning techniques and algorithms[56-62] for reconstruction attacks. This 
+paper investigates the mechanism of federation learning, the structural hierarchy of con-
+volutional neural networks (CNNs), and the deep gradient leakage (DLG) algorithm 
+that does not rely on the original dataset for the attack. 
+In this paper, the cat and dog classification dataset is selected as the training model 
+for federation learning, and LeNet, one of the models in CNN, is used for data training. 
+The python language and various libraries in PyCharm are used to complete the recon-
+struction attack based on gradient features, and the original attack algorithm is im-
+proved to make the effect of the restored original image better, which proves that the 
+federation learning gradient has the risk of information leakage. 
+
+20 
+5.2 
+Deficiencies and problems 
+In this paper, the gradient-based attack is implemented for the gradient in the federal 
+learning system using relevant techniques, but some problems are found in the imple-
+mentation and testing sessions of the attack, which need continuous improvement and 
+optimization. 
+(1) When trying to restore high-resolution images, the attack algorithm is not stable 
+enough, the convergence speed is too slow, and the restoration effect is not good. 
+(2) When the attack algorithm is applied to images containing only two colors (such 
+as black and white) with large differences, it may fail to converge or converge poorly, 
+and the images have a large number of random pixel points. 
+(3) The attack algorithm can only do one gradient input to restore an original image 
+for the time being, and cannot input multiple gradients to restore multiple images at the 
+same time. 
+(4) The current algorithm still has problems such as the applicability is not wide 
+enough, and it cannot attack the training model of text class and so on. 
+5.3 
+Outlook for follow-up work 
+Federation learning system will be more widely used in future artificial intelligence 
+technology, although it is not yet seen in some industries, but because of its high effi-
+ciency, it must be used more in the future to bring more convenient and fast life to 
+human beings. The research in this paper raises certain questions about the confidenti-
+ality of federal learning, and this attack algorithm can be further studied and optimized 
+in depth subsequently.  
+(1) The DLG algorithm can restore most of the images at present, but there are still 
+some problems, and the follow-up work hopes to continue to improve this algorithm, 
+and improve the convergence speed and accuracy of the restoration of the algorithm. 
+(2) Different training set categories and training set sizes may affect the training ef-
+fect and attack effect of the CNN network, which can be supplemented with different 
+categories of images to strengthen the attack algorithm. 
+(3) This attack algorithm temporarily cannot attack multiple images in batch, and the 
+attack speed is slow, which can be further improved to enhance the efficiency. 
+Reference 
+[1] 
+McMahan, H.B., Moore, E., Ramage, D., Hampson, S., et al. “Communication efficient learning of deep net-
+works from decentralized data”, Proceedings of the 20th International Conference on Artificial Intelligence 
+and Statistics, 2017, PMLR 54:1273-1282.  
+[2] 
+Jochems, A., et al.: Developing and validating a survival prediction model for NSCLC patients through dis-
+tributed learning across 3 countries. Int. J. Radiat. Oncol. Biol. Phys. 99(2), 344–352 (2017)  
+[3] 
+Yang, Q., Liu, Y., Chen, T., Tong, Y.: Federated machine learning: concept and applications. ACM Trans. Intell. 
+Syst. Technol. 2019, (TIST) 10(2), 1–19  
+[4] 
+Krizhevsky, A.: Learning multiple layers of features from tiny images. Citeseer, Technical report ,2009. 
+[5] 
+Yang Q, Liu Y, Chen T J, et al. Federated Machine Learning[J]. ACM Transactions on Intelligent Systems and 
+Technology, 2019, 10(2): 1-19. 
+
+21 
+[6] 
+Z Xiao, D Xiao, V Havyarimana, H Jiang, D Liu, D Wang, F Zeng. Toward accurate vehicle state estimation 
+under non-Gaussian noises[J]. IEEE Internet of Things Journal, 2019, 6(6): 10652-10664. 
+[7] 
+P Zhang, X Chen, X Ma, Y Wu, H Jiang, D Fang, Z Tang, Y Ma. SmartMTra: Robust indoor trajectory tracing 
+using smartphones[J]. IEEE Sensors Journal, 2017, 17(12): 3613-3624. 
+[8] 
+G. Liu, C. Wang, K. Peng, H. Huang, Y. Li and W. Cheng, "SocInf: Membership Inference Attacks on Social 
+Media Health Data With Machine Learning," in IEEE Transactions on Computational Social Systems, Oct. 
+2019, vol. 6, no. 5, pp. 907-921. 
+[9] 
+Pan X, Zhang M, Yan Y, Zhu J, Yang M. Exploring the Security Boundary of Data Reconstruction via Neuron 
+Exclusivity Analysis[J]. arXiv:2010.13356 [cs, stat], 2021. 
+[10] Zhao Q, Zhao C, Cui S, Jing S, Chen Z. PrivateDL: Privacy-preserving collaborative deep learning against 
+leakage from gradient sharing[J]. International Journal of Intelligent Systems, 2020, 35(8): 1262–1279. 
+[11] Z. Wang， M. Song， Z. Zhang， Y. Song， Q. Wang and H. Qi, “Beyond Inferring Class Representatives: 
+User-Level Privacy Leak from Federated Learning”, IEEE INFOCOM 2019 - IEEE Conference on Computer 
+Communications,2019, pp. 2512-2520. 
+[12] Jie Li, Fanzi Zeng, Zhu Xiao, Hongbo Jiang, Zhirun Zheng, Wenping Liu, Ju Ren. Drive2friends: Inferring 
+social relationships from individual vehicle mobility data[J]. IEEE Internet of Things Journal, 2020, 7(6): 
+5116-5127. 
+[13] L. Melis, C. Song, E. De Cristofaro and V. Shmatikov, “Exploiting Unintended Feature Leakage in Collabora-
+tive Learning,” 2019 IEEE Symposium on Security and Privacy (SP), 2019, pp. 691-706. 
+[14] M. Nasr, R. Shokri and A. Houmansadr, “Comprehensive Privacy Analysis of Deep Learning: Passive and 
+Active White-box Inference Attacks against Centralized and Federated Learning,” 2019 IEEE Symposium on 
+Security and Privacy (SP), 2019, pp. 739-753. 
+[15] Fredrikson, M., Jha, S., Ristenpart, T., “Model inversion attacks that exploit confidence information and basic 
+countermeasures”, Proceedings of the 22nd ACMSIGSAC Conference on Computer and Communications Se-
+curity, 2015, pp. 1322–1333. 
+[16] B. Hitaj, G.Ateniese, Fernando, “Deep Models Under the GAN: Information Leakage from Collaborative Deep 
+Learning”, Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Secu-
+rity,2017,pp. 603–618. 
+[17] Salem A, Zhang Y, Humbert M, et al. “Model and Data Independent Membership Inference Attacks and De-
+fenses on Machine Learning Models”, 2019 Network and Distributed System Security Symposium, 2019, pp. 
+1-15. 
+[18] Jiahui Geng, Yongli Mou, Feifei Li, Qing Li, Oya Beyan. Towards General Deep Leakage in Federated Learn-
+ing[J/OL]. arXiv:2110.09074[cs], 2021[2022-01-04]. 
+[19] Zhao, B., Mopuri, K.R., Bilen, H.: iDLG: improved deep leakage from gradients. 2020, arXiv preprint 
+arXiv:2001.02610  
+[20] Geiping, J.; Bauermeister, H.; Dr¨oge, H.; and Moeller, M. “Inverting Gradients - How easy is it to break 
+privacy in federated learning? ”, Advances in Neural Information Processing Systems 33,NeurIPS ,2020. 
+[21] Hongxu Yin, Arun Mallya, Arash Vahdat, Jose M. Alvarez, Jan Kautz, Pavlo Molchanov. “See Through Gra-
+dients: Image Batch Recovery via GradInversion”, Proceedings of the IEEE/CVF Conference on Computer 
+Vision and Pattern Recognition (CVPR), 2021, pp. 16337-16346 
+[22] T Liu, JCS Lui, X Ma, H Jiang. Enabling relay-assisted D2D communication for cellular networks: Algorithm 
+and protocols[J]. IEEE Internet of Things Journal, 2018, 5(4): 3136-3150. 
+[23] D Wang, J Fan, Z Xiao, H Jiang, H Chen, F Zeng, K Li. Stop-and-wait: Discover aggregation effect based on 
+private car trajectory data[J]. IEEE transactions on intelligent transportation systems, 2018, 20(10): 3623-3633. 
+[24] H Jiang, W Liu, D Wang, C Tian, X Bai, X Liu, Y Wu, W Liu. CASE: Connectivity-based skeleton extraction 
+
+22 
+in wireless sensor networks[C]//IEEE INFOCOM 2009. IEEE, 2009: 2916-2920. 
+[25] Ji S, Xu W, Yang M, et al.3D Convolutional Neural Networks for Human Action Recognition[J]. IEEE Trans-
+actions on Pattern Analysis & Machine Intelligence,2013,35(1):221-231 
+[26] Tran D, Bourdev L, Fergus R, et al. “Learning Spatiotemporal Features with 3D Convolutional Network”. 
+Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2015, pp. 4489-4497. 
+[27] X Ma, H Wang, H Li, J Liu, H Jiang. Exploring sharing patterns for video recommendation on YouTube-like 
+social media[J]. Multimedia Systems, 2014, 20(6): 675-691. 
+[28] Shokri, R., Stronati, M., Song, C., Shmatikov, V.“Membership inference attacks against machine learning mod-
+els”,2017 IEEE Symposium on Security and Privacy (SP), 2017, pp. 3–18 IEEE 
+[29] Yang, D., Zhang, D., Yu, Z., Yu, Z.: Fine-grained preference-aware location search leveraging crowdsourced 
+digital footprints from LBSNs. In: Proceedings of the 2013 ACM International Joint Conference on Pervasive 
+and Ubiquitous Computing, 2013, pp.479–488  
+[30] Huang, G.B., Ramesh, M., Berg, T., Learned-Miller, E.: Labeled faces in the wild: a database for studying face 
+recognition in unconstrained environments. University of Massachusetts, Amherst, Technical Report 07-49, 
+October 2007 
+[31] Goodfellow, I., et al.: Generative adversarial nets. In: Advances in Neural Information Processing Systems, 
+2014, pp. 2672–2680 
+[32] Geiping, J., Bauermeister, H., Dr¨oge, H., Moeller, M.: Inverting gradients-how easy is it to break privacy in 
+federated learning? 2020, arXiv preprint arXiv:2003.14053  
+[33] Zhu, L., Liu, Z., Han, S. “Deep leakage from gradients”, Annual Conference on Neural Information Processing 
+Systems, (NeurIPS) (2019) 
+[34] Ping Zhao, Hongbo Jiang, John C. S. Lui, Chen Wang, Fanzi Zeng, Fu Xiao, and Zhetao Li. P3-LOC: A Pri-
+vacy-Preserving Paradigm-Driven Framework for Indoor Localization. IEEE/ACM Transactions on Network-
+ing (ToN), vol. 26, no. 6, pp. 2856-2869, 2018.  
+[35] Hongbo Jiang, Yuanmeng Wang, Ping Zhao, Zhu Xiao. A Utility-Aware General Framework with Quantifiable 
+Privacy Preservation for Destination Prediction in LBSs. IEEE/ACM Transactions on Networking (ToN), vol. 
+29, no. 5, pp. 2228-2241, 2021.  
+[36] Ping Zhao, Jiaxin Sun, and Guanglin Zhang. DAML: Practical Secure Protocol for Data Aggregation based on 
+Machine Learning. ACM Transactions on Sensor Networks, vol. 16, no. 4, pp. 1-18, 2020.  
+[37] Ping Zhao, Wuwu Liu, Guanglin Zhang, Zongpeng Li, and Lin Wang. Preserving Privacy in WiFi Localization 
+with Plausible Dummy Locations. IEEE Transactions on Vehicular Technology, vol. 69, no. 10, pp. 11909-
+11925, 2020.  
+[38] Guanglin Zhang, Anqi Zhang, Ping Zhao, and Jiaxin Sun. Lightweight Privacy-Preserving Scheme in WiFi 
+Fingerprint-Based Indoor Localization. IEEE Systems Journal, vol. 14, no. 3, pp. 4638-4647, 2020. 
+[39] Ping Zhao, Jie Li, Fanzi Zeng, Fu Xiao, Chen Wang, Hongbo Jiang. ILLIA: Enabling k-Anonymity-based 
+Privacy Preserving against Location Injection Attacks in continuous LBS Query. IEEE Internet of Things Jour-
+nal, vol. 5, no. 2, pp. 1033–1042, 2018. 
+[40] Ping Zhao, Hongbo Jiang, Jie Li, Fanzi Zeng, Xiao Zhu, Kun Xie, and Guanglin Zhang. Synthesizing Privacy 
+Preserving Traces: Enhancing Plausibility with Social Networks. IEEE/ACM Transactions on Networking 
+(ToN), vol. 27, no. 6, pp. 2391 – 2404, 2019.  
+[41] Ping Zhao, Jiawei Tao, Guanglin Zhang. Deep Reinforcement Learning-based Joint Optimization of Delay and 
+Privacy 
+in 
+Multiple-User 
+MEC 
+Systems. 
+IEEE 
+Transactions 
+on 
+Cloud 
+Computing, 
+DOI: 
+10.1109/TCC.2022.3140231, 2022. 
+[42] Ping Zhao, Hongbo Jiang, Jie Li, Zhu Xiao, Daibo Liu, Ju Ren, Deke Guo. Garbage in, Garbage out: Poisoning 
+Attacks Disguised with Plausible Mobility in Data Aggregation. IEEE Transactions on Network Science and 
+
+23 
+Engineering (TNSE), vol. 8, no. 3, pp. 2679-2693, 2021. 
+[43] Ping Zhao, Xiaohui Zhao, Daiyu Huang, H. Huang. Privacy-Preserving Scheme against Location Data Poi-
+soning Attacks in Mobile-Edge Computing. IEEE Transactions on Computational Social Systems, vol. 7, no. 
+3, pp. 818-826, 2020.  
+[44] Ping Zhao, Chen Wang, and Hongbo Jiang. On the Performance of k -Anonymity against Inference Attacks 
+with Background Information. IEEE Internet of Things Journal, vol. 6, no. 1, pp. 808–819, 2018.  
+[45] Guanglin Zhang, Sifan Ni, and Ping Zhao. Enhancing Privacy Preservation in Speech Data Publishing. IEEE 
+Internet of Things Journal, vol. 7, no. 8, pp. 7357-7367, 2020.  
+[46] Guanglin Zhang, Anqi Zhang, Ping Zhao. LocMIA: Membership Inference Attacks against Aggregated Loca-
+tion Data. IEEE Internet of Things Journal, vol. 7, no. 12, pp. 11778-11788, 2020. 
+[47] Guanglin Zhang, Sifan Ni and Ping Zhao. Learning-based Joint Optimization of Energy-Delay and Privacy in 
+Multiple-User Edge-Cloud Collaboration MEC Systems. IEEE Internet of Things Journal, doi: 
+10.1109/JIOT.2021.3088607, 2021.  
+[48] P. Zhao, Z. Cao, J. Jiang and F. Gao, "Practical Private Aggregation in Federated Learning Against Inference 
+Attack," in IEEE Internet of Things Journal, 2022, doi: 10.1109/JIOT.2022.3201231. 
+[49] Hongbo Jiang, Yu Zhang, Zhu Xiao, Ping Zhao and Arun Iyengar. An Empirical Study of Travel Behavior 
+Using Private Car Trajectory Data. IEEE Transactions on Network Science and Engineering, vol. 8, no. 1, pp. 
+53-64, 2021. 
+[50] H Jiang, W Liu, D Wang, C Tian, X Bai, X Liu, Y Wu, W Liu. Connectivity-based skeleton extraction in 
+wireless sensor networks[J]. IEEE Transactions on Parallel and Distributed Systems, 2009, 21(5): 710-721. 
+[51] H Jiang, S Jin, C Wang. Parameter-based data aggregation for statistical information extraction in wireless 
+sensor networks[J]. IEEE Transactions on Vehicular Technology, 2010, 59(8): 3992-4001. 
+[52] Yennun Huang, Yih-Farn Chen, Rittwik Jana, Hongbo Jiang, Michael Rabinovich, Amy Reibman, Bin Wei, 
+Zhen Xiao. Capacity analysis of MediaGrid: a P2P IPTV platform for fiber to the node (FTTN) networks[J]. 
+IEEE Journal on Selected Areas in Communications, 2007, 25(1): 131-139. 
+[53] W Liu, D Wang, H Jiang, W Liu, C Wang. Approximate convex decomposition based localization in wireless 
+sensor networks[C]//2012 Proceedings IEEE INFOCOM. IEEE, 2012: 1853-1861. 
+[54] C Tian, H Jiang, X Liu, X Wang, W Liu, Y Wang. Tri-message: A lightweight time synchronization protocol 
+for high latency and resource-constrained networks[C]//2009 IEEE International Conference on Communica-
+tions. IEEE, 2009: 1-5. 
+[55] Y Huang, Z Xiao, D Wang, H Jiang, D Wu. Exploring individual travel patterns across private car trajectory 
+data[J]. IEEE Transactions on Intelligent Transportation Systems, 2019, 21(12): 5036-5050. 
+[56] K Chen, C Wang, Z Yin, H Jiang, G Tan. Slide: Towards fast and accurate mobile fingerprinting for Wi-Fi 
+indoor positioning systems[J]. IEEE Sensors Journal, 2017, 18(3): 1213-1223. 
+[57] H Huang, H Yin, G Min, H Jiang, J Zhang, Y Wu. Data-driven information plane in software-defined network-
+ing[J]. IEEE Communications Magazine, 2017, 55(6): 218-224. 
+[58] S Wang, A Vasilakos, H Jiang, X Ma, W Liu, K Peng, B Liu, Y Dong. Energy efficient broadcasting using 
+network coding aware protocol in wireless ad hoc network[C]//2011 IEEE International Conference on Com-
+munications (ICC). IEEE, 2011: 1-5. 
+[59] H Jiang, Z Ge, S Jin, J Wang. Network prefix-level traffic profiling: Characterizing, modeling, and evalua-
+tion[J]. Computer Networks, 2010, 54(18): 3327-3340. 
+[60] H Jiang, A Iyengar, E Nahum, W Segmuller, A Tantawi, CP Wright. Load balancing for SIP server clus-
+ters[C]//IEEE INFOCOM 2009. IEEE, 2009: 2286-2294. 
+[61] H Jiang, P Zhao, C Wang. RobLoP: Towards robust privacy preserving against location dependent attacks in 
+continuous LBS queries[J]. IEEE/ACM Transactions on Networking, 2018, 26(2): 1018-1032. 
+
+24 
+[62] H Jiang, J Cheng, D Wang, C Wang, G Tan. Continuous multi-dimensional top-k query processing in sensor 
+networks[C]//2011 Proceedings IEEE INFOCOM. IEEE, 2011: 793-801. 
+ 
+

5tE4T4oBgHgl3EQf1g0M/content/tmp_files/2301.05290v1.pdf.txt

@@ -0,0 +1,1569 @@
+EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
+CERN-EP-2022-280
+2023/01/16
+CMS-HIN-21-006
+K0
+S and Λ (Λ) two-particle femtoscopic correlations in PbPb
+collisions at √sNN = 5.02 TeV
+The CMS Collaboration
+Abstract
+Two-particle correlations are presented for K0
+S, Λ, and Λ strange hadrons as a func-
+tion of relative momentum in lead-lead collisions at a nucleon-nucleon center-of-mass
+energy of 5.02 TeV. The dataset corresponds to an integrated luminosity of 0.607 nb−1
+and was collected using the CMS detector at the CERN LHC. These correlations are
+sensitive to quantum statistics and to final-state interactions between the particles.
+The source size extracted from the K0
+SK0
+S correlations is found to decrease from 4 to
+1 fm in going from central to peripheral collisions. Strong interaction scattering pa-
+rameters (i.e., scattering length and effective range) are determined from the ΛK0
+S and
+ΛΛ (including their charge conjugates) correlations using the Lednick´y–Lyuboshitz
+model and are compared to theoretical and other experimental results.
+Submitted to Physics Letters B
+© 2023 CERN for the benefit of the CMS Collaboration. CC-BY-4.0 license
+arXiv:2301.05290v1  [nucl-ex]  12 Jan 2023
+
+
+1
+1
+Introduction
+Two-particle correlations in relative momentum, so-called femtoscopic correlations, arising
+from relativistic heavy ion collisions provide a powerful tool for studying both the quark-gluon
+plasma (QGP) that is created in the collisions, and the subsequent interactions of the emitted
+particles [1]. All two-particle correlations are affected by final-state interaction (FSI) effects,
+and correlations of identical particles are also sensitive to the constraints of quantum statistics
+(QS). The correlations among the neutral K0
+S, Λ, and Λ particles, collectively referred to as V0
+particles, are of special interest. First, they can be used to determine the space-time extent of
+the QGP. In addition, information can be extracted about the strong-interaction scattering pa-
+rameters, i.e., the scattering length and the effective range, that is impossible to obtain from
+currently achievable scattering experiments [2–6]. Because of their relatively heavy mass and
+the absence of a Coulomb interaction, femtoscopy based on K0
+S particles supplements the more
+commonly studied pion and charged kaon pairs [7]. The results from ΛΛ correlation studies
+can help constrain baryon-baryon and, more specifically, hyperon-hyperon interaction models
+that are used, for example, in modeling the composition of neutron stars [8–10].
+Regarding the scattering parameters, of particular interest is establishing whether the interac-
+tion between two Λ particles allows for the existence of the H-dibaryon, a bound state with
+quantum numbers I = 0, JP = 0+, S = −2. In 1977, R. L. Jaffe predicted the existence of such a
+six-quark (uuddss) state having a mass about 81 MeV below the threshold of twice the Λ mass
+by considering the strong attraction resulting from color magnetic interactions [11]. Although
+a double hypernucleus,
+6
+ΛΛHe, was subsequently observed in the NAGARA event from the
+E313 hybrid emulsion experiment at KEK [12, 13], the observed ΛΛ binding energy was not
+consistent with the conjectured H-dibaryon [14]. A study of ΛΛ correlations may provide ad-
+ditional information on whether the baryon-baryon interaction can lead to the formation of the
+conjectured H-dibaryon.
+Recently, the ALICE Collaboration reported on ΛK correlations in lead-lead (PbPb) collisions
+at a center-of-mass energy per nucleon pair of √sNN = 2.76 TeV [15]. According to their find-
+ings, the strong force is repulsive in ΛK+ interactions, yet attractive in ΛK− interactions. For
+the ΛK0
+S pairs, the uncertainty of the ALICE results does not permit a definite conclusion
+on whether the associated strong interaction is repulsive or attractive. A more precise mea-
+surement of ΛK0
+S correlations should improve our understanding of the strong interaction in
+baryon-meson systems.
+This Letter presents K0
+SK0
+S, ΛK0
+S, and ΛΛ femtoscopic correlations as a function of relative
+momentum in PbPb collisions at √sNN = 5.02 TeV, using data recorded by the CMS experi-
+ment during the 2018 LHC run. The K0
+SK0
+S correlations are measured in six centrality intervals
+within the 0–60% range, where centrality refers to the percentage of the total inelastic hadronic
+nucleus-nucleus cross section [16], and 0% corresponds to the maximum overlap of the col-
+liding nuclei. The K0
+SK0
+S, ΛK0
+S, and ΛΛ correlations are measured in an integrated centrality
+range (0–80%), with the ΛΛ femtoscopic correlation measured in PbPb collisions at the LHC
+for the first time. The source size and strong interaction parameters are determined using the
+Lednick´y–Lyuboshitz (LL) model [17]. Unless otherwise indicated, all measurements include
+the charge conjugate states, so ΛK0
+S and ΛΛ include ΛK0
+S and ΛΛ, respectively. Tabulated
+results are provided in the HEPData record for this analysis [18].
+
+2
+2
+Experimental setup and data sample
+The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diam-
+eter, providing a magnetic field of 3.8 T. Within the solenoid volume there is a silicon pixel
+and strip tracker, a lead tungstate crystal electromagnetic calorimeter, and a brass and scintil-
+lator hadron calorimeter, each composed of a barrel and two endcap sections. The silicon pixel
+detector [19] is composed of 1856 silicon pixel modules distributed in four 54 cm long bar-
+rel layers at radii of 2.9–16.0 cm plus three pairs of endcap disks covering radii of 4.5–16.1 cm
+at longitudinal distances of 31–51 cm from the origin. The 15 148 silicon strip module are ar-
+ranged in 10 barrel layers at radii of 20–116 cm plus 3 pairs of small and 9 pairs of large endcap
+disk layers. Charged particles of pseudorapidity |η| < 3 are reconstructed with the combined
+system. For particles with transverse momentum of 1 < pT < 10 GeV, the track resolutions
+are typically 1.5% in pT and 20–75 µm in the transverse impact parameter [20]. The barrel and
+endcap detectors are extended to the forward region with two calorimeters which use steel as
+the absorber and quartz fibers as the sensitive material. These hadron forward (HF) calorime-
+ters are located 11.2 m from the interaction region, one on each side, and provide coverage in
+the range 3.0 < |η| < 5.2. These detectors are segmented into multiple 0.175×0.175 (∆η×∆φ)
+“towers”, where φ is azimuthal angle in radians. Muons are measured in gas-ionization detec-
+tors embedded in the steel flux-return yoke outside the solenoid. Events of interest are selected
+using a two-tiered trigger system. The first level, composed of custom hardware processors,
+uses information from the calorimeters and muon detectors to select events at a rate of around
+100 kHz within a fixed latency of about 4 µs [21]. The second level, known as the high-level
+trigger, consists of a farm of processors running a version of the full event reconstruction soft-
+ware optimized for fast processing, and reduces the event rate to around 1 kHz before data
+storage [22]. A more detailed description of the CMS detector, together with a definition of the
+coordinate system used and the relevant kinematic variables, can be found in Ref. [23].
+With an integrated luminosity of 0.607 nb−1 [24, 25], this analysis uses 4.27 × 109 minimum bias
+events that are triggered by requiring signals above the readout threshold of 3 GeV in each of
+the HF calorimeters [22]. Background events due to beam-gas interactions and non-hadronic
+collisions are filtered offline by applying the procedure described in Ref. [26]. The events used
+in this analysis are required to have at least one primary interaction vertex determined using
+two or more tracks [27] within a distance of 15 cm from the center of the nominal interaction
+point along the beam axis and to have at least two calorimeter towers in each HF detector with
+energy deposits of more than 4 GeV per tower. The shapes of the clusters in the pixel detector
+are required to be compatible with those expected in PbPb collisions in order to suppress the
+contamination from events with multiple collisions [28]. The combined trigger and offline
+selection efficiency for inelastic events is greater than 95%. The event centrality is obtained from
+the transverse energy deposited in both HF calorimeters, using the methodology described in
+Ref. [29]. The analysis makes use of a minimum bias Monte Carlo PbPb sample, based on the
+HYDJET 1.9 [30] event generator with a full detector simulation using GEANT4 [31].
+3
+Reconstruction of K0
+S and Λ candidates
+The K0
+S and Λ candidates, denoted as V0 candidates, used in this study are reconstructed as in
+previous CMS analyses [32–34]. The V0 candidates are found by combining oppositely charged
+tracks that pass criteria based on the “loose” selection discussed in Ref. [27]. The charged tracks
+are assumed to be π+π− in K0
+S reconstruction and π−p in Λ reconstruction. For the latter, the
+higher momentum track is assumed to be a proton since the proton carries nearly all of the mo-
+mentum in the Λ decay. Each of the oppositely charged tracks must have hits in at least three
+
+3
+layers of the silicon tracker, and both tracks must have transverse and longitudinal impact pa-
+rameter significances (defined as the parameter value divided by its uncertainty) with respect
+to the primary vertex greater than 1. The two tracks are fitted to a common vertex and the χ2
+per degree of freedom (dof) from the fit must be less than 7. The distance of closest approach
+between the two tracks is required to be less than 1 cm. As a consequence of the long lifetime of
+K0
+S and Λ particles, the significance of the V0 decay length, which is the three-dimensional dis-
+tance between the primary and V0 vertices divided by its uncertainty, is required to be greater
+than 2.5 to reduce combinatorial background contributions. To remove K0
+S candidates misiden-
+tified as Λ particles and vice versa, the Λ (K0
+S) candidates must have a corresponding ππ (pπ)
+mass more than 14 (7) MeV (corresponding to approximately 3 times the average resolution)
+away from the world-average value [35] of the K0
+S (Λ) mass. The angle θ between the V0 mo-
+mentum vector and the vector connecting the primary and V0 vertices is required to satisfy
+cos θ > 0.999. This reduces the contribution from nuclear interactions, random combinations
+of tracks, and Λ particles originating from weak decays of Ξ and Ω particles.
+Further selection of V0 candidates is performed with a boosted decision tree (BDT) [36]. The se-
+lection is optimized separately for K0
+S and Λ candidates. The discriminating variables include:
+the collision centrality, the V0 candidate pT and rapidity (y), the distance of closest approach
+of the track pair, the three-dimensional decay length and significance, cos θ, and the V0 ver-
+tex fit χ2. The included variables related to the V0 daughters are pT, uncertainty in pT, η, the
+number of hits in the silicon tracker, the number of pixel detector layers with hits, and the
+transverse and longitudinal impact parameter significances with respect to the primary ver-
+tex. The BDT training is performed using the simulated minimum bias sample separated into
+the signal and background subsamples using the generator-level information. The K0
+S mesons
+are selected with 1 < pT < 8.5 GeV and |y| < 1, while the Λ baryons are required to have
+1.8 < pT < 8.5 GeV and |y| < 1. The minimum pT and maximum y requirements are used to
+reduce background while the maximum pT requirement is to reduce contributions from jets.
+The combined V0 reconstruction and selection efficiencies are strongly dependent on the cen-
+trality of the event and the pT of the V0. Integrating over the selected pT ranges, the combined
+efficiencies from the most central to peripheral PbPb collisions are 1–3% for K0
+S and 1–2% for
+Λ. The V0 reconstruction algorithm does not prevent a track from being used for more than
+one V0. While this is normally an infrequent occurrence, selecting pairs of V0 particles close
+together in phase space makes it a significant contribution. To resolve this problem, for each
+correlation measurement, a check of each pair of V0 candidates is performed and if two V0 can-
+didates are found to share one or both daughter tracks, one of the V0 candidates is randomly
+selected to be removed from the event.
+Fits to the invariant mass spectrum are performed using a sum of three Gaussian functions
+with a common mean to describe the signal distribution and a fourth-order polynomial to de-
+scribe the background. These empirical functions were chosen to provide a good description of
+the data. Peak and sideband invariant mass regions are defined to select events dominated by
+signal and background, respectively. Defining σ as the average resolution based on the Gaus-
+sian sum, the peak regions are selected to be within ±2σ from the nominal V0 mass and are
+given by 486 < M(π+π−) < 509 MeV and 1111.5 < M(pπ−) < 1120.4 MeV for K0
+S and Λ
+candidates, respectively. The sideband regions are selected to be more than 4σ from the nom-
+inal V0 mass and are given by 23.5 < |M(π+π−) − 497.5| < 62.5 MeV for K0
+S candidates and
+1080 < M(pπ−) < 1107.5 MeV together with 1124.2 < M(pπ−) < 1160 MeV for Λ candidates.
+Examples of invariant mass distributions for π+π− and pπ− pairs, and their corresponding
+fits in the 0–80% centrality range, are shown in Fig. 1.
+
+4
+0.45
+0.5
+0.55
+ invariant mass (GeV)
+−
+π
++
+π
+6
+10
+7
+10
+8
+10
+Candidates / (0.5 MeV)
+Data
+Fit
+Background
+Peak region
+Sideband region
+)
+-1
+ = 5.02 TeV (0.607 nb
+NN
+s
+PbPb, 
+CMS
+S
+0
+K
+Centrality: 0-80%
+ < 8.5 GeV
+T
+1 < p
+|y| < 1
+1.08
+1.1
+1.12
+1.14
+1.16
+ invariant mass (GeV)
++
+π
+p
++ 
+−
+π
+p
+5
+10
+6
+10
+Candidates / (0.5 MeV)
+Data
+Fit
+Background
+Peak region
+Sideband region
+)
+-1
+ = 5.02 TeV (0.607 nb
+NN
+s
+PbPb, 
+CMS
+Λ
+ + 
+Λ
+Centrality: 0-80%
+ < 8.5 GeV
+T
+1.8 < p
+|y| < 1
+Figure 1: The invariant mass of K0
+S (left) and Λ (right), and their corresponding fits in the 0–80%
+centrality range. The circles are the data, and the fit is shown with a solid (red) line for the total
+fit, and a dashed (green) line for the background fit. The vertical dashed-dotted (pink) lines
+indicate the peak region and the vertical dashed (blue) lines indicate the sideband regions.
+4
+Analysis method
+The two-particle correlation is constructed as
+Cobs(qinv) = N Aobs(qinv)
+Bobs(qinv) ,
+(1)
+where Cobs(qinv) is the observed normalized pair yield, corrected for detector effects, as a func-
+tion of the invariant relative momentum qinv, defined as [1]
+qinv =
+�
+−QµQµ,
+Qµ = (k1 − k2)µ − (k1 − k2)µPµ
+PµPµ
+Pµ,
+(2)
+where P = k1 + k2, and k1 and k2 are the four momenta of the V0 particles. Note that for two
+particles of the same mass, the second term of Qµ is zero.
+The distribution Aobs(qinv) is the signal distribution that contains femtoscopic correlations
+formed by pairing the selected V0 particles from a given event. The reference distribution
+Bobs(qinv) is used to correct for phase space effects, largely removing artifacts due to detector
+non-uniformities in the Aobs(qinv) distribution. The Bobs(qinv) distribution is constructed by
+mixing the V0 particles from different events [37]. In this procedure, the V0 particle from one
+event is paired with V0 particles from 30 different events. To ensure that the 30 events used
+in the mixing are similar to the signal events, the centrality and primary vertex of each mixed
+event must be within 5% and 2 cm, respectively, of those in the corresponding signal event.
+The normalization factor N is the ratio of the number of pairs in the reference distribution
+to that in the signal distribution. Because of the background in the peak region of the invari-
+ant mass distributions, the measured signal distribution (Aobs(qinv)) contains contributions
+from signal-signal (Ass(qinv)), signal-background (Asb(qinv)), and background-background
+(Abb(qinv)) correlations. The measured Aobs(qinv) distribution can be written as
+Aobs(qinv) = f ssAss(qinv) + f sbAsb(qinv) + f bbAbb(qinv).
+(3)
+
+5
+The distributions Asb(qinv) and Abb(qinv) are obtained from the peak-sideband and sideband-
+sideband combinations, respectively. The small amount of background (signal) contamination
+in the signal (sideband) region has a negligible effect on the shape of Asb(qinv) (Abb(qinv)). All
+distributions, Aobs(qinv), Asb(qinv), and Abb(qinv) are normalized to unity. The parameters, f ss,
+f sb, and f bb are the signal-signal, signal-background, and background-background fractions,
+extracted using an invariant mass fit based on combinatorial analyses with
+f ss =
+(s
+2)
+(s+b
+2 )
+,
+f bb =
+(b
+2)
+(s+b
+2 )
+, and
+f sb = 1 − f ss − f bb,
+(4)
+where (n
+2) = n(n−1)
+2
+is the binomial coefficient, which returns the number of ways that a pair can
+be chosen from n objects. The quantities s and b in the binomial coefficients are the number of
+signal and background particles, respectively, obtained by integrating the appropriate function
+from the fit to the invariant mass distribution.
+Once we have all the distributions (Aobs(qinv), Asb(qinv), and Abb(qinv)) and the parameters
+( f ss, f sb, and f bb), the Ass(qinv) distribution can be extracted using Eq. (3), with
+Ass(qinv) =
+�
+Aobs(qinv) − f sb(Asb(qinv)) − f bb(Abb(qinv))
+�
+/ f ss.
+(5)
+The same procedure is followed for the reference distribution. After extracting the Ass(qinv)
+and Bss(qinv) distributions, the correlation distribution is calculated as
+Css(qinv) = N Ass(qinv)
+Bss(qinv) .
+(6)
+While the Css(qinv) distribution is corrected for detector effects and non-V0 backgrounds, it
+still includes non-femtoscopic background correlations, such as those associated with elliptic
+flow [38], minijets [7], resonance decays [7], and energy-momentum conservation [39]. The
+non-femtoscopic background contribution is modeled using an empirically determined double
+Gaussian function
+Ω(qinv) = N
+�
+1 + α1e−q2
+invR2
+1
+� �
+1 − α2e−q2
+invR2
+2
+�
+,
+(7)
+where N, α1, α2, R1, and R2 are fit parameters. This function was selected for its reproduction
+of the distributions in both real data at high qinv and simulated data that do not include the
+correlations being measured.
+Fits are performed to the Css(qinv) distributions to extract the source size and strong interaction
+scattering parameters. As the V0 particles are neutral, the Coulomb interaction is absent. How-
+ever, the correlations are sensitive to QS and FSI effects, with s-wave interactions assumed to
+dominate for the small relative momenta of the particle pairs analyzed. The correlation distri-
+bution for all pairs (K0
+SK0
+S, ΛK0
+S, and ΛΛ) is interpreted in the LL model. This model relates the
+two-particle correlation function to the source size and also takes into account FSI effects [17].
+The general correlation function is
+Ctotal(qinv) =
+�
+1 + λ
+�
+CQS(qinv) + CFSI(qinv)
+��
+Ω(qinv),
+(8)
+
+6
+where CQS(qinv) is the QS function and CFSI(qinv) is the FSI function. The parameter λ is re-
+ferred to as the incoherence parameter. In the absence of FSI effects, λ equals unity for a
+perfectly incoherent Gaussian source. Effects such as resonance decay violate the incoherent
+source assumption and can lead to deviations of the λ parameter from the unity. Its value
+can also be affected by non-Gaussian components of the correlations function and by the FSI
+between particles.
+Neglecting CP violation, the K0
+SK0
+S system can be written as
+|K0
+SK0
+S⟩ = 1
+2
+�
+|K0K0⟩ + |K0K0⟩ + |K0K0⟩ + |K0K0⟩
+�
+.
+(9)
+It can be shown [17, 40] that the resulting correlations follow Bose–Einstein quantum statistics,
+with
+CQS(qinv) = e(−q2
+invR2
+inv),
+(10)
+where the source radius Rinv reflects the size of the region over which the particles are emitted.
+The FSI for the K0
+SK0
+S correlations is modeled by [17, 40]
+CFSI(qinv) = 1
+2
+�����
+f (k)
+Rinv
+����
+2
++ 4ℜ f (k)
+√πRinv
+F1(qinvRinv) − 2ℑ f (k)
+Rinv
+F2(qinvRinv)
+�
+,
+(11)
+where
+k = qinv/2,
+F1(z) = 1
+2e−z2 � z
+0 ex2dx, and
+F2(z) = 1 − e−z2
+z
+.
+(12)
+The function f (k) is the K0K0 s-wave scattering amplitude, with real and imaginary parts ℜ f (k)
+and ℑ f (k), respectively. This amplitude is dominated by the near-threshold s-wave isoscalar
+resonance f0(980) and the s-wave isovector resonance a0(980), with the total scattering ampli-
+tude given by an average of these contributions: f (k) = ( ff0(980)(k) + fa0(980)(k))/2. The indi-
+vidual resonance amplitudes depend on the resonance mass mr, with r = f0(980) or a0(980),
+the kaon mass mK, and the resonance couplings γr (γ′
+r) to the K0K0 (ππ for f0(980) and π0η
+for a0(980) channels. Then, fr(k) = γr/
+�
+m2
+r − ζ − iγrk − iγ′
+rk′
+r
+�
+, where ζ = 4(m2
+K + k2) and
+k′
+r denotes the momentum in the second (ππ or π0η) decay channel with the corresponding
+partial width Γ′ = γ′
+rk′
+r/mr (more details can be found in Ref. [40]). The scattering amplitude
+is calculated using the resonance mass and the coupling parameters from Refs. [41–44], taken
+from row C of Table 1 of Ref. [40].
+For the correlations involving Λ baryons, the CQS(qinv) and CFSI(qinv) functions are [17]
+CQS(qinv) = αe(−q2
+invR2
+inv),
+and
+CFSI(qinv) = (1 + α)
+�
+1
+2
+| f (k)|2
+R2
+inv
+�
+1 −
+1
+2√π
+d0
+Rinv
+� + 2ℜ f (k)
+√πRinv
+F1(qinvRinv) − ℑ f (k)
+Rinv
+F2(qinvRinv)
+�
+,
+(13)
+where α = −1/2 for ΛΛ correlations for two identical fermions and α = 0 for ΛK0
+S corre-
+lations as there are no QS effects for non-identical particles [17]. The scattering amplitude
+
+7
+f (k) is parameterized by a complex scattering length (f0) and an effective range (d0) with
+f (k) = [1/ f0 + d0k2/2 − ik]−1 [17]. The imaginary part of f0 is responsible for inelastic pro-
+cesses (annihilation). For an attractive interaction that is not strong enough to produce a bound
+state, the real part of f0 is positive, while a repulsive interaction corresponds to a negative ℜ f0
+of the order of the range of the repulsive potential. In the presence of a bound state, ℜ f0 is also
+negative, but with a much larger magnitude. The femtoscopic sign convention and notation
+for the scattering length differ from those used in nuclear physics, where the corresponding
+scattering length a0 = − f0. As the ΛK0
+S and ΛΛ correlations each have only one spin state that
+contributes to the s-wave scattering, Eq. (13) suffices to describe the FSI effects.
+Fits to the correlation distribution of all the pairs were performed using Eq. (8) with the non-
+femtoscopic background parameters (N, α1, α2, R1, and R2) treated as free parameters. For
+K0
+SK0
+S correlations, the parameters of interest are Rinv and λ, with the scattering amplitude
+based on previous measurements [41–44]. The ΛK0
+S and ΛΛ correlations include additional
+parameters: d0, ℜ f0, and ℑ f0. The ℑ f0 term for ΛΛ correlations is set to zero since there are no
+baryon-baryon annihilation processes.
+Histograms of the correlation distributions are generated in the range 0 < qinv < 6 GeV with
+0.02 GeV wide bins for the K0
+SK0
+S and ΛK0
+S correlations and 0.04 GeV wide bins for the ΛΛ cor-
+relations. The fits exclude the first qinv bin to avoid a potential bias from the method used to
+address cases where V0 candidates share daughter tracks. Studies using simulated events in-
+dicate that only this first bin is affected by this remediation. Least-square fits are performed to
+the experimental data with the uncertainties in the fit parameters calculated using the MINOS
+technique [45]. Examples of correlation measurements and their fits and corresponding χ2/dof
+values, are presented in Figs. 2 and 3. The K0
+SK0
+S correlations, shown in Fig. 2, are independently
+fitted for each of the six centrality bins with 0 < kT < 2.5 GeV, where kT ≡ |⃗pT,1 + ⃗pT,2|/2 is
+the average transverse momentum of the particle pair. While the LL model assumes a Gaus-
+sian source function, results from charged-particle correlations have demonstrated that this
+assumption breaks down for peripheral collisions [46]. This is likely the cause of the poor fit
+at low qinv for centralities above 40%. The ΛK0
+S (left) and ΛΛ (right) correlations, shown in
+Fig. 3, involve fewer events and, therefore, only a single fit is performed for each, with the data
+integrated over the centrality range 0–80% and with no restriction on kT.
+5
+Systematic uncertainties
+The systematic uncertainties for the fit parameters are based on the changes found in the pa-
+rameter values after individually varying each of the analysis criteria, as discussed below. In
+cases with more than one variation for a single source, the maximum deviation from the nom-
+inal value is used. The total systematic uncertainty is obtained by adding the uncertainties
+from each source in quadrature. The BDT discriminant is varied so as to adjust the signal-to-
+background ratio, with the signal yield changing by ±15% in the process. The nominal method
+to account for V0 candidates sharing daughter tracks is to remove one of the V0 candidates at
+random, which is then not used by any pair. Two alternative approaches are used, one in which
+both V0 candidates are removed and another in which, for events with multiple V0 candidate
+pair combinations, only the pairs in which the two particles share a daughter are removed.
+The systematic uncertainties related to V0 signal and background modeling are investigated by
+varying the background shape from a fourth- to a third-order polynomial and the signal shape
+from a sum of three Gaussian functions to a sum of two or four Gaussian functions. An alterna-
+tive non-femtoscopic background function Ω(qinv) = N(1 + Be−|qinv/σ|2)(1 + ϵqinv) is used to
+assess the uncertainty associated with the choice of the non-femtoscopic background function.
+
+8
+)
+-1
+ = 5.02 TeV (0.607 nb
+NN
+s
+PbPb, 
+CMS
+0-10%
+: 287
+2
+χ
+0
+S
+K
+0
+S
+K
+dof: 292
+30-40%
+: 306
+2
+χ
+0
+S
+K
+0
+S
+K
+dof: 292
+10-20%
+: 323
+2
+χ
+0
+S
+K
+0
+S
+K
+dof: 292
+40-50%
+: 334
+2
+χ
+0
+S
+K
+0
+S
+K
+dof:292
+20-30%
+: 312
+2
+χ
+0
+S
+K
+0
+S
+K
+dof: 292
+50-60%
+: 353
+2
+χ
+0
+S
+K
+0
+S
+K
+dof: 292
+ < 8.5 GeV
+T
+1 < p
+ < 2.5 GeV
+T
+0 < k
+Data
+Full fit
+Nonfemto
+0
+2
+4
+6
+ (GeV)
+inv
+q
+1
+1.5
+2
+)
+inv
+(q
+ss
+C
+0
+0.1
+0.2
+0.3
+0.4
+ (GeV)
+inv
+q
+1
+1.5
+2
+)
+inv
+(q
+ss
+C
+: 19
+2
+χ
+# bins: 19
+0
+2
+4
+6
+ (GeV)
+inv
+q
+1
+1.5
+2
+)
+inv
+(q
+ss
+C
+0
+0.1
+0.2
+0.3
+0.4
+ (GeV)
+inv
+q
+0.8
+1
+1.2
+1.4
+)
+inv
+(q
+ss
+C
+: 25
+2
+χ
+# bins: 19
+0
+2
+4
+6
+ (GeV)
+inv
+q
+1
+1.5
+2
+)
+inv
+(q
+ss
+C
+0
+0.1
+0.2
+0.3
+0.4
+ (GeV)
+inv
+q
+1
+1.5
+)
+inv
+(q
+ss
+C
+: 43
+2
+χ
+# bins: 19
+0
+2
+4
+6
+ (GeV)
+inv
+q
+1
+1.5
+2
+)
+inv
+(q
+ss
+C
+0
+0.1
+0.2
+0.3
+0.4
+ (GeV)
+inv
+q
+1
+1.5
+)
+inv
+(q
+ss
+C
+: 18
+2
+χ
+# bins: 19
+0
+2
+4
+6
+ (GeV)
+inv
+q
+1
+1.5
+2
+)
+inv
+(q
+ss
+C
+0
+0.1
+0.2
+0.3
+0.4
+ (GeV)
+inv
+q
+1
+1.5
+2
+2.5
+)
+inv
+(q
+ss
+C
+: 40
+2
+χ
+# bins: 19
+0
+2
+4
+6
+ (GeV)
+inv
+q
+1
+1.5
+2
+)
+inv
+(q
+ss
+C
+0
+0.1
+0.2
+0.3
+0.4
+ (GeV)
+inv
+q
+1
+1.5
+)
+inv
+(q
+ss
+C
+: 13
+2
+χ
+# bins: 19
+Figure 2: The correlation distributions and fits for K0
+SK0
+S pairs in different centrality ranges,
+starting from 0–10% centrality to 50–60% centrality, with 0 < kT < 2.5 GeV. In each plot, the
+red circles are the data, the blue solid line is the fit using Eq. (8), and the green dotted line is
+the non-femtoscopic background from Eq. (7). The χ2 and dof values are for the full qinv range.
+The insert plots show the data and the fit for the qinv < 0.4 GeV region, with the χ2 and number
+of bins evaluated in that region.
+0
+1
+2
+3
+4
+5
+6
+ (GeV)
+inv
+q
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+)
+inv
+(q
+ss
+C
+Data
+Full fit
+Non-femto
+Data
+Full fit
+Non-femto
+)
+-1
+ = 5.02 TeV (0.607 nb
+NN
+s
+PbPb, 
+CMS
+S
+0
+K
+Λ
+⊕
+S
+0
+K
+Λ
+0-80%
+ < 8.5 GeV
+Λ
+/
+Λ
+T
+1.8 < p
+ < 8.5 GeV
+S
+0
+K
+T
+1 < p
+T
+all k
+: 349
+2
+χ
+dof: 289
+0
+0.1
+0.2
+0.3
+0.4
+ (GeV)
+inv
+q
+0.8
+0.9
+1
+1.1
+)
+inv
+(q
+ss
+C
+: 21
+2
+χ
+# bins: 19
+0
+1
+2
+3
+4
+5
+6
+ (GeV)
+inv
+q
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+)
+inv
+(q
+ss
+C
+Data
+Full fit
+Nonfemto
+Data
+Full fit
+Nonfemto
+)
+-1
+ = 5.02 TeV (0.607 nb
+NN
+s
+PbPb, 
+CMS
+Λ
+Λ
+⊕
+Λ
+Λ
+0-80%
+ < 8.5 GeV
+T
+1.8 < p
+T
+all k
+: 124
+2
+χ
+dof: 140
+0
+0.1
+0.2
+0.3
+0.4
+ (GeV)
+inv
+q
+0.4
+0.6
+0.8
+1
+)
+inv
+(q
+ss
+C
+: 5
+2
+χ
+# bins: 9
+Figure 3: The correlation distributions and fits for ΛK0
+S (left) and ΛΛ (right) pairs with 0–80%
+centrality and no restriction on kT. In each plot, the red circles are the data, the blue solid line is
+the fit using Eq. (8), and the green dotted line is the non-femtoscopic background from Eq. (7).
+The χ2 and dof values are for the full qinv range. The insert plots show the data and the fit for
+the qinv < 0.4 GeV region, with the χ2 and number of bins evaluated in that region.
+
+9
+The selection requirements used to construct the mixed event sample are varied to require
+centrality matching of 3 and 7% instead of the nominal 5% and the primary vertex position
+matching with 1 and 3 cm instead of the nominal 2 cm. The effect of the centrality resolution
+has been checked and found to be negligible. The peak region requirement is changed from
+<2.0σ to <1.5σ and <2.5σ and the sideband region selection from >4.0σ to >3.5σ and >4.5σ.
+The upper limit of the qinv fit ranges is changed by ±1 GeV and the lower limit is changed to
+include the first bin. At low pT, the tracking efficiency is strongly dependent on pT. There-
+fore, the simulated sample is used to explore possible effects of the tracking efficiency. Based
+on these studies, it is found that the V0 reconstruction efficiency for the detection of two V0
+particles is well described by taking the product of the efficiency for each V0. It is also found
+that the fit results are only weakly affected by the V0 reconstruction efficiency. This is under-
+stood as a consequence of the signal and reference samples being similarly affected by the V0
+efficiency. Differences in the Monte Carlo and experimental pT spectra could influence the can-
+cellation of efficiency-dependent effects in the signal and background correlations. Therefore,
+a systematic uncertainty for the efficiency is assessed by comparing the results for the default
+simulated sample to one in which the V0 pT distribution is reweighted to match the data. For
+the K0
+SK0
+S correlations, an additional systematic uncertainty is found from varying the mass and
+coupling parameters for the f0(980) and a0(980) resonances by using rows A, B, and D of Table
+1 of Ref. [40]. The systematic uncertainties are summarized in Table 1.
+Table 1: Summary of absolute systematic uncertainties in K0
+SK0
+S, ΛK0
+S and ΛΛ correlation mea-
+surements. The values for Rinv, d0, ℜ f0, and ℑ f0 are in fm.
+Uncertainty source
+K0
+SK0
+S
+ΛK0
+S
+ΛΛ
+Rinv
+λ
+Rinv
+d0
+ℜ f0 ℑ f0
+λ
+Rinv
+d0
+ℜ f0
+λ
+BDT cut
+0.04–0.18 0.01–0.04
+0.19 0.75 0.10 0.07 0.03
+0.06 0.43 0.05 0.31
+Duplicate V0 removal
+0.06–0.40 0.01–0.08
+0.35 0.92 0.10 0.19 0.11
+0.01 1.14 0.05 0.14
+Mass fit function
+0.00–0.01 0.00–0.01
+0.09 0.05 0.01 0.03 0.03
+0.02 0.04 0.01 0.02
+Non-femtoscopic func. 0.02–0.16 0.01–0.12
+0.02 0.17 0.05 0.07 0.03
+0.02 1.02 0.14 0.93
+Reference sample
+0.03–0.08 0.01–0.05
+0.22 0.48 0.12 0.12 0.03
+0.10 1.12 0.20 0.76
+Peak region
+0.00–0.07 0.01–0.02
+0.43 0.10 0.05 0.17 0.08
+0.22 1.21 0.08 0.35
+Sideband region
+0.00–0.03 0.00–0.01
+0.02 0.02 0.01 0.01 0.00
+0.01 0.03 0.01 0.04
+Fitting range
+0.01–0.11 0.01–0.04
+0.20 0.18 0.03 0.08 0.04
+0.04 1.79 0.20 0.60
+Efficiency
+0.03–0.03 0.01–0.01
+0.08 0.29 0.06 0.09 0.03
+0.02 0.05 0.03 0.04
+f0(980)/a0(980) param. 0.07–0.39 0.03–0.05
+—
+—
+—
+—
+—
+—
+—
+—
+—
+Total uncertainty
+0.29–0.47 0.08–0.16
+0.69 1.34 0.21 0.32 0.16
+0.25 2.91 0.34 1.43
+6
+Results
+The size of the particle emitting source Rinv and the λ parameter extracted from the K0
+SK0
+S cor-
+relations for 0 < kT < 2.5 GeV are shown as a function of centrality in Fig. 4. It is observed that
+the Rinv value decreases from central to peripheral events, as expected from a simple geometric
+picture of the collisions. Over the full centrality range of 0–80%, Rinv = 3.30 ± 0.10 (stat) ±
+0.37 (syst) fm. The transverse mass can be calculated as mT =
+√
+(minv/2)2 + k2
+T, where minv
+is the invariant mass of the two-particle system [15]. The average ⟨mT⟩ is evaluated from the
+transverse mass distribution using two-particle pairs with qinv < 0.4 GeV, accounting for back-
+ground using the binomial analysis as done for the qinv distributions. Our results for Rinv agree
+with the ALICE K0
+SK0
+S results from PbPb collisions at √sNN = 2.76 TeV at a similar mT value [47].
+The λ parameter is seen to decrease from about 0.45 to 0.25 as the collisions become more pe-
+
+10
+ripheral. This decrease could arise from a relative increase in the contribution from resonance
+decays or a source function that becomes increasingly non-Gaussian as the collisions become
+more peripheral. The assumption of a Gaussian source function in the LL model may also be
+responsible for the relatively poor fits at low qinv for the most peripheral collisions, as seen in
+Fig. 2.
+0
+10
+20
+30
+40
+50
+60
+0
+1
+2
+3
+4
+5
+)
+-1
+ = 5.02 TeV (0.607 nb
+NN
+s
+PbPb, 
+CMS
+ < 8.5 GeV
+T
+1 < p
+ < 2.5 GeV
+T
+0 < k
+Centrality (%)
+(fm)
+inv
+R
+0
+S
+K
+0
+S
+K
+0
+10
+20
+30
+40
+50
+60
+0
+0.2
+0.4
+0.6
+0.8
+1
+)
+-1
+ = 5.02 TeV (0.607 nb
+NN
+s
+PbPb, 
+CMS
+ < 8.5 GeV
+T
+1 < p
+ < 2.5 GeV
+T
+0 < k
+Centrality (%)
+λ
+0
+S
+K
+0
+S
+K
+Figure 4: The Rinv (left) and λ parameter (right) as a function of centrality. For each data point,
+the line and shaded area indicate the statistical and systematic uncertainty, respectively.
+Table 2 includes the extracted Rinv and λ parameters as well as ⟨mT⟩ for K0
+SK0
+S, ΛK0
+S, and ΛΛ
+combinations in the 0–80% centrality range. A significant decrease is seen in Rinv as the ⟨mT⟩
+increases. Qualitatively similar results have been found, both for a given pair type in bins of
+mT and when comparing multiple pair types [1]. Because of the different minimum pT require-
+ments for K0
+S and Λ particles, the variation in ⟨mT⟩ includes both ⟨pT⟩ and particle mass differ-
+ences. The anticorrelation of Rinv and ⟨mT⟩ has been interpreted as indicating the presence of
+an expanding source [1].
+Table 2 also includes the strong interaction scattering parameters d0, ℜ f0, and ℑ f0 obtained
+from the ΛK0
+S and ΛΛ correlations. Figure 5 shows d0 and ℑ f0 versus ℜ f0 in the left and
+right panels, respectively, with the current results shown as red stars and squares for ΛK0
+S and
+ΛΛ, respectively. The displayed uncertainties are one-dimensional and are not based on a
+two-dimensional contour.
+Table 2: Extracted values of the Rinv, ℜ f0, ℑ f0, d0, λ, and ⟨mT⟩ parameters from the K0
+SK0
+S, ΛK0
+S,
+and ΛΛ combinations in the 0–80% centrality range. The first and second uncertainties are
+statistical and systematic, respectively.
+Parameter
+K0
+SK0
+S
+ΛK0
+S
+ΛΛ
+Rinv (fm)
+3.30 ± 0.10 ± 0.37
+2.1+1.4
+−0.5 ± 0.7
+1.3+0.4
+−0.2 ± 0.3
+ℜ f0 (fm)
+—
+−0.76+0.29
+−0.19 ± 0.21
+0.74+0.59
+−0.16 ± 0.34
+ℑ f0 (fm)
+—
+−0.07+0.48
+−0.11 ± 0.32
+—
+d0 (fm)
+—
+2.3+0.7
+−0.5 ± 1.3
+4.2+5.7
+−2.1 ± 2.9
+λ
+0.38 ± 0.02 ± 0.08
+0.34+0.41
+−0.12 ± 0.16
+1.5+1.2
+−1.1 ± 1.4
+⟨mT⟩ (GeV)
+1.53
+2.09
+2.60
+
+11
+2
+−
+1.5
+−
+1
+−
+0.5
+−
+0
+0.5
+1
+1.5
+2
+15
+−
+10
+−
+5
+−
+0
+5
+10
+15
+CMS
+AA collisions
+ (fm)
+0
+ f
+ℜ
+ (fm)
+0
+d
+Λ
+Λ
+CMS (5.02 TeV): 
+S
+0
+K
+Λ
+CMS (5.02 TeV): 
+Λ
+Λ
+STAR (200 GeV): 
+S
+0
+K
+Λ
+ALICE (2.76 TeV): 
+Λ
+Λ
+PRC 91, 024916: 
+Λ
+Λ
+PRC 66, 024007: 
+Λ
+Λ
+NPA 707, 491: 
+1.5
+−
+1
+−
+0.5
+−
+0
+0.5
+0.5
+−
+0
+0.5
+1
+CMS
+AA collisions
+ (fm)
+0
+ f
+ℜ
+ (fm)
+0
+ f
+ℑ
+S
+0
+K
+Λ
+CMS (5.02 TeV): 
+S
+0
+K
+Λ
+ALICE (2.76 TeV): 
+Figure 5: The measured values of d0 versus ℜ f0 (left) and ℑ f0 versus ℜ f0 (right) from this
+analysis along with other measurements and predictions as described in the text. For each
+data point, the lines and the boxes indicate the (one-dimensional) statistical and systematic
+uncertainties, respectively.
+The negative value of ℜ f0 observed for the ΛK0
+S correlations, combined with its relatively small
+magnitude, suggests a repulsive ΛK0
+S interaction. The uncertainty associated with the ℑ f0
+value for the ΛK0
+S correlations prevents any claim concerning possible inelastic processes. The
+value of ℜ f0 found for ΛK0
+S correlations differs from that reported by the ALICE Collaboration
+(teal diamonds) [15], which is also for PbPb collisions but at √sNN = 2.76 TeV. The uncertainties
+are too large to determine if d0 and ℑ f0 also differ between the two results.
+The positive ℜ f0 value obtained for the ΛΛ correlations suggests an attractive interaction
+that is not strong enough to produce a bound state such as the H-dibaryon [9, 48]. This re-
+sult disagrees with the finding from the STAR Collaboration in gold-gold (AuAu) collisions at
+√sNN = 200 GeV (blue circle). The negative ℜ f0 value of −1.10 ± 0.37 (stat)+0.68
+−0.08 (syst) fm found
+by STAR, combined with its magnitude, imply a repulsive interaction. It is noted, however,
+that a theoretical study of the STAR data which considers collective flow and feed-down effects
+(shown as a shaded region at d0 ≈ 5 fm, ℜ f0 ≈ 0.9 fm) suggests that these data are consistent
+with the ΛΛ interaction being attractive [9]. An exclusion plot by the ALICE Collaboration
+for the ΛΛ scattering parameters obtained using the ΛΛ correlations from pp collisions at
+√s = 7 and 13 TeV, as well as pPb collisions at √sNN = 5.02 TeV, also suggests an attractive
+interaction [10]. In addition, our results are consistent with two theoretical calculations (black
+triangles) that reproduce the ΛΛ binding energy of
+6
+ΛΛHe, as extracted from the NAGARA
+event [49, 50].
+7
+Summary
+The K0
+SK0
+S, ΛK0
+S, and ΛΛ femtoscopic correlations are studied using lead-lead (PbPb) collision
+data at a center-of-mass energy per nucleon pair of √sNN = 5.02 TeV, collected by the CMS Col-
+laboration. This is the first report on ΛΛ correlations in PbPb collisions at the CERN LHC. The
+source size Rinv and the incoherence parameter λ were extracted for K0
+SK0
+S correlations in six
+centrality bins covering the 0–60% range. The value of Rinv decreases from 4 to 1 fm going from
+central to peripheral collisions and agrees with results from the ALICE Collaboration at a sim-
+ilar transverse mass. Along with the Rinv and λ parameters, the strong interaction scattering
+parameters, i.e., the complex scattering length and effective range, were extracted from ΛK0
+S
+and ΛΛ correlations in the 0–80% centrality range. These scattering parameters indicate that
+
+12
+the ΛK0
+S interaction is repulsive and that the ΛΛ interaction is attractive. The scattering param-
+eters obtained from ΛK0
+S correlations differ from those reported by the ALICE Collaboration.
+The positive real scattering length obtained from the ΛΛ correlation disfavors the existence
+of a bound H-dibaryon state. The ΛΛ scattering parameters help to constrain baryon-baryon
+and, more specifically, hyperon-hyperon interaction models. These measurements provide an
+additional input to understand the nature of the strong interaction between pairs of strange
+hadrons.
+References
+[1] M. A. Lisa, S. Pratt, R. Soltz, and U. Wiedemann, “Femtoscopy in relativistic heavy ion
+collisions: Two decades of progress”, Ann. Rev. Nucl. Part. Sci. 55 (2005) 357,
+doi:10.1146/annurev.nucl.55.090704.151533, arXiv:nucl-ex/0505014.
+[2] V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester, and J. J. de Swart, “Partial-wave
+analysis of all nucleon-nucleon scattering data below 350 MeV”, Phys. Rev. C 48 (1993)
+792, doi:10.1103/PhysRevC.48.792.
+[3] J. J. de Swart and C. Dullemond, “Effective range theory and the low energy
+hyperon-nucleon interactions”, Anna. Phys. 19 (1962) 458,
+doi:10.1016/0003-4916(62)90185-9.
+[4] R. Engelmann, H. Filthuth, V. Hepp, and E. Kluge, “Inelastic Σ− p-interactions at low
+momenta”, Phys. Lett. 21 (1966) 587, doi:10.1016/0031-9163(66)91310-2.
+[5] F. Eisele et al., “Elastic Σ± p scattering at low energies”, Phys. Lett. B 37 (1971) 204,
+doi:10.1016/0370-2693(71)90053-0.
+[6] B. Sechi-Zorn, B. Kehoe, J. Twitty, and R. A. Burnstein, “Low-energy Λ-proton elastic
+scattering”, Phys. Rev. 175 (1968) 1735, doi:10.1103/PhysRev.175.1735.
+[7] CMS Collaboration, “Bose–Einstein correlations in pp, pPb, and PbPb collisions at
+√sNN = 0.9–7 TeV”, Phys. Rev. C 97 (2018) 064912,
+doi:10.1103/PhysRevC.97.064912, arXiv:1712.07198.
+[8] J. Schaffner-Bielich, M. Hanauske, H. St¨ocker, and W. Greiner, “Phase transition to
+hyperon matter in neutron stars”, Phys. Rev. Lett. 89 (2002) 171101,
+doi:10.1103/PhysRevLett.89.171101, arXiv:astro-ph/0005490.
+[9] K. Morita, T. Furumoto, and A. Ohnishi, “ΛΛ interaction from relativistic heavy-ion
+collisions”, Phys. Rev. C 91 (2015) 024916, doi:10.1103/PhysRevC.91.024916,
+arXiv:1408.6682.
+[10] ALICE Collaboration, “Study of the Λ-Λ interaction with femtoscopy correlations in pp
+and pPb collisions at the LHC”, Phys. Lett. B 797 (2019) 134822,
+doi:10.1016/j.physletb.2019.134822, arXiv:1905.07209.
+[11] R. L. Jaffe, “Perhaps a stable dihyperon”, Phys. Rev. Lett. 38 (1977) 195,
+doi:10.1103/PhysRevLett.38.195.
+[12] H. Takahashi et al., “Observation of a
+6
+ΛΛHe double hypernucleus”, Phys. Rev. Lett. 87
+(2001) 212502, doi:10.1103/PhysRevLett.87.212502.
+
+References
+13
+[13] K. Nakazawa and H. Takahashi, “Experimental study of double-Λ hypernuclei with
+nuclear emulsion”, Prog. Theor. Phys. Supplement 185 (2010) 335,
+doi:10.1143/PTPS.185.335.
+[14] Belle Collaboration, “Search for an H-dibaryon with a mass near 2mΛ in Υ(1S) and
+Υ(2S) decays”, Phys. Rev. Lett. 110 (2013) 222002,
+doi:10.1103/PhysRevLett.110.222002, arXiv:1302.4028.
+[15] ALICE Collaboration, “ΛK femtoscopy in Pb-Pb collisions at √sNN = 2.76 TeV”, Phys.
+Rev. C 103 (2021) 055201, doi:10.1103/PhysRevC.103.055201,
+arXiv:2005.11124.
+[16] C. Loizides, J. Kamin, and D. d’Enterria, “Improved Monte Carlo Glauber predictions at
+present and future nuclear colliders”, Phys. Rev. C 97 (2018) 054910,
+doi:10.1103/PhysRevC.97.054910, arXiv:1710.07098.
+[17] R. Lednick´y and V. L. Lyuboshitz, “Final state interaction effect on pairing correlations
+between particles with small relative momenta”, Sov. J. Nucl. Phys. 35 (1982) 770.
+[18] HEPData record for this analysis, 2022. doi:10.17182/hepdata.133573.
+[19] Tracker Group of the CMS Collaboration, “The CMS phase-1 pixel detector upgrade”,
+JINST 16 (2021) P02027, doi:10.1088/1748-0221/16/02/P02027,
+arXiv:2012.14304.
+[20] CMS Collaboration, “Track impact parameter resolution for the full pseudorapidity
+coverage in the 2017 dataset with the CMS phase-1 pixel detector”, CMS Detector
+Performance Note CMS-DP-2020-049, 2020.
+[21] CMS Collaboration, “Performance of the CMS level-1 trigger in proton-proton collisions
+at √s = 13 TeV”, JINST 15 (2020) P10017, doi:10.1088/1748-0221/15/10/P10017,
+arXiv:2006.10165.
+[22] CMS Collaboration, “The CMS trigger system”, JINST 12 (2017) 01020,
+doi:10.1088/1748-0221/12/01/P01020, arXiv:1609.02366.
+[23] CMS Collaboration, “The CMS experiment at the CERN LHC”, JINST 3 (2008) S08004,
+doi:10.1088/1748-0221/3/08/S08004.
+[24] CMS Collaboration, “Precision luminosity measurement in proton-proton collisions at
+√s = 13 TeV in 2015 and 2016 at CMS”, Eur. Phys. J. C 81 (2021) 800,
+doi:10.1140/epjc/s10052-021-09538-2, arXiv:2104.01927.
+[25] CMS Collaboration, “CMS luminosity measurement using nucleus-nucleus collisions at
+√sNN = 5.02 TeV in 2018”, CMS Physics Analysis Summary CMS-PAS-LUM-18-001, 2022.
+[26] CMS Collaboration, “Charged-particle nuclear modification factors in PbPb and pPb
+collisions at √sNN = 5.02 TeV”, JHEP 04 (2017) 039, doi:10.1007/JHEP04(2017)039,
+arXiv:1611.01664.
+[27] CMS Collaboration, “Description and performance of track and primary-vertex
+reconstruction with the CMS tracker”, JINST 9 (2014) P10009,
+doi:10.1088/1748-0221/9/10/p10009, arXiv:1405.6569.
+
+14
+[28] CMS Collaboration, “Transverse-momentum and pseudorapidity distributions of
+charged hadrons in pp collisions at √s = 0.9 and 2.36 TeV”, JHEP 02 (2010) 041,
+doi:10.1007/JHEP02(2010)041, arXiv:1002.0621.
+[29] CMS Collaboration, “Observation and studies of jet quenching in PbPb collisions at
+nucleon-nucleon √sNN = 2.76 TeV”, Phys. Rev. C 84 (2011) 024906,
+doi:10.1103/PhysRevC.84.024906, arXiv:1102.1957.
+[30] C. Gale, S. Jeon, and B. Schenke, “Hydrodynamic modeling of heavy ion collisions”, Int.
+J. Mod. Phys. A 28 (2013) 1340011, doi:10.1142/S0217751X13400113,
+arXiv:1301.5893.
+[31] GEANT4 Collaboration, “GEANT4—a simulation toolkit”, Nucl. Instrum. Meth. A 506
+(2003) 250, doi:10.1016/S0168-9002(03)01368-8.
+[32] CMS Collaboration, “Strange hadron collectivity in pPb and PbPb collisions”, 2022.
+arXiv:2205.00080.
+[33] CMS Collaboration, “Strange particle production in pp collisions at √s = 0.9 and 7 TeV”,
+JHEP 05 (2011) 064, doi:10.1007/JHEP05(2011)064, arXiv:1102.4282.
+[34] CMS Collaboration, “Long-range two-particle correlations of strange hadrons with
+charged particles in pPb and PbPb collisions at LHC energies”, Phys. Lett. B 742 (2015)
+200, doi:10.1016/j.physletb.2015.01.034, arXiv:1409.3392.
+[35] Particle Data Group Collaboration, “Review of particle physics”, Prog. Theor. Exp. Phys.
+2022 (2022) 083C01, doi:10.1093/ptep/ptac097.
+[36] H. Voss, A. H¨ocker, J. Stelzer, and F. Tegenfeldt, “TMVA, the toolkit for multivariate data
+analysis with ROOT”, in XIth International Workshop on Advanced Computing and Analysis
+Techniques in Physics Research (ACAT), p. 40. 2007. arXiv:physics/0703039.
+doi:10.22323/1.050.0040.
+[37] G. I. Kopylov, “Like particle correlations as a tool to study the multiple production
+mechanism”, Phys. Lett. B 50 (1974) 472, doi:10.1016/0370-2693(74)90263-9.
+[38] A. Kisiel, “Non-identical particle correlation analysis in the presence of non-femtoscopic
+correlations”, Acta Phys. Polon. B 48 (2017) 717, doi:10.5506/APhysPolB.48.717.
+[39] ALICE Collaboration, “pp, p-Λ, and Λ-Λ correlations studied via femtoscopy in pp
+reactions at √s = 7 TeV”, Phys. Rev. C 99 (2019) 024001,
+doi:10.1103/PhysRevC.99.024001, arXiv:1805.12455.
+[40] STAR Collaboration, “Neutral kaon interferometry in Au+Au collisions at
+√sNN = 200 GeV”, Phys. Rev. C 74 (2006) 054902,
+doi:10.1103/PhysRevC.74.054902, arXiv:nucl-ex/0608012.
+[41] A. D. Martin and E. N. Ozmutlu, “Analyses of KK production and scalar mesons”, Nucl.
+Phys. B 158 (1979) 520, doi:10.1016/0550-3213(79)90180-9.
+[42] A. Antonelli, “Radiative φ decays”, eConfC 020620 (2002) THAT06,
+doi:10.48550/ARXIV.HEP-EX/0209069, arXiv:hep-ex/0209069.
+[43] N. N. Achasov and V. V. Gubin, “Analysis of the nature of the φ → γπη and φ → γπ0π0
+decays”, Phys. Rev. D 63 (2001) 094007, doi:10.1103/PhysRevD.63.094007,
+arXiv:hep-ph/0101024.
+
+References
+15
+[44] N. N. Achasov and A. V. Kiselev, “New analysis of the KLOE data on the φ → ηπ0γ
+decay”, Phys. Rev. D 68 (2003) 014006, doi:10.1103/PhysRevD.68.014006,
+arXiv:hep-ph/0212153.
+[45] F. James and M. Roos, “Minuit: A system for function minimization and analysis of the
+parameter errors and correlations”, Comput. Phys. Commun. 10 (1975) 343,
+doi:10.1016/0010-4655(75)90039-9.
+[46] PHENIX Collaboration, “L´evy-stable two-pion Bose–Einstein correlations in
+√sNN = 200 GeV Au+Au collisions”, Phys. Rev. C 97 (2018) 064911,
+doi:10.1103/PhysRevC.97.064911, arXiv:1709.05649.
+[47] ALICE Collaboration, “One-dimensional pion, kaon, and proton femtoscopy in Pb-Pb
+collisions at √sNN = 2.76 TeV”, Phys. Rev. C 92 (2015) 054908,
+doi:10.1103/PhysRevC.92.054908, arXiv:1506.07884.
+[48] STAR Collaboration, “ΛΛ correlation function in Au+Au collisions at √sNN = 200 GeV”,
+Phys. Rev. Lett. 114 (2015) 022301, doi:10.1103/PhysRevLett.114.022301,
+arXiv:1408.4360.
+[49] E. Hiyama et al., “Four-body cluster structure of A = 7–10 double-Λ hypernuclei”, Phys.
+Rev. C 66 (2002) 024007, doi:10.1103/physrevc.66.024007,
+arXiv:nucl-th/0204059.
+[50] I. N. Filikhin and A. Gal, “Faddeev–Yakubovsky calculations for light ΛΛ hypernuclei”,
+Nucl. Phys. A 707 (2002) 491, doi:10.1016/s0375-9474(02)01008-4,
+arXiv:nucl-th/0203036.
+

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+arXiv:2301.11963v1  [hep-th]  27 Jan 2023
+On 10 dimensional Exceptional Drinfel’d Algebras
+Sameer Kumar1, Edvard T. Musaev2
+Moscow Institute of Physics and Technology,
+Institutskii pereulok 9, Dolgoprudny, 141700, Russia
+Abstract
+Based on the Mubarakzyanov’s classification of four-dimensional real Lie Algebras,
+we classify ten-dimensional Exceptional Drinfel’d Algebras (EDA). The classifica-
+tion is restricted to EDAs whose maximal isotropic (geometric) subalgebras cannot
+be represented as a product of a 3D Lie algebra and a 1D abelian factor. We show
+that all obtained EDAs are inequivalent and conclude that there are no Nambu-Lie
+U-dualities between 11D supergravity backgrounds within 10D EDAs.
+[email protected]
+[email protected]
+
+1
+Introduction
+String theory is a background-dependent theory meaning that dynamics of the string is
+defined on a fixed background of space-time fields including the metric, the dilaton, Kalb-
+Ramond 2-form field, and Ramond-Ramond p-form fields. The moduli space of these vacua
+appears to be highly degenerate due to duality symmetries of string theory. Some of them, such
+as (abelian) T-dualities are exact perturbative symmetries of the superstring partition function
+at all orders in α′ and gs [1–3]. This implies that physics of the string does does not change if the
+underlying space-time background is transformed by T-duality. Given a non-abelian algebra of
+isometries of a string background, abelian T-duality transformation rules can be generalized to
+what is called non-abelian T-duality (NATD) [4]. In contrast to the abelian case NATD is not
+an exact quantum symmetry of the conformal theory due to problems with definition of winding
+modes [5]. However, the NATD transformation map can be corrected to be a valid symmetry
+at the leading order in α′ [6,7]. Using the notion of non-commutative currents, the non-abelian
+T-duality transformations can be extended to Poisson-Lie T-dualities that are symmetries of
+string theory in the same sense [8,9]. While abelian T-duality starts from a background with
+certain abelian isometries and preserves them, non-abelian T-duality breaks the non-abelian
+algebra of initial isometries naively preventing from performing the inverse transformation.
+The algebraic structure behind non-abelian T-duality symmetries, that is classical Drinfeld
+algebras, reveals that the initial isometry becomes hidden inside the algebra. More specifically
+classical Drinfeld algebra D is defined in terms of Manin triple (D, g, ˜g), where D is a Lie algebra
+with non-degenerate quadratic form η, and g and ˜g are subalgebras maximally isotropic with
+respect to the form. The algebra g is commonly referred to as the geometric subalgebra, and
+is responsible for the background space, i.e.
+a group manifold or a coset space, while ˜g is
+commonly referred to as the dual algebra and it is responsible for conservation laws of the
+sigma model. To illustrate that, denote fabc and ˜fabc as structure constants of the algebras - g
+and ˜g, respectively. Then the following holds,
+[va, vb] = fab
+cvc,
+dJa = ˜fa
+bcJb ∧ Jc.
+(1.1)
+Here, vectors va define action of G = exp g on itself or on a coset space as δxi = vaiǫa, where
+xi denote coordinates on the group (coset) manifold. Noether currents Ja = Ja idxi satisfy
+the non-commutative conservation law.
+When ˜fabc = 0, the currents are conserved in the
+usual sense. Non-abelian T-duality simply maps g ↔ ˜g, hence vanishing ˜fabc get replaced by
+2
+
+non-vanishing fabc and the conservation law becomes non-commutative. The initial isometry
+becomes hidden in g′ = ˜g and is no longer manifest. In this language the condition for classical
+equations of motion for the string to satisfy is simply the Leibniz identity
+[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]],
+X, Y ∈ D.
+(1.2)
+Here, the brackets are given by the following relations in terms of the generators (Ta, ˜T a) =
+bas D:
+[Ta, Tb] = fab
+cTc,
+[ ˜T a, ˜T a] = fc
+ab ˜T c,
+[ ˜T a, Tb] = ˜fc
+ab ˜T c + fab
+cTc.
+(1.3)
+In terms of structure constants, Leibniz identity is equivalent to Jacobi identities for fabc and
+˜fabc along with the following mixed identity
+˜fl
+jkfmi
+l + ˜fm
+klfli
+j + ˜fi
+jlflm
+k + ˜fm
+jlfil
+k + ˜fi
+klflm
+j = 0.
+(1.4)
+For a review of the algebraic construction behind Poisson-Lie T-dualities see [10], for a review of
+applications of NATD see [11,12], for formulation of Poisson-Lie T-dualities in the supergravity
+language see [13,14], for geometric aspects see [15,16]
+In the most general case when both sets of structure constants are non-zero, one is able
+to define the so-called Poisson-Lie duality transformations. When dim g = d, these are such
+O(d, d) maps CAB that preserve the structure of classical Drinfel’d double:
+TA → CA
+BTB,
+TA = (Ta, ˜T a).
+(1.5)
+There is a distinguished set of such transformations called Poisson-Lie (PL) T-dualities (plu-
+ralities) when the map CAB relates different realization of the same Drinfeld algebra. The
+simplest example is the swapping g ↔ ˜g. For lower dimensional Lie algebras full classification
+of all possible Poisson-Lie T-dualities or likewise of all equivalent Manin triples is available [17].
+This is based on classification of all possible dual algebras ˜g for each g belonging to the Bianchi
+classification of three-dimensional real Lie algebras (for more on classification of Lie Algebras,
+see for example [18]). More generally, one may have maps CAB that relate different Drinfeld
+algebras, for example, Yang-Baxter deformations that draw the interest since they preserve
+integrability of the underlying sigma-model [19].
+When extending abelian T-duality symmetries by S-dualities that are non-perturbative
+3
+
+transformations exchanging gs with g−1
+s , one arrives at U-duality transformations that are
+symmetries of M-theory. Speaking more concretely, U-duality is a symmetry of classical field
+equations of 11D supergravity compactified on a d-torus. These are known as Cremmer-Julia
+symmetries and are given by the exceptional groups Ed(d) [20,21]. In M-theory, whose low-energy
+approximation is given by 11D supergravity, U-duality can be thought of as symmetries of BPS
+states [22] or in terms of a Buscher-like procedure for M2-brane wrapping a 4-torus [23,24]. The
+algebraic structure behind Poisson-Lie T-dualities can be extended to the so-called Exceptional
+Drinfeld Algebras (EDA), that include the usual abelian U-dualities (Cremmer-Julia symme-
+tries) [25–27]. Keeping the more detailed description of EDAs to the next section, we mention
+that these are Leibniz algebras with generators TA on which exceptional group Ed(d) acts in the
+same sense as the orthogonal group O(d, d) acts on generators of the classical Drinfeld double.
+Nambu-Lie U-dualities are then transformations that preserve the structure of the EDA. What
+differs these from the PL T-duality case is that there is no naturally defined analogue of the
+swapping g ↔ ˜g, simply due to the following two facts: i) dimension of the geometric subalgebra
+g of an EDA is never half of dimension of the EDA itself, ii) orthogonal completion of g inside
+the EDA is not an algebra. For this reason, searching for pairs of 11D geometries related by a
+Nambu-Lie U-duality is an extremely complicated task for a general EDA. At the moment few
+examples of such dualities between 11D backgrounds and solutions to Type IIB supergravity
+equations are known [28, 29]. In [30] a general procedure has been suggested similar to the
+natural swapping g ↔ ˜g based on external automorphisms of Ed(d) group. Further it has been
+used to generate few examples of mutually dual backgrounds in [31].
+In this work, we elaborate further on the results of [30,31] that in particular state that there
+are no non-abelian U-dualities in the defined sense between 11D background. The narrative
+we follow is along the same lines as [32] where a full classification of 6D Exceptional Drinfeld
+Doubles based on 3D geometric algebras has been presented. Starting from the classification
+of four-dimensional real Lie algebras [33], we construct all possible EDAs for a representative
+of each class. For each pair of such obtained EDAs we search for an SL(5) transformation
+relating them, that would mean existence of a Nambu-Lie U-duality between backgrounds that
+geometrically realize the corresponding geometric algebras g. Restricting ourselves to only such
+4D real Lie algebras that do not contain a 1d (abelian) factor we find no such transformations.
+The restriction is motivated by the interest only in dualities between 11D background as maps
+from 11D→IIB are known.
+The paper is structured as follows. In the beginning of Section 2 we briefly review the con-
+struction of Exceptional Drinfel’s Algebras. In Section 2.1 we discuss the geometric realization
+of EDAs and Nambu-Lie U-dualities. In Section 2.2, we present classification of 10D EDAs,
+4
+
+given the conditions stated in the preceding section and state the main results of the paper
+2
+Exceptional Drinfel’d Algebras
+Before proceeding with the classification of 10d EDAs, let us briefly review the algebraic
+construction following [25, 26]. We will be focusing on the 10d case where generators of the
+exceptional Drinfeld algebra ED4 are collected into the 10-dimensional representation of the
+SL(5) group basED4 = {TAB}, where A, B = 1, . . . , 5. Multiplication table is then given by
+TAB ◦ TCD = i
+2FAB,CD
+GHTGH.
+(2.1)
+The structures constants FAB,CDGH are defined by the following relations
+FAB,CD
+GH = 4FAB,[C
+[GδH]
+D]
+(2.2)
+FAB,C
+D = 1
+2ǫABCGHZGHD + 1
+2δD
+[ASB]C + 1
+3δD
+[AτB]C + 1
+6δD
+C τAB,
+(2.3)
+where τ is antisymmetric and S is a symmetric tensor, while Z[ABC] = 0. For the algebra to be an
+EDA, components of the constants ZABC, SAB and τAB under decomposition SL(5) ←֓ GL(4)
+must be defined as
+Zabc = 1
+6ǫabcdfde
+e + 1
+4ǫabeffef
+c,
+S5a = fab
+b − 3Za,
+τ5a = 9
+2Za − 1
+2fab
+b
+Z5[a,b] = 1
+6
+˜fc
+abc,
+Sab = 1
+3
+˜f(a
+cdeǫb)cde,
+τab = −1
+6
+˜f[a
+cdeǫb]cde
+Zab,5 = −Z5a,b + Z5b,a.
+(2.4)
+The constants FAB,CD have the same structure as the embedding tensor of [34], and in this
+language the above construction implies that only the geometric flux (anholonomy coefficients)
+and Q-flux are turned on. The former is given by the structure constants fabc of the geometric
+subalgebra g and the latter is given by ˜fabcd. The algebra is Leibniz with the fundamental
+identity given by the quadratic relations analogous to those of 7d maximal gauged SUGRA [34]:
+2F G
+AB[CF I
+GD],H − F I
+ABGF G
+CDH + F G
+ABHF I
+CDG = 0.
+(2.5)
+5
+
+In terms of structure constants fabc and dual constants fabcd, the conditions become
+6ff[a
+[c ˜fb]
+de]f + fab
+f ˜ff
+cde − 1
+3
+˜f[a
+cdefb]f
+f = 0
+˜fc
+abcfbd
+d = 0,
+fde
+a ˜f bde
+c
+− 1
+3
+˜fc
+abdfde
+e = 0
+˜fc
+abg ˜fg
+def − 3 ˜fc
+g[de ˜fg
+f]ab = 0.
+(2.6)
+The last of the above equations is also referred to as the dual Jacobi condition, just as the
+dual conditions in the Manin triples. It describes the internal (isolated) relations between the
+structure constants of the dual algebra.
+As in the case of Classical Drinfeld Algebra, in general, there might exist multiple equivalent
+choices of the geometric subalgebra g inside an EDA. Proper generalization of the isometry
+condition to the case of exceptional structures has been given in [25,26] and can be written as
+follows
+ǫABCDETAB ⊗ TCD
+����
+g⊗g
+= 0.
+(2.7)
+In other words, for a given EDA, its geometric subalgebra g is spanned by such a subset of the
+whole set of generators {TAB} that satisfy the above condition. For Classical Drinfel’d Double
+the condition is ηABTA ⊗ TB = 0, implying that one may, for example, take bas g = {Ta},
+or bas g = { ˜T a}. For EDAs, one choice is self-evident - bas g = {T5a}, while presenting an
+alternative choice is usually a hard task. This implies that there is no natural generalization
+of the Non-Abelian T-duality transformation swapping g ↔ ˜g in the case of EDAs, although
+certain progress in defining an analogue of these swappings has been done in [30,31].
+2.1
+Geometric realization and dualities
+The algebraic structure of EDAs stands behind Nambu-Lie U-dualities of supergravity so-
+lutions. These can map solutions to 11D supergravity equations into each other or into Type
+IIB supergravity equations. Such duality transformations map the group manifolds correspond-
+ing to different choices of the geometric subalgebra g into each other. For more detailed and
+concrete algorithm of constructing mutually dual backgrounds see [31]. Below, we will briefly
+recall the overall construction and highlight relations to Exceptional Field Theory (ExFT) that
+provide convenient variables for writing such duality maps [35, 36]. These are Ed(d)-covariant
+field theories defined in 11-dimensional space-time with an explicit split - 11 = D + d. The
+D-dimensional space-time is usually referred to as the external, the d-dimensional space is
+6
+
+usually referred to as internal, although no compactification is assumed. In the d = 4 case
+relevant to the present discussion, field content of the theory includes the external metric gµν,
+ten vector fields AµMN, five 2-form fields BµνM, and 14 scalar fields parametrized by a coset
+element MMN ∈ SL(5)/SO(5). The indices µ = 0, . . . , 6 parameterize directions of the external
+space-time whereas the indices M, N = 1, . . . , 5 belong to the 5 of SL(5). For more details of
+the construction see [37]. Here we are interested in the special case where all fields transform-
+ing in irreps of SL(5) can be decomposed in terms of matrices EABMN (generalized vielbeins)
+geometrically realizing an EDA. In compact notation one writes
+[EAB, ECD] = FAB,CD
+EFEEF,
+(2.8)
+where the constants FAB,CDEF are precisely the structure constants of the EDA and the brackets
+denote the so-called generalized Lie derivative of ExFT.
+Generalized vielbeins are parametrized by fields of 11D supergravity in the 11 = 7 + 4 split
+transforming as scalars under 7-dimensional diffeomorphisms. Introducing a unity matrix MAB
+compose
+MMN,KL = 2EMN
+ABEKL
+CDMACMBD = MMKMNL − MMLMNK.
+(2.9)
+The symmetric matrix
+mMN = e− φ
+2
+�
+|g|− 1
+2gij
+Vi
+Vj
+|g|
+1
+2(1 + V 2)
+�
+(2.10)
+is then defined in terms of the 4d metric gmn on the group manifold, the vector V m = 1
+3!ǫmnklCnkl
+and a scalar field eφ = |g7|1/7 which is the determinant |g7| of external 7 dimensional space.
+The metric gmn on the group manifold is defined as usual in terms of Maurer-Cartan forms.
+Let g ∈ G = exp g be an element of the group G whose Lie algebra is g, then 1-forms on the
+group manifold g−1dg ∈ g. In components we have
+g−1dg = rm
+aTadxm,
+(2.11)
+where xm are some coordinates on the group manifold.
+Given an EDA and a choice of the isotropic subalgebra g one can explicitly construct
+the corresponding generalized vielbein. A step-by-step algorithm of this procedure based on
+constructing adjoint action of eh ∈ G for some h ∈ g on an element of EDA can be found
+in [38]. An alternative choice of the isotropic subalgebra, if exists, is related to the given one
+by an SL(5) transformation
+T ′
+AB = CA
+CCB
+DTCD.
+(2.12)
+7
+
+If this transformation respects the structure of EDA, then the alternative isotropic subalgebra
+is spanned by T ′
+5a. Structure constants of the EDA then transform as
+F ′
+A′B′,C′D′ = CA′ACB′BCC′CCD
+D′FAB,C
+D.
+(2.13)
+Note that not any such matrix corresponds to a Nambu-Lie U-duality transformation. Indeed,
+one can always perform a GL(4) transformation on generators of a given algebra g thus changing
+explicit realization of the corresponding EDA. Two EDA’s related by such transformation then
+correspond to 11D backgrounds related by a coordinate transformation. Another trivial choice
+is
+CA
+B =
+�
+14×4
+λm
+0
+1
+�
+,
+(2.14)
+that corresponds to simply a gauge transformation of the 3-form Cmnk. To avoid counting
+of EDA’s related by a rotation of the basis of their isotropic subalgebras we first classify
+Exceptional Drinfeld Algebras using classification of 4D real Lie Algebras.
+2.2
+Classification of 10 dimensional EDA’s
+The main goal of this work is to investigate relations between 10d EDAs that correspond
+to Nambu-Lie U-duality transformations of 11-dimensional supergravity backgrounds. For this
+purpose, we start with a classification of 10D EDAs of certain class based on the classification of
+4-dimensional real Lie Algebras by Mubarakzyanov [33] (for a review in English see [18]). Since
+explicit examples of Nambu-Lie U-dualities between 11D and Type IIB backgrounds are known
+in the literature, we are interested here only in EDAs constructed on 4d real Lie Algebras g4
+that cannot be decomposed into a sum g4 = g4 ⊕ g1, where g3 is a 3d Lie algebra and g1 is
+1-dimensional Abelan factor. We list all relevant 4d real Lie Algebras in Table 1.
+g4,1
+[T2, T4] = T1
+[T3, T4] = T2
+g4,5
+[T1, T4] = AT1
+[T2, T4] = BT2
+[T3, T4] = CT3
+ABC̸= 0
+g4,9
+[T2, T3] = T1
+[T1, T4] = 2AT1
+[T2, T4] = AT2 − T3
+[T3, T4] = T2 + AT3
+A > 0
+g4,2
+[T2, T4] = βT1
+[T2, T4] = T2
+[T3, T4] = T2 + T3
+g4,6
+[T1, T4] = AT1
+[T2, T4] = BT2 − T3
+[T3, T4] = T2 + BT3
+A > 0
+g4,10
+[T1, T3] = T1
+[T2, T3] = T2
+[T1, T4] = −T2
+[T2, T4] = T1
+8
+
+g4,3
+[T1, T4] = T1
+[T3, T4] = T2
+g4,7
+[T2, T3] = T1
+[T1, T4] = 2T1
+[T2, T4] = T2
+[T3, T4] = T2 + T3
+2g2,1
+[T1, T2] = T1
+[T3, T4] = T3
+g4,4
+[T1, T4] = T1
+[T2, T4] = T1 + T2
+[T3, T4] = T2 + T3
+g4,8
+[T2, T3] = T1
+[T1, T4] = (1 + β)T1
+[T2, T4] = T2
+[T3, T4] = βT3
+β ∈ [−1, 1]
+Table 1: Classification of 4-dimensional indecomposable
+real Lie algebras g4,n with n = 1, . . . , 10. The algebra
+2g2,1 is decomposable, however does not have a u(1) fac-
+tor.
+To arrive at the corresponding classification of 10d EDAs, we solve quadratic constraints
+for each class in the table above to find all possible sets of the dual structure coefficients ˜fdabc.
+To solve the equations we use mathematical software Mathematica , that gives us all the 4
+dimensional EDAs in the chosen class.
+The result is listed in Table 2, where only unique
+combinations of indices are explicitly given in the coefficients of the underlying algebra. The
+rest of the indices are obtained by the antisymmetric property of the structure coefficients.
+EDA
+Structure Constants ˜f abcd
+g4,1
+1.
+˜f 1232 = ˜f 1344, ˜f 1242 = ˜f 1343
+˜f 1234 =
+˜f 1233 ˜f 1344 − ˜f 1244 ˜f 1344
+2 ˜f 1343
+˜f 1243 = ( ˜f 1244 − ˜f 1233) ˜f 1343
+2 ˜f 1344
+2.
+˜f 1232 = − ˜f 1344, ˜f 1244 = ˜f 1233, ˜f 2344 = ˜f 1231
+3.
+˜f 1242 = ˜f 1343 ˜f 1244 = ˜f 1233
+4.
+˜f 1244 = ˜f 1233, ˜f 2344 = ˜f 1231
+g4,2
+5.
+˜f 1244 = −1
+3 (1 + 2β) ˜f 1233
+6
+˜f 2344 =
+1
+3β(β − 4) ˜f 1231
+g4,3
+7.
+˜f 2344 = 1
+3 ˜f 1231
+g4,4
+8.
+˜f 1244 = − ˜f 123
+3
+g4,5
+9.
+˜f 1244 = −2A−2B+C
+3C
+˜f 1233
+9
+
+10.
+˜f 1344 =
+1
+3B(2A − B + 2C) ˜f 1232
+11.
+˜f 2344 =
+1
+3A(A − 2B − 2C) ˜f 1231
+g4,6
+12.
+˜f 2344 =
+1
+3A(A − 2B − 2C) ˜f 1231
+g4,7
+13.
+˜f 1244 = −5
+3 ˜f 1233
+g4,8
+14.
+˜f 1244 = − 1
+3β(4 − β) ˜f 1233
+15.
+˜f 1344 = 1
+3(1 + 4B) ˜f 1232
+g4,9
+˜f abcd = 0 or imaginary
+g4,10
+˜f abcd = 0
+2g2,1
+16.
+˜f 1234 = ˜f 1232, ˜f 1342 = ˜f 1344
+Table 2: All possible structure constant of 10d EDAs for
+each g4,n with n = 1, . . . , 10 and 2g2,1. The constants
+A, B, C, β are the same as in the previous table.
+Hence, given we are interested only in real non-trivial EDAs, we end up with 16 examples.
+A natural question would be: whether there exists a pair of EDAs in this set that are equivalent
+up to an SL(5) transformation. This would mean that the same EDA can be generated by two
+4d Lie Algebras that belong to different classes. In the supergravity language this would mean
+existence of a Nambu-Lie U-duality between 11D backgrounds geometrically realizing this pair
+of 4d Lie algebras. Result of our calculations is that there are no such pairs. To arrive at this
+statement we used Mathematica software and explicitly solve equations on components of the
+matrix CAB for each pair of 16 algebras with no further restrictions on the coefficients. This
+means, that although in Table 2 we list algebras as though all explicitly written dual structure
+constants are non-vanishing, our code does not assume that [39].
+3
+Discussion
+In this work we obtain a classification of 10-dimensional EDA based on the classification of
+4-dimensional real Lie Algebras by Mubarakzyanov [33]. We intentionally restrict only to such
+4d algebras that cannot be decomposed into a 3d algebra and a 1d abelian factor, i.e, we are
+interested in Nambu-Lie U-dualities between 11d backgrounds, rather than dualities between
+11D and Type IIB solutions. More specifically, we look only at EDAs whose isotropic (geomet-
+ric) subalgebra is given by g4,n with n = 1, ..., 10 and 2g2,1 in terms of the Mubarakzyanov’s
+classification. Given these restrictions the classification of EDAs is summarized in Table 2,
+10
+
+where 16 non-trivial EDAs are listed in terms of dual structure constants ˜fabcd.
+The important question we were interested in is whether there exists a Nambu-Lie U-duality
+between 11D solutions and supergravity equations. Equivalently, in the algebraic language:
+whether any of the sixteen exceptional Drinfeld algebras are equivalent up to an SL(5) trans-
+formation? For that we computed the explicit form of all possible transformations between all
+possible pairs of EDA listed in Table 2 of the form (2.13). In our findings, we discovered that
+none of the EDA pairs except the (ED2, ED4) possess transformation matrices, taking the basis
+of one EDA to another, with a non-zero determinant. Moreover, the transformation relating the
+aforesaid algebras - ED2 and ED4 is simply a GL(4) transformation rotating the basis. Hence,
+these two solutions are equivalent and their geometric realizations can be mapped into each
+other by a 4D coordinate transformation. Hence, there are no Nambu-Lie U-dualities inside
+SL(5) exceptional Drinfeld algebras relating 11D backgrounds. Note however, this does not
+rule out transformations between 11D and Type IIB backgrounds, explicit examples of which
+are known [28, 29]. Previously in [30] the same has been shown for transformations involving
+external automorphisms of the algebra sl(5), suggested as the natural analogue of Non-Abelian
+T-duality transformations. Here we complete the statement.
+There are further directions to extend this work. The most obvious task is to complete
+the classification including all 4D real Lie algebras and list sets of EDA’s mutually Nambu-
+Lie U-dual. Less straightforward is to increase the dimension of the geometric subalgebra g
+by one and consider 16D Exceptional Drinfeld Algebras. Unfortunately, there is no ready to
+use classification of 5D real Lie algebras, but certain restricted classifications are present in
+the literature. Some useful examples can be found in [40–43], for a review see [44]. Another
+interesting direction of further research is to list those EDAs from our classification that can be
+obtained as generalized Yang-Baxter deformations of the trivial EDA when all dual structure
+constants are zero. In other words, to answer the question: for which algebras in Table 2 dual
+structre constants can be represented in the form
+˜fa
+bcd = re[bcfae
+d],
+(3.1)
+where rabc is completely antisymmetric. In the case of classical Drinfeld algebras such trans-
+formations are known to preserve integrability of the 2d sigma-model on the corresponding
+background.
+There is no analogous statement for 3d sigma-models describing membranes
+propagating on 11d supergravity backgrounds. However, such defined generalized Yang-Baxter
+deformations are of certain interest (see [45] for a review).
+11
+
+Acknowledgments
+This work has been supported by the Foundation for the Advancement of Theoretical
+Physics and Mathematics “BASIS”, grant No 21-1-2-3-1 and by Russian Ministry of Education
+and Science.
+References
+[1] T. H. Buscher, “A Symmetry of the String Background Field Equations,” Phys. Lett.
+B194 (1987) 59–62.
+[2] T. H. Buscher, “Path Integral Derivation of Quantum Duality in Nonlinear Sigma
+Models,” Phys. Lett. B201 (1988) 466–472.
+[3] A. Giveon, M. Porrati, and E. Rabinovici, “Target space duality in string theory,”
+Phys.Rept. 244 (1994) 77–202, arXiv:hep-th/9401139 [hep-th].
+[4] X. C. de la Ossa and F. Quevedo, “Duality symmetries from nonAbelian isometries in
+string theory,” Nucl. Phys. B 403 (1993) 377–394, arXiv:hep-th/9210021.
+[5] A. Giveon and M. Rocek, “On Nonabelian duality,” Nucl.Phys. B421 (1994) 173–190,
+arXiv:hep-th/9308154 [hep-th].
+[6] F. Hassler and T. Rochais, “α′-Corrected Poisson-Lie T-Duality,” Fortsch. Phys. 68
+no. 9, (2020) 2000063, arXiv:2007.07897 [hep-th].
+[7] R. Borsato and L. Wulff, “Quantum Correction to Generalized T Dualities,” Phys. Rev.
+Lett. 125 no. 20, (2020) 201603, arXiv:2007.07902 [hep-th].
+[8] C. Klimcik and P. Severa, “Dual nonAbelian duality and the Drinfeld double,” Phys.
+Lett. B 351 (1995) 455–462, arXiv:hep-th/9502122.
+[9] C. Klimˇc´ık and P. Severa, “Poisson-Lie T duality and loop groups of Drinfeld doubles,”
+Phys. Lett. B372 (1996) 65–71, arXiv:hep-th/9512040 [hep-th].
+[10] C. Klimcik, “Poisson-Lie T duality,” Nucl. Phys. B Proc. Suppl. 46 (1996) 116–121,
+arXiv:hep-th/9509095.
+12
+
+[11] K. Sfetsos, “Recent developments in non-Abelian T-duality in string theory,” Fortsch.
+Phys. 59 (2011) 1149–1153, arXiv:1105.0537 [hep-th].
+[12] D. C. Thompson, “An Introduction to Generalised Dualities and their Applications to
+Holography and Integrability,” PoS CORFU2018 (2019) 099, arXiv:1904.11561
+[hep-th].
+[13] F. Hassler, “Poisson-Lie T-Duality in Double Field Theory,” Phys. Lett. B 807 (2020)
+135455, arXiv:1707.08624 [hep-th].
+[14] S. Demulder, F. Hassler, and D. C. Thompson, “An invitation to Poisson-Lie T-duality
+in Double Field Theory and its applications,” PoS CORFU2018 (2019) 113,
+arXiv:1904.09992 [hep-th].
+[15] M. Bugden, “Non-abelian T-folds,” JHEP 03 (2019) 189, arXiv:1901.03782 [hep-th].
+[16] L. Hlavat´y and I. Petr, “T-folds as Poisson-Lie plurals,” arXiv:2004.08387 [hep-th].
+[17] L. Hlavaty and L. Snobl, “Classification of 6-dimensional manin triples,”
+arXiv:math/0202209.
+[18] R. O. Popovych, V. M. Boyko, M. O. Nesterenko, and M. W. Lutfullin, “Realizations of
+real low-dimensional Lie algebras,” J. Phys. A 36 (2003) 7337–7360,
+arXiv:math-ph/0301029.
+[19] C. Klimˇc´ık, “On integrability of the Yang-Baxter sigma-model,” J. Math. Phys. 50
+(2009) 043508, arXiv:0802.3518 [hep-th].
+[20] E. Cremmer and B. Julia, “The SO(8) supergravity,” Nucl.Phys. B159 (1979) 141.
+[21] E. Cremmer, B. Julia, H. Lu, and C. Pope, “Dualization of dualities. 1.,” Nucl.Phys.
+B523 (1998) 73–144, arXiv:hep-th/9710119 [hep-th].
+[22] N. Obers and B. Pioline, “U duality and M theory,” Phys.Rept. 318 (1999) 113–225,
+arXiv:hep-th/9809039 [hep-th].
+[23] M. Duff, “Duality rotations in string theory,” Nucl.Phys. B335 (1990) 610.
+[24] M. Duff and J. Lu, “Duality rotations in membrane theory,” Nucl.Phys. B347 (1990)
+394–419.
+13
+
+[25] Y. Sakatani, “U-duality extension of Drinfel’d double,” PTEP 2020 no. 2, (2020)
+023B08, arXiv:1911.06320 [hep-th].
+[26] E. Malek and D. C. Thompson, “Poisson-Lie U-duality in Exceptional Field Theory,”
+JHEP 04 (2020) 058, arXiv:1911.07833 [hep-th].
+[27] E. Malek, Y. Sakatani, and D. C. Thompson, “E6(6) exceptional Drinfel’d algebras,”
+JHEP 01 (2021) 020, arXiv:2007.08510 [hep-th].
+[28] C. D. A. Blair, D. C. Thompson, and S. Zhidkova, “Exploring Exceptional Drinfeld
+Geometries,” JHEP 09 (2020) 151, arXiv:2006.12452 [hep-th].
+[29] C. D. A. Blair and S. Zhidkova, “Generalised U-dual solutions in supergravity,” JHEP
+05 (2022) 081, arXiv:2203.01838 [hep-th].
+[30] E. T. Musaev, “On non-abelian U-duality of 11D backgrounds,” arXiv:2007.01213
+[hep-th].
+[31] E. T. Musaev and Y. Sakatani, “Non-Abelian U duality at work,” Phys. Rev. D 104
+no. 4, (2021) 046015, arXiv:2012.13263 [hep-th].
+[32] L. Hlavaty, “Classification of 6D Leibniz algebras,” PTEP 2020 no. 7, (2020) 071B01,
+arXiv:2003.06164 [hep-th].
+[33] G. Mubarakzyanov, “On solvable Lie algebras,” Izv. Vys. Ucheb. Zaved. Matematika 1
+(32) (1963) 114–123.
+[34] H. Samtleben and M. Weidner, “The Maximal D=7 supergravities,” Nucl.Phys. B725
+(2005) 383–419, arXiv:hep-th/0506237 [hep-th].
+[35] O. Hohm and H. Samtleben, “U-duality covariant gravity,” JHEP 1309 (2013) 080,
+arXiv:1307.0509 [hep-th].
+[36] O. Hohm and H. Samtleben, “Exceptional Form of D=11 Supergravity,” Phys.Rev.Lett.
+111 (2013) 231601, arXiv:1308.1673 [hep-th].
+[37] E. T. Musaev, “Exceptional field theory: SL(5),” JHEP 02 (2016) 012,
+arXiv:1512.02163 [hep-th].
+[38] Y. Sakatani, “Type II DFT solutions from Poisson-Lie T-duality/plurality,”
+arXiv:1903.12175 [hep-th]. [PTEP,073B04(2019)].
+14
+
+[39] S. Kumar and E. T. Musaev, “eda10,” 1, 2023. https://github.com/emusaev/eda10.
+[40] I. Darijani and H. Usefi, “Classification of 5-dimensional restricted Lie algebras over
+perfect fields, I,” Journal of Algebra 464 (2016) 97–140, arXiv:1412.8377.
+[41] A. Andrada and A. Fino, “A class of Sasakian 5-manifolds,” Transformation Groups 14
+(2009) 493–512, arXiv:0807.1800 [math.dg].
+[42] L. Bieszk, “Classification of Five-Dimensional Lie Algebras of Class T2,” Demonstratio
+Mathematica 30 no. 2, (1997) 403–424. https://doi.org/10.1515/dema-1997-0219.
+[43] G. Calvaruso and A. Perrone, “Five-dimensional paracontact Lie algebras,” Differential
+Geometry and its Applications 45 (2016) 115–129.
+https://www.sciencedirect.com/science/article/pii/S0926224516000024.
+[44] L. B. Prieto, E. M. F. Martel, J. N. Vald´es, and ´A. F. Tenorio, “A historical review of
+the classifications of Lie algebras,” Revista De La Union Matematica Argentina 54
+(2013) 75–99.
+[45] K. Gubarev and E. Musaev, “Integrability structures in string theory,”
+arXiv:2301.06486 [hep-th].
+15
+

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@@ -0,0 +1,362 @@
+Decomposition of the static potential in SU(3) gluodynamics
+V. G. Bornyakov
+NRC “Kurchatov Institute” - IHEP, Protvino, 142281 Russia
+I. Kudrov
+NRC “Kurchatov Institute” - IHEP, Protvino, 142281 Russia,
+Moscow Institute of Physics and Technology, Institutskii per. 9, 141700, Dolgoprudny, Russia
+After fixing the Maximal Abelian gauge in SU(3) lattice gluodynamics we decompose the non-
+abelian gauge field into the Abelian field created by Abelian monopoles and the modified nonabelian
+field with monopoles removed. We then calculate respective static potentials in the fundamental
+representation and show that the sum of these potentials approximates the nonabelian static poten-
+tial with good precision at all distances considered. Comparison with other ways of decomposition
+is made.
+PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.Aw
+Keywords: gauge field theory, confinement, monopoles, maximal Abelian gauge
+I.
+INTRODUCTION
+We study numerically the lattice SU(3) gluodynamics in the Maximal Abelian gauge (MAG) and consider decom-
+position of the lattice gauge field Uµ(x) ∈ SU(3)
+Uµ(x) = U mod
+µ
+(x)U mon
+µ
+(x)
+(1)
+where U mon
+µ
+(x) is the component of the gauge field due to Abelian monopoles (to be defined later) and U mod
+µ
+(x)
+is respectively the monopoleless component which we call a modified gauge field. By modification we understand
+removal of the Abelian monopoles.
+This kind of decomposition was studied before in SU(2) gluodynamics in [1]. It was shown that while the monopole
+component U mon
+µ
+(x) is reproducing the linear part of the static potential, the monopoleless component U mod
+µ
+(x)
+produces purely Coulomb potential and their sum provides a good approximation of the original unaltered static
+potential at all distances:
+V (R) ≈ Vmod(R) + Vmon(R)
+(2)
+Recently, in [2] it was shown that this approximation becomes better when the lattice spacing is decreased leaving a
+possibility that relation (2) becomes exact in the continuum limit. It was also shown that (2) is satisfied in SU(2)
+QCD as well. In the present work we extend the study of the decomposition (1) to the more realistic case - SU(3)
+gluodynamics.
+It is well known [3–7] that after performing the Abelian projection in the MAG [8, 9], the Abelian string tension
+calculated from the Abelian static potential is very close to the nonabelian string tension. This observation, confirmed
+in gluodynamics and in QCD, supports the concept of the Abelian dominance (for a review see e.g. [10]). It was further
+discovered [5, 11, 12] that the so called monopole static potential also has string tension close to the nonabelian one.
+These observations are in agreement with conjecture that monopole degrees of freedom are responsible for confinement
+[13].
+The interesting question is what is the role of the other, i.e. monopoleless degrees of freedom. The results obtained
+in SU(2) gluodynamics suggest that they are responsible for the Coulomb part of the static potential both at small
+and large distances. This suggests that while at small distances U mod
+µ
+(x) gives perturbative contribution into the
+static potential, it provides nonperturbative contribution at large distances.
+It is worth to note that the gauge covariant decomposition was introduced in Refs. [14] and [15] and developed
+further in Refs. [16–18], see for review [19]. The numerical results demonstrating analogues of the Abelian dominance
+and the monopole dominance within this approach were obtained in [20]. It would be interesting to check if the
+decomposition into the monopole and monopoleless components works in this approach.
+The decomposition different from eq. (1) was considered in SU(3) gluodynamics after fixing MAG [7]. The usual
+coset decomposition of the gauge field into the Abelian and the off-diagonal components was used:
+Uµ(x) = U offd
+µ
+(x)U Abel
+µ
+(x)
+(3)
+arXiv:2301.03076v1  [hep-lat]  8 Jan 2023
+
+2
+and respective decomposition for the static potential was verified:
+V (R) ≈ Voffd(R) + VAbel(R).
+(4)
+We will compare our results for decomposition (4) with results of Ref. [7] in section III.
+The decomposition similar to (2) for the static potential in the maximal center gauge
+V (R) ≈ Vcent(R) + Vmod,cent(R)
+(5)
+corresponding to the decomposition of the gauge field into the center and modified (vortex free) components:
+Uµ(x) = U mod,cent
+µ
+(x)U cent
+µ
+(x) .
+(6)
+was first checked long ago in SU(2) gluodynamics [1] and was studied recently in QCD [21]. We will comment on
+these numerical results later in section III.
+II.
+DECOMPOSITION OF THE GAUGE FIELD
+We consider the SU(3) lattice gluodynamics after fixing MAG. We use the definition of MAG introduced for lattice
+SU(N) theory in [22] and later specified for the SU(3) group in [23]. The MAG is fixed by maximizing the functional
+F =
+1
+8 V
+�
+x,µ
+�
+|U (11)
+µ
+(x)|2 + |U (22)
+µ
+(x)|2 + |U (33)
+µ
+(x)|2 − 1
+�
+(7)
+with respect to local gauge transformations g of the lattice gauge field,
+Uµ(x) → U g
+µ(x) = g(x)†Uµ(x)g(x + ˆµ) .
+(8)
+To fix MAG, the simulated annealing algorithm with three random gauge copies was used. This algorithm was first
+used to fix MAG in the SU(2) case [5] and then extended to the SU(3) group in [24]. The details of implementation
+of the simulated annealing algorithm in the case of SU(3) gauge group can be found in [25]. For the gauge fixing
+functional F we obtained the average value < F >= 0.73388(1) to be compared with < F > 0.7322(2) quoted in [26].
+The larger is the value of the maximized functional the better is the gauge fixing. The difference in < F > is due
+to the Gribov copies effects and implies that there might be substantial difference between our results and results of
+Ref. [26] for gauge dependent quantities like Abelian or monopole string tension as discussed in details in [25].
+The Abelian projection means coset decomposition (3) of the nonabelian lattice gauge field Uµ(x) ∈ SU(3) into the
+Abelian field U Abel
+µ
+(x) ∈ U(1) × U(1) and the coset field U offd
+µ
+(x) ∈ SU(3)/U(1) × U(1). The Abelian field U Abel
+µ
+(x)
+is determined as
+U Abel
+µ
+(x) = diag
+�
+u(1)
+µ (x), u(2)
+µ (x), u(3)
+µ (x)
+�
+,
+(9)
+where
+u(a)
+µ (x) = eiθ(a)
+µ
+(x)
+(10)
+with
+θ(a)
+µ (x) = arg (Uµ(x))a − 1
+3
+3
+�
+b=1
+arg(Uµ(x))b
+��
+mod 2π
+(11)
+such that
+θ(a)
+µ (x) ∈ [−4
+3π, 4
+3π] .
+(12)
+This definition of Abelian projection uµ(x) maximizes the expression |Tr
+�
+U †
+µ(x)uµ(x)
+�
+|2 [27]. The Abelian gauge
+fields can in turn be decomposed into monopole (singular) and photon (regular) parts:
+θ(a)
+µ (x) = θ(a) mon
+µ
+(x) + θ(a) ph
+µ
+(x) ,
+(13)
+
+3
+The monopole part is defined by [28]:
+θ(a) mon
+µ
+(x) = 2π
+�
+y
+D(x − y)∂−
+α m(a)
+αµ(y) ,
+(14)
+where integers m(a)
+µν (x) denote the singular part of the Abelian plaquettes (Dirac plaquettes), ∂−
+α is the backward
+lattice derivative, and D(x) denotes the lattice Coulomb propagator. Then U mon
+µ
+(x) introduced in (1) is defined as
+U mon
+µ
+(x) = diag
+�
+eiθ(1) mon
+µ
+(x), eiθ(2) mon
+µ
+(x), eiθ(3) mon
+µ
+(x)�
+.
+(15)
+III.
+THE STATIC POTENTIAL DECOMPOSITION
+We calculated R × T rectangular Wilson loops W(R, T), Wmon(R, T) and Wmod(R, T) using lattice gauge fields
+Uµ(x), U mon
+µ
+(x), U mod
+µ
+(x) introduced above. To extract respective static potentials V (R), Vmon(R) and Vmod(R) the
+APE smearing [29] has been employed. Computations were done with the Wilson lattice action at β = 6.0 on 244
+lattices using 5000) statistically independent configurations. The lattice spacing at this value of the bare coupling
+constant is defined by a/r0 = 0.186(4) [30], where r0 = 0.5fm is called Sommer parameter. In Fig. 1 our results
+are presented. One can see that similar to the SU(2) gluodynamics [2] the monopole potential Vmon(R) is almost
+perfectly linear with small curvature at small distances while the modified field potential Vmod(R) is well described
+by the Coulomb potential. It is also seen that the relation (2) is valid at all distances with most essential discrepancy
+of about 10% at large distances. It is clear that this discrepancy is mostly due to rather low slope of Vmon(R). As we
+have already mentioned it was found in SU(2) gluodynamics that with decreasing of the lattice spacing agreement in
+relation (2) improves substantially. This should be checked in SU(3) gluodynamics in future.
+FIG. 1: Comparison of the nonabelian potential V (R) (filled circles) with the sum Vmod(R)+Vmon(R) (filled triangles). Vmod(r)
+(empty circles) and Vmon(r) (filled inverted triangles) are also depicted. The solid curves show the fits to the Cornell potential.
+Next we come to our results for decomposition (4) considered in Ref. [7]. These results are presented in Fig. 2.
+In this figure we compare the original potential V (R) with the decompositions (2) and (4). One can see that the
+decomposition (2) clearly works better. Both decompositions should be checked in the continuum limit. In any case,
+our results clearly contradict to the conclusion made in Ref. [7] that the relation (4) is satisfied very nicely already at
+β = 6.0. Comparing our results presented here with results we obtained on 164 lattices we found that finite volume
+effects are small. We should note that in [7] slightly different procedure of the Abelian projection was used but as
+it was claimed in Ref. [31] the differences between these two procedures of Abelian projection are negligible. Thus
+our understanding is that the reason for such discrepancy with results of Ref. [7] is the difference in the gauge fixing
+quality. It was shown in the past that the Gribov copy effects for gauge non-invariant quantities might be quite
+substantial [25, 31].
+
+SU3
+mon
+mod
+mon+mod
+5
+4 
+V(R)
+3
+0
+0.5
+1.0
+1.5
+2.0
+R/ro4
+potential
+σr2
+0
+α
+r0V0
+V (R)
+1.34(2) -0.34(1) 3.61(2)
+Vmon(R)
+0.99(1) 0.09(1) -0.36(2)
+Vmod(R)
+0
+-0.42(1) 4.22(1)
+Vmon(R) + Vmod(R) 0.94(1) -0.39(1) 3.98(2)
+TABLE I: Parameters of the potentials obtained by fits to function V0 − α/R + σR.
+FIG. 2: Comparison of the nonabelian potential V (R) (filled circles) with the sum Vmod(R)+Vmon(R) (filled inverted triangles)
+and the sum VAbel(R) + Voffd(R) (filled squares). Voffd(r) (filled triangles) and VAbel(r) (empty circles) are also depicted. The
+solid curves show the fits to the Cornell potential.
+Next we wish to make a remark about the decomposition (5). This decomposition was first studied in [1] using
+the Direct Central gauge in SU(2) gluodynamics. It was concluded that this decomposition holds with substantially
+less precision than the decomposition (2), first of all, due to the low string tension provided by the center vortex
+component. To queue the problem of low string tension obtained after the center projection the new approach to
+the definition of the center gauge was formulated and successfully applied to SU(2) gluodynamics in [32]. Recently
+this decomposition was studied in the lattice QCD with light quarks [21]. It was found that in this theory the center
+projected string tension is in a very good agreement with the physical string tension. On the other hand, the modified
+component U mod,cent
+µ
+(x) produces the static potential which is not compatible with the Coulomb potential. Thus, the
+decomposition fails to work at small distances.
+IV.
+CONCLUSIONS
+There are a few suggestions for the decomposition of the gauge field into components describing (mostly) either
+infrared or ultra-violate physics. These include the gauge covariant Cho decomposition [14–18], two decompositions
+in the MAG, eqs. (1) and (3) and one decomposition in the maximal center gauge (6). In this work we extended
+our study of the decomposition in MAG into the monopole and the modified (monopoleless) components in SU(2)
+gluodynamics and SU(2) QCD [2] to the case of SU(3) gluodynamics. We presented our results for one lattice spacing
+to demonstrate that the decomposition works quite well. Our results obtained in [2] for SU(2) gluodynamics give
+hope that the decomposition (1) will work even better when the lattice spacing will be decreased.
+Our results for another MAG decomposition, 3, contradict to results from Ref. [7] and also indicate that the
+decomposition 1 is superior. There is another reason to consider the decomposition 1 better motivated physically
+than the decomposition 3. The decompositon 1 separates out the monopole component U mon
+µ
+(x) which is responsible
+
+SU(3)
+mon+mod
+abel
+offdiag
+abel+offdiag
+5
+4 
+V(R)
+3
+2
+1
+0
+0.5
+1.0
+1.5
+2.0
+R/ro5
+for the linear part of the static potential as well as for the chiral symmetry breaking. The modified (monopoleless)
+component U mod
+µ
+(x) produces purely Coulomb potential which is in agreement with the original Coulomb part both
+at small and large distances. At the same time in the decomposition 3 the Coulomb part is distributed in an unnatural
+way between two components: Abelian and off-diagonal.
+It is clear that the study of both decompositions at varying lattice spacing are necessary to understand their fate
+in the continuum limit. This is the subject of our future work.
+[1] V. G. Bornyakov, M. I. Polikarpov, G. Schierholz, et al, Nucl. Phys. B Proc. Suppl. 153, 25-32 (2006) [arXiv:hep-
+lat/0512003 [hep-lat]].
+[2] V. G. Bornyakov, I. Kudrov and R. N. Rogalyov, Phys. Rev. D 105, no.5, 054519 (2022) [arXiv:2101.04196 [hep-lat]].
+[3] T. Suzuki and I. Yotsuyanagi, Phys. Rev. D 42 (1990) 4257.
+[4] S. Hioki, S. Kitahara, S. Kiura, et al, Phys. Lett. B 272, 326 (1991) [Erratum-ibid. B 281, 416 (1992)].
+[5] G. S. Bali, V. Bornyakov, M. Muller-Preussker and K. Schilling, Phys. Rev. D 54 (1996) 2863.
+[6] V. Bornyakov and M. Muller-Preussker, Nucl. Phys. Proc. Suppl. 106, 646 (2002).
+[7] N. Sakumichi and H. Suganuma, Phys. Rev. D 90 (2014) no.11, 111501.
+[8] A. S. Kronfeld, M. L. Laursen, G. Schierholz and U. J. Wiese, Phys. Lett. B 198, 516 (1987).
+[9] G. ’t Hooft, Nucl. Phys. B 190 (1981) 455.
+[10] M. N. Chernodub and M. I. Polikarpov, in ”Confinement, Duality and Non-perturbative Aspects of QCD”, p.387, Plenum
+Press, 1998, hep-th/9710205; R. W. Haymaker, Phys. Rept. 315 (1999) 153; J. Greensite, Prog. Part. Nucl. Phys. 51
+(2003), 1
+[11] H. Shiba and T. Suzuki, Phys. Lett. B 333, 461 (1994).
+[12] J. D. Stack, S. D. Neiman and R. J. Wensley, Phys. Rev. D 50, 3399 (1994).
+[13] G. ’t Hooft, in High Energy Physics, ed. A. Zichichi, EPS International Conference, Palermo (1975); S. Mandelstam, Phys.
+Rept. 23, 245 (1976).
+[14] Y. M. Cho, Phys. Rev. D 21 (1980), 1080 doi:10.1103/PhysRevD.21.1080
+[15] Y.S. Duan and M.L. Ge, Sinica Sci., 11, 1072–1081 (1979).
+[16] L. Faddeev and A.J. Niemi, Phys. Rev. Lett. 82, 1624–1627 (1999); L.D. Faddeev and A.J. Niemi, Nucl. Phys. B776, 38-65
+(2007).
+[17] S.V. Shabanov, Phys. Lett. B458, 322–330 (1999); S.V. Shabanov, Phys. Lett. B463, 263–272 (1999).
+[18] K. I. Kondo, T. Murakami and T. Shinohara, Prog. Theor. Phys. 115 (2006), 201-216; S. Kato, K. I. Kondo, T. Murakami,
+A. Shibata, T. Shinohara and S. Ito, Phys. Lett. B 632 (2006), 326-332.
+[19] K. I. Kondo, S. Kato, A. Shibata and T. Shinohara, Phys. Rept. 579 (2015), 1-226.
+[20] S. Kato, K. I. Kondo and A. Shibata, Phys. Rev. D 91 (2015) no.3, 034506.
+[21] J. C. Biddle, W. Kamleh and D. B. Leinweber, Phys. Rev. D 106 (2022) no.5, 054505 doi:10.1103/PhysRevD.106.054505
+[arXiv:2206.00844 [hep-lat]].
+[22] A. S. Kronfeld, G. Schierholz and U. J. Wiese, Nucl. Phys. B 293 (1987), 461-478.
+[23] F. Brandstater, U. J. Wiese and G. Schierholz, Phys. Lett. B 272 (1991), 319-325.
+[24] V. Bornyakov, G. Schierholz and T. Streuer, Nucl. Phys. B Proc. Suppl. 106 (2002), 676-678 doi:10.1016/S0920-
+5632(01)01813-8 [arXiv:hep-lat/0111018 [hep-lat]].
+[25] V. G. Bornyakov et al. [DIK], Phys. Rev. D 70 (2004), 074511 doi:10.1103/PhysRevD.70.074511 [arXiv:hep-lat/0310011
+[hep-lat]].
+[26] H. Ohata and H. Suganuma, Phys. Rev. D 102 (2020) no.1, 014512.
+[27] V. G. Bornyakov et al. [DIK], Phys. Rev. D 71 (2005), 114504 doi:10.1103/PhysRevD.71.114504 [arXiv:hep-lat/0401014
+[hep-lat]].
+[28] J. Smit and A. van der Sijs, Nucl. Phys. B 355 (1991), 603-648
+[29] M. Albanese et al. [APE], Phys. Lett. B 192 (1987), 163-169
+[30] S. Necco and R. Sommer, Nucl. Phys. B 622 (2002), 328-346.
+[31] J. D. Stack, W. W. Tucker and R. J. Wensley, Nucl. Phys. B 639 (2002), 203-222 doi:10.1016/S0550-3213(02)00537-0
+[arXiv:hep-lat/0110196 [hep-lat]].
+[32] R. Golubich and M. Faber, Particles 3 (2020) no.2, 444-455.
+

6dE1T4oBgHgl3EQfTQOX/content/tmp_files/load_file.txt

@@ -0,0 +1,412 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf,len=411
+page_content='Decomposition of the static potential in SU(3) gluodynamics  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Bornyakov  NRC “Kurchatov Institute” - IHEP, Protvino, 142281 Russia  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kudrov  NRC “Kurchatov Institute” - IHEP, Protvino, 142281 Russia,  Moscow Institute of Physics and Technology, Institutskii per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 9, 141700, Dolgoprudny, Russia  After fixing the Maximal Abelian gauge in SU(3) lattice gluodynamics we decompose the non-  abelian gauge field into the Abelian field created by Abelian monopoles and the modified nonabelian  field with monopoles removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' We then calculate respective static potentials in the fundamental  representation and show that the sum of these potentials approximates the nonabelian static poten-  tial with good precision at all distances considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Comparison with other ways of decomposition  is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  PACS numbers: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='Ha, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='Gc, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='Aw  Keywords: gauge field theory, confinement, monopoles, maximal Abelian gauge  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  INTRODUCTION  We study numerically the lattice SU(3) gluodynamics in the Maximal Abelian gauge (MAG) and consider decom-  position of the lattice gauge field Uµ(x) ∈ SU(3)  Uµ(x) = U mod  µ  (x)U mon  µ  (x)  (1)  where U mon  µ  (x) is the component of the gauge field due to Abelian monopoles (to be defined later) and U mod  µ  (x)  is respectively the monopoleless component which we call a modified gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' By modification we understand  removal of the Abelian monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  This kind of decomposition was studied before in SU(2) gluodynamics in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' It was shown that while the monopole  component U mon  µ  (x) is reproducing the linear part of the static potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' the monopoleless component U mod  µ  (x)  produces purely Coulomb potential and their sum provides a good approximation of the original unaltered static  potential at all distances:  V (R) ≈ Vmod(R) + Vmon(R)  (2)  Recently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' in [2] it was shown that this approximation becomes better when the lattice spacing is decreased leaving a  possibility that relation (2) becomes exact in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' It was also shown that (2) is satisfied in SU(2)  QCD as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' In the present work we extend the study of the decomposition (1) to the more realistic case - SU(3)  gluodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  It is well known [3–7] that after performing the Abelian projection in the MAG [8, 9], the Abelian string tension  calculated from the Abelian static potential is very close to the nonabelian string tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' This observation, confirmed  in gluodynamics and in QCD, supports the concept of the Abelian dominance (for a review see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' It was further  discovered [5, 11, 12] that the so called monopole static potential also has string tension close to the nonabelian one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  These observations are in agreement with conjecture that monopole degrees of freedom are responsible for confinement  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  The interesting question is what is the role of the other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' monopoleless degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The results obtained  in SU(2) gluodynamics suggest that they are responsible for the Coulomb part of the static potential both at small  and large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' This suggests that while at small distances U mod  µ  (x) gives perturbative contribution into the  static potential, it provides nonperturbative contribution at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  It is worth to note that the gauge covariant decomposition was introduced in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [14] and [15] and developed  further in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [16–18], see for review [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The numerical results demonstrating analogues of the Abelian dominance  and the monopole dominance within this approach were obtained in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' It would be interesting to check if the  decomposition into the monopole and monopoleless components works in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  The decomposition different from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' (1) was considered in SU(3) gluodynamics after fixing MAG [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The usual  coset decomposition of the gauge field into the Abelian and the off-diagonal components was used:  Uµ(x) = U offd  µ  (x)U Abel  µ  (x)  (3)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='03076v1  [hep-lat]  8 Jan 2023  2  and respective decomposition for the static potential was verified:  V (R) ≈ Voffd(R) + VAbel(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  (4)  We will compare our results for decomposition (4) with results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [7] in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  The decomposition similar to (2) for the static potential in the maximal center gauge  V (R) ≈ Vcent(R) + Vmod,cent(R)  (5)  corresponding to the decomposition of the gauge field into the center and modified (vortex free) components:  Uµ(x) = U mod,cent  µ  (x)U cent  µ  (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  (6)  was first checked long ago in SU(2) gluodynamics [1] and was studied recently in QCD [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' We will comment on  these numerical results later in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  DECOMPOSITION OF THE GAUGE FIELD  We consider the SU(3) lattice gluodynamics after fixing MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' We use the definition of MAG introduced for lattice  SU(N) theory in [22] and later specified for the SU(3) group in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The MAG is fixed by maximizing the functional  F =  1  8 V  �  x,µ  �  |U (11)  µ  (x)|2 + |U (22)  µ  (x)|2 + |U (33)  µ  (x)|2 − 1  �  (7)  with respect to local gauge transformations g of the lattice gauge field,  Uµ(x) → U g  µ(x) = g(x)†Uµ(x)g(x + ˆµ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  (8)  To fix MAG, the simulated annealing algorithm with three random gauge copies was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' This algorithm was first  used to fix MAG in the SU(2) case [5] and then extended to the SU(3) group in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The details of implementation  of the simulated annealing algorithm in the case of SU(3) gauge group can be found in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' For the gauge fixing  functional F we obtained the average value < F >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='73388(1) to be compared with < F > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='7322(2) quoted in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  The larger is the value of the maximized functional the better is the gauge fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The difference in < F > is due  to the Gribov copies effects and implies that there might be substantial difference between our results and results of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [26] for gauge dependent quantities like Abelian or monopole string tension as discussed in details in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  The Abelian projection means coset decomposition (3) of the nonabelian lattice gauge field Uµ(x) ∈ SU(3) into the  Abelian field U Abel  µ  (x) ∈ U(1) × U(1) and the coset field U offd  µ  (x) ∈ SU(3)/U(1) × U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The Abelian field U Abel  µ  (x)  is determined as  U Abel  µ  (x) = diag  �  u(1)  µ (x), u(2)  µ (x), u(3)  µ (x)  �  ,  (9)  where  u(a)  µ (x) = eiθ(a)  µ  (x)  (10)  with  θ(a)  µ (x) = arg (Uµ(x))a − 1  3  3  �  b=1  arg(Uµ(x))b  ��  mod 2π  (11)  such that  θ(a)  µ (x) ∈ [−4  3π, 4  3π] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  (12)  This definition of Abelian projection uµ(x) maximizes the expression |Tr  �  U †  µ(x)uµ(x)  �  |2 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The Abelian gauge  fields can in turn be decomposed into monopole (singular) and photon (regular) parts:  θ(a)  µ (x) = θ(a) mon  µ  (x) + θ(a) ph  µ  (x) ,  (13)  3  The monopole part is defined by [28]:  θ(a) mon  µ  (x) = 2π  �  y  D(x − y)∂−  α m(a)  αµ(y) ,  (14)  where integers m(a)  µν (x) denote the singular part of the Abelian plaquettes (Dirac plaquettes), ∂−  α is the backward  lattice derivative, and D(x) denotes the lattice Coulomb propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Then U mon  µ  (x) introduced in (1) is defined as  U mon  µ  (x) = diag  �  eiθ(1) mon  µ  (x), eiθ(2) mon  µ  (x), eiθ(3) mon  µ  (x)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  (15)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  THE STATIC POTENTIAL DECOMPOSITION  We calculated R × T rectangular Wilson loops W(R, T), Wmon(R, T) and Wmod(R, T) using lattice gauge fields  Uµ(x), U mon  µ  (x), U mod  µ  (x) introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' To extract respective static potentials V (R), Vmon(R) and Vmod(R) the  APE smearing [29] has been employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Computations were done with the Wilson lattice action at β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='0 on 244  lattices using 5000) statistically independent configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The lattice spacing at this value of the bare coupling  constant is defined by a/r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='186(4) [30], where r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='5fm is called Sommer parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 1 our results  are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' One can see that similar to the SU(2) gluodynamics [2] the monopole potential Vmon(R) is almost  perfectly linear with small curvature at small distances while the modified field potential Vmod(R) is well described  by the Coulomb potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' It is also seen that the relation (2) is valid at all distances with most essential discrepancy  of about 10% at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' It is clear that this discrepancy is mostly due to rather low slope of Vmon(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' As we  have already mentioned it was found in SU(2) gluodynamics that with decreasing of the lattice spacing agreement in  relation (2) improves substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' This should be checked in SU(3) gluodynamics in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 1: Comparison of the nonabelian potential V (R) (filled circles) with the sum Vmod(R)+Vmon(R) (filled triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Vmod(r)  (empty circles) and Vmon(r) (filled inverted triangles) are also depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The solid curves show the fits to the Cornell potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  Next we come to our results for decomposition (4) considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' These results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  In this figure we compare the original potential V (R) with the decompositions (2) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' One can see that the  decomposition (2) clearly works better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Both decompositions should be checked in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' In any case,  our results clearly contradict to the conclusion made in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [7] that the relation (4) is satisfied very nicely already at  β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Comparing our results presented here with results we obtained on 164 lattices we found that finite volume  effects are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' We should note that in [7] slightly different procedure of the Abelian projection was used but as  it was claimed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [31] the differences between these two procedures of Abelian projection are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Thus  our understanding is that the reason for such discrepancy with results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [7] is the difference in the gauge fixing  quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' It was shown in the past that the Gribov copy effects for gauge non-invariant quantities might be quite  substantial [25, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  SU3  mon  mod  mon+mod  5  4  V(R)  3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='0  R/ro4  potential  σr2  0  α  r0V0  V (R)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='34(2) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='34(1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='61(2)  Vmon(R)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='99(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='09(1) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='36(2)  Vmod(R)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='42(1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='22(1)  Vmon(R) + Vmod(R) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='94(1) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='39(1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='98(2)  TABLE I: Parameters of the potentials obtained by fits to function V0 − α/R + σR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 2: Comparison of the nonabelian potential V (R) (filled circles) with the sum Vmod(R)+Vmon(R) (filled inverted triangles)  and the sum VAbel(R) + Voffd(R) (filled squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Voffd(r) (filled triangles) and VAbel(r) (empty circles) are also depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The  solid curves show the fits to the Cornell potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  Next we wish to make a remark about the decomposition (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' This decomposition was first studied in [1] using  the Direct Central gauge in SU(2) gluodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' It was concluded that this decomposition holds with substantially  less precision than the decomposition (2), first of all, due to the low string tension provided by the center vortex  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' To queue the problem of low string tension obtained after the center projection the new approach to  the definition of the center gauge was formulated and successfully applied to SU(2) gluodynamics in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Recently  this decomposition was studied in the lattice QCD with light quarks [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' It was found that in this theory the center  projected string tension is in a very good agreement with the physical string tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' On the other hand, the modified  component U mod,cent  µ  (x) produces the static potential which is not compatible with the Coulomb potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Thus, the  decomposition fails to work at small distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  CONCLUSIONS  There are a few suggestions for the decomposition of the gauge field into components describing (mostly) either  infrared or ultra-violate physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' These include the gauge covariant Cho decomposition [14–18], two decompositions  in the MAG, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' (1) and (3) and one decomposition in the maximal center gauge (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' In this work we extended  our study of the decomposition in MAG into the monopole and the modified (monopoleless) components in SU(2)  gluodynamics and SU(2) QCD [2] to the case of SU(3) gluodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' We presented our results for one lattice spacing  to demonstrate that the decomposition works quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Our results obtained in [2] for SU(2) gluodynamics give  hope that the decomposition (1) will work even better when the lattice spacing will be decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  Our results for another MAG decomposition, 3, contradict to results from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [7] and also indicate that the  decomposition 1 is superior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' There is another reason to consider the decomposition 1 better motivated physically  than the decomposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The decompositon 1 separates out the monopole component U mon  µ  (x) which is responsible  SU(3)  mon+mod  abel  offdiag  abel+offdiag  5  4  V(R)  3  2  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='0  R/ro5  for the linear part of the static potential as well as for the chiral symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' The modified (monopoleless)  component U mod  µ  (x) produces purely Coulomb potential which is in agreement with the original Coulomb part both  at small and large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' At the same time in the decomposition 3 the Coulomb part is distributed in an unnatural  way between two components: Abelian and off-diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  It is clear that the study of both decompositions at varying lattice spacing are necessary to understand their fate  in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' This is the subject of our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Bornyakov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Polikarpov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Schierholz, et al, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 153, 25-32 (2006) [arXiv:hep-  lat/0512003 [hep-lat]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Bornyakov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kudrov and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rogalyov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 105, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='5, 054519 (2022) [arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='04196 [hep-lat]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Suzuki and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Yotsuyanagi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 42 (1990) 4257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Hioki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kitahara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kiura, et al, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 272, 326 (1991) [Erratum-ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 281, 416 (1992)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Bali, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Bornyakov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Muller-Preussker and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Schilling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 54 (1996) 2863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [6] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Bornyakov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Muller-Preussker, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 106, 646 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [7] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Sakumichi and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Suganuma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 90 (2014) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='11, 111501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kronfeld, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Laursen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Schierholz and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Wiese, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 198, 516 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' ’t Hooft, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 190 (1981) 455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Chernodub and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Polikarpov, in ”Confinement, Duality and Non-perturbative Aspects of QCD”, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='387, Plenum  Press, 1998, hep-th/9710205;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Haymaker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 315 (1999) 153;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Greensite, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 51  (2003), 1  [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Shiba and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Suzuki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 333, 461 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Stack, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Neiman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Wensley, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 50, 3399 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' ’t Hooft, in High Energy Physics, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Zichichi, EPS International Conference, Palermo (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Mandelstam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 23, 245 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [14] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Cho, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 21 (1980), 1080 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='1080  [15] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Duan and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Ge, Sinica Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=', 11, 1072–1081 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Faddeev and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Niemi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 82, 1624–1627 (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Faddeev and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Niemi, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B776, 38-65  (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Shabanov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B458, 322–330 (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Shabanov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B463, 263–272 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [18] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kondo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Murakami and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Shinohara, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 115 (2006), 201-216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kondo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Murakami,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Shibata, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Shinohara and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Ito, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 632 (2006), 326-332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [19] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kondo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kato, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Shibata and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Shinohara, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 579 (2015), 1-226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kato, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kondo and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Shibata, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 91 (2015) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='3, 034506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Biddle, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kamleh and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Leinweber, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 106 (2022) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='5, 054505 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='054505  [arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='00844 [hep-lat]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Kronfeld, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Schierholz and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Wiese, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 293 (1987), 461-478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [23] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Brandstater, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Wiese and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Schierholz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 272 (1991), 319-325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [24] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Bornyakov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Schierholz and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Streuer, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' 106 (2002), 676-678 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='1016/S0920-  5632(01)01813-8 [arXiv:hep-lat/0111018 [hep-lat]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [25] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Bornyakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [DIK], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 70 (2004), 074511 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='074511 [arXiv:hep-lat/0310011  [hep-lat]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Ohata and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Suganuma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 102 (2020) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='1, 014512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [27] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Bornyakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [DIK], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D 71 (2005), 114504 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='114504 [arXiv:hep-lat/0401014  [hep-lat]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [28] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Smit and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' van der Sijs, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 355 (1991), 603-648  [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Albanese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' [APE], Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 192 (1987), 163-169  [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Necco and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Sommer, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 622 (2002), 328-346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Stack, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Tucker and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Wensley, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' B 639 (2002), 203-222 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='1016/S0550-3213(02)00537-0  [arXiv:hep-lat/0110196 [hep-lat]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='  [32] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Golubich and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content=' Faber, Particles 3 (2020) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}
+page_content='2, 444-455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQfTQOX/content/2301.03076v1.pdf'}

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+arXiv:2301.01714v1  [quant-ph]  1 Jan 2023
+Canonical steering ellipsoids of pure symmetric multiqubit states with two distinct
+spinors and volume monogamy of steering
+B. G. Divyamani,1 I. Reena,2 Prasanta K. Panigrahi,3 A. R. Usha Devi,2, 4 and Sudha5, 4, ∗
+1Tunga Mahavidyalaya, Thirthahalli-577432, Karnataka, India
+2Department of Physics, Bangalore University, Bangalore-560 056, India
+3Department of Physical Sciences, Indian Institute of Science Education
+and Research Kolkata, Mohanpur-741246, West Bengal, India
+4Inspire Institute Inc., Alexandria, Virginia, 22303, USA.
+5Department of Physics, Kuvempu University, Shankaraghatta-577 451, Karnataka, India
+(Dated: January 5, 2023)
+Quantum steering ellipsoid formalism provides a faithful representation of all two-qubit states and
+helps in obtaining correlation properties of the state through the steering ellipsoid. The steering
+ellipsoids corresponding to the two-qubit subsystems of permutation symmetric N-qubit states is
+analysed here. The steering ellipsoids of two-qubit states that have undergone local operations on
+both the qubits so as to bring the state to its canonical form are the so-called canonical steering
+ellipsoids.
+We construct and analyze the geometric features of the canonical steering ellipsoids
+corresponding to pure permutation symmetric N-qubit states with two distinct spinors. Depending
+on the degeneracy of the two spinors in the pure symmetric N-qubit state, there arise several families
+which cannot be converted into one another through Stochastic Local Operations and Classical
+Communications (SLOCC). The canonical steering ellipsoids of the two-qubit states drawn from
+the pure symmetric N-qubit states with two distinct spinors allow for a geometric visualization of
+the SLOCC-inequivalent class of states. We show that the states belonging to the W-class correspond
+to oblate spheroid centered at (0, 0, 1/(N −1)) with fixed semiaxes lengths 1/
+√
+N − 1 and 1/(N −1).
+The states belonging to all other SLOCC inequivalent families correspond to ellipsoids centered at
+the origin of the Bloch sphere. We also explore volume monogamy relations of states belonging to
+these families, mainly the W-class of states.
+PACS numbers: 03.65.Ud, 03.67.Bg
+I.
+INTRODUCTION
+The Bloch sphere representation of a single qubit contains valuable geometric information needed for quantum
+information processing tasks. A natural generalization and an analogous picture for a two-qubit system is provided by
+the quantum steering ellipsoid [1–3] and is helpful in understanding correlation properties such as quantum discord [4,
+5], volume monogamy of steering [2, 3] etc., Quantum steering ellipsoid is the set of all Bloch vectors to which one
+party’s qubit could be ‘steered’ when all possible measurements are carried out on the qubit belonging to other party.
+The volume of the steering ellipsoids [1] corresponding to the two-qubit subsystems of an N-qubit state, N > 3, capture
+monogamy properties of the state effectively [2, 3] and provides insightful information about two-qubit entanglement.
+While the quantum steering ellipsoid [1–3] is the set of all Bloch vectors of first qubit steered by local operations
+on second qubit, the so-called canonical steering ellipsoid [6–8] is the steering ellipsoid of a two-qubit state that has
+attained a canonical form under suitable SLOCC operations on both the qubits. It has been shown that the SLOCC
+canonical forms of a two-qubit state can either be a Bell diagonal form or a nondiagonal one (when the two-qubit
+state is rank-deficient) [6, 8]. The canonical steering ellipsoids corresponding to the two-qubit states can thus have
+only two distinct forms and provide a much simpler geometric picture representing the set of all SLOCC equivalent
+two-qubit states.
+The canonical steering ellipsoids corresponding to the two-qubit subsystems of pure three-qubit permutation sym-
+metric states are analyzed in Ref. [9].
+It has been shown that [9] the two SLOCC inequivalent families of pure
+three-qubit permutation symmetric states, the W-class of states (with two distinct spinors) and the GHZ class of
+states (with three distinct spinors) correspond to distinct canonical steering ellipsoids. While an ellipsoid centered at
+the origin of the Bloch sphere is the canonical steering ellipsoid for the GHZ class of states, an oblate spheroid with
+its center shifted along the z-axis is the one for W-class of states. Using these, the volume monogamy relations are
+established and the obesity of the steering ellipsoids is made use of to obtain expressions for concurrence of states
+belonging to these two SLOCC inequivalent families in Ref. [9].
+∗ [email protected]
+
+2
+In this paper, we continue with the work in Ref. [9], authored by some of us, and extend the analysis to a class
+of N-qubit pure states which are symmetric under exchange of qubits. Through the SLOCC canonical forms of the
+two-qubit reduced state, extracted from pure symmetric multiqubit states with two distinct spinors and the Lorentz
+canonical forms of their real representative, we examine the features of canonical steering ellipsoids associated with
+them. We identify the special features of the canonical steering ellipsoid representing N-qubit states of the W-class
+and these features distinguish this class from all other SLOCC inequivalent families of pure symmetric N-qubit states.
+We discuss the volume monogamy of steering for pure permutation symmetric N-qubit states and obtain the volume
+monogamy relation satisfied by W-class of states. An expression for obesity of the steering ellipsoid and thereby an
+expression for concurrence of two-qubit subsystems of N-qubit states belonging to the W-class is obtained.
+Contents of this paper are organized as follows: In Sec.II, we give a brief review on SLOCC classification of pure
+permutation symmetric multiqubit states based on Majorana representation [10, 12, 13] and obtain the two-qubit
+subsystems of the states belonging to SLOCC inequivalent families of pure symmetric multiqubit states with two
+distinct spinors. Sec. III provides an outline of the real matrix representation of a two-qubit density matrix and their
+Lorentz canonical forms under SLOCC transformation of the two-qubit density matrix. We also obtain the Lorentz
+canonical forms of two-qubit subsystems corresponding to SLOCC inequivalent families, in Sec. III. In Sec.IV, we
+analyse the nature of steering ellipsoids associated with the distinct Lorentz canonical forms obtained in Sec. III. The
+volume monogamy of steering for pure symmetric multiqubit states with two distinct spinors is discussed along with
+illustration for W-class of states, in Sec. V. Summary of our results is presented in Sec. VI.
+II.
+MAJORANA GEOMETRIC REPRESENTATION OF PURE SYMMETRIC N-QUBIT STATES
+WITH TWO DISTINCT SPINORS
+Ettore Majorana, in his novel 1932 paper [10] proposed that a pure spin j = N
+2 quantum state can be represented
+as a symmetrized combination of N constituent spinors as follows:
+|Ψsym⟩ = N
+�
+P
+ˆP {|ǫ1, ǫ2, . . . ǫN⟩},
+(1)
+where
+|ǫl⟩ = (cos(αl/2) |0⟩ + sin(αl/2) |1⟩) eiβl/2,
+l = 1, 2, . . . , N.
+(2)
+The symbol ˆP corresponds to the set of all N! permutations of the spinors (qubits) and N corresponds to an overall
+normalization factor. The name Majorana geometric representation is owing to the fact that it leads to an intrinsic
+geometric picture of the state in terms of N-points on the unit sphere. In fact, the spinors |ǫl⟩, l = 1, 2, . . . , N
+of (2) correspond geometrically to N points on the unit sphere S2, with the pair of angles (αl, βl) determining the
+orientation of each point on the sphere.
+The pure symmetric N-qubit states characterized by two distinct qubits are given by [11–13],
+|DN−k,k⟩ = N
+�
+P
+ˆP {| ǫ1, ǫ1, . . . , ǫ1
+�
+��
+�
+N−k
+; ǫ2, ǫ2, . . . , ǫ2
+�
+��
+�
+k
+⟩}.
+(3)
+Here one of the spinors say |ǫ1⟩ occurs N − k times whereas the other spinor |ǫ2⟩ occurs k times in each term of the
+symmetrized combination. Under identical local unitary transformations, the pure symmetric N qubit states with
+two distinct spinors can be brought to the canonical form [13],
+|DN−k,k⟩ ≡
+k
+�
+r=0
+β(k)
+r
+����
+N
+2 , N
+2 − r
+�
+,
+k = 1, 2, 3, . . .
+�N
+2
+�
+(4)
+β(k)
+r
+= N
+�
+N!(N − r)!
+r!
+ak−r br
+(N − k)!(k − r)!,
+0 ≤ a < 1,
+b =
+�
+1 − a2.
+(5)
+Notice that
+�� N
+2 , N
+2 − r
+�
+, r = 0, 1, 2 . . . k are the Dicke states, which are common eigenstates of the collective angular
+momentum operators J2 and Jz. They are the basis states of the N + 1 dimensional symmetric subspace of collective
+angular momentum space. The states |DN−k,k⟩ (see (4), (5)) are characterized by only one real parameter ‘a’ and
+thus form one parameter family of states {DN−k,k}.
+It is important to notice that [13] in the family {DN−k,k}, different values of k, (k = 1, 2, 3, . . .
+� N
+2
+�
+), correspond to
+different SLOCC inequivalent classes. That is, a state |DN−k,k⟩ cannot be converted into |DN−k′,k′⟩, k ̸= k′ through
+
+3
+any choice of local unitary (identical) transformations. In fact, different values of k lead to different degeneracy
+configurations [13] of the two spinors |ǫ1⟩, |ǫ2⟩ in the state |DN−k,k⟩. When k = 1, one gets the W-class of states
+{DN−1,1} where one of the qubits say |ǫ1⟩ repeats only once in each term of the symmetrized combination (see (3))
+and the other qubit |ǫ2⟩ repeating N − 1 times. The N-qubit W-state
+|WN⟩ =
+1
+√
+N
+[|000 . . .1⟩ + |000 . . .10⟩ + · · · + |100 . . .00⟩] ≡
+����
+N
+2 , N
+2 − 1
+�
+(6)
+belongs to the family {DN−1,1} and hence the name W-class of states.
+A.
+Two-qubit reduced density matrices of the states |DN−k, k⟩
+The two-qubit marginal ρ(k) corresponding to any random pair of qubits in the pure symmetric N-qubit state
+|DN−k, k⟩ ∈ {DN−k,k} is obtained by tracing over the remaining N − 2 qubits in it. In Ref. [15], it has been shown,
+using the algebra of addition of angular momenta, j1 = 1 (corresponding to two-qubit marginal) and j2 = (N − 2)/2,
+that the two-qubit reduced density matrix ρ(k) has the form
+ρ(k) =
+
+
+
+
+A(k)
+B(k)
+B(k)
+C(k)
+B(k)
+D(k)
+D(k)
+E(k)
+B(k)
+D(k)
+D(k)
+E(k)
+C(k)
+E(k)
+E(k)
+F (k)
+
+
+
+ .
+(7)
+The elements A(k), B(k), C(k), D(k), E(k) and F (k) are real and are explicitly given by [15]
+A(k) =
+k
+�
+r=0
+�
+βk
+r
+�2 �
+c(r)
+1
+�2
+, B(k) =
+1
+√
+2
+k−1
+�
+r=0
+β(k)
+r β(k)
+r+1 c(r)
+1 c(r+1)
+0
+C(k) =
+k−2
+�
+r=0
+β(k)
+r
+β(k)
+r+2 c(r)
+1 c(r+2)
+−1
+, D(k) = 1
+2
+k
+�
+r=1
+�
+β(k)
+r
+�2 �
+c(r)
+0
+�2
+(8)
+E(k) =
+1
+√
+2
+k−1
+�
+r=0
+β(k)
+r
+β(k)
+r+1 c(r)
+0 c(r+1)
+−1
+,
+F (k) =
+k
+�
+r=0
+�
+β(k)
+r
+�2 �
+c(r)
+−1
+�2
+.
+where, β(k)
+r
+are given as functions of the parameter ‘a’ in (5) and
+c(r)
+1
+=
+�
+(N − r)(N − r − 1)
+N(N − 1)
+,
+c(r)
+−1 =
+�
+r (r − 1)
+N(N − 1),
+c(r)
+0
+=
+�
+2r (N − r)
+N(N − 1)
+(9)
+are the Clebsch-Gordan coefficients c(r)
+m2 = C
+� N
+2 − 1, 1, N
+2 ; m − m2, m2, m
+�
+, m =
+N
+2 − r, m2 = 1, 0, −1 [16]. In
+particular, for W-class of states i.e., when k = 1, we have
+ρ(1) = TrN−2 (|DN−1, 1⟩⟨DN−1, 1|)
+=
+��
+β(1)
+0
+�2
++
+�
+β(1)
+1
+c(1)
+1
+�2�
+|1, 1⟩⟨1, 1|
++
+�
+β(1)
+1
+c(1)
+0
+�2
+|1, 0⟩⟨1, 0| + β(1)
+0 β(1)
+1
+c(1)
+0 |1, 1⟩⟨1, 0|
++β(1)
+0 β(1)
+1
+c(1)
+0 |1, 0⟩⟨1, 1|
+(10)
+Here (see (5)) we have β(1)
+0
+= NN a, β(1)
+1
+= N
+�
+N(1 − a2) with N =
+1
+√
+N 2 a2+N(1−a2) and the associated non-zero
+Clebsch-Gordan coefficients (see (9)) are given by
+c(1)
+1
+=
+�
+N − 2
+N
+,
+c(1)
+0
+=
+�
+2
+N .
+(11)
+
+4
+In the standard two-qubit basis {|0A, 0B⟩, |0A, 1B⟩, |1A, 0B⟩, |1A, 1B⟩}, the two-qubit density matrix ρ(1) drawn from
+the states |DN−1,1⟩ take the form
+ρ(1) =
+
+
+
+
+A(1)
+B(1)
+B(1)
+0
+B(1)
+D(1)
+D(1)
+0
+B(1)
+D(1)
+D(1)
+0
+0
+0
+0
+0
+
+
+
+
+(12)
+where
+A(1) = N 2a2 + (N − 2)(1 − a2)
+N 2 a2 + N(1 − a2)
+,
+B(1) =
+a
+√
+1 − a2
+1 + a2(N − 1),
+D(1) =
+1 − a2
+N 2 a2 + N(1 − a2),
+(13)
+In a similar manner, the two-qubit subsystems of pure symmetric N-qubit states |DN−k,k⟩ belonging to each SLOCC
+inequivalent family {DN−k, k}, k = 2, 3, . . . ,
+� N
+2
+�
+can be obtained as a function of N and ‘a’ using Eqs. (7), (8), (9).
+As is shown in Refs. [8, 9], the real representative Λ(k) of the two-qubit subsystem ρ(k) and its Lorentz canonical form
+�Λ(k) are essential in obtaining the geometric visualization of the states |DN−k,k⟩ for all k. We thus proceed to obtain
+Λ(k) and its Lorentz canonical form �Λ(k) in the following.
+III.
+THE REAL REPRESENTATION OF ρ(k) AND ITS LORENTZ CANONICAL FORMS
+The real representative Λ(k) of the two-qubit state ρ(k) is a 4 × 4 real matrix with its elements given by
+Λ(k)
+µ ν = Tr
+�
+ρ(k) (σµ ⊗ σν)
+�
+(14)
+That is, Λ(k)
+µ ν, µ, ν = 0, 1, 2, 3 are the coefficients of expansion of ρ(k), expanded in the Hilbert-Schmidt basis {σµ⊗σν}:
+ρ(k) = 1
+4
+3
+�
+µ, ν=0
+Λ(k)
+µ ν (σµ ⊗ σν) ,
+(15)
+Here, σi, i = 1, 2, 3 are the Pauli spin matrices and σ0 is the 2 × 2 identity matrix;
+σ0 =
+�
+1 0
+0 1
+�
+,
+σ1 =
+�
+0 1
+1 0
+�
+,
+σ2 =
+�
+0 −i
+i
+0
+�
+,
+σ3 =
+�
+1
+0
+0 −1
+�
+.
+(16)
+It can be readily seen that (see (14), (15)) the real 4 × 4 matrix Λ(k) has the form
+Λ(k) =
+
+
+
+1
+r1
+r2
+r3
+s1 t11 t12 t13
+s2 t21 t22 t23
+s3 t31 t32 t33
+
+
+ ,
+(17)
+where r = (r1, r2, r3)T , s = (s1, s2, s3)T are Bloch vectors of the individual qubits and T = (tij) is the correlation
+matrix;
+ri = Λ(k)
+i 0 = Tr
+�
+ρ(k) (σi ⊗ σ0)
+�
+(18)
+sj = Λ(k)
+0 j = Tr
+�
+ρ(k) (σ0 ⊗ σj)
+�
+(19)
+tij = Λ(k)
+i j = Tr
+�
+ρ(k) (σi ⊗ σj)
+�
+,
+i, j = 1, 2, 3.
+(20)
+For a symmetric two-qubit density matrix, the Bloch vectors r and s are identical and hence ri = si, i = 1, 2, 3; From
+the structure of ρ(k) in (7) and using (18), (19), (20) we obtain the general form of the real matrix Λ(k) as
+Λ(k) =
+
+
+
+
+
+
+
+1
+2(B(k)+E(k))
+A(k)+2D(k)+F (k)
+0
+A(k)−F (k)
+A(k)+2D(k)+F (k)
+2(B(k)+E(k))
+A(k)+2D(k)+F (k)
+2(C(k)+D(k))
+A(k)+2D(k)+F (k)
+0
+2(B(k)−E(k))
+A(k)+2D(k)+F (k)
+0
+0
+2(D(k)−C(k))
+A(k)+2D(k)+F (k)
+0
+A(k)−F (k)
+A(k)+2D(k)+F (k)
+2(B(k)−E(k))
+A(k)+2D(k)+F (k)
+0
+1 −
+4D(k)
+A(k)+2D(k)+F (k)
+
+
+
+
+
+
+
+.
+(21)
+The elements of Λ(k), for different k, can be evaluated using (8), (9)):
+
+5
+A.
+Lorentz canonical forms of Λ(k)
+Under SLOCC transformation, the two-qubit density matrix ρ(k) transforms to �ρ(k),
+ρ(k) −→ �ρ(k) = (A ⊗ B) ρ(k) (A† ⊗ B†)
+Tr
+�
+ρ(k) (A† A ⊗ B† B)
+�.
+(22)
+Here, A, B ∈ SL(2, C) denote 2 × 2 complex matrices with unit determinant. A suitable choice of A and B takes the
+two-qubit density matrix ρ(k) to its canonical form �ρ(k).
+Under the transformation ρ(k) −→ �ρ(k) (22) of the two-qubit state, its real representative Λ(k) transforms as [8, 9]
+Λ(k) −→ �Λ(k) =
+LA Λ(k) LT
+B
+�
+LA Λ(k) LT
+B
+�
+00
+.
+(23)
+Here LA, LB ∈ SO(3, 1) are 4 × 4 proper orthochronous Lorentz transformation matrices [17] corresponding respec-
+tively to A, B ∈ SL(2, C) and the superscript ‘T ’ denotes transpose operation. The Lorentz canonical form �Λ(k)
+of Λ(k) and thereby the SLOCC canonical form of the two-qubit density matrix ρ(k) (see (22)) can be obtained by
+constructing the 4 × 4 real symmetric matrix Ω(k) = Λ(k) G
+�
+Λ(k)�T , where G = diag (1, −1, −1, −1) denotes the
+Lorentz metric. Using the defining property [17] LT G L = G of Lorentz transformation L, it can be seen that Ω(k)
+undergoes a Lorentz congruent transformation under SLOCC (upto an overall factor) [8] as
+Ω(k) → �Ω(k)
+A = �Λ(k) G
+�
+�Λ(k)�T
+= LA Λ(k) LT
+B G LB Λ(k)T LT
+A
+= LA Ω(k) LT
+A.
+(24)
+It has been shown in Ref. [8] that �Λ(k) can either be a real 4 × 4 diagonal matrix or a nondiagonal matrix with only
+one off-diagonal element, depending on the eigenvalues, eigenvectors of G Ω(k) = G
+�
+Λ(k) G
+�
+Λ(k)�T �
+.
+(i) The diagonal canonical form �Λ(k)
+Ic results when the eigenvector X0 associated with the highest eigenvalue λ0 of
+G Ω(k) obeys the Lorentz invariant condition XT
+0 G X0 > 0. The diagonal canonical form �Λ(k)
+Ic is explicitly given
+by
+Λ(k) −→ �Λ(k)
+Ic =
+LA1 Λ(k) LT
+B1
+�
+LA1 Λ(k) LT
+B1
+�
+00
+= diag
+�
+1,
+�
+λ1
+λ0
+,
+�
+λ2
+λ0
+, ±
+�
+λ3
+λ0
+�
+,
+(25)
+where λ0 ≥ λ1 ≥ λ2 ≥ λ3 > 0 are the non-negative eigenvalues of G Ω(k).
+The Lorentz transformations
+LA1, LB1 ∈ SO(3, 1) in (25) respectively correspond to SL(2, C) transformation matrices A1, B1 which take the
+two-qubit density matrix ρ(k) to its SLOCC canonical form �ρ(k)
+Ic through the transformation (22). The diagonal
+form of �Λ(k)
+Ic readily leads, on using (15), to Bell-diagonal form
+�ρ(k)
+Ic = 1
+4
+
+σ0 ⊗ σ0 +
+�
+i=1,2
+�
+λi
+λ0
+σi ⊗ σi ±
+�
+λ3
+λ0
+σ3 ⊗ σ3
+
+
+(26)
+as the canonical form of the two-qubit state ρ(k).
+(ii) The Lorentz canonical form of Λ(k) turns out to be a nondiagonal matrix (with only one nondiagonal element)
+given by
+Λ(k) −→ �Λ(k)
+IIc =
+LA2 Λ(k) LT
+B2
+�
+LA2 Λ(k) LT
+B2
+�
+00
+=
+
+
+
+1
+0
+0
+0
+0
+a1
+0
+0
+0
+0
+−a1
+0
+1 − a0
+0
+0
+a0
+
+
+
+(27)
+
+6
+when the non-negative eigenvalues of GΩ(k) are doubly degenerate with λ0 ≥ λ1 and the eigenvector X0
+belonging to the highest eigenvalue λ0 satisfies the Lorentz invariant condition XT
+0 G X0 = 0. In Ref. [8], it
+has been shown that when the maximum amongst the doubly degenerate eigenvalues of GΩ(k) possesses an
+eigenvector X0 satisfying the condition XT
+0 G X0 = 0, the real symmetric matrix Ω(k) = Λ(k)G
+�
+Λ(k)�T attains
+the nondiagonal Lorentz canonical form given by
+Ω(k)
+IIc = �Λ(k)
+IIc G
+�
+�Λ(k)
+IIc
+�T
+= LA2 Ω(k) LT
+A2
+=
+
+
+
+φ0
+0
+0
+φ0 − λ0
+0
+−λ1
+0
+0
+0
+0
+−λ1
+0
+φ0 − λ0
+0
+0
+φ0 − 2λ0
+
+
+ .
+(28)
+The parameters a0, a1 in (27) are related to the eigenvalues λ0, λ1 of GΩ(k) and the 00th element of �Λ(k)
+IIc, the
+canonical form of Ω(k) (see (28)). It can be seen that [8]
+a0 = λ0
+φ0
+,
+a1 =
+�
+λ1
+φ0
+,
+where φ0 =
+�
+Ω(k)
+IIc
+�
+00 =
+��
+LA2 Λ(k) LT
+B2
+�
+00
+�2
+.
+(29)
+The Lorentz matrices LA2, LB2 ∈ SO(3, 1) correspond to the SL(2,C) transformations A2, B2 which take the
+density matrix ρ(k) to its SLOCC canonical form ρ(k)
+IIc through the transformation (22). The nondiagonl canonical
+form �Λ(k)
+IIc leads to the SLOCC canonical form �ρ(k)
+IIc of the two-qubit density matrix ρ(k) on using (15);
+�ρ(k)
+IIc = 1
+2
+
+
+
+1
+0
+0
+a1
+0
+1 − a0 0
+0
+0
+0
+0
+0
+a1
+0
+0 a0
+
+
+ .
+(30)
+B.
+Lorentz canonical form of Λ(1) corresponding to W-class of states
+Using the explicit structure of the two-qubit state ρ(1) given in (12), (13), its real representative Λ(1) is obtained
+as (see (14))
+Λ(1) =
+
+
+
+
+
+
+1
+2a
+√
+1−a2
+1+a2(N−1)
+0
+1 +
+2a2
+1+a2(N−1) − 2
+N
+2a
+√
+1−a2
+1+a2(N−1)
+2(1−a2)
+N(1+a2(N−1))
+0
+2a
+√
+1−a2
+1+a2(N−1)
+0
+0
+2(1−a2)
+N(1+a2(N−1))
+0
+1 +
+2a2
+1+a2(N−1) − 2
+N
+2a
+√
+1−a2
+1+a2(N−1)
+0
+1 +
+4a2
+1+a2(N−1) − 4
+N
+
+
+
+
+
+
+=
+�
+Λ(1)�T
+.
+(31)
+We now construct the 4 × 4 symmetric matrix Ω(1) and obtain
+Ω(1) = Λ(1) G
+�
+Λ(1)�T
+= Λ(1) G Λ(1)
+= χ
+
+
+
+N − 1
+0
+0
+N − 2
+0
+−1
+0
+0
+0
+0
+−1
+0
+N − 2
+0
+0
+N − 3
+
+
+ ,
+χ =
+�
+2(1 − a2)
+N (1 + a2(N − 1))
+�2
+.
+(32)
+The eigenvalues of the matrix G Ω(1), G = diag (1, −1, −1, −1) are readily seen to be four-fold degenerate and are
+given by
+λ0 = λ1 = λ2 = λ3 = χ =
+�
+2(1 − a2)
+N (1 + a2(N − 1))
+�2
+.
+(33)
+
+7
+It can be seen that X0 = (1, 0, 0, −1) is an eigenvector of G Ω(1) belonging to the four-fold degenerate eigenvalue λ0
+and obeys the Lorentz invariant condition XT
+0 G X0 = 0. We notice here that Ω(1) is already in the canonical form
+(28). On comparing (32) with (28), we get
+φ0 = (Ω(1))00 = (N − 1)χ.
+(34)
+On substituting the parameters a0, a1 (See (29), (33), (34) in (27), we arrive at the Lorentz canonical form of the real
+matrix Λ(1) as
+�Λ(1) =
+
+
+
+
+1
+0
+0
+0
+0
+1
+√N−1
+0
+0
+0
+0
+−
+1
+√N−1
+0
+N−2
+N−1
+0
+0
+1
+N−1
+
+
+
+ .
+(35)
+It can be readily seen that �Λ(1), the Lorentz canonical form corresponding to the W-class of states, is independent of
+the parameter ‘a’.
+C.
+Lorentz canonical form of Λ(k), k = 2, 3, . . . ,
+� N
+2
+�
+The real representative Λ(k) given in (21) can readily be evaluated for different values of k (k = 2, 3, . . . ,
+� N
+2
+�
+) on
+using (8), (9). We then construct the real symmetric matrix Ω(k) = Λ(k) G
+�
+Λ(k)�T for k = 2, 3, . . . ,
+� N
+2
+�
+and observe
+that GΩ(k) = GΛ(k) G (Λ(k))
+T has non-degenerate eigenvalues λ0 ̸= λ1 ̸= λ2 ̸= λ3 when k = 2, 3, . . . ,
+� N
+2
+�
+and the
+highest eigenvalue λ0 possesses a positive eigenvector X0 satisfying the relation XT
+0 G X0 > 0. The Lorentz canonical
+form �Λ(k), k = 2, 3, . . . ,
+� N
+2
+�
+, is thus given by the diagonal matrix (see (25)).
+�Λ(k) = diag
+�
+1,
+�
+λ1/λ0,
+�
+λ2/λ0, ±
+�
+λ3/λ0
+�
+.
+The eigenvalues λµ, (µ = 0, 1, 2, 3) of GΩ(k) are dependent on the parameters ‘a’, k and N characterizing the state
+|DN−k, k⟩, when k takes any of the integral values greater than 1 and less than
+� N
+2
+�
+. Hence the canonical form �Λ(k),
+k = 2, 3, . . . ,
+� N
+2
+�
+is different for different states |DN−k, k⟩ unlike in the case of �Λ(1), the canonical form of W-class of
+states, which depends only on the number of qubits N.
+IV.
+GEOMETRIC REPRESENTATION OF THE STATES |DN−k,k⟩
+In this section, based on the two different canonical forms of Λ(k) obtained in Section III, we find the nature of
+canonical steering ellipsoids associated with the pure symmetric multiqubit states |DN−k,k⟩ belonging to SLOCC
+inequivalent families {DN−k, k}. To begin with, we give a brief outline [8, 9] of obtaining the steering ellipsoids of a
+two-qubit density matrix ρ(k) based on the form of its real representative Λ(k).
+In the two-qubit state ρ(k), local projective valued measurements (PVM) Q > 0, Q = �3
+µ=0 qµ σµ, q0 = 1,
+�3
+i=1 q2
+i
+= 1 on Bob’s qubit leads to collapsed state of Alice’s qubit characterized by its Bloch-vector pA =
+(p1, p2, p3)T through the transformation [8]
+(1, p1, p2, p3)T = Λ(k) (1, q1, q2, q3)T ,
+q2
+1 + q2
+2 + q2
+3 = 1.
+(36)
+Notice that the vector qB = (q1, q2, q3)T , q2
+1 + q2
+2 + q2
+3 = 1 represents the entire Bloch sphere and the steered Bloch
+vectors pA of Alice’s qubit constitute an ellipsoidal surface EA| B enclosed within the Bloch sphere. When Bob employs
+convex combinations of PVMs i.e., positive operator valued measures (POVMs), to steer Alice’s qubit, he can access
+the points inside the steering ellipsoid. Similar will be the case when Bob’s qubit is steered by Alice through local
+operations on her qubit.
+For the Lorentz canonical form �Λ(k)
+Ic (see (25)) of the two-qubit state �ρ(k)
+Ic , Eq. (36) leads to
+p1 =
+�
+λ1
+λ0
+q1,
+p2 =
+�
+λ2
+λ0
+q2,
+p3 = ±
+�
+λ3
+λ0
+q3,
+(37)
+
+8
+as steered Bloch points pA of Alice’s qubit. They are seen to obey the equation
+λ0 p2
+1
+λ1
++ λ0 p2
+2
+λ2
++ λ0 p2
+3
+λ3
+= 1
+(38)
+of an ellipsoid with semiaxes (
+�
+λ1/λ0,
+�
+λ2/λ0,
+�
+λ3/λ0) and center (0, 0, 0) inside the Bloch sphere q2
+1 +q2
+2 +q2
+3 = 1.
+We refer to this as the canonical steering ellipsoid representing the set of all two-qubit density matrices which are on
+the SLOCC orbit of the state �ρ(k)
+Ic (see (22)).
+For the second Lorentz canonical form �ΛIIc (see (27)) we get the coordinates of steered Alice’s Bloch vector pA, on
+using (36);
+p1 = a1 q1,
+p2 = −a1q2,
+p3 = (1 − a0) + a0q3,
+q2
+1 + q2
+2 + q2
+3 = 1
+(39)
+and they satisfy the equation
+p2
+1
+a2
+1
++ p2
+2
+a2
+1
++ (p3 − (1 − a0))2
+a2
+0
+= 1.
+(40)
+Eq. (40) represents the canonical steering spheroid (traced by Alice’s Bloch vector pA) inside the Bloch sphere with
+its center at (0, 0, 1 − a0) and lengths of the semiaxes given by a1 =
+�
+λ1/φ0, a0 =
+�
+λ2/φ0 (see (29)). In other
+words, a shifted spheroid inscribed within the Bloch sphere, represents two-qubit states that are SLOCC equivalent
+to �ρ(k)
+IIc.
+A.
+Canonical steering ellipsoids of W-class of states
+We have seen in Sec. III B that the Lorentz canonical form of Λ(1), the real representative of the symmetric two-
+qubit state ρ(1) drawn from the W-class of states |DN−1,1⟩ has a nondiagonal form given in Eq. (35). On comparing
+(35) with the canonical form in (27), we get
+a1 =
+1
+√
+N − 1,
+a0 =
+1
+N − 1.
+(41)
+From (40) and the discussions prior to it, it can be readily seen that the quantum steering ellipsoid associated with �Λ(1)
+in (35) is a spheroid centered at (0, 0,
+1
+N−1) inside the Bloch sphere, with fixed semiaxes lengths (
+1
+√N−1,
+1
+√N−1,
+1
+N−1)
+(see Fig. 1). It is interesting to note that the Lorentz canonical form �Λ(1) is not dependent on the state parameter
+‘a’, 0 ≤ a < 1 and hence all states |DN−1, 1⟩ in the family {DN−1, 1} are represented by an oblate spheroid, all its
+parameters such as center, semiaxes, volume etc., dependent only on the number of qubits N.
+FIG. 1. (Colour online) Steering spheroids inscribed within the Bloch sphere representing the Lorentz canonical form �Λ(1)
+(see (35)) of W-class of states |DN−1,1⟩ for N = 4 and N = 20. The spheroids are centered (0, 0,
+1
+N−1) and the length of the
+semi-axes are given by (
+1
+√N−1,
+1
+√N−1,
+1
+N−1).
+
+9
+B.
+Canonical steering ellipsoids of the states |DN−k,k⟩, k = 2, 3, . . . ,
+� N
+2
+�
+As is seen in Sec. III C, the Lorentz canonical form of Λ(k), k = 2, 3, . . . ,
+� N
+2
+�
+, the real representative of the two-
+qubit states ρ(k) drawn from the pure symmetric N-qubit states |DN−k,k⟩, has the diagonal form (see (25)). The
+values of λ0, λ1, λ2, λ3, the eigenvalues of the matrix G Ωk can be evaluated for each value of k, k = 2, 3, . . . ,
+� N
+2
+�
+for a chosen N. From (38) and the discussions therein, it follows that the canonical steering ellipsoids of the states
+|DN−k,k⟩, k = 2, 3, . . . ,
+� N
+2
+�
+is an ellipsoid centered at the origin of the Bloch sphere with lengths of the semiaxes
+given by
+�
+λ1/λ0,
+�
+λ2/λ0,
+�
+λ3/λ0. The eigenvalues λµ, µ = 0, 1, 2, 3 of GΩ(k) depend on the parameter ‘a’ also,
+unlike in the case of W-class of states where they depend only on N, the number of qubits. Thus each state |DN−k,k⟩
+belonging to the family {DN−k, k}, k = 2, 3, . . . ,
+� N
+2
+�
+is represented by an ellipsoid whose semiaxes depend on the
+values of k, N and ‘a’. In Fig. 2 and Fig. 3 the canonical steering ellipsoids for some chosen values of k, N and ‘a’
+are shown.
+FIG. 2. (Colour online) Steering ellipsoids centered at the origin of the Bloch sphere representing the Lorentz canonical form
+of pure symmetric multiqubit states |DN−k,k⟩ (see (3)) when (i) N = 10, k = 2, a = 0.2 and (ii) N = 10, k = 5, a = 0.2.
+FIG. 3. (Colour online) Steering ellipsoids representing the Lorentz canonical form of pure symmetric multiqubit states |DN−k,k⟩
+(see (3)) when (i) N = 100, k = 2, a = 0.2 and (ii) N = 100, k = 5, a = 0.1.
+V.
+VOLUME MONOGAMY RELATIONS FOR PURE SYMMETRIC MULTIQUBIT STATES |DN−k,k⟩
+Monogamy relations restrict shareability of quantum correlations in a multipartite state. They find potential ap-
+plications in ensuring security in quantum key distribution [18, 19]. Milne et. al. [2, 3] introduced a geometrically
+intuitive monogamy relation for the volumes of the steering ellipsoids representing the two-qubit subsystems of mul-
+tiqubit pure states, which is stronger than the well-known Coffman-Kundu-Wootters monogamy relation [20]. In this
+
+10
+section we explore how volume monogamy relation [2] imposes limits on the volumes of the quantum steering ellip-
+soids representing the two-qubit subsystems ρ(k) = TrN−2 [|DN−k,k⟩⟨DN−k,k|] of pure symmetric multiqubit states
+|DN−k,k⟩.
+For the two-qubit state ρAB(= ρ(k)) (see (15)), we denote by EA| B, the quantum steering ellipsoid containing all
+steered Bloch vectors of Alice when Bob carries out local operations on his qubit. The volume of EA| B is given by [1]
+VB|A =
+�4π
+3
+�
+| det Λ|
+(1 − r2)2 ,
+(42)
+where r2 = r · r = r2
+1 + r2
+2 + r2
+3 (see (18)). As the steering ellipsoid is constrained to lie within the Bloch sphere, i.e.,
+VB|A ≤ Vunit = (4π/3), one can choose to work with the normalized volumes vA|B =
+VA|B
+4π/3 , the ratio of the volume of
+the steering ellipsoid to the volume of a unit sphere.
+The volume monogamy relation satisfied by a pure three-qubit state shared by Alice, Bob and Charlie is given
+by [1–3]
+�
+VA|B +
+�
+VC|B ≤
+�
+4π
+3 .
+(43)
+where VA|B, VC|B are respectively the volumes of the ellipsoids corresponding to steered states of Alice and Charlie
+when Bob performs all possible local measurements on his qubit. The normalized form of the volume monogmay
+relation (43) turns out to be
+√vA|B + √vC|B ≤ 1,
+(44)
+where vA|B =
+VA|B
+4π/3 are the normalized volumes.
+The monogamy relation (44) is not, in general, satisfied by mixed three-qubit states [3] and it has been shown that
+�
+vA|B
+� 2
+3 +
+�
+vC|B
+� 2
+3 ≤ 1,
+(45)
+is the volume monogamy relation for pure as well as mixed three-qubit states [3].
+As there are 1
+2(N − 2)(N − 1) three qubit subsystems in a N-qubit state, each of which obey monogamy relation
+(45), on adding these relations and simplifying, one gets [3]
+�
+vA|B
+� 2
+3 +
+�
+vC|B
+� 2
+3 +
+�
+vD|B
+� 2
+3 + · · · ≤ N − 1
+2
+.
+(46)
+The relation (46) is the volume monogamy relation satisfied by pure as well as mixed N-qubit states . For N = 3, it
+reduces to (45).
+For multiqubit states that are invariant under exchange of qubits, vA|B = vC|B = vD|B = · · · = vN where vN denotes
+the normalized volume of the steering ellipsoid corresponding to any of the N − 1 qubits, the steering performed by,
+say Nth qubit. Eq. (46) thus reduces to
+(N − 1) (vN)
+2
+3 ≤ N − 1
+2
+=⇒ (vN)
+2
+3 ≤ 1
+2
+(47)
+implying that (vN)
+2
+3 ≤ 1
+2 is the volume monogamy relation for permutation symmetric multiqubit states.
+A.
+Volume monogamy relations governing the W-class of states {DN−1,1}
+On denoting the normalized volume of a steering ellipsoid corresponding to the states |DN−1,1⟩ by v(1)
+N , we have
+(see (42))
+v(1)
+N = | det Λ(1)|
+(1 − r2)2 ,
+(48)
+where Λ(1) is given in (31) and
+r1 =
+2a
+√
+1 − a2
+1 + a2(N − 1),
+r2 = 0,
+r3 = 1 +
+2a2
+1 + a2(N − 1) − 2
+N
+(49)
+
+11
+Under suitable Lorentz transformations, the real matrix Λ(1) (see (31)) associated with the state ρ(1)
+2
+gets transformed
+to its Lorentz canonical form �Λ(1) (see (35)). It follows that (see (29), (33))
+�
+LA Λ(1) LT
+B
+�
+00 =
+�
+φ0 = 2
+√
+N − 1
+�
+1 − a2
+N(1 + (N − 1) a2)
+�
+.
+(50)
+Using the property det LA = det LB = 1 of orthochronous proper Lorentz transformations [17] and substituting
+| det �Λ(1)| =
+1
+(N−1)2 in (23), we obtain
+| det �Λ(1)| =
+1
+(N − 1)2 = | det LA| | det LB|
+����det
+� Λ(1)
+√φ0
+����� = | det Λ(1)|
+φ2
+0
+.
+(51)
+Eq. (51) leads to | det Λ(1)| = φ2
+0| det �Λ(1)|. The normalized volume v(1)
+N
+of the steering ellipsoid corresponding to
+W-class of states thus becomes (see (48))
+v(1)
+N = | det �Λ(1)|
+φ2
+0
+(1 − r2)2
+(52)
+From (49) and (50) it readily follows that φ2
+0 = (1 − r2)2 and hence (see (52)) the simple form for the normalized
+volume of the corresponding steering ellipsoid associated with the two-qubit state ρ(1) turns out to be
+v(1)
+N =
+φ2
+0
+(N − 1)2 (1 − r2)2 =
+1
+(N − 1)2 .
+(53)
+The volume monogamy relation
+�
+v(1)
+N
+� 2
+3 ≤ 1
+2 (see (47)) takes the form
+�
+1
+(N − 1)2
+�2/3
+≤ 1
+2 =⇒ 2(N − 1)
+−4
+3 ≤ 1
+(54)
+and is readily satisfied for any N ≥ 3 as can be seen in Fig.5.
+10
+20
+30
+40
+50
+0.0
+0.1
+0.2
+0.3
+0.4
+N
+(N-1)
+-4
+3
+FIG. 4. (Colour online) The LHS of the monogamy relation 2(N − 1)
+−4
+3 ≤ 1 is seen to be less than 1 for the states |DN−1, 1⟩
+for any N ≥ 3.
+B.
+Relation between obesity of steering ellipsoids and concurrence
+We recall here that the obesity O(ρAB) = | det Λ|1/4 of the quantum steering ellipsoid [2] depicting a two-qubit state
+ρAB is an upper bound for the concurrence C(ρAB):
+C(ρAB) ≤ O(ρAB) = | det Λ|1/4.
+(55)
+
+12
+Furthermore, if ρAB −→ �ρAB = (A ⊗ B)ρAB (A† ⊗ B†)/(Tr(A† A ⊗ B†B)ρAB], A, B ∈ SL(2, C) it follows that [2]
+O(ρAB)
+C(ρAB) = O(�ρAB)
+C(�ρAB).
+(56)
+We make use of the relation (56) to obtain a relation for concurrence [21] of a pair of qubits in the symmetric N-qubit
+pure states |DN−k,k⟩, k = 1, 2, . . . ,
+� N
+2
+�
+. For the states |DN−1,1⟩ belonging to W-class, we readily get (see (31), (35))
+det Λ(1) =
+�
+2(1 − a2)
+N(1 + a2(N − 1))
+�4
+,
+det �Λ(1) =
+�
+1
+N − 1
+�2
+(57)
+and thereby the obesities O(ρ(1)), O(�ρ(1)):
+O(ρ(1)) =
+2(1 − a2)
+N(1 + a2(N − 1)),
+O(�ρ(1)) =
+1
+√
+N − 1
+(58)
+As the concurrence of the state �ρ(1) turns out to be
+C(�ρ(1)) = O(�ρ(1)) =
+1
+√
+N − 1
+(59)
+we obtain (see (56),(59))
+C(ρ(1)) = O(ρ(1)) =
+2(1 − a2)
+N(1 + a2(N − 1)).
+(60)
+The value of concurrence in (60) matches exactly with that obtained [21] using C(ρ(1)) = max(0, µ1 − µ2 − µ3 − µ4)
+where µ1 ≥ µ2 ≥ µ3 ≥ µ4 are square-roots of the eigenvalues of the matrix R = ρ(1) (σ2 ⊗ σ2) ρ(1)∗ (σ2 ⊗ σ2).
+We have seen that the state |DN−1, 1⟩ reduces to W-state when a = 0 and hence for the N-qubit W-state, concurrence
+of any pair of qubits is given by C(ρ(1)
+W ) = 2
+N .
+VI.
+SUMMARY
+In this work we have analyzed the canonical steering ellipsoids and volume monogamy relations of the pure symmetric
+N-qubit states characterized by two distinct Majorana spinors. We have shown that the entire W-class of states has a
+geometric representation in terms of a shifted spheroid inscribed inside the Bloch sphere. The center of the spheroid,
+the length of its semiaxes and its volume are shown to be dependent only on the number of qubits N and hence all
+states in the N-qubit W-class are characterized by a single spheroid, shifted along the polar axis of the Bloch sphere.
+All other families of pure symmetric N-qubit states with two distinct spinors which are SLOCC inequivalent to the
+W-class are geometrically represented by ellipsoids centered at the origin. A discussion on volume monogamy relations
+applicable to identical subsystems of a pure N-qubit symmetric state is given here and a volume monogamy relation
+applicable for W-class of states is obtained. A relation connecting concurrence of the two-qubit state and obesity
+of the associated quantum steering ellipsoid with its canonical counterparts is made use of to obtain concurrence of
+the states belonging to W-class. It would be interesting to examine the features of canonical steering ellipsoids and
+volume monogamy relations for the SLOCC inequivalent families of pure symmetric multiqubit states with more than
+two distinct spinors; in particular, the class of pure symmetric N-qubit states belonging to GHZ-class (with three
+distinct spinors).
+ACKNOWLEDGEMENTS
+BGD thanks IASC-INSA-NASI for the award of Summer Research Fellowship-2022, during this work.
+Sudha,
+ARU and IR are supported by the Department of Science and Technology (DST), India through Project No.
+DST/ICPS/QUST/2018/107.
+[1] Jevtic, S., Pusey, M. F., Jennings, D. and Rudolph, T.: Quantum Steering Ellipsoids. Phys. Rev. Lett. 113, 020402 (2014).
+
+13
+[2] Milne, A., Jevtic, S., Jennings, D., Wiseman, H., and Rudolph, T.: Quantum steering ellipsoids, extremal physical states
+and monogamy. New J. Phys. 16, 083017 (2014).
+[3] Cheng, S., Milne, A., Hall, M. J. W., Wiseman, H. M.: Volume monogamy of quantum steering ellipsoids for multiqubit
+systems. Phys. Rev. A., 94, 042105 (2016).
+[4] Shi, M., Jiang, F., Sun, C., and Du, J.: “Geometric picture of quantum discord for two-qubit quantum states,” New J.
+Phys. 13, 073016 (2011).
+[5] Shi, M., Yang, W., Jiang, F., and Du, J.: Quantum discord of two-qubit rank-2 states. J.Phys.A: Math Theor. 44, 415304
+(2011).
+[6] Verstraete, F., Dehaene, J., DeMoor, B.,: Local filtering operations on two qubits. Physical Review A., 64, 010101(R)
+(2001).
+[7] Verstraete, F., Quantum entanglement and quantum information,Ph.D. thesis, Katholieke Universiteit Leuven, 2002.
+[8] Sudha, Karthik, H. S., Pal, R., Akhilesh, K. S., Ghosh, S., Mallesh, K. S., Usha Devi, A. R.: Canonical forms of two-qubit
+states under local operations. Phys.Rev.A, 102, 052419 (2020).
+[9] Anjali, K, Reena, I, Sudha, Divyamani, B. G., Karthik, H. S., Mallesh, K. S., and Usha Devi, A. R., Geometric picture
+for SLOCC classification of pure permutation symmetric three-qubit states, Quantum Inf Proc. 21, 326 (2022).
+[10] Majorana, E.: Atomi Orientati in Campo Magnetico Variabile, Nuovo Cimento 9, 43 (1932).
+[11] Bastin, T., Krins, S., Mathonet, P., Godefroid, M., Lamata, L., and Solano E.: Operational Families of Entanglement
+Classes for Symmetric N-Qubit States, Phys. Rev. Lett. 103, 070503 (2009).
+[12] Mathonet, P., Krins, S., Godefroid, M., Lamata, L., Solano, E., and Bastin, T.: Entanglement equivalence of N-qubit
+symmetric states, Phys. Rev. A81, 052315 (2010).
+[13] Usha Devi, A. R., Sudha, Rajagopal, A. K.: Majorana representation of symmetric multiqubit states, Quantum Inf. Proc.
+11 685 (2012)
+[14] Sudha, Usha Devi, A. R., Rajagopal, A. K.: Monogamy of quantum correlations in three-qubit pure states, Phys. Rev. A,
+85, 012103 (2012).
+[15] Akhilesh, K. S., Divyamani, B. G., Sudha, Usha Devi, A. R., and Mallesh, K. S., Spin squeezing in symmetric multiqubit
+states with two non-orthogonal Majorana spinors, Quantum Inf. Proc. 18, 144 (2019).
+[16] Varshalovich, D. A., Moskalev, A. N. and Khersonskii, V. K.,: Quantum Theory of Angular Momentum, World Scientific,
+Singapore (1988).
+[17] Srinivasa Rao, K. N.: The Rotation and Lorentz groups and their representations for physicists, Wiley Eastern, New Delhi
+(1988).
+[18] Tehral, B. M.: Is entanglement monogamous? IBM J. Res. & Dev. 48, 71 (2004).
+[19] Paw�lowski, M.: Security proof for cryptographic protocols based only on the monogamy of Bell’s inequality violations.
+Phys. Rev. A. 82, 032313 (2010).
+[20] Coffman, V., Kundu, J., Wootters, W. K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000).
+[21] Wootters, W. K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998).
+

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+1 
+ 
+Spinel Cu-Mn-Cr Oxide Nanoparticle-Pigmented 
+Solar Selective Coatings Maintaining >94% 
+Efficiency at 750ºC 
+Can Xu, Xiaoxin Wang, and Jifeng Liu* 
+Thayer School of Engineering, Dartmouth College, 14 Engineering Drive, Hanover, New 
+Hampshire 03755, USA 
+*Corresponding Author: [email protected] 
+ABSTRACT  
+High-temperature concentrating solar power (CSP) system is capable of harvesting and storing 
+solar energy as heat towards cost-effective dispatchable solar electricity. Solar selective coating is 
+a critical component to boost its efficiency by maximizing solar absorptance and minimizing 
+thermal emittance losses. However, maintaining a high solar-thermal conversion efficiency >90% 
+for long-term operation at ≥750ºC remains a significant challenge. Herein, we report spray-coated 
+spinel Cu-Mn-Cr oxide nanoparticle-pigmented solar selective coatings on Inconel tube sections 
+maintaining ≥94% efficiency at 750ºC and ≥92.5% at 800ºC under 1000x solar concentration after 
+60 simulated day-night thermal cycles in air, each cycle comprising 12h at 750ºC/800ºC and 12h 
+cooling to 25ºC. The solar spectral selectivity is intrinsic to the band-to-band and d-d transitions 
+
+2 
+ 
+of non-stoichiometric spinel Cu-Mn-Cr oxide nanoparticles by balancing the lattice site inversion 
+of Cu2+ and Mn3+ on tetrahedral vs. octahedral sites. This feature offers a large fabrication tolerance 
+in nanoparticle volume fraction and coating thickness, facilitating low-cost and scalable spray-
+coated high-efficiency solar selective absorbers for high-temperature CSP systems.  
+Key Words: Concentrating solar power, Solar selective absorber, Spinel oxide nanoparticle; Ionic 
+site inversion; d-d transition,   
+TOC GRAPHICS 
+ 
+ 
+ 
+
+14um
+After60day-night cycles
+13
+11
+between750Cand252C
+9
+6
+UV-vis)
+500μm
+PigmentedNP
+Siliconeresin
+Coating
+μm
+um
+Substra.e
+400
+500
+300
+400
+300
+200 100
+200
+100
+Cu-Mn-Cr Oxide NPs, 750C Cycling
+%
+100
+95
+SpinelCu-Mn-Croxidepigmentnanoparticles(NPs)
+90
+---SolarAbsorptance
+85
+.-ThermalEmittance
+80
+-Thermal Efficiency
+75
+@750, 1000x solar
+(111)
+70
+concentration
+d=0.481nm
+65
+60
+55
+0
+20
+40
+60
+50nm
+nm
+# of simulated day/nights3 
+ 
+1. Introduction 
+Recent years have seen a rapid growth in solar energy, currently supplying 2.8% of electricity in 
+the U.S. as the third largest renewable source after wind and hydropower. 1 Concentrating solar 
+power (CSP) system utilizes reflective mirrors (“collectors”) to concentrate solar irradiation and 
+heat up working fluids (e.g. molten salts) to high temperatures. 2 A great advantage compared to 
+photovoltaic (PV) system is that CSP is capable of storing the solar-thermal energy for >10 hours 
+so as to meet the peak hours of electricity consumption towards dispatchable solar electricity. 3 
+Solar selective coating, a critical component to boost the solar-to-thermal energy conversion 
+efficiency (ηtherm) by maximizing solar spectral absorption and minimizing infrared (IR) thermal 
+emittance losses, can reduce the levelized cost of energy (LCOE) of CSP by >12% 4 for ηtherm>90%. 
+Further increasing ηtherm to 95% is projected to achieve >17% LCOE cost reduction, which strongly 
+supports the goal of achieving $0.05/kWh solar electricity by 2030. 3 Based on Carnot Theorem, 
+the operation temperature of Generation 3 CSP system is being increased to 750ºC with a solar 
+concentration of ~1000x to achieve >50% overall power cycle efficiency in solar electricity 
+production. 5 Therefore, it is highly desirable to develop cost-effective and highly scalable solar 
+selective coatings that can maintain ηtherm ~95% at 750ºC in air. 
+However, currently commercial solar coating products are unable to maintain ηtherm >90% at 
+operating temperatures >700ºC in air. 6 The benchmark Pyromark 2500 coating suffers from a high 
+thermal emittance of εtherm~87% and pigment particle phase instability at 750°C, 7 limiting its ηtherm 
+to ~88% at 1000x solar concentration after operating at 750°C for 300 h. 8,9 Various transition 
+metal oxide solar absorbers have been investigated, and some of them demonstrate excellent 
+thermal stability at 750°C, 10,11 yet the lack of spectral selectivity limits their maximal ηtherm to 
+~91% at >700°C. While dielectric selective absorber coatings based on nitrides and oxides could 
+
+4 
+ 
+sustain high temperature in air and achieve some degree of solar selectivity, 12,13 the principles of 
+their optical design requires relatively expensive vacuum deposition for stringent thickness control.  
+Our previous optical design 14 based on Lorentz-Mie scattering theory and Four-Flux model has 
+found it feasible to achieve good solar selectivity in nanoparticle (NP)-pigmented silicone coatings 
+for NPs with diameters <40 nm and steep optical transition near the optimal cut-off wavelength 
+(λcut) between the solar spectrum and the thermal radiation spectrum. For Generation 3 CSP 
+systems, λcut=2475 nm for 1000x solar concentration at 750ºC. Spinel (AB2O4) NPs with intrinsic 
+high-temperature thermal stability is a promising candidate since their manifold compositions and 
+cationic valences enable a high degree of freedom to tailor the optical properties. Previously, we 
+have demonstrated MnFe2O4 NP-pigmented solar selective coatings maintaining ηtherm ~89% 
+under 1000x solar concentration after serving at 750ºC in air for 700h. 15 To further enhance the 
+performance, in this Paper we demonstrate spinel Cu-Mn-Cr oxide NP-pigmented silicone solar 
+selective coatings on Inconel tube sections with a higher solar absorbance αsolar=98.2% and a 
+notably reduced thermal emittance εtherm=59.4% compared to benchmark Pyromark 2500 
+(αsolar=96.0%; εtherm=89.5%). To the best of our knowledge, this performance leads to a record-
+high ηtherm=94.50.2% for 1000x solar concentration at 750ºC in air. These coatings also maintain 
+ηtherm≥94% at 750ºC and ηtherm ≥92.5% at 800ºC after 60 simulated day (750ºC/800 ºC 12h) and 
+night (25 ºC 12h) cycles in air without degradation in surface morphology or phase stability. The 
+solar spectral selectivity is intrinsic to the band-to-band and d-d transitions of non-stoichiometric 
+spinel Cu-Mn-Cr oxide NPs. This feature offers a large fabrication tolerance in nanoparticle 
+volume fraction and coating thickness, greatly facilitating high-efficiency solar selective absorbers 
+layers via low-cost and highly scalable spray coating for high-temperature CSP systems.  
+
+5 
+ 
+2. Results and Discussions 
+ 
+Figure 1. (a) XRD pattern of the synthesized spinel Cu-Mn-Cr oxide NPs. A small amount of 
+Mn2O3 is also identified. (b) shows a TEM image of the NPs, and (c) zooms into one of the NPs 
+in the red box shown in (b). 
+Structural and Compositional Analyses of Spinel Cu-Mn-Cr Oxide NPs. We utilized co-
+precipitation method to synthesize spinel oxide NPs with Cu:Mn:Cr=1:3:1 from  nitrate salt 
+precursors (see Supporting Information Section 1). Such a nonstoichiometric ratio is selected to 
+optimize the solar selective absorption by engineering the valences of cations and cationic site 
+distribution to our advantage, as will be discussed later. These NPs are further annealed at 550°C 
+to enhance the crystallinity. Energy dispersive X-ray spectroscopy (EDS) analysis shows 
+Cu:Mn:Cr =1.00:3.33:1.17 in the synthesized NPs, close to the targeted ratio. Figure 1a shows the 
+X-ray diffraction (XRD) pattern of the synthesized NPs loaded on Si(100) substrate. Most of the 
+peaks are attributed to spinel structure with a lattice constant of  a=8.307 Å, between those of cubic 
+CuMn2O4 (a=8.331 Å) and CuCr2O4 (a=8.270 Å) 16 as expected for spinel Cu-Mn-Cr oxide. A 
+couple of small peaks from Mn2O3 are also observed, consistent with the phase diagram reported 
+for Cu-Mn spinel oxides with Cu:Mn <1. 17 Based on XRD relative intensity ratio (RIR) analysis, 
+the molar ratio of Cu-Mn-Cr spinel oxide to Mn2O3 is 1:(0.142 0.018). Figures 1b and 1c further 
+
+(a)
+Spinel
+(b)
+(c)
+(311)
+Mn203
+Intensity
+si (400)
+(511)
+(440)
+(111)
+(220)
+(400)
+(422)
+(533)
+d=0.481nm
+(111)
+(222)
+50nm
+nm
+20
+30
+40
+50
+60
+70
+80
+2 theta (degree)6 
+ 
+show transmission electron microscopy (TEM) images of the NPs. The average NP diameter is 
+313.6 nm (see the NP size histogram in Supporting Information Figure S1). The interplanar 
+spacing of {111} planes is 4.81 Å in the high-resolution TEM image in Figure 1c, fully consistent 
+with the XRD results. 
+Furthermore, X-ray photoelectron spectroscopy (XPS) analyses provide the valences and the 
+corresponding percentages of the cations, as summarized in Table 1 (see Supporting Information 
+Section 3 for data analyses).  Notably, the Cu+:Cu2+ ratio is as high as 4:1. It has been reported that 
+Cu+ on tetrahedral A sites in spinel oxides can be stabilized by Cu2+ and Mn4+ on octahedral B 
+sites 17.  According to ligand field theory (LFT) and the octahedral site preference energy (OSPE) 
+18,19 , Cu+(3d10) and Mn2+(3d5) prefer tetrahedral sites while Cu2+, Mn3+ and Cr3+/Mn4+ are 
+sequentially more energetically favored to take the octahedral sites. Therefore, considering the 
+cation valance distribution, OSPE, and overall charge balance, the detailed formula can be written 
+as (Cu+0.48Mn2+0.52)Td,A(Cu2+0.12Mn2+0.07Mn3+0.65Mn4+0.46Cr3+0.70)oh,BO3.89 assuming OSPE fully 
+applies. Here, the subscripts “Td,A” and “Oh,B” stand for tetrahedral A site and octahedral B site, 
+respectively. Characteristic vibration modes corresponding to octahedral and tetrahedral sites in 
+spinel structures are also observed in Fourier transform IR spectroscopy (FTIR) and detailed in the 
+Supporting Information (Figure S3).  On the other hand, even though OSPE is highly effective in 
+predicting cation lattice sites, we note that ~30% site inversion of Mn3+ from octahedral to 
+tetrahedral sites has been reported in closely related CuMn2O4 spinel structures in order to dilute 
+the Jahn-Teller effect. 20 Such Mn3+ site inversion can lead to strong and broad absorption bands 
+in the near infrared (NIR) regimes at 1000-2000 nm wavelength. 21,22 Furthermore, the deficiency 
+of oxygen compared to stoichiometric AB2O4 indicates oxygen vacancies, likely on the surface of 
+the NPs as has been reported in other spinel NP systems. 23 Oxygen vacancies are known to interact 
+
+7 
+ 
+with transition metal cations and further enhanced the NIR absorption. 24 These influences on 
+optical properties will be discussed next.  
+Table 1. Cu:Mn:Cr atomic ratios and corresponding cationic valences/percentages in the synthesized NPs 
+comprising both spinel Cu-Mn-Cr oxide and Mn2O3 at a ratio of 1:0.142  
+Atomic Species 
+Atomic Ratio  
+Ionic Valence/Molar Percentage 
+Cu 
+1.00 
+Cu+: 80% 
+Cu2+: 20% 
+Mn 
+3.33 
+Mn2+: 30% 
+Mn3+: 47% 
+Mn4+: 23% 
+Cr 
+1.17 
+Cr3+:100% 
+       
+ 
+ 
+Figure 2. (a) Indirect bandgap Tauc plot of the spinel Cu-Mn-Cr oxide NPs (b) Absorption 
+spectrum beyond the indirect bandgap and the corresponding Gaussian peaks fitting. The two 
+
+(a)
+(b)
+103
+x104
+Indirect gap absorption subtracted
+IndirectBandgapTaucPlot
+Overall fitting
+Individual Gaussian peakfitting
+8.
+1.0
+α(cm")
+6
+0.95eV
+4
+1.95,eV
+0.5
+2
+0.0-
+fo
+0.51.01.52.02.53.03.5
+4.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Photon Energy (eV)
+Photon Energy (eV)
+(c)
+(d)
+SolarAbsorptanceContours
+Thermal EfficiencyContours
+15
+15
+αsolar (%)
+Thickness (um)
+Ntherm (%)
+92.83
+94
+92
+95
+93.12
+90
+98.67
+.34
+88
+10
+10
+90
+86
+93.41
+9o.
+Coating
+28
+84
+93.70
+98.02
+85
+7397.
+93.99
+82
+97.70
+5
+5
+5
+10
+15
+5
+10
+15
+NPVolumeConcentration(%)
+NPVolumeConcentration(%)8 
+ 
+absorption bands peaked at 0.95 eV and 1.95 eV are mainly contributed by Cu2+ and Mn3+ d-d 
+transitions on tetrahedral and octahedral sites, respectively. (c) and (d) show theoretical modeling 
+of solar absorptance and thermal efficiency contour maps as a function of NP volume fraction and 
+coating thickness when dispersed in silicone matrix.  
+Optical Properties. Critical to the optical performance is the absorption spectrum of Cu-Mn-Cr 
+oxide NPs, as shown in the Tauc plot in Figure 2a. It reveals an indirect gap of 1.68 0.04 eV. 
+Remarkably, the absorption beyond the indirect gap is extended all the way to 0.5 eV with 
+absorption coefficients >104 cm-1 to cover the entire solar spectrum. To single out the absorption 
+spectrum beyond the indirect bandgap, we subtract the indirect bandgap absorption (as derived 
+from the Tauc plot) from Figure 2a and show the result in Figure 2b. Two broad Gaussian 
+absorption bands peaked at 0.95 and 1.95 eV are clearly identified, typical of d-d transitions 
+between the split d levels of transition metal ions induced by tetrahedral or octahedral ligand 
+(crystal) field. 21 The energy ratio of these two peaks is 0.48, very close to the expected ratio of 
+4/9 for tetrahedral site vs. octahedral site d-d transition energies in spinel structures. 25 Therefore, 
+the 0.95 eV and 1.95 eV absorption bands are attributed to tetrahedral and octahedral site d-d 
+transitions, respectively. Note that in most of these d-d transitions, the excited electrons still remain 
+localized to the transition metal ion instead of becoming free electrons in the conduction band, 
+therefore the transition energy is lower than the bandgap. The 0.95 eV peak is between the 
+tetrahedral site Cu2+ absorption band peaked at ~0.8 eV 25–27 and that of Mn3+ peaked at ~1.2 eV, 
+21,22 suggesting both types of ions on tetrahedral sites have contributed to this NIR d-d absorption 
+band. Typically, the oscillation strength of tetrahedral d-d transition is stronger than their 
+octahedral counterparts due to broken inversion symmetry that enables various spin-forbidden 
+transitions.21,25 In our case, on the other hand, the octahedral d-d absorption at 1.95 eV is ~2x 
+
+9 
+ 
+stronger than tetrahedral absorption at 0.95 eV. This result indicates that a relatively small fraction 
+of Cu2+ and Mn3+ cations occupy tetrahedral sites compared to octahedral sites, consistent with the 
+prediction of OSPE discussed earlier and similar to the case of CuMn2O4 in terms of a small 
+fraction of site inversion.20  Such a distribution is beneficial to solar selectivity, where a decrease 
+in absorption beyond the optimal cut-off wavelength of 2475 nm (hv<0.5 eV) is needed, as 
+mentioned earlier. Since Cr3+ and Mn4+ have the strongest tendency to compete for octahedral sites 
+based on OSPE, tuning their percentages may further optimize Cu2+ and Mn3+ site inversion for 
+better solar selectivity. In addition, oxygen vacancies, as identified in our previous analyses, 
+further lower the ligand symmetry to enhance the oscillation strength of d-d optical absorption 
+especially in the NIR solar spectral regime at 0.5-1 eV, as has been reported in spinel ZnFe2O4 
+system.28 Therefore, these two factors work synergistically to enhance the overall solar selectivity.  
+Optical Modeling of Spinel Cu-Mn-Cr Oxide NP-Pigmented Solar Selective Coatings. Based 
+on the absorption spectra of the spinel Cu-Mn-Cr oxide NPs, we further modelled the spectrally 
+integrated solar absorptance αSolar and the thermal efficiency ηtherm for 1000x solar concentration 
+at 750ºC using Lorentz-Mie scattering theory and four-flux radiative model, as detailed in Refs. 
+14 and 15. Figure 2c and d show αSolar and ηtherm contour maps, respectively, as a function of NP 
+volume fraction and coating thickness when dispersed in silicone matrix for the solar absorber 
+coating. The intrinsic solar spectral selectivity of the NPs and their nanoscale diameters (d=313.6 
+nm) lead to a large tolerance in coating thickness and NP volume fraction, such that αSolar>97.4% 
+and ηtherm>93.6% can be achieved even if the NP volume fraction varies between 7-13 vol.% and 
+the coating thickness varies between 6-15 μm. Such a high fabrication tolerance greatly facilitates 
+low-cost and highly scalable spray-coated solar selective absorbers. The green stars and red error 
+bars in Figures 2c and d reflect the experimentally measured coating thickness and NP volume 
+
+10 
+ 
+fraction variations in our spray-coated Inconel tube section samples. From the modeling, we expect 
+αSolar98% and ηtherm94%.As will be detailed next, these theoretically modelled values indeed 
+agree very well with the experimental results. 
+Characterization and Optical Performance of the Spray-Coated Solar Selective Coating. 
+Figure 3a shows a photograph of an Inconel 625 tube section (outer diameter=76 mm) coated with 
+the spinel Cu-Mn-Cr oxide NP-pigmented silicone solar selective absorber and annealed in air at 
+750ºC for 24h. Detailed coating procedure is discussed in Section 1 of the Supporting Information. 
+Figure 3b shows a digital optical microscopy image of the surface profile, indicating a surface 
+undulation of ~6 μm with sporadic humps reaching >10 μm in height. A focus ion beam (FIB) 
+cross-sectional cutting is made on a relatively flat and thin region, revealing an average thickness 
+of 5.5 μm as shown in Figure 3c. Using this flat region as a calibration and the surface profile in 
+Figure 3b, we determined that overall, the coating thickness is 8.5 3 μm. The volume fraction of 
+the spinel Cu-Mn-Cr oxide NPs is estimated to be 10 3 vol.%, as detailed in Section 5 of the 
+Supporting Information. These parameters allow us to model the expected performance of the 
+coating, as shown in Figures 2c and 2d. The XRD data in Figure 3d shows that the spinel structure 
+is well maintained and the crystallinity gets better (narrower diffraction peaks) after 750ºC 
+annealing for 24h in air.  
+Remarkably, Figure 3e and f demonstrate that the new spinel Cu-Mn-Cr oxide NP-pigmented 
+solar selective coating improves the solar absorptance αSolar from 96.0% to 98.2% compared to 
+benchmark Pyromark 2500, while simultaneously the thermal emittance εtherm is drastically 
+reduced from 89.5% to 59.4%. Correspondingly, ηtherm is notably increased from 90.40.3% to 
+94.50.2%. As will be further discussed in Table 2, to the best of our knowledge, so far this is the 
+
+11 
+ 
+highest optical-to-thermal conversion efficiency for air-stable solar selective coatings operating at 
+750ºC. 
+ 
+Figure 3. (a) A photograph of an Inconel 625 tube section (outer diameter=76 mm) coated with 
+spinel Cu-Mn-Cr oxide NP-pigmented silicone solar selective absorber layer. (b) An optical 
+topography map of the solar selective coating, including a 3D view and a top view in the inset. (c) 
+FIB cross-section of the coating in a relatively flat and thin region in (b) for thickness calibration. 
+(d) XRD pattern of the coated sample after annealing for 24 h at 750ºC in air compared to that of 
+the as-synthesized particles. (e) and (f) show solar absorptance and thermal emittance spectra of 
+the Cu-Mn-Cr oxide NP-pigmented silicone solar selective coating compared to benchmark 
+Pyromark 2500.  
+ 
+Endurance Testing. We further performed extensive thermal cycling for endurance testing of 
+these high efficiency solar selective coatings. Each simulated day/night thermal cycle includes 12h 
+annealing at 750ºC or 800ºC and 12h cooling to 25ºC. Here thermal cycles at 800ºC (i.e., 50ºC 
+
+(a)
+(b)
+(c)
+11
+Pt
+6
+Solar coating
+4
+200μm
+Cr,
+InconelSubstrate
+10cm
+300
+400
+500
+400
+300200100
+200
+10μm
+100
+(d)
+(e)
+(f) 100
+750°C24h
+Spinel
+100
+as-synthesized
+Mn203
+(%)
+(311)
+Inconel625
+80
+Pyromark2500
+80
+Thermal
+larAbsorptance
+Cu-Mn-Cr oxide
+Emittance
+EmittanceLoss
+Intensity
+(220)
+60
+100
+60
++（111)
+(222)
+(440)
+(422)
+98
+小
+15
+SolarAbsorptance
+40
+96
+40
+Thermal
+(400)
+20
+92
+20
+Pyromark2500
+Sol
+Cu-Mn-Croxide
+901
+500
+1000
+1500
+2000
+2500
+20
+30
+40
+50
+0
++0
+60
+500
+1000
+1500
+2000
+2500
+3000
+6000
+9000
+12000
+2 theta (degree)
+Wavelength(nm)
+Wavelength(nm)12 
+ 
+higher than the working temperature) are conducted to further confirm thermal stability and to 
+investigate possible degradation mechanisms at an accelerated rate. Solar coatings on Inconel 625 
+tube sections (with outer diameter=76 mm as shown in Figure 3) are annealed for up to 60 
+simulated day/night cycles in air. Comparing the surface morphology from scanning electron 
+microscopy (SEM) images shown in Figures 4a-c, we find no deterioration in coating integrity or 
+appreciable changes in morphology after 60 cycles between 750ºC/800ºC and 25ºC. The 
+micropores on the surface of the samples help to accommodate volume changes upon thermal 
+cycling, thereby stabilizing the coating against thermal stress. Such surface texture also helps to 
+reduce surface reflectance and enhance solar absorption, similar to the case of PV cells. XRD 
+analyses further show that the spinel Cu-Mn-Cr oxide NPs are thermodynamically stable upon 
+thermal cycling (see Section 6 of the Supporting Information). Figures 4d-f show the evolution of 
+solar absorptance spectra, thermal emittance spectra, and spectrally integrated solar 
+absorptance/thermal emittance/thermal efficiency vs. the number of thermal cycles between 750ºC 
+and 25ºC. The solar absorptance decreases only very slightly after 60 thermal cycles between 
+750ºC and 25ºC, while the thermal emittance fluctuates around 60% during the cycling, 
+maintaining a record-high thermal efficiency ηtherm>94% during the entire thermal cycles. Similar 
+data shown in Figures 4g-i indicate a more noticeable decrease in solar absorptance when the high 
+temperature cycles are increased to 800ºC. Even so, a high ηtherm=92.80.3% is still maintained 
+after 60 thermal cycles between 800ºC and 25ºC. The increasingly wavy solar absorption spectra 
+upon 800ºC/25ºC thermal cycling with reduced solar absorptance closely resemble the behavior 
+of CuCr2O4 NP-pigmented coatings (intentionally synthesized for comparison; see Section 7 of 
+Supporting Information) as well as previous literature on CuCr2O4 due to Cr3+ d-d absorption bands. 
+29 In fact, these peaks and valleys blueshift towards those of CuCr2O4 after more cycles. This result 
+
+13 
+ 
+suggests Cr diffusion and substitution into the Cu-Mn-Cr spinel oxide NPs from the Inconel 
+substrate as the key mechanism for the slight solar absorptance degradation upon 800ºC/25ºC 
+thermal cycling. This is indeed confirmed by detailed cross-sectional EDS mapping before and 
+after the thermal cycling, as detailed in Section 7 of the Supporting Information. Therefore, 
+limiting Cr diffusion into the coating could further improve the endurance of the solar selective 
+coating. A possible approach is to pre-oxidize the Inconel substrate to form a Cr2O3 layer first 
+before spray-coating, which has proved to be an effective approach to address the Cr diffusion 
+issue in our previous work. 15 
+ 
+Figure 4. SEM surface morphology of the coatings on Inconel 625 tube sections (76 mm outer 
+diameter) after (a) 750ºC 24h annealing; (b) 750ºC 24h annealing plus 60 simulated day-night 
+
+750°c.24h
+500um
+After60 cyclesbetween
+500μm
+After60cycles between
+500um
+750°C（12h）/25°C(12h）
+800°c（12h)/25°c（12h）
+Opticalperformanceafter750°C/25°cThermal Cycling
+(d)
+[e)100
+100
+24h
+Absorptance (%)
+80
+Cycle10
+80
+90.
+cycle 30
+(%) 
+SolarAbsorptance
+cycle60
+Thermal Emittance
+60
+66
+Emittance
+60
+80.
+Thermal Efficiency
+86
+40
+40
+70.
+97
+24h
+20
+96
+20
+cycle10
+60
+cycle30
+95
+500
+1000
+1500
+2000
+cycle60
+2500
+0.
+0.
+50
+500
+1000
+1500
+2000
+2500
+3000
+6000
+9000
+12000
+10
+20
+30
+40
+50
+60
+Wavelength (nm)
+Wavelength (nm)
+#of Simulated Day-Night Cycles
+Optical performance after 800°C/25°C Thermal Cycling
+(g) 100
+(h) 100
+(0) 100
+%
+24h
+Absorptance (%)
+80
+cycle 10
+80
+90-
+cycle30
+(%) 
+Solar Absorptance
+cycle 60
+Thermal Emittance
+60
+Emittance
+100
+60
+80.
+Thermal Efficiency
+98
+40
+40
+70 
+96
+24h
+20
+94
+20
+cycle10
+60-
+cycle 30
+92
+500
+1000
+1500
+2000
+2500
+cycle60
+0-
+50.
+500
+1000
+1500
+2000
+2500
+3000
+6000
+9000
+12000
+0
+10
+20
+30
+40
+50
+60
+Wavelength(nm)
+Wavelength (nm)
+#of Simulated Day-Night Cycles14 
+ 
+cycles between 750 ºC and 25 ºC; (c) 750ºC 24h annealing plus 60 simulated day-night cycles 
+between 800 ºC and 25 ºC. (d)-(f) show the evolution of solar absorptance spectra, thermal 
+emittance spectra, and spectrally integrated solar absorptance/thermal emittance/thermal 
+efficiency vs. the number of thermal cycles between 750ºC and 25ºC. (g)-(i) show similar data for 
+thermal cycles between 800ºC and 25ºC 
+Table 2 compares the efficiency and endurance of our coating with some recent work. Most of the 
+previous solar coatings lack spectral selectivity with a thermal emittance ~90%, limiting their 
+thermal efficiency to ηtherm~90.5%. We have notably improved ηtherm to >94% by engineering and 
+balancing the intrinsic NIR vs. visible d-d absorption bands of Cu2+ and Mn3+ on tetrahedral vs. 
+octahedral sites of spinel structure. The thermal emittance is drastically reduced to ~60% while a 
+high solar absorptance of ~98% is maintained.  To the best of our knowledge, this is the highest 
+efficiency demonstrated and maintained so far for 750ºC endurance testing in air. Optimizing the 
+extent of lattice site inversion of Cu2+ and Mn3+ on tetrahedral vs. octahedral sites by fine-tuning 
+Cr3+ and Mn4+ percentages may further improve the spectral selectivity towards ηtherm>95%. The 
+intrinsic solar spectral selectivity of the spinel Cu-Mn-Cr oxide NPs also enables an excellent 
+fabrication margin for cost-effective, highly scalable spray coating, as preliminary demonstrated 
+on a 48-inch-long tube shown in Figure S8 of the Supporting Information. 
+Table 2 High-temperature solar selective coatings with reported endurance test 
+Material System 
+Substrate 
+Fabrication Method 
+ηstart 
+ηend 
+T (℃) 
+Endurance in 
+Air (h/℃) 
+Refs. 
+Cu0.15Co2.84O4-SPB-SiO2 
+Inconel 625 
+Spray Coating 
+0.904 
+0.903 
+750 
+1000/750 
+Ref.30 
+Cu1.5Mn1.5O4-SPB-SiO2 
+Inconel 625 
+Spray Coating 
+0.909 
+0.905 
+750 
+1000/750 
+Ref.30 
+Porous Cu0.5Cr1.1Mn1.4O4-SiO2 
+Haynes 230 
+Spray Coating 
+0.903 
+0.902 
+800 
+2000/800 
+Ref.11 
+
+15 
+ 
+Cu0.86Cr0.14Mn1.5Fe0.5O4-SiO2 
+Inconel 617 
+Spray Coating 
+≤ 0.917 
+≤ 0.894 
+750 
+1300/800 
+Ref.31 
+TiN/AlCrSiO(two nano-multilayers) 
+/AlCrSiO(amorphous) 
+SS 
+Cathode Arc Ion 
+Plating 
+≤ 0.908 
+≤ 0.867 
+750 
+200/700 
+Ref.32 
+Spinel Cu-Mn-Cr oxide NP-silicone 
+Inconel 625 
+Spray Coating 
+0.945 
+0.942 
+750 
+60 thermal 
+cycles 
+750ºC/25 ºC 
+This 
+work- 
+0.937 
+0.928 
+800 
+60 thermal 
+cycles 
+800ºC/25 ºC 
+This 
+work- 
+ 
+ηstart: efficiency as deposited; ηend: efficiency after annealing;  
+T: temperature at which thermal efficiency is evaluated. 
+Note that Refs. 31 and 32 only reported thermal emittance at 80ºC instead of high 
+temperatures >700ºC. We therefore estimated the upper limit of the thermal efficiency at high 
+temperatures in these cases using 80ºC thermal emittance values, considering thermal emittance 
+typically increases at higher temperatures. 
+3. Conclusions 
+In conclusion, we demonstrate spray-coated spinel Cu-Mn-Cr oxide NP-pigmented solar 
+selective coating that maintains ηtherm >94% upon 60 simulated day-night cycles between 750ºC 
+and 25ºC in air. The spectral selectivity is intrinsic to the band-to-band and d-d transitions of 
+these non-stoichiometric spinel NPs, where Cu2+ and Mn3+ on tetrahedral sites (through spinel 
+site inversion) contribute to the NIR absorption band to cover the entire solar spectrum up to 
+2500 nm wavelength. This feature offers a large fabrication tolerance in NP volume fraction and 
+coating thickness, greatly facilitating high-efficiency, high-temperature solar selective absorbers 
+layers via low-cost and highly scalable spray coating for Generation 3 high-temperature CSP 
+systems.  
+
+16 
+ 
+ASSOCIATED CONTENT 
+Supporting Information includes experimental methods, nanoparticle size (diameter) histogram, 
+XPS data analyses, FTIR data analyses, volume fraction determination, XRD analyses during 
+thermal endurance tests, optical spectra and EDS mapping for interdiffusion investigation, and 
+solar selective coating on a 48-inch-long tube. 
+AUTHOR INFORMATION 
+Corresponding Author 
+Jifeng Liu − Thayer School of Engineering, Dartmouth College, 14 Engineering Drive, Hanover, 
+New Hampshire 03755, United States; Email: [email protected] 
+ORCID: 0000-0003-4379-2928 
+Authors 
+Can Xu − Thayer School of Engineering, Dartmouth College, 14 Engineering Drive, Hanover, 
+New Hampshire 03755, United States; https://orcid.org/0000-0001-5306-5367; 
+Xiaoxin Wang − Thayer School of Engineering, Dartmouth College, 14 Engineering Drive, 
+Hanover, New Hampshire 03755, United States; 
+Notes 
+The authors declare no competing financial interest. 
+ 
+ACKNOWLEDGMENTS 
+This project is funded by U.S. Department of Energy, Solar Energy Technologies Office, under 
+the award number DE-EE-0008530. We would like to thank Dr. Maxime J. Guinel at Dartmouth 
+
+17 
+ 
+College and Dr. Jules Gardener at Harvard University for their support with electron microscopy 
+analyses, and Dr. Min Li at Yale University for support with XPS.   
+REFERENCES 
+(1)  
+EIA. What is U.S. electricity generation by energy source? 
+https://www.eia.gov/tools/faqs/faq.php?id=427&t=3 (accessed 2022 -04 -13). 
+(2)  
+Jacobson, M. Z.; Delucchi, M. A. Providing All Global Energy with Wind, Water, and 
+Solar Power, Part I: Technologies, Energy Resources, Quantities and Areas of 
+Infrastructure, and Materials. Energy Policy 2011, 39 (3), 1154–1169. 
+https://doi.org/10.1016/j.enpol.2010.11.040. 
+(3)  
+Murphy, C.; Sun, Y.; Cole, W.; Maclaurin, G.; Turchi, C.; Mehos, M.; Murphy, C.; Sun, 
+Y.; Cole, W.; Maclaurin, G.; Turchi, C.; Mehos, M. The Potential Role of Concentrating 
+Solar Power within the Context of DOE’s 2030 Solar Cost Targets; Golden, Colorado, 
+2019. 
+(4)  
+Boubault, A.; Ho, C. K.; Hall, A.; Lambert, T. N.; Ambrosini, A. Levelized Cost of 
+Energy (LCOE) Metric to Characterize Solar Absorber Coatings for the CSP Industry. 
+Renew. Energy 2016, 85, 472–483. https://doi.org/10.1016/j.renene.2015.06.059. 
+(5)  
+Mehos, M.; Turchi, C.; Vidal, J.; Wagner, M.; Ma, Z.; Ho, C.; Kolb, W.; Andraka, C.; 
+Kruizenga, A. Concentrating Solar Power Gen3 Demonstration Roadmap; Golden, 
+Colorado, 2017. https://doi.org/10.2172/1338899. 
+
+18 
+ 
+(6)  
+Suman, S.; Khan, M. K.; Pathak, M. Performance Enhancement of Solar Collectors - A 
+Review. Renew. Sustain. Energy Rev. 2015, 49, 192–210. 
+https://doi.org/10.1016/j.rser.2015.04.087. 
+(7)  
+Ho, C. K.; Ambrosini, A.; Bencomo, M.; Hall, A. C.; Lambert, T. N.; Mahoney, A. R. 
+Characterization of Pyromark 2500 for High-Temperature Solar Receivers. In the ASME 
+2012 6th International Conference on Energy Sustainability; San Diego, CA, 2012. 
+(8)  
+Ambrosini, A. High‐Temperature Solar Selective Coating Development for Power Tower 
+Receivers https://www.energy.gov/sites/prod/files/2016/08/f33/R 5 - Ambrosini_CSP-
+Program-Summit-SSC Presentation_to DOE.pdf. 
+(9)  
+Ambrosini, A.; Lambert, T. N.; Boubault, A.; Hunt, A.; Davis, D. J.; Adams, D.; Hall, A. 
+C. Thermal Stability of Oxide-Based Solar Selective Coatings for CSP Central Receivers. 
+In Proceedings of the ASME 2015 9th International Conference on Energy Sustainability; 
+San Diego, California, 2015; pp 1–10. https://doi.org/10.1115/ES2015-49706. 
+(10)  Moon, J.; Kyoung Kim, T.; VanSaders, B.; Choi, C.; Liu, Z.; Jin, S.; Chen, R. Black 
+Oxide Nanoparticles as Durable Solar Absorbing Material for High-Temperature 
+Concentrating Solar Power System. Sol. Energy Mater. Sol. Cells 2015, 134, 417–424. 
+https://doi.org/10.1016/j.solmat.2014.12.004. 
+(11)  Rubin, E. B.; Chen, Y.; Chen, R. Optical Properties and Thermal Stability of Cu Spinel 
+Oxide Nanoparticle Solar Absorber Coatings. Sol. Energy Mater. Sol. Cells 2019, 195 
+(January), 81–88. https://doi.org/10.1016/j.solmat.2019.02.032. 
+
+19 
+ 
+(12)  Cao, F.; McEnaney, K.; Chen, G.; Ren, Z. A Review of Cermet-Based Spectrally 
+Selective Solar Absorbers. Energy Environ. Sci. 2014, 7 (5), 1615–1627. 
+https://doi.org/10.1039/c3ee43825b. 
+(13)  Ambrosini, A. High-Temperature Solar Selective Coating Development for Power Tower 
+Receivers 
+http://energy.gov/sites/prod/files/2014/01/f7/csp_review_meeting_042413_ambrosini.pdf. 
+(14)  Wang, X.; Yu, X.; Fu, S.; Lee, E.; Kekalo, K.; Liu, J. Design and Optimization of 
+Nanoparticle-Pigmented Solar Selective Absorber Coatings for High-Temperature 
+Concentrating Solar Thermal Systems. J. Appl. Phys. 2018, 123 (3). 
+https://doi.org/10.1063/1.5009252. 
+(15)  Wang, X.; Lee, E.; Xu, C.; Liu, J. High-Efficiency, Air-Stable Manganese–Iron Oxide 
+Nanoparticle-Pigmented Solar Selective Absorber Coatings toward Concentrating Solar 
+Power Systems Operating at 750 °C. Mater. Today Energy 2021, 19. 
+https://doi.org/10.1016/j.mtener.2020.100609. 
+(16)  Brik, M. G.; Suchocki, A.; Kamińska, A. Lattice Parameters and Stability of the Spinel 
+Compounds in Relation to the Ionic Radii and Electronegativities of Constituting 
+Chemical Elements. Inorg. Chem. 2014, 53 (10), 5088–5099. 
+https://doi.org/10.1021/ic500200a. 
+(17)  Vandenberghe, R. E.; Robbrecht, G. G.; Brabers, V. A. M. On the Stability of the Cubic 
+Spinel Structure in the System Cu-Mn-O. Mater. Res. Bull. 1973, 8 (5), 571–579. 
+https://doi.org/10.1016/0025-5408(73)90134-7. 
+
+20 
+ 
+(18)  Liu, Y.; Ying, Y.; Fei, L.; Liu, Y.; Hu, Q.; Zhang, G.; Pang, S. Y.; Lu, W.; Mak, C. L.; 
+Luo, X.; Zhou, L.; Wei, M.; Huang, H. Valence Engineering via Selective Atomic 
+Substitution on Tetrahedral Sites in Spinel Oxide for Highly Enhanced Oxygen Evolution 
+Catalysis. J. Am. Chem. Soc. 2020, 141 (20), 8136–8145. 
+https://doi.org/10.1021/jacs.8b13701. 
+(19)  Navrotsky, A.; Kleppa, O. J. The Thermodynamics of Cation Distributions in Simple 
+Spinels. J. Inorg. Nucl. Chem. 1967, 29 (11), 2701–2714. https://doi.org/10.1016/0022-
+1902(67)80008-3. 
+(20)  Shoemaker, D. P.; Li, J.; Seshadri, R. Unraveling Atomic Positions in an Oxide Spinel 
+with Two Jahn-Teller Ions: Local Structure Investigation of CuMn2O4. J. Am. Chem. Soc. 
+2009, 131 (32), 11450–11457. https://doi.org/10.1021/ja902096h. 
+(21)  Lakshmi, S.; Endo, T.; Siva, G. Electronic (Absorption) Spectra of 3d Transition Metal 
+Complexes. Adv. Asp. Spectrosc. 2012, 3–48. https://doi.org/10.5772/48089. 
+(22)  Bosi, F.; Hålenius, U.; Andreozzi, G. B.; Skogby, H.; Lucchesi, S. Structural Refinement 
+and Crystal Chemistry of Mn-Doped Spinel: A Case for Tetrahedrally Coordinated Mn3+ 
+in an Oxygen-Based Structure. Am. Mineral. 2007, 92 (1), 27–33. 
+https://doi.org/10.2138/am.2007.2266. 
+(23)  Huang, R.; Ikuhara, Y. H.; Mizoguchi, T.; Findlay, S. D.; Kuwabara, A.; Fisher, C. A. J.; 
+Moriwake, H.; Oki, H.; Hirayama, T.; Ikuhara, Y. Oxygen-Vacancy Ordering at Surfaces 
+of Lithium Manganese(III,IV) Oxide Spinel Nanoparticles. Angew. Chemie - Int. Ed. 
+2011, 50 (13), 3053–3057. https://doi.org/10.1002/anie.201004638. 
+
+21 
+ 
+(24)  Eppstein, R.; Caspary Toroker, M. On the Interplay Between Oxygen Vacancies and 
+Small Polarons in Manganese Iron Spinel Oxides. ACS Mater. Au 2022. 
+https://doi.org/10.1021/acsmaterialsau.1c00051. 
+(25)  Le Nestour, A.; Gaudon, M.; Villeneuve, G.; Daturi, M.; Andriessen, R.; Demourgues, A. 
+Defects in Divided Zinc-Copper Aluminate Spinels: Structural Features and Optical 
+Absorption Properties. Inorg. Chem. 2007, 46 (10), 4067–4078. 
+https://doi.org/10.1021/ic0624064. 
+(26)  Buvaneswari, G.; Aswathy, V.; Rajakumari, R. Comparison of Color and Optical 
+Absorbance Properties of Divalent Ion Substituted Cu and Zn Aluminate Spinel Oxides 
+Synthesized by Combustion Method towards Pigment Application. Dye. Pigment. 2015, 
+123, 413–419. https://doi.org/10.1016/j.dyepig.2015.08.024. 
+(27)  F, R. A.; B, F.; S, H.; H, U.; Aldo, B.; Orabona, E.; Bari, I.-; Università, S.; Moro, P. A.; 
+Roma, I.-. Cation Ordering over Short-Range and Long-Range Scales in the MgAl2O4-
+CuAl2O4 Series. 2021, 97 (March), 1821–1827. 
+(28)  Zviagin, V.; Sturm, C.; Esquinazi, P. D.; Grundmann, M.; Schmidt-Grund, R. Control of 
+Magnetic Properties in Spinel ZnFe2O4 Thin Films through Intrinsic Defect 
+Manipulation. J. Appl. Phys. 2020, 128 (16). https://doi.org/10.1063/5.0019712. 
+(29)  Youn, Y.; Miller, J.; Nwe, K.; Hwang, K. J.; Choi, C.; Kim, Y.; Jin, S. Effects of Metal 
+Dopings on CuCr2O4 Pigment for Use in Concentrated Solar Power Solar Selective 
+Coatings. ACS Appl. Energy Mater. 2019, 2 (1), 882–888. 
+https://doi.org/10.1021/acsaem.8b01976. 
+
+22 
+ 
+(30)  Karas, D. E.; Byun, J.; Moon, J.; Jose, C. Copper-Oxide Spinel Absorber Coatings for 
+High-Temperature Concentrated Solar Power Systems. Sol. Energy Mater. Sol. Cells 
+2018, 182 (March), 321–330. https://doi.org/10.1016/j.solmat.2018.03.025. 
+(31)  Noč, L.; Ruiz-Zepeda, F.; Merzel, F.; Jerman, I. High-Temperature “Ion Baseball” for 
+Enhancing Concentrated Solar Power Efficiency. Sol. Energy Mater. Sol. Cells 2019, 200 
+(April). https://doi.org/10.1016/j.solmat.2019.109974. 
+(32)  Yang, D.; Zhao, X.; Liu, Y.; Li, J.; Liu, H.; Hu, X.; Li, Z.; Zhang, J.; Guo, J.; Chen, Y.; 
+Yang, B. Enhanced Thermal Stability of Solar Selective Absorber Based on Nano-
+Multilayered AlCrSiO Films. Sol. Energy Mater. Sol. Cells 2020, 207 (December 2019), 
+110331. https://doi.org/10.1016/j.solmat.2019.110331. 
+ 
+
+1 
+ 
+ 
+Supporting Information 
+Spinel Cu-Mn-Cr Oxide Nanoparticle-Pigmented 
+Solar Selective Coatings Maintaining >94% 
+Efficiency at 750ºC 
+Can Xu, Xiaoxin Wang, and Jifeng Liu* 
+Thayer School of Engineering, Dartmouth College, 14 Engineering Drive, Hanover, New 
+Hampshire 03755, USA 
+*Corresponding Author: [email protected] 
+1. Experimental Methods 
+Synthesis. To obtain Cu, Mn and Cr oxide nanoparticle precursors, copper nitrate 
+(Cu(NO3)2·3H2O, Fisher Scientific, C99.9), manganese nitrate (Mn(NO3)2·4H2O, Sigma-Aldrich, 
+C99.5) and chromium nitrate (Cr(NO3)3·9H2O, Sigma-Aldrich, C99.8), were dissolved in 
+deionized (DI) water at a molar ratio of 1:3:1 at room temperature with an initial pH value between 
+2 and 3. The aqueous solution was under vigorous magnetic stirring for homogeneity and an excess 
+amount of appropriate base solution, sodium hydroxide (NaOH(aq), 50%, Sigma-Aldrich, dilute 
+
+2 
+ 
+to 10M) as we selected, was steadily added dropwise for precipitation until the pH of the solution 
+was adjusted to around 12 for a full precipitation. The stirring was kept for 1h after the 
+precipitation. The precipitated material was rinsed 5 times and then dried at 120 ºC overnight. 
+After a rough grinding step, it was further calcined at 550°C for 5h and finally ground into fine 
+powders. 
+Solar Selective Coating Fabrication. To obtain spraying precursors, 4 wt.% synthesized 
+nanoparticles were well dispersed in xylene diluted silicone resin (BLUESIL RES 6406XM) with 
+a ratio of 1:10 for an appropriate viscosity. The precursors were put in an ultrasonic bath for 30 
+min and then sprayed onto Inconel 625 substrate on the hot plate with a surface temperature of 
+about 180 ºC. The sample was settled on the hotplate for 5 min and then cooled down to room 
+temperature. Subsequently, the sample was put into a muffle furnace (Thermo scientific) for the 
+following heat treatment. The sample was first heated under 250 ºC for 2h and then ramped up to 
+750 ºC at a rate of 9.1C/min. After dwelling for 24h and cooling back to room temperature, the 
+nanoparticle-pigmented silicone solar selective coating was formed.  
+Thermal test. A total of 60 simulated day-night thermal cycles (total annealing time of 720h) were 
+conducted for stability tests at 750 ºC and 800 ºC, respectively. Each separate cycle consisted of 
+10 days. In each day, the sample was heated up to the target temperature at a rate of 9.1C/min and 
+dwelled for 12h. Then the sample was cooled down to room temperature until the start of the next 
+day. 
+Characterization. XRD patterns were recorded on a Rigaku 007 X-ray Diffractometer (Cu Kα1 
+line, λ = 1.54059 Å) operating at 40 kV/300 mA and in a 2theta angular range of 10–90 degree 
+with a velocity of 2 degree/min and a step size of 0.02 degree. Chemical composition was 
+
+3 
+ 
+determined by X-ray Photoelectron Spectroscopy (XPS, PHI VersaProbe II, from West Campus 
+Materials Characterization Core at Yale University) and Energy dispersive spectroscopy (EDS, 
+EDAX Si (Li) detector with Genesis software). Scanning electron images were obtained by using 
+a TESCAN SEM operating at 20 kV in secondary electron (SE) mode and a FEI Helios 5CX 
+DualBeam SEM equipped with FIB was utilized for specimen preparation for cross section views. 
+Tecnai F20 (200 keV) TEM was utilized to collect transmission electron images and SAED 
+patterns. Elemental distribution was measured via JEOL 2010 FEG - TEM/STEM equipped with 
+an EDS detector (from Center for Nanoscale Systems at Harvard University). Vibrational signals 
+of bonding as well as reflectance in the mid infrared (MIR) region (λ=2.5 ~15 μm) were carried 
+out by a Jasco 4100 Fourier transformation IR (FTIR) spectrometer equipped with a Pike IR 
+integrating sphere in the range from 400 to 4000 cm-1. Jasco V-570 ultraviolet/visible/near infrared 
+(UV/Vis/NIR) spectrometer equipped with a Jasco ISN-470 integrating sphere was used to 
+characterize optical performance in the ultraviolet, visible and infrared regime, ranging from 200 
+nm to 2500 nm. The visualization of the surface roughness was obtained via Keyence VHX-700 
+Digital Microscope. 
+The solar-to-thermal energy conversion efficiency of the solar selective coatings is given by 
+𝜂𝑡ℎ𝑒𝑟𝑚 = 𝐹𝑂𝑀 =
+∫(1−𝑅(𝜆))𝐼(𝜆)𝑑𝜆−1
+𝐶[∫(1−𝑅(𝜆))𝐵(𝜆,𝑇)𝑑𝜆] 
+∫ 𝐼(𝜆)𝑑𝜆
+= 𝛼𝑠𝑜𝑙𝑎𝑟 −
+𝜀𝑡ℎ𝑒𝑟𝑚𝜎𝑇4
+𝐶𝐼𝑠𝑜𝑙𝑎𝑟  ,      (1) 
+where 𝑅(𝜆) is the spectral reflectance of the solar selective coating at wavelength 𝜆 , 𝐼(𝜆) 
+represents the AM 1.5 solar spectral irradiance per square meter at wavelength 𝜆,  𝐼𝑠𝑜𝑙𝑎𝑟 =
+1000 𝑊/𝑚2 is the solar power density integrated from the spectral radiance 𝐼(𝜆), 𝐵(𝜆, 𝑇) is the 
+spectral blackbody thermal emission at 𝜆 and 𝑇,  𝛼solar is the overall spectrally normalized solar 
+
+4 
+ 
+absorbance, 𝜀𝑡ℎ𝑒𝑟𝑚 is the overall thermal emittance at T, and 𝜎 is the Stefan-Boltzmann constant 
+of 5.67 × 10−8
+𝑊
+𝑚2
+𝐾4 , and C=1000 is the solar concentration ratio of power tower CSP systems. 
+2. Nanoparticle Size (Diameter) Histogram 
+22
+24
+26
+28
+30
+32
+34
+36
+38
+40
+0
+5
+10
+15
+20
+25
+Frequency (%)
+Size (nm)
+ 
+Figure S1. Nanoparticle size distribution histogram. The average diameter is 313.6 nm. 
+3. X-ray Photoelectron Spectroscopy (XPS) Data Analyses 
+Figure S2.a shows a survey spectrum of as-synthesized Cu-Mn-Cr oxide nanoparticles. Cu, 
+Mn, Cr and O are detected from the surface. Figure S2.b shows the binding energies of the Cu 2p 
+core levels. Two sharp peaks at about 932 eV and 952 eV are observed, corresponding to the 
+significantly split spin-orbit components Cu 2p3/2 and Cu 2p1/2 respectively. Shake-up structures 
+at about 945 eV and 963 eV are satellite features of Cu with oxidation states. Such two satellite 
+peaks reveal the existence of Cu(II) while the relatively low intensity indicates possible mixed 
+states of Cu(I) and Cu(II).  
+ 
+
+5 
+ 
+ 
+ 
+Figure S2. XPS spectra of Cu, Mn, and Cr ions in the synthesized spinel oxide nanoparticles. 
+To further investigate the oxidation states and their compositions, principal Cu LMM Auger 
+peaks (Figure S2.c) are collected as well. One deconvoluted peak with a binding energy of 916.5 
+eV is assigned to Cu(I) 1 and the other one at 918.4 eV is assigned to Cu(II) 2 with an error of 1 
+eV. The modified Auger parameters are calculated and compared to minimize the effect of 
+charging of non-conducting specimens during the measurement. Cu in CuCr2O4 has an Auger 
+parameter of nearly 1853 eV 3, close to 1852.7 eV, the value we obtained for Cu(II). Biesinger 4 
+analyzed XPS data of various copper-containing species in previously published literature and 
+summarized their Auger parameters. Our Cu(I) has an Auger parameter of 1848.6 eV, which lies 
+in the range of several Cu(I) involved materials. In this case, Cu(I) and Cu(II) are present 
+
+(a)
+(b)
+(c)
+Survey
+Cu2p
+CuLMM
+Cu2p12
+Cu2p3/2
+2p
+Mn 2p
+cu
+01s
+Intensity(a.u.)
+Intensity(a.u.)
+Intensity(a.u.)
+Cr 2p
+Mn
+1000
+800
+600
+400
+200
+915916917918919920921
+B.E.(eV)
+B.E.(eV)
+K.E.(eV)
+(d)
+(e)
+(f)
+Mn2p
+Mn3s
+Cr2p
+Cr2P3/2
+Mn 2p3/2
+Intensity(a.u.)
+Mn2p1/2
+Intensity(a.u.)
+Intensity(a.u.)
+Cr 2P1/2
+657654651648645642639
+92
+90
+88
+86
+84
+82
+594591588585582579576573
+B.E.(eV)
+B.E.(eV)
+B.E.(eV)6 
+ 
+simultaneously with a ratio of about 4 : 1, taking into account both the percentage of peak areas 
+and Relative Sensitivity Factors (RSF). 
+Mn 2p (Figure S2.d) and Mn 3s (Figure S2.e) data are collected for quantification and 
+qualification of the chemical species in the specimen. In the 2p spectrum, two peaks at about 643.0 
+eV and 654.5 eV are assigned to the Mn 2p3/2 and Mn 2p1/2 components. An extremely broad and 
+weak satellite peak observed at around 649.1 eV is the feature of Mn(II). Three deconvolved peaks 
+represent Mn(II), Mn(III) and Mn(IV) with increasing values of binding energies. Mn 3s spectrum 
+distinguishes Mn oxidation states in a more straightforward way. Based on the photoemission final 
+states with the s electrons parallel or antiparallel to the 3d spin 5, the fewer and fewer 3d unpaired 
+electrons in Mn(II), Mn(III) and Mn(IV) lead to the smaller and smaller magnitudes of peak 
+splitting, from around 6.0 eV to 5.4 eV and to 4.7 eV for oxides 6. This assists in identifying the 
+oxidation states and calculating the percentage respectively by taking the energy difference as a 
+new constraint during the curve fitting. Eventually, Mn(II), Mn(III) and Mn(IV) are confirmed 
+coexisting in our specimen with a percentage of 30%, 47% and 23%. Figure 2.f shows the Cr 2p 
+spectrum, with Cr 2p3/2 peak at around 577.5 eV and Cr 2p1/2 peak at around 587.3 eV. Cr in 
+CuCr2O4 is located at 577.3 eV 2 and that suggests the existence of Cr(III) 7,8. Three sub-peaks 
+with the same full width at half maximum (FWHM) around 1.979eV are deconvoluted for Cr 2p3/2 
+peak, showing the multiplet structure 9 due to the coupling effect between the unpaired core 
+electron and unpaired electrons in the outer shell 10 in Cr(III). 
+ 
+ 
+ 
+ 
+
+7 
+ 
+4. Fourier Transform Infrared Spectroscopy (FTIR) 
+750
+700
+650
+600
+550
+500
+450
+400
+74
+76
+78
+Transmittance (%)
+Wavenumber (cm
+-1)
+Mn3+/Cr3+in either 
+tetrahedron or octahedron
+Mn3+/Cr3+ in
+[MO6] octahedron
+Tetrahedral+
+Octahedral 
+modes
+ 
+Figure S3. FTIR spectrum of the Cu-Mn-Cr oxide nanoparticles showing characteristic spinel 
+features. 
+Characteristic IR absorption peaks are observed at ~616, 505, and 420 cm-1, close to the 
+previous reports on spinel CuMn2O4 11, ZnMn2O4 12, and MnCr2O4 13. According to Ref. 13, the 
+peak at ~616 cm-1 corresponds to trivalent cation vibrations in the [MO6] octahedron. In our case 
+this peak is asymmetric, which can be induced by the coexistence of Mn3+ and Cr3+ as well as 
+valance 2+ and 4+ cations on the octahedral sites, as revealed by the XRD, EDS and XPS analyses 
+in the main text. The second peak at 505 cm-1 is attributed to trivalent ions, either on tetrahedral 
+or octahedral sites. The last peak at 420 cm-1 is a complex vibrational mode involving both 
+tetrahedral and octahedral sites.  
+ 
+ 
+ 
+
+8 
+ 
+5. Nanoparticle Volume Fraction Estimation 
+ 
+Figure S4. EDS mapping of Mn in a cross-sectional FIB lamellar of the coating. The lamellar is 
+~150 nm thick. 
+Since Mn is only present in the pigment nanoparticles and not in the silicone matrix, the 
+Inconel substrate, or the TEM grid (which is used to mount the FIB sample of the coating), we can 
+use Mn as a characteristic element of the NPs to derive the corresponding volume fraction by 
+analyzing the EDS mapping of Mn (Figure S4). We utilized Image J to obtain the area fraction of 
+Mn in the selected area. As shown in Figure S4, the yellow dots represent Mn signals and a brighter 
+color reveals a higher Mn intensity in that region. The original figure was firstly converted to an 
+8-bit binary image and a threshold ranging from 51 to 255 was chosen to select the Mn pixels. Mn 
+is determined to take 48.0% of the entire selected area.  
+Assuming spherical nanoparticle approximation and no overlapping of the nanoparticles in 
+the vertical direction, the volume fraction could be expressed as the following equation: 
+𝑓 =
+𝑉𝑀𝑛
+𝑉𝑡𝑜𝑡𝑎𝑙 = 
+𝐴∗𝑥
+𝜋𝑟2×4
+3𝜋𝑟3
+𝐴×𝑡
+= 
+4
+3 (
+𝑟
+𝑡) ∗ 𝑥 ,  
+ 
+ 
+ 
+ (2) 
+
+500nm9 
+ 
+where 𝑥 is the area fraction, 𝑟=15.51.8 nm is the radius of nanoparticles (see the histogram in 
+Figure S1) and 𝑡 = 150 𝑛𝑚 is the thickness of the FIB processed lamellar. With this equation, a 
+volume fraction of f ~7 vol. % was derived, which could be regarded as the lower limit since 
+nanoparticles do overlap in reality.  
+Another estimation was conducted according to the parameters used during the fabrication 
+process, including the weight percentage of nanoparticles in the precursor, the density of each 
+nonvolatile component, and the average coating thickness of 8.5 μm (as discussed in the main 
+text). Assuming no excessive loss of nanoparticles during the coating process and annealing 
+process, we obtain an upper limit of volume fraction f ~13 vol. %. Taking the average between the 
+lower limit estimated by EDS Mn area mapping and the upper limit from chemical precursor ratios, 
+it is reasonable to consider the volume fraction f=10±3 vol. % for comparison with theoretical 
+modeling.  
+6. XRD Analyses during Thermal Endurance Tests 
+XRD measurements were taken every 10 day-night annealing cycles, and Figures S5a and 5c 
+were plotted with each intermediate state during the whole thermal cycle procedure at 750 ºC and 
+800 ºC separately to show the deviation for several important peaks. Generally, the cycled coating 
+shows similar X-ray diffraction patterns while minor shifts occur. The peak position of the 
+characteristic spinel (311) peak is closely examined as shown in Figure S5.b and 5d for 750 ºC and 
+800 ºC, respectively. It originally lies at 35.78° and tends to shift as the thermal cycle starts. It 
+stabilizes at 35.62° for 750 ºC and 35.56° for 800 ºC after 20 day-night simulated cycles. Based 
+on Bragg’s equation, the peak shift towards a smaller diffraction angle refers to a larger interplanar 
+spacing, which means the lattice expands slightly.  
+
+10 
+ 
+ 
+ 
+Figure S5. XRD patterns during thermal endurance tests at (a) 750 ºC and (b) 800 ºC, with close 
+position examination of spinel (311) peak for samples through thermal cycles at (c) 750 ºC and 
+(d) 800 ºC, respectively. 
+From previous work published by Mikhail G. Brik 14, the lattice constants of CuMn2O4 and 
+CuCr2O4 are 8.33 Å and 8.27 Å, while that of MnCr2O4 is 8.437 Å. A variation from 8.410 Å to 
+8.474 Å depending on different milling hours was reported by R. N. Bhowmik15. Considering the 
+valences and atom position distributions discussed in the previous section, as more Cr ions dope 
+into the spinel system and take the octahedral site, Cu and Mn ions tend to sit in the tetrahedral 
+site, making it closer to the structure of CuCr2O4 and MnCr2O4. Taking into account the lattice 
+parameters mentioned above and ion radius data obtained from WebElements 16, it is reasonable 
+
+(a)
+(q)
+Spinel
+Mn,03
+小Inconel625
+533
+cycle60
+(400)
+(311)
+(111)
+077
+cycle50
+cycle40
+Intensity (a.u.)
+cycle30
+Intensity (a.u.)
+cycle 20
+cycle10
+24h
+20
+30
+40
+50
+60
+70
+80
+34.5
+35.0
+35.5
+36.0
+36.5
+2 theta (degree)
+2 theta (degree)
+(c)
+(d)
+cycle60
+cycle50
+cycle40
+cycle 30
+Intensity
+cycle20
+Intensity
+cycle 10
+24h
+20
+30
+40
+50
+60
+70
+80
+34.5
+35.0
+35.5
+36.0
+36.5
+2 theta (degree)
+2 theta (degree)11 
+ 
+to observe the lattice expansion. Mn ions are partially released to form Mn2O3, which is in 
+agreement with a higher Mn2O3/spinel ratio (from 0.142:1 to 0.466:1, stabilizing after 30 day-night 
+thermal cycles at 750 ºC) derived from XRD result. 
+7. Optical Spectra Evolution and EDS Mapping for Interdiffusion Investigation 
+upon Thermal Cycling 
+500
+1000
+1500
+2000
+2500
+85
+90
+95
+100
+Absorptance (%)
+Wavelength (nm)
+ 24h
+ cycle 10
+ cycle 20
+ cycle 30
+ cycle 40
+ cycle 50
+ cycle 60
+ CuCr2O4
+  
+Figure S6. UV-vis-NIR absorption spectra of Cu-Mn-Cr oxide nanoparticle pigmented solar 
+selective coating during thermal endurance tests at 800 ºC compared with as-coated CuCr2O4 
+nanoparticle pigmented coating. 
+ 
+Figure S7. Cross-section STEM image and EDS mapping result of cross-sectional FIB cut 
+specimens of Cu-Mn-Cr oxide nanoparticle pigmented solar selective coating (a) before and (b) 
+
+Ni
+Mn
+wrl
+um
+um
+1um
+Ni
+Mn
+2μm
+2um
+2um
+2um12 
+ 
+after 60 day-night thermal cycles at 750 ºC. Part of the coating was damaged during the FIB milling 
+processed. 
+Detailed investigation in the interface between the nanoparticle-pigmented coating and the 
+Inconel alloy substrate was conducted by observing the cross-sections of the specimen processed 
+by FIB milling. According to the STEM image shown in Figure S7a, an oxide layer of around 100 
+nm was formed after 24h annealing at 750 ºC. Further EDS mapping reveals that the oxide layer 
+mainly consists of Cr based oxides. As an obvious comparison in Figure S7b, a much thicker oxide 
+layer was observed for the sample that has completed 60 day-night simulated cycles at 750 ºC. It 
+approximately increases to 1 μm thick after long-time thermal cycles. This is a common oxide 
+scale when oxidizing Inconel alloys. 
+EDS analyses of the coating were also conducted, and an increasing Cr concentration with 
+thermal cycling was detected. Near the surface of the coatings, the Cr concentration increased by 
+58% after 60 day-night cycles at 750 ºC, and 117% after 60 day-night cycles at 800 ºC. This clearly 
+demonstrates that Cr atoms have diffused from the Inconel substrate through the solar coating and 
+emerged at the upper layers of the coating. The observed Cr diffusion into the coating is fully 
+consistent with the optical absorption spectrum evolution shown in Figure S6 and the 
+corresponding discussions in the main text. 
+ 
+ 
+ 
+ 
+
+13 
+ 
+8. Spray-Coated Solar Selective Coating on 48-Inch-Long Tube 
+ 
+Figure S8. A photo of spinel Cu-Mn-Cr oxide nanoparticle pigmented solar selective coating on 
+a 48-inch-long tube prepared by spray coating method.  
+ 
+Reference 
+(1)  
+Losev, A.; Rostov, K.; Tyuliev, G. Electron Beam Induced Reduction of CuO in the 
+Presence of a Surface Carbonaceous Layer: An XPS/HREELS Study. Surf. Sci. 1989, 213 
+(2–3), 564–579. https://doi.org/10.1016/0039-6028(89)90313-0. 
+(2)  
+Capece, F. M.; Castro, V. Di; Furlani, C.; Mattogno, G.; Fragale, C.; Gargano, M.; Rossi, 
+M. “Copper Chromite” Catalysts: XPS Structure Elucidation and Correlation with 
+Catalytic Activity. J. Electron Spectros. Relat. Phenomena 1982, 27 (2), 119–128. 
+https://doi.org/10.1016/0368-2048(82)85058-5. 
+(3)  
+NIST X-ray Photoelectron Spectroscopy Database, NIST Standard Reference Database 
+Number 20, National Institute of Standards and Technology, Gaithersburg MD, 20899 
+(2000) http://dx.doi.org/10.18434/T4T88K (accessed 2021 -10 -13). 
+
+14 
+ 
+(4)  
+Biesinger, M. C. Advanced Analysis of Copper X-Ray Photoelectron Spectra. Surf. 
+Interface Anal. 2017, 49 (13), 1325–1334. https://doi.org/10.1002/sia.6239. 
+(5)  
+Waskowska, A.; Gerward, L.; Olsen, J. S.; Steenstrup, S.; Talik, E. CuMn2O4: Properties 
+and the High-Pressure Induced Jahn-Teller Phase Transition. J. Phys. Condens. Matter 
+2001, 13 (11), 2549–2562. https://doi.org/10.1088/0953-8984/13/11/311. 
+(6)  
+Junta, J. L.; Hochella, M. F. Manganese (II) Oxidation at Mineral Surfaces: A 
+Microscopic and Spectroscopic Study. Geochim. Cosmochim. Acta 1994, 58 (22), 4985–
+4999. https://doi.org/10.1016/0016-7037(94)90226-7. 
+(7)  
+Biesinger, M. C.; Payne, B. P.; Grosvenor, A. P.; Lau, L. W. M.; Gerson, A. R.; Smart, R. 
+S. C. Resolving Surface Chemical States in XPS Analysis of First Row Transition Metals, 
+Oxides and Hydroxides: Cr, Mn, Fe, Co and Ni. Appl. Surf. Sci. 2011, 257 (7), 2717–
+2730. https://doi.org/10.1016/j.apsusc.2010.10.051. 
+(8)  
+Mohamed, R. M.; Kadi, M. W. Generation of Hydrogen Gas Using CuCr2O4-g-C3N4 
+Nanocomposites under Illumination by Visible Light. ACS Omega 2021, 6 (6), 4485–
+4494. https://doi.org/10.1021/acsomega.0c06193. 
+(9)  
+Oladipo, A. A. Rapid Photocatalytic Treatment of High-Strength Olive Mill Wastewater 
+by Sunlight and UV-Induced CuCr2O4@CaFe–LDO. J. Water Process Eng. 2021, 40 
+(January), 2–6. https://doi.org/10.1016/j.jwpe.2021.101932. 
+(10)  Moulder, J. F.; Stickle, W. F.; E.’Sobol, P.; Bomben, K. D. Handbook of X-Ray 
+Photoelectron Spectroscopy; Perkin-Elmer Corp, Eden Prairie, MN, 1992. 
+https://doi.org/10.1002/0470014229.ch22. 
+
+15 
+ 
+(11)  Patra, P.; Naik, I.; Kaushik, S. D.; Mohanta, S. Crossover from Meta-Magnetic State to 
+Spin-Glass Behaviour upon Ti-Substitution for Mn in CuMn2O4. J. Mater. Sci. Mater. 
+Electron. 2021, 33 (1), 554–564. https://doi.org/10.1063/5.0017104. 
+(12)  Ghannam, M. M.; Heiba, Z. K.; Sanad, M. M. S.; Mohamed, M. B. Functional Properties 
+of ZnMn2O4/MWCNT/Graphene Nanocomposite as Anode Material for Li-Ion Batteries. 
+Appl. Phys. A Mater. Sci. Process. 2020, 126 (5), 1–9. https://doi.org/10.1007/s00339-
+020-03513-6. 
+(13)  Allen, G. C.; Paul, M. Chemical Characterization of Transition Metal Spinel-Type Oxides 
+by Infrared Spectroscopy. Appl. Spectrosc. 1995, 49 (4), 451–458. 
+https://doi.org/10.1366/0003702953964372. 
+(14)  Brik, M. G.; Suchocki, A.; Kamińska, A. Lattice Parameters and Stability of the Spinel 
+Compounds in Relation to the Ionic Radii and Electronegativities of Constituting 
+Chemical Elements. Inorg. Chem. 2014, 53 (10), 5088–5099. 
+https://doi.org/10.1021/ic500200a. 
+(15)  Bhowmik, R. N.; Ranganathan, R.; Nagarajan, R. Lattice Expansion and Noncollinear to 
+Collinear Ferrimagnetic Order in a MnCr2O4 Nanoparticle. Phys. Rev. B - Condens. 
+Matter Mater. Phys. 2006, 73 (14), 1–9. https://doi.org/10.1103/PhysRevB.73.144413. 
+(16)  WebElements https://www.webelements.com (accessed 2022 -01 -22). 
+

FNFRT4oBgHgl3EQfyzjn/vector_store/index.pkl

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