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+Draft version January 13, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+SN 2020bio: A Double-peaked Type IIb Supernova with Evidence of Early-time Circumstellar
+Interaction
+C. Pellegrino,1, 2 D. Hiramatsu,3, 4 I. Arcavi,5, 6 D. A. Howell,1, 2 K. A. Bostroem,7, ∗ P. J. Brown,8, 9 J. Burke,1, 2
+N. Elias-Rosa,10, 11 K. Itagaki,12 H. Kaneda,13 C. McCully,1, 2 M. Modjaz,14 E. Padilla Gonzalez,1, 2 and
+T. A. Pritchard15
+1Las Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA
+2Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA
+3Center for Astrophysics |Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138-1516, USA
+4The NSF AI Institute for Artificial Intelligence and Fundamental Interactions
+5The School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
+6CIFAR Azrieli Global Scholars Program, CIFAR, Toronto, Canada
+7Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195-0002, USA
+8Department of Physics and Astronomy, Texas A&M University, 4242 TAMU, College Station, TX 77843, USA
+9George P. and Cynthia Woods Mitchell Institute for Fundamental Physics & Astronomy, College Station, TX 77843, USA
+10INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy
+11Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Barcelona, Spain
+12Itagaki Astronomical Observatory, Yamagata, Yamagata 990-2492, Japan
+13Kaneda Astronomical Observatory, Sapporo, Hokkaido 005-0862, Japan
+14Department of Astronomy, University of Virginia, Charlottesville, VA 22904
+15Department of Physics, New York University, New York, NY 10003, USA
+Submitted to ApJ
+ABSTRACT
+We present photometric and spectroscopic observations of SN 2020bio, a double-peaked Type IIb
+supernova (SN) discovered within a day of explosion, primarily obtained by Las Cumbres Observatory
+and Swift. SN 2020bio displays a rapid and long-lasting initial decline throughout the first week of its
+light curve, similar to other well-studied Type IIb SNe. This early-time emission is thought to originate
+from the cooling of the extended outer envelope of the progenitor star that is shock-heated by the SN
+explosion. We compare SN 2020bio to a sample of other double-peaked Type IIb SNe to investigate its
+progenitor properties. Analytical model fits to the early-time emission give progenitor radius (≈ 100–
+1500 R⊙) and H-rich envelope mass (≈ 0.01–0.5 M⊙) estimates that are consistent with other Type IIb
+SNe. However, SN 2020bio displays several peculiarities, including: 1) weak H spectral features and
+narrow emission lines indicative of pre-existing circumstellar material; 2) an underluminous secondary
+light curve peak which implies a small amount of synthesized 56Ni (MNi ≈ 0.02 M⊙); and 3) low-
+luminosity nebular [O I] features. These observations are more consistent with a lower-mass progenitor
+(MZAMS ≈ 12 M⊙) that was stripped of most of its H envelope before exploding. This study adds to
+the growing diversity in the observed properties of Type IIb SNe and their progenitors.
+Keywords: Circumstellar matter(241) — Core-collapse supernovae(304) — Supernovae(1668)
+1. INTRODUCTION
+Corresponding author: Craig Pellegrino
+[email protected]
+∗ LSST Catalyst Fellow
+While the majority of stars with initial masses ≳ 8 M⊙
+end their lives as H-rich core-collapse supernovae (SNe;
+e.g., Janka 2012), some massive stars lose their outer H
+and even He envelopes and explode as stripped-envelope
+SNe (SESNe; e.g., Filippenko 1997; Gal-Yam 2017). A
+small but growing number of SNe have been observed
+arXiv:2301.04662v1  [astro-ph.HE]  11 Jan 2023
+
+2
+Pellegrino et al.
+with spectra that show similarities to both these classes
+(Smith et al. 2011). Classified as Type IIb SNe (SNe
+IIb), their spectra have H features at early times that
+gradually give way to He features, indicating that their
+progenitors were partially stripped of their outer en-
+velopes before exploding (Woosley et al. 1994).
+It is unclear what mechanisms are responsible for this
+mass loss. Common hypotheses include stellar winds,
+binary interaction, or late-stage stellar instabilities (see
+e.g., Smith 2014, for a review).
+Recent studies have
+shown that mass loss is common during the late stages
+of massive star evolution, as inferred from early-time
+observations of core-collapse SNe (e.g., Ofek et al. 2014;
+Bruch et al. 2021; Strotjohann et al. 2021). A signif-
+icant fraction of core-collapse SNe show signatures of
+pre-existing circumstellar material (CSM) in their early-
+time spectra, obtained days after their estimated explo-
+sion epochs. This CSM is the material shed by the pro-
+genitor star in the months to years before core-collapse.
+As the SN shock breaks out of the expanding ejecta the
+resulting X-ray and ultraviolet (UV) flash may ionize
+the surrounding CSM, producing narrow spectral fea-
+tures as the CSM cools and recombines (e.g., Fassia et
+al. 2001; Yaron et al. 2017).
+Interaction between the
+SN ejecta and CSM can also influence the early-time
+light-curve evolution (Morozova et al. 2018).
+Some SNe IIb are observed to have double-peaked
+light curves, with rapidly-fading luminosities during the
+first several days after explosion before the radioactive
+decay of 56Ni synthesized during the explosion causes a
+re-brightening that lasts for several weeks. The early-
+time emission is thought to be the cooling of the ex-
+tended envelope of the progenitor star that is heated
+by the SN shock (Soderberg et al. 2012). This shock-
+cooling emission (SCE) has only been extensively ob-
+served in a handful of cases, including SN 1993J (e.g.,
+Woosley et al. 1994; Richmond et al. 1994), SN 2011dh
+(e.g., Arcavi et al. 2011; Ergon et al. 2014), SN2013 df
+(e.g., Morales-Garoffolo et al. 2014; Van Dyk et al. 2014),
+SN 2016gkg (Arcavi et al. 2017), SN 2017jgh (Armstrong
+et al. 2021), and ZTF18aalrxas (Fremling et al. 2019),
+among others. Most of these objects are nearby and had
+follow-up observations scheduled hours after explosion,
+which proved crucial to observing the rapidly-evolving
+SCE. These studies have found that SNe IIb are con-
+sistent with the explosions of stars with extended outer
+envelopes, with the duration of the SCE dependent on
+the extent of this envelope (Soderberg et al. 2012).
+Numerical and analytical models of SCE can comple-
+ment pre-explosion imaging in determining the progen-
+itors of these objects.
+Several models have been suc-
+cessful in reproducing the observed early-time evolution
+across all wavelengths.
+Piro (2015, hereafter P15) is
+one of the first to present a one-zone analytical descrip-
+tion of the cooling of an extended low-mass envelope
+shock-heated by the explosion of a compact massive
+core. Piro et al. (2021, hereafter P21) extend this to
+a two-zone model in order to better capture the emis-
+sion from the outermost material in extended envelopes.
+Sapir & Waxman (2017, hereafter SW17) calibrate ear-
+lier models by Rabinak & Waxman (2011)—that depend
+on the precise density structure of the outer material—to
+numerical simulations for several days after explosion.
+Comparing observed SCE to analytical and numeri-
+cal models is one of the only ways of directly measuring
+the radii and stellar structure of core-collapse progen-
+itors from SN observations. This has been done for a
+handful of SNe IIb as well as SNe of other subtypes,
+including stripped-envelope Type Ib SNe (e.g., Modjaz
+et al. 2009; Yao et al. 2020), short-plateau Type II SNe
+(Hiramatsu et al. 2021), and exotic Ca-rich transients
+(e.g., Jacobson-Gal´an et al. 2020, 2022). Analytical and
+numerical modeling of double-peaked SNe IIb generally
+yield large radii progenitors (≈ 100–500 R⊙) with low-
+mass (≈ 10−2–10−1 M⊙) extended envelopes (Piro et
+al. 2021, and references therein). These properties are
+usually in agreement with those of SNe IIb progeni-
+tors from pre-explosion Hubble Space Telescope images,
+which have revealed them to be supergiants (Aldering
+et al. 1994; Maund et al. 2011; Van Dyk et al. 2014).
+In some cases, however, the progenitor radii estimated
+from SCE modeling are in tension with those measured
+from direct imaging (e.g., Arcavi et al. 2017; Tartaglia
+et al. 2017, in the case of SN 2016gkg;). Potential bi-
+nary companions to the progenitor, which have been
+observed or inferred in a handful of cases (e.g., Maund
+et al. 2004; Benvenuto et al. 2013) can further compli-
+cate direct imaging estimates when the individual binary
+members are unresolvable.
+Here we present photometric and spectroscopic ob-
+servations of SN 2020bio, an SN IIb showing remark-
+ably strong SCE, obtained by Las Cumbres Observatory
+(LCO) through the Global Supernova Project (GSP).
+LCO extensively observed SN 2020bio from hours to
+≈ 160 days after explosion, providing a detailed look
+into the full evolution of a double-peaked SN IIb. In
+this work, we analyze its light curve evolution, spectral
+features, and fit analytic models to its full light-curve
+evolution to estimate the radius, mass, and structure of
+its progenitor star. We also compare its bolometric light
+curve and spectra to numerical models in order to infer
+its progenitor mass and the properties of its circumstel-
+lar environment.
+
+The Double-peaked Type IIb SN 2020bio
+3
+This paper is organized as follows. In Section 2 we
+describe the discovery and follow-up observations of
+SN 2020bio.
+We present its full light curve and spec-
+tral time series in Section 3 and compare observations
+to analytical and numerical models in Section 4.
+Fi-
+nally, in Section 5 we discuss the potential progenitor
+properties of SN 2020bio given the presented evidence.
+2. DISCOVERY AND DATA DESCRIPTION
+SN 2020bio was discovered by Koichi Itagaki on UT
+2020 January 29.77 at the Itagaki Astronomical Obser-
+vatory at an unfiltered Vega magnitude of 16.7. Analy-
+sis of an image of the same field by the ATLAS survey
+on the previous night yields a nondetection at c-band
+magnitude 18.7. Soon after discovery rapid photometric
+and spectroscopic follow-up observations were requested
+by the GSP through the Las Cumbres global network of
+telescopes. The GSP also triggered its Swift Key Project
+(1518618: PI Howell) to obtain daily UV and optical
+photometry. A classification spectrum obtained on the
+2.0m Liverpool Telescope on 2020 January 31.19—ap-
+proximately 1.5 days after the first detection—shows a
+blue continuum superimposed with a narrow Hα emis-
+sion feature and a broad possible He I λ 5876˚A feature,
+consistent with a young core-collapse SN (Srivastav et
+al. 2020).
+SN 2020bio exploded at right ascension 13h55m37s.69
+and declination +40°28′39′′.1 in the spiral galaxy NGC
+5371 at redshift z = 0.008533 (Springob et al. 2005).
+The distance to NGC 5371 is uncertain due to its low
+redshift. We adopt the mean of several distances mea-
+sured using the method of Tully & Fisher (1977), which
+gives d = 29.9 ± 5.1 Mpc (values from the NASA Ex-
+tragalactic Database1). Using the Schlafly & Finkbeiner
+(2011) dust map calibrations, we estimate a Galactic
+line-of-sight extinction to SN 2020bio EMW (B − V ) =
+0.008 mag. Given the location of SN 2020bio with re-
+spect to its host galaxy, we also estimate host extinc-
+tion using the Na I D equivalent widths measured in a
+high-resolution spectrum of the SN. From the conver-
+sions presented in Poznanski et al. (2012), we estimate
+Ehost(B −V ) = 0.068 ± 0.038 mag for a total extinction
+E(B − V ) = 0.076 ± 0.038 mag. The photometry of
+SN 2020bio presented throughout this work is corrected
+for this mean total extinction.
+LCO photometric follow-up commenced less than a
+day after discovery. UBgVri-band images were obtained
+by the Sinistro and Spectral cameras mounted on LCO
+1.0m and 2.0m telescopes, respectively, located at Mc-
+1 https://ned.ipac.caltech.edu/
+Table 1. UV and Optical Photometry
+JD
+Filter
+Magnitude
+Uncertainty
+Source
+2458878.27
+Clear
+16.77
+0.15
+Itagaki
+2458878.33
+Clear
+16.55
+0.15
+Itagaki
+2458878.39
+Clear
+16.51
+0.15
+Itagaki
+2458879.27
+Clear
+16.22
+0.15
+Itagaki
+2458880.26
+Clear
+16.49
+0.15
+Itagaki
+2458881.25
+Clear
+16.68
+0.15
+Itagaki
+2458882.18
+Clear
+16.82
+0.15
+Itagaki
+2458883.26
+Clear
+16.85
+0.15
+Itagaki
+2458878.85
+UVW2
+13.56
+0.04
+Swift
+2458879.89
+UVW2
+14.59
+0.05
+Swift
+This table will be made available in its entirety in machine-
+readable format.
+Donald Observatory, Teide Observatory, and Haleakala
+Observatory.
+Data were reduced using lcogtsnpipe
+(Valenti et al. 2016) which extracts point-spread func-
+tion magnitudes after calculating zero-points and color
+terms (Stetson 1987). UBV -band photometry was cali-
+brated to Vega magnitudes using Landolt standard fields
+(Landolt 1992) while gri-band photometry was cali-
+brated to AB magnitudes (Smith et al. 2002) using Sloan
+Digital Sky Survey (SDSS) catalogs. As SN 2020bio ex-
+ploded coincident with its host galaxy, to remove host
+galaxy light we performed template subtraction using
+the HOTPANTS (Becker 2015) algorithm and template
+images obtained after the SN had faded. Unfiltered im-
+ages were obtained with the Itagaki Astronomical Ob-
+servatory (Okayama and Kochi, Japan) 0.35 m tele-
+scopes + KAF-1001E (CCD). Using our custom soft-
+ware, the photometry was extracted after host subtrac-
+tion and calibrated to the V-band magnitudes of
+45
+field stars from the Fourth US Naval Observatory CCD
+Astrograph Catalog (Zacharias et al. 2013).
+UV and optical photometry were obtained with the
+Ultraviolet and Optical Telescope (UVOT; Roming et
+al. 2005) on the Neil Gehrels Swift observatory (Gehrels
+et al. 2004). Swift data were reduced using a custom
+adaptation of the Swift Optical/Ultraviolet Supernova
+Archive (Brown et al. 2014) pipeline with the most re-
+cent calibration files and the zeropoints of Breeveld et
+al. (2011). Images from the final epoch, obtained after
+the SN had sufficiently faded, were used as templates
+to subtract the host galaxy light. All Swift photometry
+is calibrated to Vega magnitudes. The entire UV and
+optical data sets from LCO, Itagaki, and Swift UVOT
+are given in Table 1.
+
+4
+Pellegrino et al.
+58880
+58900
+58920
+58940
+58960
+58980
+59000
+59020
+59040
+MJD
+10
+12
+14
+16
+18
+20
+22
+24
+Apparent Magnitude + Offset
+58877.0
+58880.5
+58884.0
+10
+12
+14
+16
+18
+20
+UVW2 - 4
+UVM2 - 3
+UVW1 - 2
+U - 1
+B
+g + 1
+V + 2
+r + 3
+i + 3.5
+Clear + 2
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Days From Discovery
+-22
+-20
+-18
+-16
+-14
+-12
+-10
+-8
+Absolute Magnitude + Offset
+Figure 1. The full extinction-corrected light curves of SN 2020bio. Photometry in different filters have been offset for clarity.
+Unfiltered photometry from the Itagaki Astronomical Observatory is included as clear points and calibrated to the V -band.
+The inset focuses on the rapidly-evolving shock-cooling emission.
+LCO spectra were obtained by the FLOYDS spectro-
+graph on the 2.0m Faulkes Telescope North at Haleakala
+Observatory.
+Spectra cover a wavelength range of
+3500–10,000 ˚A at a resolution R ≈ 300-600.
+Data
+were reduced using the floydsspec pipeline2, a custom
+pipeline which performs cosmic ray removal, spectrum
+extraction, and wavelength and flux calibration.
+We
+also present one spectrum obtained by the B&C spectro-
+graph on the 2.3m Bok Telescope at Steward Observa-
+tory, two spectra obtained by the Blue Channel Spectro-
+graph on the 6.5m MMT at the Fred Lawrence Whipple
+Observatory, and one spectrum obtained by the Optical
+System for Imaging and low-Intermediate-Resolution In-
+2 https://github.com/svalenti/FLOYDS pipeline/
+tegrated Spectroscopy spectrograph on the 10.4m Gran
+Telescopio Canarias. Details of all these spectra are pre-
+sented in Table 2.
+3. PHOTOMETRIC AND SPECTRAL ANALYSIS
+3.1. Light Curve and Color Evolution
+In Figure 1 we show the full LCO and Swift extinction-
+corrected light curve of SN 2020bio, from detection to
+≈ 160 days after explosion. The discovery and subse-
+quent follow-up photometry from Itagaki are included
+as “Clear” data points. The inset shows in greater de-
+tail the early-time evolution of the SCE, focusing on the
+first week after discovery. The most distinctive feature
+of the light curve is the luminous and rapidly-declining
+SCE at early times. The peak SCE luminosity exceeds
+that of the secondary peak ≈ 15 days later, but SCE
+only dominates the light curve during the first several
+
+The Double-peaked Type IIb SN 2020bio
+5
+-2
+0
+2
+UVW2 - B
+-2
+0
+2
+UVW2 - V
+-2
+0
+2
+UVM2 - B
+-2
+0
+2
+UVM2 - V
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Days Since Discovery
+-2
+0
+2
+UVW1 - B
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Days Since Discovery
+-2
+0
+2
+UVW1 - V
+SW17 Model
+SN 2010jr
+SN 2011dh
+SN 2013df
+SN 2016gkg
+SN 2020bio
+Figure 2. Swift colors of SN 2020bio compared with those of other SNe IIb with early-time Swift observations. We also include
+the best fit SW17 model from Section 4 for comparison. SN 2020bio was bluer at earlier phases than the other SNe IIb. Data
+for these comparison SNe were obtained from the following sources: Arcavi et al. (2011) (SN 2011dh); Morales-Garoffolo et al.
+(2014) (SN 2013df); Arcavi et al. (2017) (SN 2016gkg); this work (SNe 2010jr and 2020bio).
+days. Over this time the light curve falls by ≈ 4 mag
+in the first week, making this phase difficult to observe
+without rapid multi-wavelength follow-up.
+After ≈ 4 days from discovery the slope of the light
+curve decline changes as the luminosity from 56Ni de-
+cay begins to dominate the light curve.
+After about
+a week the light curve re-brightens and reaches a sec-
+ondary maximum ≈ 15 days after discovery. From this
+point the emission settles onto the radioactive decay tail,
+powered by 56Co decay, for the remainder of the obser-
+vations. The secondary peak and overall late-time light
+curve is relatively dim, peaking at M ≈ -14 mag in the
+V -band, hinting at a small amount of 56Ni synthesized
+in the explosion.
+In Figure 2 we compare the early-time Swift UV-
+optical colors of SN 2020bio to those of other SNe IIb
+with observed SCE in the UV. All dates are given with
+respect to the time of discovery and corrected for extinc-
+tion according to the published values for each object.
+SN 2020bio has both the earliest observations relative to
+discovery and the bluest colors throughout its evolution
+compared to the other objects. While objects such as
+SN 2010jr and SN 2016gkg have more densely-sampled
+light curves, their observations began later and their col-
+ors evolved redward faster compared to SN 2020bio.
+Of the 6 colors plotted, SN 2020bio is exceptionally
+blue in the UVM2-B and UVM2-V colors, particularly
+in the earliest epochs.
+We plot a representative SCE
+model color curve from Section 4.2 in each panel for
+comparison. SN 2020bio is bluer than the model, which
+more accurately reproduces the color evolution of the
+other SNe IIb up to several days after the discovery.
+This may be evidence for another luminosity contribu-
+tion besides SCE, as we discuss in Section 5.
+3.2. Spectral Comparison
+Spectral coverage of SN 2020bio began fewer than 2
+days after the first detection—approximately 3 days
+since the estimated explosion time (Section 4.2)—and
+continued for 201 days. We plot the full spectral series
+
+6
+Pellegrino et al.
+Table 2. Log of Spectroscopic Observations
+Date of Observation
+Days Since Discovery
+Facility/Instrument
+Exposure Time (s)
+Wavelength Range (˚A)
+2020-01-31 04:27:31
+1
+LT/SPRAT
+1200
+4000–7925
+2020-02-03 14:32:18
+4
+LCO/FLOYDS-N
+1800
+3500–10,000
+2020-02-05 12:19:05
+6
+LCO/FLOYDS-N
+1800
+3500–10,000
+2020-02-15 09:35:59
+16
+Bok/B&C
+600
+3850–7500
+2020-02-18 12:32:26
+19
+MMT/Blue Channel
+300
+5700–7000
+2020-02-24 13:00:37
+25
+LCO/FLOYDS-N
+1800
+3500–10,000
+2020-03-03 10:49:44
+33
+LCO/FLOYDS-N
+2700
+3500–10,000
+2020-03-22 14:22:56
+52
+LCO/FLOYDS-N
+3600
+3500–10,000
+2020-03-30 14:20:34
+60
+LCO/FLOYDS-N
+3600
+3500–10,000
+2020-04-16 11:12:12
+77
+LCO/FLOYDS-N
+3600
+3500–10,000
+2020-04-27 12:09:24
+88
+LCO/FLOYDS-N
+3600
+3500–10,000
+2020-08-18 22:02:01
+201
+GTC/OSIRIS
+1500
+3600–7808
+Note—All spectra will be made publicly-available on WiseRep (Yaron & Gal-Yam 2012).
+in Figure 3. The earliest spectrum of SN 2020bio, re-
+ported to the Transient Name Server (Srivastav et al.
+2020), shows a hot blue continuum superimposed with
+emission lines.
+We identify narrow features of H and
+Mg I as well as a potential weak, broad feature of He I
+λ5876 ˚A. These lines are consistent with flash-ionized
+features observed in other core-collapse SNe, which is
+evidence of nearby CSM lost by the progenitor star.
+After about a week post explosion, absorption fea-
+tures begin to develop in the spectra. We identify lines
+of He, O, and Ca. We also note persistent narrow H
+emission features that last for several weeks. To deter-
+mine if these features are produced by interaction with
+CSM or by host galaxy emission, we fit the narrow Hα
+emission line with a Gaussian function to estimate its
+full-width at half-maximum (FWHM). The results are
+shown in Figure 4. In our earliest spectrum we estimate
+a FWHM of the Hα line of 1500 km s−1, greater than
+the average widths of host galaxy emission lines, while
+our spectrum obtained roughly two weeks after discov-
+ery has a FWHM of ≈ 350 km s−1, more consistent with
+host-galaxy emission at this resolution. The latter value
+is also consistent with the FWHMs we measure for the
+nearby host-dominated [N II] λ 6583 line throughout the
+first several weeks. Therefore, we conclude that circum-
+stellar interaction likely contributes to the H emission
+during the first ≈ 2 weeks after explosion.
+An absorption feature blueward of the rest-frame Hα
+line matches He I λ 6678˚A absorption blueshifted by ≈
+7500 km s−1, which is commonly noted to cause “flat-
+topped” Hα emission profiles in other SNe IIb (e.g., Fil-
+ippenko et al. 1993). In general, the absorption features
+in the SN 2020bio spectra are shallower than those of the
+other SNe IIb, particularly SN 2011dh. Interaction with
+CSM can produce absorption features that are weaker
+and shallower than expected, which has been noted in
+the spectra of SN 1993J and SN 2013df (Fremling et al.
+2019).
+To
+further
+investigate
+the
+differences
+between
+SN 2020bio and other SNe IIb, we plot comparison spec-
+tra just after explosion (top), after two weeks (middle),
+and three weeks (bottom) after explosion in Figure 5.
+Among this sample, SN 2020bio is the only object to
+show narrow features indicative of pre-existing CSM at
+early times, despite similar phase coverage of the other
+SNe IIb. This likely reflects differences in their circum-
+stellar environments—if the narrow lines were formed
+from the expanding outer envelopes of the progenitor,
+they should be ubiquitous among SNe IIb at this phase.
+Instead, the presence of narrow H and Mg lines in the
+earliest spectrum of SN 2020bio more likely points to
+confined CSM formed from material stripped from the
+progenitor star.
+Differences persist weeks after the estimated explo-
+sion times.
+While the other SNe IIb have developed
+broad Hα and Hβ emission features, these same lines
+are weaker in SN 2020bio. This could be partly caused
+by He I λ 6678˚A absorption, which has an absorption
+trough coincident with the Hα flux when blueshifted by
+≈ 7500 km s−1. Another possibility is that the H emis-
+sion from SN 2020bio is inherently weaker than in other
+SNe IIb, which may be the case if the progenitor lost
+more of its outer H envelope than the progenitors of the
+other SNe IIb did. Weak H emission, combined with the
+
+The Double-peaked Type IIb SN 2020bio
+7
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+Rest-frame Wavelength (Å)
+Normalized F  + Constant
+1d
+4d
+6d
+16d
+19d
+25d
+33d
+52d
+60d
+77d
+88d
+201d
+H
+He I
+Mg I
+O III
+Ca II
+Figure 3.
+The full spectral time series of SN 2020bio.
+Phases with respect to the detection epoch are given above
+each spectrum. Notable spectral features are identified with
+dashed lines.
+The first spectrum is the publicly-available
+classification spectrum retrieved from the Transient Name
+Server.
+observed CSM features, point to a scenario in which the
+progenitor of SN 2020bio underwent enhanced mass-loss,
+shedding almost all of its outer H layer before explod-
+ing. If this is the case, such a progenitor scenario to
+SN 2020bio is unique among other well-studied SNe IIb.
+4. LIGHT-CURVE MODELING AND
+PROGENITOR INFERENCE
+4.1. Shock-cooling Model Descriptions
+A variety of analytical and numerical models of SCE
+have been developed in recent years. Here we consider
+6400
+6500
+6600
+6700
+Rest-frame Wavelength (Å)
+Normalized F  + Constant
+1d
+4d
+6d
+16d
+Figure 4. Gaussian fits to the Hα emission line in the early-
+time spectra of SN 2020bio. Phases relative to discovery are
+given above each spectrum. The dashed line shows the rest-
+frame Hα wavelength. The FWHMs decrease over time, evi-
+dence that circumstellar interaction contributes to the emis-
+sion profile.
+3 analytical models that are commonly used to fit the
+early-time emission of core-collapse SNe. The P15 model
+extends the formalism of Nakar & Piro (2014) to repro-
+duce the full shock-cooling peak.
+It assumes a lower
+mass extended envelope without assuming its specific
+density structure. On the other hand, SW17 calibrates
+to the numerical models of Rabinak & Waxman (2011)
+and assumes specific polytropic indices for the extended
+envelope. The methodology used to fit these models to
+the data and derive resulting blackbody properties are
+presented in Arcavi et al. (2017).
+More recently, Piro et al. (2021) developed another
+analytical model to better reproduce the early SCE ob-
+served in a variety of transients (e.g., Arcavi et al. 2017;
+Yao et al. 2020). They assume a two-zone extended en-
+velope in homologous expansion and calculate the emis-
+sion from this shocked material. This method begins by
+assuming extended material in homologous expansion
+separated into two regions—an outer density profile de-
+scribed by ρ ∝ r−n, where n ≈ 10, and an inner region
+
+8
+Pellegrino et al.
+Table 3. SCE Model Parameters
+Model
+Renv (R⊙)
+Menv (10−2 M⊙)
+va (104 km s−1)
+t0 (days)
+χ2 / d.o.f.
+P15
+510+30
+−30
+1.14+0.02
+−0.02
+1.67+0.02
+−0.01
+0.67+0.02
+−0.02
+21.6
+P21
+1700+85
+−95
+1.60+0.03
+−0.02
+1.36+0.01
+−0.02
+0.98+0.01
+−0.01
+21.1
+SW17 (n=3/2)
+160+12
+−10
+47.12+0.96
+−0.92
+1.69+0.04
+−0.04
+0.26+0.04
+−0.04
+8.7
+SW17 (n=3)
+220+19
+−15
+322.60+6.10
+−6.20
+1.60+0.04
+−0.04
+0.25+0.04
+−0.04
+8.7
+aThe characteristic velocity for P15 and P21 and the shock velocity for SW17.
+1.5d
+2d
+3d
+2d
+2d
+17d
+16d
+17d
+13d
+17d
+SN 2020bio
+SN 1993J
+SN 2013df
+SN 2016gkg
+SN 2011dh
+4000
+5000
+6000
+7000
+8000
+9000
+Rest-frame Wavelength (Å)
+26d
+25d
+25d
+21d
+25d
+Normalized F  + Constant
+Figure 5.
+Spectra of SN 2020bio compared with spectra
+of other SNe IIb at similar phases. Phases with respect to
+the estimated explosion time are given above each spectrum
+and notable spectral features are identified with red (H) and
+blue (He) vertical lines at their rest-frame wavelengths. The
+spectra of SN 2016gkg are unpublished spectra obtained by
+LCO while the other comparison spectra were retrieved from
+WiseRep (Yaron & Gal-Yam 2012).
+with ρ ∝ r−d, where δ ≈ 1.1. Assuming a transitional
+velocity vt between the inner and outer regions of the
+extended material, the time for the diffusion front to
+reach this transition is given by
+td =
+� 3κKMe
+(n − 1)vtc
+�1/2
+(1)
+where K = (n−3)(3−δ)
+4π(n−δ) , κ is the optical opacity, and Me
+is the mass of the extended material. The luminosity
+from the cooling of the extended material is then defined
+piecewise for times before and after this diffusion time:
+L(t) ≈ π(n − 1)
+3(n − 5)
+cRev2
+t
+κ
+�td
+t
+�4/(n−2)
+, t ≤ td
+(2)
+and
+L(t) ≈ π(n − 1)
+3(n − 5)
+cRev2
+t
+κ
+exp
+�
+−1
+2
+�t2
+t2
+d
+− 1
+��
+, t ≥ td
+(3)
+To fit the photometry in each band, we assume that
+the material radiates as a blackbody at some photo-
+spheric radius rph. The photosphere reaches the transi-
+tion between the two regions at a time
+tph =
+� 3κKMe
+2(n − 1)v2
+t
+�1/2
+(4)
+and the time evolution of the photospheric radius is
+given relative to this characteristic time:
+rph(t) =
+�tph
+t
+�2/(n−1)
+vtt, t ≤ tph
+(5)
+and
+rph(t) =
+� δ − 1
+n − 1
+� t2
+t2
+ph
+− 1
+�
++ 1
+�−1/(δ−1)
+vtt, t ≥ tph (6)
+In addition, we attempt to fit the analytical models
+of Shussman et al. (2016), which are calibrated to nu-
+merical simulations from shock breakout to recombina-
+tion. However, these model fits are unable to reproduce
+the rapidly-declining shock-cooling emission in all fil-
+ters during the week after explosion. It is possible this
+
+The Double-peaked Type IIb SN 2020bio
+9
+0
+2
+4
+6
+8
+10
+Days from Discovery
+10
+12
+14
+16
+18
+20
+22
+Apparent Magnitude + Offset
+UVW2 - 3.5
+UVM2 - 3
+UVW1 - 2
+U - 1
+B
+g + 1
+V + 2
+r + 3
+i + 3.5
+0
+2
+4
+6
+8
+10
+Days from Discovery
+10
+12
+14
+16
+18
+20
+22
+SW17
+(n=3/2)
+SW17
+(n=3)
+Figure 6. Shock-cooling fits to the early-time photometry of SN 2020bio using the models of (left) P15 and P21; and (right)
+SW17, assuming a constant optical opacity appropriate for solar-composition material. Photometry in each band has been offset
+for clarity. Itagaki discovery photometry has been included in the V -band fits.
+shortcoming is due to an unphysical application of the
+model—which is calibrated to numerical simulations of
+red supergiants—to the early light curve of SN 2020bio,
+which likely had a different progenitor structure. De-
+tailed comparisons between numerical models of SNe IIb
+and the Shussman et al. (2016) models are beyond the
+scope of this work.
+4.2. Best-fit Analytic Models
+We fit each model to the early-time photometry of
+SN 2020bio. For the SW17 model we consider two poly-
+tropic indices (n = 3/2 and n = 3), appropriate for con-
+vective and radiative envelopes, respectively. Only data
+taken up to 3.5 days after discovery are fit, as this is the
+time when SCE dominates the luminosity over radioac-
+tive decay (see Section 4.3 for a quantitative treatment
+of the 56Ni light curve). Additionally, we ensure that the
+phases we fit fall within the validity range of each model.
+In each case we fit for the progenitor extended envelope
+radius, Renv, the envelope mass, Menv, either the char-
+acteristic velocity or the shock velocity v of the outer
+material, and the offset time since explosion t0. We use
+the emcee package (Foreman-Mackey et al. 2013) to per-
+form Markov Chain Monte Carlo fitting of each model,
+initializing 100 walkers with 1000 burn-in steps and run-
+ning for an additional 1000 steps after burn-in. For each
+step, the total luminosity is computed using the analyt-
+ical model formalism, and the luminosity within each
+filter is compared to the observed photometry assuming
+a blackbody spectral energy distribution (SED). We fit
+each model assuming an optical opacity κ = 0.34 cm2
+g−1, consistent with solar composition material.
+The best-fit models to the multi-band SCE light
+curves are shown in Figure 6, and best-fit parameters are
+given in Table 3 with corner plots shown in Appendix
+A. The Itagaki discovery data that capture the rise are
+calibrated to the V -band. We find that all the mod-
+els fit the early-time data well, reproducing the rapid
+rise, luminous peak, and subsequent decline in all filters.
+Quantitatively the SW17 model for convective envelopes
+(n = 3/2) has the lowest reduced χ2 value, indicating
+the model most closely matches the observations. On
+the other hand, the best-fit envelope mass for the SW17
+model with a radiative (n = 3) envelope is larger than
+the total ejecta mass, estimated in Section 4.3. There-
+fore, we do not consider this model representative of the
+progenitor of SN 2020bio.
+Based on the unusual properties of SN 2020bio com-
+pared to other SNe IIb, including its weak H spectral
+features and faint secondary light-curve peak, we test
+whether a lower-opacity envelope better reproduces the
+observed SCE. This could be the case if the progeni-
+tor star was almost completely stripped of its outer H
+envelope. We perform the same fitting routine but fix
+the opacity κ = 0.20 cm2 g−1 for H-poor material. We
+find no differences in goodness of fits for each model
+between the two chosen opacities—both the H-rich and
+
+10
+Pellegrino et al.
+0
+20
+40
+60
+80
+100
+Days Since Discovery
+1040
+1041
+1042
+Pseudo-Bolometric Luminosity (erg s−1)
+E = 0.9 × 1051 erg
+MNi = 0.015 M⊙
+MNi = 0.017 M⊙
+MNi = 0.019 M⊙
+MNi = 0.020 M⊙
+SN2020bio
+Figure 7. Numerical MESA and STELLA model light curves of
+SN 2020bio for varying MNi. Both the secondary light-curve
+peak and late-time light-curve slope are best reproduced with
+≈ 0.02 M⊙ of 56Ni synthesized in the explosion.
+H-poor envelopes produce similarly good fits. However,
+there are differences in the fitted parameters between
+the best-fit models.
+In the H-rich case, the envelope
+radii and masses from the best-fit SW17 model are con-
+sistent with those estimated for other SNe IIb (i.e. radii
+of ≈ 1×1013 cm and masses of 10−3–10−2 M⊙). In the
+H-poor case, however, the radii are smaller (≈ 3×1012
+cm) and the envelope masses are larger (≈ 10−1 M⊙).
+These values are more consistent with those estimated
+for Type Ib and Ca-rich transients with observed SCE
+(e.g., Yao et al. 2020; Jacobson-Gal´an et al. 2022).
+4.3. Bolometric Luminosities and Numerical Modeling
+SCE dominates the total luminosity only for several
+days after explosion. The rest of the light curve is pow-
+ered by the radioactive decay of 56Ni and its children
+isotopes. Using our multi-band coverage of SN 2020bio
+for ≈ 160 days after explosion, we construct a pseudo-
+bolometric light curve to fit for the amount of 56Ni pro-
+duced in the explosion. For epochs with observations
+in more than 3 filters, we extrapolate the SED out to
+the blue and red edges of the U - and i-band filters,
+respectively, using a univariate spline. We choose to ex-
+trapolate the (extinction-corrected) photometry rather
+than fit a blackbody SED because the spectra are not
+representative of a blackbody throughout the object’s
+evolution.
+To infer the properties of the pre-explosion progeni-
+tor as well as the explosion itself, we compare numerical
+MESA (Paxton et al. 2011, 2013, 2015, 2018, 2019) and
+STELLA (Blinnikov et al. 1998, 2000, 2006) model ex-
+plosions to our pseudo-bolometric light curve. We begin
+with a MESA progenitor with MZAMS = 15 M⊙ and evolve
+it to a final mass of 4.8 M⊙. At explosion the progenitor
+has a H-rich envelope radius of 280 R⊙ and mass of 0.10
+M⊙, in agreement with values we find from our best-
+fit SCE models. The explosion energy and ejecta mass
+are fixed at 0.9 × 1051 erg and 2.9 M⊙, respectively,
+and the mass of 56Ni (MNi) is varied between 0.015 and
+0.020 M⊙. These explosion models are then run through
+STELLA in order to reproduce the bolometric luminosity
+evolution. For more information, see Hiramatsu et al.
+(2021).
+The resulting model light curves are shown in Fig-
+ure 7, compared with the pseudo-bolometric light curve
+of SN 2020bio. We find decent qualitative agreement be-
+tween the numerical models and the observed light-curve
+evolution, particularly at later times.
+The secondary
+light-curve peak and late-time light-curve slope are well
+reproduced by an explosion which synthesizes ≈ 0.02
+M⊙ of 56Ni.
+The secondary light-curve peak may be
+overproduced, but the exact peak luminosity and time
+of peak are uncertain given the gap in our observational
+coverage.
+Interestingly, however, the peak luminosity of the SCE
+is not reproduced by these models. It may be that the
+treatment of the SN shock and the subsequent cool-
+ing of the outer envelope is too complex to fully sim-
+ulate within these models.
+On the other hand, it is
+possible that an additional powering mechanism con-
+tributes to the early-time evolution.
+To test this, we
+explore how the addition of different mass-loss rates
+and timescales to the models affects the early-time light
+curve through short-lived circumstellar interaction. To
+the best-fit MESA model we attach a wind density profile
+ρCSM(r) = ˙Mwind/4πr2vwind, where vwind = 10 km s−1.
+These CSM models are shown in Figure 8. We find that
+the best-fit models have a confined CSM with masses of
+1 × 10−3 – 1 × 10−2 M⊙ lost by the progenitor within
+the last several months before explosion. This hints that
+circumstellar interaction may contribute to the rapidly-
+fading early-time emission of SN 2020bio and possibly
+other SNe IIb. If this is the case, then the information
+estimated through SCE model fits may not be truly rep-
+resentative of the true nature of their progenitors.
+The values inferred from this numerical modelling,
+particularly the 56Ni mass, are on the low end of the
+distribution of values estimated for other well-studied
+SNe IIb. SNe IIb with double-peaked light curves typi-
+cally display secondary radioactive decay-powered peaks
+equally or more luminous than the peak of the SCE,
+implying a greater amount of 56Ni synthesized. Stud-
+
+The Double-peaked Type IIb SN 2020bio
+11
+0
+3
+6
+9
+12
+Days Since Discovery
+1041
+1042
+Pseudo-Bolometric Luminosity (erg s−1)
+MNi = 0.019 M⊙
+vwind = 10 km s−1
+˙Mwind = 0.1 M⊙ yr−1
+twind = 0.1 yr
+˙Mwind = 0.1 M⊙ yr−1
+twind = 0.01 yr
+˙Mwind = 0.1 M⊙ yr−1
+twind = 1.0 yr
+˙Mwind = 0.01 M⊙ yr−1
+twind = 0.1 yr
+˙Mwind = 1.0 M⊙ yr−1
+twind = 0.1 yr
+˙Mwind = 1.0 M⊙ yr−1
+twind = 0.01 yr
+SN2020bio
+Figure
+8.
+Numerical MESA and STELLA circumstellar
+interaction-powered model light curves of SN 2020bio at early
+times. Different color curves correspond to models with vary-
+ing mass-loss rates and timescales. The early-time emission
+excess is best reproduced with 0.001-0.01 M⊙ of CSM.
+ies using samples of these objects have found average
+56Ni masses of ≈ 0.10 – 0.15 M⊙ and average ejecta
+masses of ≈ 2.2 – 4.5 M⊙ (Lyman et al. 2016; Pren-
+tice et al. 2016; Taddia et al. 2018), in better agreement
+with ejecta parameters of other stripped-envelope and
+H-rich core-collapse SNe.
+However, rare cases of un-
+derluminous SNe IIb with low inferred MNi have been
+discovered (e.g., Nakaoka et al. 2019; Maeda et al. 2023).
+These objects have light curves that appear transitional
+between standard SNe II-P and SNe IIb, which differ
+from the observed photometric evolution of SN 2020bio.
+On the other hand, in the case of SN 2018ivc, both a
+low 56Ni mass (MNi ≤ 0.015M⊙) and progenitor mass
+(MZAMS ≲ 12M⊙) are inferred (Maeda et al. 2023). It is
+possible that other SNe IIb with little synthesized 56Ni
+may be undercounted due to their rapidly-fading or un-
+derluminous light curves. Maeda et al. (2023) also con-
+cluded that the light curve of SN 2018ivc was powered at
+least in part by circumstellar interaction. Sustained cir-
+cumstellar interaction has been inferred for other SNe
+IIb, either through late-time spectral features (Maeda
+et al. 2015; Fremling et al. 2019) or through X-ray and
+radio observations (Fransson et al. 1996).
+It may be
+that the mechanism that produced the confined CSM
+inferred from our numerical models of SN 2020bio, and
+possibly that seen in the case of SN 2018ivc, points to
+more extreme mass-loss than found in other SNe IIb.
+4.4. Comparison to Nebula Spectra Models
+A trend between an increasing amount of synthesized
+O and increasing core-collapse SN progenitor mass has
+been extensively studied (e.g., Woosley & Heger 2007).
+Jerkstrand et al. (2015) use this relationship to calibrate
+the [O I] λλ 6300,6364 luminosity, normalized by the
+radioactive decay luminosity at the same phase, with
+numerical models of SNe IIb progenitors (see Eq.
+1
+of Jerkstrand et al. 2015). The authors consider mod-
+els with zero-age main-sequence masses between 12 M⊙
+and 17 M⊙. Comparing the observed normalized [O I]
+luminosity for a handful of SNe IIb, such as SN 1993J,
+SN 2008ax, and SN 2011dh, to these models allows for a
+direct estimate of their progenitor masses—all of which
+fall in the range of masses modeled.
+Here we reproduce this analysis using a nebular spec-
+trum of SN 2020bio, obtained 201 days after the esti-
+mated explosion, shown in Figure 3. We estimate the
+luminosity from the [O I] λλ 6300,6364 emission doublet
+in the same way as Jerkstrand et al. (2015)—assuming
+the width of the feature to be 5000 km s−1, we estimate
+the continuum by finding the minimum flux redward
+and blueward of this width and calculate the luminosity
+within the continuum-subtracted feature. We normal-
+ize this luminosity using the luminosity of 56Ni decay,
+assuming the best-fit MNi from Section 4.3.
+The
+normalized
+luminosity
+at
+201
+days
+is
+Lnorm(t=201) = 9×10−4 ± 2×10−5. This value is lower
+than any of the numerical models analyzed by Jerk-
+strand et al. (2015), implying a progenitor mass ≤ 12
+M⊙.
+A low progenitor mass for SN 2020bio can also
+be inferred from the ratio of the [Ca II] λλ 7311, 7324
+to [O I] λλ 6300, 6364 fluxes.
+A higher ratio implies
+a lower-mass progenitor, with SNe IIb from literature
+having values ≲ 1 throughout their nebular phases (e.g.,
+Terreran et al. 2019; Hiramatsu et al. 2021). Using the
+same procedure as above, we estimate a [Ca II] to [O I]
+ratio of 1.34 ± 0.03—again pointing to a low-mass
+progenitor star.
+Based on its low synthesized 56Ni mass and nebular
+spectral features, we conclude that SN 2020bio was likely
+the core-collapse of a star with a lower mass than the
+progenitors of most other SNe IIb.
+5. DISCUSSION AND CONCLUSIONS
+We have presented rapid multi-band photometric and
+spectroscopic observations of SN 2020bio, a Type IIb SN
+with luminous and rapidly-evolving SCE, beginning ≤ 1
+day after explosion. Compared with other well-observed
+SNe IIb, SN 2020bio has the bluest colors at early times
+as well as unique spectral features with signatures of
+pre-existing CSM. Fitting analytical models of SCE to
+
+12
+Pellegrino et al.
+the early-time light curve gives progenitor radii on the
+order of 100 R⊙ – 500 R⊙ and envelope masses of 0.01
+M⊙ – 0.5 M⊙ for our best-fit models, which are slightly
+greater than values derived for other SNe IIb progenitors
+using the same methods (e.g., SN 2016gkg; Arcavi et al.
+2017). The weak secondary peak powered by radioac-
+tive decay is evidence of relatively little 56Ni synthe-
+sized, MNi ≈ 0.02 M⊙, which is in tension with average
+MNi estimates from samples of other SNe IIb. Numer-
+ical modeling of the progenitor explosion within con-
+fined circumstellar material is consistent with the ob-
+served light curve, showing that circumstellar interac-
+tion is likely needed to reproduce the complete pseudo-
+bolometric light curve. Finally, comparing the nebular
+spectra to numerical models implies a progenitor mass
+≤ 12 M⊙.
+It is difficult to explain all these peculiar features of
+SN 2020bio in one consistent model. The combination of
+its blue colors, early-time spectral features, and numer-
+ical modeling points to interaction with confined H-rich
+CSM that was stripped from the progenitor’s outer enve-
+lope during the months prior to explosion. The best-fit
+progenitor parameters, particularly the large envelope
+radius and low envelope mass, may suggest an inflated
+progenitor undergoing enhanced mass-loss immediately
+before exploding. However, the very low 56Ni and ejecta
+masses inferred from the later-time light curve, as well as
+the nebular spectroscopy, point to a lower-mass progeni-
+tor. It is possible that SN 2020bio was the collapse of an
+unusually low-mass core within a dense CSM produced
+from its lost H layers. Such extensive mass-loss likely re-
+quires interaction with a binary companion, as inferred
+for other SNe IIb (e.g., Maund et al. 2004; Benvenuto
+et al. 2013). Interaction between the SN ejecta and this
+CSM explains the blue colors and narrow H spectral
+features at early times while the small 56Ni mass and
+nebular spectrum indicate a low zero-age main-sequence
+mass.
+This interaction can lead to an over-estimated
+progenitor radius—if the CSM was near enough to the
+progenitor, we may have observed the shock-cooling of
+this extended CSM instead of the outer envelope of the
+progenitor.
+In the future, more detailed models and
+multi-wavelength observations, particularly in the radio
+and X-rays, will be needed to infer SNe IIb progenitor
+mass-loss rates and CSM masses.
+Given the weak H spectral features when compared to
+spectra of other SNe IIb, SN 2020bio may be an inter-
+mediary object between the Type IIb and Ib subclasses,
+representing a progenitor that was recently stripped al-
+most entirely of its H-rich envelope.
+Transitional ob-
+jects between SNe IIb and SNe Ib have been observed
+(Prentice & Mazzali 2017) and can be explained by dif-
+ferent amounts of H remaining in the outer envelope at
+the time of explosion. More difficult to explain are the
+small 56Ni and ejecta masses, which are lower than those
+measured for both SNe IIb and SNe Ib (e.g., Taddia et
+al. 2018). Some objects that exist in the literature with
+both low ejecta and 56Ni masses and observed SCE are
+peculiar SNe Ib as well as Ca-rich transients. However,
+it is difficult to reconcile the photospheric-phase spec-
+tra of SN 2020bio, which are most similar to those of
+other SNe IIb, with the spectra of these objects, which
+are often used to argue for a degenerate or ultra-stripped
+progenitor (Yao et al. 2020; Jacobson-Gal´an et al. 2022).
+Instead, it is more likely that SN 2020bio had a massive
+star progenitor more similar to the progenitors of other
+SNe IIb based on their similar spectral features.
+This study contributes to the overall diversity in the
+progenitors of SNe IIb. More systematic studies of SNe
+with observed SCE will be needed to search for simi-
+larities and differences in their progenitor systems. In
+particular, this work shows the importance of rapid,
+multi-wavelength follow-up of these objects. It is par-
+ticularly important to better understand the number of
+SNe IIb with weak secondary light-curve peaks, such as
+SN 2020bio.
+These objects may have later-time (≥ 5
+days) luminosity below the detection threshold of cur-
+rent all-sky surveys as well as rapid early-time emis-
+sion which evolves too quickly to be extensively followed.
+Therefore we may be under-counting the rates of core-
+collapse, stripped-envelope SNe with low 56Ni and ejecta
+masses. A better understanding of their progenitors will
+be important for exploring the low-mass end of core-
+collapse SNe.
+This work made use of data from the Las Cumbres Ob-
+servatory network.
+The LCO group is supported by
+AST-1911151 and AST-1911225 and NASA Swift grant
+80NSSC19k1639. I.A. is a CIFAR Azrieli Global Scholar
+in the Gravity and the Extreme Universe Program and
+acknowledges support from that program, from the Eu-
+ropean Research Council (ERC) under the European
+Union’s Horizon 2020 research and innovation program
+(grant agreement No. 852097), from the Israel Science
+Foundation (grant No. 2752/19), from the United States
+- Israel Binational Science Foundation (BSF), and from
+the Israeli Council for Higher Education Alon Fellow-
+ship.
+Software:
+Astropy
+(Astropy
+Collaboration
+et
+al.
+2018),
+emcee
+(Foreman-Mackey
+et
+al.
+2013),
+lcogtsnpipe (Valenti et al. 2016), Matplotlib (Hunter
+2007), MESA (Paxton et al. 2011, 2013, 2015, 2018, 2019),
+
+The Double-peaked Type IIb SN 2020bio
+13
+Numpy (Harris et al. 2020), STELLA (Blinnikov et al.
+1998, 2000, 2006)
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+The Double-peaked Type IIb SN 2020bio
+15
+APPENDIX
+A. CORNER PLOTS
+In Figures A1, A2, A3, and A4 we present distributions of the fitted parameters of the models detailed in Section
+4.2.
+1.05
+1.10
+1.15
+1.20
+1.25
+Menv (10−2 M ⊙ )
+1.60
+1.64
+1.68
+1.72
+1.76
+v (104 km s−1)
+400
+480
+560
+640
+Renv (R ⊙ )
+0.60
+0.64
+0.68
+0.72
+0.76
+t0 (days)
+1.05
+1.10
+1.15
+1.20
+1.25
+Menv (10−2 M ⊙ )
+1.60
+1.64
+1.68
+1.72
+1.76
+v (104 km s−1)
+0.60
+0.64
+0.68
+0.72
+0.76
+t0 (days)
+Figure A1. Corner plots showing the fitted parameter distributions for the P15 model.
+
+16
+Pellegrino et al.
+1.50
+1.56
+1.62
+1.68
+1.74
+Menv (10−2 M ⊙ )
+1.300
+1.325
+1.350
+1.375
+1.400
+v (104 km s−1)
+1400
+1600
+1800
+2000
+Renv (R ⊙ )
+0.9775
+0.9790
+0.9805
+0.9820
+t0 (days)
+1.50
+1.56
+1.62
+1.68
+1.74
+Menv (10−2 M ⊙ )
+1.300
+1.325
+1.350
+1.375
+1.400
+v (104 km s−1)
+0.9775
+0.9790
+0.9805
+0.9820
+t0 (days)
+Figure A2. Same as Figure A1, but for the P21 model.
+
+The Double-peaked Type IIb SN 2020bio
+17
+44
+46
+48
+50
+Menv (10−2 M ⊙ )
+1.5
+1.6
+1.7
+1.8
+1.9
+v (104 km s−1)
+125
+150
+175
+200
+Renv (R ⊙ )
+0.08
+0.16
+0.24
+0.32
+0.40
+t0 (days)
+44
+46
+48
+50
+Menv (10−2 M ⊙ )
+1.5
+1.6
+1.7
+1.8
+1.9
+v (104 km s−1)
+0.08
+0.16
+0.24
+0.32
+0.40
+t0 (days)
+Figure A3. Same as Figure A1, but for the SW17 (n=3/2) model.
+
+18
+Pellegrino et al.
+300
+315
+330
+345
+Menv (10−2 M ⊙ )
+1.4
+1.5
+1.6
+1.7
+v (104 km s−1)
+160
+200
+240
+280
+Renv (R ⊙ )
+0.1
+0.2
+0.3
+0.4
+t0 (days)
+300
+315
+330
+345
+Menv (10−2 M ⊙ )
+1.4
+1.5
+1.6
+1.7
+v (104 km s−1)
+0.1
+0.2
+0.3
+0.4
+t0 (days)
+Figure A4. Same as Figure A1, but for the SW17 (n=3) model.
+

-NAzT4oBgHgl3EQfSvt8/content/tmp_files/2301.01237v1.pdf.txt

@@ -0,0 +1,1928 @@
+Safe Path following for Middle Ear Surgery
+Bassem Dahroug1, Brahim Tamadazte2, and Nicolas Andreff1
+1Bassem Dahroug and Nicolas Andreff are with FEMTO-ST, AS2M,
+Univ. Bourgogne Franche-Comt´e,
+CNRS/ENSMM, 25000 Besan¸con, France
+2Brahim Tamadazte is with Sorbonne Universit´e,
+CNRS UMR 7222, INSERM U1150, ISIR, F-75005, Paris, France.
+[email protected]
+January 4, 2023
+Abstract
+This article formulates a generic representation of a path-following
+controller operating under contained motion, which was developed in
+the context of surgical robotics. It reports two types of constrained
+motion: i) Bilateral Constrained Motion, also called Remote Center
+Motion (RCM), and ii) Unilaterally Constrained Motion (UCM). In
+the first case, the incision hole has almost the same diameter as the
+robotic tool, while in the second state, the diameter of the incision
+orifice is larger than the tool diameter. The second case offers more
+space where the surgical instrument moves freely without constraints
+before touching the incision wall.
+The proposed method aims to combine two tasks that must oper-
+ate hierarchically: i) respect the RCM or UCM constraints formulated
+by equality or inequality, respectively, and ii) perform a surgical as-
+signment, e.g., scanning or ablation expressed as a 3D path-following
+task. The proposed methods and materials were successfully tested
+first on our simulator that mimics realistic conditions of middle ear
+surgery, then on an experimental platform. Different validation sce-
+narios were carried out experimentally to assess quantitatively and
+qualitatively each developed approach. Although ultimate precision
+was not the goal of this work, our concept is validated with enough
+accuracy (≤ 100µm) for the ear surgery.
+keywords: Medical Robotics, Constrained motion, Path Follow-
+ing, Visual Servoing.
+1
+arXiv:2301.01237v1  [cs.RO]  3 Jan 2023
+
+1
+INTRODUCTION
+Surgical robots are gaining more popularity due to their advantages for both
+the patient and the physician [11,37,40]. It is particularly valid for so-called
+Minimally-Invasive Surgery (MIS) approaches. For instance, a laparoscopy
+or keyhole surgery [23] performs incision around 10mm. It is tiny compared
+to the larger incisions needed in laparotomy (open surgery). Another sit-
+uation where the surgical instruments could be inserted through a natural
+orifice (e.g., mouth, nasal clefts, urethra, anus) to reach the targeted organ.
+In both cases, the entry space (i.e., the incision hole or the natural orifice)
+restricts the surgical tool motion, consequently the surgeon’s hands and the
+robot carrying the instrument [7].
+This article mainly discusses two types of constrained motion that result
+directly from MIS procedures:
+1. Remote Center Motion (RCM), also known as fulcrum effect, implies
+the incision hole has almost the same diameter as that of the surgical
+tool [2];
+2. Unilaterally Center Motion (UCM) implies the incision diameter size
+is bigger than that of the tool, offering more freedom for the tool
+motion [12].
+The first type of motion was initially achieved by designing a particu-
+lar robotic structure that imposes the constrained motion mechanically [2,
+21, 29]. The RCM dictates that the center-line of the surgical tool is al-
+ways coincident with the center point of the incision orifice (trocar point).
+Consequently, the linear movement of the tool is prohibited along two axes.
+The main advantage of RCM mechanisms is to reduce the risk of damag-
+ing the trocar wall because their kinematic structures ensure the pivoting
+motion. Their modest controller is also easy to implement. However, this
+kind of mechanism is restricted to a unique configuration and cannot provide
+enough flexibility for shifting the location of the trocar point.
+An alternative solution proposes a software RCM for overcoming the
+previous problem by guiding a general-purpose robot with the advantage of
+being flexible enough for achieving complex tasks [6]. This solution is con-
+venient for diverse medical applications (e.g., laparoscopic [32] and eye [28]
+surgeries). However, we claim that the RCM approach is not the best choice
+for other surgery types (e.g., ear, nose, mouth, knee arthroscopy). In latter
+cases, the orifice diameter is generally bigger than the tool diameter. Con-
+sequently, the RCM controller imposes too strong limitations on the tool
+2
+
+motion. Indeed, the RCM is a mathematical equality constraint (i.e., the
+distance between the tool body and the center point of the incision orifice
+must be equal to zero). As such, RCM motion can be named as a bilaterally
+constrained motion. On the contrary, UCM is a weaker restriction since the
+unilateral constraints are inequality equations (i.e., the latter distance could
+be greater or less than zero) [19].
+In the literature, the term forbidden-region virtual fixtures [1] are used
+for collaboration tasks where the user can either manipulate a robotic de-
+vice [5] or telemanipulate a master device [33].
+These fixtures could be
+defined as geometric forms [12,39] or vector field [26] around the tool. Then
+a kinematic control [12] or dynamic one [26, 38, 39] is applied to guide the
+robot during the desired task.
+The theoretical contribution of this article lies in the improvement of the
+generic formulation of constrained motion. It has the objective to achieve a
+velocity controller that can maintain the RCM or UCM depending on the
+configuration of the surgical procedure. Besides that, it reveals a new path-
+following controller integrated with a task-hierarchy controller for imposing
+a priority between the RCM/UCM and the path-following tasks.
+Nevertheless, the technical contribution lies in the assessment of such
+approaches. Therefore, we developed a simulator including surgical tools and
+a numerical twin mimicking the middle ear cavity. Based on the auspicious
+evaluation, we also carried out a pre-clinical setup that takes up the diverse
+components of the simulator to assess the proposed methods experimentally.
+Various scenarios are also implemented to accomplish these evaluations. The
+obtained performances in terms of behavior and accuracy are promising.
+The remainder of the article is organized as follows. Section 2 presents
+the clinical needs and challenges. The methodology followed to design the
+proposed controllers will be discussed in Section 3. After that, Section 4
+focuses on both the numerical and experimental validations of the proposed
+approaches. Ultimately, Section 5 presents the conclusion and perspectives.
+2
+MEDICAL MOTIVATIONS
+2.1
+Treated Disease
+The work discussed in this article represents a part of a long-term project.
+It deals with the development of a robotic system that is dedicated to
+cholesteatoma surgery. The system will aim to achieve an MIS within the
+middle ear cavity by passing through the external ear canal or an incision
+orifice made on the mastoid portion.
+3
+
+Cholesteatoma is a frequent disease that invades the middle ear. It in-
+fects the middle ear by introducing abnormal skin (lesional tissue) in the
+middle ear-cavity. The most common explanation [31] is due to the immi-
+gration of the epidermal cells, which are the cells type in the external ear
+canal, and cover up the mucosa of the middle ear cavity, as shown in Fig.1.
+These cells gradually proliferate within the temporal bone and destroy the
+adjacent bony structures.
+Figure 1: Evolution of cholesteatoma disease within the middle ear, which
+is located behind the tympanic membrane.
+The evolution of cholesteatoma is life-threatening in the long run. The
+complications can be classified as follows [3]: i) destruction of the ossicular
+chain, ii) facial paralysis, iii) labyrinthitis, iv) extracranial complications,
+and v) intracranial complications. It can notice the irreversible effects that
+cholesteatoma can cause in a patient. Despite that, there is no drug therapy
+for the treatment. The only solution is surgical intervention.
+2.2
+Current Surgical Procedure
+As claimed above, the only treatment for cholesteatoma is a surgical pro-
+cedure. It aims to eradicate all cholesteatoma tissue and reconstruct the
+anatomy of the middle ear [18].
+For reaching the middle-ear cavity, the surgeon often drills the temporal
+bone behind the auricular, as shown in Fig. 2. This surgical procedure is
+called mastoidectomy where the surgeon maintains the wall of the exter-
+nal ear canal. This technique creates an incision that forms a triangular
+(around 40 × 40 × 30mm) with a depth of about 30mm. The latter pro-
+4
+
+pdemni
+Mucosa
+Demis
+Submucosa
+Muscularis
+Retraction and perforation of the tympanic
+membrane
+Normal tympanic
+Cholestetoma
+membraneFigure 2: Mastoidectomy procedure with canal-wall-up indicates that the
+external ear canal is preserved. (a) side view of the mastoidectomy tunnel
+and (b) top view of the mastoidectomy tunnel.
+cedure can also become more invasive by sacrificing the posterior portion
+of the external ear canal (i.e., canal-wall-down). Furthermore, even if the
+surgical orifice is relatively large, the surgical procedure remains complex
+and requires high expertise and dexterity from the surgeon. Also, even with
+an experienced clinician in the cholesteatoma case, the clinical outcomes
+remain unsatisfactory in terms of effectiveness. Indeed, there is a high risk
+that the cholesteatoma could regrow a few months after the surgical inter-
+vention. It occurs due to residual cholesteatoma cells. Consequently, 10 to
+40% of patients perform more than one surgery to get definitively over this
+disease [4].
+Due to the complexity of the temporal bone cavity, the surgeon mainly
+faces numerous difficulties during the procedure (Fig.3): i) lack of ergonomy
+of the tools; ii) limited field of view of the oto-microscope (the surgeon can-
+not visualize the lateral regions hidden (blind spots) in the middle ear cavity)
+and iii) access with the conventional rigid instruments requires considerable
+expertise to handle.
+Therefore, it is increasingly important to overcome the previous problems
+and evolve this procedure towards less invasive.
+It implies reducing the
+incision orifice size, improving the cholesteatoma ablation efficiency, and
+avoiding the current high surgical recurrence rate for this kind of surgery.
+3
+METHODOLOGY
+This section begins by presenting a brief summary of the new surgical pro-
+tocol associated with the robotic system. After that, it discusses the hier-
+archical controller for managing simultaneously the various tasks. It then
+5
+
+Wall of external
+Wallof
+earcanal
+external
+earcanal
+Removed
+bone
+Cholesteatoma
+Cholesteatoma
+(a)
+(b)Figure 3: Conceptual scheme to demonstrate the ”blind spot” during the
+cholesteatoma surgery.
+explains separately the path-following, the RCM, and the UCM controllers.
+3.1
+New Surgical Protocol
+In collaboration with surgeons experts in middle ear surgery, especially
+cholesteatoma treatment, we have attempted to set up a new and more
+efficient surgical protocol reported in [11].
+Firstly, the idea is to make
+cholesteatoma surgery less invasive compared to the traditional one. Thus, a
+macro-micro robotic system should pass through a millimetric incision made
+behind the ear (in the mastoid portion) to access the middle ear cavity [35].
+Secondly, cholesteatoma surgery needs to be more efficient by eliminating
+the residual cases. This second objective can be accomplished by removing
+a large part of the cholesteatoma tissue using rigid miniature mechanical
+resection tools. After that, a bendable actuated tool [16,36] could be used
+to guide a laser fiber. This fiber carbonizes the residual cholesteatoma (re-
+sulting from the mechanical resection phase) [22].
+Both mechanical resection and laser ablation should be performable ei-
+ther in automatic or semi-automatic mode. While the mechanical resection
+does not require high accuracy, the laser ablation requires higher precision
+since the residual cholesteatoma cells can be a few tens of micrometers in
+size. Therefore, the contributions of robotics and vision-based control are
+essential to fundamental this kind of task. In this work, we investigated
+6
+
+Microscope
+Wall of the external
+Incision of
+the mastoidectomy
+ear canal
+Outside
+blind spot
+Within
+blind spot
+Suction tool
+Cholesteatoma
+Knifethe use of path-following control schemes under constrained motion (due to
+the incision orifice) to carry out the notions requested by the cholesteatoma
+removal (i.e., mechanical resection and laser ablation).
+3.2
+Task Hierarchical Controller
+A surgical procedure can be considered as a set of sequential or overlap-
+ping sub-tasks. The hierarchical methods ensure the execution of several
+tasks simultaneously. Consequently, the required tasks do not enter into
+conflict [13,34]. In the case of cholesteatoma surgery, various sub-tasks can
+be involved during the procedure, such as constraint enforcement (RCM
+or UCM) and ablation tools for the pathological tissues. Therefore, these
+sub-tasks must be carried out according to a defined hierarchical scheme.
+To express a controller that manages simultaneous sub-tasks, let us start
+by assuming that a generic sub-task (˙ei ∈ Rmi) given by
+˙ei = Li eve,
+where i=1,2,...,j
+(1)
+where eve ∈ se(3) is the end-effector twist velocity to be computed in the
+end-effector frame Fe, and Li ∈ Rmi×n is the interaction matrix which
+relates the vector eve to the error ˙ei.
+The inverse solution of the previous equation is not guaranteed since the
+interaction matrix Li could be non-square, and the matrix rank is locally
+deficient. Thanks to the least-square method, an approximate solution can
+be found by minimizing ∥˙ei − Li eve∥ over eve, and using numerical proce-
+dures (such as QR or SVD). The formal result of it can be simply written
+as eve = L†
+i ˙ei, where L†
+i is the pseudo-inverse of Li. If Li does not have
+full rank then it has at least one singular vector z1, located in its null-space
+(Liz1 = 0).
+The vector z1 is also described as the null space of ei, be-
+cause any twist vector parallel to z1 will leave ei unchanged. Therefore, the
+projection gradient general form [27] is given by
+eve = L†
+1˙e1 + (I − L†
+1L1)z1
+(2)
+In order to define z1, let us first consider a secondary sub-task ˙e2 =
+L2 eve. Since the control vector must include the first sub-task, equation
+(2) is injected in the latter expression, resulting in
+˙e2 = L2
+�
+L†
+1˙e1 + (I − L†
+1L1)z1
+�
+= L2L†
+1˙e1 + L2(I − L†
+1L1)
+�
+��
+�
+˜L2
+z1
+(3)
+7
+
+From the previous equation, the vector z1 is deduced as
+z1 = ˜L†
+2(˙e2 − L2L†
+1˙e1) + (I − ˜L†
+2˜L2)z2
+(4)
+with another criteria vector z2 which is projected in the null-space of the
+secondary sub-task. By introducing (4) in (2), a recursive form of the pro-
+jection gradient is obtained as
+eve = L†
+1 ˙e1 + (I − L†
+1L1)
+�
+˜L†
+2(˙e2 − L2L†
+1 ˙e1) + (I − ˜L†
+2˜L2)z2
+�
+= L†
+1 ˙e1 + (I − L†
+1L1)˜L†
+2(˙e2 − L2L†
+1 ˙e1)
++ (I − L†
+1L1)(I − ˜L†
+2˜L2)z2
+(5)
+The right-hand side of the previous equation can further be simplified as [24]
+eve = L†
+1˙e1 + ˜L†
+2(˙e2 − L2L†
+1˙e1)
+.
+(6)
+The latter equation finds a solution to satisfy both sub-tasks ˙e1 and ˙e2.
+It also ensures a form of hierarchy/priority between them. The analytical
+expression of each sub-task with its Li is presented in the coming sections.
+3.3
+6D Approach Controller
+This section is dedicated to mathematically describing how to control the
+tool-tip for regulating its position and orientation with respect to a reference
+frame, e.g., the orifice frame Fr. This task is applied when the tool locates
+outside the incision orifice, and its pose must be adjusted with respect to
+the orifice before it starts another task inside the orifice.
+To do this, a traditional 3D position-based visual servo [8] is applied.
+The feature vector s
+=
+(rtt, θ rut) is defined as the pose vector which
+describes the tool-tip frame Ft with respect to the orifice frame Fr. This
+vector gathers the translation t of the tool-tip and its rotation θu in form
+of angle/axis parameterization. The desired feature vector s∗ = (0, 0) is
+set to a zero vector since it is required to make coincident the frame Ft with
+Fr. Thus, the approach task error eapp is deduced as the difference between
+the current features vector and the desired one, i.e.,
+eapp = s − s∗
+(7)
+The time variation of the latter error is related to the spatial velocity of
+the tool-tip tvt by the interaction matrix L3D ∈ R6×6 as
+˙eapp = L3D tvt
+(8)
+8
+
+where tvt = (tvt,t ω) gathers the instantaneous linear and angular velocities
+of the tool-tip. Since the desired feature vector equals to 06×1, then the
+interaction matrix L3D is determined by
+L3D =
+� −I3×3
+03×3
+03×3
+Lθu
+�
+(9)
+where I3×3 is a 3 × 3 identity matrix, 03×3 is a 3 × 3 zero matrix, and Lθu
+is given by [25]
+Lθu = I3×3 − θ
+2 [u]× +
+�
+1 − sinc θ
+sinc2 θ
+2
+�
+[u]2
+×
+(10)
+in which sinc x is the sinus cardinal.
+Finally, the spatial velocity tvt is determined for ensuring an exponential
+decoupled reduction of the error (i.e., ˙e = −λe) as
+tvt = −γL−1
+3Deapp
+(11)
+where γ is a gain coefficient, and L−1
+3D is the inverse of the interaction matrix
+since it is square and has a closed-form inverse [25].
+The command velocity of the robot end-effector eve = eVt tvt is deduced
+by the following twist matrix
+eVt =
+�
+eRt
+[ett]×
+eRt
+03×3
+eRt
+�
+(12)
+since the tool body is rigid and the transformation between the end-effector
+frame Fe and the tool-tip frame Ft is fixed. Finally, the controller stability
+was demonstrated in [25] to be globally exponentially stable.
+3.4
+3D Path-Following Controller
+This section will focus on a generic modelling of a 3D path-following scheme.
+The advantage of using such as controller is the separation between i) the
+geometric curve (desired path Sp) which is planned by the surgeon based on
+pre-operative images, and ii) the advance speed (vtis) of the tool-tip along
+the desired path which is controlled by the surgeon during the operation. In
+this manner, the collaboration surgeon/robot ensures that the robot guides
+the tool along the path while the surgeon controls the robot progression
+without planning the robot velocity direction.
+9
+
+Figure 4: Orthogonal projection of the tool-tip onto a geometric curve.
+Fig. 4 depicts the surgical instrument and its reference frames with re-
+spect to the desired path Sp. By projecting the tool-tip Ot onto the reference
+path, the resultant orthogonal distance dpf is considered as the error (i.e.,
+lateral deviation) which must be controlled to zero. Therefore, the 3D vec-
+tor distance between the tool-tip Ot and the projection point pp′ calculated
+as
+dpf = Ot − pp′.
+(13)
+In order to express the command velocity, the time-derivative of (13)
+provides the tool-tip velocity vt as discussed in [10]
+˙dpf =
+�
+�I3×3 −
+kpk⊤
+p
+1 − d⊤
+pf
+�
+Cp(sp) × kp
+�
+�
+� vt
+(14)
+where Cp(sp) is the path curvature in function of the path curve length, kp
+is the unit-vector of the instantaneous tangential vector (Fig. 4).
+At this stage, it requires to choose the adequate velocity of the tool-tip
+vt in the latter equation to ensure that the lateral error dpf is regulated
+to zero while progressing along the path. An intuitive solution consists of
+decomposing the control velocity into two orthogonal components (Fig. 5): i)
+the advance velocity (vadv) along the path, and ii) the return velocity (vret)
+for regulating the tool deviation from the reference path.
+The previous
+10
+
+desired 3D path
+mk-1
+tool body
+mk
+mk+1
+M
+mk+2Figure 5: Representation of the different velocities involved in the path-
+following controller.
+concept is formulated as follows:
+vt = αkp
+����
+vadv
++ βdpf
+� �� �
+vret
+.
+(15)
+The tuning coefficients of the controller α and β allow adjusting the
+priority between the advance and return velocities, respectively.
+Besides
+that, the controller stability demonstrated in [10] shows that α should be
+a positive scalar while β must be a negative scalar to ensure the system
+stability.
+The choice of these gain factors can be imposed by a function of a con-
+stant velocity vtis > 0 that depends on the interaction between the surgical
+tool and the lesional tissue.
+This velocity could be tuned easily by the
+surgeon before or during the intervention. Therefore, (15) yields
+v2
+tis
+����
+=∥vt∥2
+= α2∥kp∥2
+� �� �
+=1
++ β2∥dpf∥2
+�
+��
+�
+=∥vret∥2
+.
+(16)
+The gain factor α is thus determined as
+α =
+� �
+v2
+tis − ∥vret∥2
+∥vret∥2 < v2
+tis
+0
+∥vret∥2 > v2
+tis
+.
+(17)
+If the tool is not far from the reference path, the first condition in (17) is
+selected. Otherwise, the priority is returning the tool-tip to the reference
+path, and the advance velocity is null (i.e., second condition in (17)).
+11
+
+desired 3D path
+tool body
+mk
+ret
+adv
+mk+1The latter strategy proposed in [10] applies a constant value for the gain
+factor β. However, this section presents a new formulation of β to make
+the controller sensitive to the path curvature. Thus, it is calculated by the
+following equation
+β = β′
+�
+1 + sign
+�
+d⊤
+pf (Cp(sp) × kp)
+� �
+1 − eγc∥Cp(sp)∥� �
+(18)
+where β′ is a negative gain for returning to path, sign(•) is a sign function
+to determine the direction along the reference path, and γc is a negative
+gain for sensing the amount of path curvature.
+The ratio between the gain factors (i.e., vtis and β′) forms an acceptable
+error band around the reference path. For instance, if β′ is higher than vtis,
+then the error band will be small. On the contrary, in the case where vtis
+is bigger than β′, then the error band will be large since the priority is to
+advance along the reference path. The effect of this ratio is presented in
+section 4.
+Furthermore, the control velocity of the tool-tip (15) could be repre-
+sented with respect to any desired frame.
+Note that if the end-effector
+frame is selected, then the end-effector twist velocity eve is related to the
+linear velocity of the tool-tip evt as
+evt = [I3×3
+− [eet]×]
+�
+��
+�
+Lpf∈R3×6
+� eve
+eωe
+�
+�
+��
+�
+eve
+(19)
+whereby [eet]× is the skew-symmetric matrix associated to the vector eet,
+and Lpf is the interaction matrix related to the path-following task.
+Finally, the control velocity for the path-following task is deduced as
+eve = L†
+pf
+evt
+.
+(20)
+3.5
+Bilateral Constrained Motion Controller
+As claimed above, the resection/ablation task is performed in a minimally
+invasive procedure. Therefore, the robot should perform the surgical task
+under the constraints of the incision point. This section begins with the
+description of RCM (bilateral constraints), while the following section de-
+scribes the UCM (unilateral constraints). The RCM imposes that the center-
+line of tool body St should be coincident with the point Or. Simultaneously,
+the tool-tip must follow the desired path inside the incision orifice.
+12
+
+Figure 6: Geometric scheme of the bilateral linear error drcm.
+Fig. 6 shows a straight tool which is located far from the center-point
+of incision orifice Or. The previous works [10,12] built the controller based
+on the angular error between the vectors et′ and er while the proposed
+controller in this section is based on the linear error drcm. This new choice
+offers the controller to become independent of the tool shape. Let us imagine
+that the tool-tip position in Fig. 6 is fixed in space, but its length can change.
+In the case of angular error, when the tool length increases, the error reduces
+its value.
+However, the linear error stays constant when the tool length
+changes. Therefore, the new choice grants better numerical computing.
+The error drcm is deduced by the orthogonal projection of the point Or
+onto the tool body St. The point pt′ is resultant from the latter projection
+that is calculated as follows
+ept′ =
+euet eu⊤
+et
+eer
+(21)
+whereby euet is the unit vector of et expressed in Fe, and eer represents the
+vector between both points Oe and Or which is expressed in Fe.
+In case the surgical tool is curved, the point pt′ is determined by dis-
+cretizing the tool body. Then the closest point onto the tool body is located.
+After that, the orthogonal projection is performed with respect to this point
+and the previous one on the tool center-line. Thus, the error drcm is de-
+duced as
+drcm =
+eOr − ept′
+.
+(22)
+13
+
+center of the incision hole
+er
+et'
+tool body
+Pt
+rcm
+X
+M
+incision wallThe controller task is to find the spatial velocity of the robot end-effector
+eve for eliminating the rate-of-change of the bilateral linear error drcm.
+Thereby, the time-derivative of the latter equation results in
+˙drcm =
+evr − evt′
+(23)
+where evt′ is the linear velocity of the projected point pt′ along the tool
+body, and evr is the linear velocity of the trocar point described in Fe.
+Indeed, the velocity of the projected point depends on the movement of the
+tool body with respect to the trocar point. Hence, this velocity is computed
+as [12]
+evt′ =
+ekt ekT
+t
+1 + dTrcm(Ct(st) × ekt)
+evr
+(24)
+whereby Ct(st) is the tool curvature in the function of its arc length, and
+ekt is the instantaneous tangential unit-vector onto the tool curve/shape.
+Since the calculation is done in the perspective of the end-effector frame
+Fe, it implies that this frame is fixed, and the other ones are dynamic with
+respect to it. Consequently, the incision orifice virtually moves, and its linear
+velocity evr is related to the spatial velocity of the robot end-effector thanks
+to the following formula
+evr =
+�
+I3×3
+− [eOr]×
+�
+�
+��
+�
+Lr∈R3×6
+eve
+.
+(25)
+By injecting the latter equation in (24) then the resultant in (23), the
+time-derivative of the error drcm equals to
+˙drcm =
+�
+I3 −
+ekt ekT
+t
+1 + dTrcm(Ct(st) × ekt)
+� �
+I3×3
+− [eOr]×
+�
+�
+��
+�
+Lrcm∈R3×6
+eve
+(26)
+where Lrcm is the interaction matrix which relates between the end-effector
+velocity eve and the rate-of-change of the error drcm.
+Furthermore, a linearized proportional controller is applied to reduce
+the bilateral linear error in an exponential decay form. It defines the control
+velocity of the end-effector as
+eve = −λ L†
+rcm drcm.
+(27)
+whereby λ is a positive gain which allows tuning the rate of exponential
+decay, and L†
+rcm is the pseudo-inverse of the interaction matrix Lrcm.
+14
+
+Finally, the RCM task can be combined as the highest priority with the
+path-following task as the secondary criteria. The hierarchical controller
+deduces the control velocity, by replacing the equations (27) and (20) in
+equation (6), as
+eve = −λL†
+rcmdrcm + ˜L†
+pf
+�
+evt + λLpfL†
+rcmdrcm
+�
+,
+with
+˜Lpf = Lpf
+�
+I − L†
+rcmLrcm
+�
+.
+(28)
+In the opposite case, the hierarchical controller sets the path-following task
+(20) as the highest priority while the RCM task (27) as the secondary one.
+The control velocity is deduced from equation (6) as
+eve = L†
+pf
+evt − ˜L†
+rcm
+�
+λdrcm + LrcmL†
+pf
+evt
+�
+,
+(29)
+with
+˜Lrcm = Lrcm
+�
+I − L†
+pfLpf
+�
+.
+(30)
+3.6
+Unilaterally Constrained Motion Controller
+This section continues with the design of the path-following controller under
+unilateral constraints. Notice that the UCM task assumes the incision orifice
+is larger than the tool diameter. Consequently, it imposes on the tool-tip
+to follow the incision/ablation path while the tool body is free to move
+within the incision orifice as long as it does not damage the orifice wall.
+Therefore, the formulation of the previous section needs to extend to satisfy
+the unilateral constraints.
+Fig. 7(left image 1) shows how the point pt′ is orthogonally projected
+onto the orifice wall in order to determine the closest point ph′ on the orifice
+wall Sh. The distance between the latter two points forms the vector error
+ducm which can be defined as (left image 2 of Fig. 7)
+ducm =
+et′r
+����
+=drcm
+− eh′r
+����
+=dwall
+.
+(31)
+The question now is how to maintain the value of the error ducm greater
+or equal to zero. For security issues, three regions are defined around the
+projected point ph′, as shown in the left image of Fig. 7:
+1. critical zone (dark red circle) which its border is defined by a minimal
+distance dmin;
+15
+
+Figure 7: Geometric modelling of the unilateral linear error ducm.
+2. dangerous zone (light green circle) which its border is defined by a
+maximal distance dmax; and
+3. safe zone which is the remain region outside the dangerous zone.
+When the Euclidean norm ∥ducm∥ is larger than the ”dangerous” dis-
+tance dmax, the tool can follow the reference path without any constraints
+since its location is in the safe zone.
+However, an admittance control is
+activated, which is composed of a virtual damper µobs, when the tool body
+passes the dangerous zone border. Indeed, the admittance control imposes
+unilateral constraint towards the safe point ps by generating a compensation
+velocity in the opposite direction to the orifice wall.
+By differentiating equation (31) with respect to time for deducing the
+velocity twist of the end-effector, it becomes equal to
+˙ducm = ( evr − evt′)
+�
+��
+�
+˙drcm
+− ( evr − evh′)
+�
+��
+�
+˙dwall
+=
+evh′ − evt′
+(32)
+The velocity of the projected point ph′ is deduced in the same way as
+equation (24)
+evh′ =
+ekh ekT
+h
+1 + dTucm (Ch(sh) × ekh)
+evt′
+.
+(33)
+16
+
+Ph
+tool body
+(1)
+centerofthe
+dangerous zone
+incision hole
+et'
+incision wall
+ducm
+d
+rcm
+d
+11DM,
+30
+critical zone
+25mm
+(2)where Ch(sh) is the orifice curvature in function of its arc length, and ekh is
+the instantaneous tangential unit-vector onto the orifice curve. In another
+perspective, the latter equation describes how the projection of the point
+pt′ onto the geometric curve of the orifice wall Sh evolves with time.
+The velocity evt′ is deduced by combining equations (24) and (25)
+evt′ =
+ekt ekT
+t
+1 + dTrcm (Ct(st) × ekt)
+�
+I3×3
+− [eOr]×
+�
+�
+��
+�
+Lvt′ ∈R3×6
+eve
+.
+(34)
+Replacing equations (33) and (34) in (32) yields
+˙ducm =
+�
+ekh ekT
+h
+1 + dTucm (Ch(sh) × ekh) − I3×3
+�
+Lvt′
+�
+��
+�
+Lucm∈R3×6
+eve
+(35)
+whereas Lucm is the interaction matrix that relates the twist end-effector
+with the rate of change of the error ducm.
+Thereby, the control velocity of the UCM task is defined as
+eve = −µobsλL†
+ucmducm
+.
+(36)
+The damping coefficient µobs changes following a sigmoid function that
+depends on the vector ducm. It means that the gain µobs reaches its mini-
+mal value when ducm is higher than the safe distance dmax, where the tool
+location in the dangerous zone. However, µobs gradually increases until it
+reaches its maximal value when ducm is smaller than the critical distance
+dmin, where the tool location in the critical zone. This behaviour is modeled
+as
+µobs =
+σmax
+1 + e
+�
+σstep
+�
+∥ducm∥−σmin
+��
+(37)
+where σmax, σmin and σstep are tunable parameters for modifying the sigmoid
+form.
+Finally, the path-following task can be combined as the highest priority
+with the UCM task as the secondary criteria. The hierarchical controller
+deduces the control velocity, by replacing the equations (36) and (20) in
+equation (6), as
+eve = L†
+pf
+evt − ˜L†
+ucm
+�
+µobsλducm + LucmL†
+pf
+evt
+�
+,
+with
+˜Lucm = Lucm
+�
+I − L†
+pfLpf
+�
+.
+(38)
+17
+
+4
+VALIDATION
+This section discusses several scenarios to evaluate qualitatively and quan-
+titatively the proposed methods and materials. The developed controllers
+were first tested using our simulator framework and then in an experimental
+set-up that takes up the various components of the simulator.
+4.1
+Implementation Issues
+This part begins by converting the patient’s ear to its numerical-twin and
+then its 3D printed-twin. The first step to accomplish this job is the scan of
+the patient’s ear during the preoperative phase for getting DICOM (Digital
+Imaging and Communications in Medicine) images, as depicted in Fig. 8.
+The DICOM images are handled by the software 3D Slicer which converts
+these images to a 3D surface model after a segmentation process.
+Prior
+works were done in relation to this subject for achieving an automated seg-
+mentation process (e.g., [15, 30]). However, the segmentation process that
+we have done manually is not automated since this is not the focus of this
+article. In the future, we believe that our segmentation process needs to be
+done again in an automated manner for efficiency.
+The 3D Slicer software exports the segmentation results as STL files for
+each anatomical structure. Afterward, the software MeshLab treats the STL
+files for smoothing the surface and reducing the number of vertices and faces
+to cut down the final STL file size. This step produces the numerical-twin
+of the patient’s ear.
+The next step creates the 3D printed-twin for conducting the experimen-
+tal validation. Indeed, a simplified version of the numerical-twin is imported
+in Solidworks for i) adding some thickness to the middle ear cavity and ii)
+creating the incision orifice through the mastoid.
+After that, the planning stage of the desired path within the middle
+ear cavity begins. The path planning step can be optimized (e.g., [14,20]).
+However, this step was done manually on Solidworks to generate text files
+that contain the geometry of the reference path and the orifice wall as a
+sequence of 3D points. These files are inputs for the controller. This step
+should be investigated in the future and add to the adequate functions in
+the simulator.
+Fig. 9 presents the proposed control architect with the TCP/IP com-
+munication. This architect allows easy interchangeability between the real-
+system (robot) and its numerical-twin (simulator). The latter figure (the
+red block at the left-hand side) also shows that the implemented controller
+18
+
+Figure 8: The steps done to achieve a numerical and physical model of the
+middle ear cavity.
+is firstly initialized with the end-effector and the incision orifice poses, ⋆Te
+and ⋆Tr respectively. These poses must be described in the same frame
+(e.g., the world frame Fw or the camera Fc). Indeed, the tool geometry
+19
+
+DICOM images of the patient's ear
+Preoperative scanning for the patient
+Surface structure (numerical ear twin) of the patient's ear
+Ossicles
+Inner ear
+Temporal bone
+Middle ear
+External ear
+Chorda
+cavity
+canal
+External ear canal
+tympani nerve
+3D printed ear twin version for
+experimental validation
+Preoperative planning phase
+Red region
+will be preserved
+Mastoidectomy
+Temporal bone
+(Canal wall up)
+Green region
+External ear canal
+will be removedFigure 9: Block diagram of the TCP/IP communication between the client
+(proposed controller) and the server (simulator or robot) or vice-versa.
+20
+
+Robot
+Contro
+connection
+unit
+Simulator
+Robot control (case 2)
+Control Unit
+Robot control (case 1)
+Simulator controlSt is defined with respect to the end-effector frame Fe while the reference
+path Sp and the orifice wall Sh are described in the incision orifice frame
+Fr. Furthermore, the controllers should be initialized by the different gain
+coefficients before the control-loop starts.
+The hierarchy controller arranges throughout the control-loop the prior-
+ity between the different tasks (i.e., the approach task, the path-following
+task, and the RCM/UCM constraints). Indeed, the control-loop is mainly
+divided into three phases:
+1. the outside phase: the tool corrects its initial pose with respect to the
+incision orifice. This stage applies the approach task for regulating: i)
+the tool-tip position to the point located before the orifice center point,
+and ii) the tool-tip rotation as the rotation of the orifice reference
+frame. This manoeuvrer is performed to ensure some security for the
+next phase;
+2. the transition phase: the tool-tip passes the center point of the inci-
+sion orifice. The RCM controller could oscillate when the trocar point
+is close to the tool-tip. These oscillations are generated because the
+controller computes large rotation displacement, due to the lever phe-
+nomena, for compensating the rotation error. Thus, the trocar point
+is virtually moved to the first point on the reference path.
+Conse-
+quently, the tool body can rotate about this new point. This virtual
+trocar point moves towards the orifice frame while the tool-tip ad-
+vances along the reference path;
+3. the inside phase: the tool-tip follows the desired path while the tool
+body is constrained by the orifice wall or the orifice center point.
+Therefore, the output of this block is the spatial velocity of the end-effector
+expressed in its frame (eve) while its inputs are the instantaneous poses of
+the end-effector and the incision orifice (⋆Te and ⋆Tr). The question now
+is: what is the observation frame?
+In the simulator case (the blue block at the right-hand side of Fig. 9), it is
+straightforward since the user initializes the poses with respect to the world
+frame Fw of the virtual scene. Thus, the spatial velocity eve is transformed
+to wve then it is integrated over the sample time Te to deduce the new pose
+of the end-effector. Consequently, the tool pose is updated in the virtual
+scene, and this new pose is sent back to the control unit block for computing
+a new iteration.
+There are two options for designing the control architect in the exper-
+imental case. The first one consists of using an exteroceptive sensor (e.g.,
+21
+
+camera) for estimating the required poses. This option is depicted in the
+green block of Fig. 9 named Robot control (case 1). The input of this block
+is the spatial velocity eve that is transformed to deduce the angular velocity
+of each joint ˙q with the help of the inverse differential kinematic model to
+move mechanical structure of the robot. This motion is observed from the
+camera frame Fc in order to estimate the new pose of the end-effector and
+that of the orifice. These poses are the output of this block which are sent
+back to the control unit block for calculating a new iteration.
+However,
+this option is uneasy for implementation since it needs a particular setup to
+accurately track both the end-effector and the orifice [17].
+The second option is more fundamental than the first one. It is also
+presented in the green block of Fig. 9 named Robot control (case 2). It uses
+the proprioceptive sensors of the robot and its forward geometric model to
+estimate the end-effector pose. Despite that, this option requires performing
+a registration process [9, 17] between the robot and the orifice before the
+control-loop. After that, the robot works blindly, and the user assumes that
+the orifice does not move during the control-loop.
+The simulator is implemented in C++. It uses Eigen library for linear
+algebra (e.g., vectors, matrices, numerical solvers) and PCL (Point Cloud
+Library) for visualizing the STL parts and converting them to point clouds.
+This conversion is done to initialize the collision detection that is accom-
+plished by VCollide library. Finally, ViSP library is used for manipulating
+the camera images throughout the experimental work.
+4.2
+Numerical Validation
+A numerical simulator was developed, as the first step, to validate the func-
+tioning of the diverse methods before physical implementation. It simulates
+the geometric motion of the surgical tool through the incision orifice and
+the middle ear cavity. The software interchangeability of the simulator and
+the physical set-up allowed us also to tune the controller parameters before
+the experimental validation. Therefore, this part presents three scenarios
+for the demonstration:
+• scenario 1 performs the path-following task without any constraint
+applied on the tool motion.
+It demonstrates the effect of the gain
+coefficients vtis and β in equations (16) and (18), respectively, on the
+performance of the path-following controller;
+• scenario 2 performs the path-following task with RCM constraints. It
+simulates the drilling of a minimal invasive tunnel (i.e., conical tunnel)
+22
+
+through the mastoid portion to reach the middle ear cavity;
+• scenario 3 assumes the surgeon performed a standard mastoidectomy.
+It simulates an inspection/resection task performed under the UCM
+constraints.
+4.2.1
+Simulation of the path-following task without constraints
+Throughout this first trial, the value of vtis = 4mm/second in equation (16)
+remains constant during all tests. Besides that, the same reference path is
+tested during this trial, and it is defined as a spiral curve.
+Figure 10: The effect of the ratio between vdes and β′ on the path-following
+error dpf with a zoom and magnification on the orange region.
+The first group of tests keeps the value of γc in equation (18) constant
+while decreasing the value of β′ which its value varies from −4 to −16.
+Fig. 10 shows the influence of the gain coefficient β′ on the path-following
+error dpf. Indeed, this error computed as in equation (13). The ripples
+appearing in this figure represent the linear error between the projected
+point pt′ and the closest point on the reference path pp′. An orange rectangle
+appeared in this figure for zooming on one of these ripples. One can observe
+that the error reduced as designed exponentially.
+The latter figure also demonstrates that the best ratio between β′ and
+vtis should be greater than −2 (the saddle-brown line with star markers),
+and less than or equal −3 (the olive line with square markers). If the ratio is
+less than or equal to −1, the controller response is relatively slow, and there
+is a steady-state error (the maroon line with round markers in Fig. 10). On
+23
+
+0.8
+0.040
+Udes = 4.0, β = -4.0, %c = -1.0
+0.7
+0.035
+Udes = 4.0, β = -8.0, %c = -1.0
+Udes = 4.0, β =-12.0, %c = -1.0
+Udes = 4.0, β = -16.0, % = -1.0
+0.6
+0.030
+0.025
+0.5
+0.020
+(mm)
+0.4
+ldpfll
+0.015
+0.3
+0.010
+0.005
+0.2
+0.000
+1200
+1210
+1220
+1230
+1240
+1250
+1260
+1270
+1280
+1290
+0.1
+-
+1000
+2000
+3000
+4000
+5000
+Iterationsthe opposite, if the ratio is higher than or equal to −4, the system begins
+to oscillate (having over-shoots). However, the controller reduces the error
+faster than the previous cases (the sea-green line with triangular markers in
+Fig. 10).
+The second group of tests chose a constant ratio −2 while decreasing
+the value of γc from −2 to −16. This group shows that the best value of
+γc is to be near from β′. If γc is higher than β′, the system begins to have
+over-shoots, but it reduces faster the path-following error.
+4.2.2
+Simulation of a robotic drilling task under RCM constraint
+The surgeon perforates manually until now the mastoid portion in the tem-
+poral bone for reaching the middle ear cavity. The resultant mastoidectomy
+orifice is invasive. Thereby, a less invasive tunnel is proposed in this trial.
+Besides that, the drilling procedure becomes automated so that the surgeon
+can concentrate on other essential tasks. Indeed, this drilling procedure is
+achieved by merging the approach task, the 3D path-following task, and the
+RCM task.
+(a)
+(b)
+Figure 11: Numerical validation of the 3D path-following under a RCM
+constraint (see Extension 2). (a) The tool pose with respect to the desired
+path. (b) Sequence of zoom images during the tool motion.
+24
+
+Tool body
+Incision center
+point
+3D pathFig. 11 depicts the tool motion throughout the drilling procedure. The
+subplot (a) draws the tool geometry and its poses at different instances (or-
+ange straight-lines). It also shows the drilling path defined as a combination
+of spiral and linear portions (sea-green dotted-line). One can view that the
+tool body is always coincident with the orifice center point. The subplot
+(b1) shows the path done by the tool-tip (dodger-blue line) to accomplish
+the outside phase by i) approaching towards the point located before the
+orifice center point, and ii) regulating the rotation of the tool-tip frame to
+be as that of the orifice reference frame. The subplot (b2) depicts an in-
+stantaneous zoom on the tool pose during the inside phase to visualize the
+RCM effect.
+Figure 12: The approach task error eapp, where the left column is the linear
+error and the right column represents the angular error.
+The approach task error eapp computed in equation (11) is visualized in
+Fig. 12 which depicts the linear errors in the column and the angular errors
+in the right one. Over this period, the error is reduced in an exponential
+form as planned.
+At the end of the latter period, the transition phase starts.The task-
+hierarchical controller becomes active, and it arranges the path-following
+task as the highest priority while the RCM task is the second one. The
+errors of these tasks presented in the left columns of Fig. 14 and 13 which
+are obtained from equations (13) and (22) for the path-following and RCM
+errors, respectively. One can observe a peak appeared around 4 seconds in
+the path-following figure due to the initial error when the controller becomes
+activated. Then, it attenuates the error until it attains stability. Further-
+more, one can visualize in the RCM figure that three peaks appeared at the
+end of this phase. This behaviour happened due to the movement of the
+virtual trocar point.
+25
+
+20
+12
+Eappr
+Eappr
+15
+Eappy
+Eappy
+10
+Eapp:
+Eapp:
+10
+8
+eapp
+5
+(mm)
+(deg)
+6
+ddpa
+Eappe
+.5
+4
+.10
+2
+0
+-20
+0.5
+1.0
+1.52.02.5
+3.0
+3.5
+4.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Time (second)
+Time (second)Figure 13: The RCM task error drcm, where the left column shows the error
+evolution during the transition phase while the right column presents the
+error during the inside phase.
+Figure 14: The path-following task error dpf, where the left column shows
+the error evolution during the transition phase while the right column
+presents the error during the inside phase.
+After the previous period, the inside phase starts where the hierarchical
+controller modifies the priority by setting the RCM task as the highest one
+while the path-following is the secondary one. The RCM task error drcm was
+26
+
+outside/transition phases
+inside phase
+0.20
+0.015
+drcms
+0.15
+drcmy
+0.010
+drcm
+drcm:
+0.10
+drcm
+Idrcml
+0.05
+(mm)
+0.005
+0.00
+0.000
+d
+-0.05
+-0.10
+-0.005
+-0.15
+-0.20
+-0.010,
+1
+2
+3
+5
+6
+7
+8
+10
+20
+30
+ 40
+50
+60
+Time (second)
+Time (second)outside/transition phases
+inside phase
+0.25
+0.08
+drcms
+0.06
+0.20
+drcm
+0.04
+Idpf ll
+0.15
+0.02
+(mm)
+TAAAA
+0.10
+0.00
+dpf
+0.02
+0.05
+drcmr
+-0.04
+drcmy
+0.00
+-0.06
+II dp ll
+-0.05
+-0.08
+1
+2
+3
+4
+5
+6
+7
+8
+10
+20
+30
+40
+50
+60
+Time (second)
+Time (second)computed as 0.002 ± 0.002 mm (mean error ± STD (STandard Deviation)
+error), as shown in the right column of Fig. 13, while the path-following error
+dpf was 0.008±0.009 mm, as shown in the right column of Fig. 14. The gain
+values used for this trail were equal to λ = 1, γ = 1, vtis = 4 mm/second,
+β′ = −10, γc = −10 and Te = 0.008 second.
+4.2.3
+Simulation of an ablation/excision surgical task under UCM
+constraint
+In this trial, the incision orifice size is larger than the instrument diameter.
+The tool is consequently subject to the UCM for providing more freedom to
+the tool movements inside the incision orifice. This behaviour is shown in
+Fig. 15a where the orifice wall is represented by the red surface. The latter
+figure also presents the curved tool employed during this trial which performs
+an ablation or scanning process. The desired 3D path is thus composed of a
+linear portion to reach the middle ear cavity and a spiral curve to simulate
+the required surgical task. This selected path can reach some regions where
+a straight tool cannot attain (see Extension 4 to visualize the collision of
+the latter one with the orifice wall).
+The subplot (b1) of Fig. 15b indicates the path done by the tool during
+the outside phase. It also presents an instantaneous pose of the tool body
+throughout the transition phase.
+As explained in the previous trial, the
+proposed controller executes the same tasks over these two phases. Subplot
+(b2) presents the tool motion during the inside phase, where the dangerous
+and critical zones are represented by the green and red circles, respectively.
+The center point of these circles corresponds to the point ph′ obtained by
+projecting pt′ onto the orifice wall Sh.
+Throughout the inside phase, the hierarchical controller combines the
+UCM task with the path-following task as described in (38). Fig. 16 shows
+the UCM task error ducm which is deduced as in equation (31).
+It also
+presents the boundaries of the critical and dangerous zones. One can observe
+that the error ducm begins with a considerable value, compared to the error
+drcm, since the previous phase delivers the tool to the center point of the
+incision orifice. Then, the error ducm reduced, while the error drcm increased
+because the tool approached the incision wall to follow the reference path.
+However, the error ducm did not exceed the dmin, which implies the tool
+body did not enter the critical zone.
+Fig. 17 presents the path-following error dpf during the inside phase. It
+was measured was 0.005 ± 0.006 mm. The gain values used for this trail
+were equal to λ
+=
+0.8, γ
+=
+0.8, vtis
+=
+4 mm/second, β′
+=
+− 10,
+27
+
+(a)
+(b)
+Figure 15: Numerical validation of the 3D path-following under a UCM
+constraint (see Extension 3). (a) The tool pose with respect to the desired
+path. (b) Sequence of zoom images during the tool motion.
+Figure 16: The UCM task error ducm during the inside phase along side the
+error drcm.
+28
+
+reference curve
+actual curve
+tool body
+orificewall5
+I drcm l
+Ild ucmll
+dmas
+4
+dmin
+Safe zone
+3
+dangerous
+zone
+Critical
+zone
+0
+10
+15
+20
+25
+Time (second)Figure 17: The path-following task error dpf during the inside phase.
+γc =
+− 10 and Te = 0.008 second.
+4.3
+Experimental Validation
+This part is devoted to the physical implementation of the blocks Robot
+control that is shown in Fig. 9. Its physical correspondence is presented in
+Fig. 18. The robotic work-cell in the latter figure consists of:
+• a serial robot from Universal Robot (UR3) with ±0.03 mm pose re-
+peatability. It communicates with the proposed controller via TCP/IP
+for receiving the command velocity of the end-effector. It also sends
+the end-effector pose to the controller if the block Robot control (case
+1) is required to be executed;
+• a monocular camera from Guppy (with image size 640 × 420 pixels)
+and an optical objective lens from Computar with distortion (model
+MLM3X-MP) are used for the control purpose. This optical system
+tracks and estimates the poses of the end-effector and the incision
+orifice.
+It then sends these poses to the proposed controller if the
+block Robot control (case 2) is needed to be executed;
+29
+
+0.06
+urcm
+0.04
+dpf 
+0.02
+(mm)
+WAWWAAMWWM
+dpf
+0.00
+-0.02
+-0.04
+10
+15
+20
+25
+Time (second)• two visualization cameras provide other views for recording the mul-
+timedia videos.
+Figure 18: Configuration of the experimental setup.
+The numerical twin of the ear model shown previously in Fig. 8 is mod-
+ified for implementing its 3D printed twin. This modification holds up the
+(a)
+(b)
+Figure 19: The printed ear model used during the different tests. (a) The
+different parts of the ear model and the rigid tools. (b) After assembling
+the different parts.
+30
+
+Robot
+controller
+End-effector
+Tool body
+Visualization
+Control
+cameras
+Incision orifice-
+cameramastoidectomy orifice with the middle ear cavity and a planar grid/marker.
+Fig. 19 presented the fabricated parts before and after the assembly, along-
+side the rigid tools used during the validation tests.
+The trials of this part have the objective to evaluate the performance
+of the path-following controller under constraints. Therefore, a curved tool
+follows the same planned path, one time under the RCM constraint and the
+second time under the UCM constraint.
+4.3.1
+Path-Following under RCM Constraint
+(a)
+(b)
+Figure 20: Experimental validation of the 3D path-following under a RCM
+constraint (see Extension 5). (a) The tool pose with respect to the desired
+path. (b) Sequence of zoom images during the tool motion.
+Fig. 20 presents the desired path (sea-green dotted line), the resultant
+motion of the curved tool (orange line), and the path done by tool-tip
+(dodger-blue line). One can observe in Fig. 20b(1) that the tool approaches
+to the incision orifice by executing the controller given in equation (11). The
+approach task error eapp computed from equation (7). Fig. 21 presents the
+latter error and it converges toward zero by the end of this phase.
+Afterward, the transition phase starts so that the tool passes the center
+point of the incision orifice, as explained previously. The hierarchical con-
+troller (equation 29) arranges the path-following task as the highest priority
+while the RCM task is the second one. This behaviour is demonstrated in
+the left column of Fig. 22-23, where the hierarchical controller has been ac-
+tivated around 4 second. One can visualize that the RCM task error drcm
+31
+
+reference curve
+actualcurve
+tool bodyFigure 21: The approach task error eapp, where the left column is the linear
+error and the right column represents the angular error.
+has some steps due to the movements of the virtual trocar point while the
+path-following error dpf maintained its value around zero.
+Figure 22: The RCM task error drcm, where the left column shows the
+error evolution during the outside/transition phases while the right column
+presents the error during the inside phase.
+When the tool passes the center point of the incision orifice, the inside
+phase begins. The hierarchical controller (equation 28) modifies its priorities
+by setting the RCM task as the highest one and the path-following as the
+32
+
+25
+50
+linear
+angular
+20
+40
+(mm)
+15
+10
+20
+5
+10
+0.5
+1.0
+1.52.0
+2.5
+3.0
+3.5
+4.0
+0.5
+1.0
+1.52.02.5
+3.0
+3.5
+4.0
+Time (second)
+Time (second)outside/transition phases
+inside phase
+3.0
+0.7
+0.6
+2.5
+0.5
+2.0
+(u)
+0.4
+1.5
+0.3
+1.0
+0.2
+0.5
+0.1
+0.0.
+0.0
+0
+2
+4
+6
+8
+10
+12
+14
+16
+5
+20
+25
+30
+40
+45
+50
+Time (second)
+Time (second)Figure 23: The path-following task error dpf, where the left column shows
+the error evolution during the outside/transition phases while the right col-
+umn presents the error during the inside phase.
+second one. The system performances during the inside phase are shown
+in the right columns of Fig. 22-23. During this phase, the RCM task error
+drcm measured as 0.06 ± 0.05mm (mean error ± standard deviation (STD)
+error) while the path-following error dpf was 0.05 ± 0.03mm.
+A exteroceptive sensor used to close the control loop, as presented in
+Fig. 9 by the block Robot control (case 2). Besides that, the gain values
+used in this experiment were equal to λ = 1, γ = 1, vtis = 0.5 mm/second,
+β′ = −1.25, γc = −10 and Te = 0.008 second.
+Another trial was conducted for testing the block Robot control (case 1)
+by using the proprioceptive sensor in the control loop. The system perfor-
+mances are better than the exteroceptive test (see test 2 in Table 1). The
+errors drcm and dpf are reduced to almost half. It implies that our vision
+system needed amelioration in terms of accuracy.
+From the surgeon’s perspective, it is required to target the residual cells
+of cholesteatoma. It implies that the robot should detect/remove a human
+cell whose size is around 0.1mm. The proposed controller reached the re-
+quirements since the error dpf is smaller than the human cell size. Besides
+that, the surgical tool does not damage the entry orifice (patient’s head).
+By increasing the tool velocity vtis = 2 mm/second and maintain the
+same ratio β′/vtis = −2, the system performances deteriorated as expected.
+The errors drcm and dpf are almost increase by half (see tests 2 and 4 in
+33
+
+outside/transition phases
+inside phase
+0.30
+0.20
+0.25
+0.15
+0.20
+(mm)
+0.15
+0.10
+0.10
+0.05
+0.05
+0.00
+0.00
+0
+2
+4
+6
+8
+10
+12
+14
+16
+15
+20
+30
+3540
+45
+50
+Time (second)
+Time (second)Table 1). Therefore, the choice of the gain coefficients effects the system per-
+formances.
+4.3.2
+Path Following under UCM Constraint
+This second trial assumes the same conditions as the previous one. It in-
+volves the same curved tool and the desired path. However, this trial im-
+posed a unilateral constraint on the tool motion. Consequently, the tool can
+leave the center point of the incision orifice and move near the orifice wall.
+This behaviour is demonstrated in Fig. 24. The sub-figure (b1) of the lat-
+ter figure shows the path done by the tool-tip during the outside/transition
+phases, while the sub-figure (b2) presents the tool-tip path during the inside
+phase. The dangerous and critical regions are presented by the green and
+red circles in the latter sub-figure.
+(a)
+(b)
+Figure 24: Experimental validation of the 3D path-following under a UCM
+constraint (see Extension 6). (a) The tool motion during the different phases.
+(b) Sequence of zoom images during the tool motion.
+Throughout the inside phase, the hierarchical controller arranges the
+different tasks as explained in section 3.6. The highest priority is the path-
+following task when the tool is located in the safe zone. However, the highest
+priority changes to the UCM task when the tool body passes the danger
+zone. The system performances are shown in Fig. 25-26. One can observe
+from the UCM task error ducm (Fig. 25) that the tool body is maintained in
+34
+
+referencecurve
+actualcurve
+tool body
+orificewallFigure 25: The UCM task error ducm during the inside phase.
+Figure 26: The path-following task error dpf during the inside phase.
+the dangerous zone since the error ducm changes its value between dmax and
+dmin. Besides that, the path-following error dpf (Fig. 26) was 0.05±0.03mm
+(mean error ± STD error) and its median error was 0.05mm.
+A exteroceptive sensor used as the feedback sensor. Additionally, the
+gain values used for this second trial were equal to λ = 1, γ = 1, vtis =
+0.5 mm/second, β′ = −1.25, γc = −10 and Te = 0.008 second.
+The error dpf of this trial remains almost the same as the previous trial.
+It implies that the UCM constraint does not deteriorate the path-following
+error. Indeed, it provides the surgical tool to move with more liberty in
+order to take advantage of the large size of the entry orifice.
+35
+
+5
+Idrcm
+I d ucm ll
+dmax
+4
+dmin
+Safe zone
+(mm)
+3
+dangerous
+zone
+Critical
+zone
+0
+15
+20
+25
+30
+35
+40
+Time (second)inside phase
+0.18
+0.16
+0.14
+0.12
+(mm)
+0.10
+0.08
+d
+0.06
+0.04
+0.02
+0.0Q
+10
+15
+20
+25
+30
+35
+40
+45
+Time (second)N°
+constraint
+feedback
+type of
+error
+mean (∥e∥) ± STD
+1
+RCM
+exteroceptive
+drcm
+dpf
+0.06±0.05
+0.05±0.02
+2
+RCM
+exteroceptive
+drcm
+dpf
+0.15±0.06
+0.08±0.05
+3
+RCM
+proprioceptive
+drcm
+dpf
+0.02±0.05
+0.02±0.01
+4
+RCM
+proprioceptive
+drcm
+dpf
+0.03±0.08
+0.03±0.02
+5
+UCM
+exteroceptive
+drcm
+dpf
+3.30±0.93
+0.05±0.03
+6
+UCM
+exteroceptive
+drcm
+dpf
+3.30±0.93
+0.09±0.06
+7
+UCM
+proprioceptive
+drcm
+dpf
+2.74±0.77
+0.02±0.01
+8
+UCM
+proprioceptive
+drcm
+dpf
+2.69±0.67
+0.03±0.02
+Table 1: Summary of different trials achieved with the curved tool during
+the experimental tests.
+∥e∥ (in mm) is the absolute average of the linear error along x − y − z axes,
+and STD is the related standard deviation (in mm).
+Results obtained with the following parameters: λ = 1, vtis = 0, 5 mm/s,
+and Te = 0, 008 second. The while trials applied β′ = −1.25, while the blue
+ones applied β′ = −5.
+5
+CONCLUSION AND FUTURE WORK
+This article discussed the design of an original controller for guiding a rigid
+instrument under constrained motions such as RCM or UCM. The proposed
+methodology allows a generic formulation, in the same controller, two tasks:
+i) the constrained motion (RCM or UCM), and ii) a revisited 3D path-
+following scheme by increasing the sensitivity to the path complexity (e.g.,
+curvature radius) and then reducing the path-following error. To manage
+the achievement of two or more tasks without conflicts, we also implemented
+a task prioritizing paradigm. Consequently, the developed control scheme
+can be integrated easily with various robotic systems without an accurate
+knowledge of the robot inverse kinematics.
+36
+
+Experimental validation was also successfully conducted using a 6-DoF
+robotic system. The obtained results are promising in terms of behavior
+and precision. These performances, even if they meet the specifications of
+the targeted middle ear surgery, may be considered improvements.
+The
+positioning error depends directly on the registration process that is not
+treated optimally in this work. Furthermore, the pose estimation of the tool-
+tip was done based on a geometric model of the instrument. Its estimation
+could be another source of error. Thus, it would be interesting to find out
+another method for estimating the tool shape and the pose of its tip.
+The forthcoming work will implement the discussed methods in a clinical
+context using a realistic phantom and a human cadaver. Besides that, a force
+control could be added to increase the robot sensitivity to its environment
+and increase the level of security.
+ACKNOWLEDGMENTS
+This work was supported by the Inserm ROBOT Project: ITMO Cancer
+no 17CP068-00.
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+Adaptive Quantum Amplitude Estimation
+Xi Lu1 and Hongwei Lin1, ∗
+1School of Mathematical Science, Zhejiang University, Hangzhou, 310027, China
+The maximum likelihood amplitude estimation (MLAE) algorithm is a practical solution to the
+quantum amplitude estimation problem, which has a theoretically quadratic speedup over classical
+Monte Carlo method. However, we find that MLAE is not unbiased, which is one of the major causes
+of its inaccuracy. We propose an adaptive quantum amplitude estimation (AQAE) algorithm by
+choosing MLAE parameters adaptively to avoid critical points. We also do numerical experiments
+to show that our algorithm is approximately unbiased and more efficient than MLAE.
+I.
+INTRODUCTION
+Quantum computing is an emerging subject that studies faster solutions on quantum computers over clas-
+sical ones. Early quantum algorithms have achieved astonishing speedups over known classical algorithms,
+such as the quadratic speedup of Grover’s search [1], and the exponential speedup of Shor’s integer factor-
+ization [2]. Later algorithms like quantum approximate optimization algorithms (QAOA) [3–5], variational
+quantum eigen solver (VQE) [6, 7] and quantum neural networks (QNN) [8, 9] also shows great potentials in
+quantum computing.
+The amplitude estimation problem [10] is one of the most fundamental problems in quantum computing, a
+quantum variant of the classical Monte Carlo problem. Let A be any quantum algorithm that performs the
+following unitary transformation,
+A |00 · · · 0⟩ =
+√
+1 − a |ψ0⟩ |0⟩ + √a |ψ1⟩ |1⟩ = cos φ |ψ0⟩ |0⟩ + sin φ |ψ1⟩ |1⟩ .
+(1)
+The goal of amplitude estimation problem is to estimate a. It is derived from the well-known phase estimation
+problem, and has been widely applied in quantum chemistry [11–13] and machine learning [14, 15] in recent
+studies.
+The earliest solution [10] is a combination of quantum phase estimation and Grover’s search. There are some
+later researches [16–19] that improve the robustness of phase estimation. The modified Grover’s operator [20]
+is an approach that is designed to perform robustly under depolarizing noise. However, most of the recent
+researches study amplitude estimation algorithms without the use of phase estimation, since it is believed that
+the controlled amplification operations required by phase estimation can be different to implement on noise
+intermediate-scale quantum devices. The maximum likelihood amplitude estimation (MLAE) [21] algorithm is
+an approach without phase estimation, which is proved to have an error convergence O(N −1) asymptotically
+when using an exponential incremental sequence (EIS), which is quadratically faster than O(N −1/2) for
+classical Monte Carlo algorithm. The error convergence O(N −1) is also known as the Heisenberg limit [22].
+There is a variant of MLAE [23] that is built for noisy devices without estimating the noise parameters.
+The iterative quantum amplitude estimation (IQPE) [24] is another approach without phase estimation,
+which is proved rigorously to achieve a quadratic speedup up to a double-logarithmic factor compared to
+classical Monte Carlo (MC) estimation. The variational amplitude estimation [25] is a variational quantum
+algorithm based on constant-depth quantum circuits that also outperforms MC. There are also several other
+approaches [26, 27].
+In this paper, we dive further into MLAE. In more precise experiments we find that the MLAE algorithm
+is not unbiased, and the bias behaves periodically with respect to the ground truth a, as shown in Fig. 1.
+Moreover, statistics theories show that the variance of any estimation ˜a follows the Cram´er-Rao inequality [28],
+E[(˜a − a)2] ≥ [1 + b′(a)]2
+F(a)
++ b(a)2,
+(2)
+∗ [email protected]
+arXiv:2301.00528v1  [quant-ph]  2 Jan 2023
+
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.002
+0.000
+0.002
+0.004
+0.006
+0.008
+Bias
+RMSE
+CRLB
+FIG. 1: The bias and root of mean squared error (RMSE) of MLAE, for different a. The unbiased
+Cram´er-Rao lower bound (CRLB) is the ideal distribution of RMSE, which is equal to Eq. (2) where
+b(a) = 0.
+where b(a) = E[˜a − a] is the bias, and the Fisher information F is defined as,
+F(a) = E
+��∂ ln L(a)
+∂a
+�2�
+,
+(3)
+where L is the likelihood function of MLAE. An estimation is fully efficient [29] if it is unbiased and saturates
+the Cram´er-Rao inequality. From Fig. 1, we can see that MLAE is approximately unbiased and close to the
+unbiased Cram´er-Rao lower bound in most area, except some periodical small intervals. We improve MLAE
+and propose the adaptive quantum amplitude estimation (AQAE) algorithm in this paper by introducing an
+adaptive rule to avoid these small intervals, and show that our estimation algorithm is approximately efficient
+with numerical experiments.
+II.
+PRELIMINARY
+Most amplitude estimation algorithms are based on a general procedure called amplitude amplification [10],
+which performs the transformation
+QmA |00 · · · 0⟩ = cos[(2m + 1)φ] |ψ0⟩ |0⟩ + sin[(2m + 1)φ] |ψ1⟩ |1⟩ ,
+(4)
+where
+Q = A(2 |00 · · · 0⟩⟨00 · · · 0| − I)A−1(I ⊗ Z).
+(5)
+By measuring the last qubit with respect to the computational basis we obtain one with probability
+sin2[(2m + 1)φ], and zero with probability cos2[(2m + 1)φ]. Such amplitude amplification process requires
+(2m + 1) calls to the oracle A.
+The MLAE algorithm requires parameters {mk, Rk}K
+k=1. For each k the state QmkA |00 · · · 0⟩ is measured
+for Rk times. Let hk be the number of ones in all Rk measurement results. The final estimation ˜a is obtained
+by maximizing the likelihood function
+L(a) :=
+K
+�
+k=1
+ℓk(φ),
+(6)
+where a ≡ sin2 φ, and
+ℓk(φ) :=
+�
+sin2(Mkφ)
+�hk �
+cos2(Mkφ)
+�Rk−hk ,
+(7)
+
+3
+1( )
+2( )
+3( )
+4( )
+5( )
+FIG. 2: An illustration of how MLAE works. The curves illustrate the function ℓk(φ) for each k. Here
+M1 = 1, Mk = 2k + 1(k = 2, 3, 4, 5).
+where Mk ≡ 2mk + 1.
+The Fig. 2 illustrates how MLAE works. Generally the function ℓk(φ) has Mk peaks. For M1 = 1, there is
+a single smooth peak in the likelihood function ℓ1(φ). For bigger Mks, the peaks are sharper and thus have
+better estimation ability, but there is more than one peak. So we cannot get more accurate estimation with
+ℓk(φ) alone. The MLAE algorithm combines the information of ℓk(φ) for different Mks by multiplying all
+those likelihood functions, thus obtaining a likelihood function L that has only one sharp peak.
+By calculation the Fisher information of MLAE is [21],
+F(a) =
+1
+a(1 − a)
+�
+k
+RkM 2
+k.
+(8)
+In most application problems the major complexity lies in the oracle A itself. Therefore, the time cost of
+MLAE is,
+N =
+�
+k
+RkMk.
+(9)
+The original article about MLAE algorithm [21] presents two strategies of choosing parameters,
+• Linear Incremental Sequence (LIS): mk = k − 1 and Rk = R for k = 1, 2, · · · , K, which has error
+convergence ε ∼ N −3/4;
+• Exponential Incremental Sequence (EIS): m1 = 0, mk = 2k−2(k = 2, 3, · · · , K) and Rk = R(k =
+1, 2, · · · , K), which has error convergence ε ∼ N −1.
+As MLAE is approximately unbiased and saturates the Cram´er-Rao inequality in most area, the RMSE
+has the same error convergence as F−1/2. The MLAE algorithm with EIS fixes R1 = · · · = RK = R, and
+chooses M1 = 1, Mk = 2k−1 + 1(k ≥ 2), then N = O(R · 2K) and F−1/2 = O(R−1/2 · 2−K) = O(N −1), which
+is quadratically faster than MC and reaches the Heisenberg limit. But in reality, the existence of the bias
+term in Eq. (2) has a significant impact and violates the quadratic speedup, as is shown by the numerical
+experiments in the next section.
+III.
+THEORY AND ALGORITHM
+In the beginning of this section, we set up a model for the bias of MLAE. We call,
+�
+sin2
+� j
+m
+π
+2
+�����j = 1, 2, · · · , m − 1
+�
+(10)
+
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.002
+0.000
+0.002
+0.004
+0.006
+0.008
+Bias
+RMSE
+CRLB
+FIG. 3: The bias and RMSE of MLAE with parameters K = 5, R = 32. The vertical black dashed lines are
+the critical points of order M5 = 24 + 1 = 17.
+the critical points of order m. In MLAE, consider two values on each side of some critical point sin2(jπ/2MK),
+namely a± = sin2(φ±) = sin2(jπ/2MK ± ε). It is harder for the likelihood function Eq. (6) to tell apart
+a± = sin2(φ±) = sin2(jπ/2MK ± ε) when ε is small, as ℓK(φ+) = ℓK(φ−), and thus they can only be told
+apart by other terms {ℓk(φ)}K−1
+k=1 that is less sharp than ℓK(φ). As a result, MLAE has a positive bias when
+a− is the ground truth, and has a negative bias when a+ is the ground truth. It should be mentioned that
+other smaller Mks can also bring bias around their critical points, which is anyway not so obvious as MK.
+Our theory concludes that MLAE has obvious bias in the intervals centered at each critical point of order
+MK, as shown in Fig. 3.
+By this observation, it is hard to flatten the bias curve on the whole interval [0, 1] with a fixed parameter
+set {Mk, Rk}K
+k=1. The intuition of our AQAE algorithm is that we can adaptively and randomly choose the
+next Mk with the hope of staying away from the critical points of order Mk, namely {sin2(jπ/2Mk) : j =
+1, 2, · · · , Mk − 1}.
+Define the score function,
+s(M; ˆa) = sin2(2M ˆφ),
+(11)
+where ˆa ≡ sin2(ˆφ), which is close to zero when ˆa is close to a critical point of order M.
+Suppose we have already performed amplitude amplification procedures with parameters {Mk, Rk}K′
+k=1,
+and have got the results {hk}K′
+k=1. The Bayes theory tells us the posterior probability density distributions
+of a is,
+ρ(ˆa) ∝
+K′
+�
+k=1
+ℓk(ˆφ).
+(12)
+Combining the score function Eq. (11), we define the weight function about M as the expectation of the
+score function,
+w(M) = Eˆa[s(M; ˆa)] ∝
+� 1
+0
+s(M; ˆa)
+�
+�
+K′
+�
+k=1
+ℓk(ˆφ)
+�
+� d ˆa.
+(13)
+Similar to the score function, if ˆa is distributed mostly around some critical point of M, then w(M) is
+small. The weight function is our guidance for the adaptive choice of the subsequent parameters {Mk, Rk}.
+To avoid critical points, the key idea of AQAE is that the smaller w(Mk) is, the smaller Rk will be.
+The Eq. (4) enables us to generate a 0-1 distribution random variable with p(1) = sin2[Mφ] for any odd
+number M. An important thing for AQAE is that we should generalize it to the even case. From Eq. (1) we
+
+5
+have,
+cos φA−1(|ψ0⟩ |0⟩) + sin φA−1(|ψ1⟩ |1⟩) = |00 · · · 0⟩ .
+(14)
+By the orthogonality of A−1 we know that,
+|ψ′⟩ := sin φA−1(|ψ0⟩ |0⟩) − cos φA−1(|ψ1⟩ |1⟩),
+(15)
+is orthogonal to |00 · · · 0⟩. That is, if we measure all qubits of |ψ′⟩ under the computational basis, we will
+certainly get results that contain one. Moreover,
+A−1 |ψ0⟩ |0⟩ = cos φ |00 · · · 0⟩ + sin φ |ψ′⟩ ,
+(16)
+A−1 |ψ1⟩ |1⟩ = sin φ |00 · · · 0⟩ − cos φ |ψ′⟩ .
+(17)
+Define,
+Q′ = A−1(I ⊗ Z)A(2 |00 · · · 0⟩⟨00 · · · 0| − I).
+(18)
+Then,
+Q′ |00 · · · 0⟩ =A−1(I ⊗ Z)A |00 · · · 0⟩
+=A−1(I ⊗ Z)(cos φ |ψ0⟩ |0⟩ + sin φ |ψ1⟩ |1⟩)
+=A−1(cos φ |ψ0⟩ |0⟩ − sin φ |ψ1⟩ |1⟩)
+= cos φ(cos φ |00 · · · 0⟩ + sin φ |ψ′⟩) − sin φ(sin φ |00 · · · 0⟩ − cos φ |ψ′⟩)
+= cos(2φ) |00 · · · 0⟩ + sin(2φ) |ψ′⟩ ,
+(19)
+and,
+Q′ |ψ′⟩ =A−1(I ⊗ Z)A(− |ψ′⟩)
+=A−1(I ⊗ Z)(− sin φ |ψ0⟩ |0⟩ + cos φ |ψ1⟩ |1⟩)
+=A−1(− sin φ |ψ0⟩ |0⟩ − cos φ |ψ1⟩ |1⟩)
+= − sin φ(cos φ |00 · · · 0⟩ + sin φ |ψ′⟩) − cos φ(sin φ |00 · · · 0⟩ − cos φ |ψ′⟩)
+= − sin(2φ) |00 · · · 0⟩ + cos(2φ) |ψ′⟩ .
+(20)
+Therefore, Q′ is a rotation by angle 2φ in the plane spanned by |00 · · · 0⟩ and |ψ′⟩. We can deduce that,
+Q′m |00 · · · 0⟩ = cos(2mφ) |00 · · · 0⟩ + sin(2mφ) |ψ′⟩ .
+(21)
+By measuring all qubits under the computational basis we obtain all zero with probability cos2(2mφ), and
+results containing one with probability sin2(2mφ). The extended amplitude amplification process requires
+2m calls to the oracle A.
+In summary, no matter M is odd or even, we can obtain a random variable rM with 0-1 distribution where
+p(1) = sin2(Mφ), with a cost of M oracle calls to the oracle A. When M is odd, we measure the last qubit
+of the state Q(M−1)/2A |00 · · · 0⟩, and obtain one with probability sin2(Mφ). When M is even, we measure
+all qubits of the state Q′M/2 |00 · · · 0⟩, and the probability that the results contain one is sin2(Mφ). For
+convenience, we use the terminology measuring rM to mean that we use the procedure above to obtain a
+random variable of 0-1 distribution with p(1) = sin2(Mφ). The extended amplitude amplification is crucial
+
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+0.0025
+0.0030
+0.0035
+Bias
+RMSE
+CRLB
+FIG. 4: The bias, RMSE and CRLB for AQAE, with parameters K = 5 and R = 32. The CRLB is
+calculated as the average value of F−1/2 = [
+1
+a(1−a)
+�
+k RkM 2
+k]−1/2, according to Eq. (8).
+to our proposed algorithm.
+Algorithm 1: Adaptive Quantum Amplitude Estimation (AQAE)
+Input
+: K: Number of iterations; R: Number of measurements in each iteration;
+Output: ˜a: Estimation of a;
+1 Set M1 = 1 and R1 = R;
+2 Measure r1 for R times and let h1 be the number of ones;
+3 for i = 2..K do
+4
+Calculate the weights {w(m)}2i−1
+m=2i−1;
+5
+Set Mm = m and Rm = 0 for m = 2i−1, · · · , 2i − 1;
+6
+for j = 1..R do
+7
+Draw a random sample mj from {2i−1, · · · , 2i − 1}, with probabilities w(mj)/ �2i−1
+m=2i−1 w(m);
+8
+Increase Rmj by one;
+9
+end
+10
+Measure rk for Rk times and let hk be the number of ones;
+11 end
+12 Calculate ˜a using MLE.
+Our algorithm is shown in Alg. 1. First we set M1 = 1 and R1 = R, and measure the state Eq. (1)
+directly for R times to obtain h1. In the second iteration, we set M2 = 2 and M3 = 3, and compute w(2)
+and w(3). We draw R samples from {2, 3} with probabilities
+�
+w(2)
+w(2)+w(3),
+w(3)
+w(2)+w(3)
+�
+, and set R2, R3 to be
+the number of 2 and 3 in the outcome, respectively. In the third iteration we run the same procedure for
+M4 = 4, M5 = 5, M6 = 6, M7 = 7. After all K iterations, we apply the MLE to obtain the result ˜a.
+We carry out several numerical experiments to show the efficiency of AQAE algorithm. All the quantum
+outputs in the experiments are obtained by sampling the theoretic distribution functions. First, in comparison
+to Fig. 1, the bias and RMSE curve for AQAE is shown in Fig. 4. We find that the bias intensity of AQAE
+is much lower than MLAE. Besides, the RMSE curve is smooth and close to the average CRLB curve.
+To illustrate how the parameters chosen by AQAE vary with different as, we make statistics for two typical
+as, as shown in Fig. 5. In Fig. 5 (a) all even Mks are chosen less frequently then odd Mks, since a is a critical
+point of order 2. In Fig. 5 (b) all Mks that are multiples of 3 are chosen less frequently since a is a critical
+point of order 3. This set of experiments show that AQAE can effectively avoid critical points.
+Finally, we compare different amplitude estimation algorithms and take the time cost into consideration.
+In this experiment we uniformly randomly draw 216 samples in the interval [0, 1] as a, and compare the
+error behavior with respect to the time cost. For Monte Carlo (MC) estimation, suppose the state Eq. (1) is
+prepared for R times, and by measuring the last qubit the result 1 is obtained for h times, then the estimation
+to a is given by ˆa = h/R. The time cost for MC is N = R, as each preparation of the state Eq. (1) requires
+
+7
+0
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+25
+30
+(a) When a = sin2(π/4) = 0.5, a critical point of order 2.
+0
+5
+10
+15
+20
+25
+30
+0
+5
+10
+15
+20
+25
+30
+(b) When a = sin2(π/6) = 0.25, a critical point of order 3.
+FIG. 5: The average Rk (y-axis) for each Mk (x-axis) chosen by AQAE when K = 5 and R = 32. The Mks
+that are multiples of 2 in (a) or multiples of 3 in (b) are labelled orange.
+102
+103
+104
+Time Cost
+10
+3
+10
+2
+RMSE
+MC
+QPE
+UQPE
+IQAE
+MLAE
+AQAE
+FIG. 6: The error behavior (y-axis) with respect to the time cost N (x-axis).
+one call to the oracle A. For MLAE and AQAE, both algorithms require two parameters K and R, which
+is chosen by pre-calculation that has the minimal RMSE among several parameter pairs with approximate
+time cost. The time cost of MLAE is N = �
+k RkMk = R(2K + K − 2). Since the parameter set {Mk, Rk}
+is not fixed in AQAE, we calculate the average value of �
+k RkMk chosen in numerical experiments as its
+time cost. The quantum phase estimation (QPE) based amplitude estimation requires a parameter t as the
+number of controlled qubits [30], with time cost N = �t−1
+j=0 2j = 2t − 1. An efficient way to reduce the
+RMSE of QPE is to repeat for R times and use MLE to give the final estimation. The unbiased quantum
+phase estimation (UQPE) [19] is an unbiased variant of QPE. The time cost for both QPE and UQPE in
+our experiments is N = R(2t − 1). In our experiments we fix R = 4 and let t vary. For IQPE [24], we use
+Clopper-Pearson confidence interval method, fix α = 0.05, Nshots = 100 and let ϵ vary. The results are shown
+in Fig. 6. The MC algorithm have an error convergence of O(N −1/2), while all other algorithms have an
+
+8
+asymptotic O(N −1) error convergence. The UQPE performs the best among those algorithms. If we limit
+the comparison in algorithms without phase estimation, as they are more likely to be implemented widely in
+recent years, then our AQAE algorithm outperforms other algorithms.
+IV.
+CONCLUSION
+The maximum likelihood amplitude estimation (MLAE) algorithm is a practical solution to the quan-
+tum amplitude estimation problem, which has a theoretically quadratic speedup over classical Monte Carlo
+method. We find that MLAE behaves efficient, i.e. unbiased and saturates the Cram´er-Rao inequality in
+most area except some periodical small intervals. We analyze how the bias occurs around the so-called crit-
+ical points, and propose an adaptive quantum amplitude estimation (AQAE) algorithm by choosing MLAE
+parameters adaptively to avoid critical points. In the end, we do numerical experiments among some ampli-
+tude estimation algorithms, including Monte Carlo estimation, quantum phase estimation and its unbiased
+variant, iterative quantum amplitude estimation, maximum likelihood amplitude estimation and our adaptive
+amplitude estimation. We show that our algorithm outperforms the original MLAE obviously, and it behaves
+the best among all algorithms without phase estimation.
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+

6dAyT4oBgHgl3EQfpfjS/content/tmp_files/load_file.txt

@@ -0,0 +1,355 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf,len=354
+page_content='Adaptive Quantum Amplitude Estimation  Xi Lu1 and Hongwei Lin1, ∗  1School of Mathematical Science, Zhejiang University, Hangzhou, 310027, China  The maximum likelihood amplitude estimation (MLAE) algorithm is a practical solution to the  quantum amplitude estimation problem, which has a theoretically quadratic speedup over classical  Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' However, we find that MLAE is not unbiased, which is one of the major causes  of its inaccuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We propose an adaptive quantum amplitude estimation (AQAE) algorithm by  choosing MLAE parameters adaptively to avoid critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We also do numerical experiments  to show that our algorithm is approximately unbiased and more efficient than MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  INTRODUCTION  Quantum computing is an emerging subject that studies faster solutions on quantum computers over clas-  sical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Early quantum algorithms have achieved astonishing speedups over known classical algorithms,  such as the quadratic speedup of Grover’s search [1], and the exponential speedup of Shor’s integer factor-  ization [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Later algorithms like quantum approximate optimization algorithms (QAOA) [3–5], variational  quantum eigen solver (VQE) [6, 7] and quantum neural networks (QNN) [8, 9] also shows great potentials in  quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  The amplitude estimation problem [10] is one of the most fundamental problems in quantum computing, a  quantum variant of the classical Monte Carlo problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Let A be any quantum algorithm that performs the  following unitary transformation,  A |00 · · · 0⟩ =  √  1 − a |ψ0⟩ |0⟩ + √a |ψ1⟩ |1⟩ = cos φ |ψ0⟩ |0⟩ + sin φ |ψ1⟩ |1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (1)  The goal of amplitude estimation problem is to estimate a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' It is derived from the well-known phase estimation  problem, and has been widely applied in quantum chemistry [11–13] and machine learning [14, 15] in recent  studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  The earliest solution [10] is a combination of quantum phase estimation and Grover’s search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' There are some  later researches [16–19] that improve the robustness of phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The modified Grover’s operator [20]  is an approach that is designed to perform robustly under depolarizing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' However, most of the recent  researches study amplitude estimation algorithms without the use of phase estimation, since it is believed that  the controlled amplification operations required by phase estimation can be different to implement on noise  intermediate-scale quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The maximum likelihood amplitude estimation (MLAE) [21] algorithm is  an approach without phase estimation, which is proved to have an error convergence O(N −1) asymptotically  when using an exponential incremental sequence (EIS), which is quadratically faster than O(N −1/2) for  classical Monte Carlo algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The error convergence O(N −1) is also known as the Heisenberg limit [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  There is a variant of MLAE [23] that is built for noisy devices without estimating the noise parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  The iterative quantum amplitude estimation (IQPE) [24] is another approach without phase estimation,  which is proved rigorously to achieve a quadratic speedup up to a double-logarithmic factor compared to  classical Monte Carlo (MC) estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The variational amplitude estimation [25] is a variational quantum  algorithm based on constant-depth quantum circuits that also outperforms MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' There are also several other  approaches [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  In this paper, we dive further into MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' In more precise experiments we find that the MLAE algorithm  is not unbiased, and the bias behaves periodically with respect to the ground truth a, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Moreover, statistics theories show that the variance of any estimation ˜a follows the Cram´er-Rao inequality [28],  E[(˜a − a)2] ≥ [1 + b′(a)]2  F(a)  + b(a)2,  (2)  ∗ hwlin@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='cn  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='00528v1  [quant-ph]  2 Jan 2023  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
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+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='008  Bias  RMSE  CRLB  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 1: The bias and root of mean squared error (RMSE) of MLAE, for different a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The unbiased  Cram´er-Rao lower bound (CRLB) is the ideal distribution of RMSE, which is equal to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (2) where  b(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  where b(a) = E[˜a − a] is the bias, and the Fisher information F is defined as,  F(a) = E  ��∂ ln L(a)  ∂a  �2�  ,  (3)  where L is the likelihood function of MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' An estimation is fully efficient [29] if it is unbiased and saturates  the Cram´er-Rao inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 1, we can see that MLAE is approximately unbiased and close to the  unbiased Cram´er-Rao lower bound in most area, except some periodical small intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We improve MLAE  and propose the adaptive quantum amplitude estimation (AQAE) algorithm in this paper by introducing an  adaptive rule to avoid these small intervals, and show that our estimation algorithm is approximately efficient  with numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  PRELIMINARY  Most amplitude estimation algorithms are based on a general procedure called amplitude amplification [10],  which performs the transformation  QmA |00 · · · 0⟩ = cos[(2m + 1)φ] |ψ0⟩ |0⟩ + sin[(2m + 1)φ] |ψ1⟩ |1⟩ ,  (4)  where  Q = A(2 |00 · · · 0⟩⟨00 · · · 0| − I)A−1(I ⊗ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (5)  By measuring the last qubit with respect to the computational basis we obtain one with probability  sin2[(2m + 1)φ], and zero with probability cos2[(2m + 1)φ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Such amplitude amplification process requires  (2m + 1) calls to the oracle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  The MLAE algorithm requires parameters {mk, Rk}K  k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' For each k the state QmkA |00 · · · 0⟩ is measured  for Rk times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Let hk be the number of ones in all Rk measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The final estimation ˜a is obtained  by maximizing the likelihood function  L(a) :=  K  �  k=1  ℓk(φ),  (6)  where a ≡ sin2 φ, and  ℓk(φ) :=  �  sin2(Mkφ)  �hk �  cos2(Mkφ)  �Rk−hk ,  (7)  3  1( )  2( )  3( )  4( )  5( )  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 2: An illustration of how MLAE works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The curves illustrate the function ℓk(φ) for each k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Here  M1 = 1, Mk = 2k + 1(k = 2, 3, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  where Mk ≡ 2mk + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 2 illustrates how MLAE works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Generally the function ℓk(φ) has Mk peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' For M1 = 1, there is  a single smooth peak in the likelihood function ℓ1(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' For bigger Mks, the peaks are sharper and thus have  better estimation ability, but there is more than one peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' So we cannot get more accurate estimation with  ℓk(φ) alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The MLAE algorithm combines the information of ℓk(φ) for different Mks by multiplying all  those likelihood functions, thus obtaining a likelihood function L that has only one sharp peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  By calculation the Fisher information of MLAE is [21],  F(a) =  1  a(1 − a)  �  k  RkM 2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (8)  In most application problems the major complexity lies in the oracle A itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Therefore, the time cost of  MLAE is,  N =  �  k  RkMk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (9)  The original article about MLAE algorithm [21] presents two strategies of choosing parameters,  Linear Incremental Sequence (LIS): mk = k − 1 and Rk = R for k = 1, 2, · · · , K, which has error  convergence ε ∼ N −3/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Exponential Incremental Sequence (EIS): m1 = 0, mk = 2k−2(k = 2, 3, · · · , K) and Rk = R(k =  1, 2, · · · , K), which has error convergence ε ∼ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  As MLAE is approximately unbiased and saturates the Cram´er-Rao inequality in most area, the RMSE  has the same error convergence as F−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The MLAE algorithm with EIS fixes R1 = · · · = RK = R, and  chooses M1 = 1, Mk = 2k−1 + 1(k ≥ 2), then N = O(R · 2K) and F−1/2 = O(R−1/2 · 2−K) = O(N −1), which  is quadratically faster than MC and reaches the Heisenberg limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' But in reality, the existence of the bias  term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (2) has a significant impact and violates the quadratic speedup, as is shown by the numerical  experiments in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  THEORY AND ALGORITHM  In the beginning of this section, we set up a model for the bias of MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We call,  �  sin2  � j  m  π  2  �����j = 1, 2, · · · , m − 1  �  (10)  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='008  Bias  RMSE  CRLB  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 3: The bias and RMSE of MLAE with parameters K = 5, R = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The vertical black dashed lines are  the critical points of order M5 = 24 + 1 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  the critical points of order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' In MLAE, consider two values on each side of some critical point sin2(jπ/2MK),  namely a± = sin2(φ±) = sin2(jπ/2MK ± ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' It is harder for the likelihood function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (6) to tell apart  a± = sin2(φ±) = sin2(jπ/2MK ± ε) when ε is small, as ℓK(φ+) = ℓK(φ−), and thus they can only be told  apart by other terms {ℓk(φ)}K−1  k=1 that is less sharp than ℓK(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' As a result, MLAE has a positive bias when  a− is the ground truth, and has a negative bias when a+ is the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' It should be mentioned that  other smaller Mks can also bring bias around their critical points, which is anyway not so obvious as MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Our theory concludes that MLAE has obvious bias in the intervals centered at each critical point of order  MK, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  By this observation, it is hard to flatten the bias curve on the whole interval [0, 1] with a fixed parameter  set {Mk, Rk}K  k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The intuition of our AQAE algorithm is that we can adaptively and randomly choose the  next Mk with the hope of staying away from the critical points of order Mk, namely {sin2(jπ/2Mk) : j =  1, 2, · · · , Mk − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Define the score function,  s(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' ˆa) = sin2(2M ˆφ),  (11)  where ˆa ≡ sin2(ˆφ), which is close to zero when ˆa is close to a critical point of order M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Suppose we have already performed amplitude amplification procedures with parameters {Mk, Rk}K′  k=1,  and have got the results {hk}K′  k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The Bayes theory tells us the posterior probability density distributions  of a is,  ρ(ˆa) ∝  K′  �  k=1  ℓk(ˆφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (12)  Combining the score function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (11), we define the weight function about M as the expectation of the  score function,  w(M) = Eˆa[s(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' ˆa)] ∝  � 1  0  s(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' ˆa)  �  �  K′  �  k=1  ℓk(ˆφ)  �  � d ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (13)  Similar to the score function, if ˆa is distributed mostly around some critical point of M, then w(M) is  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The weight function is our guidance for the adaptive choice of the subsequent parameters {Mk, Rk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  To avoid critical points, the key idea of AQAE is that the smaller w(Mk) is, the smaller Rk will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (4) enables us to generate a 0-1 distribution random variable with p(1) = sin2[Mφ] for any odd  number M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' An important thing for AQAE is that we should generalize it to the even case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (1) we  5  have,  cos φA−1(|ψ0⟩ |0⟩) + sin φA−1(|ψ1⟩ |1⟩) = |00 · · · 0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (14)  By the orthogonality of A−1 we know that,  |ψ′⟩ := sin φA−1(|ψ0⟩ |0⟩) − cos φA−1(|ψ1⟩ |1⟩),  (15)  is orthogonal to |00 · · · 0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' That is, if we measure all qubits of |ψ′⟩ under the computational basis, we will  certainly get results that contain one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Moreover,  A−1 |ψ0⟩ |0⟩ = cos φ |00 · · · 0⟩ + sin φ |ψ′⟩ ,  (16)  A−1 |ψ1⟩ |1⟩ = sin φ |00 · · · 0⟩ − cos φ |ψ′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (17)  Define,  Q′ = A−1(I ⊗ Z)A(2 |00 · · · 0⟩⟨00 · · · 0| − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (18)  Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Q′ |00 · · · 0⟩ =A−1(I ⊗ Z)A |00 · · · 0⟩  =A−1(I ⊗ Z)(cos φ |ψ0⟩ |0⟩ + sin φ |ψ1⟩ |1⟩)  =A−1(cos φ |ψ0⟩ |0⟩ − sin φ |ψ1⟩ |1⟩)  = cos φ(cos φ |00 · · · 0⟩ + sin φ |ψ′⟩) − sin φ(sin φ |00 · · · 0⟩ − cos φ |ψ′⟩)  = cos(2φ) |00 · · · 0⟩ + sin(2φ) |ψ′⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (19)  and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Q′ |ψ′⟩ =A−1(I ⊗ Z)A(− |ψ′⟩)  =A−1(I ⊗ Z)(− sin φ |ψ0⟩ |0⟩ + cos φ |ψ1⟩ |1⟩)  =A−1(− sin φ |ψ0⟩ |0⟩ − cos φ |ψ1⟩ |1⟩)  = − sin φ(cos φ |00 · · · 0⟩ + sin φ |ψ′⟩) − cos φ(sin φ |00 · · · 0⟩ − cos φ |ψ′⟩)  = − sin(2φ) |00 · · · 0⟩ + cos(2φ) |ψ′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (20)  Therefore, Q′ is a rotation by angle 2φ in the plane spanned by |00 · · · 0⟩ and |ψ′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We can deduce that,  Q′m |00 · · · 0⟩ = cos(2mφ) |00 · · · 0⟩ + sin(2mφ) |ψ′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  (21)  By measuring all qubits under the computational basis we obtain all zero with probability cos2(2mφ), and  results containing one with probability sin2(2mφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The extended amplitude amplification process requires  2m calls to the oracle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  In summary, no matter M is odd or even, we can obtain a random variable rM with 0-1 distribution where  p(1) = sin2(Mφ), with a cost of M oracle calls to the oracle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' When M is odd, we measure the last qubit  of the state Q(M−1)/2A |00 · · · 0⟩, and obtain one with probability sin2(Mφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' When M is even, we measure  all qubits of the state Q′M/2 |00 · · · 0⟩, and the probability that the results contain one is sin2(Mφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' For  convenience, we use the terminology measuring rM to mean that we use the procedure above to obtain a  random variable of 0-1 distribution with p(1) = sin2(Mφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The extended amplitude amplification is crucial  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='0035  Bias  RMSE  CRLB  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 4: The bias, RMSE and CRLB for AQAE, with parameters K = 5 and R = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The CRLB is  calculated as the average value of F−1/2 = [  1  a(1−a)  �  k RkM 2  k]−1/2, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  to our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Algorithm 1: Adaptive Quantum Amplitude Estimation (AQAE)  Input  : K: Number of iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' R: Number of measurements in each iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Output: ˜a: Estimation of a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  1 Set M1 = 1 and R1 = R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  2 Measure r1 for R times and let h1 be the number of ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  3 for i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='.K do  4  Calculate the weights {w(m)}2i−1  m=2i−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  5  Set Mm = m and Rm = 0 for m = 2i−1, · · · , 2i − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  6  for j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='.R do  7  Draw a random sample mj from {2i−1, · · · , 2i − 1}, with probabilities w(mj)/ �2i−1  m=2i−1 w(m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  8  Increase Rmj by one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  9  end  10  Measure rk for Rk times and let hk be the number of ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  11 end  12 Calculate ˜a using MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Our algorithm is shown in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' First we set M1 = 1 and R1 = R, and measure the state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (1)  directly for R times to obtain h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' In the second iteration, we set M2 = 2 and M3 = 3, and compute w(2)  and w(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We draw R samples from {2, 3} with probabilities  �  w(2)  w(2)+w(3),  w(3)  w(2)+w(3)  �  , and set R2, R3 to be  the number of 2 and 3 in the outcome, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' In the third iteration we run the same procedure for  M4 = 4, M5 = 5, M6 = 6, M7 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' After all K iterations, we apply the MLE to obtain the result ˜a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  We carry out several numerical experiments to show the efficiency of AQAE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' All the quantum  outputs in the experiments are obtained by sampling the theoretic distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' First, in comparison  to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 1, the bias and RMSE curve for AQAE is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We find that the bias intensity of AQAE  is much lower than MLAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Besides, the RMSE curve is smooth and close to the average CRLB curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  To illustrate how the parameters chosen by AQAE vary with different as, we make statistics for two typical  as, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 5 (a) all even Mks are chosen less frequently then odd Mks, since a is a critical  point of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 5 (b) all Mks that are multiples of 3 are chosen less frequently since a is a critical  point of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' This set of experiments show that AQAE can effectively avoid critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  Finally, we compare different amplitude estimation algorithms and take the time cost into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  In this experiment we uniformly randomly draw 216 samples in the interval [0, 1] as a, and compare the  error behavior with respect to the time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' For Monte Carlo (MC) estimation, suppose the state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (1) is  prepared for R times, and by measuring the last qubit the result 1 is obtained for h times, then the estimation  to a is given by ˆa = h/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The time cost for MC is N = R, as each preparation of the state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (1) requires  7  0  5  10  15  20  25  30  0  5  10  15  20  25  30  (a) When a = sin2(π/4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='5, a critical point of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  0  5  10  15  20  25  30  0  5  10  15  20  25  30  (b) When a = sin2(π/6) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='25, a critical point of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 5: The average Rk (y-axis) for each Mk (x-axis) chosen by AQAE when K = 5 and R = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The Mks  that are multiples of 2 in (a) or multiples of 3 in (b) are labelled orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  102  103  104  Time Cost  10  3  10  2  RMSE  MC  QPE  UQPE  IQAE  MLAE  AQAE  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 6: The error behavior (y-axis) with respect to the time cost N (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  one call to the oracle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' For MLAE and AQAE, both algorithms require two parameters K and R, which  is chosen by pre-calculation that has the minimal RMSE among several parameter pairs with approximate  time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The time cost of MLAE is N = �  k RkMk = R(2K + K − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Since the parameter set {Mk, Rk}  is not fixed in AQAE, we calculate the average value of �  k RkMk chosen in numerical experiments as its  time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The quantum phase estimation (QPE) based amplitude estimation requires a parameter t as the  number of controlled qubits [30], with time cost N = �t−1  j=0 2j = 2t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' An efficient way to reduce the  RMSE of QPE is to repeat for R times and use MLE to give the final estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The unbiased quantum  phase estimation (UQPE) [19] is an unbiased variant of QPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The time cost for both QPE and UQPE in  our experiments is N = R(2t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' In our experiments we fix R = 4 and let t vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' For IQPE [24], we use  Clopper-Pearson confidence interval method, fix α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='05, Nshots = 100 and let ϵ vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The results are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The MC algorithm have an error convergence of O(N −1/2), while all other algorithms have an  8  asymptotic O(N −1) error convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The UQPE performs the best among those algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' If we limit  the comparison in algorithms without phase estimation, as they are more likely to be implemented widely in  recent years, then our AQAE algorithm outperforms other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  CONCLUSION  The maximum likelihood amplitude estimation (MLAE) algorithm is a practical solution to the quan-  tum amplitude estimation problem, which has a theoretically quadratic speedup over classical Monte Carlo  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We find that MLAE behaves efficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' unbiased and saturates the Cram´er-Rao inequality in  most area except some periodical small intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We analyze how the bias occurs around the so-called crit-  ical points, and propose an adaptive quantum amplitude estimation (AQAE) algorithm by choosing MLAE  parameters adaptively to avoid critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' In the end, we do numerical experiments among some ampli-  tude estimation algorithms, including Monte Carlo estimation, quantum phase estimation and its unbiased  variant, iterative quantum amplitude estimation, maximum likelihood amplitude estimation and our adaptive  amplitude estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' We show that our algorithm outperforms the original MLAE obviously, and it behaves  the best among all algorithms without phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
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+page_content='  [19] Xi Lu and Hongwei Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Unbiased quantum phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='00231, Quantum Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' (in  press), 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  9  [20] Shumpei Uno, Yohichi Suzuki, Keigo Hisanaga, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, and Naoki  Yamamoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Modified grover operator for quantum amplitude estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' New Journal of Physics, 23(8):083031,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [21] Yohichi Suzuki, Shumpei Uno, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, and Naoki Yamamoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Am-  plitude estimation without phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=', 19(2):1–17, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [22] Alicja Dutkiewicz, Barbara M Terhal, and Thomas E O’Brien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Heisenberg-limited quantum phase estimation of  multiple eigenvalues with few control qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Quantum, 6:830, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [23] Tomoki Tanaka, Shumpei Uno, Tamiya Onodera, Naoki Yamamoto, and Yohichi Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Noisy quantum ampli-  tude estimation without noise estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' A, 105:012411, Jan 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [24] Dmitry Grinko, Julien Gacon, Christa Zoufal, and Stefan Woerner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Iterative quantum amplitude estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  NPJ Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=', 7:1–6, 3 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [25] Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini, and Michael Lubasch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Variational quantum amplitude  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Quantum, 6:670, March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [26] Scott Aaronson and Patrick Rall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Quantum approximate counting, simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' In Symposium on Simplicity in  Algorithms, pages 24–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' SIAM, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [27] Kouhei Nakaji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Faster amplitude estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='02417, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Kullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Certain inequalities in information theory and the cramer-rao inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' The Annals of Mathematical  Statistics, 25(4):745–751, 1954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [29] Ronald A Fisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
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+page_content=' Philosophical transactions of the  Royal Society of London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Series A, containing papers of a mathematical or physical character, 222(594-604):309–  368, 1922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content='  [30] Michael A Nielsen and Isaac L Chuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Quantum computation and quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}
+page_content=' Cambridge University  Press, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQfpfjS/content/2301.00528v1.pdf'}

7dAzT4oBgHgl3EQfgPzo/content/tmp_files/2301.01467v1.pdf.txt

@@ -0,0 +1,1188 @@
+arXiv:2301.01467v1  [cond-mat.supr-con]  4 Jan 2023
+Nodeless superconductivity in noncentrosymmetric LaRhSn
+Z. Y. Nie,1, 2 J. W. Shu,1, 2 A. Wang,1, 2 H. Su,1, 2 W. Y. Duan,1, 2 A. D. Hillier,3 D. T.
+Adroja,3, 4 P. K. Biswas,3 T. Takabatake,1, 5 M. Smidman,1, 2, ∗ and H. Q. Yuan1, 2, 6, †
+1Center for Correlated Matter and School of Physics, Zhejiang University, Hangzhou 310058, China
+2Zhejiang Province Key Laboratory of Quantum Technology and Device,
+Department of Physics, Zhejiang University, Hangzhou 310058, China
+3ISIS Facility, STFC Rutherford Appleton Laboratory,
+Harwell Science and Innovation Campus, Oxfordshire, OX11 0QX, United Kingdom
+4Highly Correlated Matter Research Group, Physics Department,
+University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa
+5Department of Quantum Matter, AdSE, Hiroshima University, Higashi-Hiroshima 739-8530, Japan
+6State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310058, China
+(Dated: January 5, 2023)
+The superconducting order parameter of the noncentrosymmetric superconductor LaRhSn is
+investigated by means of low temperature measurements of the specific heat, muon-spin relax-
+ation/rotation (µSR) and the tunnel-diode oscillator (TDO) based method.
+The specific heat
+and magnetic penetration depth [λ(T )] show an exponentially activated temperature dependence,
+demonstrating fully gapped superconductivity in LaRhSn. The temperature dependence of λ−2(T )
+deduced from the TDO based method and µSR show nearly identical behavior, which can be well
+described by a single-gap s-wave model, with a zero temperature gap value of ∆(0) = 1.77(4)kBTc.
+The zero-field µSR spectra do not show detectable changes upon cooling below Tc, and therefore
+there is no evidence for time-reversal-symmetry breaking in the superconducting state.
+PACS number(s):
+I.
+INTRODUCTION
+Noncentrosymmetric superconductors (NCS) have at-
+tracted considerable interest, since in the absence of
+inversion symmetry, an antisymmetric potential gradi-
+ent gives rise to an antisymmetric spin-orbit coupling
+(ASOC). The ASOC lifts the two-fold spin degeneracy
+of the electronic bands, potentially allowing for uncon-
+ventional superconducting properties such as the admix-
+ture of spin-singlet and spin-triplet pairing states [1, 2].
+In the noncentrosymmetric heavy fermion superconduc-
+tor CePt3Si, measurements of the magnetic penetration
+depth, thermal conductivity and specific heat showed
+the presence of line nodes in the energy gap [3–5], and
+nodal superconductivity was subsequently found in other
+NCS, such as Li2Pt3B [6, 7], Y2C3 [8], K2Cr3As3 [9, 10],
+and ThCoC2 [11]. However, many NCS are found to be
+fully gapped superconductors, such as Mo3Al2C [12, 13],
+RT Si3 (R = La, Sr, Ba, Ca; T = transition metal) [14–
+18], BiPd [19, 20], Re6T [21–23], La7T3 [24, 25], BeAu
+[26] and PbTaSe2 [27–29]. Even though some of these
+systems have been found to have multiple superconduct-
+ing gaps, many NCS show evidence for single gap s-
+wave superconductivity, indicating negligible contribu-
+tions from a spin-triplet pairing component.
+The pre-
+dominance of such s-wave superconductivity even in sys-
+tems with strong ASOC has posed the question as to
+what conditions are required to give rise to mixed parity
+pairing. In addition, even in NCS exhibiting unconven-
+tional properties, unambiguosly demonstrating the pres-
+ence of singlet-triplet mixing remains challenging, and
+obtaining direct evidence may require probing associated
+topological superconducting phenomena such as gapless
+edge modes and Majorana modes [30, 31].
+Time reversal symmetry breaking (TRSB) has been
+observed in the superconducting states of some weakly
+correlated NCS, such as LaNiC2 [32], La7T3 [24, 25], and
+several Re-based superconductors [21, 33, 34]. TRSB has
+primarily been revealed by muon-spin relaxation mea-
+surements, which detect the spontaneous appearance of
+small magnetic fields in the superconducting state, even
+in the absence of external applied fields [35].
+In most
+cases, such systems have been found to have nodeless
+superconducting gaps, which has often been difficult to
+reconcile with the unconventional nature of the pairing
+state implied by TRSB. On the other hand, different be-
+havior was recently found in the weakly correlated NCS
+CaPtAs, where there is evidence for both nodal supercon-
+ductivity and TRSB [36, 37]. Consequently, it is impor-
+tant to survey a wide range of different classes of NCS,
+so as to look for novel behaviors arising from ASOC, as
+well as to reveal the origin of any time reversal symmetry
+breaking and to understand its relationship to the broken
+inversion symmetry.
+LaRhSn crystallizes in the noncentrosymmetric hexag-
+onal ZrNiAl-type structure (space group P¯62m) dis-
+played in the inset of Fig. 1, where the rare-earth atoms
+form a distorted kagome lattice.
+Compounds in this
+family with a magnetic rare-earth atom have been ex-
+tensively studied due to the interplay of strong elec-
+tronic correlations and frustrated magnetism [38–40],
+while several other systems with nonmagnetic rare-earth
+elements are superconductors. For example, Sc(Ir,Rh)P,
+LaRhSn, LaPdIn are superconductors with relatively low
+transition temperatures Tc [41–44], while (Zr,Hf)RuP,
+
+2
+μ
+FIG. 1. (Color online) Temperature dependence of the electri-
+cal resistivity ρ(T ) of LaRhSn from room temperature down
+to 0.5 K. The insets show ρ(T ) near the superconducting tran-
+sition, and the crystal structure of LaRhSn.
+ZrRu(As,Si) and Mo(Ni,Ru)P have Tc’s over 10 K [45–
+49], where the higher Tc values may be a consequence
+of the phonon spectra and electron-phonon coupling
+strengths [47, 50, 51]. In this article, we study the order
+parameter of LaRhSn via measurements of the electronic
+specific heat and magnetic penetration depth, where the
+latter is probed using both the tunnel-diode oscillator
+(TDO) based method and muon-spin rotation (µSR).
+The experimental results obtained by various techniques
+can be consistently described by a single-gap s-wave
+model corresponding to weak electron-phonon coupling.
+In addition, zero-field µSR measurements do not exhibit
+detectable changes below Tc, and therefore there is no
+evidence for TRSB in the superconducting state.
+II.
+EXPERIMENTAL DETAILS
+Single crystals of LaRhSn were synthesized using the
+Czochralski method, as described in Ref 52. The specific
+heat was measured in a Quantum Design Physical Prop-
+erty Measurement System (PPMS) with a 3He insert.
+The resistivity ρ(T ) was measured in a 3He cryostat from
+room temperature down to 0.5 K, using a standard four-
+probe method. µSR measurements were performed using
+the MuSR spectrometer at the ISIS pulsed muon source
+of the Rutherford Appleton Laboratory, UK [53, 54]. The
+µSR experiments were conducted in transverse-field (TF)
+and zero-field (ZF) configurations, so as to probe the flux
+line lattice (FLL) and the presence or absence of time-
+reversal symmetry breaking, respectively. Powdered sin-
+gle crystals of LaRhSn were mounted on a high-purity
+silver sample holder, which was mounted on a dilution
+refrigerator, with a temperature range from 0.05 K to
+2.5 K. With an active compensation system, the stray
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+ 
+ 
+ s-wave
+ 
+(0) = 1.76 k
+B
+T
+c
+C
+el 
+/
+n
+T (K)
+0
+2
+4
+6
+8
+0
+20
+40
+C/T (mJ mole
+-1
+ K
+-2
+)
+T (K)
+C/T=
+n
++
+T
+ 2
++
+T
+ 4
+FIG. 2. (Color online) Temperature dependence of the elec-
+tronic specific heat as Cel(T )/γnT of LaRhSn, where the solid
+line represents fitting with a single-gap s-wave model. The in-
+set displays the total specific heat C(T )/T , where the dashed
+line represents the fitting to the normal state contribution.
+magnetic field at the sample position can be canceled to
+within 1 µT. TF-µSR experiments were carried out in
+several fields up to 60 mT.
+The shift of the magnetic penetration depth from the
+zero-temperature value ∆λ(T ) = λ(T ) − λ(0) was mea-
+sured down to 0.3 K in a 3He cryostat, using a tunnel-
+diode oscillator (TDO) based method [55–57], with an
+operating frequency of 7 MHz and a noise level of 0.1 Hz.
+Samples with typical dimensions of 550 × 450 × 300 µm3,
+were mounted on a sapphire rod. The generated ac field is
+about 2 µT, which is much smaller than the lower critical
+field Hc1, ensuring that the sample remains in the Meiss-
+ner state. ∆λ(T ) is proportional to the frequency shift
+from zero temperature ∆f(T ), i.e., ∆λ(T ) = G∆f(T ),
+where G is the calibration factor determined from the
+geometry of the coil and sample [56].
+III.
+RESULTS
+A.
+Electrical resistivity and specific heat
+The single crystals of LaRhSn were characterized by
+measurements of the electrical resistivity and specific
+heat. Figure 1 displays the electrical resistivity ρ(T ) from
+room temperature down to 0.5 K, which exhibits metallic
+behavior in the normal state. The inset shows ρ(T ) at
+low temperatures, where there is a sharp superconduct-
+ing transition at around 2.0 K.
+The inset of Figure. 2 displays the total specific heat
+C(T )/T of LaRhSn in zero field, where there is a clear
+superconducting transition with a midpoint Tc = 1.9 K,
+in line with the behavior of ρ(T ). In the normal state, the
+specific heat data are fitted by C(T )/T = γn+βT 2+δT 4,
+
+BP3
+0.5
+1.0
+1.5
+2.0
+2.5
+0.0
+0.5
+1.0
+1.5
+2.0
+ 
+ 
+ 0.25 T
+ 0.20 T
+ 0.17 T
+ 0.15 T
+ 0.13 T
+ 0.10 T
+ 0.08 T
+ 0.05 T
+ 0.02 T
+ 0.00 T
+C
+el 
+/
+n
+T
+T (K)
+(a)
+0.0
+0.5
+1.0
+1.5
+0.0
+0.1
+0.2
+B
+c2
+ (T)
+T (K)
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+(b)
+ This work
+          (0.38K)
+ MgB
+2
+ LaNiC
+2
+ Re
+24
+Nb
+5
+ 
+ 
+0.38K
+(B) /
+n
+B/B
+c2
+(0)
+FIG. 3. (Color online) (a) Temperature dependence of the
+electronic specific heat as Cel/γnT of LaRhSn under vari-
+ous applied fields.
+The inset displays the temperature de-
+pendence of the upper critical field Bc2(T ), derived from the
+specific heat measurements, where the solid line represents
+fitting with the WHH model where Bc2(0) = 0.219(2) T. (b)
+Field dependence of the residual Sommerfeld coefficient plot-
+ted as γ0.38K(B)/γn versus B/Bc2(0) for LaRhSn, Re24Nb5
+[34], MgB2 [58] and LaNiC2 [59]. The dashed and dashed-
+dotted lines correspond to the expected behaviors of nodal
+and single-gap s-wave superconductivity, respectively.
+with γn = 11.15(4) mJ mole−1 K−2, β = 0.410(6) mJ
+mole−1 K−4 and δ = 0.87(1) µJ mole−1 K−6. Here γn is
+the normal state Sommerfeld coefficient, and the latter
+two terms represent the phonon contribution. The De-
+bye temperature θD is estimated to be 241(1) K using
+θD = (12π4Rn/5β)1/3, where R = 8.31 J mole−1 K−1
+is the molar gas constant and n = 3 is the number of
+atoms per formula unit. The electron-phonon coupling
+constant λel-ph can be approximated via
+λel-ph =
+1.04 + µ∗ln(
+θD
+1.45Tc )
+(1 − 0.62µ∗)ln(
+θD
+1.45Tc ) − 1.04.
+(1)
+Using the typical values for µ∗ of 0.1 – 0.15, λel-ph = 0.47
+– 0.57 are obtained, close to the derived values for
+isostructural LaPdIn [44], indicating weakly coupled su-
+perconductivity in LaRhSn. In addition, the value of γn
+is very similar to that of LaPdIn, but larger than the
+values for LuPdIn and LaPtIn which are not supercon-
+ducting down to at least 0.5 K [44]. This is consistent
+with the magnitude of the density of states at the Fermi
+level playing an important role in giving rise to super-
+conductivity in this family of compounds.
+The main panel of Fig. 2 shows the low tempera-
+ture electronic specific heat Cel(T )/γnT , from which the
+phonon contribution has been subtracted. In the super-
+conducting state, the entropy S can be calculated by [60]
+S = − 3γn
+π3
+� 2π
+0
+� ∞
+0
+[flnf + (1 − f)ln(1 − f)]dεdφ, (2)
+where the f(E, T ) = [1+exp(E/kBT )]−1 is the Fermi-
+Dirac distribution function. Here, E =
+�
+ε2 + ∆2
+k, where
+∆k(T ) = ∆(T )gk is the superconducting gap function.
+Therefore, the electronic specific heat of superconducting
+state can be obtained by Cel = T dS/dT . In the case of
+a single-gap s-wave model, there is no angle dependent
+component (gk = 1), and ∆(T ) was approximated by [61]
+∆(T ) = ∆(0)tanh
+�
+1.82 [1.018 (Tc/T − 1)]0.51�
+,
+(3)
+where ∆(0) is the zero-temperature superconducting gap
+magnitude. As shown by the solid line in Fig. 2, the zero
+field Cel/γnT can be well described by this single-gap
+s-wave model, with ∆(0) = 1.76(1)kBTc.
+Upon applying a magnetic field, the bulk supercon-
+ducting transition is shifted to lower temperatures and
+is completely suppressed at about 0.25 T (see Fig.
+3 (a)).
+The inset displays the extracted upper crit-
+ical field Bc2(T ) and the corresponding fitting using
+the Werthamer-Helfand-Hohenberg (WHH) model [62],
+with a zero temperature upper critical field Bc2(0) =
+0.219(2) T. Using λ(0) =
+�
+Φ0Bc2(0)/√24γn∆(0) [63],
+where the units of Bc2(0) and γn
+are gauss and
+ergs cm−3 K−2, respectively, a penetration depth at zero
+temperature λ(0) = 244(1) nm is estimated using ∆(0) =
+1.76(1)kBTc. Combined with a Ginzburg-Landau (GL)
+coherence length of ξGL =
+�
+Φ/2πBc2(0) = 38.7(2) nm,
+the GL parameter κ is estimated to be 6.30(4), indicating
+that LaRhSn is a type-II superconductor. Using the val-
+ues of λ(0)=244(1) nm, a residual normal state resistivity
+ρ0 = 25 µΩ cm and γn = 11.15(4) mJ mole−1K−2, the
+mean free path ℓ and BCS coherence length ξBCS are es-
+timated to be ℓ=17.91(8) nm and ξBCS=43.8(2) nm [64].
+The mean free path ℓ is smaller than ξBCS, indicating
+that the sample is in the dirty limit.
+
+4
+0
+2
+4
+6
+8
+10
+12
+0.0
+0.1
+0.2
+ 
+ 
+ 2.5 K
+ 0.1K
+ 2.5 K fit
+ 0.1 K fit
+Asymmetry
+Time (
+s)
+FIG. 4. (Color online) ZF-µSR spectra of LaRhSn at 2.5 K
+(T > Tc) and 0.1 K (T < Tc). The solid lines show the results
+from fitting using Eq. 4.
+Figure 3 (b) displays the field dependence of the Som-
+merfeld coefficient value at 0.38 K, normalized by its
+value in the normal-state, i.e., γ0.38K(B)/γn. It can be
+seen that γ0.38K(B)/γn shows a nearly linear field depen-
+dence, being similar to the fully gapped superconductor
+Re24Nb5 [34]. On the other hand, γ0.38K(B)/γn clearly
+deviates from the square-root field dependence (dashed
+line) expected for line nodal superconductors, as well as
+the typical behaviors of the multiband superconductors
+MgB2 [58] and LaNiC2 [59]. Note that γ0.38K(B)/γn of
+LaRhSn are determined from the specific heat at the low-
+est measured temperature, and therefore even in zero-
+field the data have a finite value.
+B.
+µSR measurements
+Figure 4 displays the zero-field (ZF) µSR spectra col-
+lected at 2.5 K (T > Tc) and 0.1 K (T < Tc). These
+are fitted with a damped Gaussian Kubo-Toyabe (KT)
+function
+GZF(t) = A
+�1
+3 + 2
+3(1 − δ2t2)exp
+�
+−δ2t2
+2
+��
+exp(−Λt)+Abg,
+(4)
+where A is the initial asymmetry, and Abg corresponds
+to the time independent background term from muons
+stopping in the silver sample holder.
+δ and Λ are
+the Gaussian and Lorentzian relaxation rates, respec-
+tively.
+Upon fitting with Eq.
+4, δ = 0.086(3) µs−1
+and Λ = 0.0134(11) µs−1 were obtained at 2.5 K, while
+δ = 0.082(3) µs−1 and Λ = 0.0157(10) µs−1 at 0.1 K.
+Therefore, we find no evidence for TRSB in the super-
+conducting state of LaRhSn, and these results suggest
+that any spontaneous internal fields should be no larger
+than 6.6 µT, which is smaller than the corresponding
+fields in other reported TRSB superconductors [35].
+Transverse-field µSR (TF-µSR) measurements were
+carried out in the mixed state with applied fields in the
+range 40 mT to 60 mT, where the data were collected
+0
+2
+4
+6
+8
+-0.2
+0.0
+0.2
+ 
+ 
+ Time (
+s)
+Asymmetry
+(b) 0.05 K
+-0.2
+0.0
+0.2
+ 
+ 
+ 
+(a) 2.5 K
+FIG. 5.
+(Color online) Transverse field µSR spectra of
+LaRhSn at (a) 2.5 K (T > Tc) and (b) 0.05 K (T < Tc)
+in an applied field of 40 mT. The solid lines show the results
+of fitting with Eq. 5
+upon field-cooling in order to probe a well-ordered flux-
+line lattice (FLL). The results at 2.5 K and 0.05 K in a
+field of 40 mT are displayed in Fig. 5. The significant
+increase of the depolarization rate corresponds to the in-
+homogeneous field distribution in the sample, character-
+istic of the formation of a FLL. The TF-µSR asymmetry
+were fitted to the sum of oscillations damped by Gaussian
+decaying functions
+GTF(t) =
+n
+�
+i=1
+Aicos(γµBit + φ)e−(σit)2/2 + ABG, (5)
+where Ai is the amplitude of the oscillating compo-
+nent, which precesses about a local field Bi with a com-
+mon phase offset φ and a Gaussian decay rate σi, while
+γµ/2π = 135.5 MHz/T and ABG are the muon gyromag-
+netic ratio and background term, respectively. The asym-
+metry can be well fitted with three oscillatory compo-
+nents (n = 3), where σ3 was fixed to zero, corresponding
+to muons stopping in the silver sample holder. Figure
+6(a) displays the temperature dependence of σ(T ) ob-
+tained following the multiple-Gaussian method described
+in Ref 65. Here, the first and second moment of the field
+distribution are calculated as
+⟨B⟩ =
+n−1
+�
+i=1
+Ai Bi
+A1 + · · ·An−1
+,
+(6)
+⟨B2⟩ =
+n−1
+�
+i=1
+Ai
+A1 + · · ·An−1
+[(σi/γµ)2 +[Bi −⟨B⟩]2], (7)
+and σ = γµ
+�
+⟨B2⟩. The relaxation rate in the normal
+state is ascribed to a temperature independent contri-
+bution arising from quasistatic nuclear moments, with a
+nuclear dipolar relaxation rate σN
+= 0.0851(27) µs−1.
+
+5
+40
+50
+60
+70
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ 
+ 
+sc
+ (
+s
+-1
+)
+Field (mT)
+ 0.1 K
+ 0.3 K
+ 0.5 K
+ 0.7 K
+ 0.9 K
+ 1.0 K
+ 1.1 K
+ 1.2 K
+ 1.3 K
+ 1.4 K
+ 1.5 K
+(b)
+0
+1
+2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+  40 mT
+  45 mT
+  50 mT
+  60 mT
+ 
+ 
+ (
+s
+-1
+)
+T (K)
+N
+ = 0.0851 
+s
+-1
+(a)
+FIG. 6. (Color online) (a) Temperature dependence of the
+Gaussian relaxation rate of the TF-µSR spectra in different
+applied fields between 40 mT and 60 mT. (b) Field depen-
+dence of the superconducting contribution to the TF-µSR re-
+laxation rate σsc at various temperatures, where the solid lines
+correspond to fitting using Eq. 8.
+The superconducting component of the variance σsc is
+calculated as σsc =
+�
+σ2 − σ2
+N, and its field dependence
+is displayed in Fig. 6(b) for several temperatures.
+For small applied fields and large κ, σsc is field inde-
+pendent and proportional to λ−2, which is not applicable
+for the current measurements of LaRhSn. On the other
+hand, for κ ≥ 5 and 0.25/κ1.3 ≤ b ≤ 1, σsc may be ap-
+proximated by [66]
+σsc = 4.854 × 104 1
+λ2 (1 − b)[1 + 1.21(1 −
+√
+b)3],
+(8)
+where b = B/Bc2 is the applied field normalized by the
+upper critical field. Since the κ of LaRhSn was deter-
+mined to be about 6.30(4), the measurements of LaRhSn
+are within the applicability of Eq. 8. Therefore by fixing
+Bc2(T ) to the bulk values derived from the specific heat
+0.4
+0.6
+0.8
+1.0
+0
+10
+20
+ 
+ 
+ s-wave
+ ~T
+ 4.4
+ ~T
+ 2
+ (nm)
+T (K)
+1
+2
+0
+2
+4
+6
+8
+f (kHz)
+T (K)
+FIG. 7. (Color online) The change of magnetic penetration
+depth ∆λ(T ) of LaRhSn at low temperatures. The solid red,
+dashed blue and dashed-dotted magenta lines represent fitting
+to an s-wave model, and power-law dependences ∼ T 4.4 and
+∼ T 2, respectively.
+The inset displays the frequency shift
+∆f(T ) from 2.5 K down to 0.3 K, where there is a sharp
+superconducting transition at around Tc = 2 K.
+in Fig. 3, the temperature dependence of λ−2(T ) can be
+obtained from fitting with Eq. 8 [Fig. 6(b)], and the re-
+sults are shown in Fig. 8, together with the TDO results
+described in following section.
+C.
+TDO measurements and superfluid density
+analysis
+Figure 7 shows the penetration depth shift ∆λ(T ) of
+LaRhSn at low temperatures, with a calibration factor
+G = 14.2 ˚A/Hz. The inset displays the frequency shift
+∆f(T ) from 2.5 K down to the base temperature of 0.3 K,
+where a sharp superconducting transition is observed at
+Tc = 2 K, in accordance with other measurements. Upon
+further cooling, ∆λ(T ) flattens at the lowest measured
+temperatures, indicating fully gapped superconductivity
+in LaRhSn. For an s-wave superconductor, the temper-
+ature dependence of ∆λ(T ) for T ≪ Tc can be approxi-
+mated by
+∆λ(T ) = λ(0)
+�
+π∆(0)
+2kBT exp
+�
+−∆(0)
+kBT
+�
+.
+(9)
+As shown by the solid line, the experimental data be-
+low Tc/3 can be well described by the s-wave model with
+∆(0) = 1.80(1)kBTc, where λ(0) = 227.9 nm was fixed
+to the value derived from TF-µSR. The data were also
+fitted by a power law dependence ∆λ(T ) ∝ T n, from
+0.3 K up to 0.75 K. A large exponent of n = 4.4 is ob-
+tained, which is much larger than two, excluding nodal
+superconductivity in LaRhSn.
+
+6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+5
+10
+15
+20
+ 
+ 
+ TDO
+ 
+SR
+ s-wave (clean)
+ s-wave (dirty)
+ d-wave
+ p-wave
+-2
+ (
+m
+-2
+)
+T/T
+c
+FIG. 8. (Color online) Temperature dependence of λ−2(T ) as
+a function of the normalized temperature T/Tc. The data are
+derived from measurements using the TDO based method and
+TF-µSR measurements, which correspond to the empty circle
+and solid symbols, respectively. The lines show the results
+from fitting with different models for the gap structure.
+To further characterize the superconducting pairing
+state of LaRhSn, the temperature dependence of λ−2(T )
+was analyzed, which is proportional to the superfluid den-
+sity ρs(T ) as ρs(T ) = [λ(0)/λ(T )]2. Figure 8 displays
+λ−2(T ) as a function of the reduced temperature T/Tc,
+where the data are derived from both the TDO and TF-
+µSR measurements, which show nearly identical behav-
+ior. Since the previous analysis suggested that the sample
+is in the dirty limit, the results from TF-µSR were fitted
+with the following expression for a dirty s-wave model
+[67]
+ρs(T ) = ∆(T )
+∆(0) tanh
+� ∆(T )
+2kBT
+�
+.
+(10)
+As shown by the dashed line in Fig.
+8, the dirty s-
+wave model can well describe the experimental data, with
+λ(0) = 227.9(9) nm and ∆(0) = 1.77(4)kBTc. The data
+were also analyzed using the clean limit expression
+ρs(T ) = λ−2(T )
+λ−2(0) = 1 + 2
+�� ∞
+∆k
+EdE
+�
+E2 − ∆2
+k
+∂f
+∂E
+�
+FS
+,
+(11)
+where a clean single gap s-wave model can also fit
+the data well, yielding a larger gap value of ∆(0) =
+2.05(3)kBTc. Here the gap value obtained from the dirty
+s-wave model is in very good agreement to those derived
+from the analysis of specific heat and low temperature
+∆λ(T ), while the clean limit value is considerably larger,
+which is in-line with the previous dirty limit calculation.
+We note that due to the samples being in the dirty limit,
+TABLE I. Superconducting parameters of LaRhSn, where the
+parentheses with C and µSR denote the results from the spe-
+cific heat and µSR, respectively.
+Property
+Unit
+Value
+Tc
+K
+1.9
+Bc2(0)
+T
+0.219(2)
+γn
+mJ mole−1K−2
+11.15(4)
+ΘD
+K
+241(1)
+λel−ph
+0.47-0.57
+ξGL
+nm
+38.7(2)
+ℓ
+nm
+17.91(8)
+ξBCS
+nm
+43.8(2)
+λ0(C)
+nm
+244(1)
+λ0(µSR)dirty
+nm
+227.9(9)
+κ(C)
+6.30(4)
+κ(µSR)dirty
+5.89(4)
+∆(0)(C)
+kBTc
+1.76(1)
+∆(0)(µSR)dirty
+kBTc
+1.77(4)
+we cannot exclude an anisotropic superconducting gap
+in LaRhSn, since impurity scattering can suppress any
+gap anisotropy.
+On the other hand, as also shown in
+Fig. 8, a d-wave model with gk = cos 2φ and p-wave
+model with gk = sin θ (φ= azimuthal angle, θ= polar
+angle) cannot account for the data, further indicating a
+lack of nodal superconductivity in LaRhSn. Meanwhile,
+the value of λ(0) obtained from µSR experiments is very
+close to that from specific heat results. Using this value of
+λ(0) = 227.9(9)nm, κ = 5.89(4) is estimated, which cor-
+responds well to the value from the specific heat analysis.
+The obtained superconducting parameters of LaRhSn are
+displayed in Table I. Therefore, the results of specific
+heat, TDO-based measurements and µSR can all be con-
+sistently described by a single-gap s-wave model with a
+gap magnitude very close to that of weak-coupling BCS
+theory, and there is no evidence for time-reversal sym-
+metry breaking below Tc.
+IV.
+SUMMARY
+In summary, we have studied the order parameter of
+the noncentrosymmetric superconductor LaRhSn. Both
+the specific heat and magnetic penetration depth show
+exponentially activated behavior at low temperatures,
+providing strong evidence for fully gapped superconduc-
+tivity. λ−2(T ) derived from the TDO based method and
+TF-µSR, as well as the specific heat can be consistently
+well described by a single-gap s-wave model, with a gap
+magnitude very close to that of weak coupling BCS the-
+ory. Together with findings for LaPdIn [44] and ZrRuAs
+[47], our results suggest that fully gapped s-wave super-
+conductivity, together with a lack of evidence for time
+
+7
+reversal symmetry breaking, are consistent common fea-
+tures of weakly correlated NCS with the ZrNiAl-type
+structure and there is a lack of significant singlet-triplet
+mixing.
+ACKNOWLEDGMENTS
+This work was supported by the National Key R&D
+Program of China (Grant No. 2017YFA0303100), the
+Key R&D Program of Zhejiang Province, China (Grant
+No. 2021C01002), the National Natural Science Foun-
+dation of China (Grant No. 11874320, No. 11974306
+and No. 12034017), and the Zhejiang Provincial Natu-
+ral Science Foundation of China (R22A0410240). D.T.A.
+would like to thank the Royal Society of London for Ad-
+vanced Newton Fellowship founding between UK and
+China.
+Experiments at the ISIS Pulsed Neutron and
+Muon Source were supported by a beamtime alloca-
+tion from the Science and Technology Facilities Council
+(Grant No. RB2010190 [54])
+∗ Corresponding author: [email protected]
+† Corresponding author: [email protected]
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+

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+DEEP LEARNING BASED MULTI-LABEL IMAGE CLASSIFICATION OF PROTEST
+ACTIVITIES
+Yingzhou Lu, Kosaku Sato, Jialu Wang
+Electrical and Computer Engineering Department
+Virginia Tech
+Arlington,VA 22203, USA
+George Washington University
+Washington, DC 20052, USA
+Email: [email protected], [email protected], [email protected]
+ABSTRACT
+With the rise of internet technology amidst increasing ur-
+banization rates, sharing information has never been easier,
+thanks to globally-adopted platforms for digital communi-
+cation.
+The resulting output of massive amounts of user-
+generated data can be used to enhance our understanding
+of significant societal issues, particularly for urbanizing ar-
+eas. In order to better analyze protest behavior, we enhanced
+the GSR dataset and manually labeled all the images. We
+used deep learning techniques to analyze social media data
+to detect social unrest through image classification, which
+performed well in predicting multi-attributes. Then, we used
+map visualization to display protest behaviors across the
+country.
+Index Terms— Machine Learning, Deep Learning, Im-
+age Classification, Multi-Label Classification, Social Media
+1. INTRODUCTION
+The study of protest activities plays a profound role in so-
+ciologists and scholars’ studying citizens’ political behavior.
+With the advancement of social media networks, people now
+share an unprecedented amount of user-generated content in
+the form of text, images, and videos on the web. Classifi-
+cation of social media data not only helps in understanding
+online behavior, but also elucidates significant priorities of
+urban populations that carry real-life consequences. Using
+social media data, we focuses on social unrest in the form of
+public protest images, specifically for Latin American coun-
+tries.
+The traditional approach to the study of social media
+dataset focused on using natural language processing to mon-
+itor how hashtags and links are used by the user and the
+propagation of those items to other users. However, these ap-
+proaches may not effectively capture some important features
+or details of protest activities. For instance, we may be in-
+terested in knowing details such as whether there was a large
+crowd involved in the protest, if polices were present, or what
+the demographics (young or adults) of protesters carrying a
+sign. Our approach uses image processing to capture those
+features of the protest activitiesFu et al. (2021).
+We took several approaches in image classification of
+social media data: our initial approach is to utilize traditional
+machine learning methods such as Support Vector Machine
+(SVM)Weston et al. (1999) and a deep learning method like
+Convolutional Neural Networks (CNNs)Krizhevsky et al.
+(2012) which have shown some advantages in large-scale im-
+age and video analysis. Traditional machine learning method,
+such as SVM, can be used to classify images with good accu-
+racy; however, as the volume of data and number of classes
+for recognition increases, the deep learning approaches be-
+comes the more advanced approach for object recognition.
+2. LITERATURE REVIEW
+As our objective of our model is to detect protest activi-
+ties using the image, the preliminary work relevant to our
+study is the EMBERS system by Naren, Patrick and et
+elRamakrishnan et al. (2014). The EMBERS system con-
+tinuously monitor the social media dataset such as Twitter,
+Facebook, news pages, and use data mining to process the
+trend to predict the protest activities in South America re-
+gions. Their planned protest model based on custom multi-
+lingual lexicon matching predicted the protest activities with
+precision and recall rate of 0.69 and 0.82 respectively. How-
+ever, their approach does not capture the additional features
+of the protests such as demography.
+Moreover, image classification is classical topic in com-
+puter vision area which aims to predict and assign each given
+image a specific label from several categories.
+However,
+background clutter, occlusion and variation in image scale
+make the computer vision tasks more challenging. The tra-
+ditional approaches to perform image classification includes
+k-nearest neighbor and SVM algorithmsYi et al. (2018).
+k-nearest neighbor is one of the simplest classification al-
+gorithm that aims at labeling an image based on the best fit
+result, but the model is usually not robust to noise or im-
+balanced class datasetZhang et al. (2006). Similarly, SVM,
+which is originally proposed as a binary classification by
+Cortes and VapnikCortes and Vapnik (1995) is another clas-
+sical approach to perform classification and it has shown a
+better performance than the k-nearest neighbor in some ap-
+arXiv:2301.04212v1  [cs.CV]  10 Jan 2023
+
+Fig. 1. Labeled GSR images in GSR dataset
+plicationsBoiman et al. (2008). Furthermore, our approach
+adopts a deep learning based on CNNs based on the study
+of the visual cortex of the human brain which have shown
+a great success recently in many computer vision applica-
+tionsAghdam and Heravi.
+3. OSI DATASET
+The OSI (Open Source Indicators) dataset, provided by com-
+puter science department from Virginia Tech, is MITRE’s
+gold standard report (GSR) of protests organized by survey-
+ing newspapers for civil unrest reports.
+The dataset also
+contains large samples of non-protest images that were col-
+lected in the process. There are 48,713 images in GSR and
+40,647 non-protest images.
+High-confidence images indi-
+cate a top image among multiple images embedded in the
+articles.
+High-confidence images within datasets indicate
+a top image among multiple images embedded in the arti-
+clesJayachandra et al. (2020). High-confidence GSR images
+relevant to GSR articles based on social protest total 7,884,
+and low-confidence GSR images total 40,829.
+3.1. Details of the Image Labels
+Table 1 shows the visual attributes that characterize the
+protests which we used to label each image from the GSR
+dataset. Out of 40,647, a total of 9,504 images were hand-
+picked to train and test our prediction models by excluding
+bad data points that are obviously irrelevant to the social ac-
+tivities that we are interested in detectingYang et al. (2020).
+The Annotated images consist of 327 fire, 1,943 flag, 7,347
+large crowd, 248 other, 2,159 police, 4,462 sign, and 1,233
+student images. Fig. 1 contains sample images with their
+class labels. Each image has a label with vector of length
+7 that has ”0”s and ”1”s corresponding to the index number
+of visual attributes. The ”Other” label was inferred from the
+absence of positive class labels across categories.
+3.2. Challenges of the Dataset
+There are a few inherent challenges in our training dataset.
+First, some attributes of protest are commonly shared with
+Fig. 2. Sample protest images from training set
+Fig. 3. Sample non-protest images from training set
+Table 1. Visual attributes of protest images
+Class label
+Description
+Sample size
+0
+Fire
+Presence of active fire
+327
+1
+Flag
+Presence of flag
+1943
+2
+Large Crowd
+Presence of roughly more than 20 people
+7347
+3
+Other
+None of the above or the below
+248
+4
+Police
+Presence of police
+2159
+5
+Sign
+Presence of a protest sign
+4462
+6
+Student
+Presence of young students
+1233
+non-protest images. For instance, Figure 2 shows a sample
+of protest images used to train the machine learning and deep
+learning models. Images with a protest class label often de-
+picted fire, police, handwritten signs, and large crowds. How-
+ever, Figure 3 shows a sample of non-protest images in which
+large crowds are also frequently seen attribute while it also
+comprised of a variety of other objects (such as animals or
+soccer players). Second, we have imbalanced dataset where
+it does not have exactly equal number of instances in each
+class. This issue is mainly defined by the specific subject or
+attribute we set up in our problem. The training of the classi-
+fiers in imbalanced dataset can cause the trained model clas-
+sifying images as images of majority class most of the times
+and under-represent the minority class.
+3.3. Image Augmentation for Imbalanced Dataset
+One way to balance the imbalanced classes is to use image
+augmentation. Data augmentation is a method to artificially
+increase sample size of the training images through various
+pre-processing or combinations of multiple pre-processing of
+the image such as adding noise, flipping and re-scaling. Ta-
+ble 2 summarizes the different techniques we adopted to per-
+form image augmentation for minority classes. Image aug-
+mentation has been considered as a promising method to im-
+prove the performance of prediction model.
+For instance,
+adding noise to our observation can help make the prediction
+Table 2. Increased sample size after image augmentation
+Class label
+Image transformation used
+New sample size
+Fire
+Flipping, Scaling, Translation, Noise,
+Affine Transform, Perspective Transform,
+Intensity, Contrast, Filters, Crop, Shear
+4,578
+Flag
+Flipping, Noise, Affine Transform
+5,829
+Large Crowd
+–
+7347
+Other
+Flipping, Scaling, Translation, Noise,
+Affine Transform, Perspective Transform,
+Intensity, Contrast, Filters, Crop, Shear
+3,472
+Police
+Affine Transform, Noise
+6,477
+Sign
+–
+4462
+Student
+Flipping, Noise, Affine Transform, Crop
+6,165
+
+GSR Image with Labels
+['0", "1', '1', "0", '0', "1', "0'
+["0", "1', '1', '0', 0', '1',"1']
+["0",'0', "1', "0", "','0", "0]
+["0", "0', "0', "1', 0', '0", "0"Fig. 4. Example images from the dataset after transformation
+model more robust in the face of social media dataset and pre-
+vents it from overfitting Perez and Wang (2017). Moreover,
+Python’s scikit-image library has the full list of the avail-
+able image transformation but we have adopted the thirteen
+of them. Those are horizontal and vertical flipping, affine
+transfomration, perspective transformation, rescaling, crop-
+ping, blurring, changing contrast and intensity, gaussian and
+exposure filter, translation, and shearing. Fig. 4 shows the
+example of such transformations of the images. As a result
+of implementing image augmentation, we were able to signif-
+icantly improve the sample size of the minority classes; for
+instance, the sample size of ’fire’ class increased from 327 to
+4,578 as seen in Table 2. Also, as an alternative approach,
+we considered oversampling using Synthetic Minority Over-
+sampling Technique (SMOTE) Chawla et al. (2002). How-
+ever, we believe that image augmentation can create more
+variation in training images to prevent over fitting, and hence
+we did not utilize SMOTE in this paper.
+4. MULTI-LABEL IMAGE CLASSIFICATION
+Multi-label learning is a form of supervised learning where
+the classification algorithm learns from a set of images in
+which an image can belong to one or more classes. The goal
+of multi-label image classification is to predict a set of class
+labels for the input image.
+A more generalized approach
+is multi-class learning where each image is limited to one
+correct class label. Multi-label classification and prediction
+is more practical since the many real world problems involve
+multiple objects belonging to different categories.
+Multi-
+label classification is also applicable various domains such
+as text, video, and scene classification. For a typical multi-
+label image, objects of different categories in each image are
+located at varying positions with differing scale, zoom, size,
+and poseWon et al. (2017). For example, two images labeled
+as ’police’ and ’fire’ may have different spatial arrangements
+of identified objects. Although factors such as differing ar-
+rangements or occlusion can contribute to the inaccuracy of
+multi-label classification, we expect reasonable results with a
+sufficiently large dataset. Details of implementation of multi-
+label classification for SVM and CNNs will be discussed in
+the following sections.
+5. APPROACH OF IMAGE CLASSIFICATION
+In our approach, we utilized multi-label SVM and CNNs to
+detect protest attributes in image classification. First, multi-
+label SVM will be explained in details. In our proposal, SVM
+is a baseline model to evaluate the performance of the pre-
+diction model using CNNs. As the problem requires a large
+dataset for training and good accuracy of classifying many
+protest attributes, we believe that CNNs will perform better
+than SVM.
+5.1. Support Vector Machine
+SVM is originally proposed as a binary classification by
+Cortes and VapnikCortes and Vapnik (1995), but the model
+has been extended to apply to multi-label classification prob-
+lemsWeston and Watkins (1998).
+One-vs.-All: One-vs.-All is a classical approach to solve
+k-class pattern recognition problem. It involves training a sin-
+gle binary classifier per class, with the samples of one class
+as positive samples while other samples are set as negative.
+More specifically, using this method, n-th classifier finds a
+hyperplane between class n and the rest of the classesWeston
+and Watkins (1998). A point where the distance from the
+margin is maximal is assigned to the class. We aim at de-
+tecting seven classes so that this strategy requires the training
+of seven different SVMs. During testing the models, all clas-
+sifiers would vote ’true’ by predicting that a testing sample
+belongs to their class. In the end of testing, a sample is classi-
+fied by the ensemble as the class that has the highest number
+of votes. One-vs.-All is widely used in multi-label classifica-
+tion.
+Weighting Hyper-Parameter for SVM: Imbalanced
+dataset causes misclassification of images that belonging to
+the minority class impacted more heavily than that of the ma-
+jority class because the frequency of the minority class is rare
+compared to that of the majority class. In order to mitigate
+over fitting of training classifier resulted from the imbalanced
+data, we propose the modification of hyperparameter C in
+SVM’s objective function which determines the penalty for
+misclassifying the objects. Instead of defaulting C to be one,
+Ck belonging to class k will have different values as shown
+in (4).
+Ck = C ·
+n
+knj
+(1)
+As you can see, the updated Ck value will be inversely propor-
+tional to instances of j class in order to increase Ck value for
+the minority class in order to mitigate under-representation
+issue. k is the number of class and j is the sample size be-
+longing to the class.
+5.2. Convolutional Neural Networks
+5.2.1. Architecture
+CNNs consist of input, convolution, activation function, pool-
+ing, deep layers, and output layers. Throughout the training
+of the network, the parameters are updated except for the ones
+between convolution and pooling. There are some important
+properties of the convolution layer. Some patterns are smaller
+
+Original Image
+Contrast Adjusted
+Noise Added
+ELPAQUETE
+ELPAOUETE
+Cropped
+Horizontal Flip
+Vertical Flip
+ESEKKON
+PAQUETE
+ETBHOREIE
+CIEN
+NOAENERKOthan the entire image so that the image can be subsampled to
+reduce the image size; this is to train fewer parameters in the
+neural network. The same patterns can appear in different re-
+gions so that the same set of parameters can be used to reduce
+computation.
+Convolutional Layers: Convolution of a filter on an input
+image is a point-wise multiplication operation. The activation
+function, which in our case is Rectified Linear Unit (ReLU)
+activation, is applied on each image separately in an element-
+wise fashion to create activation maps based on outputs of the
+convolution Aghdam and Heravi.
+Activation Function: Nair and Hinton introduced the
+non-saturating nonlinearity f(x) = max(0, x), also known as
+the ReLU, which has gained popularity in the deep-learning
+community because of its fast computing time Krizhevsky
+et al. (2012).Hence, our model applied the ReLU nonlinear-
+ity function to the output of every convolutional and fully-
+connectedKrizhevsky et al. (2012).
+Pooling Layer: In our model, we applied a 2x2 filter size
+with a 2-length stride after each layer with the option of max-
+pooling. Max-pooling applies the filters and the stride to the
+input and returns the maximum value, dropping the non-max
+values in each sub-region that convolution is applied.
+Fully Connected Layers: The fully connected layer uses
+those inputs to produce N-dimensional vectors, where N is
+the number of classes needed for prediction.
+Loss Function: In our model, we used sigmoid cross-
+entropy for multi-label classification. Cross-entropy is used
+to define the loss function in training the network in which
+the model is penalized if it estimates a low probability for the
+target class Nielsen (2015).
+J(θ) = − 1
+m
+m
+�
+i=1
+K
+�
+k=1
+[yi
+k log(ˆpi
+k)]
+(2)
+For the loss function, we used Adaptive Moment Estima-
+tion (ADAM) Kingma and Ba (2014). ADAM keeps track of
+a learning rate for each network weight and computes indi-
+vidual adaptive learning rates for different parameters based
+on estimates of first and second moments of the gradients
+Kingma and Ba (2014). Since ADAM is an adaptive rate
+learning algorithm, it requires less tuning. The default learn-
+ing rate 0.001 is often used to support usability of the algo-
+rithm.
+Table 3 shows our CNNs architecture. There are three
+convolutional layers with ReLU, and max-pooling is used to
+down-sample the image after each convolutional layer. The
+filter size for max-pooling is 2 × 2 so that output image of the
+max-pooling is half the size of the input. The convolutional
+layers uses the filter size of 3 × 3 while length of stride equal
+to 2. The first fully connected layer (FC1) is a vector with a
+length of 1024, and the second fully connected layer (FC2)
+is a vector of length 7 which is the number of class labels to
+predict visual attributes of protest images.
+5.2.2. Evaluation Method of Multi-label Classification
+Evaluation of multi-label classification has a notion of being
+partially correct. One way to evaluate the classification is
+label-set based accuracy or exact match that considers par-
+tially correct as incorrect. On the other hand, evaluation of
+label-based accuracy is carried out on a per label basisChen
+Table 3. Architecture of Multi-Label Classification CNN
+Layer
+Feature Map
+Feature Size
+Filter Size
+Stride
+Pad
+Activation
+FC2
+-
+7
+-
+-
+-
+Sigmoid
+FC1
+-
+1024
+-
+-
+-
+ReLU
+Max-Pooling
+128
+4x4
+2x2
+2
+Same
+-
+Conv3
+128
+7x7
+3x3
+2
+Same
+ReLU
+Max-Pooling
+64
+14x14
+2x2
+2
+Same
+-
+Conv2
+64
+28x28
+3x3
+2
+Same
+ReLU
+Max-Pooling
+32
+56x56
+2x2
+2
+Same
+-
+Conv1
+32
+112x112
+3x3
+2
+Same
+ReLU
+Input
+3(RGB)
+224x224
+-
+-
+-
+-
+et al. (2021). The calculation method of label-set accuracy,
+where a predicted set of labels ˆy must exactly match the
+ground truth y, is shown in equation (3) Read et al. (2011);
+0/1 loss dictates that any label vector not predicted perfectly
+will be given a zero score.
+0/1
+loss = 1 − 1
+N
+N
+�
+i=1
+1yi= ˆ
+yi
+(3)
+Label-based accuracy is more lenient approach to evaluate the
+performance since it does not consider multi-label problem as
+a whole. When each label has a separate binary evaluation, we
+have hamming loss which is shown in the following equation:
+Hamming
+Loss = 1 −
+1
+NL
+N
+�
+i=l
+L
+�
+j=l
+1yi= ˆ
+yi
+(4)
+We adapted both approaches to evaluate the performance of
+our multi-label classifier.
+5.2.3. Threshold Selection
+The fully connected layer represents a vector containing prob-
+ability for each class. The threshold function can be used to
+obtain a multi-label prediction ˆy. Specifically, we used the
+Matthews Correlation Coefficient (MCC) which is an evalu-
+ation metric of binary classification. The MCC is a correla-
+tion coefficient for ground truth versus predictions and varies
+between -1 and 1, where 1 represents a perfect prediction
+Gorodkin (2004). The MCC is given by the following equa-
+tion (5).
+MCC =
+Tp × Tn − Fp × Fn
+�
+(Tp + Fp)(Tp + Fn)(Tn + Fp)(Tn + Fn)
+(5)
+For multi-label classification, the MCC is defined in terms
+of a confusion Matrix C for K classes in the equation (6) .
+MCC =
+c × s − �K
+k pkxtk
+�
+(s2 − �K
+k p2
+k)(s2 − �K
+k t2
+k)
+(6)
+The values tk = �K
+i Cik is the number of times class k
+truly happened. pk = �K
+i Cki is the number of times class
+k was predictedTian et al. (2021). c = �K
+i Ckk is the total
+number of samples correctly predicted. s = �K
+i
+�K
+j Cij is
+the total number of samples.
+
+5.3. Advantages and Disadvantages of SVM and CNN
+In image classification, there are advantages and disadvan-
+tages for both SVM and CNNs. Theoretically, SVM is very
+good at finding the margin and hyperplane for classification,
+and it is very robust for high dimensional dataZhang et al.
+(2006). However, SVM model is sensitive to noise, for exam-
+ple, if there is a noise in background or a visible object in one
+image is occluded or partially blocked by scenes in other, it
+will have a negative impact on the performance of the clas-
+sification model Cortes and Vapnik (1995). Moreover, since
+one-vs-all involves training a binary classifier for all classes,
+computation time can be very expensive.
+On the other hand, the main advantage of CNNs is that
+it can be used to extract important image features with a suf-
+ficiently large datasetLeCun et al. (2015). The performance
+of CNNs classifier largely depend on the size of the dataset.
+The bottle necks in training the CNNs model are computa-
+tional time and memory used to retain activation from for-
+ward pass and error gradients computation when dataset is
+very largeSun et al. (2008). However, an efficient parallel
+computation with a help of GPU or training a model in mini
+batches can be used to mitigate those issues to some degrees.
+Also, there are many parameters for CNNs that need to be
+set by the users in order to train a robust and good prediction
+model.
+6. EXPERIMENT
+We implemented K-class SVM and CNNs using scikit-learn
+and TensorFlow libraries in Python 3.5. In the experiment,
+we used our personal server desktop which runs on Windows
+with 2 Intel Xeon E5-2630 V3 CPU 2.4 GHz and 8 small
+cores with RAM size of 64GB. First, we split the sample im-
+ages via image augmentation from Table 2 into training and
+testing (the ration of 80% and 20% respectively) and then
+we re-sized each image to 224x224 with 3 color channels.
+However, the implementation of SVM in scikit-learn does not
+adopt online learning so that we had to down-sample the sam-
+ple images from 31,472 to 12,000 to avoid memory limit er-
+ror. On the other hand, with a help of mini-batches, we used
+all of the data points without any down-sampling for CNNs
+model.
+We trained a baseline SVM using One-vs.-All method.
+Our final setting of SVM consisted of max iterations to be
+ran as 4000 to ensure it converges. Also, we experimented
+weighting of hyperparameter in equation (1) but it did not im-
+prove the result so our final setting of the weight parameters
+for each classes are set to 1. For CNNs, we used the learning
+rate of 0.001 which is a standard rate and mini batch size of
+202 images with numbers of batches as 125.
+7. RESULTS AND DISCUSSION
+7.1. Evaluation
+Evaluation was conducted on each class. Since we have the
+manual label as ground truth, we calculated accuracy, the pre-
+cision rate, recall rate, and F1 score for each class respec-
+tively. Specifically, the following is the equations we used to
+calculate each evaluation criteria: Precision rate=
+T P
+T P +F P ,
+Recall rate=
+T P
+T P +F N , Accuracy=
+T P +T N
+T P +T N+F P +F N .
+F1
+score is the harmonic mean of precision rate and recall rate.
+Table 4. Evaluation of SVM Model Prediction
+Fire
+Flag
+Large Crowd
+Other
+Police
+Sign
+Student
+Accuracy (%)
+88
+74
+60
+89
+77
+63
+79
+Precision (%)
+53
+53
+63
+53
+52
+55
+50
+Recall (%)
+35
+22
+60
+24
+30
+39
+19
+F1 Score (%)
+42
+31
+61
+33
+38
+45
+27
+Fig. 5. SVM vs CNN Precision Rate
+Fig. 6. SVM vs CNN Recall Rate
+7.2. Result
+Training SVM model with one-vs.-all method took longer
+than 12 hours and consistently consumed 70-90% of avail-
+able memory on our machine whereas the CNNs model only
+took less than half of the training time with much less con-
+sumption of memory with a help of mini-batch. Therefore,
+we were able to obtain the results by CNNs easier and faster
+than the SVM. Table 4 and 5 shows the accuracy, precison,
+recall, and F1 score of each predicted lables for SVM and
+CNNs respectively. We also plotted the precision and recall
+of the two models side by side in histogram in Fig. 5 and 6
+to compare the performance of the two models. As you can
+see, their overall performance is comparable to each other but
+recall using CNNs is slightly better than that of SVM.
+For the evaluation of CNNs prediction, We calculated
+the best threshold using MCC to transform the probability of
+each label from the fully connected layer into the 7 predicted
+class labels: the calculated thresholds are 0.2, 0.4, 0.7, 0.6,
+0.5, 0.4, 0.5 for ’fire’, ’flag’, ’large crowd’, ’other’, ’police’,
+’sign’, and ’student’ respectively.
+Prediction accuracy per
+Table 5. Evaluation of CNNs Model Prediction
+Fire
+Flag
+Large Crowd
+Other
+Police
+Sign
+Student
+Accuracy (%)
+91
+72
+71
+89
+76
+61
+76
+Precision (%)
+76
+48
+74
+52
+45
+51
+32
+Recall (%)
+67
+36
+73
+37
+32
+46
+23
+F1 Score (%)
+71
+41
+73
+43
+38
+49
+27
+
+SVM
+■CNNs
+76
+74
+63
+53
+55
+53
+53
+52
+52
+48
+5051
+45
+32
+FIRE
+FLAG
+LARGE
+OTHER
+POLICE
+SIGN
+STUDENT
+CROWDSVMCNNs
+73
+67
+60
+46
+37
+39
+35
+36
+30 32
+22
+24
+23
+19
+FIRE
+FLAG
+LARGE
+OTHER
+POLICE
+SIGN
+STUDENT
+CROWDFig. 7. Image of a burning vehicle with police in background
+(top); image of bikers and a flag (bottom)
+label reached almost 77% on average. Fig. 7 shows sample
+test images of ’fire’ and ’police’ on the left and ’large crowd’
+and ’police’ on the right. Our CNNs model predicted them
+correctly but when we evaluated our classifier model with a
+large dataset, we found that our label-set accuracy was very
+low around 20% due to the challenges of exact matching on a
+multi-label classifier.
+7.3. Future work
+From the experiment, we learnt that we were able to get rea-
+sonable performance using both SVM and CNNs model to
+predict each class label separately but the bottom line perfor-
+mance of our prediction model is still not desirable: our goal
+is to increase the accuracy of exact matching. Therefore, Fu-
+ture work can be done in following aspects. First, we can
+apply state-of-the-art algorithm like Generative Adversarial
+Network to generate more training samples, which would be
+helpful to prevent over fitting. Second, we modify the equa-
+tion for SVM to enhance the classifier, and improve the deep
+learning modelYan et al. (2018). Moreover, The main limi-
+tation of our image classification approach is that it does not
+consider the credibility of the source in decision making, and
+hence requires assessment of the social media source or of
+each image posted on the web. Also, there are privacy protec-
+tion concern in using both social media and image dataChai
+and Nayak (2018). In other further research, we will merge
+image and text data like article headlines and descriptions
+associated with each image which should help improve the
+performance of prediction model. We will conduct privacy
+protection procedure such as Randomized Response Chai and
+Nayak (2019) to the data. Then, we can evaluate our model
+using the OSI database as well as social media such as Twit-
+ter to determine the level of generalization our model may be
+able to achieve.
+8. CONCLUSION
+Our paper demonstrates a rapid means of image augmenta-
+tion and identifying key aspects of protest activity from pub-
+licly available image streams, using open source software.
+Although there are additional work that need to be done to
+improve our classifier models, our approach creates greater
+opportunities for the collection of such data to enable work
+for public good. While traditional efforts to monitor violence
+and protests may largely be hampered by linguistic barriers
+and reporting delays, images streams from social media pro-
+vide a language-agnostic means of assessing such threats. By
+demonstrating that we were able to get reasonable prediction
+accuracy of key aspects of protest images using SVM and
+CNNs, we hope to enable its application to improve moni-
+toring of social unrest activities within unstable regionsSun
+et al. (2021).
+ACKNOWLEDGMENT
+We thank Virginia Tech CS department for providing us with
+the OSI dataset.
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+Spatial scales of COVID-19 transmission in Mexico
+Brennan Klein∗1,2, Harrison Hartle1, Munik Shrestha1, Ana Cecilia Zenteno3,
+David Barros Sierra Cordera4, José R. Nicolas-Carlock5, Ana I. Bento6,
+Benjamin M. Althouse7,8, Bernardo Gutierrez9,10,11,
+Marina Escalera-Zamudio9,11, Arturo Reyes-Sandoval12,13,
+Oliver G. Pybus9,14,18, Alessandro Vespignani1,2,
+Jose Alberto Diaz-Quiñonez*†15, Samuel V. Scarpino*‡1,16,17, and
+Moritz U.G. Kraemer*§9,18
+1Network Science Institute, Northeastern University, Boston, Massachusetts, USA
+2Laboratory for the Modeling of Biological & Socio-technical Systems,
+Northeastern University, Boston, Massachusetts, USA
+3Massachusetts General Hospital, Boston, Massachusetts, USA
+4Instituto Mexicano del Seguro Social, Ciudad de México, México
+5Instituto de Investigaciones Jurídicas, Universidad Nacional Autónoma de México,
+Ciudad de México, México
+6Department of Epidemiology and Biostatistics, School of Public Health,
+Indiana University, Bloomington, Indiana, USA
+7Information School, University of Washington, Seattle, Washington, USA
+8Department of Biology, New Mexico State University, Las Cruces, New Mexico, USA
+9Department of Biology, University of Oxford, Oxford, UK
+10School of Biological & Environmental Sciences,
+Universidad San Francisco de Quito, Quito, Ecuador
+11Consorcio Mexicano de Vigilancia Genómica
+12The Jenner Institute, University of Oxford, Oxford, UK
+13Instituto Politécnico Nacional, IPN, Ciudad de México, México
+14Department of Pathobiology and Population Science, Royal Veterinary College, London, UK
+15Instituto de Ciencias de la Salud, Universidad Autónoma del Estado de Hidalgo,
+Pachuca, Hidalgo, México
+16Institute for Experiential AI, Northeastern University, Boston, Massachusetts, USA
+17Santa Fe Institute, Santa Fe, New Mexico, USA
+18Pandemic Sciences Institute, University of Oxford, UK
+February 1, 2023
+∗[email protected]
+†[email protected]
+‡[email protected]
+§[email protected]
+1
+arXiv:2301.13256v1  [physics.soc-ph]  30 Jan 2023
+
+Abstract
+During outbreaks of emerging infectious diseases, internationally connected cities
+often experience large and early outbreaks, while rural regions follow after some delay
+[1–6]. This hierarchical structure of disease spread is influenced primarily by the mul-
+tiscale structure of human mobility [7–9]. However, during the COVID-19 epidemic,
+public health responses typically did not take into consideration the explicit spatial
+structure of human mobility when designing non-pharmaceutical interventions (NPIs).
+NPIs were applied primarily at national or regional scales [10]. Here we use weekly
+anonymized and aggregated human mobility data and spatially highly resolved data
+on COVID-19 cases, deaths and hospitalizations at the municipality level in Mexico
+to investigate how behavioural changes in response to the pandemic have altered the
+spatial scales of transmission and interventions during its first wave (March - June
+2020). We find that the epidemic dynamics in Mexico were initially driven by SARS-
+CoV-2 exports from Mexico State and Mexico City, where early outbreaks occurred.
+The mobility network shifted after the implementation of interventions in late March
+2020, and the mobility network communities became more disjointed while epidemics
+in these communities became increasingly synchronised. Our results provide actionable
+and dynamic insights into how to use network science and epidemiological modelling
+to inform the spatial scale at which interventions are most impactful in mitigating the
+spread of COVID-19 and infectious diseases in general.
+Table 1: Policy summary
+Background
+The establishment, persistence and growth rates of COVID-19 mainly depend
+on human mobility and mixing. However, current approaches attempting to
+limit transmission have been primarily based on administrative boundaries
+instead of the natural scales of human mobility.
+Main findings
+& limitations
+Using aggregated and anonymized human mobility and detailed COVID-19
+case data, we find that the scales of human mixing shift during the pandemic
+and that transmission is highly clustered amongst mobility communities.
+Policy
+implications
+Structuring interventions based on spatial mobility may be more effective com-
+pared to interventions based on administrative boundaries. Future pandemic
+control interventions should consider empirical human mobility networks when
+designing interventions.
+2
+
+1
+Introduction
+The transmission of infectious diseases is highly heterogeneous. Differences in population
+structure, the landscape of prior immunity, and environmental factors, result in differences
+in the timing of outbreaks, their magnitude, and duration [2, 3, 9, 11–20].
+For infec-
+tious diseases, one principal component determining the spatial structure of outbreaks is
+the frequency of interactions between susceptible and infectious individuals within and be-
+tween regions. In most geographies, public health decision-making authority follows political
+boundaries. However, from an epidemiological perspective, the relevant spatial units may
+not strictly follow political boundaries but rather human mixing [8, 14, 21]. Evaluating
+the spatial structure of COVID-19 transmission remains important in determining optimal
+interventions (non-pharmaceutical and/or vaccination) to reduce transmission and limit the
+risk of resurgence of cases [22–25].
+During the first half of 2020, Mexico experienced one of the largest SARS-CoV-2 epi-
+demics worldwide, with more than 600,000 cases and 65,000 confirmed deaths reported be-
+tween February and September 2020 [26] (Fig. 1a). The epidemic wave peaked in May in
+the largest metropolitan areas of Mexico City and the State of Mexico and later ignited
+epidemics in all other states [27], peaking between June and July 2020 (Fig. 1b). Here we
+combine municipality level epidemiological data with weekly anonymized aggregated human
+mobility data at the same scale, to characterise the spatial scales of the Mexican COVID-19
+pandemic and their implications for the implementation of spatially targeted interventions.
+2
+Results
+2.1
+Spatial expansion of COVID-19 in Mexico
+In Mexico, the spatial range of transmission expanded rapidly after reports of the earliest
+cases in March 2020, with over 700 municipalities reporting transmission by July 2020 (out of
+2,448, Fig. 1c). During April and May the risk of positive RTq-PCR confirmed cases amongst
+men aged 30-69 was 1.4 times higher than between July 1 and September 1 (Fig. 1d,e),
+indicating that the epidemic spread initially within and through these age groups (Extended
+Data Figure A.1).
+This dynamic trend in the demographics of cases is similar to that
+observed in other countries during the early stages of the pandemic [28, 29].
+States that experienced early transmission were the state of Mexico and Mexico City
+(Fig. 1b) [27]. Due to the centrality of Mexico City connecting people from abroad (in-
+ternational arrivals) and within Mexico we hypothesise that human mobility from these
+states was a key driver of the spread of COVID-19 in Mexico. Using anonymized, opt-in
+and aggregated human movement data from mobile phones (Materials and Methods) we
+find that case growth rates across Mexican states were well predicted by a lagged model
+of human movements from the State of Mexico and Mexico City between March and May
+2020 (Fig. 2c, conditional R2 = 0.62; see Materials & Methods). Further, we observe that
+the share of overall relative human mobility to and from Mexico and Mexico City increased
+3
+
+Apr.
+May
+Jun.
+Jul.
+Aug.
+Sep.
+0
+2
+4
+6
+8
+10
+12
+14
+Total reported cases as of Sept. 1, 2020:
+(b) Daily new cases per 100,000, state level (7-day rolling avg.)
+Apr.
+May
+Jun.
+Jul.
+Aug.
+Sep.
+0
+200
+400
+600
+800
+(c)
+Municipalities reporting cases (2,448 total)
+Apr.
+May
+Jun.
+Jul.
+Aug.
+Sep.
+0%
+10%
+20%
+30%
+40%
+(d)
+"early"
+"late"
+Percent of new cases (7-day rolling avg.)
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+Early: April 1 - May 1
+Late: June 30 - Aug. 30
+F: under 30
+F: under 30
+F: 30-49
+F: 30-49
+F: 50-69
+F: 50-69
+F: over 70
+F: over 70
+M: under 30
+M: under 30
+M: 30-49
+M: 30-49
+M: 50-69
+M: 50-69
+M: over 70
+M: over 70
+(e)
+Relative risk ("early" vs. "late" periods)
+10
+100
+1000
+10000
+Cases per 100,000
+(as of September 1)
+Figure 1: Epidemiological situation of COVID-19 in Mexico. (a) Map of cumulative
+cases per 100,000 people, as of September 1, 2020. (b) Timeline of new cases per 100,000
+population at the state level (7-day rolling average), highlighting the 15 states with the most
+severe cumulative outbreaks. (c) Number of municipalities that reported confirmed cases of
+COVID-19 through time. (d) Age and sex distributions of confirmed COVID-19 cases across
+Mexico, highlighting “early” and “late” periods during which the relative risk of infections
+were calculated. (e) Age and sex relative risk ratios of infection, comparing the early vs.
+late periods from panel (d).
+markedly during that period (Fig. 2b) when overall human mobility between states declined
+(Fig. 2b, Extended Data Figure A.2 showing state level data on change in human mobility).
+This points towards a change in the network structure of human mobility in Mexico, as
+documented in some other countries [30, 31]. Overall transmission, and the importance of
+Mexico City driving the epidemic, declined after the implementation of NPIs through May
+2020. However, after the lifting of physical distancing measures on June 1st (see table of
+documented changes in NPIs, Table A.1), case growth rates in the country increased again as
+4
+
+a function of mobility from Mexico City, in line with models predicting that lifting lockdowns
+can lead to reseeding of transmission chains from larger to smaller cities where epidemics
+were successfully controlled (Fig. 2b, Table A.1, [7]).
+Variation in weekly new cases within each state in Mexico are generally well predicted
+by cases in Mexico City weighted by human mobility except for Baja California, More-
+los, Chihuahua, Oaxaca, and Chiapas (Extended Data Figure A.3). We hypothesise that
+epidemics there were possibly seeded from other countries (USA and Guatemala); further
+SARS-CoV-2 genomic analyses of unbiased collections of samples will be needed to confirm
+the SARS-CoV-2 lineage dynamics in these states [27, 32–36]. Human mobility data showing
+cross border (US to Mexico) movements indicate higher overall mobility to bordering states
+in Mexico and growth rates in US-Mexico border states appear higher in the period between
+24 May - 28 June 2020 (Extended Data Figures A.4, A.5, A.6). The high mobility during
+that phase resulted in larger case numbers in states bordering the US when compared to
+other states in Mexico (Extended Data Figure A.5).
+2.2
+The scales of COVID-19 transmission
+It is well known that reductions in mobility (a proxy for reductions in population mixing)
+have reduced the transmission of COVID-19 within a location [38]. However, it remains un-
+clear how structural changes to the mobility network (shifts in the frequency and intensity
+of mobility within and among regions) have impacted COVID-19 dynamics empirically [30,
+31, 39–41]. Our underlying hypothesis is that more tightly connected communities exhibit
+more synchronised epidemic dynamics and, conversely, that more disjointed individual com-
+munities have less synchronised epidemics and their epidemics are more likely to fade out
+[4–6] (here, communities are equivalent to municipalities and synchrony is defined as the
+similarity among communities in weekly case growth rates [42]). Both processes have critical
+implications for disease mitigation and eliminations locally, and at a country level [7, 43–47].
+The Mexican government announced stringent physical distancing policies on March 30th,
+2020 which resulted in marked changes in the mobility network (Fig. 2a, Table A.1).
+To quantify the degree to which mobility patterns are structured by geopolitical bound-
+aries, we use a community detection algorithm that groups municipalities based on their
+movement patterns [48]. Specifically, we aim to identify groups of municipalities such that
+movements between municipalities within the same group, i.e., community, are more fre-
+quent than movements to other municipalities in other communities. Community detection
+is often accomplished via modularity maximization [49]; however, these approaches neglect
+information about the flow of mobility through the network. Instead, we leverage the map
+equation via an algorithm called InfoMap [48]. The InfoMap algorithm utilises an informa-
+tion theoretic approach to derive expected connectivity patterns if the observed flows were
+entirely determined by a random walk process. For this study, InfoMap is ideal because
+it is conceptually related to infectious disease transmission models, which often also utilise
+stochastic processes [50].
+The aim is to identify municipalities where frequent interactions between individuals
+occur, such that the detected communities approximate the spatial scales of disease trans-
+5
+
+03-01
+04-12
+05-24
+07-05
+08-16
+20%
+40%
+60%
+80%
+100%
+(b)
+Percent of typical mobility
+(total across all of Mexico)
+03-01
+04-12
+05-24
+07-05
+08-16
+0.004
+0.002
+0.000
+0.002
+0.004
+0.006
+(c)
+Coefficients of case growth rate
+and mobility from Mexico City
+03-01
+04-12
+05-24
+07-05
+08-16
+18%
+19%
+20%
+21%
+22%
+23%
+24%
+(d)
+Dynamics of states' outgoing
+mobility to Mexico City
+Community size distribution
+(n = 16, using Infomap)
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+Figure 2: Human mobility and transmission of COVID-19 in Mexico. (a) Pre-
+pandemic average of the inter-municipality mobility network, coloured by network commu-
+nity (detected using the Infomap algorithm). Mobility flow data is based on the aggregated
+Google Mobility Research dataset (see Materials & Methods). (b) Percent of typical weekly
+mobility nationwide (typical refers to mobility between January 12 and February 29, 2020).
+(c) Evolution of the coefficients of mobility flow from Mexico City in (lagged) correlations
+with state-level case rates across the country, highlighting the key role that mobility from
+Mexico City played in the early stage of the epidemic. (d) Average fraction of total outgoing
+mobility from each state that is to Mexico City (black) and the median entropy of states’
+distributions of outgoing mobility. Error bands correspond to 95% confidence intervals.
+mission (i.e., communities in which it is assumed that infection spreads via contacts within
+a relatively homogeneously mixing population [51]). Accounting for spatial heterogeneity is
+6
+
+Network of average mobility flow:
+(a)
+2020-01-12 t0 2020-02-23
+Administrative
+(state) boundary
+ Example:
+Chiapas
+Network community03-01
+03-29
+04-26
+05-24
+06-21
+07-19
+08-16
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+(c)
+Standard deviation of municipality growth rates
+within grouping (lower values: higher synchrony)
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+Growth rate (reported cases)
+0.11
+Nayarit
+Morelos
+Michoacán de Ocampo
+México
+Jalisco
+Hidalgo
+Guerrero
+Guanajuato
+Durango
+Distrito Federal
+Chihuahua
+Chiapas
+Colima
+Coahuila de Zaragoza
+Campeche
+Baja California Sur
+Baja California
+Aguascalientes
+0.36
+0.26
+0.69
+0.43
+0.29
+0.30
+0.44
+0.31
+0.83
+0.42
+0.57
+0.47
+0.44
+0.37
+0.39
+0.48
+0.47
+...
+...
+...
+...
+0.465
+(mean)
+ Std. dev. of growth rates 
+(d)
+Example: Variance in growth rates (2020-04-19)
+municipalities grouped by administrative boundaries
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+Growth rate (reported cases)
+0.38
+Comm. 01
+0.54
+Comm. 02
+0.35
+Comm. 03
+0.45
+Comm. 04
+0.50
+Comm. 05
+0.43
+Comm. 06
+0.30
+Comm. 07
+0.49
+Comm. 08
+0.37
+Comm. 09
+0.31
+Comm. 10
+0.59
+Comm. 11
+0.00
+Comm. 12
+0.00
+Comm. 13
+0.00
+Comm. 14
+0.00
+Comm. 15
+0.00
+Comm. 16
+0.00
+Comm. 17
+0.00
+Comm. 18
+...
+...
+...
+...
+0.223
+(mean)
+ Std. dev. of growth rates 
+(e)
+Example: Variance in growth rates (2020-04-19)
+municipalities grouped by network communities
+Figure 3: Network structure determines the synchrony of epidemics. (a) Grouping
+of municipalities based on the state administrative boundaries. Shaded municipalities are re-
+moved from downstream analyses as they could not be assigned a movement community (see
+Materials & Methods). (b) Example grouping of municipalities based on human movement
+data and a community detection algorithm [37] (Materials and Methods). Colours indicate
+movement communities. Grey municipalities have limited recorded movements and could not
+be assigned to a community and were consequently excluded from analysis. (c) Synchrony of
+weekly growth rates of epidemics across municipalities as measured by the pairwise standard
+error between growth rates. The lower the error, the more synchronised epidemics are. Blue
+line shows grouping by network communities, and orange shows groupings by state admin-
+istrative boundaries. The green dashed line shows the nationwide trend in reported cases
+during this period. For a visual intuition of the differences in within-community standard
+deviations of growth rates, see Extended Data Figure A.9.
+known to be important for assessing strategies for interventions [2], especially in areas that
+have marked differences in urban and rural areas [52]. Using this algorithm, we identify 16
+communities before the first cases of COVID-19 were detected in Mexico (Fig. 3b). Com-
+munity size and organisation changed following the announcement of the lockdown (March
+23 and 30, 2020) in Mexico and communities generally became smaller (fewer municipalities
+within each community (Extended Data Figures A.7 and A.8 show the communities for each
+week during the study period). At the peak of the lockdown, we identified approximately 60
+movement communities (a 4-fold increase from the baseline period).
+More specifically, there are two notable shifts in the network following the introduction of
+NPIs. First, more communities are identified but importantly the size of these communities
+shrinks disproportionately so that one community expands (Mexico City) and many very
+7
+
+b
+Example network grouping
+(infomap)a)
+Administrative grouping
+(states)small ones emerge (Fig. 2d). Further, as a result of the lockdown human movements across
+municipalities decline more rapidly than movements within a community with one important
+exception: Mexico City. There we observe that the ratio of within municipality movements
+declines at a similar rate than movements across municipalities (Extended Data Figure A.2)
+further proving its central importance in the mobility network in Mexico.
+We then compared the weekly infection incidence growth rates within each community
+and contrasted them to growth rates under a scenario in which municipalities are grouped
+based on state boundaries (black lines, Fig. 3a,b). As expected, we find that epidemics in
+municipalities that are grouped by human mobility were more synchronised compared to
+those grouped by state (Fig. 3c; see Extended Data Figure A.9 for an illustration of the
+variance in municipality epidemic growth rates for several example groups of municipalities
+defined by administrative or network boundaries).
+The synchrony among municipalities
+within each community were maximised in April and May 2020, a period when cases were
+rapidly rising across the country. After June, epidemics that are grouped by movement are
+still more synchronised, but the differences with groupings by state appear to be smaller
+(Fig. 3c). This later period (June to October 2020) is a time when Mexico City appears
+to also lose importance in seeding the epidemic across the country, and local factors (e.g.,
+population size) become more important in determining the epidemic trajectory [53]. These
+results are expected as local factors become more influential in determining disease dynamics
+(population size, local mixing) and that the importance of continued virus re-importations
+wanes through time [33].
+3
+Discussion & Limitations
+We present a generalisable approach for understanding the spatial structure of transmission
+of COVID-19 and other emerging infectious diseases by accounting for the variations of
+the human mobility network. We aimed to differentiate the transmission dynamics at a
+level defined by administrative boundaries from that defined by simple community detection
+algorithms that are applied to aggregated anonymized weekly human mobility data. We
+find that as human mobility network structures change, so does to spatial transmission.
+Incorporating these findings into real-world public health decision-making may result in
+more effective strategies to control an epidemic [54–57].
+The European Commission for
+example published a report on Mobility Functional Areas (MFAs) which were informed by
+mobile phone data but the adoption of these recommendations remained sparse [55].
+Our model and results are only as accurate as the data that go into them. The Mexican
+COVID-19 database may suffer from underreporting due to testing shortages, changing case
+definitions and spatial heterogeneity in reporting [58, 59]. For example, relatively few cases
+were reported from Oaxaca (Fig. 1a) which may be due to barriers to access to testing [60].
+Future extensions of the model and as the pandemic continues will need to take into account
+high-resolution SARS-CoV-2 cross-immunity. Further, our model is based on higher level
+descriptions of the population (raw case data and population level human movement data)
+and these do not capture the high contact heterogeneity within each municipality (e.g., de-
+8
+
+mographic heterogeneity and assortative mixing) shown to be important in the transmission
+of COVID-19 [61]. Contact patterns may differ significantly by age group, employment sta-
+tus and other factors not accounted for in this work. We did however observe heterogeneity
+in the demographic makeup of cases during the earlier phases of the Mexican COVID-19
+pandemic.
+Further, results should be interpreted in light of important limitations related to the
+human mobility data. First, the Google mobility data is limited to smartphone users who
+have opted into Google’s Location History feature, which is off by default. These data may
+not be representative of the population as whole, and furthermore their representativeness
+may vary by municipality.
+Importantly, these limited data are only viewed through the
+lens of differential privacy algorithms, specifically designed to protect user anonymity and
+obscure fine detail.
+Mexico is composed of 31 free and sovereign states and Mexico City, united under a
+federation.
+This means that each administrative region or state is governed by its own
+constitution, although they are not completely independent of the federal jurisdiction. Fur-
+thermore, each state is divided into municipalities, the nation’s basic administrative unit,
+which possesses limited autonomy (discretionary power on how best to respond to, or apply
+a public policy). Under a serious nationwide health threat or emergency, such as a pandemic,
+the federal Ministry of Health (MoH) acquires full authority over the health policies to be
+implemented nationwide. Nevertheless, Mexican law establishes that the General Health
+Council (GHC), a collegial body that reports to the president of the republic has the char-
+acter of health authority, and can emit obligatory norms to be abided by the MoH. The
+GHC is presided by the Minister of Health, and is conformed by federal institutions (e.g.h,
+Economy, Communication & Transport) as well as academic institutions, representatives
+from pharmaceutical industry, and other health system actors [62]. Given its mandate and
+position in the Mexican health system, the GHC constitutes a promising agent to drive pub-
+lic policy outside of the margins or across geo-administrative units. Furthermore, there are
+examples of inter-state and inter-municipality coordination to resolve problems that extend
+beyond their borders such as waste management, tax, policing, and perhaps most relevant,
+health provision. It is in these contexts where evidence-based interventions on innovative
+approaches, such as the ones presented here become not only an option but a possibility,
+with greater impact in reducing transmission as compared to approaches where interven-
+tions are based on administrative boundaries. However, theory often differs from practice
+and reality brings along additional and expected factors into play (e.g., economic [63] and
+political interests) many of which are not accounted for in this work. Some state governors
+for example refused to comply with federal health policies in the early relaxation phase in
+May 2020 [64].
+Mexico has suffered a large and devastating epidemic, and we hope that our findings
+contribute to a more rational implementation of interventions in the future that can account
+for the substantial and changing spatial heterogeneity in transmission. Such analyses can
+be updated and translated to any other country in the world for which aggregated human
+mobility data is available. Future work should also focus on validating the inferred spatial
+9
+
+scales with genomic data [32, 33, 65] or other coarse-graining techniques [66, 67]. Developing
+interventions using patterns observed in empirical mobility networks must be added to the
+list of priorities for pandemic response and preparedness in the 21st century.
+4
+Materials & Methods
+Epidemiological data:
+Epidemiological data include individual level information on pa-
+tients with confirmed RTq-PCR COVID-19 infection between March - September 30th, 2020.
+Data were downloaded from http://datosabiertos.salud.gob.mx/gobmx/salud/datos_
+abiertos/datos_abiertos_covid19.zip (last accessed October 24, 2020). Data include
+information about patients demographics (age and sex) and municipality of residence. In all
+analyses we used the date of onset of symptoms.
+Population and travel data:
+Human mobility and population data were extracted at the
+municipality level based on the 2016 boundaries (INEGI 2016: https://www.inegi.org.
+mx/app/mapa/espacioydatos/default.aspx). Population data were downloaded from the
+COVID-19 indicator dataset, which was provided by INEGI (https://www.inegi.org.mx/
+investigacion/covid/).
+Aggregated and anonymised human mobility data:
+We used the Google COVID-
+19 Aggregated Mobility Research Dataset described in detail in [68, 69], which contains
+anonymized relative mobility flows aggregated over users who have turned on the Location
+History setting, which is turned off by default. This is similar to the data used to show
+how busy certain types of places are in Google Maps—helping identify when a local business
+tends to be the most crowded. The mobility flux is aggregated per week, between pairs of
+approximately 5km2 cells worldwide, and for the purpose of this study further aggregated
+for municipalities in Mexico.
+To produce this dataset, machine learning is applied to log data to automatically segment
+it into semantic trips. To provide strong privacy guarantees [70], all trips were anonymized
+and aggregated using a differentially private mechanism to aggregate flows over time (see
+https://policies.google.com/technologies/anonymization). This research is done on
+the resulting heavily aggregated and differentially private data. No individual user data was
+ever manually inspected, only heavily aggregated flows of large populations were handled. All
+anonymized trips are processed in aggregate to extract their origin and destination location
+and time. For example, if n users travelled from location a to location b within time interval
+t, the corresponding cell (a, b, t) in the tensor would be n±err, where err is Laplacian noise.
+The automated Laplace mechanism adds random noise drawn from a zero mean Laplacian
+distribution and yields (ϵ, δ)-differential privacy guarantee of ϵ = 0.66 and δ = 2.1 × 1029
+per metric. Specifically, for each week W and each location pair (A, B), we compute the
+number of unique users who took a trip from location A to location B during week W. To
+each of these metrics, we add Laplace noise from a zero-mean distribution of scale 1/0.66.
+We then remove all metrics for which the noisy number of users is lower than 100, following
+10
+
+the process described in [70], and publish the rest. This yields that each metric we publish
+satisfies (ϵ, δ)-differential privacy with values defined above. The parameter ϵ controls the
+noise intensity in terms of its variance, while δ represents the deviation from pure ϵ-privacy.
+The closer they are to zero, the stronger the privacy guarantees.
+These results should be interpreted in light of several important limitations. First, the
+Google mobility data is limited to smartphone users who have opted into Google’s Loca-
+tion History feature, which is off by default. These data may not be representative of the
+population as whole, and furthermore their representativeness may vary by location. Impor-
+tantly, these limited data are only viewed through the lens of differential privacy algorithms,
+specifically designed to protect user anonymity and obscure fine detail. Moreover, compar-
+isons across rather than within locations are only descriptive since these regions can differ
+in substantial ways.
+Timeline of interventions:
+The Mexican government has outlined four principle objec-
+tives for the control of COVID-19: a) Reduce risk of acquiring infection, b) Reduce risk of
+severe morbidity and mortality, c) Reduce risk and impact on society and d) Reduce risk
+of transmission between infectious and susceptible individuals. We collated a full list of
+interventions between February and September 2020 and details are provided in Table A.1,
+including references.
+Relative risk model:
+Following Goldstein and Lipsitch [71] we used age stratified epi-
+demiological data to assess the temporal shifts in the share of a given age group among all
+cases of infection. To do so we use the relative risk (RR) [72, 73] statistic that estimates the
+ratio of the proportion of a given age group among all detected cases of COVID-19 for a later
+time period vs. an early time period. We selected the early time period to be the month of
+April (the period right after the implementation of the lockdown) and the late period to be
+June to September. We adopted the code and model from Goldstein and Lipsitch described
+in detail [71].
+Community detection algorithm:
+Human mobility networks, based on data from mo-
+bile devices, can be used to capture important population-level trends. Microscopic descrip-
+tions often remain too complex to extract meaningful information to describe the transmis-
+sion process accurately [61]. We here use a community detection algorithm following [48] to
+identify human movement communities (basins) where within-community mobility among
+municipalities is higher than across-community mobility. We chose this community detec-
+tion algorithm as it is conceptually related to infectious disease transmission models—both
+utilising random walks.
+Municipality level case growth rates:
+To estimate the daily epidemic growth rates in
+each municipality, we fit a mixed effects GLM of log new daily case counts in sliding 7-day
+windows (fixed effect; approximately the generation time of COVID-19 in the earliest wave)
+11
+
+and a random effect for each municipality on the slope and intercept, using the R package
+lme4 v.1.1-21 [74]. Daily case counts were determined using the date of symptom onset.
+Relationship between case growth rates and mobility:
+To test for an effect of mo-
+bility from Mexico City on municipality growth rates, we fit a mixed effect GLM with log
+mobility as a fixed effect, a random effect on the intercept for each municipality and a random
+effect on the slope and intercept for log mobility each week. The conditional and marginal
+coefficient of determination, i.e., R2, were calculated using the R package MuMIn v1.471.
+[75] which implements the method developed by Nakagawa et al. 2017 [76]. Model selection
+was performed using analysis of variance for mixed effects models as implemented in the R
+package lmerTest v.3.1-3 [77].
+Additional information
+Acknowledgments:
+We thank all health care workers and those involved in the collection,
+processing and publishing COVID-19 epidemiological data from Mexico.
+Funding:
+M.U.G.K., O.G.P., B.G. acknowledge funding from the Oxford Martin School
+Pandemic Genomics programme. M.U.G.K. acknowledges funding from the European Hori-
+zon 2020 programme MOOD (grant no. #874850), the Wellcome Trust, a Branco Weiss
+Fellowship, The Rockefeller Foundation and Google.org. The contents of this publication
+are the sole responsibility of the authors and do not necessarily reflect the views of the Euro-
+pean Commission or the other funders. B.K., H.H., S.V.S., & A.V. acknowledge the support
+of a grant from the John Templeton Foundation (61780). The opinions expressed in this
+publication are those of the author(s) and do not necessarily reflect the views of the John
+Templeton Foundation.
+Author contributions:
+S.V.S., M.U.G.K. and B.K. developed the idea, planned the re-
+search and conducted analyses. A.C.Z. and D.B.S.C. collected government intervention data.
+S.V.S., M.U.G.K. and B.K. wrote the first draft of the manuscript. All authors interpreted
+the data, contributed to writing and approved the manuscript.
+Competing interests:
+We declare no conflicts of interest.
+Data and materials availability:
+Code, spatial, and epidemiological data are available
+upon publication. The Google COVID-19 Aggregated Mobility Research Dataset used for
+this study is available with permission from Google LLC. Correspondence and requests for
+materials should be addressed to B.K., J.A.D-Q., S.V.S., or M.U.G.K.
+12
+
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+
+A
+Extended Data Figures
+Figure A.1: Number of new cases per state and sex (7-day average).
+21
+
+Veracruz de Ignacio
+Ciudad de México
+Nuevo Leon
+México
+Tabasco
+de la Llave
+Puebla
+Guanajuato
+Sonora
+Female
+Female
+ Female
+Female
+emale
+Female
+400
+Male
+ Male
+Male
+ Male
+9300
+100
+0
+Coahuila de Zaragoza
+Michoacan de Ocampo
+Baja California
+sedineel
+Sinaloa
+Jalisco
+Guerrero
+Oaxaca
+ Female
+ Female
+— Female
+— Female
+— Female
+ Female
+ Female
+400
+Male
+Male
+Male
+Male
+300
+100-
+Yucatan
+Quintana Roo
+San Luis Potosi
+Hidalgo
+Chiapas
+Chihuahua
+Tlaxcala
+Morelos
+emale
+Female
+Female
+Female
+ Female
+emale
+Female
+400 -
+Male
+Male
+ Male
+Male
+Male
+Male
+Male
+00m
+ 200
+100 -
+- 0
+Campeche
+Durango
+Zacatecas
+Aguascalientes
+Baja California Sur
+Querétaro
+Nayarit
+Colima
+Female
+Female
+Female
+ Female
+Female
+Female
+Female
+ Female
+400
+Male
+ Male 
+ Male
+ Male
+Male
+ Male
+100
+1 0
+04-01 05-01 06-01 07-01
+04-01 05-01 06-01 07-01
+04-01 05-01 06-01 07-01
+04-01 05-01 06-01 07-01
+04-01 05-01 06-01 07-01
+04-01 05-01 06-01 07-01
+04-01 05-01 06-01 07-01
+04-01 05-01 06-01 07-0150%
+100%
+150%
+Percent typical
+mobility
+Distrito Federal
+within-state
+outgoing movement
+incoming movement
+México
+within-state
+outgoing movement
+incoming movement
+50%
+100%
+150%
+Percent typical
+mobility
+Guanajuato
+within-state
+outgoing movement
+incoming movement
+Nuevo León
+within-state
+outgoing movement
+incoming movement
+Veracruz
+within-state
+outgoing movement
+incoming movement
+Tabasco
+within-state
+outgoing movement
+incoming movement
+Puebla
+within-state
+outgoing movement
+incoming movement
+Tamaulipas
+within-state
+outgoing movement
+incoming movement
+50%
+100%
+150%
+Percent typical
+mobility
+Coahuila
+within-state
+outgoing movement
+incoming movement
+Sonora
+within-state
+outgoing movement
+incoming movement
+Jalisco
+within-state
+outgoing movement
+incoming movement
+San Luis Potosí
+within-state
+outgoing movement
+incoming movement
+Baja California
+within-state
+outgoing movement
+incoming movement
+Michoacán
+within-state
+outgoing movement
+incoming movement
+50%
+100%
+150%
+Percent typical
+mobility
+Sinaloa
+within-state
+outgoing movement
+incoming movement
+Guerrero
+within-state
+outgoing movement
+incoming movement
+Yucatán
+within-state
+outgoing movement
+incoming movement
+Oaxaca
+within-state
+outgoing movement
+incoming movement
+Hidalgo
+within-state
+outgoing movement
+incoming movement
+Quintana Roo
+within-state
+outgoing movement
+incoming movement
+50%
+100%
+150%
+Percent typical
+mobility
+Chihuahua
+within-state
+outgoing movement
+incoming movement
+Baja California Sur
+within-state
+outgoing movement
+incoming movement
+Querétaro de Arteaga
+within-state
+outgoing movement
+incoming movement
+Durango
+within-state
+outgoing movement
+incoming movement
+Tlaxcala
+within-state
+outgoing movement
+incoming movement
+Chiapas
+within-state
+outgoing movement
+incoming movement
+12-01
+01-26
+03-22
+05-17
+07-12
+50%
+100%
+150%
+Percent typical
+mobility
+Aguascalientes
+within-state
+outgoing movement
+incoming movement
+12-01
+01-26
+03-22
+05-17
+07-12
+Zacatecas
+within-state
+outgoing movement
+incoming movement
+12-01
+01-26
+03-22
+05-17
+07-12
+Campeche
+within-state
+outgoing movement
+incoming movement
+12-01
+01-26
+03-22
+05-17
+07-12
+Morelos
+within-state
+outgoing movement
+incoming movement
+12-01
+01-26
+03-22
+05-17
+07-12
+Nayarit
+within-state
+outgoing movement
+incoming movement
+12-01
+01-26
+03-22
+05-17
+07-12
+Colima
+within-state
+outgoing movement
+incoming movement
+Figure A.2: Weekly relative change in human mobility within each state and between
+states (incoming and outgoing) as compared to baseline.
+22
+
+Figure A.3: State-specific correlations of new reported cases (weekly) vs. mobility from
+Mexico City times new reported cases in Mexico City (weekly). States with low mobility
+and case count data coverage are included but not plotted in this figure.
+23
+
+Distrito Federal
+México
+Between 2020-03-29 and 2020-07-19
+6000 
+(yellow dots = later)
+8
+O
+.
+4000 -
+·
+2000
+.
+50 
+100
+5
+10
+15
+Tabasco
+Veracruz de Ignacio
+Guanajuato
+Pueblal
+Nuevo Leon
+Sonora
+4000 -
+de la Llavel 4000 -
+3000 -
+3000
+3000 
+.
+8
+3000
+●2000
+2000
+2000 -
+··
+8
+2000 -
+2000
+.
+.
+.
+.
+:
+1000
+·
+1000
+1000 -
+1000 -
+8
+1000
+New
+·
+o
+0.001
+0.002
+0.005
+0.010
+0.002
+0.004
+0.01
+0.02
+0.03
+0.005
+0.010
+0.001
+0.002
+0.003
+1400
+sed!inee
+Baja California
+Jalisco
+Coahuila de Zaragoza
+Sinaloa
+Guerrero
+2000 -
+1200
+1500
+1500
+.
+2000
+·1000
+9
+1000
+.
+.
+1000
+.
+1000 -
+·！
+800
+.
+1000 -
+.
+500 -
+.
+500
+600 :
+ob
+0.0006
+0.0007
+0.005
+0.010
+0.005 0.010 0.015
+0.0005
+0.0010
+0.001 0.002
+0.003
+0.002
+0.004
+0.006
+San Luis Potosi
+Oaxaca| 1500
+Michoacan de Ocampo
+Yucatan
+Quintana Roo
+Hidalgo
+2000 :
+1000
+1500
+:
+·1000-
+·
+1500 -
+750
+1000
+.o.
+1000 
+oo
+0
+0
+1000 -
+500
+.0.
+8
+500 -
+500 -
+500 -
+500 -
+New
+.
+250
+0
+0.0005 0.0010 0.0015
+0.000
+0.002
+0.004
+0.002
+0.004
+0.0025 0.0050 0.0075
+0.00
+0.01
+0.02
+0.02
+0.04
+Chihuahua
+Chiapas
+Tlaxcala
+Campeche
+Baja California Sur
+Durango
+009
+1000
+1000
+600 -
+575 -
+00.05
+750
+750 -
+.
+8
+400 -
+550 -
+500
+.
+500 -
+0.00 -
+200
+525
+250
+250
+.o
+500
+-0.05
+0.0000.002 0.004 0.006
+0.002
+0.004
+0.002
+0.004
+0.006
+0.00050
+0.00055
+0.00060
+0.000
+0.002
+0.004
+0.006
+0.05
+0.00
+0.05
+500 
+Morelos
+Querétaro de Arteaga
+Nayarit
+Zacatecas
+Aguascalientes
+Colimal
+600
+400 -
+0.05
+0.05 -
+0.05
+·
+400 -
+300 -
+8
+200
+0.00
+0.00
+0.00
+200
+New
+100
+-0.05 
+.
+←0.05 
+0.05 -
+oh
+0.0005
+0.0010
+0.02
+0.04
+0.005
+0.010
+-0.05
+0.00
+0.05
+-0.05
+0.00
+-0.05
+0.00
+0.05
+0.06
+0.05
+Movement from Mexico D.F.
+Movement from Mexico D.F.
+ Movement from Mexico D.F.
+Movement from Mexico D.F.
+Movement from Mexico D.F.
+Movement from Mexico D.F.
+times Mexico D.F. new cases
+times Mexico D.F. new cases
+times Mexico D.F. new cases
+times Mexico D.F. new cases
+times Mexico D.F. new cases
+times Mexico D.F. new casesFigure A.4: Weekly relative human mobility where the origin is the USA and the destina-
+tion are states in Mexico divided into states that share a land border, Mexico and Mexico
+City and all other states.
+24
+
+USA
+States with
+140%
+USA border
+States without
+from
+120%
+USA border
+Mexico and
+Mexico City
+mobility
+100%
+80%
+typical
+60%
+40%
+Percent of i
+20%
+0%
+12-01
+02-09
+04-19
+06-28Figure A.5: Weekly new cases per 100,000 divided into cases in Mexico City and the state
+of Mexico, states that share a land border with the USA, and all other states.
+25
+
+ rate per 100,000
+100,000
+40
+65
+States with
+USA border
+States without
+4
+USA border
+30
+new cases per
+Mexico and
+Mexico City
+20
+Weekly growth I
+Z
+10
+Weekly
+L
+0
+04-19
+06-28
+03-15
+03-15
+05-24
+04-19
+05-24
+06-28Figure A.6: Weekly number of cases among municipalities in Mexico coloured by their
+geographic position to the USA (bordering vs. not bordering) and the sum of in-municipality
+mobility × weekly new cases among origin nodes (both on the log scale).
+26
+
+8
+Municipalities without
+U.S. connections
+Sum of weekly new cases among
+(log-scaled)
+6
+Municipalities with
+U.S. connections
+4
+destination nodes (
+2 
+0
+-4
+-2
+0
+2
+4
+6
+8
+10
+12
+Sum of in-municipality mobility x sum of
+weekly new cases among origin nodes (log-scaled)Community size distribution
+Community size distribution
+Community size distribution
+Community size distribution
+Community size distribution
+Community size distribution
+Community size distribution
+Community size distribution
+Community size distribution
+Figure A.7: Four-week snapshots of mobility in Mexico. Weekly human mobility in
+Mexico at the municipality level. Thickness of lines represents intensity of relative mobility
+flow. Colours represent the membership to movement communities as estimated using the
+map equation (Materials & Methods).
+27
+
+Fr0m 2019-11-24
+(four-week total)
+15 communitiesFrom 2019-12-29
+(four-week total)
+15 communitiesFr0m 2020-01-26
+(four-week total)
+18 communitiesFr0m 2020-02-23
+(four-week total)
+18 communitiesFrom 2020-03-22
+(four-week total)
+33 communitiesFr0m 2020-04-19
+(four-week total)
+26 communitiesFr0m 2020-05-17
+(four-week total)
+22 communitiesFr0m 2020-06-14
+(four-week total)
+33 communitiesFr0m 2020-07-12
+(four-week total)
+32 communities03-01
+03-29
+04-26
+05-24
+06-21
+07-19
+08-16
+20
+30
+40
+50
+60
+Number of communities detected over time
+Figure A.8:
+Number of communities detected each week during the first wave of the
+COVID-19 epidemic in Mexico.
+28
+
+03-0103-29
+04-26
+05-24
+06-21
+07-19
+08-16
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+(c)
+Standard deviation of municipality growth rates
+within grouping (lower values: higher synchrony)
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+Growth rate (reported cases)
+0.11
+Nayarit
+Morelos
+Michoacán de Ocampo
+México
+Jalisco
+Hidalgo
+Guerrero
+Guanajuato
+Durango
+Distrito Federal
+Chihuahua
+Chiapas
+Colima
+Coahuila de Zaragoza
+Campeche
+Baja California Sur
+Baja California
+Aguascalientes
+0.36
+0.26
+0.69
+0.43
+0.29
+0.30
+0.44
+0.31
+0.83
+0.42
+0.57
+0.47
+0.44
+0.37
+0.39
+0.48
+0.47
+...
+...
+...
+...
+0.465
+(mean)
+ Std. dev. of growth rates 
+Example: Variance in growth rates (2020-04-19)
+municipalities grouped by administrative boundaries
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+Growth rate (reported cases)
+0.38
+Comm. 01
+0.54
+Comm. 02
+0.35
+Comm. 03
+0.45
+Comm. 04
+0.50
+Comm. 05
+0.43
+Comm. 06
+0.30
+Comm. 07
+0.49
+Comm. 08
+0.37
+Comm. 09
+0.31
+Comm. 10
+0.59
+Comm. 11
+0.00
+Comm. 12
+0.00
+Comm. 13
+0.00
+Comm. 14
+0.00
+Comm. 15
+0.00
+Comm. 16
+0.00
+Comm. 17
+0.00
+Comm. 18
+...
+...
+...
+...
+0.223
+(mean)
+ Std. dev. of growth rates 
+Example: Variance in growth rates (2020-04-19)
+municipalities grouped by network communities
+o
+.
+(e)
+(d)
+o
+.
+Figure A.9: For one example week (April 19, 2020), comparison of the mean standard
+deviations in municipality growth rates within either (left) administrative (states) boundaries
+or (right) network communities. In each panel, the standard deviations of municipalities’
+infection growth rates within each grouping (state vs. network community) is shown on the
+right. Figure 3c shows the average of these values over time.
+29
+
+b
+Example network grouping
+(infomap)a)
+Administrative grouping
+(states)Date
+Intervention
+March 16, 2020
+Mexican Secretariat of Public Education (SEP) suspend classes in
+schools of preschool, primary, secondary education, as well as those
+of the upper middle and higher types dependent on the SEP [1].
+March 17, 2020
+Universities begin to suspend classes and social events [2].
+March 20, 2020
+Mexican Secretariat of Public Education (SEP) cancels all civic and
+sports events [3].
+March 21, 2020
+United States - Mexico border was closed to non-essential travel but
+remained open for commerce. Closure extended until November 21,
+2020 [4].
+March 23/24, 2020
+National period of social distancing begins. Schools closed and all
+non-essential operations were closed including gatherings of 100+
+people [5].
+March 30, 2020
+National health emergency declared. Policies included: (1.) Non-
+essential services suspended. (2.) Private sector is asked to require
+employees to work from home. (3.) Sectors that kept operating nor-
+mally: government, health (public and private), public safety, social
+programs, critical infrastructure, and essential services. A full list
+of essential services can be viewed here: https://www.dof.gob.
+mx/nota_detalle.php?codigo=5590914&fecha=31/03/2020. (4.)
+People over 60 years old are urged to stay home. (5.) Public gath-
+erings of over 50 people are banned. (6.) There was no enforced
+curfew. Expiration date: April 30, 2020.
+April 5, 2020
+Hospital reconversion strategy guidelines published in order to con-
+tain nosocomial transmission [6]
+April 16, 2020
+The Federal Government announces the extension of the health
+emergency and emphasizes the need to restrict movement to and
+from areas of high transmissibility until May 30th [7].
+May 14, 2020
+Ministry of Health announces epidemiologic color-coded system to
+re-open social, educational, economic activities at state level [8].
+May 18, 2020
+First phase of the “new normality” 324 municipalities with no
+recorded COVID-19 cases are given green light to reopen businesses
+and schools [9]. Car factories were meant to reopen on June 1st,
+but began reopening on May 18th under US pressure (Factories
+remained closed from March 23rd, to May 18th).
+30
+
+June 1, 2020
+Mexico’s national period of social distancing concludes.
+A new
+color-coded system was enacted across the country to assess how
+quickly states can reopen their economies and schools: red, orange,
+yellow, and green [10].
+July 20, 2020
+Daycare centers run by the country’s social security system re-
+opened in coordination with local authority and based on color-
+coded indicators. Currently, all but 4 states have daycare centers
+open. [11]
+October 18, 2020
+The Health Ministry announced that 17 states—Mexico City
+included—were at alert level orange and 14 were at yellow. Only
+one state, Campeche, was at green [12]. In states at the orange
+level, businesses such as hotels and restaurants can reopen while
+following health protocols such as enforcing limited capacity. Yel-
+low allows for most economic activities to return to normal with
+some occupancy limits.
+October 22, 2020
+Some local governments have chosen to enact more stringent re-
+strictions than the federal government guidelines, e.g., Jalisco and
+Chihuahua [13].
+Table A.1: Timeline of government interventions in Mexico.
+31
+
+A.1
+Citation diversity statement
+Recent work has quantified bias in citation practices across various scientific fields; namely,
+women and other minority scientists are often cited at a rate that is not proportional to
+their contributions to the field [14–21]. In this work, we aim to be proactive about the
+research we reference in a way that corresponds to the diversity of scholarship in this field.
+To evaluate gender bias in the references used here, we obtained the gender of the first/last
+authors of the papers cited here through either 1) the gender pronouns used to refer to them
+in articles or biographies or 2) if none were available, we used a database of common name-
+gender combinations across a variety of languages and ethnicities. By this measure (excluding
+citations to datasets/organizations, citations included in this section, and self-citations to the
+first/last authors of this manuscript), our references contain 3% woman(first)-woman(last),
+22% woman-man, 20% man-woman, 47% man-man, 0% nonbinary, 8% man solo-author, and
+0% woman solo-author. This method is limited in that an author’s pronouns may not be
+consistent across time or environment, and no database of common name-gender pairings is
+complete or fully accurate.
+Supplemental References
+[1]
+DOF - Diario Oficial de la Federación. url: https://www.dof.gob.mx/nota_
+detalle.php?codigo=5589479&fecha=16/03/2020.
+[2]
+Coronavirus en México: universidades suspenden clases y se intensifican las acciones
+preventivas. 2020. url: https : / / www . infobae . com / america / mexico / 2020 /
+03 / 13 / coronavirus - en - mexico - universidades - suspenden - clases - y - se -
+intensifican-las-acciones-preventivas/.
+[3]
+Gobierno de México suspenderá todas las actividades escolares por coronavirus. 2020.
+url: https : / / www . latimes . com / espanol / mexico / articulo / 2020 - 03 - 14 /
+gobierno- de- mexico- suspendera- todas- las- actividades- escolares- por-
+coronavirus.
+[4]
+U.S. Embassy & Consulates in Mexico. Mexico, U.S.M. to. Travel restrictions - Fact
+sheet. 2021. url: https://mx.usembassy.gov/travel-restrictions-fact-sheet/.
+[5]
+Inicia fase 2 por coronavirus COVID-19 – Coronavirus. url: https://coronavirus.
+gob.mx/2020/03/24/inicia-fase-2-por-coronavirus-covid-19/.
+[6]
+Gobierno de México and Secretaría de Salud COVID-19. Lineamiento de Reconversión
+Hospitalaria. url: https://coronavirus.gob.mx/wp-content/uploads/2020/04/
+Documentos-Lineamientos-Reconversion-Hospitalaria.pdf.
+[7]
+Coronavirus en México: guía para entender las cuatro nuevas medidas de control y
+prevención del COVID-19 cercanas a la Fase 3. url: https://www.infobae.com/
+america/mexico/2020/04/16/coronavirus-en-mexico-guia-para-entender-
+las - cuatro - nuevas - medidas - de - control - y - prevencion - del - covid - 19 -
+cercanas-a-la-fase-3/.
+32
+
+[8]
+DOF - Diario Oficial de la Federación. url: https://dof.gob.mx/nota_detalle.
+php?codigo=5593313&fecha=14/05/2020#gsc.tab=0.
+[9]
+Conferencia 16 de mayo – Coronavirus. url: https://coronavirus.gob.mx/2020/
+05/16/conferencia-16-de-mayo-2/.
+[10]
+Subsecretaría de Prevención y Promoción de la Salud. Semáforo de riesgo epidemi-
+ológico: COVID-19: indicadores y metodología. url: https://coronavirus.gob.mx/
+wp-content/uploads/2020/06/Lineamiento_Semaforo_COVID_05Jun2020_1600.
+pdf.
+[11]
+IMSS nurseries open on July 20, after supervision of health protocols. url: http:
+//www.imss.gob.mx/prensa/archivo/202007/463.
+[12]
+COVID-19 MÉXICO Comunicado Técnico Diario - 18 Octubre 2022. url: http :
+/ / saludsinaloa . gob . mx / wp - content / uploads / 2020 / reportescovid /
+Covid19ReporteDiario18Noviembre2020.pdf.
+[13]
+State Government publishes new agreement on Red Light measures. url: https://
+chihuahua.gob.mx/contenidos/publica-gobierno-del-estado-nuevo-acuerdo-
+de-medidas-del-semaforo-rojo.
+[14]
+Perry Zurn, Danielle S. Bassett, and Nicole C. Rust. “The citation diversity statement:
+A practice of transparency, a way of life”. In: Trends in Cognitive Sciences 24.9 (2020),
+pp. 669–672. doi: 10.1016/j.tics.2020.06.009.
+[15]
+Jordan D. Dworkin, Kristin A. Linn, Erin G. Teich, Perry Zurn, Russell T. Shinohara,
+and Danielle S. Bassett. “The extent and drivers of gender imbalance in neuroscience
+reference lists”. In: Nature Neuroscience 23.8 (2020), pp. 918–926. doi: 10.1038/
+s41593-020-0658-y.
+[16]
+Paula Chakravartty, Rachel Kuo, Victoria Grubbs, and Charlton McIlwain. “#Com-
+municationSoWhite”. In: Journal of Communication 68.2 (2018), pp. 254–266. doi:
+10.1093/joc/jqy003.
+[17]
+Daniel Maliniak, Ryan Powers, and Barbara F. Walter. The gender citation gap in in-
+ternational relations. Vol. 67. 4. 2013, pp. 889–922. doi: 10.1017/S0020818313000209.
+[18]
+Michelle L. Dion, Jane Lawrence Sumner, and Sara Mc Laughlin Mitchell. “Gendered
+citation patterns across political science and social science methodology fields”. In:
+Political Analysis 26.3 (2018), pp. 312–327. doi: 10.1017/pan.2018.12.
+[19]
+Neven Caplar, Sandro Tacchella, and Simon Birrer. “Quantitative evaluation of gender
+bias in astronomical publications from citation counts”. In: Nature Astronomy 1 (2017).
+doi: 10.1038/s41550-017-0141.
+[20]
+Pierre Azoulay and Freda Lynn. “Self-citation, cumulative advantage, and gender in-
+equality in science”. In: Sociological Science 7 (2020). doi: 10.15195/v7.a7.
+[21]
+Gita Ghiasi, Philippe Mongeon, Cassidy R. Sugimoto, and Vincent Larivière. “Gender
+homophily in citations”. In: 23rd International Conference on Science and Technology
+Indicators (2018), pp. 1519–1525. url: https://hdl.handle.net/1887/65291.
+33
+

AtAyT4oBgHgl3EQfRvdA/content/tmp_files/2301.00071v1.pdf.txt

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+arXiv:2301.00071v1  [math.CO]  30 Dec 2022
+SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE
+PRINCIPLE
+M.M. SKRIGANOV
+Abstract. Stolarsky’s invariance principle, known for point distributions on
+the Euclidean spheres Sd [25], has been extended to the real RP n, complex
+CP n, and quaternionic HP n projective spaces and the octonionic OP 2 pro-
+jective plane in our previous paper [23]. Geometric features of such spaces as
+well as their models in terms of Jordan algebras have been used very essen-
+tially in the proof. In the present paper, we give a new pure analytic proof of
+Stolarsky’s invariance principle relying on the theory of spherical functions on
+compact symmetric Riemannian manifolds of rank one.
+1. Introduction and main results
+1.1 Introduction. In 1973 Kenneth B. Stolarsky [25] established the following
+remarkable formula for point distributions on the Euclidean spheres. Let Sd =
+{x ∈ Rd+1 : ∥x∥ = 1} be the standard d-dimensional unit sphere in Rd+1 with
+the geodesic (great circle) metric θ and the Lebesgue measure µ normalized by
+µ(Sd) = 1. We write C(y, t) = {x ∈ Sd : (x, y) > t} for the spherical cap of height
+t ∈ [−1, 1] centered at y ∈ Sd. Here we write (·, ·) and ∥ · ∥ for the inner product
+and the Euclidean norm in Rd+1.
+For an N-point subset DN ⊂ Sd, the spherical cap quadratic discrepancy is
+defined by
+λcap[DN] =
+� 1
+−1
+�
+Sd ( #{|C(y, t) ∩ DN} − Nµ(C(y, t)) )2 dµ(y) dt.
+(1.1)
+We introduce the sum of pairwise Euclidean distances between points of DN
+τ[DN] = 1
+2
+�
+x1,x2∈DN ∥x1 − x2∥ =
+�
+x1,x2∈DN sin 1
+2θ(x1, x2),
+(1.2)
+and write ⟨τ⟩ for the average value of the Euclidean distance on Sd,
+⟨τ⟩ = 1
+2
+��
+Sd×Sd ∥y1 − y2∥ dµ(y1) dµ(y2).
+(1.3)
+The study of the quantities (1.1) and (1.2) falls within the subjects of discrepancy
+theory and geometry of distances, see [1,6] and references therein. It turns out that
+the quantities (1.1) and (1.2) are not independent and are intimately related by the
+following remarkable identity
+γ(Sd)λcap[DN] + τ[DN] = ⟨τ⟩N 2,
+(1.4)
+2010 Mathematics Subject Classification. 11K38, 22F30, 52C99.
+Key words and phrases. Geometry of distances, discrepancies, spherical functions, projective
+spaces, Jacobi polynomials.
+
+2
+M.M. SKRIGANOV
+for an arbitrary N-point subset DN ⊂ Sd. Here γ(Sd) is a positive constant inde-
+pendent of DN,
+γ(Sd) = d √π Γ(d/2)
+2 Γ((d + 1)/2) .
+(1.5)
+The identity (1.4) is known in the literature as Stolarsky’s invariance principle.
+Its original proof given in [25] has been simplified in [7,10]. simplified in [7,10].
+In our previous paper [23] Stolarsky’s invariance principle (1.4) has been ex-
+tended to the real RP n, the complex CP n, the quaternionic HP n projective spaces,
+and the octonionic OP 2 projective plane. Geometric features of such spaces as well
+as their models in terms of Jordan algebras have been used very essentially in the
+proof. The aim of the present paper is to give an alternative pure analytic proof
+relying on the theory of spherical functions.
+1.2 Discrepancies and metrics. L1-invariance principles. Let us consider Sto-
+larsky’s invariance principle in a broader context. Let M be a compact metric
+measure space with a fixed metric θ and a finite Borel measure µ, normalized, for
+convenience, by
+diam(M, θ) = π,
+µ(M) = 1,
+(1.6)
+where diam(E, ρ) = sup{ρ(x1, x2) : x1, x2 ∈ E} denotes the diameter of a subset
+E ⊆ M with respect to a metric ρ.
+We write B(y, r) = {x ∈ M : θ(x, y) < r} for the ball of radius r ∈ I centered at
+y ∈ M and of volume v(y, r) = µ(B(y, r)). Here I = {r = θ(x1, x2) : x1, x2 ∈ M}
+denotes the set of all possible radii. If the space M is connected, we have I = [ 0, π ].
+We consider distance-invariant metric spaces. Recall that a metric space M is
+called distance-invariant, if the volume of any ball v(r) = v(y, r) is independent of
+y ∈ M. The typical examples of distance-invariant spaces are homogeneous spaces
+M = G/H with G-invariant metrics θ and measures µ.
+For an N-point subset DN ⊂ M, the ball quadratic discrepancy is defined by
+λ[ξ, DN] =
+�
+I
+�
+M
+( #{B(y, r) ∩ DN} − Nv(r)) )2 dµ(y) dξ(r),
+(1.7)
+where ξ is a finite measure on the set of radii I.
+Notice that for Sd spherical caps and balls are related by C(y, t) = B(y, r),
+t = cos r, and the discrepancies (1.1) and (1.7) are related by λcap[DN] = λ[ξ♮, DN],
+where dξ♮(r) = sin r dr, r ∈ I = [0, π].
+The ball quadratic discrepancy (1.7) can be written in the form
+λ[ξ, DN] =
+�
+x1,x2∈DN λ(ξ, x1, x2)
+(1.8)
+with the kernel
+λ(ξ, x1, x2) =
+�
+I
+�
+M
+Λ(B(y, r), x1) Λ(B(y, r), x2) dµ(y) dξ(r) ,
+(1.9)
+where
+Λ(B(y, r), x) = χ(B(y, r), x) − v(r),
+(1.10)
+and χ(E, ·) denotes the characteristic function of a subset E ⊆ M.
+For an arbitrary metric ρ on M we introduce the sum of pairwise distances
+ρ[DN] =
+�
+x1,x2∈DN ρ(x1, x2).
+(1.11)
+
+SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE
+3
+and the average value
+⟨ρ⟩ =
+�
+M×M
+ρ(y1, y2) dµ(y1) dµ(y2).
+(1.12)
+We introduce the following symmetric difference metrics on the space M
+θ∆(ξ, y1, y2) = 1
+2
+�
+I
+µ(B(y1, r)∆B(y2, r)) dξ(r)
+= 1
+2
+�
+I
+�
+M
+χ(B(y1, r)∆B(y2, r), y) dµ(y) dξ(r),
+(1.13)
+where
+B(y1, r)∆B(y2, r) = B(y1, r) ∪ B(y2, r) \ B(y1, r) ∩ B(y2, r)
+is the symmetric difference of the balls B(y1, r) and B(y2, r).
+In line with the definitions (1.11) and (1.12), we put
+θ∆[ξ, DN] =
+�
+x1,x2∈DN θ∆(ξ, x1, x2).
+and
+⟨θ∆(ξ)⟩ =
+�
+M×M
+θ∆(ξ, y1, y2) dµ(y1) dµ(y2) .
+A direct calculation leads to the following result.
+Proposition 1.1. Let a compact metric measure space M be distance-invariant,
+then we have
+λ(ξ, y1, y2) + θ∆(ξ, y1, y2) = ⟨θ∆(ξ)⟩.
+(1.14)
+In particular, we have the following invariance principle
+λ[ ξ, DN ] + θ∆[ ξ, DN ] = ⟨θ∆(ξ)⟩ N 2
+(1.15)
+for an arbitrary N-point subset DN ⊂ M.
+Proof. In view of the symmetry of the metric θ, we have
+χ(B(x, r), y) = χ(B(y, r), x) = χ0(r − θ(y, x)) ,
+(1.16)
+where χ0(·) is the characteristic function of the half-axis (0, ∞). Therefore
+χ(B(y1, r)∆B(y2, r), y) = χ(B(y1, r), y) + χ(B(y2, r), y)
+−2χ(B(y1, r) ∩ B(y2, r), y) ,
+and
+�
+M χ(B(x, r), y)dµ(x) =
+�
+M χ(B(x, r), y)dµ(y) = v(r).
+Using these relations, we obtain
+λ(ξ, x1, x2) =
+�
+I
+�
+µ(B(x1, r) ∩ B(x2, r)) − v(r)2�
+dξ(r) ,
+θ∆(ξ, y1, y2) =
+�
+I
+�
+v(r) − µ(B(y1, r) ∩ B(y2, r))
+�
+dξ(r) ,
+⟨θ∆(ξ)⟩ =
+�
+I
+�
+v(r) − v(r)2�
+dξ(r) .
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(1.17)
+These relations imply (1.14).
+□
+
+4
+M.M. SKRIGANOV
+In the case of spheres Sd, relations of the type (1.14) and (1.15) were given
+in [25]. Their extensions to more general metric measure spaces were given in [21,
+Theorem 2.1], [22, Eq. (1.30)] and [23, Proposition 1.1].
+Notice that
+χ(B(y1, r)∆B(y2, r), y) = |χ(B(y1, r), y) − χ(B(y2, r), y)| ,
+(1.18)
+and hence
+θ∆(ξ, y1, y2) = 1
+2
+�
+I
+�
+M
+|χ(B(y1, r), y) − χ(B(y2, r), y)| dµ(y) dξ(r)
+(1.19)
+is an L1-metric.
+Recall that a metric space M with a metric ρ is called isometrically Lq-embeddable
+(q = 1 or 2), if there exists a mapping ϕ : M ∋ x → ϕ(x) ∈ Lq, such that
+ρ(x1, x2) = ∥ϕ(x1)−ϕ(x2)∥Lq for all x1, x2 ∈ M. Notice that the L2-embeddability
+is stronger and implies the L1-embeddability, see [13, Sec. 6.3].
+It follows from (1.19) that the space M with the symmetric difference metrics
+θ∆(ξ) is isometrically L1-embeddable by the formula
+M ∋ x → χ(B(x, r), y) ∈ L1(M × I) ,
+(1.20)
+The identity (1.15) can be called the L1-invariance principle, while Stolarsky’s
+invariance principle (1.4) should be called the L2-invariance principle, because it
+involves the Euclidean metric. The identities of such a type including correspond-
+ingly L1 and L2 metrics could be also called weak and strong invariance principles.
+1.3 L2-invariance principles. Recall the definition and necessary facts on two-
+point homogeneous spaces. Let G = G(M) be the group of isometries of a metric
+space M with a metric θ, i.e. θ(gx1, gx2) = θ(x1, x2) for all x1, x2 ∈ M and g ∈ G.
+The space M is called two-point homogeneous, if for any two pairs of points x1,
+x2 and y1, y2 with θ(x1, x2) = θ(y1, y2) there exists an isometry g ∈ G, such that
+y1 = gx1, y2 = gx2. In this case, the group G is obviously transitive on M and
+M = G/H is a homogeneous space, where the subgroup K ⊂ G is the stabilizer of
+a point x0 ∈ M. Furthermore, the homogeneous space M is symmetric, i.e. for
+any two points y1, y2 ∈ M there exists an isometry g ∈ G, such that gy1 = y2,
+gy2 = y1.
+There is a very large number of two-point homogeneous spaces. For example, all
+Hamming spaces, known in the coding theory, are two-point homogeneous. We will
+consider compact connected two-point homogeneous spaces. The assumption that
+the space is connected turns out to be a strong restriction. All compact connected
+two-point homogeneous spaces Q = G/H are known, and by Wang’s classifications
+they are the following, see [16,17,20,29,30]:
+(i) The d-dimensional Euclidean spheres Sd = SO(d + 1)/SO(d) × {1}, d ⩾ 2,
+and S1 = O(2)/O(1) × {1}.
+(ii) The real projective spaces RP n = O(n + 1)/O(n) × O(1).
+(iii) The complex projective spaces CP n = U(n + 1)/U(n) × U(1).
+(iv) The quaternionic projective spaces HP n = Sp(n + 1)/Sp(n) × Sp(1),
+(v) The octonionic projective plane OP 2 = F4/ Spin(9).
+Here we use the standard notation from the theory of Lie groups; in particular,
+F4 is one of the exceptional Lie groups in Cartan’s classification.
+All these spaces are Riemannian symmetric manifolds of rank one.
+Geomet-
+rically, this means that all geodesic sub-manifolds in Q are one-dimensional and
+
+SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE
+5
+coincide with geodesics. From the spectral stand point, this also means that all
+operators on Q commuting with the action of the group G are functions of the
+Laplace–Beltrami operator on Q, see [16,17,29,30] for more details.
+The spaces FP n as Riemannian manifolds have dimensions d,
+d = dimR FP n = nd0,
+d0 = dimR F,
+(1.21)
+where d0 = 1, 2, 4, 8 for F = R, C, H, O, correspondingly.
+For the spheres Sd we put d0 = d by definition. Projective spaces of dimension
+d0 (n = 1) are homeomorphic to the spheres Sd0: RP 1 ≈ S1, CP 1 ≈ S2, HP 1 ≈
+S4, OP 1 ≈ S8. We can conveniently agree that d > d0 (n ⩾ 2) for projective spaces,
+while the equality d = d0 holds only for spheres. Under this convention, the dimen-
+sions d = nd0 and d0 define uniquely (up to homeomorphism) the corresponding
+homogeneous space which we denote by Q = Q(d, d0).
+We consider Q(d, d0) as a metric measure space with the metric θ and measure
+µ proportional to the invariant Riemannian distance and measure on Q(d, d0). The
+coefficients of proportionality are defined to satisfy (1.6). In what follows we always
+assume that n = 2 if F = O, since projective spaces OP n do not exist for n > 2.
+Any space Q(d, d0) is distance-invariant and the volume of balls in the space is
+given by
+v(r) = κ
+� r
+0
+(sin 1
+2u)d−1(cos 1
+2u)d0−1 du
+r ∈ [ 0, π ]
+= κ 21−d/2−d0/2
+� 1
+cos r
+(1 − z)
+d
+2 −1 (1 + z)
+d0
+2 −1 dz,
+(1.22)
+where κ = κ(d, d0) = B(d/2, d0/2)−1; B(a, b) = Γ(a)Γ(b)/Γ(a + b) and Γ(a) are
+the beta and gamma functions. Equivalent forms of (1.22) can be found in the
+literature, see, for example, [15, pp. 177–178], [17, pp. 165–168], [18, pp. 508–510].
+For even d0, the integrals (1.22) can be calculated explicitly that gives convenient
+expressions for v(r) in the case of CP n, HP n and OP 2, see, for example, [20].
+The chordal metric on the spaces Q(d, d0) is defined by
+τ(x1, x2) = sin 1
+2θ(x1, x2) =
+�
+1 − cos(x1, x2)
+2
+,
+x1, x2 ∈ Q(d, d0).
+(1.23)
+The formula (1.23) defines a metric because the function ϕ(θ) = sin θ/2, 0 ⩽ θ ⩽ π,
+is concave, increasing, and ϕ(0) = 0, that implies the triangle inequality. For the
+sphere Sd we have
+τ(x1, x2) = sin 1
+2θ(x1, x2) = 1
+2 ∥x1 − x2∥,
+x1, x2 ∈ Sd.
+(1.24)
+Lemma 1.1. The space Q(d, d0), d = nd0, can be embedded into the unit sphere
+Π : Q(d, d0) ∋ x → Π(x) ∈ Sm−1 ⊂ Rm,
+m = 1
+2(n + 1)(d + 2) − 1,
+(1.25)
+such that
+τ(x1, x2) =
+�
+d
+2(d + d0)
+�1/2
+∥Π(x1) − Π(x2)∥,
+x1, x2 ∈ Q(d, d0),
+(1.26)
+where ∥ · ∥ is the Euclidean norm in Rm.
+
+6
+M.M. SKRIGANOV
+Hence, the metric τ(x1, x2) is proportional to the Euclidean length of a segment
+joining the corresponding points Π(x1) and Π(x2) on the unit sphere. The chordal
+metric τ on the complex projective space CP n is known as the Fubini–Study metric.
+Lemma 1.1 will be proved in Section 2, and the embedding (1.25) will be de-
+scribed explicitly in terms of spherical functions on the space Q(d, d0). Note that
+the embedding (1.25) can be described in different ways, see, for example, [23,27].
+The following general result has been established in [23, Theorems 1.1 and 1.2].
+Theorem 1.1. For each space Q = Q(d, d0), we have the equality
+τ(x1, x2) = γ(Q) θ∆(ξ♮, x1, x2).
+(1.27)
+where dξ♮(r) = sin r dr, r ∈ [0, π] and
+γ(Q) =
+√π
+4 (d + d0)
+Γ(d0/2)
+Γ((d0 + 1)/2) = d + d0
+2d0
+γ(Sd0) ,
+(1.28)
+where γ(Sd0) is defined by (1.5).
+Comparing Theorem 1.1 with Proposition 1.1, we arrive to the following.
+Corollary 1.1. We have the following L2-invariance principle
+γ(Q) λ[ξ♮, DN] + τ[DN] = ⟨τ⟩N 2
+(1.29)
+for an arbitrary N-point subset DN ⊂ Q.
+The constant γ(Q) has the following geometric interpretation
+γ(Q) =
+⟨τ⟩
+⟨θ∆(ξ♮)⟩ =
+diam(Q, τ)
+diam(Q, θ∆(ξ♮)) .
+(1.30)
+Indeed, it suffices to calculate the average values (1.12) of both metrics in (1.27) to
+obtain the first equality in (1.30). Similarly, writing (1.27) for any pair of antipodal
+points x1, x2, θ(x1, x2) = π, we obtain the second equality in (1.30). The average
+value ⟨τ⟩ of the chordal metric τ can be easily calculated with the help of the
+formulas (1.12) and (1.22):
+⟨τ⟩ = B(d/2, d0/2)−1 B((d + 1)/2, d0/2) .
+(1.31)
+In the case of spheres Sd, the identity (1.29) coincides with (1.4). The identity
+(1.29) can be thought of as an extension of Stolarsky’s invariance principle to all
+projective spaces.
+Applications of L1- and L2-invariance principles and similar identities to the
+discrepancy theory, geometry of distances, and information theory have been given
+in many papers, see, for example, [1,3–10,21–23,25].
+It is worth noting that the equality (1.27) is of interest by itself.
+Since the
+integrand in (1.19) takes the values 0 and 1 only, we can write
+θ∆(ξ, y1, y2) =
+�
+θ∆
+p (ξ, y1, y2)
+�p
+, p > 0,
+(1.32)
+where
+θ∆
+p (ξ, y1, y2) =
+�1
+2
+� π
+0
+�
+Q
+|χ(B(y1, r), y) − χ(B(y2, r), y)|p) dµ(y) dξ(r)
+�1/p
+, (1.33)
+is an Lp-metric for p ⩾ 1.
+
+SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE
+7
+Comparing (1.27) and (1.32), we see that the chordal metric τ is proportional
+to the p-th power of the metric θ∆
+p (ξ♮) for all p ⩾ 1. This is a nontrivial fact. For
+example, we have for p = 2
+τ(x1, x2) = γ(Q)
+2
+� π
+0
+�
+Q
+|χ(B(x1, r), y) − χ(B(x2, r), y)|2 dµ(y) dξ♮(r),
+(1.34)
+and the equality (1.34) implies the existence of Gaussian random fields on the
+spaces Q(d, d0), see [12,15]. However, a detailed considerations of these questions
+is beyond the scope of the present paper.
+In the context of our discussion, the following open problems are of interest:
+- Do there exist measures ξ on the set of radii for spaces Q(d, d0) (for spheres Sd,
+say) other than the measure ξ♮ such that the corresponding symmetric difference
+metrics θ∆(ξ) are the L2-metrics?
+- Do there exist compact measure metric spaces other than spheres Sd and pro-
+jective spaces FP n for which the L2-invariance principle is also true?
+1.4 Proof of Theorem 1.2. In the present paper we use the theory of spherical
+functions to prove the following result.
+Theorem 1.2. The equality (1.27) is equivalent to the following series of formulas
+for Jacobi polynomials
+� 1
+−1
+�
+P (d/2,d0/2)
+l−1
+(t)
+�2
+(1 − t)d (1 + t)d0 dt
+= 2d+d0+1 (1/2)l−1
+((l − 1)!)2
+B(d + 1, d0 + 1) Tl−1(d/2, d0/2)
+(1.35)
+for all l ⩾ 1, where
+Tl−1(d/2, d0/2) ==
+Γ(d/2 + l) Γ(d0/2 + l) Γ(d/2 + d0/2 + 3/2))
+Γ(d/2 + 1) Γ(d0/2 + 1) Γ(d/2 + d0/2 + 1/2 + l) .
+(1.36)
+Here P (α,β)
+n
+(t), t ∈ [−1, 1], α > −1, β > −1, are the standard Jacobi polynomials
+of degree n normalized by
+P (α,β)
+n
+(1) =
+�α + n
+n
+�
+=
+Γ(α + n + 1)
+Γ(n + 1)Γ(α + 1) ,
+(1.37)
+and P (α,β)
+n
+can be given by Rodrigues’ formula
+P (α,β)
+n
+(t) = (−1)n
+2nn! (1 − t)−α(1 + t)−β dn
+dtn
+�
+(1 − t)n+α(1 + t)n+β�
+.
+(1.38)
+Notice that |P (α,β)
+n
+(t)| ⩽ P (α,β)
+n
+(1) for t ∈ [−1, 1]. Recall also that P (α,β)
+n
+are
+orthogonal polynomials with the following orthogonality relations
+π
+�
+0
+P (α,β)
+l
+(cos u)P (α,β)
+l′
+(cos u)(sin 1
+2u)2α+1(cos 1
+2u)2β+1 du
+= 2−α−β−1
+1
+�
+−1
+P (α,β)
+l
+(z)P (α,β)
+l′
+(z)(1 − z)α(1 + z)β dz = M −1
+l
+δll′,
+(1.39)
+
+8
+M.M. SKRIGANOV
+where M0 = B(α + 1, β + 1)−1 and
+Ml = Ml(α, β) = (2l + α + β + 1)Γ(l + 1)Γ(l + α + β + 1)
+Γ(l + α + 1)Γ(l + β + 1),
+l ⩾ 1.
+(1.40)
+All necessary facts about Jacobi polynomials can be found in [2, 26]. We also
+use the notation
+(a)0 = 1,
+(a)k = a(a + 1) . . . (a + k − 1) = Γ(a + k)
+Γ(a)
+(1.41)
+for the rising factorial powers (Pochhammer’s symbol).
+Theorem 1.2 reduces the proof of Theorem 1.1 to the proof of the formulas (1.35).
+Perhaps such formulas are known but I could not find them in the literature. For
+spheres Jacobi polynomials P (d/2,d/2)
+n
+with equal parameters coincide (up to con-
+stant factors) with Gegenbauer polynomials, and in this case very general formulas
+for weighted L2-norms of Gegenbauer polynomials are given in the paper [11].
+In the present paper we will prove the following statement.
+Lemma 1.2. For all n ⩾ 0, Re α > −1/2 and Re β > −1/2, we have
+� 1
+−1
+�
+P (α,β)
+n
+(t)
+�2
+(1 − t)2α(1 + t)2β dt
+= 22α+2β+1 (1/2)n
+(n!)2
+B(2α + 1, 2β + 1) Tn(α, β),
+(1.42)
+where
+Tn(α, β) = (α + 1)n (β + 1)n
+(α + β + 3/2)n
+= Γ(α + n + 1) Γ(β + n + 1) Γ(α + β + 3/2))
+Γ(α + 1) Γ(β + 1) Γ(α + β + 3/2 + n)
+(1.43)
+is a rational function of α and β.
+The integral (1.42) converges for Re α > −1/2 and Re β > −1/2, and represents
+in this region a holomorphic function of two complex variables. The equality (1.42)
+defines an analytic continuation of the integral (1.42) to α ∈ C and β ∈ C.
+For α = d/2, β = d0/2 and n = l − 1 the equality (1.42) coincides with (1.35).
+This proves Theorem 1.1.
+Lemma 1.2 will be proved in Section 3. The crucial point in the proof is Watson’s
+theorem on the value of hypergeometric series 3F2(1).
+2. Spherical functions. Proofs of Lemma 1.1 and Theorem 1.2
+2.1. Invariant kernels and spherical functions. The general theory of spherical
+functions on homogeneous spaces can be found in [16,17,28,30]. The homogeneous
+spaces Q(d, d0) of interest to us belong to the class of so-called commutative spaces
+or symmetric Gelfand pairs. In this case the theory becomes significantly simpler.
+For Euclidean spheres Sd this theory is well known, see, for example, [14, 19].
+However, the theory of spherical functions on general spaces Q(d, d0) is probably
+not commonly known. In this section we describe the basic facts about spherical
+functions on spaces Q(d, d0) in a form convenient for us.
+
+SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE
+9
+Let us consider the quasi-regular representation U(g)f(x) = f(g−1x), f ∈ L2(Q),
+x ∈ Q, g ∈ G, and its decomposition into the orthogonal sum
+U(g) = �
+�
+l⩾0 Ul(g),
+L2(Q) = �
+�
+l⩾0 Vl ,
+(2.1)
+of irreducible representations Ul(g) in mutually orthogonal subspaces Vl of dimen-
+sions ml < ∞.
+Let A denote the algebra of Hilbert–Schmidt operators in L2(Q) commuting
+with the representation U. Each K ∈ A is an integral operator
+Kf(x) =
+�
+Q
+K(x, y) f(y) dµ(y),
+with the invariant kernel:
+K(gx1, gx2) = K(x1, x2), x1, x2 ∈ Q, g ∈ G,
+(2.2)
+which satisfies the condition
+||K||2
+HS = Tr KK∗
+=
+�
+Q×Q
+|K(x, y)|2 dµ(x)dµ(y) =
+�
+Q
+|K(x, y)|2 dµ(x) < ∞,
+(2.3)
+where Tr denotes the trace of an operator, and the second integral is independent
+of y in view of (2.2).
+Since the space Q is two-point homogeneous, the condition (2.2) implies that the
+kernel K(x1, x2) depends only on the distance θ(x1, x2), and can be written as
+K(x1, x2) = K(θ(x1, x2)) = k(cos θ(x1, x2)), x1, x2 ∈ Q,
+(2.4)
+with function K(z), z ∈ [0, π] and k(z), z ∈ [−1, 1]. The cosine is presented here for
+convenience in further calculations. The formula (2.4) can be also written as
+K(x1, x2) = K(θ(x, x0)) = k(cos θ(x, x0)),
+(2.5)
+where x1 = g1x0, x2 = g2x0, x = g−1
+2 g1x0, g1, g2 ∈ G and x0 ∈ Q is the fixed point
+of the subgroup H. Moreover, K(hx, x0) = K(x, x0), h ∈ H. Therefore, invariant
+kernels can be thought of as functions on the double co-sets H \ G/H.
+In terms of the function K(·) and k(·), the Hilbert-Schmidt norm (2.3) takes the
+form
+||K||2
+HS =
+� π
+0
+|K(u)|2 dv(u)
+=κ
+� π
+0
+|k(cos u)|2(sin 1
+2u)d−1(cos 1
+2u)d0−1 du
+=κ 21−d/2−d0/2
+� 1
+−1
+|k(z)|2 (1 − z)
+d
+2 −1 (1 + z)
+d0
+2 −1 dz,
+(2.6)
+where v(·) is the volume function (1.22).
+We conclude from (2.2) and (2.4) that for K ∈ A its kernel K(x1, x2) =
+K(x2, x1), the value K(x, x) = k(1) is independent of x ∈ Q, and if an opera-
+tor K is self-adjoint, then its kernel is real-valued.
+
+10
+M.M. SKRIGANOV
+It follows from (2.2) and (2.4) that the algebra A is commutative. Indeed,
+(K1K2)(x1, x2) =
+�
+Q
+K1(x1, x)K2(x, x2)dµ(x)
+=
+�
+Q
+K2(x2, x)K1(x, x1)dµ(x) = (K2K1)(x2, x1) = (K2K1)(x1, x2).
+Therefore, the decomposition (2.1) is multiplicity-free, that is any two representa-
+tions Ul and Ul′, l ̸= l′, are non-equivalent, because otherwise the algebras A could
+not be commutative.
+Let Pl denote orthogonal projectors in L2(Q) onto the subspaces Vl in (2.1),
+P ∗
+l = Pl ,
+Pl Pl′ = δl,l′ Pl ,
+�
+l⩾0 Pl = I ,
+(2.7)
+where δl,l′ is Kronecker’s symbol and I is the identity operator in L2(Q). By Schur’s
+lemma, we have for K ∈ A
+Pl K Pl′ = δl,l′ cl(K) Pl, ,
+(2.8)
+where cl(K) is a constant. Calculating the trace of both sides of the equality (2.8),
+we find cl(K) = m−1
+l
+Tr KPl. Therefore, we have the expansions
+K =
+�
+l,l′⩾0 Pl K Pl′ =
+�
+l⩾0 cl(K) Pl,
+(2.9)
+with Parseval’s identity
+||K||2
+HS =
+�
+l⩾0 ml |cl(K)|2 ,
+(2.10)
+and for K1, K2 ∈ A, we have
+K1 K2 =
+�
+l⩾0 cl(K1) cl(K2) Pl,
+(2.11)
+The equality (2.10) implies that the series (2.11) converges in the Hilbert-Schmidt
+norm (2.3), while the series (2.11) converges in the norm (2.3) for the subclass of
+nuclear operators.
+Since Vl are invariant subspaces, Pl ∈ A, their kernels Pl(·, ·) are symmetric and
+real-valued, and can be written as follows
+Pl(x1, x2) = pl(cos θ(x1, x2)) =
+�ml
+1
+ψl,j(x1) ψl,j(x2),
+(2.12)
+where {ψl,j(·)}ml
+1
+is an orthonormal and real-valued basis in Vl. Hence, subspace
+Vl and irreducible representations Ul in (2.1) can be thought of as defined over the
+field of reals, this means that all representations Ul in (2.1) are of the real type.
+Using (2.12), we obtain the formulas
+||Pl||2
+HS = ml,
+Tr Pl = pl(1) = ml > 0.
+(2.13)
+Furthermore,
+Pl(x, x) = pl(1) =
+�ml
+1
+ψl,j(x)2.
+(2.14)
+is independent of x ∈ Q. Applying Cauchy-Schwartz inequality to (2.12) and taking
+(2.14) into account, we obtain the bound
+|Pl(x1, x2)| = |pl(cos θ(x1, x2))| ⩽ pl(1).
+(2.15)
+It follows from (2.14) and (2.13) that the mapping
+Πl : Q ∋ x → (m−1/2
+l
+ψl,1(x) . . . m−1/2
+l
+ψl,ml(x)) ∈ Sml−1 ⊂ Rml
+(2.16)
+
+SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE
+11
+defines an embedding of the space Q into the unite sphere in Rml.
+By definition the (zonal) spherical function are kernels of the operators Φl =
+m−1
+l
+Pl:
+Φl(x1, x2) = φl(cos θ(x1, x2)) = pl(cos θ(x1, x2))
+pl(1)
+.
+(2.17)
+From (2.14) and (2.17) we conclude that |φl(cos θ(x1, x2))| ⩽ φl(1) = 1. Comparing
+(2.13), (2.14) and (2.17), we find the formulas for dimensions
+ml = ||Φl||−2
+HS =
+�
+κ
+� π
+0
+|φl(cos u)|2(sin 1
+2u)d−1(cos 1
+2u)d0−1 du
+�−1
+=
+�
+κ 21−d/2−d0/2
+� 1
+−1
+|φl(z)|2 (1 − z)
+d
+2 −1 (1 + z)
+d0
+2 −1 dz
+�−1
+.
+(2.18)
+In terms of spherical functions the formulas (2.9) and (2.11) take the form
+k(cos θ(x1, x2)) =
+�
+l⩾0 cl(K) ml φl(cos θ(x1, x2)),
+(2.19)
+where
+cl(K) = Tr KΦl =
+�
+Q
+K(x1, x2) Φ(x1, x2) dµ(x1)dµ(x2)
+= κ
+� π
+0
+k(cos u) φl(cos u) (sin 1
+2u)d−1(cos 1
+2u)d0−1 du
+= κ 21−d/2−d0/2
+� 1
+−1
+k(z) φl(z) (1 − z)
+d
+2 −1 (1 + z)
+d0
+2 −1 dz.
+(2.20)
+and
+�
+Q
+k1(cos θ(x1, y)) k2( cos θ(y, x2)) dµ(y)
+=
+�
+l⩾0 cl(K1) cl(K2) ml φl(cos θ(x1, x2)),
+(2.21)
+for K1, K2 ∈ A. It follows from (2.11) with K1 = K and K2 = K∗ that
+||K||2
+HS =
+�
+l⩾0 ml |cl(K)|2 ,
+(2.22)
+The above facts are valid for all compact two-point homogeneous spaces. Since
+spaces Q are also symmetric Riemannian manifolds of rank one, the invariant ker-
+nels pl(cos θ(x, x0)) are eigenfunctions of the radial part of the Laplace–Beltrami
+operator on Q (in the spherical coordinates centered at x0).
+This leads to the
+following explicit formula for spherical functions
+Φ(x1, x2) = φl(cos θ(x1, x2)) = P
+( d
+2 −1, d0
+2 −1)
+l
+(cos θ(x1, x2))
+P
+( d
+2 −1, d0
+2 −1)
+l
+(1)
+,
+l ⩾ 0.
+(2.23)
+where P (α,β)
+n
+(t), t ∈ [−1, 1], are Jacobi polynomials (1.38). We refer to [15, p. 178],
+[17, Chap. V, Theorem 4.5], [18, pp. 514–512, 543–544], [28, Chapters 2 and 17]: [30,
+Theorem 11.4.21] for more details.
+From (1.37) and (1.40) we obtain
+P
+( d
+2 −1, d0
+2 −1)
+n
+(1) =
+Γ(n + d/2)
+Γ(n + 1)Γ(1 + d/2) ,
+(2.24)
+
+12
+M.M. SKRIGANOV
+and Ml = Ml(d/2 − 1, d0/2 − 1), where M0 = B(d/2, d0/2)−1 and
+Ml = (2l − 1 + (d + d0)/2)Γ(l + 1)Γ(l − 1 + (d + d0)/2)
+Γ(l + d/2)Γ(l + d0/2)
+, l ⩾ 1,
+(2.25)
+Substituting (2.23), into (2.18) and using (2.24) and (2.25), we obtain the following
+explicit formulas for dimensions of irreducible representations (2.1) : m0 = 1 and
+ml =Ml B(d/2, d0/2)
+�
+P ( d
+2 −1, d0
+2 −1)(1)
+�2
+=(2l − 1 + (d + d0)/2) Γ(l − 1 + (d + d0)/2)Γ(l + d/2)Γ(d0/2)
+Γ((d + d0)/2)Γ(l + d0/2)Γ(d/2)Γ(l + 1)
+l ⩾ 1. (2.26)
+The formulas (2.19) for invariant kernels coincide with Fourier-Jacobi expansions.
+Suppose that a function k(t), t ∈ [−1, 1], has the expansion
+k(cos r) =
+�
+l⩾0
+Ml Cl(F) P
+( d
+2 −1, d0
+2 −1)
+l
+(cos r),
+(2.27)
+with Fourier-Jacobi coefficients
+Cl(k) =
+� π
+0
+k(cos u) P
+( d
+2 −1, d0
+2 −1)
+l
+(cos u) (sin 1
+2u)d−1 (cos 1
+2u)d0−1 du,
+(2.28)
+then the corresponding invariant kernel k(cos θ(x1, x2)) has the expansion (2.19)
+with coefficients
+cl(k) = Cl(k)
+κ(d, d0)
+P
+( d
+2 −1, d0
+2 −1)
+l
+(1)
+,
+l ⩾ 0.
+(2.29)
+Lemma 2.1. (i) For the chordal metric (1.23), we have
+τ(x1, x2) = 1
+2
+�
+l⩾1 Ml Cl [ 1 − φl(x1, x2) ] ,
+(2.30)
+where
+Cl = B((d + 1)/2, l + d0/2) Γ(l + 1)−1 (1/2)l−1 P
+(( d
+2 −1, d0
+2 −1))
+l
+(1) .
+(2.31)
+(ii) For the symmetric difference metrics (1.13), we have
+θ∆(ξ, x1, x2) = κ(d, d0)
+�
+l⩾1 l−2MlAl(ξ) [ 1 − φl(x1, x2) ] ,
+(2.32)
+where
+Al(ξ) =
+� π
+0
+�
+P
+( d
+2 , d0
+2 )
+l−1
+(cos r)
+�2
+(sin 1
+2r)2d(cos 1
+2r)2d0 dξ(r).
+(2.33)
+The series (2.30) and (2.32) converge absolutely and uniformly.
+The expansions (2.30) and (2.32) have been established in [23, Lemma 4.1] and
+[22, Theorema 4.1(ii)], correspondingly.
+2.3 Proof of Lemma 1.1. Let us consider the embedding (2.16) for l = 1. From the
+formula (2.26) we find
+m1 = d(d + d0 + 2)
+2d0
+= (n + 1)(d + 2)
+2
+− 1,
+d = nd0,
+(2.34)
+and for x1, x2 ∈ Q, we have
+∥Π1(x1) − Π1(x2)∥2 = 2 − 2(Π1(x1), Π1(x2)) = 2(1 − φ1(cos θ(x1, x2)),
+(2.35)
+where ∥ · ∥ and (·, ·) are the Euclidean norm and inner product in Rm1.
+
+SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE
+13
+On the other hand, from Rodrigues’ formula (1.38) we obtain
+P
+( d
+2 −1, d0
+2 −1)
+1
+(t) = ((d + d0)t + d − d0)/4,
+P
+( d
+2 −1, d0
+2 −1)
+1
+(1) = d/2, and
+1 − t
+2
+=
+d
+d + d0
+
+1 − P
+( d
+2 −1, d0
+2 −1)
+1
+(t)
+P
+( d
+2 −1, d0
+2 −1)
+1
+(1)
+
+ .
+Therefore,
+1 − cos θ(x1, x2)
+2
+=
+d
+d + d0
+�
+1 − φ1(cos θ(x1, x2))
+�
+.
+(2.36)
+Comparing (1.23), (2.35) and (2.36), we complete the proof.
+□
+2.3 Proof of Theorem 1.2. Since zonal spherical functions are mutually orthogonal,
+we conclude from the expansions (2.30) and (2.32) that the equality (1.27) is equiv-
+alent to the formulas
+γ(Q) l−2 B(d/2, d0/2)−1 Al(ξ♮) = Cl/2 ,
+l ⩾ 1 .
+(2.37)
+The integral (2.33) with the special measure dξ♮(r) = sin r dr takes the form
+Al(ξ♮) =
+� π
+0
+�
+P
+( d
+2 , d0
+2 )
+l−1
+(cos r)
+�2
+(sin 1
+2r)2d(cos 1
+2r)2d0 sin r dr
+= 2−d−d0
+� 1
+−1
+�
+P (d/2,d0/2)
+l−1
+(t)
+�2
+(1 − t)d (1 + t)d0 dt .
+(2.38)
+Hence, the formulas (2.37) can be written as follows
+� 1
+−1
+�
+P (d/2,d0/2)
+l−1
+(t)
+�2
+(1 − t)d (1 + t)d0 dt
+= 2d+d0+1 (1/2)l−1
+((l − 1)!)2
+B(d + 1, d0 + 1) T ∗,
+(2.39)
+where
+T ∗ =
+(l!)2 B(d/2, d0/2) Cl
+4 (1/2)l−1 B(d + 1, d0 + 1) γ(Q) .
+(2.40)
+On the other hand, using (1.37) and (??), we find
+Cl = (l!)−1 (1/2)l−1
+Γ(d/2 + 1/2) Γ(l + d/2) Γ(l + d0/2)
+Γ(l + 1/2 + d/2 + d0/2) Γ(d/2)
+.
+(2.41)
+Substituting (2.41) and (1.28) into (2.40), we obtain
+T ∗ =π−1/2 (d + d0)−1
+Γ(d + d0 + 2)
+Γ(d + 1) Γ(d0 + 1) ×
+× Γ(d/2 + 1/2) Γ(l + d/2) Γ(d0/2 + 1/2) Γ(l + d0/2)
+Γ(d/2 + d0/2) Γ(l + d/2 + d0/2 + 1/2)
+.
+(2.42)
+Applying the duplication formula for gamma function
+Γ(2z) = π−1/2 22z−1 Γ(z) Γ(z + 1/2)
+(2.43)
+
+14
+M.M. SKRIGANOV
+to the first co-factor in (2.42), we find
+π−1/2 (d + d0)−1
+Γ(d + d0 + 2)
+Γ(d + 1) Γ(d0 + 1)
+=
+Γ(d/2 + d0/2) Γ(d/2 + d0/2 + 3/2)
+Γ(d/2 + 1/2) Γ(d0/2 + 1) Γ(d0/2 + 1/2) Γ(d0/2 + 1) ,
+(2.44)
+where the relation Γ(z + 1) = zΓ(z) with z = d/2 + d0/2 has been also used.
+Substituting (2.44) into (2.42), we find that T ∗ = Tl−1(d/2, d0/2).
+□
+3. Proof of Lemma 1.2
+Lemma 1.2 follows from Lemma 3.1 and Lemma 3.2 given below.
+Lemma 3.1. For all n ⩾ 0, Re α > −1/2 and Re β > −1/2, we have
+� 1
+−1
+�
+P (α,β)
+n
+(t)
+�2
+(1 − t)2α(1 + t)2β dt
+= 22α+2β+1
+(n!)2
+B(2α + 1, 2β + 1)
+Wn(α, β)
+(2α + 2β + 2)2n
+,
+(3.1)
+where
+Wn(α, β)
+=
+�2n
+k=0
+(−1)n+k
+k!
+⟨2n⟩k ⟨α + n⟩k ⟨β + n⟩2n−k (2α + 1)2n−k (2β + 1)k (3.2)
+is a polynomial of α and β.
+Proof. Using Rodrigues’ formula (1.38), we can write
+� 1
+−1
+�
+P (α,β)
+n
+(t)
+�2
+(1 − t)2α(1 + t)2β dt =
+�
+1
+2n n!
+�2
+In(α, β) .
+(3.3)
+where
+In(α, β) =
+� 1
+−1
+� dn
+dtn
+�
+(1 − t)n+α(1 + t)n+β� �2
+dt .
+(3.4)
+Integrating in (3.4) n times by part, we obtain
+In(α, β)
+= (−1)n
+� 1
+−1
+�
+(1 − t)n+α(1 + t)n+β� d2n
+dt2n
+�
+(1 − t)n+α(1 + t)n+β�
+dt ,
+(3.5)
+since all terms outside the integral vanish. By Leibniz’s rule,
+d2n
+dt2n
+�
+(1 − t)n+α (1 + t)n+β�
+=
+�2n
+k=0
+�2n
+k
+� dk
+dtk (1 − t)n+α d2n−k
+dt2n−k (1 + t)n+β ,
+
+SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE
+15
+where
+�2n
+k
+�
+= ⟨2n⟩k/k! and
+dk
+dtk (1 − t)n+α = (−1)k ⟨α + n⟩k (1 − t)n−k+α ,
+d2n−k
+dt2n−k (1 + t)n+β = ⟨β + n⟩2n−k (1 + t)−n+k+β .
+Substituting these formulas into (3.5), we obtain
+In(α, β)
+= 22α+2β+2n+1 �2n
+k=0
+(−1)n+k
+k!
+⟨2n⟩k ⟨α + n⟩k ⟨β + n⟩2n−k I(k)
+n (α, β) , (3.6)
+where
+I(k)
+n (α, β) = B(2α + 2n − k + 1, 2β + k + 1).
+(3.7)
+Here we have used the following Euler’s integral
+21−a−b
+� 1
+−1
+(1 − t)a−1 (1 + t)b−1 dt = B(a, b) = Γ(a)Γ(b)
+Γ(a + b)
+(3.8)
+with Re a > 0, Re b > 0.
+The formula (3.7) can be written as follows
+I(k)
+n (α, β) = Γ(2α + 2n − k + 1) Γ(2β + k + 1)
+Γ(2α + 2β + 2n + 2)
+=Γ(2α + 2n − k + 1)
+Γ(2α + 1)
+Γ(2β + k + 1)
+Γ(2β + 1)
+Γ(2α + 1) Γ(2β + 1)
+Γ(2α + 2β + 2)
+Γ(2α + 2β + 2)
+Γ(2α + 2β + 2n + 2)
+=(2α + 1)2n−k (2β + 1)k
+(2α + 2β + 2)2n
+B(2α + 1, 2β + 1) .
+(3.9)
+Combining the formulas (3.9), (3.6) and (3.3), we obtain (3.1).
+□
+The next Lemma 3.2 is more specific, it relies on Watson’s theorem for general-
+ized hypergeometric series, see [2,24]. We consider the series of the form
+3F2(a, b, c; d, e; z) =
+�
+k⩾0
+(a)k (b)k (c)k
+(d)k (e)k k! z ,
+(3.10)
+where neither d nor e are negative integers. The series absolutely converges for
+|z| ⩽ 1, if Re(d + e) > Re(a + b + c). The series (3.10) terminates, if one of the
+numbers a, b, c is a negative integer.
+Watson’s theorem.We have
+3F2(a,b, c; (a + b + 1)/2, 2c; 1)
+=
+Γ(1/2) Γ(c + 1/2) Γ((a + b + 1)/2) Γ(c − (a + b − 1)/2)
+Γ((a + 1)/2) Γ((b + 1)/2) Γ(c − (a − 1)/2) Γ(c − (b − 1)/2) .
+(3.11)
+provided that
+Re (2c − a − b + 1) > 0.
+(3.12)
+The condition (3.12) ensures the convergence of hypergeometric series in (3.11).
+Furthermore, this condition is necessary for the truth of equality (3.11) even in
+the case of terminated series. The proof of Watson’s theorem can be found in [2,
+Therem 3.5.5], [24, p.54, Eq.(2.3.3.13)].
+
+16
+M.M. SKRIGANOV
+Lemma 3.2. For all n ⩾ 0, α ∈ C and β ∈ C, the polynomial (3.2) is equal to
+Wn(α, β) =22n (α + 1)n (β + 1)n (α + β + 1)n
+=22n Γ(α + 1 + n) Γ(β + 1 + n) Γ(α + β + 1 + n)
+Γ(α + 1) Γ(β + 1) Γ(α + β + 1)
+.
+(3.13)
+In particular,
+Wn(α, β)
+(2α + 2β + 2)2n
+= (α + 1)n (β + 1)n
+(α + β + 3/2)n
+.
+(3.14)
+Proof. Since Wn(α, β) is a polynomial, it suffers to check the equality (3.13) for α
+and β in an open subset in C2. As such a subset we shall take the following region
+O = { α, β : Re α < 0, Re β < 0, Im α > 0, Im β > 0 }.
+(3.15)
+For α and β in O, the co-factors in terms in (3.2) may be rearranged as follows:
+⟨2n⟩k = (−1)k (−2n)k ,
+⟨α + n⟩k = (−1)k (−α − n)k ,
+⟨β + n⟩2n−k = (−1)k (−β − n)2n−k = (−β − n)2n
+(β + 1 − n)k
+,
+(2α + 1)2n−k = (−1)k(2α + 1)2n
+(−2α − 2n)k
+,
+
+
+
+
+
+
+
+
+
+
+
+
+
+(3.16)
+Here we have used the following elementary relation for the rising factorial powers
+(a)m−k =
+(−1)k (a)m
+(1 − a − m)k
+,
+m ⩾ 0 , 0 ⩽ k ⩽ m .
+(3.17)
+Substituting (3.16) into (3.2), we find that
+Wn(α, β) = (−1)n (2α + 1)2n (−β − n)2n Fn(α, β)
+= (−1)n Γ(2α + 1 + 2n) Γ(−β + n)
+Γ(2α + 1) Γ(−β − n)
+Fn(α, β) ,
+(3.18)
+where
+Fn(α, β) =
+�2n
+k=0
+(−2n)k (2β + 1)k (−α − n)k
+(β + 1 − n)k (−2α − 2n)k k!
+(3.19)
+In view of the definition (3.10), we have
+Fn(α, β) = 3F2 (−2n, 2β + 1, −α − 1; β + 1 − n, −2α − 2n; 1) .
+(3.20)
+The parameters in hypergeometric series (3.20) are identical with those in (3.11)
+for a = −2n, b = 2β + 1, c = −α − n, and in this case, (a + b + 1)/2 = 2β + 1 + n,
+2c = −2α − 2n. The condition (3.12) also holds for α and β in the region O, since
+Re (2c − a − b + 1) = Re (−2α − 2β) > 0. Therefore, Watson’s theorem (3.11) can
+be applied to obtain
+Fn(α, β) =
+Γ(1/2) Γ(−α − n − 1/2) Γ(β + 1 − n) Γ(−α − β)
+Γ(−n + 1/2) Γ(β + 1) Γ(−α + 1/2) Γ(−α − β − n) .
+(3.21)
+Substituting the expression (3.21) into (3.18) , we may write
+Wn(α, β) = c0 c1(α) c2(β) c3(α + β) ,
+(3.22)
+
+SPHERICAL FUNCTIONS AND STOLARSKY’S INVARIANCE PRINCIPLE
+17
+where
+c0 = (−1)n Γ(1/2)
+Γ(−n + 1/2) ,
+c1(α) = Γ(2α + 2n + 1) Γ(−α − n + 1/2)
+Γ(2α + 1) Γ(−α + 1/2)
+,
+c2(β) = Γ(β + 1 − n) Γ(−β + n)
+Γ(β + 1) Γ(−β − n)
+,
+c3(α + β) =
+Γ(−α − β)
+Γ(−α − β − n) .
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(3.23)
+Using the duplication formula (2.43) and reflection formulas, see [2, Sec. 1.2],
+Γ(1 − z)Γ(z) =
+π
+sin πz ,
+Γ(1/2 − z)Γ(1/2 + z) =
+π
+cos πz ,
+(3.24)
+we may rearrange the expressions in (3.23) as follows. For c0, we have
+c0 =
+(−1)n Γ(1/2)2
+Γ(−n + 1/2) Γ(n + 1/2)
+Γ(n + 1/2)
+Γ(1/2)
+= (1/2)n ,
+since Γ(1/2) = √π. For c1(α) and c2(β), we have
+c1(α) =22n Γ(α + n + 1) Γ(α + n + 1/2) Γ(−α − n + 1/2)
+Γ(α + 1) Γ(α + 1/2) Γ(−α + 1/2)
+=22n cos πα Γ(α + n + 1)
+cos π(α + n) Γ(α + 1) = 22n (−1)n (α + 1)n
+and
+c2(β) = Γ(β + 1 − n) Γ(−β + n)
+Γ(β + 1) Γ(−β − n)
+= sin π(β + n) Γ(β + 1 + n)
+sin π(β − n) Γ(β + 1)
+= (β + 1)n .
+Finally,
+c3(α + β) = sin π(α + β) Γ(α + β + 1 + n)
+sin π(α + β + n) Γ(α + β + 1) = (−1)n (α + β + 1)n .
+Substituting these expressions into (3.22), we obtain (3.13).
+It follows from (2.31) and the duplication formula (2.43) that
+(2α + 2β + 2)2n = 22n (α + β + 1)n (α + β + 3/2)n .
+(3.25)
+Using (3.13) together with (3.25), we obtain (3.14).
+□
+Now it suffers to substitute (3.14) into (3.1) to obtain the formulas (1.42). The
+proof of Lemma 1.2 is complete.
+References
+[1] J. R. Alexander, J. Beck, W. W. L. Chen, Geometric discrepancy theory and uniform dis-
+tributions, in Handbook of Discrete and Computational Geometry (J. E. Goodman and
+J. O’Rourke eds.), Chapter 10, pages 185–207, CRC Press LLC, Boca Raton, FL, 1997.
+[2] G. E. Andrews, R. Askey, R. Roy, Special functions, Cambridge Univ. Press, 2000.
+[3] A. Barg, Stolarsky’s invariance principle for finite metric spaces, Mathematika, 67(1),
+(2021), 158–186.
+[4] A. Barg, M.M. Skriganov, Bounds for discrepancies in the Hemming space, J. of Complexity,
+65, (2021), 101552.
+[5] J. Beck, Sums of distances between points on a sphere: An application of the theory of
+irregularities of distributions to distance geometry, Mathematika, 31, (1984), 33–41.
+
+18
+M.M. SKRIGANOV
+[6] J. Beck, W. W. L. Chen, Irregularities of Distribution, Cambridge Tracts in Math., vol. 89,
+Cambridge Univ. Press, 1987.
+[7] D. Bilyk, M. Lacey, One bit sensing, discrepancy, and Stolarsky principle, Sbornik Math.,
+208(6), (2017), 744–763.
+[8] D. Bilyk, F. Dai, R. Matzke,
+Stolarsky principle and energy optimization on the sphere,
+Constr. Approx., 48(1), (2018), 31–60.
+[9] D. Bilyk, R. Matzke, O. Vlasiuk, Positive definiteness and the Stolarsky principle, J. of Math.
+Analysis, 513(1), (2022), 126220.
+[10] J. S. Brauchart, J. Dick, A simple proof of Stolarsky’s invariance principle, Proc. Amer.
+Math. Soc., 141, (2013), 2085–2096.
+[11] J. S. Brauchart, P. J. Grabner, Weighted L2-norms of Gegenbauer polynomials, Aequat.
+Math., 96, (2022), 741–762.
+[12] P. Cartier, Introduction `a l’´etude des mouvements browniens `a plusieurs param`etres,
+S´eminaire de Probabilit´es V, Lectures Notes in Math., 191, Springer—Verlag, 1971.
+[13] M. M. Deza, M. Laurent, Geometry of cuts and metrics, Springer, 1997.
+[14] F. Dai, Y. Xu, Approximation theory and harmonic analysis on spheres and balls, Springer,
+2013.
+[15] R. Gangolli, Positive definite kernels on homogeneous spaces and certain stochastic processes
+related to L´evy’s Brownian motion of several parameters, Ann. Inst. Henri Poincar´e, vol. III,
+No. 2, (1967), 121–325.
+[16] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press,
+1978.
+[17] S. Helgason, Groups and geometric analysis. Integral geometry, invariant differential opera-
+tors, and spherical functions, Academic Press, 1984.
+[18] V. I. Levenshtein, Universal bounds for codes and designs, in Handbook of Coding Theory
+(V. S. Pless and W. C. Huffman eds.), Chapter 6, pages 499–648, Elsevier, 1998.
+[19] C. M¨uller. Spherical Harmonics, Lecture Notes in Math., 17. Springer, 1966.
+[20] A. V. Shchepetilov, Calculus and Mechanics on two-point homogeneous spaces, Springer,
+2006.
+[21] M. M. Skriganov, Point distributions in compact metric spaces, Mathematika, 63, (2017),
+1152–1171.
+[22] M. M. Skriganov, Point distributions in two-point homogeneous spaces, Mathematika, 65,
+(2019), 557–587.
+[23] M. M. Skriganov, Stolarsky’s invariance principle for projective spaces, J. of Complexity, 56,
+(2020), 101428.
+[24] L. J. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, 1966.
+[25] K. B. Stolarsky, Sums of distances between points on a sphere, II, Proc. Amer. Math. Soc.,
+41, (1973), 575–582.
+[26] G. Szeg˝o , Orthogonal polynomials, Amer. Math. Soc., 1950.
+[27] S. S. Tai, Minimum embeddings of compact symmetric spaces of rank one, J.Differential
+Geometry, 2, (1968), 55–66.
+[28] N. Ja. Vilenkin, A. U. Klimyk, Representation of Lie groups and special functions, vols. 1–3,
+Kluwer Acad. Pub., Dordrecht, 1991–1992.
+[29] J. A. Wolf, Spaces of constant curvature, Univ. Califormia, Berkley, 1972.
+[30] J. A. Wolf, Harmonic analysis on commutative spaces, Math. Surveys and Monographs,
+vol. 142, Amer. Math. Soc., 2007.
+St. Petersburg Department of the Steklov Mathematical Institute of the Russian
+Academy of Sciences, 27, Fontanka, St.Petersburg 191023, Russia
+Email address: [email protected]
+

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+Semileptonic D Meson Decays D → P/V/Sℓ+νℓ with the SU(3) Flavor
+Symmetry/Breaking
+Ru-Min Wang1,†,
+Yue-Xin Liu1,
+Chong Hua1,
+Jin-Huan Sheng2,§,
+Yuan-Guo Xu1,♯
+1College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
+2School of Physics and Engineering, Henan University of Science and Technology, Luoyang, Henan 471000, China
+†[email protected]
+§[email protected]
+♯[email protected]
+Many exclusive c → d/sℓ+νℓ (ℓ = e, µ, τ) transitions have been well measured, and they can be
+used to test the theoretical calculations. Motivated by this, we study the D → P/V/Sℓ+νℓ decays
+induced by the c → d/sℓ+νℓ transitions with the SU(3) flavor symmetry approach, where P denotes
+the pseudoscalar meson, V denotes the vector meson, and S denotes the scalar meson with a mass
+below 1 GeV . The different decay amplitudes of the D → Pℓ+νℓ, D → V ℓ+νℓ or D → Sℓ+νℓ
+decays can be related by using the SU(3) flavor symmetry and by considering the SU(3) flavor
+breaking. Using the present data of D → P/V/Sℓ+νℓ, we predict the not yet measured or not yet
+well measured processes in the D → P/V/Sℓ+νℓ decays. We find that the SU(3) flavor symmetry
+approach works well in the semileptonic D → P/V ℓ+νℓ decays. For the D → Sℓ+νℓ decays, only
+the decay D+
+s → f0(980)e+νe has been measured, the branching ratios of the D+
+s → f0(980)e+νe
+and D → S(S → P1P2)ℓ+νℓ decays are used to constrain the nonperturbative parameters and then
+predict not yet measured D → Sℓ+νℓ decays, in addition, the two quark and the four quark scenarios
+for the light scalar mesons are analyzed. The SU(3) flavor symmetry predictions of the D → Sℓ+νℓ
+decays need to be further tested, and our predictions of the D → Sℓ+νℓ decays are useful for probing
+the structure of light scalar mesons. Our results in this work could be used to test the SU(3) flavor
+symmetry approach in the semileptonic D decays by the future experiments at BESIII, LHCb and
+BelleII.
+I.
+Introduction
+Semileptonic heavy meson decays dominated by tree-level exchange of W-bosons in the standard model have
+attracted a lot of attention in testing the stand model and in searching for the new physics beyond the stand model.
+Many semileptonic D → P/V ℓ+νℓ decays and one D → Sℓ+νℓ decay have been observed [1], and present experimental
+measurements give us an opportunity to additionally test theoretical approaches.
+In theory, the description of semileptonic decays are relatively simple, and the weak and strong dynamics can be
+separated in these processes since leptons do not participate in the strong interaction. All the strong dynamics in
+the initial and final hadrons is included in the hadronic form factors, which are important for testing the theoretical
+calculations of the involved strong interaction. The form factors of the D decays have been calculated, for examples,
+by quark model [2–7], QCD sum rules [8], light-cone sum rules [9–11], covariant light-front quark models [12–14], and
+lattice QCD [15, 16].
+The SU(3) flavor symmetry approach is independent of the detailed dynamics offering us an opportunity to relate
+different decay modes, nevertheless, it cannot determine the sizes of the amplitudes or the form factors by itself.
+arXiv:2301.00079v1  [hep-ph]  31 Dec 2022
+
+2
+However, if experimental data are enough, one may use the data to extract the amplitudes or the form factors, which
+can be viewed as predictions based on symmetry, has a smaller dependency on estimated form factors, and can provide
+some very useful information about the decays. The SU(3) flavor symmetry works well in the b-hadron decays [17–30],
+and the c-hadron decays [29–45].
+Semileptonic decays of D mesons have been studied extensively in the standard model and its various extensions, for
+instance, in Refs. [3, 46–56]. In this work, we will systematically study the D → P/V/Sℓ+νℓ decays with the SU(3)
+flavor symmetry. We will firstly construct the amplitude relations between different decay modes of D → Pℓ+νℓ,
+D → V ℓ+νℓ or D → Sℓ+νℓ decays by the SU(3) flavor symmetry and the SU(3) flavor breaking. We use the available
+data to extract the SU(3) flavor symmetry/breaking amplitudes and the form factors, and then predict the not yet
+measured modes for further tests in experiments. The forward-backward asymmetries Aℓ
+F B, the lepton-side convexity
+parameters Cℓ
+F , the longitudinal polarizations of the final charged lepton P ℓ
+L, the transverse polarizations of the final
+charged lepton P ℓ
+T , the lepton spin asymmetries Aλ and the longitudinal polarization fractions FL of the final vector
+mesons with two ways of integration have also been predicted in the D → P/V ℓ+νℓ decays. In addition, the q2
+dependence of some differential observables for the D → P/V ℓ+νℓ decays are shown in figures.
+This paper will be organized as follows. In Sec. II, the theoretical framework in this work is presented, including the
+effective hamiltonian, the hadronic helicity amplitude relations, the observables and the form factors. The numerical
+results of the D → P/V/Sℓ+νℓ semileptonic decays will be given in Sec. III. Finally, we give the summary and
+conclusion in Sec. IV.
+II.
+Theoretical Frame
+A.
+The effective Hamiltonian
+In the standard model, the four-fermion charged-current effective Hamiltonian below the electroweak scale for the
+decays D → Mℓ+νℓ (M = P, V, S) can be written as
+Heff(c → qℓ+νℓ) = GF
+√
+2 V ∗
+cq¯qγµ(1 − γ5)c ¯νℓγµ(1 − γ5)ℓ,
+(1)
+with q = s, d.
+The helicity amplitudes of the decays D → Mℓ+νℓ can be written as
+M(D → Mℓ+νℓ) = GF
+√
+2 Vcb
+�
+mm′
+gmm′Lλℓλν
+m
+HλM
+m′ ,
+(2)
+with
+Lλℓλν
+m
+= ϵα(m) ¯νℓγα(1 − γ5)ℓ,
+(3)
+HλM
+m′
+=
+�
+�
+�
+ϵ∗
+β(m′)⟨P/S(pP/S)|¯qγβ(1 − γ5)c|D(pD)⟩
+ϵ∗
+β(m′)⟨V (pV , ϵ∗)|¯qγβ(1 − γ5)c|D(pD)⟩
+,
+(4)
+where the particle helicities λM = 0 for M = P/S, λM = 0, ±1 for M = V, λℓ = ± 1
+2 and λν = + 1
+2, as well as ϵµ(m)
+is the polarization vectors of the virtual W with m = 0, t, ±1.
+
+3
+The form factors of the D → P, D → S and D → V transitions are given by [2, 3, 13]
+�
+P(p)
+�� ¯dkγµc
+�� D(pD)
+�
+= f P
++ (q2)(p + pD)µ +
+�
+f P
+0 (q2) − f P
++ (q2)
+� m2
+D − m2
+P
+q2
+qµ,
+(5)
+�
+S(p)
+�� ¯dkγµγ5c
+�� D(pD)
+�
+= −i
+�
+f S
++(q2)(p + pD)µ +
+�
+f S
+0 (q2) − f S
++(q2)
+� m2
+D − m2
+S
+q2
+qµ
+�
+,
+(6)
+�
+V (p, ε∗)
+�� ¯dkγµ(1 − γ5)c
+�� D(pD)
+�
+=
+2V V (q2)
+mD + mV
+ϵµναβε∗νpα
+Dpβ
+−i
+�
+ε∗
+µ(mD + mV )AV
+1 (q2) − (pD + p)µ(ε∗.pD) AV
+2 (q2)
+mD + mV
+�
++iqµ(ε∗.pD)2mV
+q2 [AV
+3 (q2) − AV
+0 (q2)],
+(7)
+where s = q2 (q = pD − pM), and ε∗ is the polarization of vector meson. The hadronic helicity amplitudes can be
+written as
+H± = 0,
+(8)
+H0 = 2mDq|⃗pP |
+�
+q2
+f P
++ (q2),
+(9)
+Ht =
+m2
+Dq − m2
+P
+�
+q2
+f P
+0 (q2),
+(10)
+for D → Pℓ+νℓ decays,
+H± = 0,
+(11)
+H0 = i2mDq|⃗pS|
+�
+q2
+f S
++(q2),
+(12)
+Ht =
+im2
+Dq − m2
+S
+�
+q2
+f S
+0 (q2),
+(13)
+for D → Sℓ+νℓ decays, and
+H± = (mDq + mV )A1(q2) ∓
+2mDq|⃗pV |
+(mDq + mV )V (q2),
+(14)
+H0 =
+1
+2mV
+�
+q2
+�
+(m2
+Dq − m2
+V − q2)(mDq + mV )A1(q2) −
+4m2
+Dq|⃗pV |2
+mDq + mV
+A2(q2)
+�
+,
+(15)
+Ht = 2mDq|⃗pV |
+�
+q2
+A0(q2),
+(16)
+for D → V ℓ+νℓ decays, where |⃗pM| ≡
+�
+λ(m2
+Dq, m2
+M, q2)/2mDq with λ(a, b, c) = a2 + b2 + c2 − 2ab − 2ac − 2bc.
+B.
+Hadronic helicity amplitude relations by the SU(3) flavor symmetry
+Charmed mesons containing one heavy c quark are flavor SU(3) anti-triplets
+Di =
+�
+D0(c¯u), D+(c ¯d), D+
+s (c¯s)
+�
+.
+(17)
+
+4
+Light pseudoscalar P and vector V meson octets and singlets under the SU(3) flavor symmetry of u, d, s quarks are
+[57]
+P =
+�
+�
+�
+�
+π0
+√
+2 + η8
+√
+6 + η1
+√
+3
+π+
+K+
+π−
+− π0
+√
+2 + η8
+√
+6 + η1
+√
+3
+K0
+K−
+K
+0
+− 2η8
+√
+6 + η1
+√
+3
+�
+�
+�
+� ,
+(18)
+V
+=
+�
+�
+�
+�
+ρ0
+√
+2 + ω8
+√
+6 + ω1
+√
+3
+ρ+
+K∗+
+ρ−
+− ρ0
+√
+2 + ω8
+√
+6 + ω1
+√
+3
+K∗0
+K∗−
+K
+∗0
+− 2ω8
+√
+6 + ω1
+√
+3
+�
+�
+�
+� ,
+(19)
+where ω and φ mix in an ideal form, and the η and η′ ( ω and φ) are mixtures of η1(ω1) = u¯u+d ¯d+s¯s
+√
+3
+and η8(ω8) =
+u¯u+d ¯d−2s¯s
+√
+6
+with the mixing angle θP (θV ). η and η′ (ω and φ) are given by
+�
+� η
+η′
+�
+� =
+�
+� cosθP −sinθP
+sinθP
+cosθP
+�
+�
+�
+� η8
+η1
+�
+� ,
+�
+� φ
+ω
+�
+� =
+�
+� cosθV
+−sinθV
+sinθV
+cosθV
+�
+�
+�
+� ω8
+ω1
+�
+� ,
+(20)
+where θP = [−20◦, −10◦] and θV = 36.4◦ from Particle Data Group (PDG) [1] will be used in our numerical analysis.
+The structures of the light scalar mesons are not fully understood yet. Many suggestions are discussed, such as
+ordinary two quark states, four quark states, meson-meson bound states, molecular states, glueball states or hybrid
+states, for examples, in Refs. [58–66]. In this work, we will consider the two quark and the four quark scenarios for
+the scalar mesons below or near 1 GeV . In the two quark picture, the light scalar mesons can be written as [67]
+S =
+�
+�
+�
+�
+a0
+0
+√
+2 +
+σ
+√
+2
+a+
+0
+K+
+0
+a−
+0
+− a0
+0
+√
+2 +
+σ
+√
+2
+K0
+0
+K−
+0
+K
+0
+0
+f0
+�
+�
+�
+� .
+(21)
+The two isoscalars f0(980) and f0(500) are obtained by the mixing of σ = u¯u+d ¯d
+√
+2
+and f0 = s¯s
+�
+� f0(980)
+f0(500)
+�
+� =
+�
+� cosθS
+sinθS
+−sinθS cosθS
+�
+�
+�
+� f0
+σ
+�
+� ,
+(22)
+where the three possible ranges of the mixing angle, 25◦ < θS < 40◦, 140◦ < θS < 165◦ and
+− 30◦ < θS < 30◦
+[58, 68] will be analyzed in our numerical results. In the four quark picture, the light scalar mesons are given as [1, 69]
+σ = u¯ud ¯d,
+f0 = (u¯u + d ¯d)s¯s/
+√
+2,
+a0
+0 = (u¯u − d ¯d)s¯s/
+√
+2,
+a+
+0 = u ¯ds¯s,
+a−
+0 = d¯us¯s,
+K+
+0 = u¯sd ¯d,
+K0
+0 = d¯su¯u,
+¯K0
+0 = s ¯du¯u,
+K+
+0 = s¯ud ¯d,
+(23)
+and the two isoscalars are expressed as
+�
+� f0(980)
+f0(500)
+�
+� =
+�
+� cosφS
+sinφS
+−sinφS cosφS
+�
+�
+�
+� f0
+σ
+�
+� ,
+(24)
+where the constrained mixing angle φS = (174.6+3.4
+−3.2)◦ [59].
+
+5
+In terms of the SU(3) flavor symmetry, meson states and quark operators can be parameterized into SU(3) tensor
+forms, while the leptonic helicity amplitudes Lλℓ,λν
+m
+are invariant under the SU(3) flavor symmetry. And the hadronic
+helicity amplitude relations of the D → Mℓ+νℓ(M = P, V, S) decays can be parameterized as
+H(D → Mℓ+νℓ) = cM
+0 DiM i
+jHj,
+(25)
+where H2 ≡ V ∗
+cd and H3 ≡ V ∗
+cs are the CKM matrix elements, and cM
+0
+are the nonperturbative coefficients of the
+D → Mℓ+νℓ decays under the SU(3) flavor symmetry. Noted that the hadronic helicity amplitudes for the D → Sℓ+νℓ
+decays in Eq. (25) are given in the two quark picture of the light scalar mesons, and ones in the four quark picture
+of the light scalar mesons will be given later.
+The SU(3) flavor breaking effects mainly come from different masses of u, d and s quarks. Following Ref. [70], the
+SU(3) breaking amplitudes of the D → Mℓ+νℓ decays can be give as
+∆H(D → Mℓ+νℓ) = cM
+1 DaW a
+i M i
+jHj + cM
+2 DiM i
+aW a
+j Hj,
+(26)
+with
+W =
+�
+W i
+j
+�
+=
+�
+�
+�
+�
+1 0
+0
+0 1
+0
+0 0 −2
+�
+�
+�
+� ,
+(27)
+where cM
+1,2 are the nonperturbative SU(3) flavor breaking coefficients.
+In the four quark picture of the light scalar mesons, the hadronic helicity amplitudes of the D → Sℓ+νℓ decays
+under the SU(3) flavor symmetry are
+H(D → Sℓ+νℓ)4q = c′S
+0 DiSim
+jmHj.
+(28)
+And the corresponding SU(3) flavor breaking amplitudes of the D → Sℓ+νℓ decays are
+∆H(D → Sℓ+νℓ)4q = c′S
+1 DaW a
+i Sim
+jmHj + c′S
+2 DiSim
+amW a
+j Hj + c′S
+1 DiSim
+ja W a
+mHj.
+(29)
+In terms of the SU(3) flavor symmetry, the hadronic helicity amplitude relations for the D → Pℓ+νℓ, D → V ℓ+νℓ
+and D → Sℓ+νℓ decays are summarized in later Tab. I, Tab. IV and Tab. VIII, respectively.
+C.
+Observables for the D → Mℓ+νℓ decays
+The double differential branching ratios of the D → Mℓ+νℓ decays are [56]
+dB(D → Mℓ+νℓ)
+dq2d(cos θ)
+= τDG2
+F |Vcq|2λ1/2(q2 − m2
+ℓ)2
+64(2π)3M 3
+D(s)q2
+�
+(1 + cos2 θ)HU + 2 sin2 θHL + 2 cos θHP
++m2
+ℓ
+q2 (sin2 θHU + 2 cos2 θHL + 2HS − 4 cos θHSL)
+�
+,
+(30)
+where λ ≡ λ(m2
+Dq, m2
+M, q2), m2
+ℓ ≤ q2 ≤ (mDq − mM)2, and
+HU = |H+|2 + |H−|2,
+HL = |H0|2,
+HP = |H+|2 − |H−|2,
+HS = |Ht|2,
+HSL = ℜ(H0H†
+t ).
+(31)
+
+6
+The differential branching ratios integrated over cos θ are [56]
+dB(D(s) → Mℓ+νℓ)
+dq2
+= τDG2
+F |Vcq|2λ1/2(q2 − m2
+ℓ)2
+24(2π)3M 3
+D(s)q2
+Htotal,
+(32)
+with
+Htotal ≡ (HU + HL)
+�
+1 + m2
+ℓ
+2q2
+�
++ 3m2
+ℓ
+2q2 HS.
+(33)
+The lepton flavor universality in D(s) → Mℓ+νℓ is defined in a manner identical Rµ/e as
+Rµ/e =
+� qmax
+qmin dB(D(s) → Mµ+νµ)/dq2
+� qmax
+qmin dB(D(s) → Me+νe)/dq2 .
+(34)
+The forward-backward asymmetries are defined as [56]
+Aℓ
+F B(q2) =
+� 0
+−1 dcosθℓ
+dB(D→Mℓν)
+dq2dcosθℓ
+−
+� 1
+0 dcosθℓ
+dB(D→Mℓν)
+dq2dcosθℓ
+� 0
+−1 dcosθℓ
+dB(D→Mℓν)
+dq2dcosθℓ
++
+� 1
+0 dcosθℓ
+dB(D→Mℓν)
+dq2dcosθℓ
+(35)
+= 3
+4
+HP − 2m2
+ℓ
+q2 HSL
+Htotal
+.
+(36)
+The lepton-side convexity parameters are given by [56]
+Cℓ
+F (q2) = 3
+4
+�
+1 − m2
+ℓ
+q2
+� HU − 2HL
+Htotal
+.
+(37)
+The longitudinal polarizations of the final charged lepton ℓ are defined by [56]
+P ℓ
+L(q2) =
+(HU + HL)
+�
+1 − m2
+ℓ
+2q2
+�
+− 3m2
+ℓ
+2q2 HS
+Htotal
+,
+(38)
+and its transverse polarizations are
+P ℓ
+T (q2) = − 3πmℓ
+8
+�
+q2
+HP + 2HSL
+Htotal
+.
+(39)
+The lepton spin asymmetry in the ℓ − ¯νℓ center of mass frame is defined by [71–74]
+Aλ(q2) = dB(D → Mℓ+νℓ)[λℓ = − 1
+2]/dq2 − dB(D → Mℓ+νℓ)[λℓ = + 1
+2]/dq2
+dB(D → Mℓ+νℓ)[λℓ = − 1
+2]/dq2 + dB(D → Mℓ+νℓ)[λℓ = + 1
+2]/dq2
+(40)
+=
+Htotal − 6m2
+ℓ
+2q2 HS
+Htotal
+.
+(41)
+For the D → V ℓ+νℓ decays, the longitudinal polarization fractions of the final vector mesons are given by [56]
+FL(q2) =
+HL
+�
+1 + m2
+ℓ
+2q2
+�
++ 3m2
+ℓ
+2q2 HS
+Htotal
+,
+(42)
+then its transverse polarization fraction FT (q2) = 1 − FL(q2).
+Noted that, for q2-integration of X(q2) = Aℓ
+F B, Cℓ
+F , P ℓ
+L, P ℓ
+T , Aλ and FL, following Ref. [75], two ways of integration
+are considered. The normalized q2-integrated observables ⟨X⟩ are calculated by separately integrating the numerators
+and denominators with the same q2 bins. The “naively integrated” observables are obtained by
+X =
+1
+q2max − q2
+min
+� q2
+max
+q2
+min
+dq2X(q2).
+(43)
+
+7
+D.
+Form factors
+In order to obtain more precise observables, one also need considering the q2 dependence of the form factors for
+the D → Pℓ+νℓ, D → V ℓ+νℓ and D → Sℓ+νℓ decays. The following cases will be considered in our analysis of
+D → P/V ℓ+νℓ decays.
+C1: All form factors are treated as constants without the hadronic momentum-transfer q2 dependence, and different
+form factors are related by the SU(3) flavor symmetry, i.e., the SU(3) flavor breaking terms such as cM
+1,2 and
+c′S
+1,2,3 in later Tabs. I, IV and VIII are ignored.
+C2: With the SU(3) flavor symmetry, the modified pole model for the q2-dependence of Fi(q2) is used [76]
+Fi(q2) =
+Fi(0)
+�
+1 −
+q2
+m2
+pole
+� �
+1 − αi
+q4
+m4
+pole
+�,
+(44)
+where mpole = mD∗+ for c → dℓ+νℓ transitions and mpole = mD∗+
+s
+for c → sℓ+νℓ transitions, and αi are free
+parameters and are different for f P
++ (q2), f P
+0 (q2), V (q2), A1(q2) and A2(q2), we will take αi ∈ [−1, 1] in our
+analysis.
+C3: With the SU(3) flavor symmetry, following Ref. [2]
+Fi(q2) =
+Fi(0)
+�
+1 −
+q2
+m2
+pole
+� �
+1 − σ1i
+q2
+m2
+pole + σ2i
+q4
+m4
+pole
+�
+for f P
++ (q2) and V (q2),
+(45)
+Fi(q2) =
+Fi(0)
+�
+1 − σ1i
+q2
+m2
+pole + σ2i
+q4
+m4
+pole
+�
+for f P
+0 (q2), A1(q2) and A2(q2),
+(46)
+where σ1,2 for the D → π and D → K∗ transitions from Ref. [2] will be used in our results.
+C4: Considering the SU(3) flavor breaking terms such as cM
+1,2 and c′S
+1,2,3 in later Tabs. I, IV and VIII, the form factors
+in C3 case are used.
+As for the form factors of the D → Sℓ+νℓ decays, we find that the vector dominance model [77] and the double
+pole model [78] give the similar SU(3) flavor symmetry predictions for the branching ratios of the D → Sℓ+νℓ decays.
+The following form factors from the vector dominance model will be used in the numerical results,
+Fi(q2) =
+Fi(0)
+�
+1 − q2/m2
+pole
+�
+for f S
++(q2) and f S
+0 (q2).
+(47)
+After considering above q2 dependence, we only need to focus on the Fi(0). Since these form factors Fi(0) also
+preserve the SU(3) flavor symmetry, the same relations in Tabs. I, IV and VIII will be used for Fi(0). If considering
+the form factors ratios f+(0)/f0(0) = 1 for D → P/Sℓ+νℓ decays, rV ≡ V (0)/A1(0) = 1.46±0.07, r2 ≡ A2(0)/A1(0) =
+0.68 ± 0.06 in D0 → K∗−ℓ+νℓ decays from PDG [1] and the SU(3) flavor symmetry, there is only one free form factor
+f P,S
++
+(0) and A1(0) for the D → P/Sℓ+νℓ and D → V ℓ+νℓ decays, respectively. As a result, the branching ratios only
+depend on one form factor f P
++ (0), f S
++(0) or A1(0) and the CKM matrix element Vcq.
+
+8
+III.
+Numerical results
+The theoretical input parameters and the experimental data within the 2σ errors from PDG [1] will be used in our
+numerical results.
+A.
+D → Pℓ+νℓ decays
+Considering both the SU(3) flavor symmetry and the SU(3) flavor breaking contributions, the hadronic helicity
+amplitudes for the D → Pℓ+νℓ decays are given in Tab. I, in which we keep the CKM matrix element Vcs and Vcd
+information for comparing conveniently. In addition, H(D+
+s → π0ℓ+νℓ) are obtained by neutral meson mixing with
+δ2 = (5.18 ± 0.71) × 10−4 in Ref. [76]. From Tab. I, we can easily see the hadronic helicity amplitude relations
+of the D → Pℓ+νℓ decays.
+There are four nonperturbative parameters A1,2,3,4 in the D → Pℓ+νℓ decays with
+A1 ≡ cP
+0 + cP
+1 − 2cP
+2 , A2 ≡ cP
+0 − 2cP
+1 − 2cP
+2 , A3 ≡ cP
+0 + cP
+1 + cP
+2 and A4 ≡ cP
+0 − 2cP
+1 + cP
+2 . If neglecting the SU(3) flavor
+breaking cP
+1 and cP
+2 terms, A1 = A2 = A3 = A4 = cP
+0 , and then all hadronic helicity amplitudes are related by only
+one parameter cP
+0 .
+Many decay modes of the D → Pe+νe, Pµ+νµ decays have been measured, and the experimental data with 2σ
+errors are listed in the second column of Tab. II. One can constrain the parameters Ai by the present experimental
+data within 2σ errors and then predict other not yet measured branching ratios. Four cases C1,2,3,4 will be considered
+in our analysis. The numerical results of B(D → Pℓ+νℓ) in the C1, C2, C3 and C4 cases are given in the third, forth,
+fifth and sixth columns of Tab. II, respectively. And our comments on the results are as follows.
+• Results in C1 case:
+From the third column of Tab. II, one can see that the SU(3) flavor symmetry predictions
+of B(D → Pℓ+νℓ) in the C1 case are entirely consistent with all present experiential data. The not yet measured
+branching ratios of the D+
+s → π0e+νe, D+
+s → π0µ+νµ, D+ → η′µ+νµ and D+
+s → K0µ+νµ decays are predicted
+TABLE I: The hadronic helicity amplitudes for the D → Pℓ+ν decays including both the SU(3) flavor symmetry and the
+SU(3) flavor breaking contributions. A1 ≡ cP
+0 + cP
+1 − 2cP
+2 , A2 ≡ cP
+0 − 2cP
+1 − 2cP
+2 , A3 ≡ cP
+0 + cP
+1 + cP
+2 , A4 ≡ cP
+0 − 2cP
+1 + cP
+2 .
+A1 = A2 = A3 = A4 = cP
+0 if neglecting the SU(3) flavor breaking cP
+1 and cP
+2 terms.
+Hadronic helicity amplitudes
+SU(3) flavor amplitudes
+H(D0 → K−ℓ+νℓ)
+A1V ∗
+cs
+H(D+ → K
+0ℓ+νℓ)
+A1V ∗
+cs
+H(D+
+s → ηℓ+νℓ)
+�
+− cosθP
+�
+2/3 − sinθP /
+√
+3�
+A2V ∗
+cs
+H(D+
+s → η′ℓ+νℓ)
+�
+− sinθP
+�
+2/3 + cosθP /
+√
+3�
+A2V ∗
+cs
+H(D+
+s → π0ℓ+νℓ)
+−δ�
+− cosθP
+�
+2/3 − sinθP /
+√
+3�
+A2V ∗
+cs
+H(D0 → π−ℓ+νℓ)
+A3V ∗
+cd
+H(D+ → π0ℓ+νℓ)
+− 1
+√
+2 A3V ∗
+cd
+H(D+ → ηℓ+νℓ)
+�
+cosθP /
+√
+6 − sinθP /
+√
+3�
+A3V ∗
+cd
+H(D+ → η′ℓ+νℓ)
+�
+sinθP /
+√
+6 + cosθP /
+√
+3�
+A3V ∗
+cd
+H(D+
+s → K0ℓ+νℓ)
+A4V ∗
+cd
+
+9
+TABLE II: Branching ratios of the D → Pℓ+ν decays. †Denotes that the corresponding experimental data from PDG [1] are
+not used to constrain Ai in this case.
+Branching ratios
+Exp. data
+Ones in C1
+Ones in C2
+Ones in C3
+Ones in C4
+Previous ones
+B(D+ → K
+0e+νe)(×10−2)
+8.72 ± 0.18
+8.84 ± 0.06
+8.83 ± 0.07
+8.84 ± 0.06
+8.83 ± 0.07
+B(D+ → π0e+νe)(×10−3)
+3.72 ± 0.34
+3.75 ± 0.05
+5.40 ± 1.33†
+5.04 ± 0.12†
+3.70 ± 0.11
+B(D+ → ηe+νe)(×10−3)
+1.11 ± 0.14
+1.15 ± 0.05
+1.20 ± 0.05
+1.20 ± 0.05
+0.92 ± 0.08
+B(D+ → η′e+νe)(×10−4)
+2.0 ± 0.8
+2.59 ± 0.14
+2.22 ± 0.34
+2.09 ± 0.14
+1.50 ± 0.20
+B(D0 → K−e+νe)(×10−2)
+3.549 ± 0.052
+3.52 ± 0.02
+3.52 ± 0.03
+3.52 ± 0.03
+3.52 ± 0.02
+B(D0 → π−e+νe)(×10−3)
+2.91 ± 0.08
+2.95 ± 0.03
+4.23 ± 1.03†
+3.97 ± 0.09†
+2.89 ± 0.06
+B(D+
+s → ηe+νe)(×10−2)
+2.32 ± 0.16
+2.37 ± 0.11
+2.34 ± 0.14
+2.36 ± 0.12
+2.32 ± 0.16
+B(D+
+s → η′e+νe)(×10−3)
+8.0 ± 1.4
+9.05 ± 0.04
+8.25 ± 1.13
+8.04 ± 0.43
+8.02 ± 1.38
+B(D+
+s → K0e+νe)(×10−3)
+3.4 ± 0.8
+3.10 ± 0.08
+3.56 ± 0.39
+3.54 ± 0.12
+3.40 ± 0.80
+B(D+
+s → π0e+νe)(×10−5)
+· · ·
+1.51 ± 0.07
+2.10 ± 0.56
+1.96 ± 0.10
+1.92 ± 0.13
+2.65 ± 0.38 [76]
+B(D+ → K
+0µ+νµ)(×10−2)
+8.76 ± 0.38
+8.56 ± 0.06
+8.69 ± 0.15
+8.61 ± 0.06
+8.61 ± 0.06
+B(D+ → π0µ+νµ)(×10−3)
+3.50 ± 0.30
+3.67 ± 0.05
+5.32 ± 1.31†
+4.96 ± 0.12†
+3.64 ± 0.10
+B(D+ → ηµ+νµ)(×10−3)
+1.04 ± 0.22
+1.11 ± 0.05
+1.18 ± 0.07
+1.17 ± 0.05
+0.90 ± 0.08
+1.21 [7]
+0.75±0.15 [79]
+B(D+ → η′µ+νµ)(×10−4)
+· · ·
+2.42 ± 0.13
+2.10 ± 0.33
+1.96 ± 0.13
+1.41 ± 0.19
+2.11 [7]
+1.06±0.20 [79]
+B(D0 → K−µ+νµ)(×10−2)
+3.41 ± 0.08
+3.41 ± 0.02
+3.44 ± 0.05
+3.43 ± 0.02
+3.43 ± 0.02
+B(D0 → π−µ+νµ)(×10−3)
+2.67 ± 0.24
+2.89 ± 0.02
+4.17 ± 1.01†
+3.90 ± 0.09†
+2.85 ± 0.06
+B(D+
+s → ηµ+νµ)(×10−2)
+2.4 ± 1.0
+2.30 ± 0.10
+2.30 ± 0.17
+2.31 ± 0.12
+2.26 ± 0.16
+B(D+
+s → η′µ+νµ)(×10−2)
+1.1 ± 1.0
+0.86 ± 0.03
+0.79 ± 0.11
+0.77 ± 0.04
+0.76 ± 0.13
+B(D+
+s → K0µ+νµ)(×10−3)
+· · ·
+3.01 ± 0.08
+3.51 ± 0.38
+3.46 ± 0.11
+3.33 ± 0.78
+3.9 [7]
+3.85±0.76 [79]
+B(D+
+s → π0µ+νµ)(×10−5)
+· · ·
+1.48 ± 0.07
+2.09 ± 0.53
+1.93 ± 0.10
+1.89 ± 0.13
+B(D+
+s → π0τ +ντ)(×10−10)
+· · ·
+3.45 ± 0.21
+160.34 ± 149.53
+4.20 ± 0.26
+4.08 ± 0.34
+(27 ∼ 36) [76]
+Rµ/e(D+ → K
+0ℓ+νℓ)
+0.969
+0.984 ± 0.013
+0.974
+0.974
+Rµ/e(D+ → π0ℓ+νℓ)
+0.977
+1.009 ± 0.026
+0.984
+0.984
+Rµ/e(D+ → ηℓ+νℓ)
+0.967
+0.984 ± 0.014
+0.973
+0.973
+Rµ/e(D+ → η′ℓ+νℓ)
+0.935
+0.948 ± 0.012
+0.940
+0.940
+Rµ/e(D0 → K−ℓ+νℓ)
+0.969
+0.984 ± 0.013
+0.974
+0.974
+Rµ/e(D0 → π−ℓ+νℓ)
+0.977
+1.008 ± 0.026
+0.984
+0.984
+Rµ/e(D+
+s → ηℓ+νℓ)
+0.971
+0.987 ± 0.013
+0.976
+0.976
+Rµ/e(D+
+s → η′ℓ+νℓ)
+0.946
+0.958 ± 0.011
+0.952
+0.952
+Rµ/e(D+
+s → K0ℓ+νℓ)
+0.973
+0.992 ± 0.016
+0.978
+0.978
+Rµ/e(D+
+s → π0ℓ+νℓ)
+0.980
+1.010 ± 0.025
+0.985
+0.985
+
+10
+on the order of O(10−3 − 10−5), nevertheless, B(D+
+s → π0τ +ντ) is predicted on the order of O(10−10) due to
+its narrow phase space and (q2 − m2
+τ)2 suppression of the differential branching ratios in Eq. (32).
+• Results in C2,3 cases:
+The numerical results in C2,3 cases are similar. The experimental upper limits of
+B(D+ → π0ℓ+νℓ) and B(D0 → π−ℓ+νℓ) have not been used to constrain the predictions of B(D → Pℓ+νℓ), since
+the upper limits of the predictions of B(D+ → π0ℓ+νℓ) and B(D0 → π−ℓ+νℓ) by the SU(3) flavor symmetry
+in C2,3 cases are slightly larger than their experimental data. Other SU(3) flavor symmetry predictions are
+consistent with their experimental data within 2σ errors.
+• Results in C4 case:
+As given in the sixth column of Tab. II, if considering both the hadronic momentum-
+transfer q2 dependence of the form factors and the SU(3) flavor breaking contributions, all SU(3) flavor symmetry
+predictions are consistent with their experimental data within 2σ errors. For some decays, the errors of the
+theoretical predictions are much smaller than ones of their experimental data.
+• The previous predictions for the not yet measured branching ratios are listed in the last column of Tab. II, our
+predictions are in the same order of magnitude as previous ones for the D → Pe+νe, Pµ+νµ decays. And our
+prediction of B(D+
+s → π0τ +ντ) is one order smaller than previous one in Ref. [76].
+• In addition, the lepton flavor universality parameters Rµ/e(D → Pℓ+νℓ) are also given in Tab. II, since many
+terms are canceled in the ratios, these predictions are quite accurate, and all processes have similar results.
+For the q2 dependence of the differential branching ratios of the D → Pℓ+νℓ decays with present experimental
+bounds, we only show the not yet measured processes D+ → η′µ+νµ, D+
+s → K0µ+νµ, D+
+s → π0µ+νµ and D+
+s →
+π0τ +ντ in Fig. 1. We do not show dB(D+
+s → π0e+νe)/dq2, since it is similar to dB(D+
+s → π0µ+νµ)/dq2 in Fig. 1
+C
+1
+C
+2
+C
+3
+C
+4
+dB(D
++
+s
+0
++
+)/dq
+2
+ ( x10
+-10
+ )
+q
+2
+ 
+ 
+ 
+ 
+FIG. 1: The q2 dependence of the differential branching ratios for some D → Pℓ+νℓ with present experimental bounds.
+
+qB(D
+0.0
+S.0
+.0
+0
+文
+→>,")qd
+Se
+8(C)
+d.
+S
+3
+4
+Q
+00
+0C
+Sc
+c
+(ε)
+d.4
+a.0
+8.0
+0.10.8
+1.8
+0S.8
+3'
+001
+5
+3
+4qB(D→>K^")/qd
+0.0
+2.0
+0
+xXoX
+S3(q)
+3'S
+3'3
+3
+003
+3'4
+0
++ix
+Q
+0
+(p)
+d.0.1
+2.1
+s'o
+20
+0
+0'42.0
+1
+gB
+0.0
+2
+个←
+2.00.1
+3
+2.1
+s'o11
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+-20
+-15
+-10
+-5
+0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+-1.6
+-1.2
+-0.8
+-0.4
+0.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.0
+0.3
+0.6
+0.9
+1.2
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+-1.0
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+( d )
+( c )
+( b )
+( a )
+ 
+ 
+ e
+in unit of 10
+-6
+ 
+in unit of 10
+-2
+dA
+FB
+(D
++
+s
+K
+0
+l
++
+l
+)/dq
+2
+q
+2
+ 
+ 
+ e
+ 
+dC
+l
+F
+(D
++
+s
+K
+0
+l
++
+l
+)/dq
+2
+q
+2
+ 
+ 
+ e
+ 
+dP
+l
+L
+(D
++
+s
+K
+0
+l
++
+l
+)/dq
+2
+q
+2
+ 
+ 
+ e
+ 
+dP
+l
+T
+(D
++
+s
+K
+0
+l
++
+l
+)/dq
+2
+q
+2
+FIG. 2:
+The differential forward-backward asymmetries, differential lepton-side convexity parameters, differential longitudinal
+lepton polarizations and differential transverse lepton polarizations for the D+
+s → K0ℓ+νℓ decays in the C3 case.
+(c). From Fig. 1, one can see that present experimental measurements give quite strong bounds on the differential
+branching ratios of D+ → η′µ+νµ, D+
+s → π0µ+νµ and D+
+s → π0τ +ντ decays in the C1, C3 and C4 cases as well as
+D+
+s → K0µ+νµ decays in the C1 and C3 cases, and all predictions of the four differential branching ratios in the C2
+case have large error due to the form factor choice. Comparing with dB(D+
+s → π0µ+νµ)/dq2 in Fig. 1 (c), as shown
+in Fig. 1 (d), dB(D+
+s → π0τ +ντ)/dq2 is suppressed about the order of O(10−4) by mτ.
+The forward-backward asymmetries Aℓ
+F B, the lepton-side convexity parameters Cℓ
+F , the longitudinal polarizations
+of the final charged leptons P ℓ
+L and the transverse polarizations of the final charged leptons P ℓ
+T with two ways of
+integration for the D → Pℓ+νℓ decays could also be obtained. These predictions are very accurate, and they are
+similar to each other in the four C1,2,3,4 cases. So we only give the predictions within the C3 case in Tab. III for
+examples. From Tab. III, one can see that the predictions are obviously different between two ways of q2 integration,
+and the slight difference in the same way of q2 integration is due to the different decay phase spaces. For displaying
+the differences between the D → Pe+νe and D → Pµ+νµ decays, we take D+
+s → K0e+νe and D+
+s → K0µ+νµ
+as examples. The differential forward-backward asymmetries, the differential lepton-side convexity parameters, the
+differential longitudinal lepton polarizations and the differential transverse lepton polarizations of D+
+s → K0e+νe and
+D+
+s → K0µ+νµ decays within the C3 case are displayed in Fig. 2. And one can see that differential observables
+between ℓ = e and ℓ = µ are obviously different, specially in the low and high q2 ranges.
+
+12
+TABLE III: Quantities ⟨X⟩ and X of the D → Pℓ+ν in C3 case.
+Decay modes
+⟨Aℓ
+F B⟩
+Ae
+F B(×10−6)
+Aµ,τ
+F B(×10−2)
+⟨Cℓ
+F ⟩
+Cℓ
+F
+⟨P ℓ
+L⟩
+P ℓ
+L
+⟨P ℓ
+T ⟩
+P e
+T (×10−3)
+P µ,τ
+T
+D+ → K
+0e+νe
+−0.087
+−3.254 ± 0.001
+−1.239
+−1.500
+0.768
+1.000
+−0.273
+−2.442 ± 0.001
+D+ → π0e+νe
+−0.083
+−2.054 ± 0.000
+−1.252
+−1.500
+0.780
+1.000
+−0.260
+−1.730 ± 0.000
+D+ → ηe+νe
+−0.087
+−3.476 ± 0.001
+−1.239
+−1.500
+0.768
+1.000
+−0.273
+−2.490 ± 0.000
+D+ → η′e+νe
+−0.093
+−7.075 ± 0.003
+−1.222
+−1.500
+0.753
+1.000
+−0.290
+−3.890 ± 0.001
+D0 → K−e+νe
+−0.087
+−3.259 ± 0.001
+��1.239
+−1.500
+0.768
+1.000
+−0.273
+−2.446 ± 0.001
+D0 → π−e+νe
+−0.083
+−2.077 ± 0.000
+−1.252
+−1.500
+0.779
+1.000
+−0.260
+−1.751 ± 0.000
+D+
+s → ηe+νe
+−0.086
+−3.033 ± 0.001
+−1.242
+−1.500
+0.770
+1.000
+−0.270
+−2.300 ± 0.001
+D+
+s → η′e+νe
+−0.091
+−5.829 ± 0.003
+−1.226
+−1.500
+0.757
+1.000
+−0.286
+−3.484 ± 0.001
+D+
+s → K0e+νe
+−0.085
+−2.814 ± 0.001
+−1.245
+−1.500
+0.773
+1.000
+−0.267
+−2.118 ± 0.000
+D+
+s → π0e+νe
+−0.082
+−1.850 ± 0.001
+−1.254
+−1.500
+0.781
+1.000
+−0.258
+−1.634 ± 0.001
+D+ → K
+0µ+νµ
+−0.226
+−4.278 ± 0.001
+−0.822
+−1.352
+0.394
+0.851
+−0.655
+−0.414
+D+ → π0µ+νµ
+−0.201
+−2.810 ± 0.000
+−0.897
+−1.405
+0.462
+0.907
+−0.602
+−0.310
+D+ → ηµ+νµ
+−0.227
+−4.490 ± 0.001
+−0.819
+−1.347
+0.391
+0.846
+−0.657
+−0.419
+D+ → η′µ+νµ
+−0.263
+−8.097 ± 0.003
+−0.708
+−1.213
+0.287
+0.703
+−0.725
+−0.581
+D0 → K−µ+νµ
+−0.226
+−4.285 ± 0.001
+−0.822
+−1.352
+0.393
+0.850
+−0.656
+−0.414
+D0 → π−µ+νµ
+−0.201
+−2.844 ± 0.001
+−0.895
+−1.407
+0.461
+0.910
+−0.603
+−0.313
+D+
+s → ηµ+νµ
+−0.221
+−4.001 ± 0.001
+−0.836
+−1.364
+0.406
+0.864
+−0.646
+−0.394
+D+
+s → η′µ+νµ
+−0.254
+−6.952 ± 0.003
+−0.736
+−1.254
+0.314
+0.747
+−0.709
+−0.540
+D+
+s → K0µ+νµ
+−0.215
+−3.701 ± 0.001
+−0.856
+−1.377
+0.425
+0.879
+−0.632
+−0.367
+D+
+s → π0µ+νµ
+−0.197
+−2.571 ± 0.001
+−0.907
+−1.417
+0.472
+0.920
+−0.594
+−0.295
+D+
+s → π0τ +ντ
+−0.281
+−27.429 ± 0.105−0.211 ± 0.003−0.212 ± 0.003−0.868 ± 0.001−0.873 ± 0.001−0.447 ± 0.002−0.437 ± 0.002
+B.
+D → V ℓ+νℓ decays
+The hadronic helicity amplitudes for the D → V ℓ+νℓ decays are given in Tab. IV. There are four nonperturbative
+parameters B1,2,3,4 in the D → V ℓ+νℓ decay modes.
+If neglecting the SU(3) flavor breaking cV
+1 and cV
+2 terms,
+B1 = B2 = B3 = B4 = cV
+0 , and then all hadronic helicity amplitudes of D → V ℓ+νℓ are related by only one parameter
+cV
+0 .
+Among the D → V ℓ+νℓ decay modes, 13 branching ratios have been measured, and 2 branching ratios have been
+upper limited by the experiments. The experimental data with 2σ errors are listed in the second column of Tab. V.
+Now we use the listed experimental data to constrain the parameters Bi and then predict other not yet measured and
+not yet well measured branching ratios. The numerical results of B(D → V ℓ+νℓ) in the C1, C2, C3 and C4 cases are
+given in the third, forth, fifth and sixth columns of Tab. V, respectively.
+The results in the C1, C2 and C3 cases are very similar. Since the SU(3) flavor symmetry predictions of B(D+ →
+ωe+νe) and B(D0 → ρ−µ+νµ) are slightly larger than their experimental data within 2σ errors in the three cases, we
+
+13
+TABLE IV: The hadronic helicity amplitudes for D → V ℓ+ν decays including both the SU(3) flavor symmetry and the SU(3)
+flavor breaking contributions. B1 = cV
+0 +cV
+1 −2cV
+2 , B2 = cV
+0 −2cV
+1 −2cV
+2 , B3 = cV
+0 +cV
+1 +cV
+2 , B4 = cV
+0 −2cV
+1 +cV
+2 . If neglecting
+the SU(3) flavor breaking cV
+1 and cV
+2 terms, B1 = B2 = B3 = B4 = cV
+0 .
+Hadronic helicity amplitudes
+SU(3) IRA amplitudes
+H(D0 → K∗−ℓ+νℓ)
+B1V ∗
+cs
+H(D+ → K
+∗0ℓ+νℓ)
+B1V ∗
+cs
+H(D+
+s → φℓ+νℓ)
+�
+− cosθV
+�
+2/3 − sinθV /
+√
+3�
+B2V ∗
+cs
+H(D+
+s → ωℓ+νℓ)
+�
+− sinθV
+�
+2/3 + cosθV /
+√
+3�
+B2V ∗
+cs
+H(D0 → ρ−ℓ+νℓ)
+B3V ∗
+cd
+H(D+ → ρ0ℓ+νℓ)
+− 1
+√
+2 B3V ∗
+cd
+H(D+ → φℓ+νℓ)
+�
+cosθV /
+√
+6 − sinθV /
+√
+3�
+B3V ∗
+cd
+H(D+ → ωℓ+νℓ)
+�
+sinθV /
+√
+6 + cosθV /
+√
+3�
+B3V ∗
+cd
+H(D+
+s → K∗0ℓ+νℓ)
+B4V ∗
+cd
+do not use them to constrain the nonperturbative parameter cV
+0 . One can see that the prediction of B(D0 → ρ−µ+νµ)
+is agree with its experimental data within 3σ errors, nevertheless, the prediction of B(D+ → ωe+νe) still slightly
+larger than experimental data within 3σ errors. B(D+
+s → K∗0µ+νµ) and B(D+
+s → ωe+νe, ωµ+νµ) are predicted
+on the order of O(10−3) and O(10−5), respectively.
+And they could be measured in BESIII, LHCb and BelleII
+experiments. In the C4 case, as given in the sixth column of Tab. V, after considering both the hadronic momentum-
+transfer q2 dependence of the form factors and the SU(3) flavor breaking contributions, all SU(3) flavor symmetry
+predictions are consistent with their experimental data within 2σ errors. Among relevant not yet measured decays,
+B(D+
+s → K∗0µ+νµ) is calculated in the SM using light-cone sum rules [79] and in the relativistic quark model [7],
+B(D+
+s → K∗0µ+νµ) = (2.23 ± 0.32) × 10−3 [79] and 2.0 × 10−3 [7], and our predictions of B(D+
+s → K∗0µ+νµ) in the
+C1, C2, C3 and C4 cases are coincident with previous ones in Refs. [7, 79]. In addition, the lepton flavor universality
+parameters Rµ/e(D → V ℓ+νℓ) are also given in Tab. V. Since many terms are canceled in the ratios, these predictions
+of the lepton flavor universality parameters are quite accurate, and our predictions in all four cases are similar to each
+other.
+For the q2 dependence of the differential branching ratios of the D → V ℓ+νℓ decays with present experimental
+bounds, we only show the not yet measured processes D+ → φµ+νµ, D+
+s → ωµ+νµ and D+
+s → K∗0µ+νµ in Fig. 3.
+The differential branching ratios of D+ → φe+νe (D+
+s → ωe+νe) is similar to D+ → φµ+νµ (D+
+s → ωµ+νµ), so we
+do not shown them in Fig. 3. From Fig. 3, one can see that present experiment data give quite strong bounds on all
+differential branching ratios of D+ → φµ+νµ, D+
+s → ωµ+νµ and D+
+s → K∗0µ+νµ decays in the C1, C2 and C3 cases.
+The prediction of dB(D+ → φµ+νµ)/dq2 in the C4 case could be distinguished from ones in the C1,2,3 cases within
+the middle range of q2. And the error of dB(D+
+s → K∗0µ+νµ)/dq2 in the C4 case is obviously larger than ones in
+C1,2,3 cases.
+The forward-backward asymmetries Aℓ
+F B, the lepton-side convexity parameters Cℓ
+F , the longitudinal polarizations
+P ℓ
+L, the transverse polarizations P ℓ
+T , the lepton spin asymmetries Aλ and the longitudinal polarization fractions of the
+final vector mesons FL with two ways of integration have also been predicted in the four cases. Since many theoretical
+
+14
+TABLE V: Branching ratios of the D → V ℓ+ν within 2σ errors.
+†The experimental data of B(D+ → ωe+νe) and B(D0 →
+ρ−µ+νµ) from PDG [1] are not used in the C1,2,3 cases.
+Branching ratios
+Exp. data
+Ones in C1
+Ones in C2
+Ones in C3
+Ones in C4
+B(D+ → K
+∗0e+νe)(×10−2)
+5.40 ± 0.20
+5.44 ± 0.15
+5.42 ± 0.18
+5.36 ± 0.08
+5.44 ± 0.16
+B(D+ → ρ0e+νe)(×10−3)
+2.18+0.34
+−0.50
+2.31 ± 0.07
+2.39 ± 0.13
+2.33 ± 0.05
+1.83 ± 0.15
+B(D+ → ωe+νe)(×10−3)
+1.69 ± 0.22
+2.24 ± 0.07†
+2.33 ± 0.12†
+2.26 ± 0.04†
+1.77 ± 0.14
+B(D+ → φe+νe)(×10−7)
+< 130
+3.13 ± 0.12
+3.11 ± 0.19
+3.07 ± 0.07
+2.38 ± 0.23
+B(D0 → K∗−e+νe)(×10−2)
+2.15 ± 0.32
+2.12 ± 0.09
+2.13 ± 0.10
+2.08 ± 0.06
+2.13 ± 0.10
+B(D0 → ρ−e+νe)(×10−3)
+1.50 ± 0.24
+1.79 ± 0.08
+1.86 ± 0.11
+1.80 ± 0.06
+1.41 ± 0.13
+B(D+
+s → φe+νe)(×10−2)
+2.39 ± 0.32
+2.46 ± 0.12
+2.43 ± 0.14
+2.40 ± 0.10
+2.39 ± 0.32
+B(D+
+s → ωe+νe)(×10−5)
+< 200
+2.45 ± 0.13
+2.56 ± 0.20
+2.47 ± 0.10
+2.49 ± 0.38
+B(D+
+s → K∗0e+νe)(×10−3)
+2.15 ± 0.56
+2.17 ± 0.10
+2.25 ± 0.13
+2.17 ± 0.08
+2.15 ± 0.56
+B(D+ → K
+∗0µ+νµ)(×10−2)
+5.27 ± 0.30
+5.12 ± 0.15
+5.13 ± 0.16
+5.05 ± 0.08
+5.12 ± 0.15
+B(D+ → ρ0µ+νµ)(×10−3)
+2.4 ± 0.8
+2.19 ± 0.07
+2.29 ± 0.13
+2.22 ± 0.04
+1.74 ± 0.14
+B(D+ → ωµ+νµ)(×10−3)
+1.77 ± 0.42
+2.13 ± 0.06
+2.23 ± 0.12
+2.15 ± 0.04
+1.68 ± 0.13
+B(D+ → φµ+νµ)(×10−7)
+· · ·
+2.89 ± 0.11
+2.89 ± 0.17
+2.84 ± 0.07
+2.20 ± 0.21
+B(D0 → K∗−µ+νµ)(×10−2)
+1.89 ± 0.48
+1.99 ± 0.09
+2.01 ± 0.09
+1.96 ± 0.06
+2.01 ± 0.10
+B(D0 → ρ−µ+νµ)(×10−3)
+1.35 ± 0.26
+1.70 ± 0.07†
+1.78 ± 0.11†
+1.72 ± 0.06†
+1.34 ± 0.13
+B(D+
+s → φµ+νµ)(×10−2)
+1.9 ± 1.0
+2.30 ± 0.12
+2.29 ± 0.12
+2.25 ± 0.09
+2.24 ± 0.30
+B(D+
+s → ωµ+νµ)(×10−5)
+· · ·
+2.34 ± 0.12
+2.47 ± 0.19
+2.37 ± 0.09
+2.38 ± 0.36
+B(D+
+s → K∗0µ+νµ)(×10−3)
+· · ·
+2.06 ± 0.10
+2.15 ± 0.13
+2.07 ± 0.08
+2.05 ± 0.53
+Rµ/e(D+ → K
+∗0ℓ+νℓ)
+0.939 ± 0.001
+0.944 ± 0.004
+0.941 ± 0.001
+0.941 ± 0.001
+Rµ/e(D+ → ρ0ℓ+νℓ)
+0.950 ± 0.001
+0.956 ± 0.005
+0.952 ± 0.001
+0.952 ± 0.001
+Rµ/e(D+ → ωℓ+νℓ)
+0.950 ± 0.001
+0.956 ± 0.005
+0.952 ± 0.001
+0.952 ± 0.001
+Rµ/e(D+ → φℓ+νℓ)
+0.923 ± 0.001
+0.928 ± 0.005
+0.925 ± 0.001
+0.925 ± 0.001
+Rµ/e(D0 → K∗−ℓ+νℓ)
+0.939 ± 0.001
+0.944 ± 0.004
+0.941 ± 0.001
+0.941 ± 0.001
+Rµ/e(D0 → ρ−ℓ+νℓ)
+0.950 ± 0.001
+0.956 ± 0.005
+0.952 ± 0.001
+0.952 ± 0.001
+Rµ/e(D+
+s → φℓ+νℓ)
+0.937 ± 0.001
+0.942 ± 0.004
+0.939 ± 0.001
+0.939 ± 0.001
+Rµ/e(D+
+s → ωℓ+νℓ)
+0.957 ± 0.001
+0.963 ± 0.004
+0.959 ± 0.001
+0.959 ± 0.001
+Rµ/e(D+
+s → K∗0ℓ+νℓ)
+0.949 ± 0.001
+0.955 ± 0.005
+0.951 ± 0.001
+0.951 ± 0.001
+uncertainties are canceled in the ratios, these predictions are very accurate. These predictions are similar to each
+other in the four cases, and we only list the results in the C3 case in Tabs. VI-VII for examples. One can see that
+the predictions are obviously different between two ways of q2 integration, and they are also quite different between
+D → V e+νe and D → V µ+νµ decays.
+The differential observables of D+
+s → K∗0ℓ+νℓ decays in the C3 case are displayed in Fig. 4. One can see that,
+in the low q2 ranges, the differential observables expect dFL(D+
+s → K∗0ℓ+νℓ)/dq2 are obviously different between
+decays with ℓ = e and ℓ = µ.
+
+15
+C
+3
+C
+4
+dB(D
++
+s
+K
+*0
++
+)/dq
+2
+ ( x10
+-3
+ )
+q
+2
+FIG. 3: The q2 dependence of the differential branching ratios for some not yet measured D → V µ+νµ decays with present
+experimental bounds.
+0.0
+0.4
+0.8
+1.2
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0.0
+0.0
+0.4
+0.8
+1.2
+-2.0
+-1.5
+-1.0
+-0.5
+0.0
+0.5
+0.0
+0.4
+0.8
+1.2
+0.0
+0.4
+0.8
+1.2
+0.0
+0.4
+0.8
+1.2
+-6
+-4
+-2
+0
+0.0
+0.4
+0.8
+1.2
+0.0
+0.4
+0.8
+1.2
+0.0
+0.4
+0.8
+1.2
+0.0
+0.4
+0.8
+1.2
+( f )
+( e )
+( d )
+( c )
+( b )
+( a )
+ 
+ 
+ e
+ 
+dA
+FB
+(D
++
+s
+K
+*0
+l
++
+l
+)/dq
+2
+q
+2
+ 
+ 
+ e
+ 
+dC
+l
+F
+(D
++
+s
+K
+*0
+l
++
+l
+)/dq
+2
+q
+2
+ 
+ 
+ e
+ 
+dP
+l
+L
+(D
++
+s
+K
+*0
+l
++
+l
+)/dq
+2
+q
+2
+ 
+ 
+ e: in unit of 10
+-3
+ 
+dP
+l
+T
+(D
++
+s
+K
+*0
+l
++
+l
+)/dq
+2
+q
+2
+ 
+ 
+ e
+ 
+dA
+(D
++
+s
+K
+*0
+l
++
+l
+)/dq
+2
+q
+2
+ 
+ 
+ e
+ 
+dF
+L
+(D
++
+s
+K
+*0
+l
++
+l
+)/dq
+2
+q
+2
+FIG. 4: The differential forward-backward asymmetries, differential lepton-side convexity parameters, differential longitudinal
+lepton polarizations and differential transverse lepton polarizations for the D+
+s → K0ℓ+νℓ decays in the C3 case.
+
+d.
+S
+0'4
+a.0
+8.0cqB(D)
+0.0
+8.0
+0
+0
+2
+8←S
+3(p)
+d
+S
+a.
+e.0
+S.1C2.
+0.0
+8.0
+0
+H
+0
+X
+x"S
+3(c)
+a.0
+e.000
+S.0
+qB(D)
+S(0rx) "pbl(
+7
+e16
+TABLE VI: The forward-backward asymmetries Aℓ
+F B, the lepton-side convexity parameters Cℓ
+F , the longitudinal polarizations
+P ℓ
+L of the D → V ℓ+ν decays in the C3 case.
+Decay modes
+⟨Aℓ
+F B⟩
+Aℓ
+F B
+⟨Cℓ
+F ⟩
+Cℓ
+F
+⟨P ℓ
+L⟩
+P ℓ
+L
+D+ → K
+∗0e+νe
+−0.125 ± 0.006
+−0.190 ± 0.020
+−1.046 ± 0.019
+−0.500 ± 0.032
+0.786 ± 0.004
+1.000
+D+ → ρ0e+νe
+−0.130 ± 0.008
+−0.222 ± 0.024
+−1.052 ± 0.023
+−0.496 ± 0.041
+0.789 ± 0.004
+1.000
+D+ → ωe+νe
+−0.130 ± 0.008
+−0.220 ± 0.024
+−1.052 ± 0.023
+−0.497 ± 0.041
+0.789 ± 0.004
+1.000
+D+ → φe+νe
+−0.121 ± 0.005
+−0.164 ± 0.017
+−1.037 ± 0.015
+−0.500 ± 0.025
+0.784 ± 0.003
+1.000
+D0 → K∗−e+νe
+−0.125 ± 0.006
+−0.191 ± 0.020
+−1.046 ± 0.019
+−0.500 ± 0.032
+0.786 ± 0.004
+1.000
+D0 → ρ−e+νe
+−0.130 ± 0.008
+−0.221 ± 0.024
+−1.052 ± 0.023
+−0.497 ± 0.041
+0.789 ± 0.004
+1.000
+D+
+s → φe+νe
+−0.122 ± 0.006
+−0.176 ± 0.018
+−1.043 ± 0.016
+−0.500 ± 0.028
+0.786 ± 0.003
+1.000
+D+
+s → ωe+νe
+−0.130 ± 0.008
+−0.229 ± 0.025
+−1.057 ± 0.025
+−0.496 ± 0.044
+0.790 ± 0.004
+1.000
+D+
+s → K∗0e+νe
+−0.128 ± 0.007
+−0.207 ± 0.022
+−1.049 ± 0.021
+−0.495 ± 0.036
+0.789 ± 0.004
+1.000
+D+ → K
+∗0µ+νµ
+−0.284 ± 0.009
+−0.226 ± 0.019
+−0.466 ± 0.021
+−0.395 ± 0.028
+0.514 ± 0.017
+0.886 ± 0.002
+D+ → ρ0µ+νµ
+−0.292 ± 0.011
+−0.252 ± 0.023
+−0.491 ± 0.027
+−0.405 ± 0.037
+0.524 ± 0.020
+0.903 ± 0.002
+D+ → ωµ+νµ
+−0.292 ± 0.011
+−0.251 ± 0.022
+−0.490 ± 0.027
+−0.405 ± 0.037
+0.524 ± 0.020
+0.902 ± 0.002
+D+ → φµ+νµ
+−0.277 ± 0.008
+−0.206 ± 0.016
+−0.433 ± 0.016
+−0.376 ± 0.021
+0.503 ± 0.014
+0.864 ± 0.002
+D0 → K∗−µ+νµ
+−0.284 ± 0.009
+−0.226 ± 0.019
+−0.466 ± 0.021
+−0.395 ± 0.029
+0.514 ± 0.017
+0.886 ± 0.002
+D0 → ρ−µ+νµ
+−0.292 ± 0.011
+−0.252 ± 0.023
+−0.490 ± 0.027
+−0.405 ± 0.037
+0.524 ± 0.020
+0.902 ± 0.002
+D+
+s → φµ+νµ
+−0.277 ± 0.008
+−0.213 ± 0.017
+−0.459 ± 0.018
+−0.391 ± 0.024
+0.514 ± 0.015
+0.882 ± 0.002
+D+
+s → ωµ+νµ
+−0.291 ± 0.012
+−0.257 ± 0.024
+−0.509 ± 0.029
+−0.414 ± 0.041
+0.531 ± 0.021
+0.913 ± 0.002
+D+
+s → K∗0µ+νµ
+−0.286 ± 0.010
+−0.239 ± 0.021
+−0.485 ± 0.024
+−0.402 ± 0.033
+0.525 ± 0.018
+0.900 ± 0.002
+TABLE VII: The transverse polarizations P ℓ
+T , the lepton spin asymmetries Aλ and the longitudinal polarization fractions of
+the final vector mesons FL of the D → V ℓ+ν decays in the C3 case.
+Decay modes
+⟨P ℓ
+T ⟩
+P e
+T (×10−3)
+P µ
+T
+⟨Aλ⟩
+Aλ
+⟨FL⟩
+FL
+D+ → K
+∗0e+νe
+−0.251 ± 0.004
+−1.205 ± 0.066
+1.000
+1.000
+0.905 ± 0.010
+0.556 ± 0.014
+D+ → ρ0e+νe
+−0.249 ± 0.005
+−1.040 ± 0.072
+1.000
+1.000
+0.907 ± 0.012
+0.554 ± 0.018
+D+ → ωe+νe
+−0.249 ± 0.005
+−1.049 ± 0.073
+1.000
+1.000
+0.907 ± 0.012
+0.554 ± 0.018
+D+ → φe+νe
+−0.254 ± 0.003
+−1.417 ± 0.061
+1.000
+1.000
+0.902 ± 0.008
+0.556 ± 0.011
+D0 → K∗−e+νe
+−0.251 ± 0.004
+−1.206 ± 0.067
+1.000
+1.000
+0.905 ± 0.010
+0.556 ± 0.014
+D0 → ρ−e+νe
+−0.249 ± 0.005
+−1.045 ± 0.073
+1.000
+1.000
+0.907 ± 0.012
+0.554 ± 0.018
+D+
+s → φe+νe
+−0.251 ± 0.004
+−1.255 ± 0.060
+1.000
+1.000
+0.904 ± 0.009
+0.555 ± 0.012
+D+
+s → ωe+νe
+−0.247 ± 0.005
+−0.953 ± 0.071
+1.000
+1.000
+0.908 ± 0.013
+0.554 ± 0.020
+D+
+s → K∗0e+νe
+−0.248 ± 0.004
+−1.075 ± 0.066
+1.000
+1.000
+0.905 ± 0.011
+0.553 ± 0.016
+D+ → K
+∗0µ+νµ
+−0.454 ± 0.022
+−0.156 ± 0.012
+0.935 ± 0.005
+0.928 ± 0.002
+0.775 ± 0.019
+0.557 ± 0.014
+D+ → ρ0µ+νµ
+−0.452 ± 0.026
+−0.139 ± 0.014
+0.944 ± 0.006
+0.937 ± 0.002
+0.782 ± 0.023
+0.555 ± 0.018
+D+ → ωµ+νµ
+−0.452 ± 0.026
+−0.140 ± 0.014
+0.944 ± 0.006
+0.937 ± 0.002
+0.782 ± 0.023
+0.555 ± 0.018
+D+ → φµ+νµ
+−0.455 ± 0.018
+−0.175 ± 0.011
+0.924 ± 0.005
+0.915 ± 0.002
+0.763 ± 0.015
+0.557 ± 0.011
+D0 → K∗−µ+νµ
+−0.454 ± 0.022
+−0.156 ± 0.012
+0.935 ± 0.005
+0.927 ± 0.002
+0.775 ± 0.019
+0.557 ± 0.014
+D0 → ρ−µ+νµ
+−0.452 ± 0.026
+−0.140 ± 0.014
+0.944 ± 0.006
+0.937 ± 0.002
+0.782 ± 0.023
+0.555 ± 0.018
+D+
+s → φµ+νµ
+−0.454 ± 0.019
+−0.162 ± 0.011
+0.934 ± 0.005
+0.925 ± 0.002
+0.771 ± 0.016
+0.557 ± 0.012
+D+
+s → ωµ+νµ
+−0.452 ± 0.027
+−0.131 ± 0.014
+0.950 ± 0.005
+0.943 ± 0.002
+0.788 ± 0.024
+0.555 ± 0.019
+D+
+s → K∗0µ+νµ
+−0.451 ± 0.023
+−0.143 ± 0.012
+0.943 ± 0.005
+0.936 ± 0.002
+0.779 ± 0.021
+0.555 ± 0.016
+
+17
+C.
+D → Sℓ+νℓ decays
+For D → Sℓ+νℓ decays, the two quark and the four quark scenarios for the scalar mesons below or near 1 GeV are
+considered. The hadronic helicity amplitudes for the D → Sℓ+νℓ decays are given in Tab. VIII, in which the CKM
+matrix element Vcs and Vcd information are kept for comparing conveniently. There are four (five) nonperturbative
+parameters E1,2,3,4 (E′
+1,2,3,4,5) in the two quark (four quark) picture.
+After ignoring the SU(3) flavor breaking
+contributions, only one nonperturbative parameter E1 = E2 = E3 = E4 = cS
+0 or E′
+1 = E′
+2 = E′
+3 = E′
+4 = E′
+5 = c′S
+0
+relates all decay amplitudes in the two quark or the four quark picture, respectively.
+Unlike many measured decay modes in the D → Pℓ+νℓ and D → V ℓ+νℓ decays, among these D → Sℓ+νℓ decays,
+only D+
+s → f0(980)e+νe decay has been measured, and its branching ratio with 2σ errors is [1]
+B(D+
+s → f0(980)e+νe) = (2.3 ± 0.8) × 10−3.
+(48)
+In addition, the branching ratios of the D → P1P2ℓ+νℓ decays with the light scalar resonances can be obtained by
+using B(D → Sℓ+νℓ) and B(S → P1P2), and the detail analysis can been found in Ref. [80]. Five branching ratios
+TABLE VIII: The hadronic helicity amplitudes for D → Sℓ+ν decays including both the SU(3) flavor symmetry and the SU(3)
+flavor breaking contributions. In the two quark picture of the scalar mesons, E1 ≡ cS
+0 + cS
+1 − 2cS
+2 , E2 ≡ cS
+0 − 2cS
+1 − 2cS
+2 ,
+E3 ≡ cS
+0 + cS
+1 + cS
+2 , E4 ≡ cS
+0 − 2cS
+1 + cS
+2 . E1 = E2 = E3 = E4 = cS
+0 if neglecting the SU(3) flavor breaking cS
+1 and cS
+2 terms. In
+the four quark picture of the scalar mesons, E′
+1 ≡ c′S
+0 + c′S
+1 − 2c′S
+2 + c′S
+3 , E′
+2 ≡ c′S
+0 − 2c′S
+1 − 2c′S
+2 + c′S
+3 , E′
+3 ≡ c′S
+0 + c′S
+1 + c′S
+2 − 2c′S
+3 ,
+E′
+4 ≡ c′S
+0 + c′S
+1 + c′S
+2 + c′S
+3 , E′
+5 ≡ c′S
+0 − 2c′S
+1 + c′S
+2 + c′S
+3 , E′
+1 = E′
+2 = E′
+3 = E′
+4 = E′
+5 = c′S
+0 if neglecting the SU(3) flavor breaking
+c′S
+1 , c′S
+2 and c′S
+3 terms.
+Hadronic helicity amplitudes
+ones for two-quark scenario
+ones for four-quark scenario
+H(D0 → K−
+0 ℓ+νℓ)
+E1V ∗
+cs
+E′
+1V ∗
+cs
+H(D+ → K
+0
+0ℓ+νℓ)
+E1V ∗
+cs
+E′
+1V ∗
+cs
+H(D+
+s → f0ℓ+νℓ)
+E2V ∗
+cs
+√
+2E′
+2V ∗
+cs
+H(D+
+s → f0(980)ℓ+νℓ)
+cosθS E2V ∗
+cs
+√
+2cosφS E′
+2V ∗
+cs
+H(D+
+s → f0(500)ℓ+νℓ)
+−sinθS E2V ∗
+cs
+−
+√
+2sinφS E′
+2V ∗
+cs
+H(D0 → a−
+0 ℓ+νℓ)
+E3V ∗
+cd
+E′
+3V ∗
+cd
+H(D+ → a0
+0ℓ+νℓ)
+− 1
+√
+2E3V ∗
+cd
+− 1
+√
+2E′
+3V ∗
+cd
+H(D+ → f0ℓ+νℓ)
+0
+1
+√
+2E′
+3V ∗
+cd
+H(D+ → σℓ+νℓ)
+1
+√
+2E3V ∗
+cd
+E′
+4V ∗
+cd
+H(D+ → f0(980)ℓ+νℓ)
+1
+√
+2sinθS E3V ∗
+cd
+( 1
+√
+2E′
+3cosφS + E′
+4sinφS)V ∗
+cd
+H(D+ → f0(500)ℓ+νℓ)
+1
+√
+2cosθS E3V ∗
+cd
+(− 1
+√
+2E′
+3sinφS + E′
+4cosφS)V ∗
+cd
+H(D+
+s → K0
+0ℓ+νℓ)
+E4V ∗
+cd
+E′
+5V ∗
+cd
+
+18
+and two upper limits of B(D → Sℓ+νℓ, S → P1P2) have been measured, and the data within 2σ errors are
+B(D+
+s → f0(980)e+νe, f0(980) → π+π−) = (1.30 ± 0.63) × 10−3 [81],
+B(D+
+s → f0(980)e+νe, f0(980) → π0π0) = (7.9 ± 2.9) × 10−4 [82],
+B(D0 → a0(980)−e+νe, a0(980)− → ηπ−) = (1.33+0.68
+−0.60) × 10−4
+[1],
+B(D+ → a0(980)0e+νe, a0(980)0 → ηπ0) = (1.7+1.6
+−1.4) × 10−4
+[1],
+B(D+ → f0(500)e+νe, f0(500) → π+π−) = (6.3 ± 1.0) × 10−4
+[1],
+B(D+ → f0(980)e+νe, f0(980) → π+π−) < 2.8 × 10−5
+[83],
+B(D+
+s → f0(500)e+νe, f0(500) → π0π0) < 6.4 × 10−4 [82].
+(49)
+Two cases S1 and S2 will be considered in the D → Sℓ+νℓ decays.
+In S1 case, only experimental datum of
+B(D+
+s → f0(980)e+νe) is used to constrain one parameter cS
+0 or c′S
+0 and then predict other not yet measured branching
+ratios. The numerical results of B(D → Sℓ+ν) in S1 case are given in the 2-4th and 8th columns of Tab. IX. In the
+S2 case, the experimental data of both B(D+
+s → f0(980)e+νe) in Eq. (48) and B(D → Sℓ+νℓ, S → P1P2) in Eq. (49)
+will be used to constrain the parameter cS
+0 or c′S
+0 . The predictions of B(D → Sℓ+ν) in S2 case are listed in the 5-7th
+and 9th columns of Tab. IX. Our comments on the results in the S1,2 cases are as follows.
+• Results in the two quark picture: In the two quark picture, the three possible ranges of the mixing angle,
+25◦ < θS < 40◦, 140◦ < θS < 165◦ and −30◦ < θS < 30◦ [58, 68] have been analyzed. In S1 case, using the
+data of B(D+
+s → f0(980)e+νe), many predictions of B(D → Sℓ+ν) are obtained. As given in the 2-4th columns
+of Tab. IX, one can see that the predictions with 25◦ < θS < 40◦ are similar to ones with 140◦ < θS < 165◦,
+the predictions with −30◦ < θS < 30◦ are slightly different from the first two, and the errors of predictions are
+quite large. After adding the experimental bounds of B(D → Sℓ+νℓ, S → P1P2), as given in the 5-7th columns
+of Tab. IX, the three possible ranges of the mixing angle θS are obviously constrained, and they reduce to
+25◦ < θS < 35◦, 144◦ < θS < 158◦ and 22◦ ≤ |θS| ≤ 30◦, respectively. In addition, the error of every prediction
+become smaller by adding the experimental bounds of B(D → Sℓ+νℓ, S → P1P2).
+• Results in the four quark picture: The predictions in the four quark picture are listed in the 8-9th columns
+of Tab. IX. The majority of predictions in four quark picture are smaller than corresponding ones in two quark
+picture. Strong coupling constants g′
+4 and g4 are appeared in S → P1P2 decays with the four quark picture
+of light scalar mesons. At present, we only can determine
+�� g′
+4
+g4
+�� from the S → P1P2 decays. The results of
+involved decays with both g′
+4
+g4 > 0 and g′
+4
+g4 < 0 are given in the 9th column of Tab. IX, and one can see that,
+except B(D+
+s → f0(500)e+νe) and B(D+
+s → f0(980)µ+νµ), the other involved branching ratios are not obviously
+affected by the choice of g′
+4
+g4 > 0 or g′
+4
+g4 < 0. The errors of the branching ratio predictions are obviously reduced
+by the experimental bounds of B(D → Sℓ+νℓ, S → P1P2).
+• Comparing with previous predictions: Previous predictions are listed in the last column of Tab.
+IX.
+B(D+
+s → f0(500)e+νe), B(D+
+s → f0(500)µ+νµ) and B(D+ → f0(500)µ+νµ) are predicted for the first time. Our
+predictions of B(D+
+s → f0(980)µ+νµ), B(D+ → a0
+0e+νe), B(D+ → f0(980)e+νe), B(D+ → f0(500)e+νe) and
+B(D+ → a0
+0µ+νµ) are consistent with previous predictions in Refs. [78, 84, 85]. Our other predictions are about
+one order smaller or one order larger than previous ones in Refs. [67, 86].
+
+19
+TABLE IX:
+Branching ratios of D → Sℓ+ν decays within 2σ errors. As given in Ref. [80], g′
+4 and g4 are strong coupling constants obtained by the SU(3) flavor
+symmetry in S → P1P2 decays, adenotes the results with
+g′
+4
+g4 > 0, and bdenotes ones with
+g′
+4
+g4 < 0, †denotes the results with two quark picture, and ‡denotes the results
+with four quark picture.
+Branching ratios
+ones for 2q state in S1
+ones for 2q state in S2
+ones for 4q
+ones for 4q
+Previous ones
+[25◦, 40◦]
+[140◦, 165◦]
+[−30◦, 30◦]
+[25◦, 35◦]
+[144◦, 158◦]
+22◦ ≤ |θS| ≤ 30◦
+state in S1
+state in S2
+B(D0 → K−
+0 e+νe)(×10−3)
+3.38 ± 2.12
+3.18 ± 2.05
+2.57 ± 1.58
+3.02 ± 1.11
+3.00 ± 1.10
+2.98 ± 1.05
+1.11 ± 0.63
+1.25 ± 0.45
+0.103 ± 0.115† [67]
+B(D+ → K
+0
+0e+νe)(×10−3)
+8.66 ± 5.55
+7.99 ± 5.02
+7.02 ± 4.48
+7.74 ± 2.88
+7.78 ± 2.77
+7.68 ± 2.78
+2.85 ± 1.65
+3.36 ± 1.25
+38.8 ± 5.6† [67]
+B(D+
+s → f0(980)e+νe)(×10−3)
+2.30 ± 0.80
+2.30 ± 0.80
+2.30 ± 0.80
+2.58 ± 0.52
+2.57 ± 0.53
+2.71 ± 0.39
+2.30 ± 0.80
+2.49±0.61a
+2.54±0.56b
+2.1 ± 0.2† [78], 2+0.5†
+−0.4
+[84]
+B(D+
+s → f0(500)e+νe)(×10−3)
+6.73 ± 6.11
+5.98 ± 5.75
+3.25 ± 3.25
+1.49 ± 0.43
+1.45 ± 0.46
+1.42 ± 0.50
+0.37 ± 0.37
+0.31±0.31a
+0.17±0.17b
+B(D0 → K−
+0 µ+νµ)(×10−3)
+2.90 ± 1.84
+2.73 ± 1.77
+2.20 ± 1.36
+2.59 ± 0.97
+2.57 ± 0.96
+2.56 ± 0.92
+0.95 ± 0.54
+1.09 ± 0.39
+0.103 ± 0.115† [67]
+B(D+ → K
+0
+0µ+νµ)(×10−3)
+7.46 ± 4.81
+6.87 ± 4.33
+6.04 ± 3.88
+6.65 ± 2.52
+6.69 ± 2.43
+6.59 ± 2.43
+2.45 ± 1.43
+2.89 ± 1.09
+38.8 ± 5.6† [67]
+B(D+
+s → f0(980)µ+νµ)(×10−3)
+1.95 ± 0.70
+1.95 ± 0.70
+1.95 ± 0.69
+2.20 ± 0.45
+2.20 ± 0.45
+2.32 ± 0.33
+1.95 ± 0.70
+2.12±0.54a
+2.16±0.49b
+2.1 ± 0.2† [78]
+B(D+
+s → f0(500)µ+νµ)(×10−3)
+6.21 ± 5.66
+5.53 ± 5.32
+3.01 ± 3.01
+1.33 ± 0.39
+1.31 ± 0.43
+1.28 ± 0.46
+0.34 ± 0.34
+0.29±0.29a
+0.16±0.16b
+B(D0 → a−
+0 e+νe)(×10−5)
+9.99 ± 6.54
+9.56 ± 6.50
+8.34 ± 5.67
+9.22 ± 3.98
+9.09 ± 3.65
+9.17 ± 3.58
+3.42 ± 2.06
+4.32 ± 1.17
+16.8±1.5† [78], 40.8+13.7†
+−12.2 [86],
+24.4±3.0† [67]
+B(D+ → a0
+0e+νe)(×10−5)
+13.09 ± 8.62
+12.62 ± 8.67
+10.89 ± 7.35
+12.09 ± 5.19
+11.81 ± 4.71
+11.97 ± 4.66
+4.49 ± 2.71
+5.68 ± 1.52
+21.8±3.8† [78], 54.0+17.8†
+−15.9 [86]
+6∼8†[85], 5∼5.4‡[85]
+B(D+ → f0(980)e+νe)(×10−5)
+3.92 ± 2.92
+3.48 ± 3.13
+1.59 ± 1.59
+2.62 ± 0.82
+2.52 ± 0.94
+2.40 ± 0.80
+3.14 ± 1.98
+3.35±1.80a
+3.89±1.35b
+7.78±0.68† [78], 5.7±1.3† [87]
+0.4∼3.5†[85], 1.9∼6.3‡[85]
+B(D+ → f0(500)e+νe)(×10−4)
+4.05 ± 3.20
+4.08 ± 3.10
+4.21 ± 3.28
+2.16 ± 0.96
+2.59 ± 1.38
+2.70 ± 1.28
+4.97 ± 4.13
+4.97±3.34a
+4.95±3.36b
+0.4 ∼ 0.6†[85], 0.88 ∼ 1.4‡[85]
+B(D+
+s → K0
+0e+νe)(×10−4)
+3.73 ± 2.37
+3.41 ± 2.13
+2.99 ± 1.88
+3.35 ± 1.21
+3.32 ± 1.20
+3.35 ± 1.15
+1.25 ± 0.71
+1.43 ± 0.51
+26.5 ± 2.8† [67]
+B(D0 → a−
+0 µ+νµ)(×10−5)
+8.25 ± 5.45
+7.89 ± 5.42
+6.91 ± 4.75
+7.61 ± 3.37
+7.51 ± 3.10
+7.57 ± 3.04
+2.83 ± 1.72
+3.57 ± 0.99
+16.3 ± 1.4† [78], 24.4 ± 3.0† [67]
+B(D+ → a0
+0µ+νµ)(×10−5)
+10.83 ± 7.19
+10.44 ± 7.23
+9.04 ± 6.16
+10.00 ± 4.41
+9.76 ± 4.00
+9.89 ± 3.97
+3.73 ± 2.28
+4.69 ± 1.30
+21.2 ± 3.7† [78]
+B(D+ → f0(980)µ+νµ)(×10−5)
+3.23 ± 2.41
+2.88 ± 2.60
+1.32 ± 1.32
+2.15 ± 0.70
+2.09 ± 0.78
+1.99 ± 0.66
+2.56 ± 1.62
+2.74±1.49a
+3.20±1.14b
+7.87 ± 0.67† [78]
+B(D+ → f0(500)µ+νµ)(×10−4)
+3.69 ± 2.96
+3.71 ± 2.86
+3.84 ± 3.04
+1.92 ± 0.88
+2.32 ± 1.27
+2.42 ± 1.19
+4.54 ± 3.81
+4.52±3.10a
+4.49±3.12b
+B(D+
+s → K0
+0µ+νµ)(×10−4)
+3.28 ± 2.10
+3.00 ± 1.88
+2.62 ± 1.66
+2.94 ± 1.08
+2.91 ± 1.06
+2.94 ± 1.02
+1.10 ± 0.63
+1.26 ± 0.45
+26.5 ± 2.8† [67]
+
+20
+IV.
+Summary
+Many semileptonic D → P/V/Sℓ+νℓ decays have been measured, and these processes could be used to test the
+SU(3) flavor symmetry approach. In terms of the SU(3) flavor symmetry and the SU(3) flavor breaking, the amplitude
+relations have been obtained. Then using the present data of B(D → P/V/Sℓ+νℓ), we have presented a theoretical
+analysis of the D → P/V/Sℓ+νℓ decays. Our main results can be summarized as follows.
+• D → Pℓ+νℓ decays: Our predictions with the SU(3) flavor symmetry in the C1 case and the predictions after
+adding SU(3) flavor breaking contributions in the C4 case are quite consistent with all present experimental data
+of B(D → Pℓ+νℓ) within 2σ errors. In the C2 and C3 cases, our SU(3) flavor symmetry predictions are consistent
+with all present experimental data except B(D+ → π0ℓ+νℓ) and B(D0 → π−ℓ+νℓ), which are slight larger
+than their experiential upper limits. The not yet measured B(D+
+s → π0e+νe), B(D+ → η′µ+νµ), B(D+
+s →
+K0µ+νµ), B(D+
+s
+→ π0µ+νµ), B(D+
+s
+→ π0τ +ντ) and the lepton flavor universality parameters have been
+obtained. Moreover, the forward-backward asymmetries, the lepton-side convexity parameters, the longitudinal
+(transverse) polarizations of the final charged leptons with two ways of integration for the D → Pℓ+νℓ decays
+have been predicted. The q2 dependence of corresponding differential quantities of the D → Pℓ+νℓ decays in
+the C3 case have been displayed.
+• D → V ℓ+νℓ decays: As given in the C1, C2 and C3 cases, our SU(3) flavor symmetry predictions of B(D+ →
+ωe+νe) and B(D0 → ρ−µ+νµ) are slightly larger than its experimental upper limits, and other SU(3) flavor
+symmetry predictions are consistent with present data. After considering the SU(3) flavor breaking effects, as
+given in the C4 case, all predictions are consistent with present data. The not yet measured or not yet well
+measured branching ratios of D+ → φe+νe, D+
+s → ωe+νe, D+ → φµ+νµ, D+
+s → ωµ+νµ, and D+
+s → K∗0µ+νµ
+have been predicted. The q2 dependence of corresponding differential quantities of the D → V ℓ+νℓ decays in
+the C3 case have also been displayed.
+• D → Sℓ+νℓ decays:
+Among 18 D → Sℓ+νℓ decay modes, only B(D+
+s → f0(980)e+νe) has been measured, and
+this experimental datum has been used to constrain the SU(3) flavor symmetry parameter and then predict other
+not yet measured branching ratios. Furthermore, the relevant experimental bounds of B(D → Sℓ+νℓ, S → P1P2)
+have also been added. The two quark and the four quark scenarios for the light scalar mesons are considered,
+and the three possible ranges of the mixing angle θS in the two quark picture have been analyzed.
+The SU(3) flavor symmetry is approximate approach, and it can still provide very useful information. We have
+found that the SU(3) flavor symmetry approach works well in the semileptonic D → P/V ℓ+νℓ decays, and the SU(3)
+flavor symmetry predictions of the D → Sℓ+νℓ decays need to be further tested, and our predictions of the D → Sℓ+νℓ
+decays are useful for probing the structure of light scalar mesons. According to our predictions, some decay modes
+could be observed at BESIII, LHCb or BelleII in near future experiments.
+ACKNOWLEDGEMENTS
+The work was supported by the National Natural Science Foundation of China (12175088).
+
+21
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+

DdAzT4oBgHgl3EQfif0c/content/tmp_files/2301.01499v1.pdf.txt

@@ -0,0 +1,1739 @@
+Received: Added at production
+Revised: Added at production
+Accepted: Added at production
+DOI: xxx/xxxx
+ARTICLE TYPE
+Thermodynamic and transport properties of plasmas: low-density
+benchmarks
+G. Röpke
+Institut für Physik, Universität Rostock,
+18051 Rostock, Germany
+Correspondence
+Email: [email protected]
+Abstract
+Physical properties of plasmas such as equations of state and transport coefficients
+are expressed in terms of correlation functions, which can be calculated using various
+approaches (analytical theory, numerical simulations). The method of Green’s func-
+tions provides benchmark values for these properties in the low-density limit. For
+the equation of state and electrical conductivity, expansions with respect to density
+(virial expansions) are considered. Comparison of analytical results with numerical
+simulations is used to verify theory, to prove the accuracy of simulations, and to
+establish interpolation formulas.
+KEYWORDS:
+plasma equation of state, electrical conductivity, virial expansion, DFT-MD simulations, PIMC simula-
+tions
+1
+PLASMA PROPERTIES AND CORRELATION FUNCTIONS
+Plasmas consist of charged particles, number 푁푖 of species 푖 in the volume Ω, which interact via the Coulomb law. If we denote
+the charge of the component 푖 by 푍푖푒, we obtain (휖0 is the permittivity of the vacuum)
+푉 Coul
+푖푗
+(푟) =
+푍푖푍푗푒2
+4휋휖0푟 .
+(1)
+In general, an additional short-range interaction may occur. Examples are the homogeneous electron gas (uniform electron
+gas UEG), where the electrons move over a positively charged background to realize charge neutrality, or the two-component
+Hydrogen plasma, consisting of electrons and protons, where the particle density is 푛푒 = 푛푝 to maintain charge neutrality. In
+thermodynamic equilibrium, the state of the plasma is determined by the temperature 푇 in addition to the densities 푛푖 = 푁푖∕Ω
+of the components or the corresponding chemical potentials 휇푖. The relationships between the various state variables such as
+internal energy 푈, free energy 퐹, entropy 푆, pressure 푃 , etc. are called equations of state (EoS). All thermodynamic properties
+can be derived from a thermodynamic potential, for example is 퐹(Ω, 푁푖, 푇 ) as function of Ω, 푁푖, 푇 a thermodynamic potential.
+Statistical physics allows to calculate the thermodynamic properties from the microscopic properties, i.e. from the Hamiltonian
+퐻 = 퐻kin + 푉 , with kinetic energy 퐻kin = ∑
+푖
+∑푁푖
+푘 푝2
+푖,푘∕2푚푖 and potential energy 푉 = (1∕2) ∑
+푖,푘≠푗,푙 푉 (퐫푖,푘 − 퐫푗,푙). To calculate
+physical quantities, various expressions can be used. For example, for classical systems we can start from the well-known
+partition function 푍can(Ω, 푁푖, 푇 ) with 퐹(Ω, 푁푖, 푇 ) = −푘퐵푇 ln 푍can(Ω, 푁푖, 푇 ). For quantum systems, it is convenient to work
+with the grand canonical ensemble defined by 훽 = 1∕푘퐵푇 and the chemical potentials 휇푖. Then the single-particle distribution
+functions for the ideal quantum system (푉 = 0) have a simple form, the Fermi or Bose distribution. In second quantization,
+we introduce 푎+
+푖,푘, 푎푖,푘 as a creation or annihilation operator for particles of species 푖 in the quantum state 푘 = {ℏ퐤, 휎}, which
+arXiv:2301.01499v1  [physics.plasm-ph]  4 Jan 2023
+
+2
+denotes momentum vector and spin. The occupation number of this quantum state is given as
+푓푖,푘 = ⟨푎+
+푖,푘푎푖,푘⟩ =
+1
+푍gr.can
+Tr
+{
+푒−훽(퐻−∑
+푖 휇푖푁푖)푎+
+푖,푘푎푖,푘
+}
+,
+푍gr.can = Tr푒−훽(퐻−∑
+푖 휇푖푁푖)
+(2)
+with 퐻 as the Hamiltonian in second quantization and 푁푖 = ∑
+푘 푎+
+푖,푘푎푖,푘. The relation between the densities and the chemical
+potentials is given as follows.
+푛푖(푇 , {휇푗}) = 1
+Ω⟨푁푖⟩ = 1
+Ω
+∑
+푘
+푓푖,푘.
+(3)
+It is convenient to introduce a 휏-dependent correlation function as a generalization of (2) that contains dynamic information,
+⟨푎+
+푗,푙푒휏(퐻−∑
+푖 휇푖푁푖)푎푖,푘푒−휏(퐻−∑
+푖 휇푖푁푖)⟩ =
+∞
+∫
+−∞
+푑휔
+2휋 푒−휔휏퐼푖푘,푗푙(휔).
+(4)
+The spectral density 퐼푖푘,푗푙(휔) is related to the spectral function 퐴푖푘,푗푙(휔) = (1+푒훽휔)퐼푖푘,푗푙(휔) (Fermi statistics). An exact expression
+for the EoS is found if the spectral function is known,
+푛푖(푇 , {휇푗}) = 1
+Ω
+∑
+푘
+∞
+∫
+−∞
+푑휔
+2휋
+1
+푒훽휔 + 1퐴푖푘,푖푘(휔).
+(5)
+The spectral function which is diagonal with respect to {푖, 푘} for a homogeneous system, is related to the self-energy Σ푖,푘(푧) for
+which a systematic evaluation applying diagram techniques is possible, see1,2:
+퐴푖푘(휔) =
+2ImΣ푖,푘(휔 − 푖0)
+[휔 − 휖푖,푘 − ReΣ푖,푘(휔)]2 + [ImΣ푖,푘(휔 − 푖0)]2 ,
+(6)
+휖푖,푘 = ℏ2푘2∕2푚푖 − 휇푖 is the kinetic energy shifted by the chemical potential.
+The electrical conductivity 휎(푇 , 푛) of low-density plasmas was first calculated in the framework of kinetic theory. In a seminal
+work3, Spitzer and Härm determined 휎 of the fully ionized Hydrogen plasma by solving a Fokker-Planck equation. To calculate
+휎(푇 , 푛) in a wide range of temperature 푇 and particle density 푛, a quantum statistical many-particle theory is needed that
+describes screening, correlations, and degeneracy effects in a systematic way. A generalized linear response theory4,5,6 has been
+elaborated that expresses transport coefficients in terms of equilibrium correlation functions (fluctuation-dissipation theorems).
+An example is the Kubo formula7 which relates the transport coefficient 휎 to the electron current-current correlation function,
+휎(푇 , 푛) =
+푒2
+푚2
+푒푘퐵푇 Ω⟨푃 ; 푃 ⟩푖휖
+(7)
+with the total momentum of the electrons 푃 = ∑
+푘 ℏ푘푥푎+
+푒,푘푎푒,푘 in 푥 direction (the small ion contribution to the electrical current
+may be added). The thermodynamic correlation function is the Laplace transform of the Kubo scalar product (the particle number
+is assumed to commute with the observables),
+⟨퐴; 퐵⟩푧 =
+∞
+∫
+0
+푑푡 푒푖푧푡 1
+훽
+훽
+∫
+0
+푑휏⟨푒(푖∕ℏ)(푡−푖ℏ휏)퐻퐴푒−(푖∕ℏ)(푡−푖ℏ휏)퐻퐵⟩ .
+(8)
+For more details on generalized linear response theory and the evaluation of correlation functions using the method of thermo-
+dynamic Green’s functions, see2. For the relationship between generalized linear response theory and kinetic theory, see8 and
+references therein.
+2
+EVALUATION OF CORRELATION FUNCTIONS
+The properties of plasmas are expressed in terms of correlation functions in thermodynamic equilibrium. Examples are thermo-
+dynamic properties (2) and transport properties (7). There are several methods to calculate these correlation functions. Exact
+solutions are known only for ideal quantum gases where there is no interaction potential 푉 . The equations of state are known,
+e.g., the pressure 푃 is expressed by Fermi integrals. At fixed temperature, the equation of state for ideal classical gases 푃 = 푛푘퐵푇
+is approximated by considering the limiting case of low density. For electrical conductivity, 휎 = ∞ is obtained because of
+conservation of total momentum. The resistivity follows as 휌 = 1∕휎 = 0 for charged ideal Fermi gases.
+
+3
+Correlations appear for the plasma Hamiltonian with complete interaction 푉 . No closed-form solutions are known, and we
+must perform approximations to solve this many-body problem. Here we discuss three possibilities:
+1. Perturbation expansion with respect to 푉 . We obtain analytic expressions for arbitrary orders of 푉 in terms of nonin-
+teracting equilibrium correlation functions, which can be easily evaluated using Wick’s theorem. However, we have no
+proof of the convergence of this series expansion and no error estimate. In order to make this analytical approach more
+efficient, the method of thermodynamic Green’s functions and Feynman diagram technique were elaborated1,2,9. Conver-
+gence is improved by performing partial summations corresponding to special concepts such as the introduction of the
+quasiparticle picture (self-energy Σ), screening of the potential (polarization function Π), or formation of bound states
+(Bethe-Salpeter equation). This leads to useful results for the properties of the plasma in a wide range of 푇 and 푛. However,
+as characteristic for perturbative approaches, exact results can be found only in some limiting cases.
+2. This drawback is eliminated by numerical simulations of the correlation functions that apply to arbitrary interaction
+strength. In Born-Oppenheimer approximation, density functional theory (DFT) for the electron system with given ion
+configuration and molecular dynamics (MD) for the ion system are applied to evaluate the correlation functions. Single-
+electron states are calculated by solving the Kohn-Sham equations. The total energy is obtained from the kinetic energy
+of a non-interacting reference system, the classical electron-electron interaction, and an exchange-correlation energy that
+includes, to a certain approximation, all unknown contributions.
+The DFT-MD approach has been successfully applied to calculate the thermodynamic properties of complex materials in
+a wide range of 푇 and 푛, which will not be reported here, see, e.g.,10,11,12,13 and the references given there. For electrical
+conductivity (7), the Kubo-Greenwood formula7,14
+Re [휎(휔)] =
+2휋푒2
+3푚2
+푒휔Ω
+∑
+푘
+푤푘
+푁
+∑
+푗=1
+푁
+∑
+푖=1
+3
+∑
+훼=1
+[푓(휖푗,푘) − 푓(휖푖,푘)]|⟨Ψ푗,푘| ̂푝훼|Ψ푖,푘⟩|2훿(휖푖,푘 − 휖푗,푘 − ℏ휔)
+(9)
+was used to calculate the frequency-dependent dynamic electrical conductivity 휎(휔) in the long-wavelength
+limit16,17,18,19,20,15. Kohn-Sham wave functions Ψ푖,푘 from density functional theory calculations are used to calculate the
+transition matrix elements of the momentum operator ̂푝훼. The Fermi-Dirac distribution 푓(휖) accounts for the average
+occupation at energy 휖, and the summation over momentum space 푘 contains the 푘-point weights 푤푘.
+Due to the finite size of the simulation box, the delta function in equation (9) must be approximated by a finite-width
+Gaussian, which also prevents the direct calculation of the dc conductivity at 휔 = 0. Therefore, the dynamic conductivity
+is extrapolated to the limit 휔 → 0 by a Drude fit,
+Re [휎(휔)] =
+푛푒2휈
+휈2 + 휔2 ,
+(10)
+where 휈 is the collision frequency. Thus, the calculated direct current conductivity depends on choosing the appropriate
+width for the Gaussian and finding a suitable range for the Drude-fitting to 휎(휔) calculated from equation (9). The last
+point can be improved by using a frequency-dependent collision frequency21.
+One of the main shortcomings of the DFT-MD approach is that the many-particle interaction is replaced by a mean-field
+potential. When using product wave functions for the many-electron system, correlations are excluded. The exchange-
+correlation energy density functional reflects the Coulomb interaction to some approximation, e.g., as it exists in the
+homogeneous electron gas, but becomes problematic in the low-density limit where correlations are important.
+3. In principle, an accurate evaluation of equilibrium correlation functions is possible using path-integral Monte Carlo
+(PIMC) simulations, see22,23,24 and references therein. The shortcomings of this approach at present are the relatively
+small number of particles (a few dozen), the sign problem for fermions, and the computational challenges in accurately
+computing path integrals. Instead of using an exchange-correlation energy density functional, 푒 − 푒 collisions are treated
+accurately. However, at present accurate calculations have only been performed for the uniform electron gas model in
+which the charge-compensating ion subsystem is replaced by a homogeneously charged jellium. The results presented
+in25 are shown below in sec. 5. High-precision calculations for the two-component Hydrogen plasma would be of interest
+for both thermodynamics and transport properties.
+
+4
+3
+GREEN’S FUNCTIONS AND FEYNMAN DIAGRAMS
+In quantum statistics, the method of thermodynamic Green’s functions has been worked out to evaluate correlation functions in
+thermodynamic equilibrium. For the ideal quantum gas, in which there is no interaction, all equilibrium correlation functions
+can be calculated using Wick’s theorem. For plasmas, we can perform a power series expansion with respect to the interaction
+strength according to the Dyson series. The terms of this perturbation expansion are represented by Feynman diagrams.
+The problem of the perturbation expansion is that the convergence property remains open, and we cannot anticipate that for
+the correlation functions a power series expansion with respect to the interaction strength is possible. A predetermined wrong
+analytical behavior near the singular case of ideal gases leads to divergencies which are avoided performing partial summations
+that can modify the analytic behavior. The most important partial summations are the quasiparticle concept associated with the
+introduction of the self-energy, the screening associated with the introduction of the polarization function, and the introduction
+of bound states performing partial summation of ladder diagrams. For instance, the Bethe-Salpeter equation for the two-particle
+Green function in ladder approximation corresponds to the solution of the two-body problem.
+From classical statistics, the Mayer cluster expansion is well known for short-range potentials is well known for the partition
+function, and the virial expansion in powers in 푛 is obtained. Because of the long-range nature of the Coulomb potential, this
+expansion in powers in 푛 is not possible for plasmas, the virial coefficients are divergent. Screening, i.e. partial summation of the
+so-called ring diagrams in quantum statistics, solves this convergence problem, and the expansion in powers of 푛1∕2 is possible.
+When considering the spectral function, the contribution of the free particles is replaced by the contribution of the quasiparticles,
+with the energies containing the Debye shift. To obtain the thermodynamic potentials 퐹 or 푃 Ω from the equation of state (5)
+we must perform integration over 휇 or 푛, respectively, and logarithmic terms may appear. In particular, for the free energy of
+the Hydrogen plasma, the virial expansion reads
+퐹(푇 , Ω, 푁) = Ω푘퐵푇 {푛 ln 푛 + [3∕2 ln(2휋ℏ2∕(푚푘퐵푇 )) − 1]푛
+−퐹0(푇 )푛3∕2 − 퐹1(푇 )푛2 ln 푛 − 퐹2(푇 )푛2 − 퐹3(푇 )푛5∕2 ln 푛 − 퐹4(푇 )푛5∕2 + (푛3 ln 푛)} .
+(11)
+see9,25 where expressions for the lowest virial coefficients 퐹푖 are also given. Details on the calculation of the EoS for Coulomb
+systems can be found in Ref.9 and will not be repeated here. The virial expansion for the uniform electron gas is discussed below
+in Sec. 5.
+Perturbation expansion and partial summations also apply to the evaluation of the correlation function (7) which is related to
+the electrical conductivity. In the lowest order of perturbation theory, where interactions are neglected, the total momentum of
+the electrons is conserved. As a consequence, the expression (7) becomes divergent, the ideal plasma shows no finite value for the
+conductivity. Partial summations, in particular the self-energy and vertex corrections, lead to finite values for the conductivity,
+see26. Analytical evaluation of the Kubo formula remains difficult and cumbersome.
+In contrast, it is possible to perform a virial expansion for the inverse conductivity 푅 = 1∕휎, expressed as a correlation function
+of the stochastic forces26. A generalized linear response theory was worked out that takes into account correlation functions
+of higher moments of the occupation number distribution4. In this way the relation to the kinetic theory was shown21. These
+correlation functions are also treated by the methods of Green functions, Feynman diagram techniques and partial summations,
+so that virial expansions can be carried out.
+The dc conductivity 휎(푛, 푇 ) is usually associated with a dimensionless function 휎∗(푛, 푇 ) according to
+휎(푛, 푇 ) = (푘퐵푇 )3∕2(4휋휖0)2
+푚1∕2
+푒
+푒2
+휎∗(푛, 푇 ).
+(12)
+We consider both 휎 and 휎∗ as a function of density 푛 at fixed temperature 푇 . In the limiting case of low density, the following
+virial expansion for the inverse conductivity 휌∗(푛, 푇 ) = 1∕휎∗(푛, 푇 ) was obtained from kinetic theory and generalized linear
+response theory4,5,6:
+휌∗(푛, 푇 ) = 휌1(푇 ) ln 1
+푛 + 휌2(푇 ) + 휌3(푇 ) 푛1∕2 ln 1
+푛 + (푛1∕2),
+(13)
+which begins with a logarithmic term. Values for the virial coefficients 휌푖(푇 ) are given below in Sec. 6.
+
+5
+4
+VIRIAL PLOTS
+Equilibrium properties, such as the correlation functions considered here, depend on a limited number of state variables. For
+the Hydrogen plasma, this are the temperature 푇 and the electron number density 푛 (for charge neutral plasmas, the ion (proton)
+number density is also 푛). For the uniform electron gas, we have the same variables. Instead of the ion subsystem a homo-
+geneously charged background (jellium model) is considered to establish charge neutrality. In the case of a many-component
+plasma, the independent partial densities 푛푖 (not connected by chemical reactions and charge neutrality) of the components are
+the state variables in addition to 푇 . We focus here on the two simple cases where the state variables are 푇 , 푛, and we study
+the correlation energy ̄푉 (푇 , 푛) of the uniform electron gas and the electrical conductivity 휎(푇 , 푛) of the Hydrogen plasma, in
+particular the resistivity 푅(푇 , 푛) = 1∕휎(푇 , 푛).
+It is convenient to introduce dimensionless variables instead of 푇 , 푛. We use atomic units with the Hartree energy
+퐸Ha =
+(
+푒2
+4휋휖0
+)2 푚
+ℏ2 = 27, 21137 eV = 2 Ry
+(14)
+and the Bohr radius
+푎퐵 = 4휋휖0
+푒2
+ℏ2
+푚 = 5.2918 × 10−11 m.
+(15)
+The density in atomic units is usually represented by the radius of a sphere containing an electron,
+푟푠 =
+( 3
+4휋푛
+)1∕3 1
+푎퐵
+.
+(16)
+The temperature is related to the energy 푘퐵푇 , so that 1 eV corresponds to 11604.6 K. We denote 푇eV as 푘퐵푇 measured in units
+of eV, 푇Ha in units of 퐸Ha, and 푇Ry in units of Ry so that
+푇Ha = 푘퐵푇
+퐸Ha
+= 2푇Ry = 27, 21137 푇eV.
+(17)
+Another well-known choice of dimensionless parameters is
+Γ =
+푒2
+4휋휖0푘퐵푇
+(4휋
+3 푛
+)1∕3
+,
+Θ = 2푚푘퐵푇
+ℏ2
+(3휋2푛)−2∕3.
+(18)
+The plasma parameter Γ characterises the ratio of potential to kinetic energy in the non-degenerate case, and the electron degen-
+eracy parameter Θ characterises the range in which the electrons are degenerate. Different sets of dimensionless parameters are
+related. Thus, PIMC calculations for specific parameter values of 푟푠, Θ are discussed in the following section, the corresponding
+plasma parameters 푛, 푇 are determined as follows,
+푛 = 3
+4휋
+1
+(푟푠푎퐵)3 ,
+푘퐵푇 = 퐸Ha
+1
+2
+(9휋
+4
+)2∕3 Θ
+푟2
+푠
+(19)
+with 퐸Ha∕푘퐵 = 315777.1 K.
+The dc conductivity 휎(푛, 푇 ) is also associated with a dimensionless function 휎∗(푛, 푇 ) according to
+휎(푛, 푇 ) = (푘퐵푇 )3∕2(4휋휖0)2
+푚1∕2
+푒
+푒2
+휎∗ = 0.0258883 푇 3∕2 휎∗(Ωm K3∕2)−1 = 32405.4 푇 3∕2
+eV 휎∗(Ωm)−1 .
+(20)
+As with thermodynamic relations, the dimensionless conductivity 휎∗ can be expressed as a function of dimensionless variables
+푟푠, 푇Ha or Γ, Θ. These functions are now to be specified. Exact results are currently known only for limiting cases, in particular
+virial expansions.
+The analysis of a virial expansion is sometimes not easy because trivial terms dominate in limiting cases so that interesting
+terms remain hidden. In the example of the thermodynamic EoS considered in Sec. 5, one dominant term is the Debye shift,
+which covers the contribution of higher virial coefficients. We introduce reduced virial expansions where these exactly known
+contributions are suppressed, and quantities are introduced that anticipate a linear relationship in special cases. The virial plot
+is the representation of this asymptotic linear relationship and allows us to extrapolate virial coefficients from simulations. We
+demonstrate this procedure for two cases, the mean potential energy of the uniform electron gas in Sec. 5 and the electrical
+conductivity of the Hydrogen plasma in Sec. 6.
+
+6
+If we express 휎∗(푛, 푇 ) in terms of dimensionless parameters Γ, Θ and use the Born parameter Γ∕Θ, which is of interest in the
+range 푘퐵푇 ≫ 1 Ry, from Eq. (13) we obtain a modified virial expansion where the argument of the logarithm is dimensionless,
+1
+휎∗(Γ, Θ) = 휌∗(Γ, Θ) = ̃휌1(Γ2Θ) ln
+(Θ
+Γ
+)
++ ̃휌2(Γ2Θ) + … ,
+Γ2Θ =
+27∕3
+34∕3휋3∕3
+1
+푇Ha
+,
+Θ
+Γ =
+21∕3
+31∕3휋5∕3
+푇 2
+Ha
+푛푎3
+퐵
+(21)
+We define the reduced effective virial coefficient ̃휌eff
+2 (푇 ) according to
+̃휌eff
+2 (푛, 푇 ) =
+32405.4
+휎(푛, 푇 )[Ωm]푇 3∕2
+eV − ̃휌1(푇 ) ln
+(Θ
+Γ
+)
+,
+(22)
+with lim푛→0 ̃휌eff
+2 (푛, 푇 ) = ̃휌2(푇 ), see also Eq. (45) below. The plot of 휌∗∕ ln(Θ∕Γ) as a function of 푥 = 1∕ ln(Θ∕Γ) at given 푇 is
+called a virial plot. It directly allows the determination the virial coefficients 휌1(푇 ), 휌2(푇 ), as it is shown in Sec. 6.
+As will be demonstrated in this work, virial plots are very sensitive to diverse approaches, including the results of numerical
+simulations, in the low density domain. Since trivial dominant terms, which are known exactly, are suppressed, they have no
+effects due to possible approximations, and the extrapolation of numerical simulations into the low-density domain becomes
+immediately possible.
+5
+VIRIAL EXPANSION OF THE EOS OF THE UEG, COMPARISON WITH PIMC
+SIMULATIONS
+The problem of the second virial coefficient for the mean correlation energy ̄푉 was considered in a recent work25. There was
+a controversy about the high-temperature limit of the second virial coefficient, i.e. the term ∝ 1∕
+√
+푇 27. This controversy dis-
+appears in charge-neutral two-component plasmas, but not in the uniform electron gas (UEG), where interacting electrons are
+moving in front of a positively charged jellium-like background to neutralize the Coulomb field at large distances. Accurate
+PIMC simulations have been available at low densities and high temperatures25, so that it was possible to confirm the correct
+limiting behavior. In this section, we not only show the virial plot method to confirm the correct limiting law, but consider the
+full second virial coefficient and discuss deviations from this expansion.
+The virial expansion of the free energy 퐹(푇 , Ω, 푁) of the UEG is obtained from the general formula for a multi-component
+plasma given in9,25. The mean potential energy 푉 is determined by
+푉 (푇 , Ω, 푁) = 푒2
+휕
+휕(푒2)퐹(푇 , Ω, 푁)
+(23)
+(for the relation to the internal energy see28).
+From the virial expansion of 퐹(푇 , Ω, 푁), we get the following virial expansion of 푉
+푉
+푁푘퐵푇 = − 휅3
+8휋푛 − 휋푛휆3휏3 ln(휅휆)
+−휋푛휆3
+[
+휏
+2 −
+√
+휋
+2 (1 + ln(2))휏2 +
+(
+퐶
+2 + ln(3) − 1
+3 + 휋2
+24
+)
+휏3
++
+√
+휋
+∞
+∑
+푚=4
+(−1)푚푚
+2푚Γ(푚∕2 + 1)
+[2휁(푚 − 2) − (1 − 4∕2푚)휁(푚 − 1)] 휏푚
+]
+−휋푛휆4휏4휅 ln(휅휆) + 푉4(푇 )
+푁푘퐵푇 푛3∕2 + (푛2 ln(푛))
+(24)
+with the variables
+휅2 =
+푛푒2
+휖0푘퐵푇 ,
+휆2 =
+ℏ2
+푚푘퐵푇 ,
+휏 =
+푒2√
+푚
+4휋휖0
+√
+푘퐵푇 ℏ
+.
+(25)
+휁(푥) denotes the Riemann zeta function, and 퐶 = 0.57721 … is Euler’s constant. We express this expansion in terms of 푇 , 푛
+and introduce atomic units ℏ = 푚 = 푒2∕4휋휖0 = 1 so that 푘퐵푇 is measured in Hartree (Ha) and 푛 in electrons per 푎3
+퐵.
+The virial expansion of the specific mean potential energy 푣 = 푉 ∕푁 is as follows
+푣(푇 , 푛) = 푣0(푇 )푛1∕2 + 푣1(푇 )푛 ln (휅2휆2) + 푣2(푇 )푛 + 푣3(푇 )푛3∕2 ln (휅2휆2) + 푣4(푇 )푛3∕2 + (푛2 ln(푛)).
+(26)
+
+7
+If atomic units are used, this results in (휅2휆2 = 4휋푛∕푇 2)
+푣0(푇 ) = −
+√
+휋
+푇 1∕2 ,
+푣1(푇 ) = − 휋
+2푇 2 ,
+푣2(푇 ) = − 휋
+푇
+[
+1
+2 −
+√
+휋
+2 (1 + ln(2)) 1
+푇 1∕2 +
+(
+퐶
+2 + ln(3) − 1
+3 + 휋2
+24
+)
+1
+푇
+−
+√
+휋
+∞
+∑
+푚=4
+푚
+2푚Γ(푚∕2 + 1)
+( −1
+푇 1∕2
+)푚−1
+[2휁(푚 − 2) − (1 − 4∕2푚)휁(푚 − 1)]
+]
+,
+푣3(푇 ) = − 3휋3∕2
+2푇 7∕2 .
+(27)
+In ref.25, a virial plot was presented to study the behavior of the second virial coefficient. We consider the lowest orders of
+the virial expansion,
+푣(1)(푇 , 푛) = −
+√
+휋
+푇 1∕2 푛1∕2 −
+휋
+2푇 2 푛 ln
+(4휋푛
+푇 2
+)
+,
+(28)
+as exactly known and subtract them from the data obtained from the PIMC simulations, 푣PIMC = 푉 PIMC∕푁. These exactly
+known terms may become very large, hiding the higher virial coefficients. (Note that the logarithmic term contains a factor to
+become dimensionless. This factor can be moved to the next virial coefficient.)
+In25 we introduced the reduced potential energy (휏 = 푇 −1∕2, atomic units)
+푣red
+2 (푇 , 푛) = [푣PIMC − 푣(1)(푇 , 푛)] −푇
+휋푛 = −푇
+휋 푣2(푇 ) + (푛1∕2 ln(푛))
+= 1
+2 −
+√
+휋
+2 (1 + ln(2))휏 +
+(
+퐶
+2 + ln(3) − 1
+3 + 휋2
+24
+)
+휏2 + (휏3) + (푛1∕2 ln(푛)).
+(29)
+Table 1 PIMC calculations for the uniform electron gas: 푣PIMC and 푣red
+2 , eq. (29), for special parameter values 푟푠, Θ and the
+corresponding values of 푇 , 휏, 푛.
+푟푠
+Θ
+푣PIMC [Ha]
+푇Ha
+휏
+푣red
+2
+푇 [K]
+푛 [cm−3]
+0.5
+128
+-0.0826214
+942.891
+0.0325664
+0.453524
+2.97742e8
+1.28882e25
+64
+-0.1180456
+471.446
+0.0460558
+0.420822
+1.48871e8
+1.28882e25
+32
+-0.169272
+235.723
+0.0651327
+0.398701
+7.44354e7
+1.28882e25
+16
+-0.2423993
+117.861
+0.0921116
+0.356465
+3.72177e7
+1.28882e25
+8
+-0.3447641
+58.9307
+0.130265
+0.294433
+1.86089e7
+1.28882e25
+2
+128
+-0.0402248
+58.9307
+0.130265
+0.290766
+1.8609e7
+2.01378e23
+64
+-0.0568062
+29.4653
+0.184223
+0.257047
+9.30448e6
+2.01378e23
+32
+-0.0797147
+14.7327
+0.260531
+0.207038
+4.65224e6
+2.01378e23
+16
+-0.1101257
+7.36634
+0.368446
+0.126496
+2.32612e6
+2.01378e23
+8
+-0.1486611
+3.68317
+0.521062
+0.0596564
+1.16306e6
+2.01378e23
+20
+128
+-0.0119299
+0.589307
+1.30265
+1.50247
+186090.
+2.01378e20
+64
+-0.0160051
+0.294653
+1.84223
+3.48031
+93044.8
+2.01378e20
+32
+-0.0207112
+0.147327
+2.60531
+6.67878
+46522.4
+2.01378e20
+16
+-0.0256337
+0.0736634
+3.68446
+10.2475
+23261.2
+2.01378e20
+8
+-0.0302098
+0.0368317
+5.21062
+9.50255
+11630.6
+2.01378e20
+In Tab. 1, the parameter values of the uniform electron gas are given for which PIMC calculations were presented in Ref.25,
+together with the values for 푣red
+2
+(29). The results for 푣red
+2
+are also shown in Figs. 1, 2.
+In Fig. 1 all calculated PIMC data of25 are considered and the corresponding value of 푣red
+2
+is shown as function of 휏, see Tab.
+1. In addition, three expressions for (29) are shown: up to order 휏, i.e., 1∕2 −
+√
+휋(1 + ln(2))휏∕2, up to order 휏2, and the full 휏
+dependence. This Fig. 1 shows in which interval of 휏 the linear or quadratic approximation is applicable. The PIMC data are
+
+8
+0
+1
+2
+3
+4
+5
+6
+T
+-1/2 [Ha]
+0
+5
+10
+15
+20
+v
+red
+rs = 0.5
+rs = 2
+rs = 20
+T
+-1/2
+T
+-1
+2
+nd virial
+3
+rd virial, rs=20
+Figure 1 Reduced potential energy 푣red
+2 (푇 , 푛), Eq. (29), as function of 휏 = 1∕
+√
+푇 for different densities, 푟푠 = 0, 5; 2; 20. For
+comparison, the reduced second virial coefficient 푣red
+2 (푇 ) = −(푇 ∕휋)푣2(푇 ) [2nd virial, according Eq. (27)] as well as the lowest
+orders in 1∕푇 are shown. In addition, the curve 3rd virial given by Eq. (30) is also shown. (Atomic units are used.)
+very different. The lowest density, 푟푠 = 20, should be most relevant to the low-density limit, where higher virial coefficients are
+less important. However, the inverse temperature 휏 = 푇 −1∕2
+Ha
+is too large to reach the limit 휏 → 0. Close to this limit are PIMC
+simulation data for 푟푠 = 0.5. The relatively large density is compensated by the very high temperature, see Tab. 1.
+A part of Fig. 1 is shown enlarged in Fig. 2. It was a main result of Ref.25 to show that the PIMC simulation data confirm the
+limit 푣red
+2 (휏 = 0) = 1∕2. Linear fit to the data for 푟푠 = 0.5 is possible, and extrapolation to 푣red
+2 (휏 = 0) gives 1/2. At the same
+time, one gets an idea of the accuracy of the simulation, which shows up as scatter around the analytical behavior. The PIMC
+data for 푟푠 = 2 are not described by the linear approximation but almost well by the quadratic approximation. Finally, we have
+to make a comparison with the full second virial coefficient and will find that good agreement is obtained in all three density
+cases, given by the parameter 푟푠, only for the lowest values of 휏 (an exception is the lowest 휏 parameter calculation for 푟푠 = 2,
+which needs to be checked). As 휏 increases, the PIMC data are systematically below the second virial curve. We assume that
+the PIMC simulations are very accurate, so this deviation indicates the contribution of higher virial coefficients.
+Deviations from the second virial coefficient −(푇 ∕휋)푣2(푇 ) indicate the contribution of higher orders to the virial expansion.
+We expect a significant next order contribution from the low-density calculations, i.e., 푟푠 = 20. We consider the expression
+푣red
+2+3(푇 , 푛) = −푇
+휋
+[
+푣2(푇 ) + 푣3(푇 )푛1∕2 ln
+(4휋푛
+푇 2
+)]
+,
+(30)
+which accounts for the contribution of the third virial coefficient. For 푟푠 = 20, the data are well reproduced for the lowest values
+of 휏, see also Fig. 1. Deviations for larger 휏 indicate the contributions of higher virial coefficients.
+The deviation
+Δ푣red
+2 (푇 , 푛) = [푣PIMC − 푣(1)(푇 , 푛) − 푣2(푇 )푛] 푇
+휋푛
+(31)
+is shown in Tab. 2, together with the deviation
+Δ푣red
+3 (푇 , 푛) =
+[
+푣PIMC − 푣(1)(푇 , 푛) − 푣2(푇 )푛 − 푣3(푇 )푛3∕2 ln
+(4휋푛
+푇 2
+)] 푇
+휋푛.
+(32)
+As mentioned before, the inclusion of the third virial coefficient 푣3(푇 ) improves the agreement of the PIMC simulations with
+the virial expansion, as also shown in Fig. 1. The remaining difference Δ푣red
+3 (푇 , 푛) is related to the fourth-order and higher-order
+virial coefficient,
+푣eff
+4 (푇 , 푛) = Δ푣red
+3 (푇 , 푛)
+휋
+푇 푛1∕2 = 푣4(푇 ) + (푛1∕2 ln(푛)).
+(33)
+
+9
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+T
+-1/2 [Ha]
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+v
+red
+rs = 0.5
+rs = 2
+T
+-1/2
+T
+-1
+2
+nd virial
+Figure 2 Detail of Fig. 1.
+The fourth virial coefficient results when higher-order virial coefficients are neglected, lim푛→0 푣eff
+4 (푇 , 푛) = 푣4(푇 ). This should
+be possible in the low-density limit, where the contributions of higher orders of the density expansion become small. However,
+high-precision calculations are required to extract the higher-order coefficients, and the accuracy of the present calculations25
+is not sufficient to determine precisely the fourth- and higher-order virial coefficients. We give here only a discussion of the
+present data.
+From the virial expansion of the free energy9, the fourth virial coefficient 푣4(푇 ) contains contributions with temperature
+dependence ∝ 푇 −2 = 휏4 and higher orders in 휏, as well as contributions ∝ 푇 −7∕2. The coefficient of the 휏4 term follows as
+3휋
+√
+4휋. We expect a high-temperature limit behavior ∝ 푇 −2, and we show in Fig. 3 the quantity 푣eff
+4 (푇 , 푛) × 푇 2.
+We see that the lowest value of density, 푟푠 = 20, exhibits behavior at small 휏 values that can be compared to a curve 3휋
+√
+4휋 −
+6 × 휏3. However, the exact determination of the fourth virial coefficient 푣4(푇 ) is not possible from the available data. At the
+higher densities corresponding to smaller 푟푠, the accuracy of the numerical PIMC simulations may not be sufficient to extract
+higher-order virial coefficients. In the context of our analysis, in addition to the dependence on 푇 , the dependence on 푛 would be
+of interest to perform the virial plot as a function of 푛. Further calculations for density parameter values in the range of 푟푠 = 20
+would be required. Since we are investigating the differences between large numbers, high accuracy is necessary.
+The study of the uniform electron gas is not only of interest for the discussion of the exchange-correlation term of the energy-
+density functional in DFT calculations, for which Dornheim, Groth, and Bonitz derived analytical formulas29,30. It is also a
+prerequisite to treat the more interesting case of a two-component plasma, e.g., the Hydrogen plasma. The equation of state at
+low densities is of interest, for example, in helioseismology31, where the fourth virial coefficient 푣4(푇 ) is important32. In this
+context, the high-temperature limit of 푣red
+2 (휏 = 0) was discussed in27,25. For a discussion of the fourth virial coefficient 푣4(푇 ) of
+Hydrogen plasma, see also Alastuey and Ballenegger33,34.
+6
+VIRIAL EXPANSION OF THE INVERSE CONDUCTIVITY OF H PLASMAS,
+COMPARISON TO DFT-MD SIMULATIONS
+Numerous studies have been performed to calculate the electrical conductivity 휎(푛, 푇 ) of Hydrogen plasma in a wide range of
+parameters, a recent review can be found in Ref.35. A comparative study36 was also recently published that considered different
+approaches and showed large differences in the calculated conductivities. Analytical calculations in the framework of generalized
+
+10
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+5.5
+τ = (THa)
+-1/2
+-300
+-250
+-200
+-150
+-100
+-50
+0
+50
+v4
+eff(T,n)* (THa)
+2
+6π
+3/2− 6τ
+3
+rs=0.5
+rs=2
+rs=20
+Figure 3 Effective reduced fourth virial coefficient 푣eff
+4 (푇 , 푛) × 푇 2, Eq. (33), plotted as function of 휏 = 1∕
+√
+푇Ha for different
+densities, 푟푠 = 0, 5; 2; 20. For comparison, a curve 3휋
+√
+4휋 − 6휏3 is seen. (Atomic units used.)
+Table 2 PIMC calculations for the UEG: 푣 and 푣red. The calculation with the second virial coefficient, Eq. (32), is denoted by
+푣vir and 푣red
+vir .
+푟푠
+Θ
+푣 [Ha]
+푇Ha
+푛 푎3
+퐵
+휏
+푣red
+2
+Δ푣red
+2
+Δ푣red
+3
+0.5
+128
+-0.082621
+942.891
+1.90986
+0.0325664
+0.453524
+-0.000818266
+-0.000819682
+64
+-0.118045
+471.446
+1.90986
+0.0460558
+0.420822
+0.0132298
+0.0132228
+32
+-0.169272
+235.723
+1.90986
+0.0651327
+0.398701
+0.00992642
+0.00989306
+16
+-0.242399
+117.861
+1.90986
+0.0921116
+0.356465
+0.0181564
+0.0180015
+8
+-0.344764
+58.9307
+1.90986
+0.130265
+0.294433
+0.0360992
+0.0354136
+2
+128
+-0.040224
+58.9307
+0.0298416
+0.130265
+0.290766
+0.039767
+0.0396097
+64
+-0.056806
+29.4653
+0.0298416
+0.184223
+0.257047
+0.0194226
+0.0186676
+32
+-0.079714
+14.7327
+0.0298416
+0.260531
+0.207038
+0.0103138
+0.00680714
+16
+-0.110125
+7.36634
+0.0298416
+0.368446
+0.126496
+0.043972
+0.0284584
+8
+-0.148661
+3.68317
+0.0298416
+0.521062
+0.0596564
+0.12329
+0.0599871
+20
+128
+-0.011929
+0.589307
+0.0000298416
+1.30265
+1.50247
+0.440148
+0.0680086
+64
+-0.016005
+0.294653
+0.0000298416
+1.84223
+3.48031
+1.60138
+-0.0765312
+32
+-0.020711
+0.147327
+0.0000298416
+2.60531
+6.67878
+5.91999
+-1.155
+16
+-0.025633
+0.0736634
+0.0000298416
+3.68446
+10.2475
+19.7824
+-6.56867
+8
+-0.030209
+0.0368317
+0.0000298416
+5.21062
+9.50255
+60.1201
+-11.6087
+linear response theory were performed for simple systems such as the Hydrogen plasma. For more complex plasmas, the DFT-
+MD approach16,37,38,19 was elaborated to evaluate the Kubo-Greenwood formula. However, as discussed in21, electron-electron
+collisions are not correctly described in this approach. In a recent study15, the low-density limit of the electrical conductivity
+휎(푛, 푇 ) of Hydrogen as the simplest ionic plasma is presented as a function of temperature 푇 and particle density 푛 in terms
+
+11
+0
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
+1
+1.1
+1/ln[Θ/Γ]
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+32405.4 (T/eV)
+3/2/σ[S/m] / ln(Θ/Γ]
+tilde ρ1Spitzer
+tilde ρ1Lorentz
+Karakhtanov
+QLB, Ronald
+64
+125
+216
+95
+T = 2000 eV
+T = 200 eV
+n = 40 g/ccm
+n = 2 g/ccm
+Figure 4 Reduced resistivity ̃휌(푥, 푇 ) (42) for hydrogen plasma as a function of 푥 = 1∕ ln(Θ∕Γ): DFT-MD simulations from
+Ref.15, and Lenard-Balescu results (QLB, Ronald) of Desjarlais et al.37 and Karakhtanov40. 휌Spitzer
+1
+= 0.846 and 휌Lorentz
+1
+= 0.492
+are defined in the text. The green line represents a linear extrapolation of the converged DFT-MD results. Data are given in the
+Supplemental material of15.
+of a virial expansion of resistivity. The non-consideration of the contribution of electron-electron collisions in other transport
+coefficients such as thermopower and thermal conductivity has also been discussed recently37,39.
+The virial expansion of the dimensionless resistivity 휌∗, Eq. (13), contains the logarithmic term ln(1∕푛). To make its argument
+dimensionless we use the Born parameter, see Ref.15,
+Θ
+Γ =
+푇 2
+Ry
+푛Bohr
+(96휋5)−1∕3 ,
+(34)
+where the temperature is measured in Rydberg units, 푇Ry = 2푇Ha = 푘퐵푇 ∕13.6 eV. As discussed in Sec. 4 in connection with
+the logarithmic term, we use a modified virial expansion and rewrite (13)
+휌∗(푛, 푇 ) = ̃휌1(푇 ) ln
+(Θ
+Γ
+)
++ ̃휌2(푇 ) + … .
+(35)
+The modified virial coefficients ̃휌푖 are related to 휌푖 replacing in Eq. (35) the variables Θ, Γ by 푛, 푇 according to Eq. (34).
+Comparing with Eq. (13), ̃휌1 = 휌1 is obtained and
+̃휌2 = 휌2 + 휌1 ln[(96휋5)1∕3∕푇 2
+Ry] .
+(36)
+A highlight of plasma transport theory is that the exact value of the first virial coefficient for Coulomb systems is known from
+the seminal paper of Spitzer and Härm3,
+휌1 = ̃휌1 = 휌Spitzer
+1
+= 0.846024,
+(37)
+which does not depend on 푇 . Note that Eq. (37) accounts for the contribution of the electron-electron (푒 − 푒) interaction. In
+contrast, for the Lorentz plasma model where the 푒 − 푒 collisions are neglected so that only the electron-ion interaction is
+considered, the first virial coefficient is4
+휌Lorentz
+1
+= 1
+16(2휋3)1∕2 = 0.492126 .
+(38)
+Although 푒−푒 collisions do not contribute to a change of the total momentum of the electrons due to conservation of momentum,
+the distribution in momentum space is changed by 푒−푒 collisions ("reshaping"), and higher moments of the electron distribution
+
+12
+are not conserved by 푒−푒 collisions. The indirect influence of 푒−푒 collisions on the dc conductivity becomes clear in generalized
+linear response theory where these higher moments are considered, see4,6.
+No exact value is known for the second virial coefficient 휌2(푇 ) or ̃휌2(푇 ). It depends on the treatment of the many-body effects,
+in particular on the screening of the Coulomb potential. In a quantum statistical approach, the static (Debye) screening by
+electrons and ions should be replaced by dynamical screening. For the Hydrogen plasma considered here, the Born approximation
+for the collision integral at high temperatures 푇Ry ≫ 1 is justified. Consideration of screening in the random phase approximation
+(RPA), leads to the quantum Lenard-Balescu (QLB) expression. Thus, at very high temperatures, where the dynamically screened
+Born approximation becomes valid, we obtain the QLB result, see37,40,
+lim
+푇 →∞ ̃휌2(푇 ) = ̃휌QLB
+2
+= 0.4917 .
+(39)
+As 푇 decreases, strong binary collisions (represented by ladder diagrams) become important and must be treated in the
+calculation of the second virial coefficient ̃휌2(푇 ) beyond the Born approximation. According to Spitzer and Härm3, the classical
+treatment of strong collisions with a statically screened potential gives for 휌∗ = 1∕휎∗ the result
+휌∗
+Sp = 0.846 ln
+[3
+2Γ−3]
+.
+(40)
+Interpolation formulas have been proposed that link the high-temperature limit ̃휌QLB
+2
+with the low-temperature Spitzer
+limit45,41,42,5,43,6,4,44. Based on a T-matrix calculation in quasiclassical (Wentzel-Kramers-Brillouin, WKB) approximation45,46,
+the expression (푇eV = 푘퐵푇 ∕eV)
+̃휌2(푇eV) ≈ 0.4917 + 0.846 ln
+[
+1 + 8.492∕푇eV
+1 + 25.83∕푇eV + 167.2∕푇 2
+eV
+]
+(41)
+is a simple interpolation that combines the QLB result with the Spitzer limit in WKB approximation. However, the exact
+analytical form of the temperature dependence of the second virial coefficient ̃휌2(푇 ) remains an open problem.
+Thus, the available exact results for the virial expansion (35) of the inverse conductivity of fully ionized Hydrogen plasma are:
+(i) the value of the first virial coefficient is ̃휌1 = 0.846;
+(ii) the second virial coefficient has the high-temperature limit lim푇 →∞ ̃휌2(푇 ) = 0.4917;
+(iii) the second virial coefficient is temperature dependent, an approximation is given by Eq. (41).
+To extract the first and second virial coefficient from calculated or measured dc conductivities, we plot the expression
+̃휌(푥, 푇 ) =
+휌∗
+ln(Θ∕Γ) =
+32405.4
+휎(푛, 푇 )(Ωm)푇 3∕2
+eV
+1
+ln(Θ∕Γ)
+(42)
+as a function of 푥 = 1∕ ln(Θ∕Γ) and 푇 in Fig. 4 which is called virial plot. According to Eqs. (13), (35), the behavior of any
+isotherm (fixed 푇 ) is linear near 푛 → 0,
+̃휌(푥, 푇 ) = ̃휌1(푇 ) + ̃휌2(푇 )푥 + … ,
+(43)
+with ̃휌1(푇 ) as the value at 푥 = 0 and ̃휌2(푇 ) as the slope of the isotherm. In this way, the extraction of virial coefficients becomes
+immediately possible. For 푥 > 1∕ ln(100) = 0.217, the contributions of higher order virial coefficients have to be taken into
+account15. For fixed 푇 and low density, where 휃 ≫ 1, a classical plasma is present and the effects of degeneracy contribute to
+the higher order virial coefficients.
+In Fig. 4 two cases for the first virial coefficient 휌1 on the ordinate axis are shown, see also4,5,6:
+(i) 휌Spitzer
+1
+from kinetic theory when 푒 − 푒 collisions are taken into account,
+(ii) when 푒 − 푒-collisions are neglected, 휌Lorentz
+1
+is obtained for the Lorentz plasma model.
+Moreover, the second virial coefficient ̃휌QLB
+2
+of the Lenard-Balescu approximation. (39) is shown, which is correct in the high
+temperature limit. The QLB calculations of Desjarlais et al.37 are shown in Fig. 4. The 푒 − 푒 collisions are taken into account,
+yielding the same asymptote (푥 → 0) as in Karakhtanov40. With increasing 푥 = 1∕ ln(Θ∕Γ) small deviations from linear
+behavior are observed. When isotherms are presented, this deviation indicates the contribution of higher virial coefficients.
+Virial plots are presented in15 to investigate two problems: Which of the various approaches that give us analytical expressions
+for the electrical conductivity of Hydrogen plasmas are accurate in the low density limit? The virial expansion of the inverse
+conductivity serves as an exact benchmark for theoretical approaches, so that the accuracy and consistency of semi-empirical
+results for conductivity, such as those collected in Ref.36, can be checked. A more fundamental problem is whether numerical
+results from molecular dynamics simulations based on density functional theory (DFT-MD) correctly contain the contribution
+of electron-electron collisions. The virial plot confirms the position that DFT-MD simulations in the low-density limit describe
+a Lorentz plasma with only electron-ion collisions, the contribution of electron-electron collisions to 휌1 is missing15.
+
+13
+Here we discuss some details of the virial expansion for the inverse conductivity and the corresponding virial plots, see Fig.
+4. DFT-MD simulations are given in Ref.15, see the tables of data in the supplementary material. These data have sufficiently
+high accuracy, as can be seen from the small deviations from the fit line in Fig. 4. In addition to the precise solution of the Kubo-
+Greenwood formula, this is achieved by good control of convergence with increasing particle number, as shown by comparison
+of calculations with different numbers of particles. The number of particles must be sufficiently large to ensure convergence.
+In the parameter range considered in the figure, about 100 particles in the box are necessary to achieve convergence. Further
+calculations with 216 electrons were not possible due to limited computer capacity. For 푇 = 150 eV, even 125 electrons exceed
+the currently available computer capacity. This point was also discussed in a recent work39, where earlier calculations37 were
+improved to achieve convergence. Another problem is the determination of the value of the dc conductivity 휎(0) from the calcu-
+lation of the optical conductivity 휎(휔) at finite frequencies. Because of the discretisation in a finite box, the energy eigenvalues
+have a minimum spacing and the energy-conserving 훿 function must be smeared by a parameter 휖 to allow for transitions, see
+also section 3 above. To reach the limit 휔 → 0, an extrapolation is performed according to the Drude formula (10). This was
+discussed also in Ref.39. Instead, one can use the dynamic collision frequency to perform this extrapolation.
+The results shown in Fig. 4 allow the extraction of virial coefficients 휌1(푇 ), ̃휌2(푇 ). Compared to other approaches, including
+interpolation formulas, see15, as well as the QLB calculation, we assume that we are in the linear region of the virial curve.
+Deviations from linearity can be observed for QLB already at 푥 = 0.2, since the density is high (40 g/cm3). For DFT-MD
+simulations with density about 2 g/cm3, the deviation from linearity for the last point is observed at 푥 ≈ 1.
+As pointed out in15, the extrapolated value of 휌1 in the virial plot at 푥 = 0 points to the Lorentz value (38) but misses the
+Spitzer value (37). This means that electron-electron collisions are not considered in the DFT-MD calculations for the electrical
+conductivity. Also of interest is the value of ̃휌2(푇 ) given by the slope in the virial plot near 푥 = 0. Fitting it to the data gives a slope
+of 0.9886 for the DFT-MD calculations. This is about twice the slope ̃휌QLB
+2
+given above. From analytical approaches, it appears
+that the slope is determined by various effects such as dynamical screening and strong collisions. In the limiting case of high
+temperatures, the Born approximation should be possible, but the Coulomb potential must be replaced by a screened potential.
+Static screening of the proton scatterer with both electrons and protons would lead to the following result (퐶 = 0.57721 … is
+Euler’s constant).
+lim
+푇 →∞ ̃휌2(푇 ) = 휋3∕2
+24
+√
+2
+[11
+2 − 3퐶 + ln
+(3
+2휋2)]
+= 1.06036
+(44)
+which is close to the observed slope of the DFT-MD simulations. However, it remains unclear to what extent the screening is
+included in the simulations. We assume that the ionic structure factor, which is the ionic contribution to the screening, is well
+described, and that the electron screening is also captured by the exchange-correlation functional. However, we need to consider
+dynamical screening, a problem that has been discussed in previous work5 on virial expansion.
+We return to the long-debated question of whether or not 푒 − 푒 collisions are accounted for in the DFT-MD formalism. For
+example, it was pointed out in Ref.21 that a mean-field approach is not able to describe two-particle correlations, in particular
+푒 − 푒 collisions. However, to some approximation, the 푒 − 푒 interaction is accounted for by the exchange-correlation energy.
+DFT-MD simulations, which are mean-field theories that account for the 푒−푒 interaction only through the exchange-correlation
+part of the energy density, cannot account for the effect of 푒 − 푒 collisions on the conductivity, so that 휌1(푇 ) corresponds to the
+Lorentz plasma, but ̃휌2(푇 ) is determined by screening. The question arises to what extent dynamical screening, as implemented
+in the QLB calculations, is also described by the exchange-correlation part of the energy density functional. We would like to
+mention that in the case of thermal conductivity it has been shown that the contribution of 푒 − 푒 collisions is not taken into
+account in DFT-MD simulations37,35,39 and yields an additional term. Other approaches such as generalized linear response
+theory may be considered to indicate appropriate approaches.
+Our analysis has shown that the simulation results with virial evolution are extrapolated to the low-density region, where
+DFT-MD simulations are no longer feasible. The current simulations, while computationally expensive, are still not very close
+to 푥 = 0, so extrapolation to the 푥 = 0 limit is not very accurate. Better data for DFT-MD simulations would be of interest to
+confirm our results. Conversely, the benchmark capability of virial expansion discussed in this work can also serve as a criterion
+to verify the accuracy of numerical approaches such as DFT-MD simulations to evaluate conductivity.
+Another application of the virial plot is experiments to measure electrical conductivity. Assuming that the value 0.846024,
+Eq. (37), for 휌1 is exact, an effective second virial coefficient
+̃휌eff
+2 (푛, 푇 ) =
+32405.4
+휎(푛, 푇 )[Ωm]
+( 푇
+eV
+)3∕2
+− 0.846024 ln
+(Θ
+Γ
+)
+(45)
+
+14
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1/T [eV]
+-3
+-2
+-1
+0
+1
+2
+3
+second virial coefficient  tilde ρ2(T)
+H: Guenther, Radtke
+Ar, Xe, Ne: Ivanov
+Ar, Xe: Popovic
+Lenard-Balescu value
+ERR interpolation
+QLB/Desjarlais
+interpolation (7)
+Figure 5 Second virial coefficients ̃휌2(푇 ) and ̃휌eff
+2 (푛, 푇 ) for the dc conductivity of Hydrogen plasmas. Analytical interpolation
+formulas (41) and Ref.45 are compared with experiments of Günther and Radtke47 for H plasmas as well as of Ivanov et al.48
+and Popovic et al.49 for rare gas plasmas. The black dashed line corresponds to the high temperature limit that is given by
+the quantum Lenard-Balescu value. The broken blue line is the interpolation formula of Ref.45, the red full line represents the
+interpolation formula (41) for the second virial coefficient.
+has been introduced which gives the second virial coefficient in the low-density limit, lim푛→0 ̃휌eff
+2 (푛, 푇 ) = ̃휌2(푇 ). A dependence
+of ̃휌eff
+2 (푥, 푇 ) on density shows that higher orders of the virial expansion are relevant. We anticipate that at very high 푇 , i.e.,
+1∕푇 → 0, the Lenard-Balescu value is approximated. The deviations at increasing 1∕푇 , shown in the interpolation formula and
+the DFT-MD simulations, indicate that already below temperatures of the order of 100 eV, the effect of strong collisions beyond
+the Born approximation should be taken into account.
+Ultimately, the virial expansion (35) must be verified experimentally, but accurate data for the conductivity of Hydrogen
+plasma in the low-density limit and/or at high temperatures are scarce. Accurate conductivity data for dense Hydrogen plasma
+were derived by Günther and Radtke47. They are close to the benchmark data of the virial expansion. It should be noted that there
+are systematic errors associated with the analysis of such experiments. For example, the appearance of bound states requires
+a realistic treatment of the plasma composition and the influence of neutrals on electron mobility. Alternatively, conductivity
+measurements in highly compressed noble gas plasmas were carried out by Ivanov et al.48 and Popovic et al.49,45, but the
+interaction of the electrons with the ions deviates from the pure Coulomb potential due to the core of bound electrons. The
+corresponding virial plot is close to the data of Hydrogen plasma, see15, but requires a more detailed discussion on the role of
+bound electrons.
+It should also be mentioned that the densities are quite high, and extrapolation to zero density must be performed to obtain the
+second virial coefficient. This tendency can be seen in Fig. 5, especially for the experiments with Ar, Xe49, where low-density
+data point to ̃휌2(푇 ).
+Quantum statistical methods provide accurate values for the lowest virial coefficients, which serve as benchmarks for an-
+alytical approaches to electrical conductivity as well as for numerical results from molecular dynamics simulations based on
+density functional theory (DFT-MD) or path integral Monte Carlo (PIMC) simulations. While these simulations are well suited
+to compute 휎(푛, 푇 ) in a wide range of densities and temperatures, especially for the warm dense matter region, they become com-
+putationally expensive in the low density limit, and virial expansions can be used to compensate for this drawback. Interpolation
+formulas that take both approaches into account would be very useful for calculating the conductivity of plasmas.
+
+15
+Table 3 Experimental data for the electrical conductivity. Günther and Radtke: H47; Ivanov et al.: Ar, Xe, Ne48; Popovic et al.:
+Ar, Xe49.
+Plasma
+̂푛푒 × 1025
+푛 × 10−6
+푇 × 104
+푇
+Γ
+Θ
+1∕ ln(Θ∕Γ)
+휎 × 103
+̃휌(푥, 푇 )
+̃휌eff
+2
+[m3]
+[g/cm3]
+[K]
+[eV]
+[(Ωm)−1]
+H
+0.1
+1.67262
+1.54
+1.32706
+0.174914
+363.932
+0.130883
+6.2
+1.04579
+1.52647
+H
+0.15
+2.50893
+1.87
+1.61143
+0.164892
+337.249
+0.131177
+9.1
+0.955544
+0.835097
+H
+0.25
+4.18155
+2.15
+1.85271
+0.170041
+275.832
+0.13529
+11.4
+0.969821
+0.915228
+Ar
+2.8
+46.8334
+2.22
+1.91303
+0.36845
+56.8959
+0.198426
+19.0
+0.895459
+0.249255
+Ar
+5.5
+91.9942
+2.03
+1.74931
+0.504626
+33.1707
+0.238914
+15.5
+1.15565
+1.29607
+Ar
+8.1
+135.482
+1.93
+1.66313
+0.603878
+24.3632
+0.270456
+17.0
+1.10575
+0.960407
+Ar
+14
+234.167
+1.9
+1.63728
+0.736152
+16.6533
+0.320623
+25.5
+0.853604
+0.0237179
+Ar
+17
+284.346
+1.78
+1.53387
+0.838316
+13.7074
+0.357872
+24.5
+0.899216
+0.148701
+Xe
+25
+418.155
+3.01
+2.5938
+0.563757
+17.9242
+0.289077
+45
+0.869607
+0.081664
+Ne
+1.1
+18.3988
+1.98
+1.70622
+0.302559
+94.6027
+0.174059
+13
+0.966995
+0.695135
+Ne
+1.9
+31.7798
+1.96
+1.68899
+0.366725
+65.0509
+0.193113
+16.5
+0.832499
+-0.0699113
+air
+0.13
+2.17441
+1.1
+0.9479
+0.26726
+218.238
+0.14914
+6
+0.743367
+-0.688167
+Ar
+0.06
+1.00357
+1.64
+1.41323
+0.138532
+544.807
+0.120816
+8.3
+0.792469
+-0.443077
+Ar
+0.1
+1.67262
+1.64
+1.41323
+0.164248
+387.564
+0.128762
+7.9
+0.887358
+0.321199
+Ar
+0.13
+2.17441
+1.64
+1.41323
+0.17926
+325.373
+0.133264
+7.6
+0.954636
+0.815191
+Ar
+0.15
+2.50893
+1.64
+1.41323
+0.188017
+295.767
+0.135855
+6.4
+1.15567
+2.27941
+Xe
+0.06
+1.00357
+1.24
+1.06854
+0.18322
+411.928
+0.129569
+4.6
+1.0082
+1.25185
+Xe
+0.12
+2.00715
+1.24
+1.06854
+0.18322
+411.928
+0.13529
+4.1
+1.0082
+2.20078
+Xe
+0.07
+1.17083
+1.26
+1.08578
+0.189819
+377.693
+0.131652
+4.8
+1.00558
+1.21211
+Xe
+0.14
+2.34167
+1.26
+1.08578
+0.239157
+237.931
+0.144873
+4.4
+1.20715
+2.49289
+To obtain the correct values for the thermoelectric transport coefficients of Hydrogen plasma in the low-density limit, where
+the inclusion of 푒 − 푒 collisions is essential, different solutions can be considered. PIMC simulations, as successfully performed
+for the uniform electron gas25, should also be performed for the two-component plasmas. First steps of this ambitious project are
+recently in progress12,50. The study of such PIMC calculations with the virial plot would be of great interest. From generalized
+linear response theory, we also learn that higher order correlation functions, such as force-force correlation functions associated
+with the dynamic collision frequency, may be a suitable approach to include the contribution of 푒 − 푒 collisions in the transport
+coefficients4,5,6.
+7
+CONCLUSIONS
+We have from quantum statistics exact expressions for thermodynamic and transport properties of plasmas by equilibrium cor-
+relation functions, but the evaluation is a complex problem in many-particle physics. Numerical simulations are becoming more
+accurate as computer capacity increases. However, they need to be controlled with respect to their limits such as size effects,
+but also fundamental problems such as the correct description of electron-electron collisions in the context of DFT or the sign
+problem in PIMC. It is expected that PIMC simulations will provide an adequate description of electron-electron interactions,
+but they are currently unable to solve complex plasmas such as multiply charged ions in the low-temperature range.
+The comparison of analytical results for the virial expansion of thermodynamic properties with PIMC calculations for the
+uniform electron gas has been performed. In particular, we show that high-precission PIMC simulations confirm the correct
+form of the virial expansion, which has been debated recently. It seems to be possible to give also numerical values for higher
+virial coefficients, in particular the interesting 푛5∕2 coefficient. These values can be considered as exact results in plasma physics.
+Numerical values for higher virial coefficients would also be of great interest for transport properties.
+
+16
+Analytical theory gives us exact results in limiting cases. This can be used to obtain results for parameter ranges where
+numerical simulations are not efficient, e.g. in the low density range. Virial expansions are used to control theories and numerical
+simulations. They are of interest to construct interpolation formulas.
+It was indicated that the evaluation of the Kubo-Greenwood formula using DFT-MS simulations does not take into account
+the effects of electron-electron scattering and cannot reproduce the low-density limit of the electrical conductivity of Hydrogen
+plasmas. Similar results were recently reported by French et al.39 for other thermoelectric transport coefficients. It would be of
+interest to perform PIMC simulations that can accurately describe electron-electron collisions.
+The theory of virial expansion must be extended if the formation of bound states is of importance, i.e. for 푇 ∕푇Ha ≤ 1, see
+appendix. New approaches are needed. The approach described here is also applicable to other correlation functions such as
+the dynamic structure factor or to other transport properties such as thermal conductivity, thermopower, viscosity, and diffusion
+coefficients. Also of interest is the extension of virial expansion to elements other than Hydrogen, where different ions can be
+formed and the electron-ion interaction is no longer purely Coulombic.
+ACKNOWLEDGMENTS
+Thanks to M. Schörner, R. Redmer, M. Bethkenhagen, M. French, H. Reinholz, T. Dornheim, J. Vorberger, Z. Moldabekov, and
+W.-D. Kraeft for collaboration and discussions. This work was supported by the German Research Foundation (DFG), Grant #
+RO 905/37-1 AOBJ 655625.
+Author contributions
+This is an author contribution text. It is based on a contribution to the SCCS22 conference.
+Financial disclosure
+None reported.
+Conflict of interest
+The author declares no potential conflict of interests.
+APPENDIX
+A BOUND STATE FORMATION
+A special problem of plasmas is the formation of bound states (atoms, charged ions: clusters with a certain number of elemen-
+tary particles, i.e., nuclei and electrons) which can dominate the properties in the low-density and low-temperature region. A
+simple approach is the chemical picture9, where the bound states are considered as new constituents. The interaction between
+the different constituents is neglected except for reactive collisions. Thus, a chemical equilibrium is achieved in which the com-
+position of the plasma is described by the law of mass action. For a systematic approach including bound state formation see
+Refs.33,34 and references given there. We will not present here an exhaustive discussion of the chemical picture, but only discuss
+some aspects in the context of our work. For a recent review, see51,52, where further references can be found.
+Within the chemical picture, several issues arise that need to be discussed in order to improve this simple approximation,
+using the concept of virial expansions.
+(i) In addition to the ground state, excited states (푠) with excitation energy 퐸훼,푠 can occur, which can also be treated as new
+species. It is more convenient to introduce the intrinsic partition function of the cluster 훼, which is summed over all excited
+bound states by the statistical factor exp[−훽퐸훼,푠].
+
+17
+(ii) In addition to bound states, there are also scattering states that must be included in the calculation of virial coefficients.
+This leads to the Beth-Uhlenbeck formula, in which the scattering phase shifts appear. Sometimes resonances can appear in
+the spectrum of excited states. In the resonance gas approximation, the intrinsic partition function is improved by extending
+the summation over all excitations 푠 to the resonances in the continuum. Moreover, the contribution of scattering phase shifts
+should be included.
+iii) We arrive at higher virial coefficients and need to include density effects. In the framework of a quasiparticle approach, the
+intrinsic partition functions are calculated with shifted energies due to screening, mean-field effects, Pauli blocking and other
+effects.
+As example, let us consider the H plasma and give the intrinsic partition function in the simplest approximation
+푧H =
+∑
+푠
+2푠2푒−퐸H,푠∕푘퐵푇
+(A1)
+with the known energy levels 퐸H,푠 = −퐸Ha∕(2푠2) (퐸Ha = 27.2 eV is the Hartree energy). The factor 2푠2 denotes the degeneracy
+of the excitation 푠 including the spin factor. As specific for the Coulomb interaction, we have infinitely many bound states
+near the continuum edge for 푠 → ∞. Expression (A1) is not applicable because it is divergent. A convergent expression is the
+Planck-Brillouin-Larkin partition function, see9,
+푧H =
+∑
+푠
+2푠2
+[
+푒−퐸H,푠∕푘퐵푇 − 1 +
+퐸H,푠
+푘퐵푇
+]
+.
+(A2)
+The subtraction of 1 is explained as follows: We need to include the contribution of the scattering states which compensate for
+the most divergent term of the contribution of the bound states. For the short-range interaction, this has been discussed in detail,
+and generalized phase shifts have been introduced to avoid separating the bound and scattering parts of the intrinsic partition
+function53,54.
+More complex is the explanation of the subtraction of 퐸Ha∕(2푠2푘퐵푇 ). Because of the long-range character of the Coulomb
+interaction, phase shifts cannot be defined in the usual form, and the contribution of the scattering states is not well defined
+when scattering phase shifts are used. This fundamental problem of the Coulomb interaction is solved introducing the concept
+of screening. In the framework of a quantum statistical approach, we have to perform the partial sum of so-called ring diagrams
+and introduce quasiparticles. We must, however, avoid double counting. This has already been discussed in detail for the Hartree-
+Fock approximation55,56. Of interest is the generalization to partially ionized plasmas with multiply charged ions52.
+A systematic approach arises from consideration of the spectral function. We can identify a quasiparticle contribution and
+perform a cluster decomposition of the self-energy. For the cluster decomposition of the self-energy, we can introduce different
+channels. To avoid double counting, diagrams used for the single-particle self-energy must be subtracted from the ladder sums
+defining the cluster states.
+A related problem is the definition of the ionization degree in dense plasmas, since the separation of the bound state contri-
+bution from the intrinsic partition function is arbitrary, see57,58,59 and references given there. A possible solution would be the
+definition of the single-quasiparticle contribution which is extracted from the spectral function. Thus it can be performed by
+considering the compressibility or the dynamical conductivity.
+The inclusion of bound states and the corresponding generalization of the chemical picture, involving quasiparticle concepts
+for the free and bound states, is a difficult problem in plasma theory. Of course, at fixed temperature there is always a low-
+density limit at which bound states are dissolved (because of entropy) but this regime can be very limited, for instance it is not
+applicable to gases under normal conditions. A realistic description is often based on the chemical picture where bound states
+are considered, i.e. for temperatures below the binding energies. A generalized quasiparticle approach is well defined at low
+densities, but has to be generalized considering the spectral function (6) if densities are increasing. The formation of bound states
+is not only important for the thermodynamic properties, as discussed above for the second virial coefficient of the Hydrogen
+plasma. It also determines the transport properties, and the consideration of bound states as additional scatterers remains a
+complex problem if we want to go beyond the simple chemical picture.
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+

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+3DShape2VecSet: A 3D Shape Representation for Neural Fields and
+Generative Diffusion Models
+BIAO ZHANG, KAUST, Saudi Arabia
+JIAPENG TANG, TU Munich, Germany
+MATTHIAS NIESSNER, TU Munich, Germany
+PETER WONKA, KAUST, Saudi Arabia
+Input
+Reconstruction
+Input
+Reconstruction
+Condition
+Generation
+“the tallest chair”
+car
+Fig. 1. Left: Shape autoencoding results (surface reconstruction from point clouds) Right: the various down-stream applications of 3DShape2VecSet (from
+top to down): (a) category-conditioned generation; (b) point clouds conditioned generation (shape completion from partial point clouds); (c) image conditioned
+generation (shape reconstruction from single-view images); (d) text-conditioned generation.
+We introduce 3DShape2VecSet, a novel shape representation for neural fields
+designed for generative diffusion models. Our shape representation can en-
+code 3D shapes given as surface models or point clouds, and represents them
+as neural fields. The concept of neural fields has previously been combined
+with a global latent vector, a regular grid of latent vectors, or an irregu-
+lar grid of latent vectors. Our new representation encodes neural fields on
+top of a set of vectors. We draw from multiple concepts, such as the ra-
+dial basis function representation and the cross attention and self-attention
+function, to design a learnable representation that is especially suitable for
+processing with transformers. Our results show improved performance in
+3D shape encoding and 3D shape generative modeling tasks. We demon-
+strate a wide variety of generative applications: unconditioned generation,
+category-conditioned generation, text-conditioned generation, point-cloud
+completion, and image-conditioned generation.
+Additional Key Words and Phrases: 3D Shape Generation, 3D Shape Repre-
+sentation, Diffusion Models, Shape Reconstruction, Generative models
+Authors’ addresses: Biao Zhang, KAUST, Saudi Arabia, [email protected]; Jia-
+peng Tang, TU Munich, Germany, [email protected]; Matthias Nießner, TU Munich,
+Germany, [email protected]; Peter Wonka, KAUST, Saudi Arabia, peter.wonka@kaust.
+edu.sa.
+1
+INTRODUCTION
+The ability to generate realistic and diverse 3D content has many
+potential applications, including computer graphics, gaming, and vir-
+tual reality. To this end, many generative models have been explored,
+e.g., generative adversarial networks, variational autoencoders, nor-
+malizing flows, and autoregressive models. Recently, diffusion mod-
+els have emerged as one of the most popular method with fantastic
+results in the 2D image domain [Ho et al. 2020; Rombach et al. 2022]
+and have shown their superiority over other generative methods. For
+instance, it is possible to do unconditional generation [Karras et al.
+2022; Rombach et al. 2022], text conditioned generation [Rombach
+et al. 2022; Saharia et al. 2022], and generative image inpainting [Lug-
+mayr et al. 2022]. However, the success in the 2D domain has not
+yet been matched in the 3D domain.
+In this work, we will study diffusion models for 3D shape genera-
+tion. One major challenge in adapting 2D diffusion models to 3D is
+the design of a suitable shape representation. The design of such a
+shape representation is the major focus of our work, and we will
+discuss several design choices that lead to the development of our
+proposed representation.
+, Vol. 1, No. 1, Article . Publication date: January 2023.
+arXiv:2301.11445v1  [cs.CV]  26 Jan 2023
+
+Different from 2D images, there are several predominant ways
+to represent 3D data, e.g., voxels, point clouds, meshes, and neu-
+ral fields. In general, we believe that surface-based representations
+are more suitable for downstream applications than point clouds.
+Among the available choices, we choose to build on neural fields as
+they have many advantages: they are continuous, represent com-
+plete surfaces and not only point samples, and they enable many
+interesting combinations of traditional data structure design and
+representation learning using neural networks.
+Two major approaches for 2D diffusion models are to either use a
+compressed latent space, e.g., latent diffusion [Rombach et al. 2022],
+or to use a sequence of diffusion models of increasing resolution,
+e.g., [Ramesh 2022; Saharia et al. 2022]. While both of these ap-
+proaches seem viable in 3D, our initial experiments indicated that it
+is much easier to work with a compressed latent space. We therefore
+follow the latent diffusion approach.
+A subsequent design choice for a latent diffusion approach is to de-
+cide between a learned representation or a manually designed repre-
+sentation. A manually designed representation such as wavelets [Hui
+et al. 2022] is easier to design and more lightweight, but in many con-
+texts learned representations have shown to outperform manually
+designed ones. We therefore opt to explore learned representations.
+This requires a two-stage training strategy. The first stage is an
+autoencoder (variational autoencoder) to encode 3D shapes into a
+latent space. The second stage is training a diffusion model in the
+learned latent space.
+In the case of training diffusion models for 3D neural fields, it
+is even more necessary to generate in latent space. First, diffusion
+models often work with data of fixed size (e.g., images of a given
+fixed resolution). Second, a neural field is a continuous real-valued
+function that can be seen as an infinite-dimensional vector. For both
+reasons, we decide to find a way to encode shapes into latent space
+before all else (as well as a decoding method for reverting latents
+back to shapes).
+Finally, we have to design a suitable learned neural field rep-
+resentation that provides a good trade-off between compression
+and reconstruction quality. Such a design typically requires three
+components: a spatial data structure to store the latent information,
+a spatial interpolation method, and a neural network architecture.
+There are multiple options proposed in the literature shown in Fig. 2.
+Early methods used a single global latent vector in combination
+with an MLP network [Mescheder et al. 2019; Park et al. 2019]. This
+concept is simple and fast but generally struggles to reconstruct
+high-quality shapes. Better shape details can be achieved by using a
+3D regular grid of latents [Peng et al. 2020] together with tri-linear
+interpolation and an MLP. However, such a representation is too
+large for generative models and it is only possible to use grids of
+very low resolution (e.g., 8×8×8). By introducing sparsity, e.g., [Yan
+et al. 2022; Zhang et al. 2022], latents are arranged in an irregular
+grid. The latent size is largely reduced, but there is still a lot of
+room for improvement which we capitalize on in the design of
+3DShape2VecSet.
+The design of 3DShape2VecSet combines ideas from neural fields,
+radial basis functions, and the network architecture of attention
+layers. Similar to radial basis function representation for continuous
+functions, we can also re-write existing methods in a similar form
+(linear combination). Inspired by cross attention in the transformer
+network [Vaswani et al. 2017], we derived the proposed latent rep-
+resentation which is a fixed-size set of latent vectors. There are
+two main reasons that we believe contribute to the success of the
+representations. First, the representation is well-suited for the use
+with transformer-based networks. As transformer-based networks
+tend to outperform current alternatives, we can better benefit from
+this network architecture. Instead of only using MLPs to process
+latent information, we use a linear layer and cross-attention. Sec-
+ond, the representation no longer uses explicitly designed positional
+features, but only gives the network the option to encode positional
+information in any form it considers suitable. This is in line with our
+design principle of favoring learned representations over manually
+designed ones. See Fig. 2 e) for the proposed latent representation.
+Using our novel shape representation, we can train diffusion mod-
+els in the learned 3D shape latent space. Our results demonstrate an
+improved shape encoding quality and generation quality compared
+to the current state of the art. While pioneering work in 3D shape
+generation using diffusion models already showed unconditional
+3D shape generation, we show multiple novel applications of 3D dif-
+fusion models: category-conditioned generation, text-conditioned
+shape generation, shape reconstruction from single-view image, and
+shape reconstruction from partial point clouds.
+To sum up, our contributions are as follows:
+(1) We propose a new representation for 3D shapes. Any shape
+can be represented by a fixed-length array of latents and
+processed with cross-attention and linear layers to yield a
+neural field.
+(2) We propose a new network architecture to process shapes
+in the proposed representation, including a building block to
+aggregate information from a large point cloud using cross-
+attention.
+(3) We improve the state of the art in 3D shape autoencoding to
+yield a high fidelity reconstruction including local details.
+(4) We propose a latent set diffusion. We improve the state of
+the art in 3D shape generation as measured by FID, KID, FPD,
+and KPD.
+(5) We show 3D shape diffusion for category-conditioned gener-
+ation, text-conditioned generation, point-cloud completion,
+and image-conditioned generation.
+2
+RELATED WORK
+In this section, we briefly review the literature of 3D shape learning
+with various data representations and 3D shape generative models.
+2.1
+3D Shape Representations
+We mainly discuss the following representations for 3D shapes,
+including voxels, point clouds, and neural fields.
+Voxels. Voxel grids, extended from 2D pixel grids, simply repre-
+sent a 3D shape as a discrete volumetric grid. Due to their regular
+structure, early works take advantage of 3D transposed convolution
+operators for shape prediction [Brock et al. 2016; Choy et al. 2016;
+Dai et al. 2017; Girdhar et al. 2016; Wu et al. 2016, 2015]. A draw-
+back of the voxels-based decoders is that the computational and
+memory costs of neural networks cubicly increases with respect to
+2
+
+(x𝑖, 𝜆𝑖 )
+x
+𝜙 (x, x𝑖 )
+(a) RBF
+(b) Global Latent
+x
+(x𝑖, f𝑖 )
+(c) Latent Grid
+(x𝑖, f𝑖 )
+x
+𝜙 (x, x𝑖 )
+(d) Irregular Latent Grid
+x
+f𝑖
+𝜙 (x, f𝑖 ) = exp
+�
+q(x)⊺k(f𝑖 )/
+√
+𝑑
+�
+(e) Latent Set (Ours)
+Fig. 2. Continuous function representation. Scalars are represented with spheres while vectors are cubes. The arrows show how spatial interpolation is
+computed. x𝑖 and x are the coordinates of an anchor and a querying point respectively. 𝜆𝑖 is the SDF value of the anchor point x𝑖 in (a). f𝑖 is the associate
+feature vector located in x𝑖 in (c)(d). The queried SDF/feature of x is based on the distance function 𝜙 (x, x𝑖) in (a)(c)(d), while our proposed latent set
+representation (e) utilizes the similarity 𝜙 (x, f𝑖) between querying coordinate and anchored features via cross attention mechanism.
+# Latents
+Latent Position
+Methods
+OccNet [Mescheder et al. 2019]
+DeepSDF [Park et al. 2019]
+Single
+Global
+IM-Net [Chen and Zhang 2019]
+ConvOccNet [Peng et al. 2020]
+IF-Net [Chibane et al. 2020]
+LIG [Jiang et al. 2020]
+DeepLS [Chabra et al. 2020]
+SA-ConvOccNet [Tang et al. 2021]
+Multiple
+Regular Grid
+NKF [Williams et al. 2022]
+LDIF [Genova et al. 2020]
+Point2Surf [Erler et al. 2020]
+DCC-DIF [Li et al. 2022]
+3DILG [Zhang et al. 2022]
+Multiple
+Irregular Grid
+POCO [Boulch and Marlet 2022]
+Multiple
+Global
+Ours
+Table 1. Neural Fields for 3D Shapes. We show different types how la-
+tents are positioned.
+the grid resolution. Thus, most voxel-based methods are limited to
+low-resolution. Octree-based decoders [Häne et al. 2017; Meagher
+1980; Riegler et al. 2017b,a; Tatarchenko et al. 2017; Wang et al.
+2017, 2018] and sparse hash-based decoders [Dai et al. 2020] take
+3D space sparsity into account, alleviating the efficiency issues and
+supporting high-resolution outputs.
+Point Clouds. Early works on neural-network-based point cloud
+processing include PointNet [Qi et al. 2017a,b] and DGCNN [Wang
+et al. 2019]. These works are built upon per-point fully connected
+layers. More recently, transformers [Vaswani et al. 2017] were pro-
+posed for point cloud processing, e.g., [Guo et al. 2021; Zhang et al.
+2022; Zhao et al. 2021]. These works are inspired by Vision Trans-
+formers (ViT) [Dosovitskiy et al. 2021] in the image domain. Points
+are firstly grouped into patches to form tokens and then fed into
+a transformer with self-attention. In this work, we also introduce
+a network for processing point clouds. Improving upon previous
+works, we compress a given point cloud to a small representation
+that is more suitable for generative modeling.
+Neural Fields. A recent trend is to use neural fields as a 3d data
+representation. The key building block is a neural network which
+accepts a 3D coordinate as input, and outputs a scalar [Chen and
+Generative
+3D
+Models
+Representation
+3D-GAN [Wu et al. 2016]
+GAN
+Voxels
+l-GAN [Achlioptas et al. 2018]
+GAN★
+Point Clouds
+IM-GAN [Chen and Zhang 2019]
+GAN★
+Fields
+PointFlow [Yang et al. 2019]
+NF
+Point Clouds
+GenVoxelNet [Xie et al. 2020]
+EBM
+Voxels
+PointGrow [Sun et al. 2020]
+AR
+Point Clouds
+PolyGen [Nash et al. 2020]
+AR
+Meshes
+GenPointNet [Xie et al. 2021]
+EBM
+Point Clouds
+3DShapeGen [Ibing et al. 2021]
+GAN★
+Fields
+DPM [Luo and Hu 2021]
+DM
+Point Clouds
+PVD [Zhou et al. 2021]
+DM
+Point Clouds
+AutoSDF[Mittal et al. 2022]
+AR★
+Voxels
+CanMap [Cheng et al. 2022]
+AR★
+Point Clouds
+ShapeFormer[Yan et al. 2022]
+AR★
+Fields
+3DILG [Zhang et al. 2022]
+AR★
+Fields
+LION [Zeng et al. 2022]
+DM★
+Point Clouds
+SDF-StyleGAN [Zheng et al. 2022]
+GAN
+Fields
+NeuralWavelet [Hui et al. 2022]
+DM★
+Fields
+TriplaneDiffusion [Shue et al. 2022]⋄
+DM★
+Fields
+DiffusionSDF [Chou et al. 2022]⋄
+DM★
+Fields
+Ours
+DM★
+Fields
+★ Generative models in latent space.
+⋄ Works in submission.
+Table 2. Generative models for 3d shapes.
+Zhang 2019; Mescheder et al. 2019; Michalkiewicz et al. 2019; Park
+et al. 2019] or a vector [Mildenhall et al. 2020]. A 3D object is then
+implicitly defined by this neural network. Neural fields have gained
+lots of popularity as they can generate objects with arbitrary topolo-
+gies and infinite resolution. The methods are also called neural
+implicit representations or coordinate-based networks. For neural
+fields for 3d shape modeling, we can categorize methods into global
+methods and local methods. 1) The global methods encode a shape
+with a single global latent vector [Mescheder et al. 2019; Park et al.
+2019]. Usually the capacity of these kind of methods is limited and
+3
+
+they are unable to encode shape details. 2) The local methods use
+localized latent vectors which are defined for 3D positions defined
+on either a regular [Chibane et al. 2020; Jiang et al. 2020; Peng et al.
+2020; Tang et al. 2021] or irregular grid [Boulch and Marlet 2022;
+Genova et al. 2020; Li et al. 2022; Zhang et al. 2022]. In contrast, we
+propose a latent representation where latent vectors do not have
+associated 3D positions. Instead, we learn to represent a shape as a
+list of latent vectors. See Tab. 1.
+2.2
+Generative models.
+We have seen great success in different 2D image generative mod-
+els in the past decade. Popular deep generative methods include
+generative adversarial networks (GANs) [Goodfellow et al. 2014],
+variational autoencoers (VAEs) [Kingma and Welling 2014], nor-
+malizing flows (NFs) [Rezende and Mohamed 2015], energy-based
+models [LeCun et al. 2006; Xie et al. 2016], autoregressive models
+(ARs) [Esser et al. 2021; Van Den Oord et al. 2017] and more re-
+cently, diffusion models (DMs) [Ho et al. 2020] which are the chosen
+generative model in our work.
+In 3D domain, GANs have been popular for 3D generation [Achliop-
+tas et al. 2018; Chen and Zhang 2019; Ibing et al. 2021; Wu et al.
+2016; Zheng et al. 2022], while only a few works are using NFs [Yang
+et al. 2019] and VAEs [Mo et al. 2019]. A lot of recent work employs
+ARs [Cheng et al. 2022; Mittal et al. 2022; Nash et al. 2020; Sun et al.
+2020; Yan et al. 2022; Zhang et al. 2022]. DMs for 3D shapes are
+relatively unexplored compared to other generative methods.
+There are several DMs dealing with point cloud data [Luo and Hu
+2021; Zeng et al. 2022; Zhou et al. 2021]. Due to the high freedom
+degree of regressed coordinates, it is always difficult to obtain clean
+manifold surfaces via post-processing. As mentioned before, we
+believe that neural fields are generally more suitable than point
+clouds for 3D shape generation. The area of combining DMs and
+neural fields is still underexplored.
+The recent NeuralWavelet [Hui et al. 2022] first encodes shapes
+(represented as signed distance fields) into the frequency domain
+with the wavelet transform, and then train DMs on the frequency
+coefficients. While this formulation is elegant, generative models
+generally work better on learned representations. Some concurrent
+works [Chou et al. 2022; Shue et al. 2022] in submission also utilize
+DMs in a latent space for neural field generation. The TriplaneDif-
+fusion [Shue et al. 2022] trains an autodecoder first for each shape.
+DiffusionSDF [Chou et al. 2022] runs a shape autoencoder based on
+triplane features [Peng et al. 2020].
+Summary of 3D generation methods. We list several 3d generation
+methods in Tab. 2, highlighting the choice of generative model (GAN,
+DM, EBM, NF, or AR) and the choice of data structure to represent
+3D shapes (point clouds, meshes, voxels or fields).
+3
+PRELIMINARIES
+An attention layer [Vaswani et al. 2017] has three types of inputs:
+queries, keys, and values. Queries Q = [q1, q2, . . . , q𝑁𝑞] ∈ R𝑑×𝑁𝑞
+and keys K = [k1, k2, . . . , k𝑁𝑘 ] ∈ R𝑑×𝑁𝑘 are first compared to
+produce coefficients q⊺
+𝑗 k𝑖/
+√
+𝑑 (they need to be normalized with the
+softmax function),
+𝐴𝑖,𝑗 =
+q⊺
+𝑗 k𝑖/
+√
+𝑑
+�𝑁𝑘
+𝑖=1 exp
+�
+q⊺
+𝑗 k𝑖/
+√
+𝑑.
+�
+(1)
+The coefficients are then used to (linearly) combine values V =
+[v1, v2, . . . , v𝑁𝑘 ] ∈ R𝑑𝑣×𝑁𝑘 . We can write the output of an attention
+layer as follows,
+Attention(Q, K, V)
+=
+�o1
+o2
+· · ·
+o𝑁𝑞
+�
+∈ R𝑑𝑣×𝑁𝑞
+=
+� 𝑁𝑘
+∑︁
+𝑖=1
+𝐴𝑖,1v𝑖
+𝑁𝑘
+∑︁
+𝑖=1
+𝐴𝑖,2v𝑖
+· · ·
+𝑁𝑘
+∑︁
+𝑖=1
+𝐴𝑖,𝑁𝑞v𝑖
+�
+(2)
+Cross Attention. Given two sets A =
+�
+a1, a2, . . . , a𝑁𝑎
+�
+∈ R𝑑𝑎×𝑁𝑎
+and B =
+�
+b1, b2, . . . , b𝑁𝑏
+�
+∈ R𝑑𝑏×𝑁𝑏 , the query vectors Q are con-
+structed with a linear function q(·) : R𝑑𝑎 → R𝑑 by taking elements
+of A as input. Similarly, we construct K and V with k(·) : R𝑑𝑏 → R𝑑
+and v(·) : R𝑑𝑏 → R𝑑, respectively. The inputs of both k(·) and v(·)
+are from B. Each column in the output of Eq. (2) can be written as,
+o(a𝑗, B) =
+𝑁𝑏
+∑︁
+𝑖=1
+v(b𝑖) ·
+1
+𝑍 (a𝑗, B) exp
+�
+q(a𝑗)⊺k(b𝑖)/
+√
+𝑑
+�
+,
+(3)
+where 𝑍 (a𝑗, B) = �𝑁𝑏
+𝑖=1 exp
+�
+q(a𝑗)⊺k(b𝑖)/
+√
+𝑑
+�
+is a normalizing fac-
+tor. The cross attention operator between two sets is,
+CrossAttn(A, B) =
+�o(a1, B)
+o(a2, B)
+· · ·
+o(a𝑁𝑎, B)�
+∈ R𝑑×𝑁𝑎
+(4)
+Self Attention. In the case of self attention, we let the two sets be
+the same A = B,
+SelfAttn(A) = CrossAttn(A, A).
+(5)
+4
+LATENT REPRESENTATION FOR NEURAL FIELDS
+Our representation is inspired by radial basis functions (RBFs). We
+will therefore describe our surface representation design using RBFs
+as a starting point, and how we extended them using concepts
+from neural fields and the transformer architecture. A continuous
+function can be represented with a set of weighted points in 3D
+using RBFs:
+ˆORBF(x) =
+𝑀
+∑︁
+𝑖=1
+𝜆𝑖 · 𝜙(x, x𝑖)
+(6)
+where 𝜙(x, x𝑖) is a radial basis function (RBF) and typically repre-
+sents the similarity (or dissimilarity) between two inputs,
+𝜙(x, x𝑖) = 𝜙(∥x − x𝑖 ∥).
+(7)
+Given ground-truth occupancies of x𝑖, the values of 𝜆𝑖 can be ob-
+tained by solving a system of linear equations. In this way, we
+can represent the continuous function O(·) as a set of 𝑀 points
+including their corresponding weights,
+�
+𝜆𝑖 ∈ R, x𝑖 ∈ R3�𝑀
+𝑖=1 .
+(8)
+However, in order to retain the details of a 3d shape, we often need
+a very large number of points (e.g., 𝑀 = 80, 000 in [Carr et al. 2001]).
+4
+
+Shape Encoding (Sec. 5.1)
+Shape Decoding (Sec. 5.3)
+KL (Sec. 5.2)
+latent queries
+Point Cloud
+Position Embeddings
+Surface Sampling
+Cross Attention
+K, V
+Q
+...
+...
+latents
+KL Regularization
+Self Attention
+...
+Self Attention
+...
+· · ·
+...
+Self Attention
+Query Points
+Position Embeddings
+Cross Attention
+K, V
+Q
+Target
+· · ·
+Isosurface
+Fig. 3. Shape autoencoding pipeline. Given a 3D ground-truth surface mesh as the input, we first sample a point cloud that is mapped to positional
+embeddings and encode them into a set of latent codes through a cross-attention module (Sec. 5.1). Next, we perform (optional) compression and KL-
+regularization in the latent space to obtain structured and compact latent shape representations (Sec. 5.2). Finally, the self-attention is carried out to aggregate
+and exchange the information within the latent set. And a cross-attention module is designed to calculate the interpolation weights of query points. The
+interpolated feature vectors are fed into a fully connected layer for occupancy prediction (Sec. 5.3).
+This representation does not benefit from recent advances in repre-
+sentation learning and cannot compete with more compact learned
+representations. We therefore want to modify the representation to
+change it into a neural field.
+One approach to neural fields is to represent each shape as a
+separate neural network (making the network weights of a fixed
+size network the representation of a shape) and train a diffusion
+process as hypernetwork. A second approach is to have a shared
+encoder-decoder network for all shapes and represent each shape as
+a latent computed by the encoder. We opt for the second approach,
+as it leads to more compact representations because it is jointly
+learned from all shapes in the data set and the network weights
+themselves do not count towards the latent representation. Such a
+neural field takes a tuple of coordinates x and 𝐶-dimensional latent
+f as input and outputs occupancy,
+ˆONN(x) = NN(x, f),
+(9)
+where NN : R3 × R𝐶 → [0, 1] is a neural network. A first approach
+was to use a single global latent f, but a major limitation is the ability
+to encode shape details [Mescheder et al. 2019]. Some follow-up
+works study coordinate-dependent latents [Chibane et al. 2020; Peng
+et al. 2020] that combine traditional data structures such as regular
+grids with the neural field concept. Latent vectors are arranged in a
+spatial data structure and then interpolated (trilinearly) to obtain
+the coordinate-dependent latent fx. A recent work 3DILG [Zhang
+et al. 2022] proposed a sparse representation for 3D shapes, using
+latents f𝑖 arranged in an irregular grid at point locations x𝑖. The
+final coordinate-dependent latent fx is then estimated by kernel
+regression,
+fx = ˆFKN(x) =
+𝑀
+∑︁
+𝑖=1
+f𝑖 ·
+1
+𝑍
+�
+x, {x𝑖}𝑀
+𝑖=1
+� 𝜙(x, x𝑖),
+(10)
+where 𝑍
+�
+x, {x𝑖}𝑀
+𝑖=1
+�
+= �𝑀
+𝑖=1 𝜙(x, x𝑖) is a normalizing factor. Thus
+the representation for a 3D shape can be written as
+�
+f𝑖 ∈ R𝐶, x𝑖 ∈ R3�𝑀
+𝑖=1 .
+(11)
+After that, an MLP : R𝐶 → [0, 1] is applied to project the approxi-
+mated feature ˆFKN(x) to occupancy,
+ˆO3DILG(x) = MLP
+�
+ˆFKN(x)
+�
+.
+(12)
+Neural networks with latent sets (proposed). We initially explored
+many variations for 3D shape representation based on irregular
+and regular grids as well as tri-planes, frequency compositions, and
+other factored representations. Ultimately, we could not improve
+on existing irregular grids. However, we were able to achieve a
+significant improvement with the following chance. We aim to keep
+the structure of an irregular grid and the interpolation, but without
+representing the actual spatial position explicitly. We let the net-
+work encode spatial information. Both the representations (RBF in
+Eq. (6) and 3DILG in Eq. (10)) are composed by two parts, values and
+similarities. We keep the structure of the interpolation, but elmini-
+tate explicit point coordinates and integrate cross attention from
+Eq. (3). The result is the following learnable function approximator,
+ˆF (x) =
+𝑀
+∑︁
+𝑖=1
+v(f𝑖) ·
+1
+𝑍
+�
+x, {f𝑖}𝑀
+𝑖=1
+� 𝑒q(x)⊺k(f𝑖)/
+√
+𝑑,
+(13)
+where 𝑍
+�
+x, {f𝑖}𝑀
+𝑖=1
+�
+= �𝑀
+𝑖=1 𝑒q(x)⊺k(f𝑖)/
+√
+𝑑 is a normalizing factor.
+Similar to the MLP in Eq. 12, we apply a single fully connected layer
+to get desired occupancy values,
+ˆO(x) = FC
+�
+ˆF (x)
+�
+.
+(14)
+Compared to 3DILG and all other coordinate-latent-based methods,
+we dropped the dependency of the coordinate set {x𝑖}𝑀
+𝑖=1, the new
+5
+
+Cross Attention
+K, V
+Q
+Learnable
+(a) Learnable Queries
+Cross Attention
+K, V
+Q
+Subsample and Copy
+(b) Point Queries
+Fig. 4. Two ways to encode a point cloud. (a) uses a learnable query set;
+(b) uses a downsampled version of input point embeddings as the query set.
+representation only contains a set of latents,
+�
+f𝑖 ∈ R𝐶�𝑀
+𝑖=1 .
+(15)
+An alternative view of our proposed function approximator is to
+see it as cross attention between query points x and a set of latents.
+5
+NETWORK ARCHITECTURE FOR SHAPE
+REPRESENTATION LEARNING
+In this section, we will discuss how we design a variational autoen-
+coder based on the latent representation proposed in Sec. 4. The
+architecture has three components discussed in the following: a 3D
+shape encoder, KL regularization block, and a 3D shape decoder.
+5.1
+Shape encoding
+We sample the surfaces of 3D input shapes in a 3D shape dataset.
+This results in a point clouds of size 𝑁 for each shape, {x𝑖 ∈ R3}𝑁
+𝑖=1
+or in matrix form X ∈ R3×𝑁 . While the dataset used in the paper
+originally represents shapes as triangle meshes, our framework
+is directly compatible with other surface representations, such as
+scanned point clouds, spline surfaces, or implicit surfaces.
+In order to learn representations in the form of Eq. (15), the first
+challenge is to aggregate the information contained in a possibly
+large point cloud {x𝑖}𝑁
+𝑖=1 into a smaller set of latent vectors {f𝑖}𝑀
+𝑖=1.
+We design a set-to-set network to this effect.
+A popular solution to this problem in previous work is to divide
+the large point cloud into a smaller set of patches and to learn one
+latent vector per patch. Although this is a very well researched
+and standard component in many networks, we discovered a more
+successful way to aggregate features from a large point cloud that is
+better compatible with the transformer architecture. We considered
+two options.
+One way is to define a learnable query set. Inspired by DETR [Car-
+ion et al. 2020] and Perceiver [Jaegle et al. 2021], we use the cross
+attention to encode X,
+Enclearnable(X) = CrossAttn(L, PosEmb(X)) ∈ R𝐶×𝑀,
+(16)
+where L ∈ R𝐶×𝑀 is a learnable query set where each entry is 𝐶-
+dimensional, and PosEmb : R3 → R𝐶 is a column-wise positional
+embedding function.
+Another way is to utilize the point cloud itself. We first subsample
+the point cloud X to a smaller one with furthest point sampling,
+X0 = FPS(X) ∈ R3×𝑀. The cross attention is applied to X0 and X,
+Encpoints(X) = CrossAttn(PosEmb(X0), PosEmb(X)),
+(17)
+which can also be seen as a “partial” self attention. See Fig. 4 for
+an illustration of both design choices. Intuitively, the number 𝑀
+affects the reconstruction performance: the larger the 𝑀, the better
+reconstruction. However, 𝑀 strongly affects the training time due
+to the transformer architecture, so it should not be too large. In our
+final model, the number of latents 𝑀 is set as 512, and the number
+of channels 𝐶 is 512 to provide a trade off between reconstruction
+quality and training time.
+5.2
+KL regularization block
+Latent diffusion [Rombach et al. 2022] proposed to use a variational
+autoencoder (VAE) [Kingma and Welling 2014] to compress images.
+We adapt this design idea for our 3D shape representation and
+also regularize the latents with KL-divergence. We should note
+that the KL regularization is optional and only necessary for the
+second-stage diffusion model training. If we just want a method for
+surface reconstruction from point clouds, we do not need the KL
+regularization.
+We first linear project latents to mean and variance by two net-
+work branches, respectively,
+FC𝜇 (f𝑖) = �𝜇𝑖,𝑗
+�
+𝑗 ∈[1,2,···,𝐶0]
+FC𝜎 (f𝑖) =
+�
+log𝜎2
+𝑖,𝑗
+�
+𝑗 ∈[1,2,···,𝐶0]
+(18)
+where FC𝜇 : R𝐶 → R𝐶0 and FC𝜎 : R𝐶 → R𝐶0 are two linear
+projection layers. We use a different size of output channels 𝐶0,
+where 𝐶0 ≪ 𝐶. This compression enables us to train diffusion
+models on smaller latents of total size 𝑀 · 𝐶0 ≪ 𝑀 · 𝐶. We can
+write the bottleneck of the VAE formally, ∀𝑖 ∈ [1, 2, · · · , 𝑀], 𝑗 ∈
+[1, 2, · · · ,𝐶0],
+𝑧𝑖,𝑗 = 𝜇𝑖,𝑗 + 𝜎𝑖,𝑗 · 𝜖,
+(19)
+where 𝜖 ∼ N (0, 1). The KL regularization can be written as,
+Lreg
+�
+{f𝑖}𝑀
+𝑖=1
+�
+=
+1
+𝑀 · 𝐶0
+𝑀
+∑︁
+𝑖=1
+𝐶0
+∑︁
+𝑗=1
+1
+2
+�
+𝜇2
+𝑖,𝑗 + 𝜎2
+𝑖,𝑗 − log𝜎2
+𝑖,𝑗
+�
+.
+(20)
+In practice, we set the weight for KL loss as 0.001 and report the
+performance for different values of𝐶0 in Sec. 8.1. Our recommended
+setting is 𝐶0 = 32.
+5.3
+Shape decoding
+To increase the expressivity of the network, we add a latent learning
+network between the two parts. Because our latents are a set of
+vectors, it is natural to use transformer networks here. Thus, the
+proposed network here is a series of self attention blocks,
+{f𝑖}𝑀
+𝑖=1 ← SelfAttn(𝑙) �
+{f𝑖}𝑀
+𝑖=1
+�
+,
+for 𝑖 = 1, · · · , 𝐿.
+(21)
+The SelfAttn(·) with a superscript (𝑙) here means 𝑙-th block. The
+latents {f𝑖}𝑀
+𝑖=1 obtained using either Eq. (16) or Eq. (17) are fed into
+the self attention blocks. Given a query x, the corresponding latent
+is interpolated using Eq. (13), and the occupancy is obtained with a
+fully connected layer as shown in Eq. (14).
+6
+
+Fig. 5. KL regularization. Given a set of latents {f𝑖 ∈ R𝐶 }𝑀
+𝑖=1 obtained
+from the shape encoding in Sec. 5.1, we employ two linear projection layers
+FC𝜇, FC𝜎 to predict the mean and variance of a low-dimensional latent
+space, where a KL regularization commonly used in VAE training is applied
+to constrain the feature diversity. Then, we obtain smaller latents {z𝑖 ∈
+R𝐶0 } of size 𝑀 · 𝐶0 ≪ 𝑀 · 𝐶 via reparametrization sampling. Finally, the
+compressed latents are mapped back to the original space by FCup to obtain
+a higher dimensionality for the shape decoding in Sec. 5.3.
+Forward Diffusion Process
+Reverse Diffusion Process
+Add Noise
+Add Noise
+Add Noise
+Denoise
+Denoise
+Denoise
+Condition
+Fig. 6. Latent set diffusion models. The diffusion model operates on
+compressed 3D shapes in the form of a regularized set of latent vectors
+{z𝑖 }𝑀
+𝑖=1.
+Self Attention
+Self Attention
+· · ·
+(a) Unconditional Denoising Network
+Self Attention
+Cross Attention
+K V
+Q
+· · ·
+Condition
+(b) Conditional Denoising Network
+Fig. 7. Denoising network. Our denoising network is composed of several
+denoising layers (a box in the figure denotes a layer). The denoising layer
+for unconditional generation contains two sequential self attention blocks.
+The denoising layer for conditional generation contains a self attention
+and a cross attention block. The cross attention is for injecting condition
+information such as categories, images or partial point clouds.
+Loss. We optimize the binary cross entropy loss between our
+approximated function and the ground-truth indicator function as
+in prior works [Mescheder et al. 2019].
+Lrecon
+�
+{f𝑖}𝑀
+𝑖=1, O
+�
+= Ex∈R3
+�
+BCE
+�
+ˆO(x), O(x)
+��
+.
+(22)
+Surface reconstruction. We sample query points in a grid of res-
+olution 1283. The final surface is reconstructed with Marching
+Cubes [Lorensen and Cline 1987].
+6
+SHAPE GENERATION
+Our proposed diffusion model combines design decisions from latent
+diffusion (the idea of the compressed latent space), EDM [Karras et al.
+2022] (most of the training details), and our shape representation
+design (the architecture is based on attention and self-attention
+instead of convolution).
+We train diffusion models in the latent space, i.e., the bottleneck
+in Eq. (19). Following the diffusion formulation in EDM [Karras et al.
+2022], our denoising objective is
+En𝑖∼N(0,𝜎2I)
+1
+𝑀
+𝑀
+∑︁
+𝑖=1
+���Denoiser
+�
+{z𝑖 + n𝑖}𝑀
+𝑖=1, 𝜎, C
+�
+𝑖 − z𝑖
+���
+2
+2 ,
+(23)
+where Denoiser(·, ·, ·) is our denoising neural network, 𝜎 is the noise
+level, and C is the optional conditional information (e.g., categories,
+images, partial point clouds and texts). We denote the corresponding
+output of z𝑖 +n𝑖 with the subscript 𝑖, i.e. Denoiser(·, ·, ·)𝑖. We should
+minimize the loss for every noise level 𝜎. The sampling is done by
+solving ordinary/stochastic differential equations (ODE/SDE). See
+Fig. 6 for an illustration and EDM [Karras et al. 2022] for a detailed
+description for both the forward (training) and reverse (sampling)
+process.
+The function Denoiser(·, ·, ·) is a set denoising network (set-to-set
+function). The network can be easily modeled by a self-attention
+transformer. Each layer consists of two attention blocks. The first
+one is a self attention for attentive learning of the latent set. The
+second one is for injecting the condition information C (Fig. 7 (b))
+as in prior works [Rombach et al. 2022]. For simple information
+like categories, C is a learnable embedding vector (e.g., 55 different
+embedding vectors for 55 categories). For a single-view image , we
+use ResNet-18 [He et al. 2016] as the context encoder to extract
+a global feature vector as condition C. For text conditioning, we
+use BERT [Devlin et al. 2018] to learn a global feature vector as
+C. For partial point clouds, we use the shape encoder introduced
+in Sec. 5.1 to obtain a set of latent embeddings as C. In the case
+of unconditional generation, the cross attention degrades to self
+attention (Fig. 7 (a)).
+7
+EXPERIMENTAL SETUP
+We use the dataset of ShapeNet-v2 [Chang et al. 2015] as a bench-
+mark, containing 55 categories of man-made objects. We use the
+training/val splits in [Zhang et al. 2022]. We preprocess shapes as
+in [Mescheder et al. 2019]. Each shape is first converted to a water-
+tight mesh, and then normalized to its bounding box, from which we
+further sample a dense surface point cloud of size 50,000. To learn
+the neural fields, we randomly sample 50,000 points with occupan-
+cies in the 3D space, and 50,000 points with occupancies in the near
+surface region. For the single-view object reconstruction, we use
+the 2D rendering dataset provided by 3D-R2N2 [Choy et al. 2016],
+where each shape is rendered into RGB images of size of 224 × 224
+from 24 random viewpoints. For text-driven shape generation, we
+use the text prompts of ShapeGlot [Achlioptas et al. 2019]. For data
+preprocess of shape completion training, we create partial point
+clouds by sampling point cloud patches.
+7.1
+Baselines
+For shape auto-encoding, we conduct experiments against state-
+of-the-art methods for implicit surface reconstruction from point
+clouds. We use OccNet [Mescheder et al. 2019], ConvOccNet [Peng
+et al. 2020], IF-Net [Chibane et al. 2020], and 3DILG [Zhang et al.
+2022] as baselines. The OccNet is the first work of learning neural
+fields from a single global latent vector. ConvOccNet and IF-Net
+7
+
+OccNet
+ConvOccNet
+IF-Net
+3DILG
+Ours
+Learned Queries
+Point Queries
+table
+0.823
+0.847
+0.901
+0.963
+0.965
+0.971
+car
+0.911
+0.921
+0.952
+0.961
+0.966
+0.969
+chair
+0.803
+0.856
+0.927
+0.950
+0.957
+0.964
+airplane
+0.835
+0.881
+0.937
+0.952
+0.962
+0.969
+sofa
+0.894
+0.930
+0.960
+0.975
+0.975
+0.982
+rifle
+0.755
+0.871
+0.914
+0.938
+0.947
+0.960
+lamp
+0.735
+0.859
+0.914
+0.926
+0.931
+0.956
+mean (selected)
+0.822
+0.881
+0.929
+0.952
+0.957
+0.967
+IoU ↑
+mean (all)
+0.825
+0.888
+0.934
+0.953
+0.955
+0.965
+table
+0.041
+0.036
+0.029
+0.026
+0.026
+0.026
+car
+0.082
+0.083
+0.067
+0.066
+0.062
+0.062
+chair
+0.058
+0.044
+0.031
+0.029
+0.028
+0.027
+airplane
+0.037
+0.028
+0.020
+0.019
+0.018
+0.017
+sofa
+0.051
+0.042
+0.032
+0.030
+0.030
+0.029
+rifle
+0.046
+0.025
+0.018
+0.017
+0.016
+0.014
+lamp
+0.090
+0.050
+0.038
+0.036
+0.035
+0.032
+mean (selected)
+0.058
+0.040
+0.034
+0.032
+0.031
+0.030
+Chamfer ↓
+mean (all)
+0.072
+0.052
+0.041
+0.040
+0.039
+0.038
+table
+0.961
+0.982
+0.998
+0.999
+0.999
+0.999
+car
+0.830
+0.852
+0.888
+0.892
+0.898
+0.899
+chair
+0.890
+0.943
+0.990
+0.992
+0.994
+0.997
+airplane
+0.948
+0.982
+0.994
+0.993
+0.994
+0.995
+sofa
+0.918
+0.967
+0.988
+0.986
+0.986
+0.990
+rifle
+0.922
+0.987
+0.998
+0.997
+0.998
+0.999
+lamp
+0.820
+0.945
+0.970
+0.971
+0.970
+0.975
+mean (selected)
+0.898
+0.951
+0.975
+0.976
+0.977
+0.979
+F-Score ↑
+mean (all)
+0.858
+0.933
+0.967
+0.966
+0.966
+0.970
+Table 3. Shape autoencoding (surface reconstruction from point clouds) on ShapeNet. We show averaged metrics on all 55 categories and individual
+metrics for the 7 largest categories.
+𝑀 = 512 𝑀 = 256 𝑀 = 128 𝑀 = 64
+IoU ↑
+0.965
+0.956
+0.940
+0.916
+Chamfer ↓
+0.038
+0.039
+0.043
+0.049
+F-Score ↑
+0.970
+0.965
+0.953
+0.929
+Table 4. Results for different number of latents 𝑀 for
+shape autoencoding
+𝐶0 = 1 𝐶0 = 2 𝐶0 = 4 𝐶0 = 8 𝐶0 = 16 𝐶0 = 32 𝐶0 = 64
+IoU ↑
+0.727
+0.816
+0.957
+0.960
+0.962
+0.963
+0.964
+Chamfer ↓
+0.133
+0.087
+0.038
+0.038
+0.038
+0.038
+0.038
+F-Score ↑
+0.703
+0.815
+0.967
+0.967
+0.970
+0.969
+0.970
+Table 5. Ablation study of compression via the number of channels𝐶0 for shape
+(variational) autoencoding.
+Grid-83
+3DILG
+Ours
+𝐶0 = 8 𝐶0 = 16 𝐶0 = 32 𝐶0 = 64
+Surface-FPD ↓
+4.03
+1.89
+2.71
+1.87
+0.76
+0.97
+Surface-KPD (×103) ↓
+6.15
+2.17
+3.48
+2.42
+0.66
+1.11
+Rendering-FID ↓
+32.78
+24.83
+28.25
+27.26
+17.08
+24.24
+Rendering-KID (×103) ↓
+14.12
+10.51
+14.60
+19.37
+6.75
+11.76
+Table 6. Unconditional generation on full ShapeNet.
+PVD
+Ours
+Surface-FPD ↓
+2.33
+0.63
+Surface-KPD (×103) ↓
+2.65
+0.53
+Rendering-FID ↓
+270.64
+17.08
+Rendering-KID (×103) ↓
+281.54
+6.75
+Table 7. Unconditional generation on full ShapeNet.
+learn local neural fields based on latent vectors arranged in a regular
+grid, while 3DILG uses latent vectors on an irregular grid.
+For 3D shape generation, we compare against recent state-of-the-
+art generative models, including PVD [Zhou et al. 2021], 3DILG [Zhang
+8
+
+Input
+GT
+OccNet
+ConvONet
+IF-Net
+3DILG
+Proposed
+Learnable Queries
+Point Queries
+Fig. 8. Visualization of shape autoencoding results (surface reconstruction from point clouds from ShapeNet).
+et al. 2022], and NeuralWavelet [Hui et al. 2022]. PVD is a diffusion
+model for 3D point cloud generation, and 3DILG utilizes autore-
+gressive models. NeuralWavelet utilized diffusion models in the
+frequency domain of shapes.
+9
+
+Ours
+3DILG
+Grid-83
+PVD
+Fig. 9. Unconditional generation. All models are trained on full ShapeNet.
+airplane
+chair
+table
+car
+sofa
+3DILG
+NW
+Ours
+3DILG
+NW
+Ours
+3DILG
+NW
+Ours
+3DILG
+NW
+Ours
+3DILG
+NW
+Ours
+Surface-FID
+0.71
+0.38
+0.62
+0.96
+1.14
+0.76
+2.10
+1.12
+1.19
+2.93
+-
+2.04
+1.83
+-
+0.77
+Surface-KID (×103)
+0.81
+0.53
+0.83
+1.21
+1.50
+0.70
+3.84
+1.55
+1.87
+7.35
+-
+3.90
+3.36
+-
+0.70
+Table 8. Category conditioned generation. NW is short for NeuralWavelet. The dash sign “-” means the method NeuralWavelet does not release models
+trained on these categories.
+7.2
+Evaluation metrics
+To evaluate the reconstruction accuracy of shape auto-encoding
+from point clouds, we adopt Chamfer distance, volumetric Intersection-
+over-Union (IoU), and F-score as primary evaluation metrics. IoU
+is computed based on the occupancy predictions of 50𝑘 querying
+points sampled in 3D space. Chamfer distance and F-score are cal-
+culated between two sampled point clouds with the size of 50𝑘
+respectively from reconstructed and ground-truth surfaces. For IoU
+and F-score, higher is better, while for Chamfer, lower is better.
+To measure the mesh quality of unconditional and conditional
+shape generation, we follow [Ibing et al. 2021; Shue et al. 2022; Zhang
+et al. 2022] to adapt the Fréchet Inception Distance (FID) and Kernel
+Inception Distance (KID) commonly used to assess the image gener-
+ative models to rendered images of 3d shapes. To calculate FID and
+KID of rendered images, we render each shape from 10 viewpoints.
+The metrics are named as Rendering-FID and Rendering-KID.
+The Rendering-FID is defined as,
+Rendering-FID = ∥𝜇g − 𝜇r∥ +𝑇𝑟
+�
+Σ𝑔 + Σ𝑟 − 2(Σ𝑔Σ𝑟)1/2�
+(24)
+where 𝑔 and 𝑟 denotes the generated and training datasets respec-
+tively. 𝜇 and Σ are the statistical mean and covariance matrix of the
+feature distribution extracted by the Inception network.
+The Rendering-KID is defined as,
+Rendering-KID = MMD
+�
+1
+|R|
+∑︁
+x∈R
+max
+y∈G 𝐷(x, y)
+�2
+(25)
+where 𝐷(x, y) is a polynomial kernel function to evaluate the simi-
+larity of two samples, G and R are feature distributions of generated
+set and reference set, respectively. The function MMD(·) is Maxi-
+mum Mean Discrepancy. However, the rendering-based FID and KID
+are essentially designed to understand 3D shapes from 2D images.
+Thus, they have the inherent issue of not accurately understanding
+shape compositions in the 3D world. To compensate their draw-
+backs, we also adapt the FID and KID to 3D shapes directly. For each
+generated or groud-truth shape, we sample 4096 points (with nor-
+mals) from the surface mesh and then feed them into a pre-trained
+PointNet++ [Qi et al. 2017b] to extract a global latent vector, repre-
+senting the global structure of the 3D shape. The PointNet++ is first
+pretrained on shape classification on ShapeNet-55. As we use point
+clouds, we call the FID and KID for 3D shapes as Fréchet PointNet++
+Distance (FPD) and Kernel PointNet++ Distance (KPD). The two
+metrics are defined similarly as in Eq. (24) and Eq. (25), except that
+the features are extracted from a PointNet++ network.
+10
+
+Ours
+NW
+3DILG
+Grid-83
+Ours
+NW
+3DILG
+Grid-83
+Ours
+NW
+3DILG
+Grid-83
+Fig. 10. Category-conditional generation. From top to bottom, we show category (airplane, chair, table) conditioned generation results.
+7.3
+Implementation
+For the shape auto-encoder, we use the point cloud of size 2048 as
+input. At each iteration, we individually sample 1024 query points
+from the bounding volume ([−1, 1]3) and the other 1024 points
+from near surface region for the occupancy values prediction. The
+shape auto-encoder is trained on 8 A100, with batch size of 512
+for 𝑇 = 1, 600 epochs. The learning rate is linearly increased to
+𝑙𝑟max = 5𝑒 − 5 in the first 𝑡0 = 80 epochs, and then gradually
+decreased using the cosine decay schedule 𝑙𝑟max ∗ 0.51+𝑐𝑜𝑠 ( 𝑡−𝑡0
+𝑇 −𝑡0 )
+until reaching the minimum value of 1𝑒 − 6. The diffusion models
+are trained on 4 A100 with batch size of 256 for 𝑇 = 8, 000 epochs.
+The learning rate is linearly increased to 𝑙𝑟𝑚𝑎𝑥 = 1𝑒 − 4 in the first
+𝑡0 = 800 epochs, and then gradually decreased using the above
+mentioned decay schedule until reaching 1𝑒 − 6. We use the default
+settings for the hyperparameters of EDM [Karras et al. 2022]. During
+sampling, we obtain the final latent set via only 18 denoising steps.
+8
+RESULTS
+We present our results for multiple applications: 1) shape auto-
+encoding, 2) unconditional generation, 3) category-conditioned
+generation, 4) text-conditioned generation, 5) shape completion,
+11
+
+6) image-conditioned generation. Finally, we perform a shape nov-
+elty analysis to validate that we are not overfitting to the dataset.
+8.1
+Shape Auto-Encoding
+We show the quantitative results in Tab. 3 for a deterministic au-
+toencoder without the KL block described in Sec. 5.2. In particular,
+we show results for the largest 7 categories as well as averaged re-
+sults over the categories. The two design choices of shape encoding
+described in Sec. 5.1 are also investigated. The case of using the
+subsampled point cloud as queries is better than learnable queries in
+all categories. Thus we use subsampled point clouds in our later ex-
+periments. The visualization of reconstruction results can be found
+in Fig. 8. We visualize some extremely difficult shapes from the
+datasets (test split). These shapes often contain some thin structures.
+However, our method still performs well.
+Ablation study of the number of latents. The number 𝑀 is the
+number of latent vectors used in the network. Intuitively, a larger
+𝑀 leads to a better reconstruction. We show results of 𝑀 in Tab. 4.
+Thus, in all of our experiments, 𝑀 is set to 512. We are limited by
+computation time to work with larger 𝑀.
+Ablation study of the KL block. We described the KL block in Sec. 5.2
+that leads to additional compression. In addition, this block changes
+the deterministic shape encoding into a variational autoencoder.
+The introduced hyperparameter is 𝐶0. A smaller 𝐶0 leads to a higher
+compression rate. The choice of𝐶0 is ablated in Tab. 5. Clearly, larger
+𝐶0 gives better results. The reconstruction results of𝐶0 = 8, 16, 32, 64
+are very close. However, they differ significantly in the second stage,
+because a larger latent size could make the training of diffusion
+models more difficult. This result is very encouraging for our model,
+because it indicates that aggressively increasing the compression
+in the KL block does not decrease reconstruction performance too
+much. We can also see that compressing with the KL block by de-
+creasing 𝐶0 is much better than compressing using fewer latent
+vectors 𝑀.
+8.2
+Unconditional Shape Generation
+Comparison with surface generation. We evaluate the task of un-
+conditional shape generation with the proposed metrics in Tab. 6.
+We also compared our method with a baseline method proposed
+in [Zhang et al. 2022]. The method is called Grid-83 because the
+latent grid size is 83, which is exactly the same as in AutoSDF [Mittal
+et al. 2022]. The table also shows the results of different 𝐶0. Our
+results are best when 𝐶0 = 32 in all metrics. When 𝐶0 = 64 the
+results become worse. This also aligns with our conjecture that a
+larger latent size makes the training more difficult.
+Comparison with point cloud generation. Additionally, we compare
+our method with PVD [Zhou et al. 2021] which is a point cloud
+diffusion method. We re-train PVD using the official released code
+on our preprocessed dataset and splits. We use the same evaluation
+protocol as before but with one major difference. Since PVD can only
+generate point clouds without normals, we use another pretrained
+PointNet++ (without normals) as the feature extractor to calculate
+Surface-FPD and Surface-KPD. The Tab. 7 shows we can beat PVD
+by a large margin. Additionally, we also show the metrics calculated
+AutoSDF
+Ours
+“horizontal slats on top of back”
+“one big hole between back and seat”
+“this chair has wheels”
+“vertical back ribs”
+Fig. 11. Text conditioned generation. For each text prompt, we generate
+3 shapes. Our results (Right) are compared with AutoSDF (Left).
+on rendered images. Visualization of generated results can be found
+in Fig. 9.
+8.3
+Category-conditioned generation
+We train a category-conditioned generation model using our method.
+We evaluate our models in Tab. 8. We should note that the competitor
+method NeuralWavelet [Hui et al. 2022] trains models for categories
+separately; thus, NeuralWavelet is not a true category-conditioned
+model. We also visualize some results (airplane, chair, and table)
+in Fig. 10. Our training is more challenging, as we train on a dataset
+that is an order of magnitude larger and we train for all classes
+jointly. While NeuralWavelet already has good results, the joint
+training is necessary / beneficial for many subsequent applications.
+8.4
+Text-conditioned generation
+The results of our text-conditioned generation model can be found
+in Fig. 11. Since the model is a probabilistic model, we can sample
+shapes given a text prompt. The results are very encouraging and
+they constitute the first demonstration of text-conditioned 3D shape
+generation using diffusion models. To the best of our knowledge,
+there are no published competing methods at the point of submitting
+this work.
+8.5
+Probabilistic shape completion
+We also extend our diffusion model for probablistic shape comple-
+tion by using a partial point cloud as conditioning input. The compar-
+ison against ShapeFormer [Yan et al. 2022] is depicted in Fig. 12. As
+seen, our latent set diffusion can predict more accurate completion,
+and we also have the ability to achieve more diverse generations.
+12
+
+GT
+Condition
+ShapeFormer
+Ours
+Fig. 12. Point cloud conditioned generation. We show three generated
+results given a partial cloud. The ground-truth point cloud and the partial
+point cloud used as condition are shown in Left. We compare our results
+(Right) with ShapeFormer (Middle).
+Condition
+IM-Net
+OccNet
+Ours
+Fig. 13. Image conditioned generation. In the left column we show the
+condition image. In the middle we show results obtained by the method
+IM-Net and OccNet. Our generated results are shown on the right.
+8.6
+Image-conditioned shape generation.
+We also provide comparisons on the task of single-view 3D object
+reconstruction in Fig. 13. Compared to other deterministic methods
+including OccNet [Mescheder et al. 2019] and IM-Net [Chen and
+Zhang 2019], our latent set diffusion can not only reconstruct more
+accurate surface details, (e.g. long rods and tiny holes in the back),
+but also support multi-modal prediction, which is a desired property
+to deal with severe occlusions.
+Ref
+Gen
+Ref
+Gen
+Ref
+Gen
+Ref
+Gen
+Fig. 14. Shape generation novelty. For a generated shape, we retrieve
+the top-1 similar shape in the training set. The similarity is measured using
+Chamfer distance of sampled surface point clouds. In each pair, we show
+the retrieved shape (left) and the generated shape (right). The generated
+shapes are from our category-conditioned generation results.
+8.7
+Shape novelty analysis
+We use shape retrieval to demonstrate that we are not simply over-
+fitting to the training set. Given a generated shape, we measure the
+Chamfer distance between it and training shapes. The visualization
+of retrieved shapes can be found in Fig. 14. Clearly, the model can
+synthesize new shapes with realistic structures.
+8.8
+Limitations
+While our method shows convincing results on a variety of tasks,
+our design choices also have drawbacks that we would like to dis-
+cuss. For instance, we require a two stage training strategy. While
+this leads to improved performance in terms of generation quality,
+training the first stage is more time consuming than relying on
+manually-designed features such as wavelets [Hui et al. 2022]. In
+addition, the first stage might require retraining if the shape data in
+consideration changes, and for the second stage – the core of our
+diffusion architecture – training time is also relatively high. Overall,
+we believe that there is significant potential for future research av-
+enues to speed up training, in particular, in the context of diffusion
+models.
+9
+CONCLUSION
+We have introduced 3DShape2VecSet, a novel shape representation
+for neural fields that is tailored to generative diffusion models. To
+this end, we combine ideas from radial basis functions, previous
+neural field architectures, variational autoencoding, as well as cross
+attention and self-attention to design a learnable representation.
+Our shape representation can take a variety of inputs including
+triangle meshes and point clouds and encode 3D shapes as neu-
+ral fields on top of a set of latent vectors. As a result, our method
+demonstrates improved performance in 3D shape encoding and 3D
+shape generative modeling tasks, including unconditioned genera-
+tion, category-conditioned generation, text-conditioned generation,
+point-cloud completion, and image-conditioned generation.
+In future work, we see many exciting possibilities. Most impor-
+tantly, we believe that our model further advances the state of the
+art in point cloud and shape processing on a large variety of tasks.
+In particular, we would like to employ the network architecture of
+3DShape2VecSet to tackle the problem of surface reconstruction
+from scanned point clouds. In addition, we can see many applica-
+tions for content-creation tasks, for example 3D shape generation
+of textured models along with their material properties. Finally, we
+13
+
+would like to explore editing and manipulation tasks leveraging
+pretrained diffusion models for prompt to prompt shape editing,
+leveraging the recent advances in image diffusion models.
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