J. H. Yin ,

# Search for the double-charmonium state with $\eta_{c}J/\psi$ at Belle

0000-0002-1479-9349 Y. B. Li  0000-0002-9909-2851 E. Won  0000-0002-4245-7442
I. Adachi  0000-0003-2287-0173 H. Aihara ,  0000-0002-1907-5964 S. Al Said
0000-0002-4895-3869 D. M. Asner  0000-0002-1586-5790 T. Aushev ,
0000-0002-6347-7055 R. Ayad  0000-0003-3466-9290 V. Babu ,
0000-0003-0419-6912 Sw. Banerjee ,  0000-0001-8852-2409 P. Behera ,
0000-0002-1527-2266 K. Belous  0000-0003-0014-2589 J. Bennett
0000-0002-5440-2668 M. Bessner ,  0000-0003-1776-0439 T. Bilka ,
0000-0003-1449-6986 D. Biswas ,  0000-0002-7543-3471 D. Bodrov ,
0000-0001-5279-4787 G. Bonvicini ,  0000-0003-4861-7918 J. Borah
0000-0003-2990-1913 A. Bozek  0000-0002-5915-1319 M. Bračko
0000-0002-2495-0524 P. Branchini  0000-0002-2270-9673 T. E. Browder
0000-0001-7357-9007 A. Budano  0000-0002-0856-1131 D. Červenkov ,
0000-0002-1865-741X M.-C. Chang  0000-0002-8650-6058 B. G. Cheon ,
0000-0002-8803-4429 K. Chilikin  0000-0001-7620-2053 H. E. Cho
0000-0002-7008-3759 K. Cho  0000-0003-1705-7399 S.-K. Choi ,
0000-0003-2747-8277 Y. Choi  0000-0003-3499-7948 S. Choudhury ,
0000-0001-9841-0216 D. Cinabro  0000-0001-7347-6585 J. Cochran
0000-0002-1492-914X S. Das ,  0000-0001-6857-966X G. De Nardo ,
0000-0002-2047-9675 G. De Pietro  0000-0001-8442-107X R. Dhamija
0000-0001-7052-3163 F. Di Capua  0000-0001-9076-5936 J. Dingfelder
0000-0001-5767-2121 Z. Doležal  0000-0002-5662-3675 T. V. Dong
0000-0003-3043-1939 D. Epifanov ,  0000-0001-8656-2693 T. Ferber ,
0000-0002-6849-0427 D. Ferlewicz  0000-0002-4374-1234 B. G. Fulsom ,
0000-0002-5862-9739 V. Gaur ,  0000-0002-8880-6134 A. Giri ,
0000-0002-8895-0128 P. Goldenzweig  0000-0001-8785-847X E. Graziani ,
0000-0001-8602-5652 T. Gu ,  0000-0002-1470-6536 Y. Guan  0000-0002-5541-2278
K. Gudkova  0000-0002-5858-3187 C. Hadjivasiliou  0000-0002-2234-0001 S.
Halder ,  0000-0002-6280-494X T. Hara ,  0000-0002-4321-0417 K. Hayasaka ,
0000-0002-6347-433X H. Hayashii  0000-0002-5138-5903 D. Herrmann ,
0000-0001-9772-9989 W.-S. Hou ,  0000-0002-4260-5118 C.-L. Hsu
0000-0002-1641-430X T. Iijima ,  0000-0002-4271-711X N. Ipsita ,
0000-0002-2927-3366 A. Ishikawa  0000-0002-3561-5633 R. Itoh
0000-0003-1590-0266 M. Iwasaki  0000-0002-9402-7559 W. W. Jacobs ,
0000-0002-9996-6336 Q. P. Ji ,  0000-0003-2963-2565 S. Jia
0000-0001-8176-8545 Y. Jin  0000-0002-7323-0830 K. K. Joo  0000-0002-5515-0087
J. Kahn  0000-0002-8517-2359 A. B. Kaliyar ,  0000-0002-2211-619X T. Kawasaki
,  0000-0002-4089-5238 C. Kiesling ,  0000-0002-2209-535X C. H. Kim
0000-0002-5743-7698 D. Y. Kim  0000-0001-8125-9070 K.-H. Kim ,
0000-0002-4659-1112 Y.-K. Kim ,  0000-0002-9695-8103 H. Kindo ,
0000-0002-6756-3591 K. Kinoshita  0000-0001-7175-4182 P. Kodyš ,
0000-0002-8644-2349 A. Korobov ,  0000-0001-5959-8172 S. Korpar ,
0000-0003-0971-0968 E. Kovalenko  0000-0001-8084-1931 P. Križan
0000-0002-4967-7675 P. Krokovny ,  0000-0002-1236-4667 T. Kuhr
0000-0001-6251-8049 M. Kumar  0000-0002-6627-9708 R. Kumar
0000-0002-6277-2626 K. Kumara  0000-0003-1572-5365 T. Lam ,
0000-0001-9128-6806 J. S. Lange  0000-0003-0234-0474 S. C. Lee ,
0000-0002-9835-1006 L. K. Li ,  0000-0002-7366-1307 Y. Li ,
0000-0002-4413-6247 J. Libby ,  0000-0002-1219-3247 K. Lieret
0000-0003-2792-7511 Y.-R. Lin  0000-0003-0864-6693 D. Liventsev ,
0000-0003-3416-0056 M. Masuda ,  0000-0002-7109-5583 T. Matsuda
0000-0003-4673-570X D. Matvienko  0000-0002-2698-5448 S. K. Maurya
0000-0002-7764-5777 F. Meier  0000-0002-6088-0412 M. Merola
0000-0002-7082-8108 F. Metzner  0000-0002-0128-264X R. Mizuk ,
0000-0002-2209-6969 G. B. Mohanty  0000-0001-6850-7666 R. Mussa ,
0000-0002-0294-9071 I. Nakamura  0000-0002-7640-5456 D. Narwal ,
0000-0001-6585-7767 Z. Natkaniec  0000-0003-0486-9291 A. Natochii
0000-0002-1076-814X L. Nayak  0000-0002-7739-914X M. Nayak
0000-0002-2572-4692 N. K. Nisar ,  0000-0001-9562-1253 S. Nishida
0000-0001-6373-2346 S. Ogawa ,  0000-0002-7310-5079 H. Ono ,
0000-0003-4486-0064 P. Oskin ,  0000-0002-7524-0936 G. Pakhlova ,
0000-0001-7518-3022 S. Pardi ,  0000-0001-7994-0537 H. Park
0000-0001-6087-2052 J. Park  0000-0001-6520-0028 S.-H. Park
0000-0001-6019-6218 A. Passeri  0000-0003-4864-3411 S. Patra
0000-0002-4114-1091 S. Paul  0000-0002-8813-0437 R. Pestotnik ,
0000-0003-1804-9470 L. E. Piilonen ,  0000-0001-6836-0748 T. Podobnik
0000-0002-6131-819X E. Prencipe ,  0000-0002-9465-2493 M. T. Prim
0000-0002-1407-7450 N. Rout ,  0000-0002-4310-3638 G. Russo ,
0000-0001-5823-4393 S. Sandilya ,  0000-0002-4199-4369 A. Sangal
0000-0001-5853-349X L. Santelj  0000-0003-3904-2956 V. Savinov ,
0000-0002-9184-2830 G. Schnell ,  0000-0002-7336-3246 C. Schwanda ,
0000-0003-4844-5028 Y. Seino ,  0000-0002-8378-4255 K. Senyo
0000-0002-1615-9118 W. Shan ,  0000-0003-2811-2218 M. Shapkin
0000-0002-4098-9592 C. Sharma  0000-0002-1312-0429 J.-G. Shiu ,
0000-0002-8478-5639 E. Solovieva ,  0000-0002-5735-4059 M. Starič ,
0000-0001-8751-5944 Z. S. Stottler  0000-0002-1898-5333 M. Sumihama ,
0000-0002-8954-0585 M. Takizawa ,  0000-0001-8225-3973 K. Tanida ,
0000-0002-8255-3746 F. Tenchini ,  0000-0003-3469-9377 R. Tiwary ,
0000-0002-5887-1883 M. Uchida ,  0000-0003-4904-6168 T. Uglov ,
0000-0002-4944-1830 Y. Unno ,  0000-0003-3355-765X S. Uno ,
0000-0002-3401-0480 Y. Usov ,  0000-0003-3144-2920 S. E. Vahsen
0000-0003-1685-9824 G. Varner  0000-0002-0302-8151 A. Vinokurova ,
0000-0003-4220-8056 D. Wang ,  0000-0003-1485-2143 E. Wang
0000-0001-6391-5118 M.-Z. Wang  0000-0002-0979-8341 X. L. Wang
0000-0001-5805-1255 S. Watanuki ,  0000-0002-5241-6628 O. Werbycka ,
0000-0002-0614-8773 X. Xu ,  0000-0001-5096-1182 B. D. Yabsley
0000-0002-2680-0474 W. Yan ,  0000-0003-0713-0871 S. B. Yang
0000-0002-9543-7971 J. Yelton ,  0000-0001-8840-3346 Y. Yook
0000-0002-4912-048X C. Z. Yuan  0000-0002-1652-6686 Z. P. Zhang ,
0000-0001-6140-2044 V. Zhilich  0000-0002-0907-5565 V. Zhukova
0000-0002-8253-641X<EMAIL_ADDRESS>
###### Abstract

We measure the cross section of $e^{+}e^{-}\rightarrow\eta_{c}J/\psi$ at the
$\Upsilon(nS)(n=1$—$5)$ on-resonance and 10.52 GeV off-resonance energy points
using the full data sample collected by the Belle detector with an integrated
luminosity of $955~{}\rm fb^{-1}$. We also search for double charmonium
production in $e^{+}e^{-}\rightarrow\eta_{c}J/\psi$ via initial state
radiation near the $\eta_{c}J/\psi$ threshold. No evident signal of the double
charmonium state is found, but evidence for the
$e^{+}e^{-}\rightarrow\eta_{c}J/\psi$ process is found with a statistical
significance greater than $3.3\sigma$ near the $\eta_{c}J/\psi$ threshold. The
average cross section near the threshold is measured and upper limits of cross
sections are set for other regions.

##  1 Introduction

Early in this century, a number of exotic states, the so called "$XYZ$"
particles, have been discovered reviewXYZ  via their decays into two heavy-
flavor mesons and/or a quarkonium and one or two light hadrons. Many theories
have been advanced regarding the structure of these exotic states reviewXYZ ,
with one of the most favored being the “tetraquark" model tetraquark1 ;
tetraquark2 . Further study in both theory and experiment is needed to reveal
their nature. Vector states with $J^{PC}=1^{--}$, usually called $Y$ states,
normally have a relative large decay rate to lower charmonium states via
hidden charm transition, accompanied by light hadrons; viz $Y(4220)\to\eta
J/\psi$ Y4220etaJpsi , $Y(4220)\to\eta^{\prime}J/\psi$ Y4220etapJpsi . In Ref.
Chiu:2005ey , the authors calculated the mass of $Y(4220)$ using lattice
quantum chromodynamics (QCD) by treating this state as a molecule. The result
of this calculation, $4238\pm 31~{}{\rm MeV}/c^{2}$, is very close to the
world average PDG . Moreover, the authors predict two additional exotic states
with quark compositions of $cs\bar{c}\bar{s}$ and $cc\bar{c}\bar{c}$ with
masses of $(4450\pm 100)$ MeV/$c^{2}$ and $(6400\pm 50)$ MeV/$c^{2}$,
respectively.

Recently, BESIII reported the cross section measurements of $e^{+}e^{-}\to
K^{+}K^{-}J/\psi$ Ablikim:2018epj BESIII:2022joj . A structure is observed
with $M=4487.7\pm 13.3\pm 24.1~{}{\rm MeV}/c^{2}$, which is very close to the
above prediction of $4450~{}{\rm MeV}/c^{2}$ for $cs\bar{c}\bar{s}$. Belle
also reported a structure with a mass around $4620~{}{\rm MeV}/c^{2}$ in the
cross section measurements of $e^{+}e^{-}\to D_{s}^{+}D_{s1}(2536)^{-}+c.c.$
Belle:DsDs1  and $D_{s}^{+}D^{*}_{s2}(2573)^{-}+c.c.$ Belle:DsDs2 .
Additionally, LHCb reported the observation of a possible $cs\bar{c}\bar{s}$
state in $B^{+}\to K^{+}\phi J/\psi$ decays LHCb:phiJpsi1 LHCb:phiJpsi2 .

Furthermore, LHCb reported pronounced structures in the invariant mass
spectrum of $J/\psi$ pairs LHCb_X6900 . An enhancement in the near-
double-$J/\psi$ threshold region from $6.2$ to $6.8~{}\rm GeV$ is seen,
followed by another narrow peak around $6.9~{}\rm GeV$, dubbed $X(6900)$. A
fully-charmed compact tetraquark resonance is the most straightforward
candidate, and lower states should exist since there is a large energy gap
between this structure and the mass threshold of $cc\bar{c}\bar{c}$.

We present the results of a search for a $cc\bar{c}\bar{c}$ state with
$J^{PC}=1^{--}$, hereinafter designated $Y_{cc}$, with the Belle detector
belle  at the KEKB asymmetric-energy $e^{+}e^{-}$ collider kekb . The
integrated luminosity is $980~{}\rm fb^{-1}$, about 70% of which were
collected at the $\Upsilon(4S)$ resonance, with the rest were taken at other
$\Upsilon(1,2,3,5S)$ states or center-of-mass (c.m.) energies lower than the
$\Upsilon(4S)$ or the $\Upsilon(nS)$ peaks by tens of MeV, as well as various
c.m. energies between $10.63~{}\rm GeV$ and $11.02~{}\rm GeV$. Initial state
radiation (ISR) allows us to search for the double-charmonium state in the
near-threshold region. We first measure the cross section of
$e^{+}e^{-}\to\eta_{c}J/\psi$ on the on-resonance $\Upsilon(nS)$ energy point,
providing validation for our searching as well as a solid check for the next-
to-next-to-leading-order calculation in the nonrelativistic QCD approach NNLO-
cal . We then search for the possible $\eta_{c}J/\psi$ and $Y_{cc}$ signals in
the near-threshold region. We extrapolate the measured cross section from the
on-resonance points to the near-threshold region to check whether the possible
$\eta_{c}J/\psi$ signals here are from continuum production.

##  2 Belle detector and data samples

The Belle detector belle  is a large-solid-angle magnetic spectrometer that
consists of a silicon vertex detector (SVD), a 50-layer central drift chamber
(CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like
arrangement of time-of-flight scintillation counters (TOF), and an
electromagnetic calorimeter (ECL) consisting of CsI(Tl) crystals. All these
detector components are located inside a superconducting solenoid coil that
provides a 1.5 T magnetic field. An iron flux-return located outside of the
coil is instrumented with resistive plate chambers to detect $K^{0}_{L}$
mesons and to identify muons.

The signal Monte Carlo (MC) samples of the ISR processes and the decays of
$\eta_{c}$ and $J/\psi$ are simulated with the phokhara Rodrigo:2001kf  and
evtgen evtgen  event generators, respectively, with the decay branching
fractions and resonance parameters of $\eta_{c}$ and $J/\psi$ taken from Ref.
PDG . These events are processed by a detector simulation based on GEANT3
geant3 . The generic MC samples, corresponding to six times the integrated
luminosity of the data, of $e^{+}e^{-}\to\Upsilon(nS)$ events with subsequent
$\Upsilon(nS)$ decays and $e^{+}e^{-}\to q\bar{q}$ ($q=u,~{}d,~{}s,~{}c$)
events are used to check the backgrounds. A tool named topoana topoana  is
used to visualize the MC event types after event selection. A series of signal
MC samples is generated with different $m(Y_{cc})$ assumptions to estimate the
signal resolutions as well the c.m. energy-dependent efficiency.

##  3 Event selection

Two individual reconstruction methods are implemented in this analysis. One is
exclusive reconstruction of $\eta_{c}J/\psi$, and the other is inclusive
reconstruction using $J/\psi$ or $J/\psi\gamma_{\rm ISR}$ for $\Upsilon(nS)$
on/off resonance or near-threshold data.

Charged particle tracks are required to have impact parameters perpendicular
to and along the beam direction with respect to the interaction point of less
than 1.0 and 4.0 cm, respectively. The transverse momentum of each track is
required to be greater than 100 MeV/$c$. For particle identification (PID),
except for tracks from $K^{0}_{S}$, information from different detector
subsystems is combined to form the likelihood $\mathcal{L}_{i}$ for species
$i$, where $i=e,~{}\mu,~{}\pi$, $K$, or $p$ PID . Particles with
$\mathcal{L}_{K}/(\mathcal{L}_{K}+\mathcal{L}_{\pi})>0.6$ are regarded as
kaons, while those with a ratio below $0.4$ are identified as pions. A track
with $\mathcal{L}_{p}/(\mathcal{L}_{p}+\mathcal{L}_{\pi})>0.6$ and
$\mathcal{L}_{p}/(\mathcal{L}_{p}+\mathcal{L}_{K})>0.6$ is identified as a
proton. Distinct likelihoods are used for muon muID  and electron
identification eID : we require
$\mathcal{L}_{\mu}/(\mathcal{L}_{\mu}+\mathcal{L}_{K}+\mathcal{L}_{\pi})>0.1$
for muon candidates and $\mathcal{L}_{e}/(\mathcal{L}_{e}+\mathcal{L}_{{\rm
non}-e})>0.01$ for electron candidates. These PID requirements provide a
relatively high selection efficiency and a low misidentification rate, i.e.,
about 80% and 7%, respectively, for pions.

The photon with the largest energy in the c.m. frame in an event is taken as
the ISR photon; its energy is required to be greater than 1 GeV. $K^{0}_{S}$
candidates are reconstructed by combining two tracks of opposite charge
reconstructed using the pion hypothesis and consistent with emergence from a
displaced vertex. Combinatorial background is suppressed using a neural
network NNKs . Neutral pion candidates are reconstructed from pairs of
photons, each photon having deposited energy of at least 50 MeV in the barrel
region of the ECL (polar angle within interval
$[32.2^{\circ},~{}128.7^{\circ}]$), or at least 100 MeV in the end-caps (polar
angle within interval
$[12.4^{\circ},~{}31.4^{\circ}]\cup[130.7^{\circ},~{}155.1^{\circ}]$). The
invariant mass of the $\pi^{0}$ candidate is required to be within the
interval $[0.115,~{}0.155]~{}{\rm GeV}/c^{2}$, which encompasses a $3\sigma$
mass window around the nominal mass. A mass-constrained fit is performed to
each surviving $\pi^{0}$ candidate to improve its momentum resolution.

Lepton pairs, $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$, are used to reconstruct
$J/\psi$. To reduce the effects of bremsstrahlung and final-state radiation,
photons within a 50 mrad cone of the initial electron or positron direction
are included in the calculation of the candidate’s four-momentum. The mass
window of $J/\psi$ is optimized to be $M(e^{+}e^{-})\in[3.000,~{}3.120]~{}\rm
GeV/c^{2}$ and $M(\mu^{+}\mu^{-})\in[3.075,~{}3.125]~{}\rm GeV/c^{2}$ using
the figure of merit ${\rm FOM}=N_{\rm S}/\sqrt{N_{\rm B}+0.5}$, where $N_{\rm
S}$ is the number of events in signal MC depending on different mass window
requirements, and $N_{\rm B}$ is the number of events in generic MC. In the
exclusive reconstruction, six hadronic channels are used to reconstruct
$\eta_{c}$: $p\bar{p}$, $p\bar{p}\pi^{0}$, $K^{0}_{S}K^{\pm}\pi^{\mp}$,
$K^{+}K^{-}\pi^{0}$, $K^{+}K^{-}K^{+}K^{-}$, $2(\pi^{+}\pi^{-}\pi^{0})$. The
optimized mass window is $M(\eta_{c})\in[2.78,~{}3.08]~{}\rm GeV/c^{2}$. The
candidate with the smallest value of the mass recoiling against
$\eta_{c}J/\psi$, $M^{2}_{\rm
recoil}\equiv|p_{e^{+}e^{-}}-p(\eta_{c})-p(J/\psi)|^{2}/c^{2}$ is chosen as
the best candidate; here, $p$ is the four-momentum of the specified particle
in lab frame. In the inclusive reconstruction, the number of charged tracks is
required to be greater than four to suppress the QED backgrounds.

##  4 Data analysis

###  4.1 Double charmonium production at $\Upsilon(nS)$ on-resonance and
$\Upsilon(4S)$ off-resonance energies

In this part, we only analyze the datasets at the $\Upsilon(nS)$ resonances as
well as the $\Upsilon(4S)$ off-resonance sample, corresponding to a luminosity
of $955~{}\rm fb^{-1}$. In double charmonium production at $\Upsilon(nS)$ on-
resonance and $\Upsilon(4S)$ off-resonance energies, the square of the missing
mass in the exclusive reconstruction is required to be within the interval
$[-0.05,~{}0.08]~{}{\rm GeV}^{2}/c^{4}$ to increase the signal purity. The
invariant mass distributions of $\eta_{c}J/\psi$ in exclusive reconstruction
are shown in Fig. 1 for data from the selected energy points. The main
background are due to combinatorial $\eta_{c}$ and $J/\psi$ candidates. From a
study of sideband events in the data, no peaking background is expected.
Unbinned extended maximum likelihood fits are performed to the
$\eta_{c}J/\psi$ invariant mass distributions except for the $\Upsilon(3S)$
dataset. Signal components are described using shapes derived from MC
simulation, and smoothed using kernel estimation rookeyspdf . The background
components are described with a first order polynomial. Solely for the
$\Upsilon(3S)$ dataset, the upper limit at the 90% confidence level (C.L.) on
the signal yield is estimated trolke . Cross sections are calculated with the
formula

$\sigma=\frac{N_{\rm
sig}}{\epsilon\mathcal{L}\mathcal{B}(J/\psi\to\ell^{+}\ell^{-})\mathcal{B}(\eta_{c}\to\rm
6~{}channels)},$ (1)

where $N_{\rm sig}$ is the number of signal events, $\epsilon$ is the
reconstruction efficiency, $\mathcal{L}$ is the integrated luminosity, and
$\mathcal{B}$ includes the corresponding branching fractions. The calculated
cross sections are shown in Table 1.

Figure 1: Fit to the invariant mass of $\eta_{c}J/\psi$ at selected energy
points. Dots with error bars show the experimental data, red curves the fit
results, and blue dashed curves the background. Table 1: Signal yields and the
measured cross sections for different channels at selected c.m. energy points.
Here $\mathcal{L}$ is the integrated luminosity; $\epsilon$ is the
reconstruction efficiency; $N$ and $\sigma$ are, respectively, the signal
yields and measured cross section, where the superscript “$\rm exc/inc$"
indicates the exclusive/inclusive analysis; and $\sigma^{\rm comb}$ is the
cross section combining the two reconstruction analyses. The first
uncertainties in $\sigma^{\rm exc/inc}$ are statistical and the second are
systematic. The $\sigma^{\rm comb}$, total uncertainties are given including
the statistical and systematic.

| $\Upsilon(1S)$ | $\Upsilon(2S)$ | $\Upsilon(3S)$ | 10.52 GeV | $\Upsilon(4S)$ | $\Upsilon(5S)$
---|---|---|---|---|---|---
$\mathcal{L}$ [$\rm fb^{-1}]$ | $5.7$ | $24.9$ | $2.9$ | $89.4$ | $711.0$ | $121.4$
$N^{\rm exc}$ | $0.7^{+1.5}_{-0.9}$ | $6.2^{+3.1}_{-2.3}$ | $<1.9$ | $2.6^{+3.5}_{-2.5}$ | $45.0^{+8.9}_{-8.2}$ | $6.5^{+3.4}_{-2.7}$
$\epsilon^{\rm exc}$ | 8.3% | 6.9% | 5.7% | 5.6% | 5.6% | 5.4%
$\sigma^{\rm exc}~{}[\rm fb]$ | $57^{+122}_{-73}\pm 6$ | $140^{+70}_{-52}\pm 14$ | $<442$ | $20^{+27}_{-19}\pm 6$ | $44^{+9}_{-8}\pm 5$ | $39^{+20}_{-14}\pm 7$
$N^{\rm inc}$ | $23.7\pm 12.3$ | $62.0\pm 17.9$ | $8.5\pm 5.2$ | $94.7\pm 23.8$ | $1116.2\pm 62.9$ | $91.1\pm 21.5$
$\epsilon^{\rm inc}$ | 38.6% | 29.6% | 26.4% | 26.1% | 25.4% | 24.7%
$\sigma^{\rm inc}~{}[\rm fb]$ | $89.1\pm 46.2\pm 20.5$ | $70.1\pm 20.2\pm 8.9$ | $91.8\pm 56.2\pm 52.3$ | $33.8\pm 8.5\pm 2.8$ | $52.1\pm 2.9\pm 5.0$ | $25.4\pm 6.0\pm 2.8$
$\sigma^{\rm comb}~{}[\rm fb]$ | $78.3^{+47.5}_{-43.0}$ | $80.2\pm 20.4$ | $87.0^{+71.0}_{-59.0}$ | $32.5\pm 8.5$ | $50.2\pm 5.0$ | $27.5\pm 6.1$

The $J/\psi$ recoil-mass distributions in the inclusive reconstruction are
shown in Fig. 2, where the $J/\psi$ recoil mass is defined as $M_{\rm
recoil}(J/\psi)\equiv\sqrt{|p_{e^{+}e^{-}}-p_{J/\psi}|^{2}}/c$ and $p$ is the
four-momentum. To improve the resolution on the recoil mass, we replace the
recoil mass with $M_{\rm recoil}(J/\psi)+M(J/\psi)-m(J/\psi)$, where
$M(J/\psi)$ is the reconstructed $J/\psi$ mass and $m(J/\psi)$ is the nominal
mass from Ref. PDG . Clear $\eta_{c},~{}\chi_{c0}$, and $~{}\eta_{c}(2S)$
signals are found in the recoil-mass spectra, as in previous Belle
measurements BelleCC  BelleCC2 . Unbinned extended maximum likelihood fits are
performed to the recoil-mass spectra. The signals are described using shapes
derived from MC simulation smeared with a Gaussian. Parameters of the Gaussian
functions are free for the fit to the $\Upsilon(4S)$ on-resonance data sample,
but fixed to the values obtained in the $\Upsilon(4S)$ fit result when fitting
to other energy points. Background is described with a third-order-polynomial.
Signal yields from the fits are listed in Table 1 together with the cross
sections. The cross sections here are calculated using a formula similar to
Eq. 1 but omitting the $\eta_{c}$ branching fractions.

We combine cross sections measured from the two reconstruction methods with
the following approach. First, we extract the cross-section-dependent
likelihood distributions from the two methods. For a given cross section in
the certain data sample, the possibility that we observe the current number of
signal events is estimated according to the fits. (For the exclusive
reconstruction with $\Upsilon(3S)$ data, we assume the number of the observed
events follows the Poisson distribution.) Consequently, a cross-section-
dependent joint probability density function (PDF) is obtained. We smear the
PDF with Gaussian functions whose widths model the systematic uncertainties
that are discussed below. The peak of the PDF is taken as the nominal result,
and the positions bounding 68% of the total integrated area under the PDF are
taken as the uncertainties. The final cross section results are listed in
Table 1 and plotted in Fig. 3.

We fit the cross sections to extrapolate to the $\eta_{c}J/\psi$ threshold
region to estimate the continuum contribution if any signal is found there.
There should be two sources of $\eta_{c}J/\psi$ production: one is
$e^{+}e^{-}\to\gamma^{*}\to\eta_{c}J/\psi$ and the other is
$e^{+}e^{-}\to\Upsilon(nS)\to\gamma^{*}\to\eta_{c}J/\psi$. This mechanism
should be similar to the production of $e^{+}e^{-}\to\mu^{+}\mu^{-}$. Thus, we
estimate the fractions produced from the continuum and $\Upsilon(nS)$ assuming
that they are same as those of the measured $e^{+}e^{-}\to\mu^{+}\mu^{-}$
cross sections 12Smm . The fractions of the continuum production are estimated
as $5/6$ and $4.5/4.75$ for $\Upsilon(1S)$ and $\Upsilon(2S)$, respectively.
Since we do not find a signal in the $\Upsilon(3S)$ dataset, and the
uncertainty at this energy point is very large, we use a fraction of one at
this energy point. For the $\Upsilon(4S)$ and $\Upsilon(5S)$, we assume that
signals originate exclusively from continuum production. We fit the cross
sections of $e^{+}e^{-}\to\eta_{c}J/\psi$ from the continuum production at
$\Upsilon(1/2S)$ along with the $e^{+}e^{-}\to\eta_{c}J/\psi$ at other energy
points with the function

$\sigma=A\frac{\sqrt{2\mu\Delta M}}{(\frac{s}{s_{0}})^{n}},$ (2)

where $\mu=\frac{m(\eta_{c})m(J/\psi)}{m(\eta_{c})+m(J/\psi)},~{}\Delta
M=\sqrt{s}-m(\eta_{c})-m(J/\psi)$, $m(\eta_{c})$ and $m(J/\psi)$ are the
$\eta_{c}$ and $J/\psi$ nominal masses PDG , $s_{0}\equiv(10.58~{}{\rm
GeV}/c^{2})^{2}$ is the c.m. energy at the $\Upsilon(4S)$ resonance, $A$ and
$n$ are free parameters whose best-fit values are $A=(11.4\pm
0.9)~{}c^{2}\cdot\rm fb/GeV$, and $n=4.5\pm 1.3$.

Figure 2: Fit to the mass recoiling against $J/\psi$ at the selected energy
points. Dots with error bars show the experimental data, blue curves the fit
results, and blue long dashed curves the background. The red dashed, dotted,
and dashed-dotted curves show the signal for $\eta_{c}$, $\chi_{c0}$, and
$\eta_{c}(2S)$, respectively. Figure 3: Combined results of the measured cross
sections of $e^{+}e^{-}\to\eta_{c}J/\psi$ at the $\Upsilon(nS)$ on-resonance
and $\Upsilon(4S)$ off-resonance energy points. The green curve is the fit
result using Eq. 2. Figure 4: Simultaneous fit result to the invariant mass of
$\eta_{c}J/\psi$ (left) and the $\gamma$ recoil mass (right). In each panel,
dots with error bars are from data, the red solid curve is the best fit
result, the blue dashed curve the background component from the best fit, and
the green dotted curve is the fit result without the signal components.

Figure 5: Simultaneous fit result to the invariant mass of reconstructed $\eta_{c}$ (top) and the $\gamma_{\rm ISR}J/\psi$ recoil mass (bottom). From left to right are events with $M(\eta_{c}J/\psi)$ and $M_{\rm recoil}(\gamma)\in[6.0,~{}6.4],~{}[6.0,~{}6.5]$, and $[6.0,~{}6.6]~{}{\rm GeV}/c^{2}$. Dots with error bars are from data, the red solid curve is the best fit result, and the blue dashed curve is the background component from the best fit. Figure 6: Measured cross sections of $e^{+}e^{-}\to\eta_{c}J/\psi$ near the threshold. From left to right are the cross sections measured in different step sizes (0.4, 0.5, 0.6 ${\rm GeV}/c^{2}$). Green solid curve is the extrapolation from the fit to near-resonance energy points. The shadow area covers the $\pm 1\sigma$ range of the extrapolations.  Table 2: Measured cross sections for different $M(\eta_{c}J/\psi)$ and $M^{\rm corr}_{\rm recoil}(\gamma_{\rm ISR})$ regions. Here, $N_{\rm prod}$ is the number of produced $\eta_{c}J/\psi$ signal events, and $\sigma$ is the calculated cross section. regions (${\rm GeV}/c^{2}$) | $N_{\rm prod}~{}[\times 10^{2}]$ | $\sigma~{}[\rm pb]$
---|---|---
$[6.0,~{}6.4]$ | $13.1\pm 3.6$ | $3.3\pm 0.9\pm 0.8$
$[6.4,~{}6.8]$ | $<8.2$ | $<1.7$
$[6.8,~{}7.2]$ | $<3.9$ | $<0.7$
$[7.2,~{}7.6]$ | $<2.7$ | $<0.4$
$[7.6,~{}8.0]$ | $<2.1$ | $<0.3$
$[8.0,~{}8.4]$ | $<10.4$ | $<1.0$
$[6.0,~{}6.5]$ | $13.4\pm 4.0$ | $2.7\pm 0.8\pm 0.2$
$[6.5,~{}7.0]$ | $<6.1$ | $<1.0$
$[7.0,~{}7.5]$ | $<1.9$ | $<0.2$
$[7.5,~{}8.0]$ | $<3.8$ | $<0.4$
$[8.0,~{}8.5]$ | $<9.9$ | $<0.7$
$[6.0,~{}6.6]$ | $13.3\pm 4.2$ | $2.1\pm 0.7\pm 0.2$
$[6.6,~{}7.2]$ | $<5.0$ | $<0.6$
$[7.2,~{}7.8]$ | $<2.3$ | $<0.2$
$[7.8,~{}8.4]$ | $<7.4$ | $<0.5$

###  4.2 $e^{+}e^{-}\to\eta_{c}J/\psi$ near threshold

In this part, we use the entire $980~{}\rm fb^{-1}$ Belle dataset. For
exclusive reconstruction of $e^{+}e^{-}\to\eta_{c}J/\psi$ near-threshold
events, the squared recoil mass of the $\eta_{c}J/\psi$ system
($|p_{e^{+}e^{-}}-p_{\eta_{c}J/\psi}|^{2}/c^{2}$) is required to be within the
interval $[-1,~{}2]~{}{\rm GeV}^{2}/c^{4}$ to improve the signal purity. Here,
the mass window is larger than that in the previous section because of the
possibility of a second ISR photon. To suppress the possible background from
$\Upsilon(4S)\to B\bar{B}$, the ratio of the second to the zeroth order Fox-
Wolfram moments fwmomentum  is required to be greater than 0.13. To improve
the resolution of the $Y_{cc}$ signal in the inclusive reconstruction, we use
the corrected recoil mass $M^{\rm corr}_{\rm recoil}(\gamma_{\rm ISR})\equiv
M_{\rm recoil}(\gamma_{\rm ISR})-M_{\rm recoil}(\gamma_{\rm
ISR}J/\psi)+m(\eta_{c})$, where $M_{\rm recoil}(\gamma_{\rm ISR})$ and $M_{\rm
recoil}(\gamma_{\rm ISR}J/\psi)$ are the recoil masses of $\gamma_{\rm ISR}$
and $\gamma_{\rm ISR}J/\psi$, respectively, and $m(\eta_{c})$ is the
$\eta_{c}$ nominal mass PDG .

The invariant mass of $\eta_{c}J/\psi$ and the recoil mass of $\gamma_{\rm
ISR}$ are shown in Fig. 4. Events in common between the exclusive and
inclusive samples are removed from the inclusive reconstruction to avoid
double counting. The number of events increases near threshold in the mass
spectrum of $\eta_{c}J/\psi$, but no similar enhancement is seen in the recoil
mass of $\gamma_{\rm ISR}$. A simultaneous unbinned maximum likelihood fit for
the $\eta_{c}J/\psi$ invariant mass and $\gamma_{\rm ISR}$ recoil mass is
performed. The signal-yield fractions from the two reconstruction methods are
fixed to the corresponding branching fractions and reconstruction
efficiencies. The background shapes are parameterized with the ARGUS function,
whose parameters are obtained from the fit to the $\eta_{c}$ and $J/\psi$
sideband events. The signals are described with a Breit-Wigner function with
free mass and width convolved with the Gaussian functions from the resolution
study. The fit results are shown in Fig. 4. The significance of the Breit-
Wigner peak component is $2.1\sigma$, with mass and width of $(6267\pm
43)~{}{\rm MeV}/c^{2}$ and $(121\pm 72)~{}\rm MeV$, respectively. The signal
yields are $9\pm 4$ and $23\pm 11$ from the exclusive and inclusive methods,
respectively.

The reconstructed $\eta_{c}$ mass and $\gamma_{\rm ISR}J/\psi$ recoil mass
spectra for events satisfying $M(\eta_{c}J/\psi)$ and $M_{\rm
recoil}(\gamma)\in[6.0,~{}6.4~{}\rm or~{}6.5~{}\rm or~{}6.6]~{}{\rm
GeV}/c^{2}$ are shown in Fig. 5. Fit results to other mass regions are shown
in the Appendix. No peaking backgrounds are expected according to the study of
the generic MC sample. We perform a simultaneous unbinned maximum likelihood
fit to the reconstructed $\eta_{c}$ mass and $\gamma_{\rm ISR}J/\psi$ recoil
mass. The signal-yield fractions for the two reconstruction methods are fixed
to the products of the corresponding branching fractions and reconstruction
efficiencies obtained from simulated signal events. The signal shapes are
described using shapes derived from MC simulation and smoothed using kernel
estimation rookeyspdf , while the background is described with an ARGUS
function ARGUS  and third-order-polynomial for the exclusive and inclusive
analyses, respectively.

The fit results are shown in Fig. 5. The significances of the $\eta_{c}$
components are $3.9,~{}3.3$, and $3.5\sigma$ for events from the three mass
regions, respectively. The invariant mass of $\eta_{c}$ and $\gamma_{\rm
ISR}J/\psi$ recoil mass for the other $M(\eta_{c}J/\psi)/M_{\rm
recoil}(\gamma_{\rm ISR})$ mass regions are shown in the Appendix, as well as
the fit results. No evident signals are found in those distributions, and the
upper limits of the number of produced events in different $\eta_{c}J/\psi$
mass regions are estimated at 90% C.L. with the following method. First, the
profile likelihood distribution as a function of the number of produced events
is extracted from the fit. Then this likelihood distribution is smeared with a
Gaussian function whose width models the systematic uncertainty. This smeared
distribution is then integrated from zero to infinity. The point at which the
integral reaches 90% of the total value is taken as the upper limit. The
number of produced events and the upper limits from different
$M(\eta_{c}J/\psi)$ and $M_{\rm recoil}(\gamma_{\rm ISR})$ mass regions are
listed in Table 2.

Table 3: Summary of the systematic uncertainties (in %), excluding those for the fitting procedures. For both photon detection and total uncertainties, the main number shows the uncertainty for the $Y_{cc}$ search, and the parentheses show the uncertainty for the cross section measurements at the selected on- (off-)resonance points.  source | exclusive reconstruction | inclusive reconstruction
---|---|---
Tracking | 1.4 | 0.7
Photon detection | 0.0 | 2.0 (0.0)
PID | 9.2 | 7.2
$K_{S}$ selection | 0.3 | 0.0
$\pi^{0}$ selection | 3.5 | 0.0
$\eta_{c}$ decays | 0.9 | 0.0
$J/\psi$ decays | 0.5
Luminosity | 1.4
Generator | 1.0
Sum | 8.1 (7.8)

The effective luminosity in each mass region is calculated according to Ref.
effLumi . Using this, we calculate the cross sections near $\eta_{c}J/\psi$
mass threshold with an equation analogous to Eq. 1; these are plotted in Fig.
6 as points with errors. We extrapolate the lineshape of the measured cross
sections near $\Upsilon(nS)$ resonances according to Eq. 2, and plot it in
Fig. 6 as the solid curve for comparison with the measurements. We vary the
parameters of the extrapolations based on the uncertainties and correlations.
The range of the extrapolations are also shown on the plot. The measured cross
sections near $\eta_{c}J/\psi$ mass threshold are consistent with the
extrapolations from the $\Upsilon(nS)$ energy region according to their
uncertainty.

##  5 Systematic uncertainty

Possible sources of systematic uncertainty include tracking, ISR photon
detection, PID, $K_{S}^{0}$ reconstruction, $\pi^{0}$ reconstruction, the
fitting procedure, integrated luminosity, and the $\eta_{c}$ and $J/\psi$
branching fractions, as listed in Table 3.

The difference in tracking efficiency for tracks with momenta above
$200~{}{\rm MeV}/c^{2}$ between data and MC is $(-0.13\pm 0.30\pm 0.10)\%$ per
track. The uncertainty of ISR photon detection efficiency, studied using
radiative bhabha events, is 2.0%. This is only taken into account in the
$Y_{cc}$ search. We apply a reconstruction uncertainty of $0.35\%$ per track
in our analysis estimated by using partially reconstructed $D^{*+}\to
D^{0}\pi^{+},~{}D^{0}\to\pi^{+}\pi^{-}K_{S}^{0}$, and
$K^{0}_{S}\to\pi^{+}\pi^{-}$ events. According to the measurement of PID
efficiency using the control sample $D^{*}\to D^{0}\pi$ with $D^{0}\to
K^{-}\pi^{+}$, we assign uncertainties of 1.1% for each kaon and 0.9% for each
pion. For $K_{S}^{0}$ selection, we take 2.2% as the systematic uncertainty,
following Ref. kserr . For $\pi^{0}$ selection, the uncertainty is 2.3%
according to a study of the $\tau\to\pi\pi^{0}\nu_{\tau}$ control sample
pi0err .

Since we are using multiple channels to reconstruct $\eta_{c}$, the
uncertainties of reconstruction efficiencies from these channels are combined
using

$\delta=\frac{\sum_{i}\delta_{i}\epsilon_{i}\mathcal{B}_{i}}{\sum_{i}\epsilon_{i}\mathcal{B}_{i}},$
(3)

where $\delta_{i}$, $\epsilon_{i}$, and $\mathcal{B}_{i}$ are the systematic
uncertainty, reconstruction efficiency, and branching fraction from the $i$-th
$\eta_{c}$ channel, respectively.

For the fitting procedure, we change the fitting range and the background
function. The difference in signal yields from the nominal and alternate fits
is taken as the systematic uncertainty, as shown in the Appendix. For the fits
with no significant signals, this systematic uncertainty is included in the
upper limit estimation by taking the alternative fit result with the largest
upper limit for the signal yield.

The uncertainty of the integrated luminosity is about 1.4%. The uncertainty of
the branching fraction of $J/\psi\to\ell^{+}\ell^{-}$, 0.5%, is taken from
Ref. PDG . The uncertainty due to the branching fractions of $\eta_{c}$ decays
is studied with pseudo-experiments, in which we vary the branching fractions
of $\eta_{c}$ decays within $1\sigma$, and calculate the overall
reconstruction efficiency. After 1,000 trials, we obtain a distribution of
reconstruction efficiencies, which is subsequently fit to a Gaussian. The
width of this Gaussian is taken as the systematic uncertainty associated with
the $\eta_{c}$-decay branching fractions.

We are using the phokhara generator to simulate our signal events. The
discrepancy of the energy of ISR photons between the simulation and the
theoretical calculation ISRtheory  is less than 0.1%, with a statistical
uncertainty of less than 1.0%; thus, we take the uncertainty of the generator
phokhara to be 1.0% as a conservative value.

The total systematic uncertainty is obtained by adding the individual
components in quadrature. We use Eq. 3 to combine the systematic uncertainties
from the two methods used in the cross section measurements near threshold.

##  6 Summary

We perform the first search for a double-charmonium state with
$e^{+}e^{-}\to\eta_{c}J/\psi$ near threshold via the ISR process. No
significant signal of the double charmonium state is found in several bins of
the invariant mass of $\eta_{c}J/\psi$ (for exclusive reconstruction) and the
recoil mass of $\gamma_{\rm ISR}$ (for inclusive reconstruction). We measure
the $\eta_{c}J/\psi$ production cross sections in several bins of the
invariant mass of $\eta_{c}J/\psi$ and the recoil mass of $\gamma_{\rm ISR}$.
The cross sections for $e^{+}e^{-}\to\eta_{c}J/\psi$ nearest the threshold are
significantly larger than in neighboring bins. Evidence with a statistical
significance greater than $3.3\sigma$ is found for double charmonium
production near the $\eta_{c}J/\psi$ threshold. The cross sections of double
charmonium production at $\Upsilon(nS)$ on-resonance and $\Upsilon(4S)$ off-
resonance data samples are also measured. The cross sections are fitted with a
function $\sigma\propto 1/s^{n}$, and extrapolated to the lower
$\eta_{c}J/\psi$ mass regions, where consistency within $1\sigma$ with our
measurements in those regions is observed, albeit with relatively large
measurement uncertainties. The search for double charmonium production at
Belle II is expected to be revisited as SuperKEKB integrated luminosity
reaches a few $\rm ab^{-1}$ or more.

###### Acknowledgements.

This work, based on data collected using the Belle detector, which was
operated until June 2010, was supported by the Ministry of Education, Culture,
Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the
Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of
Nagoya University; the Australian Research Council including grants
DP210101900, DP210102831, DE220100462, LE210100098, LE230100085; Austrian
Federal Ministry of Education, Science and Research (FWF) and FWF Austrian
Science Fund No. P 31361-N36; the National Natural Science Foundation of China
under Contracts No. 11675166, No. 11705209; No. 11975076; No. 12135005; No.
12175041; No. 12161141008; Key Research Program of Frontier Sciences, Chinese
Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; Project ZR2022JQ02
supported by Shandong Provincial Natural Science Foundation; the Ministry of
Education, Youth and Sports of the Czech Republic under Contract No. LTT17020;
the Czech Science Foundation Grant No. 22-18469S; Horizon 2020 ERC Advanced
Grant No. 884719 and ERC Starting Grant No. 947006 “InterLeptons” (European
Union); the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the
Excellence Cluster Universe, and the VolkswagenStiftung; the Department of
Atomic Energy (Project Identification No. RTI 4002) and the Department of
Science and Technology of India; the Istituto Nazionale di Fisica Nucleare of
Italy; National Research Foundation (NRF) of Korea Grant Nos.
2016R1D1A1B02012900, 2018R1A2B3003643, 2018R1A6A1A06024970, RS202200197659,
2019R1I1A3A01058933, 2021R1A6A1A03043957, 2021R1F1A1060423, 2021R1F1A1064008,
2022R1A2C1003993; Radiation Science Research Institute, Foreign Large-size
Research Facility Application Supporting project, the Global Science
Experimental Data Hub Center of the Korea Institute of Science and Technology
Information and KREONET/GLORIAD; the Polish Ministry of Science and Higher
Education and the National Science Center; the Ministry of Science and Higher
Education of the Russian Federation, Agreement 14.W03.31.0026, and the HSE
University Basic Research Program, Moscow; University of Tabuk research grants
S-1440-0321, S-0256-1438, and S-0280-1439 (Saudi Arabia); the Slovenian
Research Agency Grant Nos. J1-9124 and P1-0135; Ikerbasque, Basque Foundation
for Science, Spain; the Swiss National Science Foundation; the Ministry of
Education and the Ministry of Science and Technology of Taiwan; and the United
States Department of Energy and the National Science Foundation. These
acknowledgements are not to be interpreted as an endorsement of any statement
made by any of our institutes, funding agencies, governments, or their
representatives. We thank the KEKB group for the excellent operation of the
accelerator; the KEK cryogenics group for the efficient operation of the
solenoid; and the KEK computer group and the Pacific Northwest National
Laboratory (PNNL) Environmental Molecular Sciences Laboratory (EMSL) computing
group for strong computing support; and the National Institute of Informatics,
and Science Information NETwork 6 (SINET6) for valuable network support. E.
Won is partially supported by the NRF grant 2022R1A2B5B02001535 and J. H. Yin
and E. Won are by 2019H1D3A1A01101787.

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##  Appendix A Appendix

###  A.1 Fit plots for $e^{+}e^{-}\to\eta_{c}J/\psi$ near threshold

Here we provide the fitting plots for different $M(\eta_{c}J/\psi)$ mass
regions.

####  A.1.1 Step size 400 ${\rm MeV}/c^{2}$

Figure 7: Simultaneous fit result to the invariant mass of reconstructed
$\eta_{c}$ (top) and the $\gamma J/\psi$ recoil mass (bottom). Top row, from
top to bottom, are the events from
$M(\eta_{c}J/\psi)\in[6.4,~{}6.8],~{}[6.8,~{}7.2]$, and $[7.2,~{}7.6]~{}{\rm
GeV}/c^{2}$ while bottom row are $M(\eta_{c}J/\psi)\in[7.6,~{}8.0]$, and
$[8.0,~{}8.4]~{}{\rm GeV}/c^{2}$. Dots with error bars are from data, red
solid curve is the best fit result, blue dashed curve is the background
component from the best fit, green dotted curve is the fit result without
signal components.

####  A.1.2 Step size 500 ${\rm MeV}/c^{2}$

Figure 8: Simultaneous fit result to the invariant mass of reconstructed
$\eta_{c}$ (top) and the $\gamma J/\psi$ recoil mass (right). From left to
right are the events from $M(\eta_{c}J/\psi)\in[6.5,~{}7.0]$, $[7.0,~{}7.5]$,
$[7.5,~{}8.0]$, and $[8.0,~{}8.5]~{}{\rm GeV}/c^{2}$. Dots with error bars are
from data, red solid curve is the best fit result, blue dashed curve is the
background component from the best fit, green dotted curve is the fit result
without signal components.

####  A.1.3 Step size 600 ${\rm MeV}/c^{2}$

Figure 9: Simultaneous fit result to the invariant mass of reconstructed
$\eta_{c}$ (left) and the $\gamma J/\psi$ recoil mass. From top to bottom are
the events from $M(\eta_{c}J/\psi)\in[6.6,~{}7.2],~{}[7.2,~{}7.8]$, and
$[7.8,~{}8.4]~{}{\rm GeV}/c^{2}$. Dots with error bars are from data, red
solid curve is the best fit result, blue dashed curve is the background
component from the best fit, green dotted curve is the fit result without
signal components.

###  A.2 Systematic uncertainties of fitting procedure

Here we provide the systematic uncertainties of fitting procedure from
different fits.

Table 4: Uncertainty of the cross sections for different datasets from fit procedure (in unit of %). dataset | inclusive | exclusive
---|---|---
$\Upsilon(1S)$ | 21.5 | -
$\Upsilon(2S)$ | 9.8 | 2.2
$\Upsilon(3S)$ | 56.4 | -
continuum | 8.8 | 25.4
$\Upsilon(4S)$ | 3.4 | 4.0
$\Upsilon(5S)$ | 13.5 | 16.2
Table 5: Uncertainty of the cross sections measurement for different mass regions from fit procedure (in unit of %). regions (${\rm GeV}/c^{2}$) | systematic uncertainty
---|---
$[6.0,~{}6.4]$ | 23.9
$[6.0,~{}6.5]$ | 6.0
$[6.0,~{}6.6]$ | 7.0


# Comparative analysis of the selected planet transits observed at the Kyiv
comet station and the TESS and Kepler database.

M. Lobodenko1, Ya. Pavlenko2, I. Kulyk2, A. Nahurna1, M. Solomakha1, O.
Baransky3
1 Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
2 Main Astronomical Observatory of the NAS of Ukraine, Kyiv, Ukraine
3 Kyiv Comet station Lisnyky, Kyiv, Ukraine

(lobodenkomo.at.ukr.net)

ABSTRACT

We present a comparative analysis of observations of the selected exoplanet
transits gathered at the Kyiv comet station and those from the TESS
(Transiting Exoplanet Survey Satellite) and Kepler space telescope archive.
The light curves obtained by the TESS and Kepler orbital telescopes were
processed using a program based on the Python package Lightkurve v2.3 which is
freely available in the MUST archive (Barbara A. Mikulski Archive for Space
Telescopes). The ground based observations were carried out with the 70-cm
telescope AZT-8 (Lisnyky). Photometric processing of the ground based
observation was performed by using the Muniwin program. The light curves and
parameters of the observed transits as well as the exoplanet orbital
parameters obtained from the ground based observations were published in the
ETD (Exoplanet Transit Database). The determined transit parameters were
compared with the results stored in the MUST archive. Here we presents a
comparison of the transit periods, depths, transit durations and some planet
orbital parameters obtained from two independent sets of the observations,
terrestrial and orbital, performed at different epochs.

Keywords: transits, exoplanets, TESS, Kepler, exoplanet parameters.

##  1 Introduction

There are a large number of methods for finding exoplanets. The transit method
is one of the most effective. The planet covers part of the star when it
passes over the disk of a star and the visible brightness falls (Winn, 2010).
The time sequence of the observed during a transit events allow us to see
these events. The magnitude of the fall in brightness depends on the relative
size of the star and the planet. Therefore, the light curve provides
information about the radius of the planet and some orbital parameters. But
this method has several disadvantages. First, the plane of the planet’s orbit
should be located in a such way that we can observe the passage of the planet
over the star’s disk. Secondly, the planet must be large enough to to create a
detectable drop in the star brightness. For this reason most of the planets
are found by the transit method and all the planets that we present in this
investigation are hot Jupiters.
Hot Jupiters are planets with a mass of the order of the mass of Jupiter,
which rotate close to their host stars and are always turned to them only by
the one side. Therefore the period of the planet‘s rotation is small, which
allows us to observe transits regularly and makes hot Jupiters the most
convenient targets for the observations.
The TESS and Kepler orbital telescopes conducted searches for exoplanets using
the transit method. The quality of observations obtained from the orbital
telescopes is definitely higher than the quality of observations obtained from
ground-based telescopes. We are not hindered by various atmospheric phenomena
we do not depend on the weather, time of day, phase of the Moon, etc. However,
the ground based observations allow us to gather the data over large time
span, which provide information about possible changes in transit parameters.
The best result can be achieved by combining these two types of observations.
To do this, it is necessary to determine how the results obtained from space
and ground-based observations correspond to each other, and whether ground
observations can be considered sufficiently accurate. The main goal of our
research is to determine to what extent the parameters of exoplanets obtained
from observations at the Lisnyky comet station coincide with the parameters
obtained from observations from the TESS and Kepler orbital telescopes.

##  2 The Kepler and TESS data bases

###  2.1 MAST

MAST (The Mikulski Archive for Space Telescopes) is an archive of data from
numerous space telescopes and contains data obtained in the optical,
ultraviolet and near-infrared ranges. The MAST archive also provides the
parameters of the planet transits (periods, transit depths, the phases of
transit events, and some planet orbital parameters) calculated by the TESS and
Kepler pipelines for all events surpassing some threshold. We used these
published data in order to compare with the results of our ground based
observations as well as with those transit parameters obtained from the Kepler
and TESS light curves using the program codes developed on the base of the
Python package Lightkurve v2.3.

###  2.2 Kepler and TESS space missions

The Kepler Orbital Telescope was launched by NASA on March 7, 2009 to search
for Earth-sized planets. The spacecraft repeated the path of the Earth,
revolving around the Sun. This arrangement allowed telescope to constantly
monitor one part of the sky. The field of view covered 115 square degrees near
the plane of the Milky Way. The telescope of 1,039 kilograms mass contained a
Schmidt camera with a 0.95-meter front corrector plate feeding a 1.4-meter
primary mirror. The light reflected by the mirror was collected in the main
focus, where there was a mosaic of 21 pairs of specially created astronomical
CCD matrices, capable of recording almost every incident photon. The
dimensions of the entire mosaic are approximately 30 x 30 cm and it consists
of 95 megapixels. In May 2013 the telescope’s second flywheel engine failed. A
year after the failure of the engine, the telescope began to transmit data to
Earth again. The new mission has been named K2. The telescope began to observe
a section of the sky along the ecliptic. Over 9 years of operation, the
telescope discovered more than 2,680 exoplanets, 550 of which may be rocky,
and 21 potentially habitable (Hall & Barentsen 2020a).

The TESS (Transiting Exoplanet Survey Satellite) space telescope was launched
on April 19, 2018. The mission was planned for two years, during these years
the telescope should examine the entire area of the sky. The celestial sphere
was divided into 26 observation sectors, each sector being 24° × 96° to detect
transits of previously unknown exoplanets near the closest and brightest
stars. TESS would focus on G, K, and M-type stars with apparent magnitudes
brighter than magnitude 12, and 1000 nearest red dwarfs. The rotational period
of the telescope is 13.7 days. Each sector is observed for 27.4 days. The sole
instrument on TESS is formed of four wide-angle CCD cameras. Each camera has a
16.8-megapixel detector with a low energy consumption and low noise, which was
developed in the Laboratory of Lincoln. Each camera has a 24° × 24° field of
view, a 100 mm effective pupil diameter, a lens assembly with seven optical
elements, and a band-pass range from 600 to 1000 nm (Hall & Barentsen 2020a).

###  2.3  The data processing

The program we used to process data from the TESS and Kepler orbital
telescopes was developed based on the Python package Lightkurve v2.3, which is
available in the MUST archive.

First, the program finds data for the object we are interested in the catalogs
(Hall & Barentsen 2020a) and download the data. If the observed data cover
several sectors, the light curves are stitched and normalized. The next step
is searching for and subtracting the long-period garmonical oscillations,
which can be star’s oscilations and some kind of artifacts (Hall & Barentsen
2020d, 2020e). The Lomb-Scargle periodogram is used for this(Hall & Barentsen
2020c). To detect transit events we build a periodogram, using the Box Least
Squares (BLS) method (Saunders 2020). This method is much more sensitive to
find periodic transit events (Terebizh 1992). Fig. 1 presents the periodograms
of star TIC236887394 from the TESS data base constructed with Lomb-Scargle and
BLS methods.

Figure 1: Lomb-Scargle periodogram of star TIC 236887394 from the TESS data
base Figure 2: BLS periodogram of star TIC 236887394 from the TESS data base

The program outputs are period, duration, and first epoch of the detected
transit based on the periodogram built with the BLS method for the period at
maximum power. Using these parameters we construct the folded light curve in
the phase space, which helps to notice a shifts (if any) in the moments of the
beginning and end of the transit. These displacements may indicate the
presence of one or more planets in this system. Fig.3 demonstrates the folded
phase curve for star TIC 236887394 with the considerable decrease in the star
brightness caused by the transit event.

Figure 3: Folded phase curve of star TIC 236887394

In order to find other possible transits we cut out parts of the light curve
where the first transit occurs and repeat the procedure again building
periodogram using the BLS method.

##  3 Observations obtained at the Lisnyky comet station

###  3.1  Observations

Our observations were also carried out from March 24, 2021, to February 14,
2022, at the Lisnyky comet station with the 70-centimeter AZT-8 reflector
telescope, using the R filter.
The telescope is equipped with a FLI PL4710 back illuminated CCD and UBV RI
Bessel filters. For faint objects we use a mode with 2 x 2 binning, which
gives a scale of 1.96 arcsec/pixel. The FoV of the instrument is 16 × 16
arcmin. The limiting magnitude of a 300 s exposure image is 20 mag under good
sky conditions. It is possible to reach 21.5-22 mag with 1800 s exposures. For
our observations, the exposure time varied from 10 to 30 seconds, depending on
the brightness of the observed object.
We didn’t observe one particular object all the night. We chose the star and
the time span for the expected transit event and conducted observations during
the transit only. We began shooting half an hour before the intended start
time of the event and completed half an hour after the intended completion to
identify possible displacements in the moments of the beginning and end of the
transit. Table 1 gives the list of he observed objects and some parameters of
their host stars.

###  3.2 List of observed objects and their characteristics

|  Spectral type |  Fluxes |  Distance [pc] |  Constellation
---|---|---|---|---
TrES-3 |  G |  12.4 |  231.34 |  Hercules
Kepler-17 |  G5V |  14.14 |  720.75 |  Cygnus
Qatar-1 |  K |  12.65 |  185.62 |  Draco
WASP-43 |  K7V |  12.08 |  86.75 |  Sextans
WASP-3 |  F7V |  10.52 |  231.16 |  Lyra
TrES-5 |  K |  13.77 |  360.31 |  Cygnus
Qatar-2 |  K5V |  13.45 |  181.37 |  Virgo
WASP-10 |  K5V |  12.03 |  141 |  Pegasus

###  3.3 Processing observations with Muniwin

The processing of the observations obtained in the Lisnyky comet station was
carried out using the C-Munipack package https://c-munipack.sourceforge.net/.
This program uses differential photometry: we compare the brightness of two or
more stars. One star is one which brightness must change due to the transit of
the planet through its disk, and the other (others) are reference stars which
brightness should be unchanged. First, we calibrate the images using dark,
bias, flatfield files. The next step is to set the parameters by which the
program determines which object in the image is the star. The next step is to
look for the corresponding stars on each image. After that, we select a star
through the disk of which transit should take place and the stars which
brightness should be unchanged. Reference stars cannot be variable stars. We
use the Simbad database to make sure that the selected star has constant
brightness. When the necessary stars are selected, we proceed to the light
curve construction.

Figure 4: The light curve for the star TIC 36734222 obtained using the
C-Munipack package.

The resulting light curve was uploaded to the Czech Exoplanet Transit Database
(ETD) for further processing.

###  3.4 Exoplanet Transit Database

The ETD has an algorithm for processing light curves, by which it determines
the main transit parameters from these light curves: the moments of the
beginning, end and middle of transit, the depth, duration and radius of the
planet, and the inclination of the orbit. The period of the planet is
determined from all observations published in the database.

All our observations were uploaded to this database and received data quality
2 and 3. The data quality in the ETD is estimated on a scale from 1 to 5,
where 1 is the data with the highest quality.

##  4 Results

The parameters of the exoplanet transits obtained from the ground-based
observations and determined from the data base of the orbital telescopes are
listed in Table 1.
We could not extract some parameters from our own observations, therefore they
were taken from ETD based on all observations published in this data base.
Such parameters are indicated by symbol "*". The table also comprise the
periods and transit durations calculated in this work from the Kepler and TESS
light curves.

##  5 Conclusion

The comparison of the transit parameters determined from two different data
sets, i.e., ground-based and space-based, indicates that:
1) The transit periods and durations obtained from the ground-based
observations with the small telescope agree well with those obtained from
Kepler and TESS orbital telescopes, highlighting the high accuracy of the
ground-based observations presented in this work.
2) It is shown that the average agreement of transit parameters is at the
level of 0.00001 day (transit period) and 0.1 hour (transit duration)
3) Independent processing of the light curves from the Kepler and TESS
databases indicates that the preliminary processing of the light curves (the
light curve detrending, removing of the long-periodic harmonic stellar
oscillations) can significantly affect the accuracy of the extracted transit
parameters.

Acknowledgements. YP anf IK participation was performed in the frames of the
government funding program for institutions of the National Academy of
Sciences of Ukraine (NASU) and supported by the National Research Foundation
of Ukraine (Project N 2020.02/0228). Part of the data presented in this paper
were obtained from the Mikulski Archive for Space Telescopes (MAST), which is
hosted at STScI. STScI is operated by the Association of Universities for
Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST
for non-HST data is provided by the NASA Office of Space Science via grant
NNX09AF08G and by other grants and contracts. This paper includes data
collected by the TESS mission. Funding for the TESS mission is provided by the
NASA Science Mission directorate.

##  6 References

Bunnell J.: 2018, Using Kepler Data to Plot a Light Curve, [online] Available
at: (https://github.com/spacetelescope
notebooks/blob/master/notebooks/MAST/Kepler/Kepler_Lightcurve
/kepler_lightcurve.ipynb)

Bunnell J.: 2018, Plotting Images from Kepler Target Pixel Files, [online]
Available at: (https://github.com/
spacetelescope notebooks/blob /master/notebooks/MAST/Kepler/Kepler_TPF/
kepler_tpf.ipynb)

Hall O., Barentsen G.: 2020a, Using Kepler Light Curve Products with
Lightkurve, [online] Available at:
(https://github.com/spacetelescope/notebooks/blob
/master/notebooks/MAST/Kepler/kepler_using_
light_curve_ products_with_lightkurve/kepler_using
_light_curve_products_with_lightkurve.ipynb)

Hall O., Barentsen G.: 2020b, Using Kepler Target Pixel File Products with
Lightkurve, [online] Available at:
(https://github.com/spacetelescope/notebooks/blob/master/notebooks/MAST/Kepler/
kepler_using_
target_pixel_file_products_with_lightkurve/kepler_using_target_pixel_file_products_with_lightkurve
.ipynb)

Hall O., Barentsen G.: 2020c, Measuring and Removing a Rotation Period Signal
from a Kepler Light Curve,[online] Available at:
(https://github.com/spacetelescope/notebooks/blob /master/notebooks/MAST
/Kepler/kepler_measuring _a_rotation_period/kepler_measuring_a_rotation
_period.ipynb)

Hall O., Barentsen G.: 2020d, Creating Periodograms and Identifying
Significant Peaks, [online] Available at:
(https://github.com/spacetelescope/notebooks/blob
/master/notebooks/MAST/Kepler/kepler_creating_
periodograms/kepler_creating_periodograms.ipynb)

Hall O., Barentsen G.: 2020e, How to Understand and Manipulate the Periodogram
of an Oscillating Star, [online] Available at:
(https://github.com/spacetelescope/notebooks/blob
/master/notebooks/MAST/Kepler/
kepler_how_to_ understand_and_manipulate_the_periodogram_of
_an_oscillating_star/kepler_how
_to_understand_ and_manipulate_the_periodogram_of_an_oscillating _star.ipynb)

Pavlenko Ya., Shubina O., Kulyk I., Kuznyetsova Y., Zakhozhay O., Korsun P.,
Borysenko S., Krushevska V., Andreev M.: 2021, Exocometary Activity Around
Stars at Different Evolutionary Stages: Current Issues (Main Astronomical
Observatory of the NAS of Ukraine, Kyiv, Ukraine)

Pavlenko, Ya., I. Kulyk, O. Shubina, M. Vasylenko, D. Dobrycheva, P. Korsun:
2022 (arXiv:2202.13373)
Saunders N.: 2020, Identifying Transiting Planet Signals in a Kepler Light
Curve, [online] Available at:
(https://github.com/spacetelescope/notebooks/blob
/master/notebooks/MAST/Kepler/kepler_identifying
_transiting_planet_signals/kepler_identifying_
transiting_planet_signals.ipynb)

Terebizh V.Y.: 1992, Analysis of time series in astrophysics (Moscow "Nauka"
Gl.ed.fiz.-mat.lit.)

Winn, J. N., 2010, Exoplanet Transits and Occultations, in "Exoplanets", ed.
Seager, S., pp. 55-77.

Table 1. Parameters of the selected transits obtained from the ground based
and orbital telescopes.

Kepler 17b
Observation MAST obtained with Lightkurve 2.0 Period [day] 1.4857108*
1.4857108 1.4856924 Transit duration [hour] 2.41 2.29 2.40 Depth 0.0201 [mag]
1.7765[%] Planet radius $[R_{j}]$ 1.34 1.31 Inclination [∘] 90.00 87.2 Rorbit
[AU] 0.026* 0.026

TrES-3b
Observation MAST obtained with Lightkurve 2.0 Period [day] 1.3061861*
1.3061858 1.3061269 Transit duration [hour] 1.31 1.37 1.2 Depth 0.0279 [mag]
2.7361[%] Planet radius $[R_{j}]$ 1.262 1.336 Inclination [∘] 81.76 81.93
Rorbit [AU] 0.023* 0.023

Qatar-1b
Observation MAST obtained with Lightkurve 2.0 Period [day] 1.4200246*
1.4200242 1.4202261 Transit duration [hour] 1.59 1.66 1.37 Depth 0.0203 [mag]
2.1557[%] Planet radius $[R_{j}]$ 1.093 1.143 Inclination [∘] 83.43 84.48
Rorbit [AU] 0.023* 0.023

WASP-43b
Observation MAST obtained with Lightkurve 2.0 Period [day] 0.81347414*
0.813475 0.8134242 Transit duration [hour] 1.05 1.16 1.2 Depth 0.0280 [mag]
2.5371[%] Planet radius $[R_{j}]$ 0.93 0.93 Inclination [∘] 81.18 82.15 Rorbit
[AU] 0.014* 0.014

WASP-3b
Observation MAST obtained with Lightkurve 2.0 Period [day] 1.846835* 1.846830
1.846529 Transit duration [hour] 2.22 2.79 2.40 Depth 0.0124[mag] 1.0269[%]
Planet radius $[R_{j}]$ 1.36 1.42 Inclination [∘] 81.71 84.15 Rorbit [AU]
0.0317* 0.032

TrES-5b
Observation MAST obtained with Lightkurve 2.0 Period [day] 1.48224718*
1.48224690 1.48225292 Transit duration [hour] 1.852 1.846 1.200 Depth 0.0224
[mag] 2.192 [%] Planet radius $[R_{j}]$ 1.208 1.256 Inclination [∘] 84.54
84.53 Rorbit [AU] 0.025* 0.025


# Shokurov’s index conjecture for quotient singularities

Yusuke Nakamura  Graduate School of Mathematical Sciences, the University of
Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153-8914, Japan.
<EMAIL_ADDRESS>and  Kohsuke Shibata  Department of Mathematics,
Faculty of Natural Science and Technology, Okayama University, 3-1-1,
Tsushimanaka, Kita-ku, Okayama, 700-8530, Japan.
<EMAIL_ADDRESS>
###### Abstract.

We prove Shokurov’s index conjecture for quotient singularities.

###### Key words and phrases:

minimal log discrepancy, Gorenstein index, Shokurov’s index conjecture

###### 2020 Mathematics Subject Classification:

Primary 14E18; Secondary 14E30, 14B05

##  1\. Introduction

Shokurov conjectured that the Gorenstein index of a $\mathbb{Q}$-Gorenstein
germ can be bounded in terms of its dimension and minimal log discrepancy.

######  Conjecture 1.1 (Shokurov, cf. [Kaw15]*Question 5.2).

For any $n\in\mathbb{Z}_{>0}$ and $a\in\mathbb{R}_{\geq 0}$, there exists a
positive integer $r(n,a)$ with the following condition.

  * •

If an $n$-dimensional $\mathbb{Q}$-Gorenstein variety $X$ and a closed point
$p\in X$ satisfy $\operatorname{mld}_{p}(X)=a$, then the Cartier index of
$K_{X}$ at $p$ is at most $r(n,a)$.

Conjecture 1.1 may be useful to attack the ACC conjecture (see [Sho04]) since
the ACC conjecture is known for varieties with a fixed Gorenstein index and
boundaries with fixed coefficients ([Nak16b]). Conjecture 1.1 is known to be
true in the following cases:

  1. (1.1.1)

When $(n,a)=(2,0)$, Conjecture 1.1 can be confirmed by the classification of
surface singularities (cf. [Sho93]*Corollary 5.10).

  2. (1.1.2)

When $n=2$, Conjecture 1.1 is proved by G. Chen and J. Han [CH21]*Theorem 1.4.
They proved Conjecture 1.1 for pairs (see [CH21]*Conjecture 6.3 for the
generalized formulation).

  3. (1.1.3)

When $(n,a)=(3,0)$, Conjecture 1.1 is proved by Ishii in [Ish00] and Fujino in
[Fuj01]. Fujino in [Fuj01] also proved that Conjecture 1.1 for $a=0$ and
arbitrary $n$ assuming another conjecture on some boundedness of the
birational automorphisms.

  4. (1.1.4)

When the considered $X$ is a terminal threefold, Conjecture 1.1 follows from
the classification by Kawamata [Kaw92]. J. Han, J. Liu, and Y. Luo in
[HLL]*Theorem 1.5 proved Conjecture 1.1 for pairs.

  5. (1.1.5)

When the considered $X$ is a canonical threefold, Conjecture 1.1 is proved by
Kawakita [Kaw15].

  6. (1.1.6)

When the considered $X$ has toric singularities, Conjecture 1.1 is proved by
Ambro [Amb09] (cf. [Amb16]*Section 5.4).

  7. (1.1.7)

When the considered $X$ has only quotient singularity and $a$ is sufficiently
small, Conjecture 1.1 is proved by Moraga [Mor22].

The purpose of this paper is to prove Conjecture 1.1 for quotient
singularities.

######  Theorem 1.2 ($=$ Theorem 3.4).

Conjecture 1.1 with an additional condition that $X$ in the statement has only
quotient singularity at $p$, is true.

The main idea of the proof of Theorem 1.2 is to use Jordan’s theorem (Theorem
3.2) on finite linear groups and reduce it to the abelian quotient case, where
Conjecture 1.1 is proved by Ambro (see (1.1.6) above). This idea is due to
Moraga [Mor22].

######  Acknowledgements.

The first author is partially supported by JSPS KAKENHI No. 18K13384,
22K13888, 18H01108, and JPJSBP120219935. The second author is partially
supported by JSPS KAKENHI No. 19K14496.

## Notation

  * •

We basically follow the standard notions and terminologies in [Har77] and
[Kol13].

  * •

Throughout this paper, $k$ is an algebraically closed field of characteristic
zero. We say that $X$ is a variety if $X$ is an integral scheme that is
separated and of finite type over $k$.

##  2\. Preriminaries

First, we clarify the definition of quotient singularities in this paper. For
a local ring $A$, let $\widehat{A}$ denote the completion of $A$ at the
maximal ideal.

######  Definition 2.1.

Let $X$ be a variety of dimension $n$ and let $p\in X$ be a closed point. We
say that $X$ has a quotient singularity at $p$ if there exists a finite
subgroup $G\subset\operatorname{GL}_{n}(k)$ such that
$\widehat{\mathcal{O}}_{X,p}\simeq\widehat{\mathcal{O}}_{M,x_{0}}$, where
$M=\mathbb{A}^{n}_{k}/G$ is the quotient variety obtained by the natural
linear action of $G$ on $\mathbb{A}^{n}_{k}$, and $x_{0}\in M$ is the image of
the origin of $\mathbb{A}^{n}_{k}$.

Next, we recall the definition of the Gorenstein indices and the minimal log
discrepancies for $\mathbb{Q}$-Gorenstein normal singularities.

######  Definition 2.2.

Let $X$ be a normal variety and let $p\in X$ be a closed point. Suppose that
$X$ has a $\mathbb{Q}$-Gorenstein singularity at $p$ (i.e. $rK_{X}$ is Cartier
at $p$ for some positive integer $r$).

  1. (1)

The Gorenstein index of $X$ at $p$ is the smallest positive integer $r$ such
that $rK_{X}$ is Cartier at $p$.

  2. (2)

Let $f:X^{\prime}\to X$ be a proper birational morphism from a normal variety
$X^{\prime}$ and let $E$ be a prime divisor on $X^{\prime}$. Then the log
discrepancy $a_{E}(X)$ is defined by

$a_{E}(X):=1+\operatorname{ord}_{E}(K_{X^{\prime}/X}),$

where $K_{X^{\prime}/X}:=K_{X^{\prime}}-f^{*}K_{X}$ is the relative canonical
divisor. We define the minimal log discrepancy $\operatorname{mld}_{p}(X)$ at
$p$ by

$\operatorname{mld}_{p}(X):=\inf_{c_{X}(E)=\\{p\\}}a_{E}(X)$

if $\dim X\geq 2$, where the infimum is taken over all prime divisors $E$ over
$X$ with center $c_{X}(E)=\\{p\\}$. When $\dim X=1$, we define
$\operatorname{mld}_{p}(X):=\inf_{c_{X}(E)=\\{p\\}}a_{E}(X)$ if
$\inf_{c_{X}(E)=\\{p\\}}a_{E}(X)\geq 0$, and we define
$\operatorname{mld}_{p}(X):=-\infty$ otherwise. It is known that
$\operatorname{mld}_{p}(X)\in\mathbb{Q}_{\geq 0}\cup\\{-\infty\\}$ in general
(cf. [KM98]*Corollary 2.31).

For a finite group $G\subset\operatorname{GL}_{n}(k)$ and $g\in G$, we define
$d(G)$ and $\operatorname{age}^{\prime}(g)$ as follows.

######  Definition 2.3.

Let $n$ be a positive integer and let $G\subset\operatorname{GL}_{n}(k)$ be a
finite subgroup. Let $d:=\\#G$ be the order of $G$, and let $\xi\in k$ be a
primitive $d$-th root of unity.

  1. (1)

We define a positive integer $d(G)$ by

$d(G):=\min\bigl{\\{}\ell\in\mathbb{Z}_{>0}\ \big{|}\
\text{$(\det(g))^{\ell}=1$ holds for any $g\in G$}\bigr{\\}}.$

  2. (2)

Let $g\in G$. Since $g$ has finite order, $g$ is conjugate to a diagonal
matrix $\operatorname{diag}(\xi^{e_{1}},\ldots,\xi^{e_{n}})$ with $1\leq
e_{i}\leq d$. Then, we define
$\operatorname{age}^{\prime}(g):=\sum_{i=1}^{n}\frac{e_{i}}{d}$. Note that the
age of $g$ is usually defined by
$\operatorname{age}(g):=\operatorname{age}^{\prime}(g)-\\#\\{1\leq i\leq n\mid
e_{i}=d\\}$.

Weston in [Wes91] proved that the Gorenstein index of $\mathbb{A}^{n}_{k}/G$
at the origin $x_{0}$ coincides with $d(G)$ when $G$ has no pseudo-
reflections.

######  Theorem 2.4 ([Wes91]*Theorem 2.2).

Let $n$ be a positive integer and let $G\subset\operatorname{GL}_{n}(k)$ be a
finite subgroup. Suppose that $G$ does not contain a pseudo-reflection. Let
$x_{0}\in\mathbb{A}^{n}_{k}/G$ be the image of the origin of
$\mathbb{A}^{n}_{k}$. Then, $d(G)$ coincides with the Gorenstein index of
$\mathbb{A}^{n}_{k}/G$ at $x_{0}$.

######  Remark 2.5.

Weston’s theorem [Wes91]*Theorem 2.2 treats a general setting, where the
finite group $G$ may contain a pseudo-reflection. Note that in the notation in
[Wes91]*Hypotheses 2.1, we have $d_{ij}=1$ and $g^{\prime}_{j}=g_{j}$ for any
$i$ and $j$ when $G$ does not contain a pseudo-reflection.

We can also see that the minimal log discrepancy of $\mathbb{A}^{n}_{k}/G$ at
the origin $x_{0}$ can be described by $\operatorname{age}^{\prime}(g)$.

######  Proposition 2.6.

Let $n$, $G$ and $x_{0}$ be as in Theorem 2.4. Then it follows that

$\operatorname{mld}_{x_{0}}\bigl{(}\mathbb{A}^{n}_{k}/G\bigr{)}=\min\\{\operatorname{age}^{\prime}(g)\mid
g\in G\\}.$

###### Proof.

First, note that this formula is well-known when $G$ is an abelian group (see
Remark 2.7(1)).

For $\gamma\in G$, let $\langle\gamma\rangle\subset G$ denote the subgroup
generated by $\gamma$, and let
$x_{\gamma}\in\mathbb{A}^{n}_{k}/\langle\gamma\rangle$ denote the image of the
origin of $\mathbb{A}^{n}_{k}$. Then, by [NS22]*Corollary 4.12 (see also
[NS22]*Remark 4.13), we have

$\operatorname{mld}_{x_{0}}\bigl{(}\mathbb{A}^{n}_{k}/G\bigr{)}=\min_{\gamma\in
G}\operatorname{mld}_{x_{\gamma}}\bigl{(}\mathbb{A}^{n}_{k}/\langle\gamma\rangle\bigr{)}.$

Therefore, the assertion follows from the cyclic quotient case. ∎

######  Remark 2.7.

  1. (1)

When $G$ is an abelian group, Proposition 2.6 can be proved by the toric
method. For the reader’s convenience, we give a proof below. We refer to [CLS]
for the standard notions on toric varieties.

Suppose that $G$ in Proposition 2.6 is an abelian group. Since $G$ is a finite
abelian group, its elements are simultaneously diagonalizable. Hence, we may
assume that $G$ consists of diagonal matrices from the beginning. Let
$d\in\mathbb{Z}_{>0}$ and $\xi\in k$ be as in Definition 2.3. Each $g\in G$
can be written as
$g=\operatorname{diag}\bigl{(}\xi^{e_{1}^{(g)}},\ldots,\xi^{e_{n}^{(g)}}\bigr{)}$.
We define
$u^{(g)}:=\bigl{(}e_{1}^{(g)}/d,\ldots,e_{n}^{(g)}/d\bigr{)}\in\mathbb{R}^{n}$,
and define a lattice $N\subset\mathbb{R}^{n}$ by $N:=\mathbb{Z}^{n}+\sum_{g\in
G}\mathbb{Z}u^{(g)}$. Let $u_{1},\ldots,u_{n}\in\mathbb{R}^{n}$ be the
standard basis (i.e. $u_{i}$ is the vector with $1$ in the $i$-th coordinate
and $0$’s elsewhere). Let $\sigma\subset\mathbb{R}^{n}=N_{\mathbb{R}}$ be the
cone spanned by $u_{1},\ldots,u_{n}$. Let $U_{\sigma,N}$ denote the toric
variety corresponding to $\sigma\subset N_{\mathbb{R}}$. Then we have
$U_{\sigma,N}\simeq\mathbb{A}^{n}_{k}/G$ (cf. [CLS]*Example 1.3.20).

Since $G$ does not contain a pseudo-reflection, $u_{1},\ldots,u_{n}\in N$ are
primitive vectors (i.e. there are no integer $\ell\geq 2$ and $u\in N$
satisfying $u_{i}=\ell u$). Therefore, the support function
$\varphi:\sigma\to\mathbb{R}$ of the $\mathbb{Q}$-Cartier divisor
$K_{U_{\sigma,N}}$ is given by the function satisfying
$\varphi((a_{1},\ldots,a_{n}))=\sum_{i}a_{i}$ for any
$(a_{1},\ldots,a_{n})\in\sigma\subset\mathbb{R}^{n}$. Furthermore, we have the
one-to-one correspondence between the primitive vectors $u$ in
$\operatorname{int}(\sigma)\cap N$ and the torus invariant divisors $D_{u}$
over $U_{\sigma,N}$ whose centers are the torus invariant closed point. Then,
we have $a_{D_{u}}(U_{\sigma,N})=\varphi(u)$ by [CLS]*Lemma 11.4.10.
Therefore, we get

$\operatorname{mld}_{x_{0}}\bigl{(}\mathbb{A}^{n}_{k}/G\bigr{)}=\min_{u\in\operatorname{int}(\sigma)\cap
N}\varphi(u)=\min_{u\in(0,1]^{n}\cap N}\varphi(u),$

and the right hand side is equal to $\min\\{\operatorname{age}^{\prime}(g)\mid
g\in G\\}$ since we have $\varphi(u^{(g)})=\operatorname{age}^{\prime}(g)$ for
$g\in G$.

  2. (2)

In the same setting as in Proposition 2.6, it is known that

$\operatorname{mld}\bigl{(}\mathbb{A}^{n}_{k}/G\bigr{)}=\min\\{\operatorname{age}(g)\mid
g\in G\setminus\\{1_{G}\\}\\},$

where $\operatorname{mld}(X):=\inf_{E}a_{E}(X)$ is the total log discrepancy.
This formula is proved by Yasuda [Yas06] using the motivic integration method.
Proposition 2.6 can also be directly proved in a similar way without reducing
the abelian quotient case. Indeed, it follows from [NS22]*Theorem 4.8 or
[NS2]*Theorem 7.9.

##  3\. Shokurov’s index conjecture for quotient singularities

In this section, we prove Shokurov’s index conjecture for quotient
singularities. First, note that Ambro in [Amb09] proved this conjecture for
toric varieties, in particular for abelian quotient singularities.

######  Theorem 3.1 (Ambro [Amb09]).

Conjecture 1.1 is true for toric singularities, in particular for abelian
quotient singularities.

We use Jordan’s theorem on finite linear groups.

######  Theorem 3.2 (Jordan’s theorem on finite linear groups [Jor78]).

For a positive integer $n$, there exists a positive integer $c_{n}$ with the
following condition:

  * •

Suppose that $K$ is a field of characteristic zero and that
$G\subset\operatorname{GL}_{n}(K)$ is a finite subgroup. Then, there exists an
abelian normal subgroup $H\subset G$ such that its index satisfies $[G:H]\leq
c_{n}$.

######  Remark 3.3.

Although Jordan’s theorem is usually asserted for $K=\mathbb{C}$, it holds for
any field $K$ of characteristic zero with the same bound $c_{n}$. Since $G$ is
a finite subgroup, $G\subset\operatorname{GL}_{n}(K^{\prime})$ holds for some
subfield $K^{\prime}\subset K$ with $\\#K^{\prime}=\aleph_{0}$. Since such
$K^{\prime}$ can be embedded into $\mathbb{C}$, the assertion follows from the
case where $K=\mathbb{C}$.

######  Theorem 3.4.

Conjecture 1.1 is true for quotient singularities.

###### Proof.

Let $n\in\mathbb{Z}_{>0}$ and $a\in\mathbb{R}_{\geq 0}$. Suppose that a
variety $X$ of dimension $n$ has a quotient singularity at a closed point
$p\in X$. Furthermore, assume $\operatorname{mld}_{p}(X)=a$. By the definition
of quotient singularities (Definition 2.1), there exists a finite subgroup
$G\subset\operatorname{GL}_{n}(k)$ such that
$\widehat{\mathcal{O}}_{X,p}\simeq\widehat{\mathcal{O}}_{M,x_{0}}$, where
$M=\mathbb{A}^{n}_{k}/G$ is the quotient variety and $x_{0}\in M$ is the image
of the origin of $\mathbb{A}^{n}_{k}$. Let $G_{\rm pr}\subset G$ be the
subgroup generated by the pseudo-reflections in $G$. Then, by replacing $G$
with $G/G_{\rm pr}$, we may assume that $G$ does not contain a pseudo-
reflection (cf. [Kol13]*3.18). Note that the minimal log discrepancy is
preserved under the completion (cf. [NS2]*Remark 3.1). Furthermore, the
Gorenstein index is also preserved under the completion (cf.
[dFEM11]*Proposition A.14). Therefore, we may assume
$X=M=\mathbb{A}^{n}_{k}/G$ and $p=x_{0}$ from the beginning.

By Jordan’s theorem (Theorem 3.2), we can take an abelian normal subgroup
$H\subset G$ such that $[G:H]\leq c_{n}$. By Proposition 2.6, we can take
$g\in G$ and $h\in H$ such that

$a=\operatorname{mld}_{x_{0}}\bigl{(}\mathbb{A}^{n}_{k}/G\bigr{)}=\operatorname{age}^{\prime}(g),\quad\operatorname{mld}_{x_{0}}\bigl{(}\mathbb{A}^{n}_{k}/H\bigr{)}=\operatorname{age}^{\prime}(h).$

We set $c^{\prime}:=c_{n}!$ and

$h^{\prime}:=h\cdot(ghg^{-1})\cdot(g^{2}hg^{-2})\cdots(g^{c^{\prime}-1}hg^{-(c^{\prime}-1)}).$

Since $H$ is a normal subgroup of $G$, we have $g^{i}hg^{-i}\in H$ for any
$i\geq 0$. Furthermore, since $[G:H]\leq c_{n}$, we have $g^{c^{\prime}}\in
H$, and hence
$g^{c^{\prime}}hg^{-c^{\prime}}=g^{c^{\prime}}g^{-c^{\prime}}h=h$. Therefore,
we have

$gh^{\prime}g^{-1}=(ghg^{-1})\cdot(g^{2}hg^{-2})\cdots(g^{c^{\prime}-1}hg^{-(c^{\prime}-1)})\cdot
h=h^{\prime}.$

Let $\langle g,h^{\prime}\rangle\subset G$ denote the subgroup generated by
$g$ and $h^{\prime}$. Since we have

$\operatorname{age}^{\prime}(g)=\operatorname{mld}_{x_{0}}\bigl{(}\mathbb{A}^{n}_{k}/G\bigr{)}=\min_{\alpha\in
G}\operatorname{age}^{\prime}(\alpha)\leq\min_{\alpha\in\langle
g,h^{\prime}\rangle}\operatorname{age}^{\prime}(\alpha)\leq\operatorname{age}^{\prime}(g),$

it follows that

$\operatorname{mld}_{x_{0}}\bigl{(}\mathbb{A}^{n}_{k}/\langle
g,h^{\prime}\rangle\bigr{)}=\min_{\alpha\in\langle
g,h^{\prime}\rangle}\operatorname{age}^{\prime}(\alpha)=\operatorname{age}^{\prime}(g)=a.$

Since $\langle g,h^{\prime}\rangle$ is an abelian group, by Theorem 3.1, we
can conclude that $d(\langle g,h^{\prime}\rangle)$ divides
$r^{\prime}_{1}(n,a)$ for some positive integer $r^{\prime}_{1}(n,a)$
depending only on $n$ and $a$. Since
$\det(h^{\prime})=(\det(h))^{c^{\prime}}$, the order of $\det(h)$ divides
$c_{n}!\cdot r^{\prime}_{1}(n,a)$. In particular,
$\operatorname{mld}_{x_{0}}\bigl{(}\mathbb{A}^{n}_{k}/H\bigr{)}=\operatorname{age}^{\prime}(h)$
is contained in the finite set $\frac{1}{c_{n}!\cdot
r^{\prime}_{1}(n,a)}\mathbb{Z}\cap(0,n]$. By applying Theorem 3.1 to $H$, we
can conclude that $d(H)$ divides $r^{\prime}_{2}(n,a)$ for some positive
integer $r^{\prime}_{2}(n,a)$ depending only on $n$ and $a$. Since $[G:H]\leq
c_{n}$, we have $\alpha^{c_{n}!}\in H$ for any $\alpha\in G$. Therefore,
$d(G)$ divides $c_{n}!\cdot d(H)$, and in particular, $d(G)$ divides
$c_{n}!\cdot r^{\prime}_{2}(n,a)$. We complete the proof. ∎

######  Remark 3.5.

As a corollary of Theorem 3.4, we could prove Conjecture 1.1 for surface
singularities. Note that log terminal surface singularities are quotient
singularities (cf. [Kaw84]*Corollary 1.9). More generally, G. Chen and J. Han
in [CH21]*Theorem 1.4 proved that Shokurov’s index conjecture for surface
singularities for pairs ([CH21]*Conjecture 6.3).

## References