The Belle Collaboration

#
The Study of $\gamma\gamma\to\gamma\psi(2S)$ at Belle

X. L. Wang  Key Laboratory of Nuclear Physics and Ion-beam Application (MOE)
and Institute of Modern Physics, Fudan University, Shanghai 200443, PR China
B. S. Gao  Key Laboratory of Nuclear Physics and Ion-beam Application (MOE)
and Institute of Modern Physics, Fudan University, Shanghai 200443, PR China
W. J. Zhu  Key Laboratory of Nuclear Physics and Ion-beam Application (MOE)
and Institute of Modern Physics, Fudan University, Shanghai 200443, PR China
I. Adachi  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801, Japan  SOKENDAI (The Graduate University for Advanced Studies),
Hayama 240-0193, Japan  H. Aihara  Department of Physics, University of Tokyo,
Tokyo 113-0033, Japan  S. Al Said  Department of Physics, Faculty of Science,
University of Tabuk, Tabuk 71451, Saudi Arabia  Department of Physics, Faculty
of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia  D. M. Asner
Brookhaven National Laboratory, Upton, New York 11973, USA  H. Atmacan
University of Cincinnati, Cincinnati, OH 45221, USA  V. Aulchenko  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russian Federation
Novosibirsk State University, Novosibirsk 630090, Russian Federation  T.
Aushev  Higher School of Economics (HSE), Moscow 101000, Russian Federation
R. Ayad  Department of Physics, Faculty of Science, University of Tabuk, Tabuk
71451, Saudi Arabia  V. Babu  Deutsches Elektronen–Synchrotron, 22607 Hamburg,
Germany  S. Bahinipati  Indian Institute of Technology Bhubaneswar, Satya
Nagar 751007, India  P. Behera  Indian Institute of Technology Madras, Chennai
600036, India  V. Bhardwaj  Indian Institute of Science Education and Research
Mohali, SAS Nagar, 140306, India  B. Bhuyan  Indian Institute of Technology
Guwahati, Assam 781039, India  T. Bilka  Faculty of Mathematics and Physics,
Charles University, 121 16 Prague, The Czech Republic  J. Biswal  J. Stefan
Institute, 1000 Ljubljana, Slovenia  A. Bobrov  Budker Institute of Nuclear
Physics SB RAS, Novosibirsk 630090, Russian Federation  Novosibirsk State
University, Novosibirsk 630090, Russian Federation  G. Bonvicini  Wayne State
University, Detroit, MI 48202, USA  A. Bozek  H. Niewodniczanski Institute of
Nuclear Physics, Krakow 31-342, Poland  M. Bračko  Faculty of Chemistry and
Chemical Engineering, University of Maribor, 2000 Maribor, Slovenia  J. Stefan
Institute, 1000 Ljubljana, Slovenia  M. Campajola  INFN - Sezione di Napoli,
80126 Napoli, Italy  Università di Napoli Federico II, 80126 Napoli, Italy  D.
Červenkov  Faculty of Mathematics and Physics, Charles University, 121 16
Prague, The Czech Republic  M.-C. Chang  Department of Physics, Fu Jen
Catholic University, Taipei 24205, Taiwan  V. Chekelian  Max-Planck-Institut
für Physik, 80805 München, Germany  A. Chen  National Central University,
Chung-li 32054, Taiwan  B. G. Cheon  Department of Physics and Institute of
Natural Sciences, Hanyang University, Seoul 04763, South Korea  K. Chilikin
P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow
119991, Russian Federation  H. E. Cho  Department of Physics and Institute of
Natural Sciences, Hanyang University, Seoul 04763, South Korea  K. Cho  Korea
Institute of Science and Technology Information, Daejeon 34141, South Korea
S.-K. Choi  Gyeongsang National University, Jinju 52828, South Korea  Y. Choi
Sungkyunkwan University, Suwon 16419, South Korea  S. Choudhury  Indian
Institute of Technology Hyderabad, Telangana 502285, India  D. Cinabro  Wayne
State University, Detroit, MI 48202, USA  S. Cunliffe  Deutsches
Elektronen–Synchrotron, 22607 Hamburg, Germany  S. Das  Malaviya National
Institute of Technology Jaipur, Jaipur 302017, India  G. De Nardo  INFN -
Sezione di Napoli, 80126 Napoli, Italy  Università di Napoli Federico II,
80126 Napoli, Italy  R. Dhamija  Indian Institute of Technology Hyderabad,
Telangana 502285, India  F. Di Capua  INFN - Sezione di Napoli, 80126 Napoli,
Italy  Università di Napoli Federico II, 80126 Napoli, Italy  Z. Doležal
Faculty of Mathematics and Physics, Charles University, 121 16 Prague, The
Czech Republic  T. V. Dong  Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai
200443, PR China  S. Eidelman  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090, Russian Federation  Novosibirsk State University,
Novosibirsk 630090, Russian Federation  P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Moscow 119991, Russian Federation  T. Ferber
Deutsches Elektronen–Synchrotron, 22607 Hamburg, Germany  D. Ferlewicz  School
of Physics, University of Melbourne, Victoria 3010, Australia  A. Frey  II.
Physikalisches Institut, Georg-August-Universität Göttingen, 37073 Göttingen,
Germany  B. G. Fulsom  Pacific Northwest National Laboratory, Richland, WA
99352, USA  R. Garg  Panjab University, Chandigarh 160014, India  V. Gaur
Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
N. Gabyshev  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090,
Russian Federation  Novosibirsk State University, Novosibirsk 630090, Russian
Federation  A. Garmash  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090, Russian Federation  Novosibirsk State University,
Novosibirsk 630090, Russian Federation  A. Giri  Indian Institute of
Technology Hyderabad, Telangana 502285, India  P. Goldenzweig  Institut für
Experimentelle Teilchenphysik, Karlsruher Institut für Technologie, 76131
Karlsruhe, Germany  B. Golob  Faculty of Mathematics and Physics, University
of Ljubljana, 1000 Ljubljana, Slovenia  J. Stefan Institute, 1000 Ljubljana,
Slovenia  C. Hadjivasiliou  Pacific Northwest National Laboratory, Richland,
WA 99352, USA  T. Hara  High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801, Japan  SOKENDAI (The Graduate University for Advanced
Studies), Hayama 240-0193, Japan  O. Hartbrich  University of Hawaii,
Honolulu, HI 96822, USA  K. Hayasaka  Niigata University, Niigata 950-2181,
Japan  H. Hayashii  Nara Women’s University, Nara 630-8506, Japan  M. T.
Hedges  University of Hawaii, Honolulu, HI 96822, USA  W.-S. Hou  Department
of Physics, National Taiwan University, Taipei 10617, Taiwan  C.-L. Hsu
School of Physics, University of Sydney, New South Wales 2006, Australia  T.
Iijima  Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan
Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan  K.
Inami  Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan
A. Ishikawa  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801, Japan  SOKENDAI (The Graduate University for Advanced Studies),
Hayama 240-0193, Japan  R. Itoh  High Energy Accelerator Research Organization
(KEK), Tsukuba 305-0801, Japan  SOKENDAI (The Graduate University for Advanced
Studies), Hayama 240-0193, Japan  M. Iwasaki  Osaka City University, Osaka
558-8585, Japan  Y. Iwasaki  High Energy Accelerator Research Organization
(KEK), Tsukuba 305-0801, Japan  W. W. Jacobs  Indiana University, Bloomington,
IN 47408, USA  S. Jia  Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE) and Institute of Modern Physics, Fudan University, Shanghai
200443, PR China  Y. Jin  Department of Physics, University of Tokyo, Tokyo
113-0033, Japan  C. W. Joo  Kavli Institute for the Physics and Mathematics of
the Universe (WPI), University of Tokyo, Kashiwa 277-8583, Japan  K. K. Joo
Chonnam National University, Gwangju 61186, South Korea  J. Kahn  Institut für
Experimentelle Teilchenphysik, Karlsruher Institut für Technologie, 76131
Karlsruhe, Germany  K. H. Kang  Kyungpook National University, Daegu 41566,
South Korea  T. Kawasaki  Kitasato University, Sagamihara 252-0373, Japan  C.
Kiesling  Max-Planck-Institut für Physik, 80805 München, Germany  C. H. Kim
Department of Physics and Institute of Natural Sciences, Hanyang University,
Seoul 04763, South Korea  D. Y. Kim  Soongsil University, Seoul 06978, South
Korea  S. H. Kim  Seoul National University, Seoul 08826, South Korea  Y.-K.
Kim  Yonsei University, Seoul 03722, South Korea  P. Kodyš  Faculty of
Mathematics and Physics, Charles University, 121 16 Prague, The Czech Republic
T. Konno  Kitasato University, Sagamihara 252-0373, Japan  A. Korobov  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russian Federation
Novosibirsk State University, Novosibirsk 630090, Russian Federation  S.
Korpar  Faculty of Chemistry and Chemical Engineering, University of Maribor,
2000 Maribor, Slovenia  J. Stefan Institute, 1000 Ljubljana, Slovenia  E.
Kovalenko  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090,
Russian Federation  Novosibirsk State University, Novosibirsk 630090, Russian
Federation  P. Križan  Faculty of Mathematics and Physics, University of
Ljubljana, 1000 Ljubljana, Slovenia  J. Stefan Institute, 1000 Ljubljana,
Slovenia  R. Kroeger  University of Mississippi, University, MS 38677, USA  P.
Krokovny  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090,
Russian Federation  Novosibirsk State University, Novosibirsk 630090, Russian
Federation  R. Kulasiri  Kennesaw State University, Kennesaw GA 30144, USA  M.
Kumar  Malaviya National Institute of Technology Jaipur, Jaipur 302017, India
R. Kumar  Punjab Agricultural University, Ludhiana 141004, India  K. Kumara
Wayne State University, Detroit, MI 48202, USA  A. Kuzmin  Budker Institute of
Nuclear Physics SB RAS, Novosibirsk 630090, Russian Federation  Novosibirsk
State University, Novosibirsk 630090, Russian Federation  Y.-J. Kwon  Yonsei
University, Seoul 03722, South Korea  K. Lalwani  Malaviya National Institute
of Technology Jaipur, Jaipur 302017, India  J. S. Lange  Justus-Liebig-
Universität Gießen, 35392 Gießen, Germany  I. S. Lee  Department of Physics
and Institute of Natural Sciences, Hanyang University, Seoul 04763, South
Korea  S. C. Lee  Kyungpook National University, Daegu 41566, South Korea  P.
Lewis  University of Bonn, 53115 Bonn, Germany  J. Li  Kyungpook National
University, Daegu 41566, South Korea  L. K. Li  University of Cincinnati,
Cincinnati, OH 45221, USA  Y. B. Li  Peking University, Beijing 100871, PR
China  L. Li Gioi  Max-Planck-Institut für Physik, 80805 München, Germany  J.
Libby  Indian Institute of Technology Madras, Chennai 600036, India  K. Lieret
Ludwig Maximilians University, 80539 Munich, Germany  D. Liventsev  Wayne
State University, Detroit, MI 48202, USA  High Energy Accelerator Research
Organization (KEK), Tsukuba 305-0801, Japan  C. MacQueen  School of Physics,
University of Melbourne, Victoria 3010, Australia  M. Masuda  Earthquake
Research Institute, University of Tokyo, Tokyo 113-0032, Japan  Research
Center for Nuclear Physics, Osaka University, Osaka 567-0047, Japan  T.
Matsuda  University of Miyazaki, Miyazaki 889-2192, Japan  D. Matvienko
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russian
Federation  Novosibirsk State University, Novosibirsk 630090, Russian
Federation  P.N. Lebedev Physical Institute of the Russian Academy of
Sciences, Moscow 119991, Russian Federation  M. Merola  INFN - Sezione di
Napoli, 80126 Napoli, Italy  Università di Napoli Federico II, 80126 Napoli,
Italy  F. Metzner  Institut für Experimentelle Teilchenphysik, Karlsruher
Institut für Technologie, 76131 Karlsruhe, Germany  K. Miyabayashi  Nara
Women’s University, Nara 630-8506, Japan  R. Mizuk  P.N. Lebedev Physical
Institute of the Russian Academy of Sciences, Moscow 119991, Russian
Federation  Higher School of Economics (HSE), Moscow 101000, Russian
Federation  G. B. Mohanty  Tata Institute of Fundamental Research, Mumbai
400005, India  M. Mrvar  Institute of High Energy Physics, Vienna 1050,
Austria  R. Mussa  INFN - Sezione di Torino, 10125 Torino, Italy  M. Nakao
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193,
Japan  Z. Natkaniec  H. Niewodniczanski Institute of Nuclear Physics, Krakow
31-342, Poland  A. Natochii  University of Hawaii, Honolulu, HI 96822, USA  L.
Nayak  Indian Institute of Technology Hyderabad, Telangana 502285, India  M.
Nayak  School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978,
Israel  M. Niiyama  Kyoto Sangyo University, Kyoto 603-8555, Japan  N. K.
Nisar  Brookhaven National Laboratory, Upton, New York 11973, USA  S. Nishida
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193,
Japan  K. Nishimura  University of Hawaii, Honolulu, HI 96822, USA  S. Ogawa
Toho University, Funabashi 274-8510, Japan  H. Ono  Nippon Dental University,
Niigata 951-8580, Japan  Niigata University, Niigata 950-2181, Japan  Y. Onuki
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan  P. Oskin
P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow
119991, Russian Federation  P. Pakhlov  P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Moscow 119991, Russian Federation  Moscow
Physical Engineering Institute, Moscow 115409, Russian Federation  G. Pakhlova
Higher School of Economics (HSE), Moscow 101000, Russian Federation  P.N.
Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991,
Russian Federation  T. Pang  University of Pittsburgh, Pittsburgh, PA 15260,
USA  S. Pardi  INFN - Sezione di Napoli, 80126 Napoli, Italy  H. Park
Kyungpook National University, Daegu 41566, South Korea  S.-H. Park  High
Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan  S.
Patra  Indian Institute of Science Education and Research Mohali, SAS Nagar,
140306, India  S. Paul  Department of Physics, Technische Universität München,
85748 Garching, Germany  Max-Planck-Institut für Physik, 80805 München,
Germany  T. K. Pedlar  Luther College, Decorah, IA 52101, USA  R. Pestotnik
J. Stefan Institute, 1000 Ljubljana, Slovenia  L. E. Piilonen  Virginia
Polytechnic Institute and State University, Blacksburg, VA 24061, USA  T.
Podobnik  Faculty of Mathematics and Physics, University of Ljubljana, 1000
Ljubljana, Slovenia  J. Stefan Institute, 1000 Ljubljana, Slovenia  V. Popov
Higher School of Economics (HSE), Moscow 101000, Russian Federation  E.
Prencipe  Forschungszentrum Jülich, 52425 Jülich, Germany  M. T. Prim
University of Bonn, 53115 Bonn, Germany  M. Röhrken  Deutsches
Elektronen–Synchrotron, 22607 Hamburg, Germany  A. Rostomyan  Deutsches
Elektronen–Synchrotron, 22607 Hamburg, Germany  N. Rout  Indian Institute of
Technology Madras, Chennai 600036, India  G. Russo  Università di Napoli
Federico II, 80126 Napoli, Italy  D. Sahoo  Tata Institute of Fundamental
Research, Mumbai 400005, India  S. Sandilya  Indian Institute of Technology
Hyderabad, Telangana 502285, India  A. Sangal  University of Cincinnati,
Cincinnati, OH 45221, USA  L. Santelj  Faculty of Mathematics and Physics,
University of Ljubljana, 1000 Ljubljana, Slovenia  J. Stefan Institute, 1000
Ljubljana, Slovenia  T. Sanuki  Department of Physics, Tohoku University,
Sendai 980-8578, Japan  G. Schnell  Department of Physics, University of the
Basque Country UPV/EHU, 48080 Bilbao, Spain  IKERBASQUE, Basque Foundation for
Science, 48013 Bilbao, Spain  C. Schwanda  Institute of High Energy Physics,
Vienna 1050, Austria  Y. Seino  Niigata University, Niigata 950-2181, Japan
K. Senyo  Yamagata University, Yamagata 990-8560, Japan  M. E. Sevior  School
of Physics, University of Melbourne, Victoria 3010, Australia  M. Shapkin
Institute for High Energy Physics, Protvino 142281, Russian Federation  C.
Sharma  Malaviya National Institute of Technology Jaipur, Jaipur 302017, India
C. P. Shen  Key Laboratory of Nuclear Physics and Ion-beam Application (MOE)
and Institute of Modern Physics, Fudan University, Shanghai 200443, PR China
J.-G. Shiu  Department of Physics, National Taiwan University, Taipei 10617,
Taiwan  B. Shwartz  Budker Institute of Nuclear Physics SB RAS, Novosibirsk
630090, Russian Federation  Novosibirsk State University, Novosibirsk 630090,
Russian Federation  F. Simon  Max-Planck-Institut für Physik, 80805 München,
Germany  J. B. Singh  Panjab University, Chandigarh 160014, India  A. Sokolov
Institute for High Energy Physics, Protvino 142281, Russian Federation  E.
Solovieva  P.N. Lebedev Physical Institute of the Russian Academy of Sciences,
Moscow 119991, Russian Federation  S. Stanič  University of Nova Gorica, 5000
Nova Gorica, Slovenia  M. Starič  J. Stefan Institute, 1000 Ljubljana,
Slovenia  Z. S. Stottler  Virginia Polytechnic Institute and State University,
Blacksburg, VA 24061, USA  M. Sumihama  Gifu University, Gifu 501-1193, Japan
M. Takizawa  Showa Pharmaceutical University, Tokyo 194-8543, Japan  J-PARC
Branch, KEK Theory Center, High Energy Accelerator Research Organization
(KEK), Tsukuba 305-0801, Japan  Meson Science Laboratory, Cluster for
Pioneering Research, RIKEN, Saitama 351-0198, Japan  U. Tamponi  INFN -
Sezione di Torino, 10125 Torino, Italy  F. Tenchini  Deutsches
Elektronen–Synchrotron, 22607 Hamburg, Germany  M. Uchida  Tokyo Institute of
Technology, Tokyo 152-8550, Japan  S. Uehara  High Energy Accelerator Research
Organization (KEK), Tsukuba 305-0801, Japan  SOKENDAI (The Graduate University
for Advanced Studies), Hayama 240-0193, Japan  T. Uglov  P.N. Lebedev Physical
Institute of the Russian Academy of Sciences, Moscow 119991, Russian
Federation  Higher School of Economics (HSE), Moscow 101000, Russian
Federation  Y. Unno  Department of Physics and Institute of Natural Sciences,
Hanyang University, Seoul 04763, South Korea  S. Uno  High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801, Japan  SOKENDAI (The Graduate
University for Advanced Studies), Hayama 240-0193, Japan  P. Urquijo  School
of Physics, University of Melbourne, Victoria 3010, Australia  Y. Usov  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russian Federation
Novosibirsk State University, Novosibirsk 630090, Russian Federation  R. Van
Tonder  University of Bonn, 53115 Bonn, Germany  G. Varner  University of
Hawaii, Honolulu, HI 96822, USA  A. Vossen  Duke University, Durham, NC 27708,
USA  E. Waheed  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801, Japan  C. H. Wang  National United University, Miao Li 36003, Taiwan
M.-Z. Wang  Department of Physics, National Taiwan University, Taipei 10617,
Taiwan  P. Wang  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, PR China  M. Watanabe  Niigata University, Niigata
950-2181, Japan  S. Watanuki  Université Paris-Saclay, CNRS/IN2P3, IJCLab,
91405 Orsay, France  O. Werbycka  H. Niewodniczanski Institute of Nuclear
Physics, Krakow 31-342, Poland  E. Won  Korea University, Seoul 02841, South
Korea  X. Xu  Soochow University, Suzhou 215006, China  W. Yan  Department of
Modern Physics and State Key Laboratory of Particle Detection and Electronics,
University of Science and Technology of China, Hefei 230026, PR China  S. B.
Yang  Korea University, Seoul 02841, South Korea  H. Ye  Deutsches
Elektronen–Synchrotron, 22607 Hamburg, Germany  J. H. Yin  Korea University,
Seoul 02841, South Korea  C. Z. Yuan  Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, PR China  Z. P. Zhang  Department
of Modern Physics and State Key Laboratory of Particle Detection and
Electronics, University of Science and Technology of China, Hefei 230026, PR
China  V. Zhilich  Budker Institute of Nuclear Physics SB RAS, Novosibirsk
630090, Russian Federation  Novosibirsk State University, Novosibirsk 630090,
Russian Federation  V. Zhukova  P.N. Lebedev Physical Institute of the Russian
Academy of Sciences, Moscow 119991, Russian Federation

###### Abstract

Using $980~{}$\mathrm{fb}$^{-1}$ of data at and around the
$\Upsilon(nS)(n=1,2,3,4,5)$ resonances collected with the Belle detector at
the KEKB asymmetric-energy $e^{+}e^{-}$ collider, the two-photon process
$\gamma\gamma\to\gamma\psi(2S)$ is studied from the threshold to $4.2~{}\rm
GeV$ for the first time. Two structures are seen in the invariant mass
distribution of $\gamma\psi(2S)$: one at $M_{R_{1}}=3922.4\pm 6.5\pm
2.0~{}{\rm MeV}/c^{2}$ with a width of $\Gamma_{R_{1}}=22\pm 17\pm 4~{}\rm
MeV$, and another at $M_{R_{2}}=4014.3\pm 4.0\pm 1.5~{}{\rm MeV}/c^{2}$ with a
width of $\Gamma_{R_{2}}=4\pm 11\pm 6~{}\rm MeV$; the signals are parametrized
with the incoherent sum of two Breit-Wigner functions. The first structure is
consistent with the $X(3915)$ or the $\chi_{c2}(3930)$, and the local
statistical significance is determined to be $3.1\sigma$ with the systematic
uncertainties included. The second matches none of the known charmonium or
charmoniumlike states, and its global significance is determined to be
$2.8\sigma$ including the look-elsewhere effect. The production rates are
$\Gamma_{\gamma\gamma}{\cal B}(R_{1}\to\gamma\psi(2S))=9.8\pm 3.6\pm 1.2~{}\rm
eV$ assuming $(J^{PC},|\lambda|)=(0^{++},0)$ or $2.0\pm 0.7\pm 0.2~{}\rm eV$
with $(2^{++},2)$ for the first structure and $\Gamma_{\gamma\gamma}{\cal
B}(R_{2}\to\gamma\psi(2S))=6.2\pm 2.2\pm 0.8~{}\rm eV$ with $(0^{++},0)$ or
$1.2\pm 0.4\pm 0.2~{}\rm eV$ with $(2^{++},2)$ for the second one. Here, the
first errors are statistical and the second systematic, and $\lambda$ is the
helicity.

###### pacs:

14.40.Gx, 13.25.Gv, 13.66.Bc

††preprint:  Belle Preprint 2021-08 KEK Preprint 2021-4

##  I Introduction

More than two dozen new resonances that are dubbed as $X$, $Y$ and/or $Z$
states have been found above the $D\overline{D}$ threshold since Belle
observed the $X(3872)$ (now labeled the $\chi_{c1}(3872)$ PDG ) in $B\to
K\pi^{+}\pi^{-}J/\psi$ x3872 , and this number is much larger than the
expectation from predictions of conventional quark-antiquark models. Among
these, candidates for both conventional and exotic charmonium-like states are
discussed widely review . Many puzzles arise from these $XYZ$ states, and one
of them concerns the candidates for $P$-wave triplet states near $3.9~{}{\rm
GeV}/c^{2}$, including the $X(3872)$, $Z(3930)\to D\overline{D}$ and
$X(3915)\to\omega J/\psi$ observed in two-photon collisions z3930_belle ;
z3930_babar ; z3930_lhcb ; x3915_belle ; x3915_babar , and $X^{*}(3860)\to
D\overline{D}$ observed in a full amplitude analysis of the process
$e^{+}e^{-}\to J/\psi D\overline{D}$ x3860_belle .

One of the most interesting $XYZ$ states is the $X(3872)$, which lies very
near the $D\bar{D}^{*}+c.c.$ mass threshold and is conjectured to have a large
$D\bar{D}^{*}+c.c.$ molecular component Swanson . Its large production rates
in $pp$ and $p\bar{p}$ collision experiments d0-x-ppbar ; lhcb-x-pp ; cms-x-pp
; atlas-x-pp  and the determination of its quantum number by LHCb lhcb-x-jpc
suggest that there is a conventional charmonium $\chi_{c1}(2P)$ core in its
wave function. This is supported by another study of
$X(3872)\to\gamma\psi(2S)$ by LHCb lhcb-x-gpsp . A study of the lineshape of
this state by LHCb reveals a pole structure that is compatible with a
quasibound state of $D^{0}\bar{D}^{*0}$ but allowing a quasivirtual state at
the level of $2\sigma$ x3872_lineshape . Partners of the $X(3872)$ are
suggested, and one of them is a $D^{*}\bar{D}^{*}$ loosely bound state with
quantum numbers $J^{PC}=2^{++}$ x2-guofk ; x2-width-guofk . Belle found
evidence for $X(3872)$ production in two-photon collisions belle-x-tp , thus
motivating the search for the possible $2^{++}$ partner of the $X(3872)$ in
such collisions. Such a study can provide essential information to understand
the nature of the $X(3872)$.

Concurrently, there have been many studies related to the $\chi_{cJ}(2P)$
triplet states. The $Z(3930)$ was discovered by Belle in the process
$\gamma\gamma\to D\overline{D}$, and the angular distribution was used to
identify it as the $\chi_{c2}(2P)$ state z3930_belle . The existence of
$Z(3930)$ and its angular distribution were confirmed by BaBar z3930_babar .
The $X(3915)$ was discovered by Belle x3915_belle  and a spin-parity analysis
of this state by BaBar favored the $J^{PC}=0^{++}$ quantum numbers x3915_babar
. The $X(3915)$ is a candidate of the $\chi_{c0}(2P)$ state liu-1 ; liu-2 ;
liu-3 ; liu-4 . In a recent amplitude analysis of the $B^{+}\to
K^{+}D^{+}D^{-}$ decay by LHCb ddb-lhcb , there are both $0^{++}$ and $2^{++}$
states at $m(D^{+}D^{-})\approx 3930~{}{\rm MeV}/c^{2}$. Their parameters are
determined to be $M=3923.8\pm 1.5\pm 0.4~{}{\rm MeV}/c^{2}$ and
$\Gamma=17.4\pm 5.1\pm 0.8~{}\rm MeV$ for $\chi_{c0}(3930)$ and $M=3926.8\pm
2.4\pm 0.8~{}{\rm MeV}/c^{2}$ and $\Gamma=34.2\pm 6.6\pm 1.1~{}\rm MeV$ for
$\chi_{c2}(3930)$. (Here and hereinafter, the first errors are statistical and
the second are systematic.) The $\chi_{c2}(3930)$ state is a good candidate of
$\chi_{c2}(2P)$, but this would imply that the hyperfine splitting of
$12~{}{\rm MeV}/c^{2}$ between $\chi_{c2}(2P)$ and $X(3915)$ would be only 6%
of that between $\chi_{c2}(1P)$ and $\chi_{c0}(1P)$ guofk . In contrast, an
early calculation theo  utilizing the Godfrey-Isgur relativistic potential
model GI_model  predicts a much larger mass difference of about $60~{}{\rm
MeV}/c^{2}$ theo . The $X^{*}(3860)$ observed by Belle is another candidate of
$\chi_{c0}(2P)$ but it was not seen in LHCb’s study of the $B^{+}\to
K^{+}D^{+}D^{-}$ decay ddb-lhcb . One interpretation of the $X^{*}(3860)$ is a
$D\overline{D}$ bound state close to the threshold with isospin $I=0$ wang_e .
Therefore, additional studies of the $P$-wave triplet states near $3.9~{}{\rm
GeV}/c^{2}$ are needed for a more comprehensive understanding of the $XYZ$
states and, in particular, of the $X(3872)$.

Both $0^{++}$ and $2^{++}$ states can be produced in two-photon collisions and
can decay to $\gamma\psi(2S)$ via an E1 transition. For example, the partial
widths are expected to be $\Gamma(\chi_{c0}(2P)\to\gamma\psi(2S))\approx
135~{}\rm keV$ and $\Gamma(\chi_{c2}(2P)\to\gamma\psi(2S))\approx 207~{}\rm
keV$ according to the aforementioned calculation theo . In this Article, we
report an investigation of the $\gamma\psi(2S)$ final state produced in two-
photon collisions ($e^{+}e^{-}\to e^{+}e^{-}\gamma\gamma\to
e^{+}e^{-}\gamma\psi(2S)$ or $\gamma\gamma\to\gamma\psi(2S)$ for brevity),
using data collected with the Belle detector Belle  at the KEKB asymmetric-
energy $e^{+}e^{-}$ collider KEKB . The $\psi(2S)$ is reconstructed from its
hadronic final state $\pi^{+}\pi^{-}J/\psi$ with $J/\psi$ reconstructed from a
lepton pair $\ell^{+}\ell^{-}~{}(\ell=e,\mu)$.

##  II Detector, data sample, and Monte Carlo (MC) simulation

The Belle detector is a large-solid-angle magnetic spectrometer that consists
of a silicon vertex detector, a 50-layer central drift chamber, an array of
aerogel threshold Cherenkov counters, a barrel-like arrangement of time-of-
flight scintillation counters, and an electromagnetic calorimeter (ECL)
comprised of CsI(Tl) crystals located inside a superconducting solenoid coil
that provides a 1.5T magnetic field. An iron flux return located outside of
the coil is instrumented to detect $K^{0}_{\rm L}$ mesons and to identify
muons. The origin of the coordinate system is defined as the position of the
nominal interaction point (IP). The $z$ axis is aligned with the direction
opposite the $e^{+}$ beam and is parallel to the direction of the magnetic
field within the solenoid. The $x$ axis is horizontal and points toward the
outside of the storage ring; the $y$ axis is vertical upward. The polar angle
$\theta$ and azimuthal angle $\phi$ are measured relative to the positive $z$
and $x$ axes, respectively.

The integrated luminosity of Belle data used in this analysis is
$980~{}$\mathrm{fb}$^{-1}$. About 70% of the data are collected at the
$\Upsilon(4S)$ resonance, and the rest are taken at other $\Upsilon(nS)$
($n=1$, 2, 3, or 5) states or center-of-mass energies a few tens of $\rm MeV$
below the $\Upsilon$ states. The Treps event generator TREPS  is used to
simulate the signals of $\gamma\gamma\to X\to\gamma\psi(2S)$ for optimization
of selection criteria, efficiency determination and calculation of the
luminosity function $L_{\gamma\gamma}$ of two-photon collisions in Belle data.
The major background is found to be the initial-state radiation (ISR) process
$e^{+}e^{-}\to\psi(2S)$, which has a cross section of $15.42\pm 0.12\pm
0.89~{}$\mathrm{pb}$$ in the Belle data sample belle_y4260 . There are
$0.6\times 10^{6}$ events with a $\pi^{+}\pi^{-}\ell^{+}\ell^{-}$ final state
in data, and an MC sample containing $3.8\times 10^{6}$ such events is
simulated with the Phokhara generator, which has a precision better than 0.5%
phokhara . An MC simulation using GEANT3 geant3  is used to model the
performance of the Belle detector.

##  III Selection criteria and signal reconstructions

Photon candidates are reconstructed from ECL clusters that do not match any
charged tracks; the candidate with the highest energy is selected to form the
$\gamma\psi(2S)$ final state. This energy is required to be larger than
$100~{}\rm MeV$ to suppress the background from fake photons. A candidate of
$\psi(2S)\to\pi^{+}\pi^{-}J/\psi$ with $J/\psi\to e^{+}e^{-}$ or
$\mu^{+}\mu^{-}$ is reconstructed from four well-measured charged tracks, each
having impact parameters with respect to the IP of $|dz|<5~{}$\mathrm{cm}$$
along the $z$ (positron-beam) axis and $dr<0.5~{}$\mathrm{cm}$$ in the
transverse $r$-$\phi$ plane. For a charged track, information from the
detector subsystems is combined to form a likelihood $\mathcal{L}_{i}$ for a
particle species of $i\in\\{e,~{}\mu,~{}\pi,~{}K~{}\hbox{or proton}\\}$ pid .
Tracks with
$\displaystyle\mathcal{R}_{K}=\mathcal{L}_{K}/(\mathcal{L}_{K}+\mathcal{L}_{\pi})<0.4$
are identified as pions with an efficiency of about 95%, while 6% of kaons
misidentified as pions. Similar likelihood ratios are formed for electron and
muon identification EID ; MUID . Both lepton candidates are required to have
$\mathcal{R}_{e}>0.1$ for the $J/\psi\to e^{+}e^{-}$ mode; at least one
candidate is required to have $\mathcal{R}_{\mu}>0.1$ for the
$J/\psi\to\mu^{+}\mu^{-}$ mode. For the first mode, any bremsstrahlung photons
detected in the ECL within 0.05 radians of the original lepton direction are
included in the calculation of the $e^{+}e^{-}$ invariant mass.

The invariant mass distributions of the lepton pair ($M_{\ell^{+}\ell^{-}}$)
from data are shown in Fig. 1(a), where clear $J/\psi$ signals are seen. By
fitting the $M_{\ell^{+}\ell^{-}}$ distributions with a Gaussian function for
the $J/\psi$ signal and a first-order polynomial function for background, we
obtain the $J/\psi$ mass resolutions of $\sigma_{\ell^{+}\ell^{-}}^{\rm
data}=11.0\pm 0.6~{}{\rm MeV}/c^{2}$ from data and
$\sigma_{\ell^{+}\ell^{-}}^{\rm MC}=9.4\pm 0.1~{}{\rm MeV}/c^{2}$ from signal
MC simulation. A lepton pair is regarded as a $J/\psi$ candidate if
$|M_{\ell^{+}\ell^{-}}-m_{J/\psi}|<4\sigma_{\ell^{+}\ell^{-}}$, where
$m_{J/\psi}$ is the nominal mass of $J/\psi$ PDG  and
$\sigma_{\ell^{+}\ell^{-}}=11.0~{}{\rm MeV}/c^{2}$. Events in the $J/\psi$
mass sidebands, defined as
$M_{\ell^{+}\ell^{-}}\in[m_{J/\psi}-13\sigma_{\ell^{+}\ell^{-}},m_{J/\psi}-5\sigma_{\ell^{+}\ell^{-}}]\cup[m_{J/\psi}+5\sigma_{\ell^{+}\ell^{-}},m_{J/\psi}+13\sigma_{\ell^{+}\ell^{-}}]$,
are used to study the background in the $\psi(2S)$ reconstruction. Figure 1(b)
shows the distributions of $M_{\pi^{+}\pi^{-}J/\psi}\equiv
M_{\pi^{+}\pi^{-}\ell^{+}\ell^{-}}-M_{\ell^{+}\ell^{-}}+m_{J/\psi}$ from data,
where $M_{\pi^{+}\pi^{-}\ell^{+}\ell^{-}}$ is the invariant mass of
$\pi^{+}\pi^{-}\ell^{+}\ell^{-}$. Fitting the $M_{\pi^{+}\pi^{-}J/\psi}$
distributions with a Gaussian function for $\psi(2S)$ signal and a second-
order polynomial function for the background, we obtain the $\psi(2S)$ mass
resolutions of $\sigma_{\pi^{+}\pi^{-}J/\psi}^{\rm data}=2.80\pm 0.21~{}{\rm
MeV}/c^{2}$ from data and $\sigma_{\pi^{+}\pi^{-}J/\psi}^{\rm MC}=2.52\pm
0.04~{}{\rm MeV}/c^{2}$ from the signal MC simulation. The $\psi(2S)$ signal
window is defined to be
$|M_{\pi^{+}\pi^{-}J/\psi}-m_{\psi(2S)}|<2.5\sigma_{\pi^{+}\pi^{-}J/\psi}$,
where $m_{\psi(2S)}$ is the nominal mass of $\psi(2S)$ PDG  and
$\sigma_{\pi^{+}\pi^{-}J/\psi}=2.80~{}{\rm MeV}/c^{2}$. To estimate the
background in the $\psi(2S)$ reconstruction, the sideband region is defined to
be $|M_{\pi^{+}\pi^{-}J/\psi}-m_{\psi(2S)}\pm
9\sigma_{\pi^{+}\pi^{-}J/\psi}|<3.75\sigma_{\pi^{+}\pi^{-}J/\psi}$.

Figure 1:  Invariant mass distributions of (a) $\ell^{+}\ell^{-}(l=e,\mu)$ for
the $J/\psi$ signal and (b) $\pi^{+}\pi^{-}J/\psi$ for the $\psi(2S)$ signal
in data. The shaded histogram is the background estimated from the $J/\psi$
mass sidebands. The curves show the best fit results.

The background is dominated by $e^{+}e^{-}\to\psi(2S)$ via ISR, where
$\psi(2S)$ is combined with a fake photon. Figure 2 shows the distributions of
the recoil mass squared $M_{\rm rec}^{2}(\gamma\psi(2S))$ of $\gamma\psi(2S)$.
For two-photon collision events, there may be an outgoing $e^{+}e^{-}$ pair
traveling back-to-back along the $e^{+}e^{-}$ beams so that $M_{\rm
rec}^{2}(\gamma\psi(2S))$, corresponding to the mass squared of the outgoing
$e^{+}e^{-}$ pair, tends to be large. For ISR events, the recoil of
$\gamma\psi(2S)$ is dominated by one energetic ISR photon with $E(\gamma_{\rm
ISR})>1.5~{}\rm GeV$, so $M_{\rm rec}^{2}(\gamma\psi(2S))$ is around zero. We
apply $M_{\rm rec}^{2}(\gamma\psi(2S))>10~{}({\rm GeV}/c^{2})^{2}$ to remove
most ISR events. Nevertheless, there still remain events with two ISR photons
traveling back-to-back along the $e^{+}e^{-}$ collision beams; such events
have a topology similar to two-photon collisions.

Figure 2: The distributions of recoil mass square of $\gamma\psi(2S)$. The
dots with error bars are data, the black blank histogram is the background
estimated from the $\psi(2S)$ mass sidebands, the blue histogram is the ISR MC
simulation and the green histogram is the signal MC simulation. The
normalization of ISR MC simulation is according the region $M_{\rm
rec}^{2}(\gamma\psi(2S))<10~{}({\rm GeV}/c^{2})^{2}$, and the one of the
signal MC simulation is arbitrary. The arrow shows the position of $M_{\rm
rec}^{2}(\gamma\psi(2S))=10~{}({\rm GeV}/c^{2})^{2}$.

To suppress the ISR background further, the transverse momenta of $\psi(2S)$
and $\gamma\psi(2S)$, i.e., $P^{*}_{t}(\psi(2S))$ and
$P^{*}_{t}(\gamma\psi(2S))$, calculated in the center-of-mass system (c.m.s.)
and shown in Fig. 3 (a) and (c), are used. $P^{*}_{t}(\gamma\psi(2S))$ is
small for most of the signal events, in which the outgoing $e^{+}e^{-}$ travel
along the accelerator beamline. However, $P^{*}_{t}(\psi(2S))$ could be large
if $\psi(2S)$ originates from the decay of a resonance such as $\chi_{c0}(2P)$
or $\chi_{c2}(2P)$. For the ISR events, $P_{t}^{*}(\psi(2S))$ is small since
the ISR photon(s) always travel along the accelerator beamline. We optimize
the selections of $P^{*}_{t}(\psi(2S))$ and $P^{*}_{t}(\gamma\psi(2S))$ based
on the Punzi figure of merit (FOM), defined as

${\rm FOM}\equiv\frac{\varepsilon(t)}{a/2+\sqrt{N_{\rm bkg}(t)}}$ (1)

according to Eq.(7) in Ref. Punzi . Here, $\varepsilon(t)$ is the signal
efficiency based on the selection criterion $t$, $a$ is the number of sigmas
corresponding to one-side Gaussian tests — we take $a=5$ — and $N_{\rm
bkg}(t)$ is the background estimated from the ISR events and the $\psi(2S)$
mass sidebands. The FOM and $\varepsilon(t)$ versus $P_{t}^{*}$ selections are
shown in Figs. 3(b) and 3(d). We apply $P^{*}_{t}(\psi(2S))>0.1~{}{\rm GeV}/c$
and $P^{*}_{t}(\gamma\psi(2S))<0.2~{}{\rm GeV}/c$ with selection efficiencies
of $\varepsilon^{\rm MC}(t)=(97.1\pm 0.3)\%$ and $\varepsilon^{\rm
MC}(t)=(67.8\pm 0.7)~{}\%$, respectively. There are about 150 ISR events
surviving these selection criteria with an efficiency of about $0.02\%$.

Figure 3:  The distributions of transverse momenta of $\psi(2S)$ (top row) and
$\gamma\psi(2S)$ (bottom row) in the c.m.s of $e^{+}e^{-}$ collision. In the
left panel, the dots with error bars are from the signal region, the shaded
histograms are the backgrounds estimated from the $\psi(2S)$ mass sidebands,
the black blank histograms are signal MC simulation with arbitrary
normalizations, and the blue blank histograms are ISR MC simulation. The
normalization of ISR MC simulation in (a) is according to data with
$P_{t}^{*}(\psi(2S))<0.1~{}{\rm GeV}/c^{2}$, and the one in (c) is according
to the size of ISR MC simulation sample. The right panel shows the Punzi FOMs
in dots and efficiencies in diamonds versus (b) $P_{t}^{*}(\psi(2S))$ and (d)
$P_{t}^{*}(\gamma\psi(2S))$. The arrows show the selections applied.

##  IV Invariant mass distribution of $\gamma\psi(2S)$ and two structures

Figure 4 shows the invariant mass distributions for data of $\gamma\psi(2S)$
($M_{\gamma\psi(2S)}$) in the $J/\psi\to e^{+}e^{-}$ and $\mu^{+}\mu^{-}$
modes. The distributions of the backgrounds estimated from the scaled
$\psi(2S)$ mass sidebands and ISR events simulated by Phokhara phokhara  are
also shown in Fig. 4. The ratio between data and ISR MC simulation is
$0.147\pm 0.012$ from the distributions in the region
$M_{\gamma\psi(2S)}<3.9~{}{\rm GeV}/c^{2}$, while the expected ratio is
$0.156\pm 0.009$ according to the cross section and the size of the ISR MC
sample. Figure 5 shows the signal selection efficiency and the two-photon
luminosity function $L_{\gamma\gamma}(\sqrt{s})$, which is defined as the
probability of a two-photon emission with $\gamma\gamma$ c.m.s. energy
$\sqrt{s}$ in the Belle experiment TREPS . The efficiencies for
$J^{PC}=0^{++}$ and $2^{++}$ ($|\lambda|=0$ or 2) range from $\sim 10\%$ to
$\sim 15\%$ for $M_{\gamma\psi(2S)}$ between $3.85~{}{\rm GeV}/c^{2}$ and
$4.20~{}{\rm GeV}/c^{2}$. The difference of $L_{\gamma\gamma}$ between
$0^{++}$ and $2^{++}$ is very small and thus ignored.

Figure 4: The invariant mass distributions of $\gamma\psi(2S)$ from (a)
$e^{+}e^{-}$ mode and (b) $\mu^{+}\mu^{-}$ mode. The dots with error bars are
data, and the shaded histograms are backgrounds estimated from the $\psi(2S)$
mass sidebands. The blank histograms are ISR events simulated and scaled to
the size of the Belle data sample.

Figure 5: (a) The efficiencies at different $M_{\gamma\psi(2S)}$ from MC
simulation, and (b) the two-photon luminosity function
$L_{\gamma\gamma}(\sqrt{s})$. In (a), the blue dots are the efficiencies for
$J^{PC}=0^{++}$, and the black (red) dots are the efficiencies for $2^{++}$
with helicity $|\lambda|=0$ ($|\lambda|=2$). The curves show the best fits
with polynomial functions.

The $M_{\gamma\psi(2S)}$ distribution of data after combining the $e^{+}e^{-}$
and $\mu^{+}\mu^{-}$ modes are shown in Fig. 6. Excesses around $3.92~{}{\rm
GeV}/c^{2}$ and $4.02~{}{\rm GeV}/c^{2}$ are seen. To study the excesses, a
binned extended maximum-likelihood fit is performed to the
$M_{\gamma\psi(2S)}$ mass spectra. The function used for the fit is
characterized by the sum:

$f_{\rm sum}=f_{\rm R_{1}}+f_{\rm R_{2}}+f_{\rm ISR}+f_{\rm bkg}+f_{\rm SB}.$
(2)

Here, $f_{\rm R_{1}}$ ($f_{\rm R_{2}}$) is for the structure $R_{1}$ ($R_{2}$)
near $3.92~{}{\rm GeV}/c^{2}$ ($4.02~{}{\rm GeV}/c^{2}$), $f_{\rm ISR}$ for
the ISR events, $f_{\rm SB}$ for the background in $\psi(2S)$ reconstruction,
and $f_{\rm bkg}$ for the possible additional backgrounds. The $f_{\rm SB}$
distribution is estimated from the $\psi(2S)$ mass sidebands, and its yield is
fixed in the fits. The function $f_{\rm R_{1}}$ ($f_{\rm R_{2}}$) contains the
convolution of a relativistic Breit-Wigner (BW) function with a form of
$12\pi\Gamma_{\gamma\gamma}\Gamma_{X}/((s-M^{2})^{2}+M^{2}\Gamma^{2})$ and a
Crystal Ball (CB) function CB_func  with a mass resolution of about
$7.2~{}{\rm MeV}/c^{2}$ ($8.0~{}{\rm MeV}/c^{2}$), and the parameters of CB
function are fixed according to the signal MC simulation of a resonance with a
mass near the one of $R_{1}$ ($R_{2}$) states and with no width. The resonant
parameters in the BW function, $M$, $\Gamma$, and
$\Gamma_{X}\Gamma_{\gamma\gamma}$ are the mass, the width, and the product of
partial widths of the decays to the final state $X$ ($=\gamma\psi(2S)$ here)
and $\gamma\gamma$, respectively. The product
$\Gamma_{X}\Gamma_{\gamma\gamma}$ is treated as one parameter, since it is
impossible to separate $\Gamma_{X}$ and $\Gamma_{\gamma\gamma}$ in the fits.
The efficiency curve $\varepsilon$ is shown in Fig. 5(a) and is incorporated
into $f_{\rm R_{1}}$ and $f_{\rm R_{2}}$, i.e., $f_{\rm
R}\propto\varepsilon\cdot(\rm BW\otimes CB)$. The widths $\Gamma_{R_{1}}$ and
$\Gamma_{R_{2}}$ are found to be small and thus the possible interference
between $R_{1}$ and $R_{2}$ is expected to be small and is ignored in the fit.
The histogram of $M_{\gamma\psi(2S)}$ distribution from the ISR MC simulation
is used for $f_{\rm ISR}$. There may be more subdominant sources of
background, such as high order QED processes and continuum production of
$\gamma\gamma\to\gamma\psi(2S)$, but their individual and collective
contributions are not clearly distinguishable with the current limited
statistics. A second-order polynomial function is used for $f_{\rm bkg}$, and
polynomial functions with different order are considered to estimate the
systematic uncertainty.

Figure 6: The $\gamma\psi(2S)$ invariant mass distribution and the fit result. The points with error bars show the data while the shaded histogram is the normalized background from the $\psi(2S)$ mass sidebands. The solid blue curve shows the best fit results. The red signal curves from the convolutions of BW and CB functions show the contributions from the two structures. The green blank histogram shows the component of ISR events of $e^{+}e^{-}\to\psi(2S)\to\pi^{+}\pi^{-}J/\psi$. The pink dashed line shows the possible additional background, modeled by a second-order polynomial. Resonant parameters | $J=0$ | $J=2$
---|---|---
$M_{R_{1}}$ | $3922.4\pm 6.5\pm 2.0$
$\Gamma_{R_{1}}$ | $22\pm 17\pm 4$
$\Gamma_{\gamma\gamma}{\cal B}(R_{1}\to\gamma\psi(2S))$ | $9.8\pm 3.6\pm 1.2$ | $2.0\pm 0.7\pm 0.2$
$M_{R_{2}}$ | $4014.3\pm 4.0\pm 1.5$
$\Gamma_{R_{2}}$ | $4\pm 11\pm 6$
$\Gamma_{\gamma\gamma}{\cal B}(R_{2}\to\gamma\psi(2S))$ | $6.2\pm 2.2\pm 0.8$ | $1.2\pm 0.4\pm 0.2$
Table 1: Summary of the resonant parameters determined. The units of mass
($M$), width ($\Gamma$), product of partial width and branching fraction
$\Gamma_{\gamma\gamma}{\cal B}$ are ${\rm MeV}/c^{2}$, $\rm MeV$ and $\rm eV$,
respectively. The first errors are statistical and the second are systematic.

The result from a fit in which all parameters are floated except the yield of
the $f_{\rm SB}$ component is shown Fig. 6 and Table 1. The reduced chi-
squared of the fit to the $M_{\gamma\psi(2S)}$ spectrum is
$\chi^{2}/ndf=0.69$. The signal yields are $N_{R_{1}}=31\pm 11$ events for
$R_{1}$ with $M_{R_{1}}=3922.4\pm 6.5~{}{\rm MeV}/c^{2}$ and
$\Gamma_{R_{1}}=22\pm 17~{}\rm MeV$, and $N_{R_{2}}=19\pm 7$ events for
$R_{2}$ with $M_{R_{2}}=4014.3\pm 4.0~{}{\rm MeV}/c^{2}$ and
$\Gamma_{R_{2}}=4\pm 11~{}\rm MeV$.

The production of $R_{1}$ and $R_{2}$ in two-photon collisions are studied by
determining the parameter ${\cal
B}\cdot\Gamma_{\gamma\gamma}\equiv\Gamma_{\gamma\psi(2S)}\Gamma_{\gamma\gamma}/\Gamma$
with the formula TREPS :

${\cal B}\cdot\Gamma_{\gamma\gamma}=\frac{n^{\rm sig}_{\rm fit}}{L_{\rm
tot}\cdot{\cal B}^{\rm prod}\cdot\varepsilon\cdot F(\sqrt{s},J)},$ (3)

where $n^{\rm sig}_{\rm fit}$ is the signal yield from the fit, $L_{\rm
tot}=980~{}$\mathrm{fb}$^{-1}$ is the integrated luminosity of the Belle data
sample, $J$ is the spin of a structure, and ${\cal B}^{\rm prod}$ is the
product of branching fractions ${\cal
B}(\psi(2S)\to\pi^{+}\pi^{-}J/\psi)\cdot{\cal B}(J/\psi\to
e^{+}e^{-}/\mu^{+}\mu^{-})$. Since $\Gamma_{R_{1}}$ and $\Gamma_{R_{2}}$ are
small compared to the available kinetic energy in the decays, the spin-
dependent factor is
$F(\sqrt{s},J)=4\pi^{2}(2J+1)L_{\gamma\gamma}(\sqrt{s})/s$. The best fit gives
$\Gamma_{\gamma\gamma}{\cal B}(R_{1}\to\gamma\psi(2S))=(8.2\pm 3.4)~{}\rm eV$
if $J=0$ and $(1.6\pm 0.7)~{}\rm eV$ if $J=2$ for structure $R_{1}$, and
$\Gamma_{\gamma\gamma}{\cal B}(R_{2}\to\gamma\psi(2S))=(6.8\pm 2.8)~{}\rm eV$
if $J=0$ and $(1.4\pm 0.6)~{}\rm eV$ if $J=2$ for structure $R_{2}$. The ISR
yield of $134\pm 15$ is consistent with the estimate from the ISR MC
simulation of $154\pm 10$. The mass of $R_{1}$ indicates that it is a good
candidate for $X(3915)$, $\chi_{c2}(3930)$ or an admixture of them. An
alternate fit with both structures included and $M_{R_{1}}$ and
$\Gamma_{R_{1}}$ fixed to the nominal $X(3915)$ parameters yields
$\Gamma_{\gamma\gamma}{\cal B}(X(3915)\to\gamma\psi(2S))=9.6\pm 2.9\pm
1.0~{}\rm eV$ if $J^{PC}=0^{++}$ of $1.9\pm 0.6~{}\rm eV$ if $J^{PC}=2^{++}$.
Another alternative fit with the mass and width of $R_{1}$ fixed to those of
$\chi_{c2}(3930)$ yields $\Gamma_{\gamma\gamma}{\cal
B}(\chi_{c2}(3930)\to\gamma\psi(2S))=2.2\pm 0.6\pm 0.4~{}\rm eV$ if
$J^{PC}=2^{++}$. A third alternate fit with $R_{1}$ being an admixture of
$X(3915)$ and $\chi_{c2}(3930)$ shows no notable change in the fit quality.

The local signal significance is determined to be $3.5\sigma$ for $R_{1}$ and
$3.4\sigma$ for $R_{2}$ by comparing the value of
$\Delta(-2\ln\mathcal{L})=-2\ln(\mathcal{L}_{\rm max}/\mathcal{L}_{0})$ and
the change of the number of free parameters ($N_{\rm par}$) in the fits, where
$\mathcal{L}_{\rm max}$ is the likelihood with both $R_{1}$ and $R_{2}$
included in Eq. 2, and $\mathcal{L}_{0}$ is the likelihood with only one of
$R_{1}$ or $R_{2}$ excluded. The values of $-2\ln\mathcal{L}$, $\chi^{2}/ndf$,
and $N_{\rm par}$ of all these fits are summarized in Table 2. Taking into
account the systematic uncertainties, described in Section V, the lowest
values of the local significance of $R_{1}$ is $3.1\sigma$. Since $R_{2}$ has
never been seen before, the look-elsewhere effect is assessed for it with
pseudo-experiments to check its global significance. The function for
generating pseudo-experiments is $f_{\rm toyMC}=f_{\rm R_{1}}+f_{\rm
ISR}+f_{\rm bkg}+f_{\rm SB}$ with the parameters from the nominal fit. The fit
in each pseudo-experiment is performed with the same procedures as for the
nominal fit to the actual data sample, except that the mass range of $R_{2}$
is limited to $M_{R_{2}}>3.95~{}{\rm GeV}/c^{2}$ because the region
$M_{\gamma\psi(2S)}<3.95~{}{\rm GeV}/c^{2}$ is dominated by $R_{1}$ and ISR
backgrounds. Among the $5.0\times 10^{4}$ pseudo-experiments, the number of
experiments with $\Delta(-2\ln\mathcal{L})$ of $R_{2}$ signal larger than the
one from data is 137. Therefore, the probability considering the look-
elsewhere effect is about $(2.74\pm 0.23)\times 10^{-3}$, corresponding to a
global significance of $2.8\sigma$.

Table 2: The values of $-2\ln\mathcal{L}$, $\chi^{2}/ndf$ and number of free parameters ($N_{\rm par}$) in the different fits. From left to right, the rows are the fits with no resonance included, only $R_{1}$ included, only $R_{2}$ included, both $R_{1}$ and $R_{2}$ included (nominal fit), both resonances included and the mass and width of $R_{1}$ fixed to those of $X(3915)$, and both resonances included and the mass and width of $R_{1}$ fixed to those of $\chi_{c2}(3930)$. Only the differences among $-2\ln\mathcal{L}$ are meaningful in studying the statistical significance of $R_{1}$ and $R_{2}$. — | no resonance |  $R_{1}$ only |  $R_{2}$ only | $R_{1}+R_{2}$ | $X(3915)+R_{2}$ | $\chi_{c2}(3930)+R_{2}$
---|---|---|---|---|---|---
$-2\ln\mathcal{L}$ | $-2932.2$ | $-2946.5$ | $-2946.3$ | $-2965.4$ | $-2964.8$ | $-2963.0$
$\chi^{2}/ndf$ | 0.76 | 0.74 | 0.78 | 0.70 | 0.68 | 0.68
$N_{\rm par}$ | 5 | 8 | 8 | 11 | 9 | 9

##  V Systematic uncertainties

There are systematic uncertainties in determining the resonant parameters of
the two structures. The masses and widths are determined from fitting to the
invariant mass distribution of $\gamma\psi(2S)$. In determining ${\cal
B}\cdot\Gamma_{\gamma\gamma}$ with Eq. (3), additional systematic
uncertainties from the selection efficiency, the luminosity of Belle data
sample and the branching fractions of $J/\psi$ and $\psi(2S)$ decays are taken
into account.

The uncertainties due to the fits are estimated by changing the fit range, the
number and the $f_{\rm SB}$ shape of the background in the $\psi(2S)$
reconstruction, the $f_{\rm bkg}$ shape, the bin width of the
$M_{\gamma\psi(2S)}$ distribution, the parametrization of the BW function, and
the resonant parameters of $X(3915)$ and $\chi_{c2}(3930)$. The fit range is
changed from $[3.70,4.20]~{}{\rm GeV}/c^{2}$ to $[3.725,~{}4.15]~{}{\rm
GeV}/c^{2}$. The number of $\psi(2S)$ mass sideband events is changed by
$1\sigma$, and the sideband region is changed from
$|M_{\pi^{+}\pi^{-}J/\psi}-m_{\psi(2S)}\pm
9\sigma_{\pi^{+}\pi^{-}J/\psi}|<3.75\sigma_{\pi^{+}\pi^{-}J/\psi}$ to
$|M_{\pi^{+}\pi^{-}J/\psi}-m_{\psi(2S)}\pm
8\sigma_{\pi^{+}\pi^{-}J/\psi}|<3.75\sigma_{\pi^{+}\pi^{-}J/\psi}$ to estimate
the uncertainty due to the $f_{\rm SB}$ component. Another ISR MC sample is
simulated to estimate the uncertainty from the shape of $f_{\rm ISR}$. The
alternative polynomial function for $f_{\rm bkg}$ is first-order or third-
order. The bin width is changed from $5~{}{\rm MeV}/c^{2}$ to $4~{}{\rm
MeV}/c^{2}$. The alternative formula of the resonant shape is
$\displaystyle{\rm BW}\propto(M^{2}/s)\cdot
12\pi\Gamma_{\gamma\gamma}\Gamma_{X}/((s-M^{2})^{2}+M^{2}\Gamma^{2})$. The
uncertainty from the resolution of $M_{\gamma\psi(2S)}$ is mainly related to
the reconstructed $\gamma$, and it is estimated with a sample of about $4,000$
$\gamma\gamma\to\chi_{c2}\to\gamma J/\psi$ events selected in the Belle data
sample. Fitting to $\chi_{c2}$ signals in the $M_{\gamma J/\psi}$
distributions from data and MC simulation results in the consistent value of
$10.84\pm 0.26~{}{\rm MeV}/c^{2}$ and $10.77\pm 0.22~{}{\rm MeV}/c^{2}$,
respectively. Thus, the uncertainty due to the mass resolution of
$\gamma\psi(2S)$ is expected to be very small and so is ignored. When
$M_{R_{1}}$ and $\Gamma_{R_{1}}$ are fixed to those of $X(3915)$ or
$\chi_{c2}(3930)$, their values are changed by $1\sigma$ to estimate the
related systematic uncertainties PDG . The largest differences between the
nominal fit results and those from these various fits are taken as the
systematic uncertainties of the mass, the width and the product
$\Gamma_{\gamma\gamma}{\cal B}(R\to\gamma\psi(2S))$.

Several sources of non-fit-related systematic uncertainties are considered.
The particle identification uncertainty is 2.8% pid ; EID ; MUID ; the
uncertainty of the tracking efficiency is 0.35% per track and is additive; the
uncertainty of the photon reconstruction is 2% per photon. The efficiency for
the tracks in the extreme forward and backward regions obtained from MC
simulation is found to be higher than that obtained in data according to the
study of $e^{+}e^{-}\to\psi(2S)\to\pi^{+}\pi^{-}J/\psi$ via ISR belle_y4260 ,
and appropriate corrections have been applied. The uncertainty in the
$\psi(2S)$ mass window requirement is measured to be 0.6%, while the one of
$J/\psi$ mass window is ignored. The efficiencies of the selection criteria on
$P^{*}_{t}(\psi(2S))$ and $P^{*}_{t}(\gamma\psi(2S))$ are strongly related to
the boost transformation from the lab system to the c.m.s. of $e^{+}e^{-}$
collisions. However, the related uncertainty is very small, and 1% is taken to
be a conservative estimation for the uncertainty due to the $P_{t}^{*}$
selections. The uncertainty due to the $M_{\rm rec}^{2}(\gamma\psi(2S))$
requirement is less than 0.5%. The uncertainty due to the momentum and angular
distributions of helicities 0 and 2 for $J=2$ is estimated to be 4.3% from the
Treps generator TREPS , while the one of $J=0$ is ignored with the decay to
$\gamma\psi(2S)$ isotropic and no uncertainty in helicity. Belle measures the
luminosity with 1.4% precision. The trigger efficiency for the events
surviving the selection criteria exceeds 99.4%, and so the uncertainty is
ignored. The uncertainties of the $J/\psi$ and $\psi(2S)$ decay branching
fractions taken from Ref. PDG  contribute a systematic uncertainty of 1.3%.
The statistical error in the MC determination of the efficiency is less than
0.7%.

The non-fit-related systematic uncertainties are listed in Table 3. Assuming
all the sources are independent, we add them in quadrature to obtain a total
systematic uncertainty of 6.2% (4.4%) of $J=2$ ($J=0$) in determining ${\cal
B}\cdot\Gamma_{\gamma\gamma}$, in addition to the uncertainties from the fits.

Source | Relative error (%)
---|---
— | $J=0$ | $J=2$
Particle identification | 2.8
Tracking efficiency | 1.4
Photon reconstruction | 2.0
$\psi(2S)$ mass window | 0.6
$P^{*}_{t}(\psi(2S))$ and $P_{t}^{*}(\gamma\psi(2S))$ | 1.0
$M_{\rm rec}^{2}(\gamma\psi(2S))$ | 0.5
Luminosity | 1.4
Generator | — | 4.3
Branching fractions | 1.3
Statistics of MC samples | 0.7
Sum in quadrature | 4.4 | 6.2
Table 3: The summary of systematic uncertainties besides the fits in
$\gamma\gamma\to\gamma\psi(2S)$ measurement.

##  VI Discussion on the two structures

We find evidence for the structure $R_{1}$ near $3.92~{}{\rm GeV}/c^{2}$,
which may be $X(3915)$, $\chi_{c2}(3930)$, or an admixture of them. Assuming
$R_{1}$ is $\chi_{c2}(3930)$ and taking into account
$\Gamma_{\gamma\gamma}{\cal B}(\chi_{c2}(3930)\to D\overline{D})=210\pm
40~{}\rm eV$, the ratio $R={\cal B}(\chi_{c2}(3930)\to\gamma\psi(2S))/{\cal
B}(\chi_{c2}(3930)\to D\overline{D})=0.010\pm 0.003$ is obtained. A rough
estimation shows the partial width
$\Gamma(\chi_{c2}(3930)\to\gamma\psi(2S))=(200\sim 300)~{}\rm keV$, which is
close to the predicted value of $207~{}\rm keV$ from Godfrey-Isgur
relativistic potential model theo .

It is interesting to see that the mass of $R_{2}$ agrees with the HQSS-
predicted mass ($\approx 4013~{}{\rm MeV}/c^{2}$) of the $2^{++}$ partner of
$X(3872)$ x2-guofk . The mass difference between $R_{2}$ and $X(3872)$ is
$142.6\pm 4.2~{}{\rm MeV}/c^{2}$, while that between $D^{*0}(2007)$ and
$D^{0}$ is $142.01~{}{\rm MeV}/c^{2}$. Meanwhile, the width of $R_{2}$ from
the fit coincides with the predicted width of $2-8~{}{\rm MeV}/c^{2}$ for the
$2^{++}$ partner of $X(3872)$ x2-width-guofk . Thus, $R_{2}$ may provide
important information for understanding the nature of the $X(3872)$. However,
the global significance of $R_{2}$ is only $2.8\sigma$. A much larger data
sample that will be collected by Belle II may resolve this in the near future.

##  VII Summary

The two-photon process $\gamma\gamma\to\gamma\psi(2S)$ is studied in the
$\gamma\gamma$ mass range from the threshold to $4.2~{}{\rm GeV}/c^{2}$ for
the first time with the full Belle data sample, and two structures are seen in
the invariant mass distribution of $\gamma\psi(2S)$. The first has a mass of
$M_{R_{1}}=3922.4\pm 6.5\pm 2.0~{}{\rm MeV}/c^{2}$ and a width of
$\Gamma_{R_{1}}=22\pm 17\pm 4~{}\rm MeV$ with a local statistical significance
of $3.1\sigma$ when the systematic uncertainties are included. This is close
to the mass of $X(3915)$ and $\chi_{c2}(3930)$. The second has a mass of
$M_{R_{2}}=4014.3\pm 4.0\pm 1.5~{}{\rm MeV}/c^{2}$ and a width of
$\Gamma_{R_{2}}=4\pm 11\pm 6~{}\rm MeV$, with a global statistical
significance of $2.8\sigma$. The values of $\Gamma_{\gamma\gamma}{\cal
B}(R\to\gamma\psi(2S))$ are of the order of several $\rm eV$.

###### Acknowledgements.

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 5 (SINET5) for valuable network support. We acknowledge
support from 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 DP180102629, DP170102389,
DP170102204, DP150103061, FT130100303; 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. 11435013, No.
11475187, No. 11521505, No. 11575017, No. 11675166, No. 11705209; No.
12175041; Key Research Program of Frontier Sciences, Chinese Academy of
Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; the CAS Center for Excellence in
Particle Physics (CCEPP); the Shanghai Science and Technology Committee
(STCSM) under Grant No. 19ZR1403000; the Ministry of Education, Youth and
Sports of the Czech Republic under Contract No. LTT17020; 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. 2016R1D1A1B01010135, 2016R1D1A1B02012900,
2018R1A2B3003643, 2018R1A6A1A06024970, 2018R1D1A1B07047294,
2019K1A3A7A09033840, 2019R1I1A3A01058933; 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.

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