[]S.-H. Park, \dagger\dagger$\dagger$now at Hiroshima University
\instBilbao\instBilbaoinstitutetext: University of the Basque Country UPV/EHU,
48080 Bilbao, Spain \instBonn\instBonninstitutetext: University of Bonn, 53115
Bonn, Germany \instBNL\instBNLinstitutetext: Brookhaven National Laboratory,
Upton, New York 11973, USA \instBINP\instBINPinstitutetext: Budker Institute
of Nuclear Physics SB RAS, Novosibirsk 630090, Russian Federation
\instCharles\instCharlesinstitutetext: Faculty of Mathematics and Physics,
Charles University, 121 16 Prague, The Czech Republic
\instChonnam\instChonnaminstitutetext: Chonnam National University, Gwangju
61186, South Korea \instCincinnati\instCincinnatiinstitutetext: University of
Cincinnati, Cincinnati, OH 45221, USA \instDESY\instDESYinstitutetext:
Deutsches Elektronen–Synchrotron, 22607 Hamburg, Germany
\instFlorida\instFloridainstitutetext: University of Florida, Gainesville, FL
32611, USA \instFuJen\instFuJeninstitutetext: Department of Physics, Fu Jen
Catholic University, Taipei 24205, Taiwan \instFudan\instFudaninstitutetext:
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute
of Modern Physics, Fudan University, Shanghai 200443, PR China
\instGifu\instGifuinstitutetext: Gifu University, Gifu 501-1193, Japan
\instSokendai\instSokendaiinstitutetext: SOKENDAI (The Graduate University for
Advanced Studies), Hayama 240-0193, Japan
\instGyeongsang\instGyeongsanginstitutetext: Gyeongsang National University,
Jinju 52828, South Korea \instHanyang\instHanyanginstitutetext: Department of
Physics and Institute of Natural Sciences, Hanyang University, Seoul 04763,
South Korea \instHawaii\instHawaiiinstitutetext: University of Hawaii,
Honolulu, HI 96822, USA \instKEK\instKEKinstitutetext: High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801, Japan
\instJPARC\instJPARCinstitutetext: J-PARC Branch, KEK Theory Center, High
Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
\instHSE\instHSEinstitutetext: Higher School of Economics (HSE), Moscow
101000, Russian Federation \instJuelich\instJuelichinstitutetext:
Forschungszentrum Jülich, 52425 Jülich, Germany
\instIKER\instIKERinstitutetext: IKERBASQUE, Basque Foundation for Science,
48013 Bilbao, Spain \instIISERM\instIISERMinstitutetext: Indian Institute of
Science Education and Research Mohali, SAS Nagar, 140306, India
\instIITG\instIITGinstitutetext: Indian Institute of Technology Guwahati,
Assam 781039, India \instIITH\instIITHinstitutetext: Indian Institute of
Technology Hyderabad, Telangana 502285, India \instIITM\instIITMinstitutetext:
Indian Institute of Technology Madras, Chennai 600036, India
\instIndiana\instIndianainstitutetext: Indiana University, Bloomington, IN
47408, USA \instIHEP\instIHEPinstitutetext: Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, PR China
\instProtvino\instProtvinoinstitutetext: Institute for High Energy Physics,
Protvino 142281, Russian Federation \instVienna\instViennainstitutetext:
Institute of High Energy Physics, Vienna 1050, Austria
\instNapoli\instNapoliinstitutetext: INFN - Sezione di Napoli, 80126 Napoli,
Italy \instTorino\instTorinoinstitutetext: INFN - Sezione di Torino, 10125
Torino, Italy \instJAEA\instJAEAinstitutetext: Advanced Science Research
Center, Japan Atomic Energy Agency, Naka 319-1195, Japan
\instJSI\instJSIinstitutetext: J. Stefan Institute, 1000 Ljubljana, Slovenia
\instKarlsruhe\instKarlsruheinstitutetext: Institut für Experimentelle
Teilchenphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe, Germany
\instIPMU\instIPMUinstitutetext: Kavli Institute for the Physics and
Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583,
Japan \instKAU\instKAUinstitutetext: Department of Physics, Faculty of
Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
\instKitasato\instKitasatoinstitutetext: Kitasato University, Sagamihara
252-0373, Japan \instKISTI\instKISTIinstitutetext: Korea Institute of Science
and Technology Information, Daejeon 34141, South Korea
\instKorea\instKoreainstitutetext: Korea University, Seoul 02841, South Korea
\instKyotoSangyo\instKyotoSangyoinstitutetext: Kyoto Sangyo University, Kyoto
603-8555, Japan \instKyungpook\instKyungpookinstitutetext: Kyungpook National
University, Daegu 41566, South Korea \instLAL\instLALinstitutetext: Université
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
\instLebedev\instLebedevinstitutetext: P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Moscow 119991, Russian Federation
\instLNNU\instLNNUinstitutetext: Liaoning Normal University, Dalian 116029,
China \instLjubljana\instLjubljanainstitutetext: Faculty of Mathematics and
Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
\instLMU\instLMUinstitutetext: Ludwig Maximilians University, 80539 Munich,
Germany \instLuther\instLutherinstitutetext: Luther College, Decorah, IA
52101, USA \instMNIT\instMNITinstitutetext: Malaviya National Institute of
Technology Jaipur, Jaipur 302017, India \instMaribor\instMariborinstitutetext:
University of Maribor, 2000 Maribor, Slovenia \instMPI\instMPIinstitutetext:
Max-Planck-Institut für Physik, 80805 München, Germany
\instMelbourne\instMelbourneinstitutetext: School of Physics, University of
Melbourne, Victoria 3010, Australia
\instMississippi\instMississippiinstitutetext: University of Mississippi,
University, MS 38677, USA \instMiyazaki\instMiyazakiinstitutetext: University
of Miyazaki, Miyazaki 889-2192, Japan \instMEPhI\instMEPhIinstitutetext:
Moscow Physical Engineering Institute, Moscow 115409, Russian Federation
\instNagoya\instNagoyainstitutetext: Graduate School of Science, Nagoya
University, Nagoya 464-8602, Japan \instNagoyaKMI\instNagoyaKMIinstitutetext:
Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan
\instUNapoli\instUNapoliinstitutetext: Università di Napoli Federico II, 80126
Napoli, Italy \instNara\instNarainstitutetext: Nara Women’s University, Nara
630-8506, Japan \instNCU\instNCUinstitutetext: National Central University,
Chung-li 32054, Taiwan \instNUU\instNUUinstitutetext: National United
University, Miao Li 36003, Taiwan \instTaiwan\instTaiwaninstitutetext:
Department of Physics, National Taiwan University, Taipei 10617, Taiwan
\instKrakow\instKrakowinstitutetext: H. Niewodniczanski Institute of Nuclear
Physics, Krakow 31-342, Poland \instNihonDental\instNihonDentalinstitutetext:
Nippon Dental University, Niigata 951-8580, Japan
\instNiigata\instNiigatainstitutetext: Niigata University, Niigata 950-2181,
Japan \instNovaGorica\instNovaGoricainstitutetext: University of Nova Gorica,
5000 Nova Gorica, Slovenia \instNovosibirsk\instNovosibirskinstitutetext:
Novosibirsk State University, Novosibirsk 630090, Russian Federation
\instOsakaCity\instOsakaCityinstitutetext: Osaka City University, Osaka
558-8585, Japan \instPNNL\instPNNLinstitutetext: Pacific Northwest National
Laboratory, Richland, WA 99352, USA \instPanjab\instPanjabinstitutetext:
Panjab University, Chandigarh 160014, India
\instPeking\instPekinginstitutetext: Peking University, Beijing 100871, PR
China \instPittsburgh\instPittsburghinstitutetext: University of Pittsburgh,
Pittsburgh, PA 15260, USA \instNPC\instNPCinstitutetext: Research Center for
Nuclear Physics, Osaka University, Osaka 567-0047, Japan
\instRIKENMSL\instRIKENMSLinstitutetext: Meson Science Laboratory, Cluster for
Pioneering Research, RIKEN, Saitama 351-0198, Japan
\instUSTC\instUSTCinstitutetext: Department of Modern Physics and State Key
Laboratory of Particle Detection and Electronics, University of Science and
Technology of China, Hefei 230026, PR China \instSeoul\instSeoulinstitutetext:
Seoul National University, Seoul 08826, South Korea
\instShoyaku\instShoyakuinstitutetext: Showa Pharmaceutical University, Tokyo
194-8543, Japan \instSoochow\instSoochowinstitutetext: Soochow University,
Suzhou 215006, China \instSoongsil\instSoongsilinstitutetext: Soongsil
University, Seoul 06978, South Korea
\instSungkyunkwan\instSungkyunkwaninstitutetext: Sungkyunkwan University,
Suwon 16419, South Korea \instSydney\instSydneyinstitutetext: School of
Physics, University of Sydney, New South Wales 2006, Australia
\instTabuk\instTabukinstitutetext: Department of Physics, Faculty of Science,
University of Tabuk, Tabuk 71451, Saudi Arabia
\instTata\instTatainstitutetext: Tata Institute of Fundamental Research,
Mumbai 400005, India \instTUM\instTUMinstitutetext: Department of Physics,
Technische Universität München, 85748 Garching, Germany
\instTelAviv\instTelAvivinstitutetext: School of Physics and Astronomy, Tel
Aviv University, Tel Aviv 69978, Israel \instToho\instTohoinstitutetext: Toho
University, Funabashi 274-8510, Japan \instTohoku\instTohokuinstitutetext:
Department of Physics, Tohoku University, Sendai 980-8578, Japan
\instERI\instERIinstitutetext: Earthquake Research Institute, University of
Tokyo, Tokyo 113-0032, Japan \instTokyo\instTokyoinstitutetext: Department of
Physics, University of Tokyo, Tokyo 113-0033, Japan
\instTIT\instTITinstitutetext: Tokyo Institute of Technology, Tokyo 152-8550,
Japan \instTMU\instTMUinstitutetext: Tokyo Metropolitan University, Tokyo
192-0397, Japan \instUtkal\instUtkalinstitutetext: Utkal University,
Bhubaneswar 751004, India \instVPI\instVPIinstitutetext: Virginia Polytechnic
Institute and State University, Blacksburg, VA 24061, USA
\instWayneState\instWayneStateinstitutetext: Wayne State University, Detroit,
MI 48202, USA \instYamagata\instYamagatainstitutetext: Yamagata University,
Yamagata 990-8560, Japan \instYonsei\instYonseiinstitutetext: Yonsei
University, Seoul 03722, South Korea

# Search for the dark photon in $B^{0}\to A^{\prime}A^{\prime}$,
$A^{\prime}\to e^{+}e^{-}$, $\mu^{+}\mu^{-}$, and $\pi^{+}\pi^{-}$ decays at
Belle

Y.-J. Kwon , I. Adachi,  H. Aihara, , S. Al Said  D. M. Asner,  H. Atmacan,
T. Aushev,  R. Ayad  V. Babu,  P. Behera,  J. Bennett,  M. Bessner,  V.
Bhardwaj,  B. Bhuyan  T. Bilka  J. Biswal  G. Bonvicini,  A. Bozek , M. Bračko
T. E. Browder, , M. Campajola  L. Cao  D. Červenkov,  M.-C. Chang  P. Chang
A. Chen,  B. G. Cheon,  K. Chilikin  H. E. Cho  K. Cho  S.-J. Cho  S.-K. Choi,
Y. Choi  S. Choudhury  D. Cinabro  S. Cunliffe,  S. Das,  N. Dash , G. De
Nardo , F. Di Capua,  J. Dingfelder  Z. Doležal  T. V. Dong ,, S. Eidelman, ,
D. Epifanov  D. Ferlewicz,  B. G. Fulsom,  R. Garg  V. Gaur , A. Garmash,  A.
Giri,  P. Goldenzweig  Y. Guan, , K. Gudkova  C. Hadjivasiliou, , T. Hara,  O.
Hartbrich  K. Hayasaka,  H. Hayashii  M. T. Hedges,  M. Hernandez Villanueva,
W.-S. Hou,  C.-L. Hsu,  K. Huang , T. Iijima,  K. Inami,  G. Inguglia, , A.
Ishikawa , R. Itoh  M. Iwasaki  Y. Iwasaki  W. W. Jacobs,  I. Jaegle  E.-J.
Jang  H. B. Jeon  S. Jia  Y. Jin  C. W. Joo,  K. K. Joo  K. H. Kang,  G.
Karyan,  T. Kawasaki,  C. Kiesling,  D. Y. Kim,  K.-H. Kim,  S. H. Kim,  Y.-K.
Kim,  K. Kinoshita,  P. Kodyš,  T. Konno, , S. Korpar  D. Kotchetkov , P.
Križan  R. Kroeger , P. Krokovny,  T. Kuhr,  M. Kumar,  K. Kumara,  K.
Lalwani,  I. S. Lee,  S. C. Lee,  C. H. Li,  J. Li,  L. K. Li  Y. B. Li,  L.
Li Gioi  J. Libby  K. Lieret ,$\dagger$ Z. Liptak , D. Liventsev,  T. Luo,  J.
MacNaughton  C. MacQueen, , M. Masuda  T. Matsuda ,, D. Matvienko , M. Merola,
F. Metzner  K. Miyabayashi , R. Mizuk,  G. B. Mohanty , S. Mohanty  T. Mori,
H.-G. Moser,  M. Mrvar  R. Mussa, , M. Nakao,  Z. Natkaniec,  A. Natochii  L.
Nayak  M. Nayak  M. Niiyama,  N. K. Nisar , S. Nishida  S. Ogawa, , H. Ono,
Y. Onuki  P. Oskin , P. Pakhlov, , G. Pakhlova  S. Pardi,  C. W. Park,  H.
Park  S. Patra, , S. Paul  T. K. Pedlar  R. Pestotnik,  L. E. Piilonen, , T.
Podobnik  V. Popov  E. Prencipe,  M. T. Prim  M. Ritter,  M. Röhrken  A.
Rostomyan,  N. Rout  G. Russo, , Y. Sakai,  S. Sandilya,  A. Sangal , L.
Santelj,  T. Sanuki  V. Savinov, , G. Schnell,  J. Schueler,  C. Schwanda  Y.
Seino,  K. Senyo,  M. E. Sevior,  M. Shapkin,  C. Sharma  C. P. Shen,  J.-G.
Shiu,  A. Sokolov,  E. Solovieva,  S. Stanič  M. Starič  Z. S. Stottler,  J.
F. Strube  M. Sumihama, , K. Sumisawa  T. Sumiyoshi,  W. Sutcliffe ,, M.
Takizawa,  K. Tanida,  F. Tenchini,  M. Uchida, , T. Uglov,  Y. Unno , S. Uno,
P. Urquijo  S. E. Vahsen,  R. Van Tonder,  G. Varner,  K. E. Varvell , A.
Vinokurova, ,, V. Vorobyev  E. Waheed,  C. H. Wang,  E. Wang  M.-Z. Wang  P.
Wang  M. Watanabe,  S. Watanuki,  S. Wehle,  E. Won,  X. Xu  B. D. Yabsley  W.
Yan,  S. B. Yang  H. Ye,  J. Yelton  J. H. Yin  Y. Yusa,  Z. P. Zhang, , V.
Zhilich,  V. Zhukova , V. Zhulanov

###### Abstract

We present a search for the dark photon $A^{\prime}$ in the $B^{0}\to
A^{\prime}A^{\prime}$ decays, where $A^{\prime}$ subsequently decays to
$e^{+}e^{-}$, $\mu^{+}\mu^{-}$, and $\pi^{+}\pi^{-}$. The search is performed
by analyzing $772\times 10^{6}$ $B\overline{B}$ events collected by the Belle
detector at the KEKB $e^{+}e^{-}$ energy-asymmetric collider at the
$\Upsilon(4S)$ resonance. No signal is found in the dark photon mass range
$0.01~{}\textrm{GeV}/c^{2}\leq m_{A^{\prime}}\leq 2.62~{}\textrm{GeV}/c^{2}$,
and we set upper limits of the branching fraction of $B^{0}\to
A^{\prime}A^{\prime}$ at the 90% confidence level. The products of branching
fractions, $\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})\times\mathcal{B}(A^{\prime}\to e^{+}e^{-})^{2}$ and
$\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})\times\mathcal{B}(A^{\prime}\to\mu^{+}\mu^{-})^{2}$, have
limits of the order of $10^{-8}$ depending on the $A^{\prime}$ mass.
Furthermore, considering $A^{\prime}$ decay rate to each pair of charged
particles, the upper limits of $\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$
are of the order of $10^{-8}$–$10^{-5}$. From the upper limits of
$\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$, we obtain the Higgs portal
coupling for each assumed dark photon and dark Higgs mass. The Higgs portal
couplings are of the order of $10^{-2}$–$10^{-1}$ at $m_{h^{\prime}}\simeq
m_{B^{0}}\pm 40~{}\textrm{MeV}/c^{2}$ and $10^{-1}$–$1$ at
$m_{h^{\prime}}\simeq m_{B^{0}}\pm 3~{}\textrm{GeV}/c^{2}$.

###### Keywords:

e+ e- Experiments, B physics, Beyond Standard Model, Branching fraction

††preprint:  Belle Preprint 2020-19 KEK Preprint 2020-36

##  1 Introduction

The validity of the Standard Model (SM) has been confirmed by various
experimental measurements PDG , but it is also known that the SM is incomplete
and cannot explain several phenomena occurring in nature, e.g. neutrino
oscillations neutrino ; neutrino2  and the baryon asymmetry baryo . A possible
way to explain the above problems while keeping the internal structure of the
SM unaffected is to introduce a dark sector PhysRevD.83.054005  that interacts
with the SM particles only very weakly. For example, a vector mediator of
hypothetical $U^{\prime}(1)$ gauge interaction in the dark sector, the so-
called dark photon, may interact with matter through various portals with a
small coupling strength HOLDOM1986196 ; FAYET1980285 ; FAYET1981184 . Such a
model of the dark sector with portal interaction to the SM could explain the
muon $g-2$ anomaly mug2 ; PhysRevD.80.095002 ; PhysRevD.83.101702 ;
PhysRevLett.107.011803 , baryogenesis baryodark , and high energy positron
fraction anomaly in cosmic rays ATIC ; PAMELA ; FermiLAT ; AMS ; CHEN2009255 .

Figure 1: A possible diagram of $B^{0}\to A^{\prime}A^{\prime}$ decay through
off-shell Higgs–dark Higgs mixing indicated by the shaded circle.

In this paper, we report a search for the dark photon $A^{\prime}$, in the
decays of $B^{0}$ mesons by analyzing the $e^{+}e^{-}$ collision data from the
Belle experiment. In particular, we study $B^{0}$ decays into a pair of dark
photons, $B^{0}\to A^{\prime}A^{\prime}$, which are mediated by an off-shell
dark Higgs $h^{\prime}$ PhysRevD.83.054005  (Fig. 1), wherein we scan the
$A^{\prime}$ mass range between $0.01~{}\textrm{GeV}/c^{2}$ and
$2.62~{}\textrm{GeV}/c^{2}$ in $10~{}\textrm{MeV}/c^{2}$
($m_{A^{\prime}}<1.1~{}\textrm{GeV}/c^{2}$) and $20~{}\textrm{MeV}/c^{2}$
($m_{A^{\prime}}>1.1~{}\textrm{GeV}/c^{2}$) intervals. Throughout the paper,
the charge-conjugate modes are always implied. In this paper, we restrict
ourselves to the hypothesis that all dark-sector particles coupling to
$A^{\prime}$ are heavier than $A^{\prime}$, therefore the latter can only
decay to SM particles. Moreover, we assume that the $A^{\prime}$ decays
promptly. In the kinematic range of this analysis, the allowed $A^{\prime}$
decay are to $e^{+}e^{-}$, $\mu^{+}\mu^{-}$, or hadronic final states. Lepton-
flavor-violating decays osti_1254817 ; PhysRevD.97.075022  $A^{\prime}\to
e^{\pm}\mu^{\mp}$ are not considered in this analysis.

###  1.1 Branching fraction of dark photon decay

In order to obtain $\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$ from the
analysis of the decays into the final states considered, we need to know the
branching fractions of $A^{\prime}$ to a particular final state. Below the
$\tau^{+}\tau^{-}$ threshold, the branching fraction of the dark photon that
is consistent with our hypothesis is obtained as

$\displaystyle\mathcal{B}(A^{\prime}\to\ell^{+}\ell^{-}/\pi^{+}\pi^{-})=\frac{\Gamma_{A^{\prime}\to\ell^{+}\ell^{-}/\pi^{+}\pi^{-}}}{\Gamma_{A^{\prime}\to
e^{+}e^{-}}+\Gamma_{A^{\prime}\to\mu^{+}\mu^{-}}+\Gamma_{A^{\prime}\to\textrm{hadrons}}},$
(1)

where $\ell=e~{}\textrm{or}~{}\mu$. Following Ref. PhysRevD.79.115008 , we
write down the partial widths to $\ell^{+}\ell^{-}$ and hadrons as

$\displaystyle\Gamma_{A^{\prime}\to\ell^{+}\ell^{-}}=$
$\displaystyle\frac{1}{3}\alpha\varepsilon_{\textrm{mix}}^{2}m_{A^{\prime}}\sqrt{1-{4m_{\ell}^{2}}/{m_{A^{\prime}}^{2}}}(1+2m_{\ell}^{2}/m_{A^{\prime}}^{2}),$
(2) $\displaystyle\Gamma_{A^{\prime}\to\textrm{hadrons}}=$
$\displaystyle\Gamma_{A^{\prime}\to\mu^{+}\mu^{-}}\times
R(s=m_{A^{\prime}}^{2}),$

with the square of the total center-of-mass (CM) frame energy $s$, the kinetic
mixing parameter $\varepsilon_{\textrm{mix}}$, and
$R(s)=\sum_{e^{+}e^{-}\to\textrm{hadrons}}/\sum_{e^{+}e^{-}\to\mu^{+}\mu^{-}}$
which is determined by various experiments PDG . The branching fraction of
$A^{\prime}\to\pi^{+}\pi^{-}$ is then obtained as PhysRevLett.108.211801 :

$\displaystyle\mathcal{B}(A^{\prime}\to\pi^{+}\pi^{-})=\mathcal{B}(A^{\prime}\to\textrm{hadrons})\times\sum(e^{+}e^{-}\to\pi^{+}\pi^{-})/\sum(e^{+}e^{-}\to\textrm{hadrons}).$
(3)

###  1.2 The SM expectation of $B^{0}$ decays to four charged leptons

The $B^{0}$-decay final states that we analyze are $e^{+}e^{-}e^{+}e^{-}$,
$e^{+}e^{-}\mu^{+}\mu^{-}$, $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$,
$e^{+}e^{-}\pi^{+}\pi^{-}$, and $\mu^{+}\mu^{-}\pi^{+}\pi^{-}$. In the SM,
branching fractions of $B^{0}$-meson decays to four-charged-lepton final
states are expected to be $\mathcal{O}(10^{-12})$ Danilina2018 . Due to the
low SM signal and background yields expected, these multileptonic $B$-meson
decay channels can be a sensitive probe for dark sector bosons. The LHCb
experiment has set an upper limit
$\mathcal{B}(B^{0}\to\mu^{+}\mu^{-}\mu^{+}\mu^{-})<6.9\times 10^{-10}$ at
$95\%$ confidence level (C.L.) Aaij2017  and measured
$\mathcal{B}(B^{0}\to\mu^{+}\mu^{-}\pi^{+}\pi^{-})=(2.1\pm 0.5)\times 10^{-8}$
Aaij:2014lba .

##  2 The Belle detector

Our analysis is based on the full $711~{}\textrm{fb}^{-1}$ integrated
luminosity of the $\Upsilon(4S)$ data set from the Belle detector Belle1 ;
Belle2  at KEKB $e^{+}e^{-}$ energy-asymmetric collider KEKB1 ; KEKB2 . The
Belle detector consists of seven subdetectors with $1.5~{}\textrm{T}$ magnetic
field along the beam axis. Inside the coil, there are the silicon vertex
detector, the central drift chamber (CDC), the aerogel threshold Cherenkov
counters (ACC), the time-of-flight scintillation counters (TOF), and the
electromagnetic calorimeter (ECL). In the return yoke outside the coil, a
$K_{L}^{0}$ meson and muon detector (KLM) is instrumented. We perform a blind
search in this analysis, for which we generate Monte Carlo (MC) simulation
samples using EvtGen evtgen  for event generation and GEANT3 geant3  for
detector simulation. Signal efficiencies are determined from the signal MC
set, where one million events are generated for each signal mode and dark
photon mass. The event shape and amount of the background events are studied
by using generic MC samples simulating $e^{+}e^{-}\to\Upsilon(4S)\to
B\overline{B}$ and $e^{+}e^{-}\to q\bar{q}~{}(q=u,d,s,c)$ (‘continuum’)
processes. The size of MC samples for $\Upsilon(4S)$ and continuum simulation
corresponds to 10 and 6 times that of real data, respectively.

##  3 Signal event selection

To select signal events, we retain tracks satisfying the following track
reconstruction quality requirements. Because we assume prompt dark photon
decays, all tracks are required to originate from near the interaction point
(IP). In particular, each track should satisfy the following conditions on the
impact parameters in the transverse and longitudinal directions,
$dr<0.2~{}\textrm{cm}$ and $|dz|<3.0~{}\textrm{cm}$, respectively. The impact
parameters are calculated using the beam IP and track helix, and the $z$-axis
is aligned opposite the direction of positron beam. We also require a good
track fit based upon $\chi^{2}$ per degree of freedom ($N_{\textrm{d.o.f.}}$)
by accepting only the tracks with $\chi^{2}/N_{\textrm{d.o.f.}}<5$. The
species of the charged particles are identified by considering the likelihood
ratios. Muons are identified by requiring
${\mathcal{L}_{\mu}}/({\mathcal{L}_{\mu}+\mathcal{L}_{K}+\mathcal{L}_{\pi}})>0.9$,
where the likelihood $\mathcal{L}_{j}\ (j=\mu,K,\pi)$ Abashian:2002bd  is
constructed by the hit position and penetration in the KLM. Electrons are
required to meet
${\mathcal{L}_{e}}/({\mathcal{L}_{e}+\mathcal{L}_{\textrm{not-}e}})>0.9$ where
the likelihood $\mathcal{L}_{j}\ (j=e,\textrm{not-}e)$ Hanagaki:2001fz  is
determined by $dE/dx$ from the CDC, ratio of the ECL cluster energy to the
matched track momentum, shower shape of the ECL cluster, and the ACC
photoelectron response. Charged pions and kaons are identified by the
likelihood Nakano:2002jw  using the $dE/dx$ from the CDC, the ACC
photoelectron response, and the time-of-flight information from the TOF. The
tracks with ${\mathcal{L}_{\pi}}/({\mathcal{L}_{K}+\mathcal{L}_{\pi}})>0.4$
are identified as pions. To recover energy losses by $e^{\pm}$ candidates due
to bremsstrahlung, radiative photons are added to the electron momentum if
they fall within a 0.05 radian cone around the $e^{\pm}$ direction. We require
these photons to exceed an energy threshold that depends on the ECL region:
$E_{\gamma}>50$ (barrel), $100$ (forward endcap), and $150$ (backward endcap)
MeV. The dark photon candidate is reconstructed in the following modes:
$A^{\prime}\to e^{+}e^{-}$, $\mu^{+}\mu^{-}$, and $\pi^{+}\pi^{-}$. For
$B^{0}\to e^{+}e^{-}e^{+}e^{-}$ and $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ modes, we
have an ambiguity between
$(\ell_{1}^{+}\ell_{1}^{-})(\ell_{2}^{+}\ell_{2}^{-})$ and
$(\ell_{1}^{+}\ell_{2}^{-})(\ell_{2}^{+}\ell_{1}^{-})$, where the lepton pair
from a single $A^{\prime}$ decay is indicated in parentheses. To find a single
dark photon combination per event, we choose that corresponding to the
smallest invariant mass difference of dark photon candidates, $\Delta
M_{A^{\prime}}$. Finally, $B^{0}$ candidates are reconstructed from two dark
photon candidates. To extract signal events from data, we use the following
five variables, defined in the CM frame: $M_{\textrm{bc}}$, $\Delta E$,
$E_{\textrm{miss}}$, $\Delta M_{A^{\prime}}$, and $\sum\delta M_{A^{\prime}}$.
$M_{\textrm{bc}}\equiv\sqrt{(\sqrt{s}/2)^{2}-\vec{\textrm{p}}_{B}^{2}}$ is the
beam-energy-constrained mass, where $\vec{\textrm{p}}_{B}$ is the momentum of
the reconstructed $B^{0}$. $\Delta E\equiv E_{B^{0}}-(\sqrt{s}/2)$ is the
difference between the $B^{0}$-candidate energy and the beam energy
($=\sqrt{s}/2$), and $E_{\textrm{miss}}$ is the missing energy,
$E_{\textrm{miss}}\equiv\sqrt{s}-\sum_{j}E_{j}$ where the index $j$ is for all
charged and neutral particles in the event. The missing energy is useful to
reduce combinatorial background due to multiple semileptonic decays from $b\to
c\ell^{-}\bar{\nu}_{\ell}$ and $c\to(s,d)\ell^{+}\nu_{\ell}$ for both $B$ and
$\overline{B}$. For the two dark photon candidates in an event, we calculate
$\Delta M_{A^{\prime}}\equiv|M_{A^{\prime}_{1}}-M_{A^{\prime}_{2}}|$ and
$\sum\delta
M_{A^{\prime}}\equiv|(M_{A^{\prime}_{1}}-m_{A^{\prime}}^{\textrm{bin}})+(M_{A^{\prime}_{2}}-m_{A^{\prime}}^{\textrm{bin}})|$,
where $M_{A^{\prime}_{j}}$ is the reconstructed mass of $A^{\prime}_{j}\
(j=1,2)$ and $m_{A^{\prime}}^{\textrm{bin}}$ is the nominal $A^{\prime}$ mass
for a particular bin of $m_{A^{\prime}}$. For the signal event selection, we
require $M_{\textrm{bc}}>5.27~{}\textrm{GeV}/c^{2}$ and
$E_{\textrm{miss}}<3.5~{}\textrm{GeV}$ for all modes. Considering the energy
loss from $e^{\pm}$, $\Delta E$ requirements are chosen separately for
different modes: $-0.2~{}\textrm{GeV}<\Delta E<0.05~{}\textrm{GeV}$ for
$B^{0}\to e^{+}e^{-}e^{+}e^{-}$, $-0.1~{}\textrm{GeV}<\Delta
E<0.04~{}\textrm{GeV}$ for $B^{0}\to e^{+}e^{-}\mu^{+}\mu^{-}$ and
$e^{+}e^{-}\pi^{+}\pi^{-}$, and $-0.03~{}\textrm{GeV}<\Delta
E<0.03~{}\textrm{GeV}$ for $B^{0}\to\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ and
$\mu^{+}\mu^{-}\pi^{+}\pi^{-}$. We use $\Delta M_{A^{\prime}}$ and $\sum\delta
M_{A^{\prime}}$ to set the search window for each
$m_{A^{\prime}}^{\textrm{bin}}$ and the final-state mode. The requirements on
these variables depend on both $m_{A^{\prime}}^{\textrm{bin}}$ and the number
of electrons in the final state. For
$m_{A^{\prime}}^{\textrm{bin}}>0.1~{}\textrm{GeV}/c^{2}$, we require $\Delta
M_{A^{\prime}}(\sum\delta M_{A^{\prime}})<0.06\times
m_{A^{\prime}}^{\textrm{bin}}+0.03~{}\textrm{GeV}/c^{2}$ for $B^{0}\to
e^{+}e^{-}e^{+}e^{-}$, $\Delta M_{A^{\prime}}(\sum\delta
M_{A^{\prime}})<0.03\times
m_{A^{\prime}}^{\textrm{bin}}+0.01~{}\textrm{GeV}/c^{2}$ for $B^{0}\to
e^{+}e^{-}\mu^{+}\mu^{-}$ and $e^{+}e^{-}\pi^{+}\pi^{-}$, and $\Delta
M_{A^{\prime}}(\sum\delta M_{A^{\prime}})<0.01\times
m_{A^{\prime}}^{\textrm{bin}}+0.01~{}\textrm{GeV}/c^{2}$ for
$B^{0}\to\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ and $\mu^{+}\mu^{-}\pi^{+}\pi^{-}$. The
above conditions are determined so that if we consider the distribution of
$\Delta M_{A^{\prime}}$ the upper edge of the accepted region has a value of
nearly $3$–$5\%$ of the peak value. In addition, we make use of the
approximately linear increase of the $\Delta M_{A^{\prime}}$ width as a
function of $m_{A^{\prime}}^{\textrm{bin}}$. We choose the same selection for
$\sum\delta M_{A^{\prime}}$ since the distribution is almost the same as
$\Delta M_{A^{\prime}}$. For $m_{A^{\prime}}^{\textrm{bin}}\leq
0.1~{}\textrm{GeV}/c^{2}$, we apply slightly different selection conditions
for $\Delta M_{A^{\prime}}$ and $\sum\delta M_{A^{\prime}}$, while
requirements on $M_{\textrm{bc}}$ and $\Delta E$ remain the same as for
$m_{A^{\prime}}^{\textrm{bin}}>0.1~{}\textrm{GeV}/c^{2}$. We do not use
$E_{\textrm{miss}}$ for $m_{A^{\prime}}^{\textrm{bin}}\leq
0.1~{}\textrm{GeV}/c^{2}$, because for such low-mass dark photons, little
background is expected from generic $B$ decays. For
$m_{A^{\prime}}^{\textrm{bin}}\leq 0.1~{}\textrm{GeV}/c^{2}$, the resolutions
of both $\Delta M_{A^{\prime}}$ and $\sum\delta M_{A^{\prime}}$ are nearly
independent of $m_{A^{\prime}}^{\textrm{bin}}$. Therefore, we require $\Delta
M_{A^{\prime}}<0.02~{}\textrm{GeV}/c^{2}$ and $\sum\delta
M_{A^{\prime}}<0.02~{}\textrm{GeV}/c^{2}$ for all $m_{A^{\prime}}\leq
0.1~{}\textrm{GeV}/c^{2}$. From the MC study, our $\Delta M_{A^{\prime}}$
selections in $A^{\prime}\to\mu^{+}\mu^{-}$ and $\pi^{+}\pi^{-}$ cover roughly
$\pm 2.5$ times the mass resolution. In case of $A^{\prime}\to e^{+}e^{-}$,
the mass resolution is worse, and our selections correspond to $\pm(1.7-2.5)$
times the mass resolution, depending on $m_{A^{\prime}}$. For instance, the
$M_{A^{\prime}}$ resolution of the $1.5~{}\mathrm{GeV}$ dark photon is about
$5~{}\mathrm{MeV}$ for $A^{\prime}\to\mu^{+}\mu^{-}$ or $\pi^{+}\pi^{-}$,
while for $A^{\prime}\to e^{+}e^{-}$ it is about $20~{}\mathrm{MeV}$. The
union of the search windows determined using $\Delta M_{A^{\prime}}$ and
$\sum\delta M_{A^{\prime}}$ for all $m_{A^{\prime}}^{\textrm{bin}}$ covers the
entire dark photon mass range of our study without any gap. The dominant SM
background sources for $\ell^{+}\ell^{-}$ pairs are photon conversion and
charmonium meson decays, mostly $J/\psi$ and $\psi(2S)$. To reduce the
background events from photon conversion, $e^{+}e^{-}$ pairs with
$M_{e^{+}e^{-}}<0.1~{}\textrm{GeV}/c^{2}$ are rejected when we search for
$m_{A^{\prime}}>0.1~{}\textrm{GeV}/c^{2}$. On the other hand, this veto is not
applied for the searches in the region $m_{A^{\prime}}\leq
0.1~{}\textrm{GeV}/c^{2}$. To suppress the lepton pairs from charmonium decays
such as $J/\psi$ or $\psi(2S)\to\ell^{+}\ell^{-}$, we reject two regions:
$3.00(3.05)~{}\textrm{GeV}/c^{2}<M_{e^{+}e^{-}(\mu^{+}\mu^{-})}<3.15(3.13)~{}\textrm{GeV}/c^{2}$
for $J/\psi$ and
$3.60(3.65)~{}\textrm{GeV}/c^{2}<M_{e^{+}e^{-}(\mu^{+}\mu^{-})}<3.75(3.73)~{}\textrm{GeV}/c^{2}$
for $\psi(2S)$. For the charged pion pairs, there is strong background from
light mesons, such as $K_{S}^{0}$, $\rho^{0}$, and $f_{0}(980)$. Because of
possible $K$–$\pi$ misidentification, $K^{*0}$, $\phi$ and so on are also a
source of possible background. Since production of such mesons is copious,
especially that of $\rho^{0}$ mesons, we reject the
$0.45~{}\textrm{GeV}/c^{2}<M_{\pi^{+}\pi^{-}}<1.1~{}\textrm{GeV}/c^{2}$.
Another source of pion pairs is $D^{0}$ meson. Two decay channels,
$D^{0}\to\pi^{+}\pi^{-}$ and $D^{0}\to\pi^{+}K^{-}$ are considered. A direct
$D^{0}$ veto is applied by removing $\pi^{+}\pi^{-}$ combinations which
satisfy
$1.85~{}\textrm{GeV}/c^{2}<M_{\pi^{+}\pi^{-}}<1.88~{}\textrm{GeV}/c^{2}$. The
other decay channel, $D^{0}\to\pi^{+}K^{-}$, can mimic the signal via
$K$–$\pi$ misidentification. We reject these events by discarding the
$1.85~{}\textrm{GeV}/c^{2}<M_{\pi^{+}K^{-}}<1.88~{}\textrm{GeV}/c^{2}$ mass
range. After signal selection, most of the combinatorial background is in the
$B^{0}\to\ell^{+}\ell^{-}\pi^{+}\pi^{-}$ mode, coming from the continuum
processes $e^{+}e^{-}\to q\bar{q}$ ($q=u,d,s$ or $c$). In the four-lepton
mode, on the other hand, there is almost no background left. The continuum
background is suppressed via multivariate analysis (MVA) using the Fisher
discriminant Fisher  method in the TMVA Hocker:2007ht  package. We make use of
16 event shape variables: the cosine of angle between the beam axis and
$B^{0}$ momentum ($\cos{\theta_{B}}$), the cosine of angle between the thrust
axis of the $B^{0}$ daughters and that of the rest of the event
($\cos{\theta_{\textrm{T}}}$), and the Fisher discriminant components of
modified Fox-Wolfram moments PhysRevLett.91.261801 . The MVA training is
performed for the $\ell^{+}\ell^{-}\pi^{+}\pi^{-}$ final state for each
$m_{A^{\prime}}^{\textrm{bin}}$, using the signal and continuum MC. We apply
MVA selection creteria to retain from 75% to 90% of signal and from 10% to 30%
of continuum background, depending on $m_{A^{\prime}}$ and final state.

##  4 Systematic uncertainties

Figure 2: Relative uncertainty of signal reconstruction efficiency for each
$A^{\prime}$ mass and final state.

We determine the branching fraction of $B^{0}\to A^{\prime}A^{\prime}$ as

$\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})=\frac{N_{\textrm{obs}}-N_{\textrm{bkg}}}{\epsilon\times
2\times N_{B\overline{B}}\times\mathcal{B}_{0}},$ (4)

where $\mathcal{B}_{0}$ is the branching fraction of $\Upsilon(4S)\to
B^{0}\overline{B}^{0}$, of which the current world-average value is $0.486\pm
0.006$ PDG , $N_{\textrm{obs}}$ is the yield, $N_{\textrm{bkg}}$ is the number
of expected background events determined from MC, $\epsilon$ is the signal
reconstruction efficiency considering branching fraction of $A^{\prime}$
subdecays, and $N_{B\overline{B}}=(772\pm 11)\times 10^{6}$ is the number of
$B\overline{B}$ pairs which are collected by the Belle detector. The most
important source of systematic uncertainties is the signal reconstruction
efficiency, which is obtained by MC. The sources of uncertainty include the
statistical error in the signal MC, track reconstruction efficiency, particle
identification (PID) efficiency, and uncertainties in the MVA method used to
suppress continuum background for $\ell^{+}\ell^{-}\pi^{+}\pi^{-}$ final
states. The uncertainties for $N_{B\overline{B}}$ and $\mathcal{B}_{0}$ also
contribute to systematics. The uncertainties due to background estimation are
very small compared to other systematic uncertainties. Track reconstruction
efficiency is studied using the decay chain $D^{*+}\to D^{0}\pi^{+}$,
$D^{0}\to K_{S}^{0}\pi^{+}\pi^{-}$, and $K_{S}^{0}\to\pi^{+}\pi^{-}$ where we
tag all the charged tracks in the chain but one from $K_{S}^{0}$ decays (‘test
track’) then try to find the test track. We compare the tracking efficiency
difference of the test track for both data and MC. The error is 1.4%,
independent of the dark photon mass and final state. The PID correction is
applied to each daughter electron, muon, and pion. The lepton (pion)
identification correction is studied using the difference between MC and data
for the process $\gamma\gamma\to e^{+}e^{-}/\mu^{+}\mu^{-}$ ($D^{*+}\to
D^{0}\pi^{+}_{\textrm{slow}}\to K^{-}\pi^{+}\pi^{+}_{\textrm{slow}}$), and the
errors are approximately $2\%~{}(1\%)$ per lepton (pion), with the resulting
correction factor being about $90\%$. The exact correction factor and
uncertainty depend on $m_{A^{\prime}}$ through different kinematics. The MVA
correction factor and uncertainty are studied using the control mode,
$B^{0}\to J/\psi{K}^{*0}\to e(\mu)^{+}e(\mu)^{-}\pi^{-}K^{+}$. We apply MVA
training results for the continuum suppression of
$\ell^{+}\ell^{-}\pi^{+}\pi^{-}$ modes for each assumed value of
$m_{A^{\prime}}$ to $B^{0}\to J/\psi K^{*0}$ MC and data. We then calculate
the double ratio
$({N_{\textrm{data,A}}/N_{\textrm{data,B}}})/({N_{\textrm{MC,A}}/N_{\textrm{MC,B}}})$,
where $N_{\textrm{data(MC),B}}$ and $N_{\textrm{data(MC),A}}$ are the number
of signal candidates in data(MC) before and after MVA training application,
respectively. The systematic uncertainty due to MVA training is taken from the
uncertainties in the double ratio, and these uncertainties are approximately
$2\%$ at all values of $m_{A^{\prime}}$. After multiplying all correction
factors, signal efficiencies are mostly $5-20$%. The efficiencies increase as
the $A^{\prime}$ mass approaches $0$ or $m_{B^{0}}/2$, in which case both
$e^{\pm}$ ($\mu^{\pm}$) from the $A^{\prime}$ decays are more likely to exceed
the energy threshold for ECL (KLM) detection. The summary of signal-
efficiency-related systematic uncertainties is shown in Fig. 2, and the total
systematic uncertainties are $7.5$–$10\%$ for $e^{+}e^{-}e^{+}e^{-}$ and
$\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ final states and $5$–$7.5\%$ for
$e^{+}e^{-}\mu^{+}\mu^{-}$, $e^{+}e^{-}\pi^{+}\pi^{-}$, and
$\mu^{+}\mu^{-}\pi^{+}\pi^{-}$ final states.

##  5 Results

Figure 3 shows the number of $B^{0}\to A^{\prime}A^{\prime}$ candidate events.
There are no events observed in any bin in the $e^{+}e^{-}\mu^{+}\mu^{-}$ and
$\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ mode, while we find $N_{\mathrm{obs}}\leq 2$
events for $e^{+}e^{-}e^{+}e^{-}$, $e^{+}e^{-}\pi^{+}\pi^{-}$, and
$\mu^{+}\mu^{-}\pi^{+}\pi^{-}$ modes. The yields are consistent with the
expected number of background events and we set the upper limits at 90% C.L.

Figure 3: The number of $B^{0}\to A^{\prime}A^{\prime}$ candidate events for
each final state.

For the limits of $\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$, we combine the
number of expected background events, signal candidates in data, and signal
reconstruction efficiencies of the five final states. The combined numbers of
expected background events and signal candidates in data are calculated by
simply adding the results for the individual final states. For the signal
efficiencies, we first obtain the ratio $F_{f}\equiv\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime}\to f)/\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$, where
$f$ is each final state, using Eq. (1). In case of $e^{+}e^{-}\mu^{+}\mu^{-}$,
for example, $F_{e^{+}e^{-}\mu^{+}\mu^{-}}$ is
$2\times\mathcal{B}(A^{\prime}\to
e^{+}e^{-})\times\mathcal{B}(A^{\prime}\to\mu^{+}\mu^{-})$. The graph of
$F_{f}$ is presented in Fig. 4. With this ratio $F_{f}$, the combined
efficiency is determined as $\sum_{f}\epsilon_{f}F_{f}$ where $\epsilon_{f}$
is the signal efficiency of the final state $f$. The upper limits are
calculated using the POLE program PhysRevD.67.012002 , which is based on the
Feldman-Cousins unified approach PhysRevD.57.3873 . We report the limits on
the products of branching fractions $\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})\times\mathcal{B}(A^{\prime}\to e^{+}e^{-})^{2}$ and
$\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})\times\mathcal{B}(A^{\prime}\to\mu^{+}\mu^{-})^{2}$, as
well as the limits on $\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$. For
$\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$, we use Eq. (1) to combine the
five final states. The upper limits of $\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})$ are obtained in the mass range
$0.01~{}\textrm{GeV}/c^{2}\leq m_{A^{\prime}}\leq 1.10~{}\textrm{GeV}/c^{2}$
with 10 $\textrm{MeV}/c^{2}$ bin and $1.10~{}\textrm{GeV}/c^{2}\leq
m_{A^{\prime}}\leq 2.62~{}\textrm{GeV}/c^{2}$ with $20~{}\textrm{MeV}/c^{2}$
bin.

Figure 4: $\mathcal{B}(B^{0}\to A^{\prime}A^{\prime}\to
f)/\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$ distributions for each final
state and dark photon mass. $e^{+}e^{-}\pi^{+}\pi^{-}$ and
$\mu^{+}\mu^{-}\pi^{+}\pi^{-}$ distributions are almost the same for the whole
region. $e^{+}e^{-}e^{+}e^{-}$ and $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$
distributions are the same and $e^{+}e^{-}\mu^{+}\mu^{-}$ distribution is
twice that of four-electron or four-muon final states in the region
$m_{A^{\prime}}>0.5~{}\textrm{GeV}/c^{2}$.

The obtained limits are shown in Fig. 5 as functions of $m_{A^{\prime}}$. The
limits on the products of branching fractions are $\mathcal{O}(10^{-8})$ for
both modes and in all $m_{A^{\prime}}$ bins. For $\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})$, the upper limits are
$\mathcal{O}(10^{-8})$–$\mathcal{O}(10^{-5})$. Due to the light meson veto in
the $\ell^{+}\ell^{-}\pi^{+}\pi^{-}$ final states and the large fraction of
$A^{\prime}\to\textrm{hadrons}$ in the veto region from Eq. (1), the upper
limits near the masses of $\rho^{0}$ and $\phi$ mesons are less restrictive
than others. Table 1 lists the signal efficiency, the expected number of
backgrounds and number of observed events ($N_{\mathrm{obs}}$) for some of
$m_{A^{\prime}}$.

Figure 5: Upper limits of $B^{0}\to A^{\prime}A^{\prime}$ branching fraction at 90% C.L. Table 1:  Signal efficnency, expected the number of backgrounds, yields for each $B^{0}$ final state and upper limits of $B^{0}\to A^{\prime}A^{\prime}$ branching fraction with 90% confidence interval. The table presents a part of the results for dark photons with (i) 20 $\textrm{MeV}/c^{2}$ interval in $m_{A^{\prime}}<2m_{\mu}$ region, (ii) 100 $\textrm{MeV}/c^{2}$ interval on the other region.  $m_{A^{\prime}}$ | $e^{+}e^{-}e^{+}e^{-}$ | $e^{+}e^{-}\mu^{+}\mu^{-}$ | $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ | $e^{+}e^{-}\pi^{+}\pi^{-}$ | $\mu^{+}\mu^{-}\pi^{+}\pi^{-}$ | 90% U.L.
---|---|---|---|---|---|---
$(\textrm{GeV}/c^{2})$ | Eff. (%) | $N_{\textrm{bkg}}^{\textrm{exp}}$ | Yield | Eff. (%) | $N_{\textrm{bkg}}^{\textrm{exp}}$ | Yield | Eff. (%) | $N_{\textrm{bkg}}^{\textrm{exp}}$ | Yield | Eff. (%) | $N_{\textrm{bkg}}^{\textrm{exp}}$ | Yield | Eff. (%) | $N_{\textrm{bkg}}^{\textrm{exp}}$ | Yield |  ($10^{-8}$)
0.02 | 14.82 |  0.83$\pm$0.37 | 2 | - | - | - | - | - | - | - | - | - | - | - | - | 4.55
0.04 | 14.62 |  0.00$\pm$0.17 | 1 | - | - | - | - | - | - | - | - | - | - | - | - | 3.85
0.06 | 14.40 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | - | - | - | - | - | - | 2.21
0.08 | 14.07 |  0.00$\pm$0.17 | 1 | - | - | - | - | - | - | - | - | - | - | - | - | 4.00
0.10 | 13.63 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | - | - | - | - | - | - | 2.34
0.12 | 13.66 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | - | - | - | - | - | - | 2.33
0.14 | 13.85 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | - | - | - | - | - | - | 2.30
0.16 | 13.57 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | - | - | - | - | - | - | 2.35
0.18 | 13.37 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | - | - | - | - | - | - | 2.38
0.20 | 13.25 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | - | - | - | - | - | - | 2.41
0.30 | 12.78 |  0.00$\pm$0.17 | 0 | 15.01 |  0.00$\pm$0.17 | 0 | 13.22 |  0.00$\pm$0.17 | 0 | 21.16 |  0.00$\pm$0.49 | 0 | 19.85 |  0.50$\pm$0.50 | 0 | 1.91
0.40 | 12.35 |  0.00$\pm$0.17 | 0 | 12.44 |  0.00$\pm$0.17 | 0 | 9.18 |  0.00$\pm$0.17 | 0 | 19.25 |  0.50$\pm$0.50 | 0 | 15.30 |  0.00$\pm$0.49 | 0 | 2.15
0.50 | 11.67 |  0.00$\pm$0.17 | 0 | 11.39 |  0.00$\pm$0.17 | 0 | 7.98 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | 4.39
0.60 | 11.07 |  0.10$\pm$0.19 | 0 | 10.71 |  0.00$\pm$0.17 | 0 | 7.53 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | 7.99
0.70 | 10.96 |  0.00$\pm$0.17 | 0 | 10.46 |  0.00$\pm$0.17 | 0 | 7.18 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | 35.2
0.80 | 11.39 |  0.00$\pm$0.17 | 0 | 10.54 |  0.00$\pm$0.17 | 0 | 6.97 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | 61.3
0.90 | 11.47 |  0.00$\pm$0.17 | 0 | 10.45 |  0.00$\pm$0.17 | 0 | 6.73 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | 11.0
1.00 | 11.26 |  0.00$\pm$0.17 | 0 | 10.20 |  0.00$\pm$0.17 | 0 | 6.42 |  0.00$\pm$0.17 | 0 | - | - | - | - | - | - | 8.63
1.10 | 11.10 |  0.00$\pm$0.17 | 0 | 9.91 |  0.00$\pm$0.17 | 0 | 6.27 |  0.00$\pm$0.17 | 0 | 14.73 |  0.30$\pm$0.52 | 0 | 9.87 |  0.50$\pm$0.50 | 0 | 3.30
1.20 | 11.07 |  0.00$\pm$0.17 | 0 | 9.88 |  0.00$\pm$0.17 | 0 | 6.34 |  0.00$\pm$0.17 | 0 | 14.72 |  0.25$\pm$0.51 | 1 | 9.87 |  0.00$\pm$0.49 | 0 | 8.29
1.30 | 11.22 |  0.00$\pm$0.17 | 0 | 10.10 |  0.00$\pm$0.17 | 0 | 6.40 |  0.00$\pm$0.17 | 0 | 15.08 |  0.30$\pm$0.52 | 0 | 9.81 |  0.00$\pm$0.49 | 0 | 5.95
1.40 | 11.48 |  0.00$\pm$0.17 | 0 | 10.18 |  0.10$\pm$0.19 | 0 | 6.44 |  0.00$\pm$0.17 | 0 | 15.57 |  1.00$\pm$0.55 | 1 | 9.76 |  0.00$\pm$0.49 | 0 | 14.5
1.50 | 11.75 |  0.00$\pm$0.17 | 0 | 10.37 |  0.00$\pm$0.17 | 0 | 6.45 |  0.00$\pm$0.17 | 0 | 15.63 |  1.10$\pm$0.56 | 1 | 9.94 |  0.00$\pm$0.49 | 0 | 19.3
1.60 | 12.13 |  0.00$\pm$0.17 | 0 | 10.57 |  0.00$\pm$0.17 | 0 | 6.62 |  0.00$\pm$0.17 | 0 | 15.84 |  0.85$\pm$0.53 | 0 | 9.44 |  0.00$\pm$0.49 | 0 | 11.8
1.70 | 12.34 |  0.00$\pm$0.17 | 0 | 10.86 |  0.00$\pm$0.17 | 0 | 6.78 |  0.00$\pm$0.17 | 0 | 13.74 |  0.40$\pm$0.53 | 0 | 8.68 |  0.00$\pm$0.49 | 0 | 13.8
1.80 | 12.69 |  0.00$\pm$0.17 | 0 | 11.42 |  0.00$\pm$0.17 | 0 | 7.24 |  0.00$\pm$0.17 | 0 | 13.49 |  0.40$\pm$0.53 | 0 | 8.58 |  0.00$\pm$0.49 | 0 | 11.9
1.90 | 13.06 |  0.10$\pm$0.19 | 0 | 12.02 |  0.00$\pm$0.17 | 0 | 7.94 |  0.00$\pm$0.17 | 0 | 17.73 |  0.50$\pm$0.54 | 0 | 12.91 |  0.00$\pm$0.49 | 0 | 8.86
2.00 | 13.43 |  0.25$\pm$0.22 | 0 | 13.08 |  0.00$\pm$0.17 | 0 | 8.84 |  0.00$\pm$0.17 | 0 | 19.94 |  1.32$\pm$0.57 | 0 | 14.62 |  0.10$\pm$0.50 | 0 | 6.80
2.10 | 13.90 |  0.15$\pm$0.20 | 0 | 14.39 |  0.00$\pm$0.17 | 0 | 10.52 |  0.00$\pm$0.17 | 0 | 20.10 |  0.75$\pm$0.56 | 0 | 16.28 |  0.27$\pm$0.51 | 0 | 7.61
2.20 | 14.50 |  0.20$\pm$0.22 | 0 | 16.20 |  0.10$\pm$0.19 | 0 | 12.87 |  0.00$\pm$0.17 | 0 | 17.77 |  1.30$\pm$0.61 | 0 | 17.76 |  0.00$\pm$0.49 | 0 | 6.25
2.30 | 15.32 |  0.00$\pm$0.17 | 0 | 18.47 |  0.10$\pm$0.19 | 0 | 16.01 |  0.00$\pm$0.17 | 0 | 18.05 |  2.04$\pm$0.63 | 1 | 19.74 |  0.20$\pm$0.51 | 0 | 8.38
2.40 | 16.47 |  0.20$\pm$0.22 | 0 | 20.79 |  0.00$\pm$0.17 | 0 | 19.21 |  0.00$\pm$0.17 | 0 | 19.01 |  2.05$\pm$0.66 | 2 | 20.87 |  0.72$\pm$0.52 | 0 | 10.2
2.50 | 18.15 |  0.20$\pm$0.22 | 0 | 23.24 |  0.00$\pm$0.17 | 0 | 22.40 |  0.00$\pm$0.17 | 0 | 18.73 |  2.40$\pm$0.66 | 1 | 23.08 |  0.50$\pm$0.50 | 0 | 5.20
2.60 | 21.05 |  0.00$\pm$0.17 | 0 | 26.34 |  0.10$\pm$0.19 | 0 | 26.85 |  0.00$\pm$0.17 | 0 | 22.52 |  2.25$\pm$0.79 | 0 | 25.34 |  1.52$\pm$0.74 | 0 | 2.31

The $B^{0}\to A^{\prime}A^{\prime}$ branching fraction with off-shell
$H$–$h^{\prime}$ mixing, for all but the $m_{h^{\prime}}\sim m_{B^{0}}$
region, is calculated as PhysRevD.83.054005  111B. Batell, private
communication on the numerical factor of Eq. (5) of Ref. PhysRevD.83.054005 ,
when we apply $B^{0}$-meson-related variables instead of $B_{s}$-meson and the
measured Higgs mass.,

$\displaystyle\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})\simeq 7\times
10^{-7}\times\lambda^{2}\times
V_{A^{\prime}A^{\prime}}^{1/2}\times\frac{V_{A^{\prime}A^{\prime}}+12m_{A^{\prime}}^{4}/m_{B^{0}}^{4}}{(1-m_{h^{\prime}}^{2}/m_{B^{0}}^{2})^{2}}$
(5)

where $\lambda$ is the Higgs portal coupling with a new scalar field
$H^{\prime}$ from
$\mathcal{L}_{\mathrm{Higgs}}=-\lambda(H^{\dagger}H)(H^{\prime\dagger}H^{\prime})$
and $V_{A^{\prime}A^{\prime}}=1-{4m_{A^{\prime}}^{2}}/{m_{B^{0}}^{2}}$. From
Eq. (5) and the limits on $\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$, we
determine the 90% C.L. upper limits on $\lambda$ versus $m_{A^{\prime}}$ (Fig.
6) and $m_{h^{\prime}}$ (Fig. 7). In the region where $m_{h^{\prime}}\simeq
m_{B^{0}}$, the upper limit on $\lambda$ gets as low as
$\mathcal{O}(10^{-2})$. Otherwise, the upper limits are
$\mathcal{O}(10^{-1})$–$\mathcal{O}(1)$.

Figure 6: 90% upper limits of the Higgs portal coupling ($\lambda$) versus the
dark photon mass for a $2.00$, $4.00$, $5.24$, $8.00~{}\textrm{GeV}/c^{2}$
dark Higgs. Figure 7: 90% upper limits of the Higgs portal coupling
($\lambda$) versus the dark Higgs mass for the $0.02$, $0.24$, $1.00$,
$2.00~{}\textrm{GeV}/c^{2}$ dark photon.

##  6 Conclusions

In summary, we have searched for $B^{0}\to A^{\prime}A^{\prime}$ decays for
the first time using the full data set of $772\times 10^{6}$ $B\overline{B}$
events of Belle. We restrict our study to the case where $A^{\prime}$ decays
promptly to $e^{+}e^{-}$, $\mu^{+}\mu^{-}$, or hadronic final states, and
consider five final states of $B^{0}$ which are $e^{+}e^{-}e^{+}e^{-}$,
$e^{+}e^{-}\mu^{+}\mu^{-}$, $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$,
$e^{+}e^{-}\pi^{+}\pi^{-}$, and $\mu^{+}\mu^{-}\pi^{+}\pi^{-}$. From the
branching fraction of $A^{\prime}$, the five $B^{0}$ final states are merged
to determine the branching fraction of $B^{0}\to A^{\prime}A^{\prime}$. We
find no significant signal in any assumed $A^{\prime}$ mass and decay mode, so
we determine upper limits on $\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})\times\mathcal{B}(A^{\prime}\to e^{+}e^{-})^{2}$,
$\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})\times\mathcal{B}(A^{\prime}\to\mu^{+}\mu^{-})^{2}$ and
$\mathcal{B}(B^{0}\to A^{\prime}A^{\prime})$, each at 90% C.L. The limits on
the products of branching fractions are of the order of
$\mathcal{O}(10^{-8})$, while the limits on $\mathcal{B}(B^{0}\to
A^{\prime}A^{\prime})$ are $\mathcal{O}(10^{-8})$–$\mathcal{O}(10^{-5})$. We
also set 90% C.L. upper limits on the Higgs portal coupling $\lambda$ for each
assumed value of $m_{A^{\prime}}$ and $m_{h^{\prime}}$. The upper limits on
$\lambda$ are of the order of $10^{-2}$–$10^{-1}$ at $m_{h^{\prime}}\simeq
m_{B^{0}}\pm 40~{}\textrm{MeV}/c^{2}$ and $10^{-1}$–$1$ at
$m_{h^{\prime}}\simeq m_{B^{0}}\pm 3~{}\textrm{GeV}/c^{2}$. With minor
modifications our analysis can be used to set limits on the other new physics
models which include prompt $B^{0}\to XX$ and
$X\to\ell^{+}\ell^{-}/\pi^{+}\pi^{-}$ decays. We expect to have much more
stringent results from the Belle II experiment Abe:2010gxa ; Kou:2018nap ,
with nearly two orders of magnitude increase in statistics, in the future.

###### Acknowledgements.

We thank B. Batell and M. Pospelov for the discussion of the $B^{0}\to
A^{\prime}A^{\prime}$ branching fraction which helped to finalize this paper.
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; 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 Pujiang Program under Grant No. 18PJ1401000; 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, 2018R1A4A1025334,
2019K1A3A7A09033840, 2019K1A3A7A09034999, 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.

## References

  * (1) P. A. Zyla et al. (Particle Data Group), The Review of Particle Physics, PTEP 2020 (2020) 083C01.
  * (2) Q. R. Ahmad et al. (SNO Collaboration), Measurement of the Rate of $\nu_{e}+d\to p+p+e^{-}$ Interactions Produced by ${}^{8}B$ Solar Neutrinos at the Sudbury Neutrino Observatory, Phys. Rev. Lett. 87 (2001) 071301.
  * (3) Y. Fukuda et al. (Super-Kamiokande Collaboration), Evidence for Oscillation of Atmospheric Neutrinos, Phys. Rev. Lett. 81 (1998) 1562.
  * (4) A. D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe, Sov. Phys. Usp. 34 (1991) 392.
  * (5) B. Batell, M. Pospelov and A. Ritz, Multi-lepton Signatures of a Hidden Sector in Rare B Decays $B$ Decays, Phys. Rev. D 83 (2011) 054005 [arXiv:0911.4938].
  * (6) B. Holdom, Two U(1)’s and $\epsilon$ Charge Shifts, Phys. Lett. B 166 (1986) 196.
  * (7) P. Fayet, Effects of the Spin 1 Partner of the Goldstino (Gravitino) on Neutral Current Phenomenology, Phys. Lett. B 95 (1980) 285.
  * (8) P. Fayet, On the Search for a New Spin 1 Boson, Nucl. Phys. B 187 (1981) 184.
  * (9) G. W. Bennett et al. (Muon g-2 Collaboration), Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL, Phys. Rev. D 73 (2006) 072003 [arXiv:hep-ex/0602035].
  * (10) M. Pospelov, Secluded U(1) below the weak scale, Phys. Rev. D 80 (2009) 095002 [arXiv:0811.1030].
  * (11) D. Tucker-Smith and I. Yavin, Muonic hydrogen and MeV forces, Phys. Rev. D 83 (2011) 101702 [arXiv:1011.4922].
  * (12) B. Batell, D. McKeen and M. Pospelov, New Parity-Violating Muonic Forces and the Proton Charge Radius, Phys. Rev. Lett. 107 (2011) 011803 [arXiv:1103.0721].
  * (13) N. Haba and S. Matsumoto, Baryogenesis from Dark Sector, Prog. Theor. Phys. 125 (2011) 1311 [arXiv:1008.2487].
  * (14) J. Chang et al., An excess of cosmic ray electrons at energies of 300-800 GeV, Nature 456 (2008) 362.
  * (15) O. Adriani et al. (PAMELA Collaboration), Cosmic-Ray Positron Energy Spectrum Measured by PAMELA, Phys. Rev. Lett. 111 (2013) 081102 [arXiv:1308.0133].
  * (16) M. Ackermann et al. (Fermi-LAT Collaboration), Measurement of separate cosmic-ray electron and positron spectra with the Fermi Large Area Telescope, Phys. Rev. Lett. 108 (2012) 011103 [arXiv:1109.0521].
  * (17) M. Aguilar et al. (AMS Collaboration), Electron and Positron Fluxes in Primary Cosmic Rays Measured with the Alpha Magnetic Spectrometer on the International Space Station, Phys. Rev. Lett. 113 (2014) 121102.
  * (18) C. R. Chen, F. Takahashi and T. T. Yanagida, High-energy Cosmic-Ray Positrons from Hidden-Gauge-Boson Dark Matter, Phys. Lett. B 673 (2009) 255 [arXiv:0811.0477].
  * (19) C. Y. Chen, H. Davoudiasl, W. J. Marciano and C. Zhang, Implications of a light "dark Higgs" solution to the $g_{\mu}-2$ discrepancy, Phys. Rev. D 93 (2016) 035006 [arXiv:1511.04715].
  * (20) G. Arcadi et al., Lepton Flavor Violation Induced by Dark Matter, Phys. Rev. D 97 (2018) 075022 [arXiv:1712.02373].
  * (21) B. Batell, M. Pospelov and A. Ritz, Probing a Secluded U(1) at B-factories, Phys. Rev. D 79 (2009) 115008 [arXiv:0903.0363].
  * (22) J. P. Lees et al. (BaBar Collaboration), Search for Low-Mass Dark-Sector Higgs Bosons, Phys. Rev. Lett. 108 (2012) 211801 [arXiv:1202.1313].
  * (23) A. V. Danilina and N. V. Nikitin, Four-Leptonic Decays of Charged and Neutral $B$ Mesons within the Standard Model, Phys. Atom. Nucl. 81 (2018) 347.
  * (24) R. Aaij et al. (LHCb Collaboration), Search for decays of neutral beauty mesons into four muons, JHEP 03 (2017) 001 [arXiv:1611.07704].
  * (25) R. Aaij et al. (LHCb Collaboration), Study of the rare $B_{s}^{0}$ and $B^{0}$ decays into the $\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ final state, Phys. Lett. B 743 (2015) 46 [arXiv:1412.6433].
  * (26) A. Abashian et al. (Belle Collaboration), The Belle Detector, Nucl. Instrum. Meth. A 479 (2002) 117.
  * (27) J. Brodzicka et al. (Belle Collaboration), Physics Achievements from the Belle Experiment, PTEP 2012 (2012) 04D001 [arXiv:1212.5342].
  * (28) S. Kurokawa and E. Kikutani, Overview of the KEKB accelerators, Nucl. Instrum. Meth. A 499 (2003) 1.
  * (29) T. Abe et al., Achievements of KEKB, PTEP 2013 (2013) 03A001.
  * (30) D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Methods Phys. Res., Sect. A 462 (2001) 152.
  * (31) R. Brun, F. Bruyant, M. Maire, A. C. McPherson and P. Zanarini, Geant3, CERN-DD-EE-84-1.
  * (32) A. Abashian, K. Abe, K. Abe, P. K. Behera, F. Handa, T. Iijima, Y. Inoue, H. Miyake, T. Nagamine and E. Nakano, et al. Muon identification in the Belle experiment at KEKB, Nucl. Instrum. Meth. A 491 (2002), 69-82.
  * (33) K. Hanagaki, H. Kakuno, H. Ikeda, T. Iijima and T. Tsukamoto, Electron identification in Belle, Nucl. Instrum. Meth. A 485 (2002), 490-503 [arXiv:hep-ex/0108044].
  * (34) E. Nakano, Belle PID, Nucl. Instrum. Meth. A 494 (2002), 402-408.
  * (35) R. A. Fisher, The use of multiple measurements in taxonomic problems, Annals Eugen. 7 (1936) 179.
  * (36) H. Voss, A. Hocker, J. Stelzer and F. Tegenfeldt, TMVA - Toolkit for Multivariate Data Analysis, PoS ACAT (2007) 040 [arXiv:physics/0703039].
  * (37) S. H. Lee et al. (Belle Collaboration), Evidence for $B^{0}\to\pi^{0}\pi^{0}$, Phys. Rev. Lett. 91 (2003) 261801 [arXiv:hep-ex/0308040].
  * (38) J. Conrad, O. Botner, A. Hallgren and C. Perez de los Heros, Including systematic uncertainties in confidence interval construction for Poisson statistics, Phys. Rev. D 67 (2003) 012002 [arXiv:hep-ex/0202013].
  * (39) G. J. Feldman and R. D. Cousins, A Unified approach to the classical statistical analysis of small signals, Phys. Rev. D 57 (1998) 3873 [arXiv:physics/9711021].
  * (40) T. Abe et al. (Belle II Collaboration), Belle II Technical Design Report, [arXiv:1011.0352].
  * (41) E. Kou et al. (Belle II Collaboration), The Belle II Physics Book, PTEP 2019 (2019) 123C01 [arXiv:1808.10567].