Observation of an alternative χc0 (2P) candidate in e+e- →j /ψD D OBSERVATION of AN ALTERNATIVE χc0 (2P) ... K. CHILIKIN et al.

Observation of an alternative χ

_c0

ð 2 PÞ candidate in e

⁺

e

⁻

→ J = ψD D ¯

K. Chilikin,^39,47I. Adachi,^13,10H. Aihara,⁷³S. Al Said,^67,33D. M. Asner,⁵⁸V. Aulchenko,^3,56R. Ayad,⁶⁷V. Babu,⁶⁸ I. Badhrees,^67,32 A. M. Bakich,⁶⁶V. Bansal,⁵⁸E. Barberio,⁴⁵ D. Besson,⁴⁷V. Bhardwaj,¹⁶B. Bhuyan,¹⁸J. Biswal,²⁷ A. Bobrov,^3,56A. Bondar,^3,56 A. Bozek,⁵³M. Bračko,^43,27T. E. Browder,¹²D. Červenkov,⁴ V. Chekelian,⁴⁴A. Chen,⁵⁰

B. G. Cheon,¹¹K. Cho,³⁴Y. Choi,⁶⁵D. Cinabro,⁷⁸N. Dash,¹⁷S. Di Carlo,⁷⁸ Z. Doležal,⁴ Z. Drásal,⁴ D. Dutta,⁶⁸ S. Eidelman,^3,56 H. Farhat,⁷⁸J. E. Fast,⁵⁸T. Ferber,⁷ B. G. Fulsom,⁵⁸V. Gaur,⁷⁷N. Gabyshev,^3,56A. Garmash,^3,56 R. Gillard,⁷⁸P. Goldenzweig,³⁰ J. Haba,^13,10T. Hara,^13,10 K. Hayasaka,⁵⁵W.-S. Hou,⁵²K. Inami,⁴⁹A. Ishikawa,⁷¹ R. Itoh,^13,10Y. Iwasaki,¹³W. W. Jacobs,²⁰I. Jaegle,⁸H. B. Jeon,³⁷Y. Jin,⁷³D. Joffe,³¹K. K. Joo,⁵T. Julius,⁴⁵K. H. Kang,³⁷ G. Karyan,⁷ P. Katrenko,^48,39D. Y. Kim,⁶³H. J. Kim,³⁷J. B. Kim,³⁵K. T. Kim,³⁵M. J. Kim,³⁷S. H. Kim,¹¹Y. J. Kim,³⁴

K. Kinoshita,⁶ P. Kodyš,⁴S. Korpar,^43,27D. Kotchetkov,¹²P. Križan,^40,27P. Krokovny,^3,56T. Kuhr,⁴¹R. Kulasiri,³¹ A. Kuzmin,^3,56Y.-J. Kwon,⁸⁰J. S. Lange,⁹ L. Li,⁶¹L. Li Gioi,⁴⁴J. Libby,¹⁹D. Liventsev,^77,13 M. Lubej,²⁷T. Luo,⁵⁹

M. Masuda,⁷² T. Matsuda,⁴⁶D. Matvienko,^3,56K. Miyabayashi,⁸¹H. Miyata,⁵⁵R. Mizuk,^39,47,48 G. B. Mohanty,⁶⁸ H. K. Moon,³⁵T. Mori,⁴⁹R. Mussa,²⁵E. Nakano,⁵⁷M. Nakao,^13,10T. Nanut,²⁷K. J. Nath,¹⁸Z. Natkaniec,⁵³M. Nayak,^78,13

M. Niiyama,³⁶ N. K. Nisar,⁵⁹S. Nishida,^13,10 S. Ogawa,⁷⁰S. Okuno,²⁸S. L. Olsen,²⁹H. Ono,^54,55P. Pakhlov,^39,47 G. Pakhlova,^39,48B. Pal,⁶S. Pardi,²⁴H. Park,³⁷S. Paul,⁶⁹R. Pestotnik,²⁷L. E. Piilonen,⁷⁷C. Pulvermacher,¹³M. Ritter,⁴¹

H. Sahoo,¹²Y. Sakai,^13,10M. Salehi,^42,41 S. Sandilya,⁶ L. Santelj,¹³T. Sanuki,⁷¹O. Schneider,³⁸G. Schnell,^1,15 C. Schwanda,²²Y. Seino,⁵⁵K. Senyo,⁷⁹O. Seon,⁴⁹M. E. Sevior,⁴⁵V. Shebalin,^3,56C. P. Shen,²T.-A. Shibata,⁷⁴J.-G. Shiu,⁵² A. Sokolov,²³E. Solovieva,^39,48M. Starič,²⁷T. Sumiyoshi,⁷⁵M. Takizawa,^62,14,60U. Tamponi,^25,76K. Tanida,²⁶F. Tenchini,⁴⁵

K. Trabelsi,^13,10M. Uchida,⁷⁴ S. Uehara,^13,10 T. Uglov,^39,48S. Uno,^13,10Y. Usov,^3,56C. Van Hulse,¹ G. Varner,¹² A. Vinokurova,^3,56 A. Vossen,²⁰C. H. Wang,⁵¹M.-Z. Wang,⁵²P. Wang,²¹M. Watanabe,⁵⁵Y. Watanabe,²⁸ E. Widmann,⁶⁴ E. Won,³⁵H. Yamamoto,⁷¹Y. Yamashita,⁵⁴H. Ye,⁷C. Z. Yuan,²¹Y. Yusa,⁵⁵ Z. P. Zhang,⁶¹

V. Zhilich,^3,56V. Zhulanov,^3,56 and A. Zupanc^40,27 (Belle Collaboration)

1University of the Basque Country UPV/EHU, 48080 Bilbao

2Beihang University, Beijing 100191

3Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090

4Faculty of Mathematics and Physics, Charles University, 121 16 Prague

5Chonnam National University, Kwangju 660-701

6University of Cincinnati, Cincinnati, Ohio 45221

7Deutsches Elektronen–Synchrotron, 22607 Hamburg

8University of Florida, Gainesville, Florida 32611

9Justus-Liebig-Universität Gießen, 35392 Gießen

10SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193

11Hanyang University, Seoul 133-791

12University of Hawaii, Honolulu, Hawaii 96822

13High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801

14J-PARC Branch, KEK Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801

15IKERBASQUE, Basque Foundation for Science, 48013 Bilbao

16Indian Institute of Science Education and Research Mohali, SAS Nagar 40306

17Indian Institute of Technology Bhubaneswar, Satya Nagar 751007

18Indian Institute of Technology Guwahati, Assam 781039

19Indian Institute of Technology Madras, Chennai 600036

20Indiana University, Bloomington, Indiana 47408

21Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049

22Institute of High Energy Physics, Vienna 1050

23Institute for High Energy Physics, Protvino 142281

24INFN—Sezione di Napoli, 80126 Napoli

25INFN—Sezione di Torino, 10125 Torino

26Advanced Science Research Center, Japan Atomic Energy Agency, Naka 319-1195

27J. Stefan Institute, 1000 Ljubljana

28Kanagawa University, Yokohama 221-8686

29Center for Underground Physics, Institute for Basic Science, Daejeon 34047

30Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe

31Kennesaw State University, Kennesaw, Georgia 30144

32King Abdulaziz City for Science and Technology, Riyadh 11442

33Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah 21589

34Korea Institute of Science and Technology Information, Daejeon 305-806

35Korea University, Seoul 136-713

36Kyoto University, Kyoto 606-8502

37Kyungpook National University, Daegu 702-701

38École Polytechnique Fédérale de Lausanne (EPFL), Lausanne 1015

39P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991

40Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana

41Ludwig Maximilians University, 80539 Munich

42University of Malaya, 50603 Kuala Lumpur

43University of Maribor, 2000 Maribor

44Max-Planck-Institut für Physik, 80805 München

45School of Physics, University of Melbourne, Victoria 3010

46University of Miyazaki, Miyazaki 889-2192

47Moscow Physical Engineering Institute, Moscow 115409

48Moscow Institute of Physics and Technology, Moscow Region 141700

49Graduate School of Science, Nagoya University, Nagoya 464-8602

50National Central University, Chung-li 32054

51National United University, Miao Li 36003

52Department of Physics, National Taiwan University, Taipei 10617

53H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342

54Nippon Dental University, Niigata 951-8580

55Niigata University, Niigata 950-2181

56Novosibirsk State University, Novosibirsk 630090

57Osaka City University, Osaka 558-8585

58Pacific Northwest National Laboratory, Richland, Washington 99352

59University of Pittsburgh, Pittsburgh, Pennsylvania 15260

60Theoretical Research Division, Nishina Center, RIKEN, Saitama 351-0198

61University of Science and Technology of China, Hefei 230026

62Showa Pharmaceutical University, Tokyo 194-8543

63Soongsil University, Seoul 156-743

64Stefan Meyer Institute for Subatomic Physics, Vienna 1090

65Sungkyunkwan University, Suwon 440-746

66School of Physics, University of Sydney, New South Wales 2006

67Department of Physics, Faculty of Science, University of Tabuk, Tabuk 71451

68Tata Institute of Fundamental Research, Mumbai 400005

69Department of Physics, Technische Universität München, 85748 Garching

70Toho University, Funabashi 274-8510

71Department of Physics, Tohoku University, Sendai 980-8578

72Earthquake Research Institute, University of Tokyo, Tokyo 113-0032

73Department of Physics, University of Tokyo, Tokyo 113-0033

74Tokyo Institute of Technology, Tokyo 152-8550

75Tokyo Metropolitan University, Tokyo 192-0397

76University of Torino, 10124 Torino

77Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

78Wayne State University, Detroit, Michigan 48202

79Yamagata University, Yamagata 990-8560

80Yonsei University, Seoul 120-749

81Nara Women’s University, Nara 630-8506 (Received 6 April 2017; published 16 June 2017)

We perform a full amplitude analysis of the processe^þe⁻→J=ψDD, where¯ Drefers to eitherD⁰orD^þ. A new charmoniumlike stateXð3860Þthat decays toDD¯ is observed with a significance of6.5σ. Its mass isð3862^þ26₋₃₂^þ40₋₁₃ÞMeV=c², and its width isð201^þ154₋₆₇ ^þ88₋₈₂ÞMeV. TheJ^PC¼0^þþhypothesis is favored over the2^þþhypothesis at the level of2.5σ. The analysis is based on the980fb⁻¹data sample collected by the Belle detector at the asymmetric-energye^þe⁻collider KEKB.

DOI:10.1103/PhysRevD.95.112003

I. INTRODUCTION

The charmoniumlike stateXð3915Þwas observed by the Belle Collaboration inB→J=ψωKdecays[1]; its original name was theYð3940Þ. Subsequently, it was also observed by the BABAR Collaboration in the same B decay mode [2,3]and by both Belle[4]andBABAR[5]in the process γγ →Xð3915Þ→J=ψω. The quantum numbers of the Xð3915Þ were measured in Ref. [5] to be J^PC¼0^þþ. As a result, theXð3915Þwas identified as theχ_c0ð2PÞin the 2014 Particle Data Group tables[6].

However, this identification remains in doubt. The χ_c0ð2PÞ is expected to decay strongly to DD¯ in an S-wave. Here and elsewhere, D refers to either D⁰ or D^þ. The χc0ð2PÞ→DD¯ decay mode is expected to be dominant but has not yet been observed experimentally for theXð3915Þ; in contrast, the decay mode that is observed, Xð3915Þ→J=ψω, would be Okubo-Zweig-Iizuka (OZI) [7] suppressed for theχc0ð2PÞ. Since theχc0ð2PÞ should decay toDD¯ in anS-wave, this state is naively expected to have a large width of Γ≳100MeV [8]; however, the measured Xð3915Þ width of ð205ÞMeV is much smaller. We note that there are some specific predictions for theχc0ð2PÞwidth that give small values. For example, the predicted width is 30 MeV in Ref.[9]and between 12 and ∼100MeV, depending on the χ_c0ð2PÞ mass, in Ref. [10]. If theχ_c0ð2PÞ andXð3915Þare the same state, then the partial width toJ=ψω, which has been estimated to beΓðχc0ð2PÞ→J=ψωÞ≳1MeV[8], is too large. Also, in this case, one can obtain contradictory estimates for the branching fractionBðχ_c0ð2PÞ→J=ψωÞ[11]. Furthermore, the χ_c2ð2PÞ−χ_c0ð2PÞ mass splitting would be much smaller than the prediction of potential models and smaller than the mass difference for the bottomonium states χbJð2PÞ [8,11], which is inconsistent with expectations based on the heavier bottom quark mass.

The quantum numbers of theXð3915Þwere measured in Ref. [5] to be J^PC¼0^þþ assuming that the Xð3915Þ is produced in the processγγ →Xð3915Þwith helicityλ¼2 if its quantum numbers areJ^PC¼2^þþ. A reanalysis of the data from Ref. [5], presented in Ref.[12], shows that the 2^þþassignment becomes possible if theλ¼2assumption is abandoned. As a result of these considerations, the Xð3915Þis no longer identified as theχc0ð2PÞin the 2016 Particle Data Group tables[13].

It is possible that an alternativeχ_c0ð2PÞcandidate was actually observed by Belle [14] and BABAR [15] in the processγγ →DD¯ together with theχ_c2ð2PÞ. An alternative fit to Belle andBABARdata was performed in Ref.[8]. In this fit, the nonresonantγγ →DD¯ events are attributed to a wide charmonium state with mass and widthM¼ ð3838 12ÞMeV=c² andΓ¼ ð21119ÞMeV. However, in this estimate, additional signal sources such as feed-down from γγ →DD¯ events were not taken into account; conse- quently, if the χc0ð2PÞ is really observed in the process

γγ→DD, then the parameters measured in Ref.¯ [8] are biased—theχc0ð2PÞ mass is shifted to lower values[11].

It is also possible to search for theχc0ð2PÞproduced in the processγγ →χc0ð2PÞusing final states other thanDD¯ and J=ψω. Belle searched for high-mass charmonium states in the process γγ →K⁰_SK⁰_S [16]. An excess with a marginal statistical significance of2.6σ was found in the mass region between 3.80 and3.95GeV=c².

A unique process that is suitable for a search for the χc0ð2PÞ and other charmonium states with positive C- parity is double-charmonium production in association with the J=ψ. The C-parity of the states observed with the J=ψ is guaranteed to be positive in the case of one- photon annihilation; the annihilation via two virtual pho- tons is suppressed by a factor ðα=αsÞ² [17,18]. The charmoniumlike Xð3940Þ state was observed by Belle in the inclusivee^þe⁻→J=ψX spectrum and in the process e^þe⁻ →J=ψDD¯ [19,20], and theXð4160Þwas observed in the processe^þe⁻ →J=ψDD¯[20]. Indications of aDD¯ resonance ine^þe⁻→J=ψDD¯ were reported in Ref.[20];

however, the existence of this resonance could not be established reliably.

Here, we present an updated analysis of the process e^þe⁻ →J=ψDD. The analysis is performed using the¯ 980fb⁻¹ data sample collected by the Belle detector at the asymmetric-energye^þe⁻collider KEKB[21]. The data sample was collected at or near theϒð1SÞ,ϒð2SÞ,ϒð3SÞ, ϒð4SÞ, andϒð5SÞresonances. The integrated luminosity is 1.4 times greater than the luminosity used in the previous analysis[20]. In addition, we use a multivariate method to improve the discrimination of the signal and background events and an amplitude analysis to study theJ^PCquantum numbers of theDD¯ system.

II. BELLE DETECTOR

The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel- like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux return located outside of the coil is instrumented to detect K⁰_L mesons and to identify muons (KLM). The detector is described in detail elsewhere [22]. Two inner detector configurations were used. A 2.0 cm beam pipe and a three-layer silicon vertex detector were used for the first sample of156fb⁻¹, while a 1.5 cm beam pipe, a four-layer silicon detector, and a small-cell inner drift chamber were used to record the remaining data[23].

We use a GEANT-based Monte Carlo (MC) simulation [24]to model the response of the detector, identify potential backgrounds, and determine the acceptance. The MC

c0

simulation includes run-dependent detector performance variations and background conditions. The signal MC generation is described in Sec. IV.

III. RECONSTRUCTION

We select events of the type e^þe⁻ →J=ψDD, where¯ only theJ=ψ and one of theD mesons are reconstructed;

the otherD meson is identified by the recoil mass of the ðJ=ψ; DÞsystem, which is denoted as M_rec.

All tracks are required to originate from the interaction point region: we require dr <0.2cm and jdzj<2cm, where dr and dz are the cylindrical coordinates of the point of the closest approach of the track to the beam axis (thezaxis of the laboratory reference frame coincides with the positron-beam axis).

ChargedπandKmesons are identified using likelihood ratiosR_π=K ¼L_π=ðL_πþLKÞandR_K=π¼LK=ðL_πþLKÞ, where L_π and LK are the likelihoods for π and K, respectively. The likelihoods are calculated from the combined time-of-flight information from the TOF, the number of photoelectrons from the ACC, and dE=dx measurements in the CDC. We require R_K=π>0.6 for K candidates and R_π=K >0.1 for π candidates. The K (π) identification efficiency is typically 90% (98%), and the misidentification probability for non-K(-π) background is about 4% (30%).

Electron candidates are identified as CDC charged tracks that are matched to electromagnetic showers in the ECL.

The track and ECL cluster matching quality, the ratio of the electromagnetic shower energy to the track momentum, the transverse shape of the shower, the ACC light yield, and the track’s dE=dx ionization are used in our electron identification selection criteria. A similar likelihood ratio is constructed: R_e¼Le=ðLeþLhÞ, whereLe andLh are the likelihoods for electrons and charged hadrons (π,Kand p), respectively [25]. An electron veto (R_e<0.9) is imposed on π and K candidates. The veto rejects from 4% to 11% of background events, depending on the D decay channel, while its signal efficiency is more than 99%.

Muons are identified by their range and transverse scattering in the KLM. The muon likelihood ratio is defined as R_μ¼Lμ=ðLμþLπþLKÞ, whereLμ is the likelihood for muons [26].

The π⁰ candidates are reconstructed via their decay to two photons; we require their energies to be greater than 30 MeV and their invariant mass M_γγ to satisfy jM_γγ−m_π0j<15MeV=c², where m_π0 is the nominal π⁰ mass[6]. This requirement corresponds approximately to a 3σ mass window around the nominal mass.

Candidate K⁰_S mesons are reconstructed from pairs of oppositely charged tracks (assumed to be pions) and selected on the basis of the π^þπ⁻ invariant mass, the candidateK⁰_Smomentum and decay angle, the inferredK⁰_S flight distance in thexy plane, the angle between the K⁰_S

momentum and the direction from the interaction point to theK⁰_Svertex, the shortestzdistance between the two pion tracks, their radial impact parameters, and numbers of hits in the SVD and CDC.

The J=ψ is reconstructed via its e^þe⁻ andμ^þμ⁻ decay channels. ForJ=ψ →e^þe⁻candidates, we include photons that have energies greater than 30 MeV and are within 50 mrad of a daughter lepton candidate momentum direction in the calculation of the J=ψ invariant mass.

The J=ψ daughter leptons are identified as electrons or muons (R_e>0.1 or R_μ>0.1) and not as kaons (R_K=e<0.9 and R_K=μ<0.9 for the J=ψ →e^þe⁻ and J=ψ →μ^þμ⁻ decay modes, respectively). The combined lepton-pair identification efficiency is 91% and 75%, and the fake rate is 0.07% and 0.13% for the J=ψ → e^þe⁻ andJ=ψ →μ^þμ⁻ channels, respectively. We retain candidates that satisfy jM_J=ψ−m_J=ψj<300MeV=c², where M_J=ψ is the reconstructed mass and m_J=ψ is the nominal mass[6]. The mass window is intentionally wide because this selection is at a preliminary stage and includes the events that will be used for the background determination.

TheD^þmesons are reconstructed in five decay channels:

K⁰_Sπ^þ, K⁻π^þπ^þ, K⁰_Sπ^þπ⁰, K⁻π^þπ^þπ⁰, and K⁰_Sπ^þπ^þπ⁻. TheD⁰ mesons are reconstructed in four decay channels:

K⁻π^þ, K⁰_Sπ^þπ⁻, K⁻π^þπ⁰, and K⁻π^þπ^þπ⁻. We retain candidates that satisfy −150MeV=c²< M_D−m_D<

350MeV=c², where M_D is the reconstructed mass and m_Dis the nominal mass[6]. The mass window is chosen to be asymmetric to exclude peaking backgrounds from partially reconstructed multibody D decays, for example D⁰→K⁻π^þπ⁰ or D^þ→K⁻π^þπ^þ backgrounds in the D⁰→K⁻π^þ event sample.

After the selection of the J=ψ and D candidates, we perform a mass-constrained fit to each candidate; then, the recoil massM_recis calculated. We retain the (J=ψ,D) pairs that satisfy jM_rec−m_Dj<350MeV=c². Finally, we per- form a beam-constrained fit withM_recconstrained to theD mass to improve theM_DD¯ resolution.

IV. MULTIVARIATE ANALYSIS AND GLOBAL OPTIMIZATION

A. Multivariate analysis

To improve the separation of signal and background events, we perform a multivariate analysis for each individual D decay channel followed by a global selec- tion requirement optimization. For each D channel, the global optimization includes the definition of the signal region and the requirement on the multivariate-analysis output.

The algorithm used for the multivariate analysis is the multilayer perceptron (MLP) neural network implemented in the TMVA library[27]. ForDdecay channels without a

daughterπ⁰, the neural network contains five variables: the cylindrical coordinatedrof theJ=ψvertex, the cylindrical coordinatesdranddzof theDvertex calculated relative to theJ=ψ vertex (these coordinates being sensitive to theD flight distance), the angle between the Dmomentum and the direction from theJ=ψ vertex to theDvertex, and the smaller of the two lepton-likelihood ratiosR_e orR_μof the J=ψ daughter leptons for the J=ψ →e^þe⁻ and J=ψ → μ^þμ⁻channels, respectively. If theDmeson has a daughter π⁰, three additional π⁰-related variables are used in the neural network: the minimum energy of the π⁰ daughter photons in the laboratory reference frame, theπ⁰mass, and the angle between −p⃗ _D andp⃗ _γ, wherep⃗ _D andp⃗ _γ are the momenta of the parent D and daughter γ in the π⁰ rest frame.

The signal data sample consists of Monte Carlo events generated by a combination of the PHOKHARA9 [28] and

EvtGen[29]generators. ThePHOKHARA9generator generates the initial-state radiation photon(s); the remaining gener- ation is performed by EvtGen. The background sample is taken from the data; a two-dimensional (M_J=ψ; M_D) side- band is used. The background region includes all initially selected events except the events with jM_J=ψ−m_J=ψj<

100MeV=c² andjM_D−m_Dj<50MeV=c². In addition, we select only the events with the DD¯ invariant mass M_DD¯ <6GeV=c² for both signal and background sam- ples. This requirement improves the signal and back- ground separation, because the angle between the D momentum and the direction from the J=ψ vertex to the D vertex provides more effective separation for D mesons with larger momentum. The background data sample is split randomly into training and testing parts of equal size.

Example distributions of the MLP outputvfor the signal and background samples are shown in Fig. 1. The signal events tend to have a larger value of v, while the back- ground events have smaller values ofv. Thus, the require- ments on the MLP output used in the global selection requirement optimization have the formv > v₀, wherev₀is a cutoff value.

B. Fit to the background

Before the global optimization, the events in the back- ground region are fitted to estimate the expected number of the background events in the signal region. An unbinned maximum likelihood fit in the three-dimensional parameter space ΦB ¼ ðM_J=ψ; M_D; M_recÞ is performed. The back- ground density function is

BðΦBÞ ¼P₂ðΦBÞexpð−αM_J=ψÞ þR_MDðMDÞP^D₁ðM_J=ψ; M_recÞ

þR_MJ=ψðM~_J=ψÞP^J=ψ₁ ðM_D; M_recÞ; ð1Þ

whereP₂is a three-dimensional second-order polynomial;

P^D₁ andP^J=ψ₁ are two-dimensional first-order polynomials;

αis a rate parameter;R_MD andR_MJ=ψ are the resolutions in M_D and M_J=ψ, respectively; and M~_J=ψ is the scaled J=ψ mass:

M~_J=ψ ¼m_J=ψ þSðM_J=ψ −m_J=ψÞ; ð2Þ whereSis the scaling coefficient. This scaling accounts for the fact that the MC resolution in the J=ψ mass is significantly better than that measured with data. The resolutions are determined from a fit of the distributions of M_J=ψ and M_D in the signal MC to a sum of an asymmetric Gaussian and asymmetric double-sided Crystal Ball functions [30]. Some of the coefficients of the polynomials are fixed at zero for theD channels that have a small number of the background events.

The average value of the J=ψ mass resolution scaling coefficient for all D channels is measured to be S¼0.790.02. If the D mass resolution is scaled similarly to Eq. (2), the scaling coefficient depends on the D decay channel; it is possible to determine this coefficient from data only for channels with a large number of real D mesons in the background region:

D^þ→K⁻π^þπ^þ and D⁰→K⁻π^þ. The resolutions in data and MC are found to be consistent for both D^þ → K⁻π^þπ^þ andD⁰→K⁻π^þ; thus, the resolution for the D mesons is not scaled. An example of the background fit results (for the channel D⁰→K⁻π^þ) is shown in Fig.2.

The fit quality is estimated using the mixed-sample method [31]; the confidence levels are found to be not less than 15%.

v

0 0.5 1

10−1

1

FIG. 1. Example of the MLP output (for the channel D⁰→K⁻π^þπ⁰). The red solid line is the MLP output for the signal events, and the blue dashed line is the MLP output for the background events. Both distributions are normalized so that their integrals are equal to 1.

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C. Global selection requirement optimization The global selection requirement optimization is per- formed by maximizing the value

P

iN^ðiÞ_sig

a2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

iN^ðiÞ_bg

q ; ð3Þ

where i is the index of the D decay channel, N^ðiÞ_sig is the expected number of the signal events for theith channel, N^ðiÞ_bgis the expected number of the background events in the signal region, and a¼5 is the target significance. This optimization method is based on Ref. [32]. The signal region is defined as

M_J=ψ −m_J=ψ M^ðiÞ_J=ψ

₂ þ

M_D−m_D M^ðiÞ_D

₂

<1; ð4Þ M_rec−m_D

M^ðiÞrec

<1; ð5Þ

whereM^ðiÞ_J=ψ andM^ðiÞ_D are the half-axes of the signal region ellipse andM^ðiÞrecis the half-width of the selected recoil mass region. All the values M^ðiÞ_J=ψ, M^ðiÞ_D, and M^ðiÞrec are channel dependent.

The expected number of signal events is given by

N^ðiÞ_sig¼σ^ðE_eþ^γe^<E−→J=ψ^cutÞ DD¯ðϒð4SÞÞf^ðiÞ N^ðiÞrec

N^ðiÞ_ðϒð4SÞE

γ<EcutÞ

×L_ϒð4SÞϵ^ðiÞ_S ðv^ðiÞ₀ Þ Z

SR

S^ðiÞðΦÞdΦ; ð6Þ where

σ^ðE_eþ^γe^<E−→J=ψD^cutÞ D¯ ¼σ^ðBornÞ_eþe⁻→J=ψDD¯

σ^ðE_eþ^γe^<E−→μ^cutþÞμ⁻

σ^ðBornÞ_eþe⁻→μ^þμ⁻

ð7Þ

is the cross section of the process e^þe⁻ → J=ψDD¯ðγISRÞðγISRÞ with the condition that the photon(s) energy is less than the cutoff energy and the vacuum polarization taken into account,σ^ðBornÞprocess is the Born cross section of the specified process,f^ðiÞ is the fraction of ith channel, N^ðiÞ_ðϒð4SÞE

γ<EcutÞ is the number of generated MC events at theϒð4SÞ resonance withE_γ < E_cut,N^ðiÞrec is the total number of reconstructed MC events for all beam energies,Lϒð4SÞ is the integrated luminosity at the ϒð4SÞ resonance, ϵ^ðiÞ_S ðv^ðiÞ₀ Þ is the efficiency of the requirement (v > v^ðiÞ₀ ) on the MLP outputvfor the signal events, SR is the signal region, and S^ðiÞðΦÞ is the signal probability density function (PDF), which is proportional to the product of the resolutions in M_D, M_J=ψ, and M_rec. The factor N^ðiÞrec=N^ðiÞ_ðϒð4SÞE

γ<E_cutÞ is the reconstruction efficiency corrected for the difference between the full cross section andσ^ðE_eþ^γe^<E−→J=ψD^cutÞ D¯ and for the difference between the number of events for all beam energies and at theϒð4SÞresonance.

The ratio σ^ðE_μþ^γμ^{<E− cutÞ}ðϒð4SÞÞ=σ^ðBornÞ_μþμ⁻ ðϒð4SÞÞ is taken from Ref. [33]. The ratio N^ðiÞ_ðϒð4SÞE

γ<E_cutÞ×σ^ðBornÞ_eþe⁻→μ^þμ⁻ðϒð4SÞÞ= σ^ðE_eþ^γe^<E−→μ^cutþÞμ⁻ðϒð4SÞÞdoes not depend on the cutoff energy; it is determined from a fit to its dependence on E_cut in the range between 20 and 100 MeV.

The background can be subdivided into three classes of events: those with real D mesons, those with real J=ψ mesons, and a featureless combinatorial background. These components can have a different efficiency dependence on the MLP output cutoff value v^ðiÞ₀ as well as different distributions in M_DD¯ and angular variables. To account

, GeV/c2

MD

1.8 2 2.2

2Events / 10 MeV/c

0 5 10 15 20 25 30 35

, GeV/c2 ψ

MJ/

2.8 3 3.2

2 Events / 12 MeV/c

0 5 10 15 20 25 30 35

, GeV/c2

Mrec

1.6 1.8 2 2.2

2 Events / 14 MeV/c

2 4 6 8 10 12 14 16 18 20 22

FIG. 2. Example of the results of the fit to the background events (for the channelD⁰→K⁻π^þ). The points with error bars are the data, and the solid line is the fit result. The region defined byjMJ=ψ−mJ=ψj<100MeV=c²,jMD−mDj<50MeV=c²[the region labeled (4) in Fig.3] is excluded from the fit; because of this exclusion, the projections of both the data and the fit result ontoMDandMJ=ψhave jump discontinuities.

for these differences, the background region is divided into three parts with the definition indicated in Fig. 3. The expected number of the background events is

N^ðiÞ_bg¼ 0 BB

@ I^ðSRÞsm

I^ðSRÞ_D I^ðSRÞ_J=ψ

1 CC A

T0 BB B@

I^ð1Þsm I^ð1Þ_D I^ð1Þ_J=ψ I^ð2Þ_sm I^ð2Þ_D I^ð2Þ_J=ψ I^ð3Þsm I^ð3Þ_D I^ð3Þ_J=ψ

1 CC CA

−10 BB B@

ϵ^ð1ÞB ðv^ðiÞ₀ ÞN^ðiÞð1Þ_{ðbg recÞ} ϵ^ð2ÞB ðv^ðiÞ₀ ÞN^ðiÞð2Þ_{ðbg recÞ} ϵ^ð3Þ_B ðv^ðiÞ₀ ÞN^ðiÞð3Þ_{ðbg recÞ}

1 CC CA;

ð8Þ where eachIis a background distribution integral with the superscript defining the subregion and the subscript iden- tifying the background component,N^ð_{ðbg recÞ}^iÞðjÞ is the number of reconstructed data events in the jth background sub- region, andϵ^ðjÞ_Bðv^ðiÞ₀ Þis the efficiency of the requirement on the MLP output for the background events in the jth background subregion.

Four parameters perDchannel are determined from the global selection requirement optimization. They include the definition of the signal region (M^ðiÞ_J=ψ,M^ðiÞ_D, andM^ðiÞrec) and the MLP output cutoff valuev^ðiÞ₀ . The MLP output require- ment is not used for the two low-background D decay channels (D^þ →K⁻π^þπ^þandD⁰→K⁻π^þ) because it has no effect.

The resulting signal region definition parameters vary in the range 33–65, 8–17, and38–57MeV=c²forM^ðiÞ_J=ψ,M^ðiÞ_D, and M^ðiÞrec, respectively. The finally selected data sample containsN_obs ¼103events.

After the global optimization, the background fits are performed again; only events passing the resulting MLP output selection requirementv > v^ðiÞ₀ are used. The weights of the background events for allDchannels are determined as the coefficients of the numbers of background events in the corresponding background region in Eq.(8)[since the MLP output requirement has already been applied, ϵ^ðjÞ_B ðv^ðiÞ₀ Þ ¼1]. The background event weights are then used to calculate the background distribution in M_rec. The resulting M_rec signal and background distributions are shown in Fig.4.

The expected number of background events in the signal region is N_bg¼24.91.11.6. The statistical error is calculated from the event weights; it is found to be 4.5%.

The systematic error is determined to be 6.3%, and it includes the error due to the difference of the expected number of the background events in the training and testing samples and the statistical error of the background PDF.

The overall error, including both statistical and systematic errors, is 7.7%. The expected fraction of the back- ground events in the signal region is b₀¼N_bg=N_obs¼ 0.240.03.

V. AMPLITUDE ANALYSIS FORMALISM We perform an amplitude analysis in a six-dimensional parameter space,

Φ¼ ðM_DD¯;θprod;θJ=ψ;θX;φl⁻;φDÞ; ð9Þ

, GeV/c2 ψ

MJ/

2.8 2.9 3 3.1 3.2 3.3

2 , GeV/cDM

1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2

(1) (1)

(1) (1)

(3) (3)

(2) (4) (2)

FIG. 3. Definition of the background subregions: (1) is the background region where bothDandJ=ψmesons are fake, (2) is the real D region, and (3) is the real J=ψ region. The central rectangle (4) is excluded from the background region as described in Sec.III.

, GeV/c2

Mrec

1.6 1.7 1.8 1.9 2 2.1 2.2

2 Events / 10 MeV/c

0 5 10 15 20 25

FIG. 4. The signal and background distributions inM_rec. The points with error bars are the data with the channel-dependent signal region selection requirements defined by Eq.(4)and MLP output requirements applied. The hatched histogram is the background. Note that the definition of the signal regionM_rec requirement depends on theDdecay channel.

c0

where θprod is the production angle; θJ=ψ andθX are the J=ψ andX helicity angles, respectively; andφ_l⁻ andφD

are thel⁻ andDazimuthal angles, respectively. The exact definitions of the angles are given in the Appendix. In general, the number of dimensions for the processe^þe⁻→ l^þl⁻DD¯ is

3N_f−N_c−N_r¼7; ð10Þ where N_f¼4 is the number of the final-state particles, N_c ¼4is the number of conservation relations, andN_r¼1 is the number of rotations around the beam axis. Since the leptons are produced in the decay of the narrowJ=ψ state, their invariant mass can be ignored, thereby reducing the number of dimensions by one.

The amplitude of the process e^þe⁻ →J=ψDD¯ is rep- resented by a sum of individual contributions. The default amplitude model includes a nonresonant amplitude and a new resonance X in the DD¯ system with J^PC¼0^þþ or 2^þþ. Since the X is a resonance in the DD¯ system, its quantum numbers are in the“normal”series [P¼ ð−1Þ^J];

thus, the asterisk is added, as for the mesons with open flavor. In the case of one-photon production, odd values of theXspin are forbidden byC-parity conservation because the DD¯ pair has C¼ ð−1Þ^JðXÞ.

The X signal is described by the Breit-Wigner amplitude,

A_XðMÞ ¼

pðMÞ pðmÞ

_L

F_LðMÞ

m²−M²−imΓðMÞ; ð11Þ wheremis the resonance mass,Mis the invariant mass of its daughters,pðMÞis the momentum of a daughter particle in the rest frame of the mother particle calculated at the mother’s mass M, L is the decay angular momentum, F_LðMÞis the Blatt-Weisskopf form factor, andΓðMÞis the mass-dependent width. The Blatt-Weisskopf form factor is given by [34]

F₀ðMÞ ¼1;

F₂ðMÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z²₀þ3z₀þ9 z⁹þ3zþ9 s

; ð12Þ

where z¼ ðpðMÞrÞ² (r being the hadron scale) and z₀¼ ðpðmÞrÞ². The mass-dependent width is given by

ΓðMÞ ¼Γ pðMÞ

pðmÞ _2Lþ1

m

MF²_LðMÞ; ð13Þ where Γis the resonance width.

The nonresonant amplitude is given by A_NRðM_DD¯Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

F_DD¯ðM_DD¯Þ

p ; ð14Þ

whereF_DD¯ðMÞis the nonresonant-amplitude form factor.

Three different parametrizations are used forF_DD¯ðMÞ. The default choice is a constant nonresonant amplitude:

F_DD¯ðM_DD¯Þ ¼1: ð15Þ Another parametrization is based on Ref. [35]. The matrix element for the processe^þe⁻ →ψχc is integrated over the angle, andM_DD¯ is used instead of theχcmass. The result is

F_DD¯ðM_DD¯Þ ¼ 2048 3s⁷m²_c

180224m¹⁰_c þ149504m⁸_cA þm⁶_c

44928A²þ1792 3 B

þm⁴_c

5376A³þ928 3 AB

þm²_cð222A⁴þ34A²BÞ þ

A⁵ 2 −A³B

6

; ð16Þ

where

A¼m²_J=ψþM²_DD¯ −s ð17Þ and

B¼s²−2sðm²_J=ψ þM²

DD¯Þ þ ðm²_J=ψ −M²

DD¯Þ² ð18Þ with ffiffiffi

ps

¼m_ϒð4SÞ and m_c ¼1.5GeV=c². This nonreso- nant-amplitude model is denoted hereinafter as the non- relativistic quantum chromodynamics (NRQCD) model (although, for the actual NRQCD calculation of the e^þe⁻ →ψχccross section, theχcmass is set to2m_c[35]).

The third parametrization is based on Ref. [36]. The matrix element for the processe^þe⁻ →J=ψDD¯ is propor- tional to M⁻⁴_DD¯ for 4m²_D≪M²_DD¯ ≪s; this dependence is used for the entire fitting region:

F_DD¯ðM_DD¯Þ ¼ 1

M⁴_DD¯: ð19Þ This amplitude model is identified below as the M⁻⁴_DD¯ nonresonant amplitude.

In the determination of systematic errors, we also use a model without a nonresonant contribution but with an additional form factor for the resonant amplitude[37],

A_XðMÞ→A_XðMÞ½1þβðM_DD¯ −2m_D0Þ; ð20Þ where βis a form-factor parameter.

The angle-dependent part of the amplitude is calculated using the helicity formalism (see the Appendix); the result is

A_{λbeamλJ=ψλXλll}ðΦÞ¼H_λJ=ψλXd¹_λ

beamλJ=ψ−λXðθprodÞe^{iλJ=ψφl−}

×d¹_λ

J=ψλ_llðθ_J=ψÞe^iλXφDd^JðX_λ ^Þ

X0ðθXÞ; ð21Þ and

A_λbeamλllðΦÞ ¼ X

λJ=ψ¼−1;0;1 λX¼−JðXÞ;…;JðXÞ

A_{λbeamλJ=ψλXλll}ðΦÞ; ð22Þ

where H_λJ=ψλX is the production helicity amplitude, d^jm₁m₂ðθÞ are Wigner d-functions, λ is the helicity of the particle specified by the index,λbeamis the sum of helicities of the beam electron and positron, and λ_ll is the sum of helicities of theJ=ψ decay products.

The signal density function is SðΦÞ ¼ X

λbeam¼−1;1 λll¼−1;1

jX

X

A_λbeamλllðΦÞA_XðM_DD¯Þj²; ð23Þ

whereXis any contribution to the signal (Xresonance or nonresonant amplitude). The resolution in M_DD¯ may be taken into account by substituting

SðΦÞ→ X^ng

ig¼−ng

R_MDD¯ðM_DD¯; i_gΔM_DD¯ÞΔM_DD¯

×SðM_DD¯ þi_gΔM_DD¯;…Þ; ð24Þ whereR_MD¯Dis the resolution inM_DD¯,ΔM_DD¯ is the distance between grid points, and i_g is the grid point index. The resolution in M_DD¯ is ∼12MeV=c² for an M_DD¯ range between 5 and 6GeV=c²; it is smaller for lower DD¯ masses. The resolution is much smaller than the width of theM_DD¯ structure (see Fig.6) and can be ignored for the default fit; nevertheless, it is taken into account since alternative models with relatively narrow states Xð3915Þ or χc2ð2PÞ are considered.

The construction of the likelihood function follows Ref. [38]. The function to be minimized is

F¼−2X

k

ln

ð1−bÞPSðΦkÞ

lSðΦlÞþbPBðΦkÞ

lBðΦlÞ

þðb−b₀Þ²

σ²_b0 ; ð25Þ

where b is the fraction of the background events, BðΦÞ is the background density in the signal region deter- mined from the fit to the background, b₀ is the expected background fraction, and σ²_b0 is its error. The index k runs over data events; the index l runs over MC events generated uniformly over the phase space and recon- structed using the same selection requirements as in data.

This procedure takes into account the nonuniformity of the reconstruction efficiency but requires a parametriza- tion of the background shape. For the background, the likelihood is

F¼−2X

k

w_kln

PBðΦkÞ

lBðΦlÞ

; ð26Þ where w_k is the weight of the kth background event, calculated as described in Sec. IV C.

VI. FIT TO THE DATA A. Fit to the background The background density function is given by BðΦÞ ¼P₂ðΦÞðexpð−ξ½pðM_DD¯Þ²Þ þηÞ½pðM_DD¯Þ^−ζ

½1−u₁ðcos²θprodÞ^v1½1−u₂ðcos²θJ=ψÞ^v2 ; ð27Þ whereP₂ðΦÞis a six-dimensional second-order polynomial and u₁, u₂, v₁, v₂, ξ, η, and ζ are coefficients. The projections of the fit results onto the individual variables are shown in Fig.5.

B. Fit to the signal

The results of the fit to the signal events in the default model are listed in Table I for two X quantum-number hypotheses: J^PC¼0^þþ and J^PC¼2^þþ. There are three solutions for the2^þþ hypothesis; all solutions have very similar likelihood values (jΔð−2lnLÞj<1). A signifi- cant resonance is observed for both the 0^þþ and 2^þþ hypotheses; the 0^þþ hypothesis is preferred. The sig- nificance is calculated using Wilks’s theorem [39] with 6 and 12 degrees of freedom for the 0^þþ and 2^þþ hypotheses, respectively. The global significance calcu- lated using the method described in Ref.[40] is 8.5σ for the 0^þþ hypothesis. Thus, we observe a new charmo- niumlike state that is referred to hereinafter as the Xð3860Þ. The projections of the fit results onto M_DD¯

and the angular variables for the case of the Xð3860Þ withJ^PC¼0^þþ are shown in Fig.6, and the amplitudes are shown in Table II. The helicity amplitudes H₁₀ and H₀₀ almost exactly coincide (with large error). This is consistent with pureS-wave production of the Xð3860Þ [see Eq. (A15)].

c0