Observation of a resonant structure in e + e − → K + K − π 0 π 0
M. Ablikim, M. N. Achasov, P. Adlarson, S. Ahmed, M. Albrecht, A. Amoroso, Q. An, Anita, Y. Bai, O. Bakina, R. Baldini Ferroli, I. Balossino, Y. Ban, K. Begzsuren, J. V. Bennett, N. Berger, M. Bertani, D. Bettoni, F. Bianchi, J. Biernat, J. Bloms, I. Boyko, R. A. Briere, H. Cai, X. Cai, A. Calcaterra, G. F. Cao, N. Cao, S. A. Cetin, J. F. Chang, W. L. Chang, G. Chelkov, D. Y. Chen, G. Chen, H. S. Chen, M. L. Chen, S. J. Chen, X. R. Chen, Y. B. Chen, W. Cheng, G. Cibinetto, F. Cossio, X. F. Cui, H.L.Dai, J. P. Dai, X. C. Dai, A. Dbeyssi, D. Dedovich, Z. Y. Deng, A. Denig, I. Denysenko, M. Destefanis, F. De Mori, Y. Ding, C. Dong, J. Dong, L. Y. Dong, M. Y. Dong, S. X. Du, J. Fang, S. S. Fang, Y. Fang, R. Farinelli, L. Fava, F. Feldbauer, G. Felici, C. Q. Feng, M. Fritsch, C. D. Fu, Y. Fu, X. L. Gao, Y. Gao, Y. Gao, Y. G. Gao, I. Garzia, E. M. Gersabeck, A. Gilman, K. Goetzen, L. Gong, W. X. Gong, W. Gradl, M. Greco, L. M. Gu, M. H. Gu, S. Gu, Y. T. Gu, C. Y Guan, A. Q. Guo, L. B. Guo, R. P. Guo, Y. P. Guo, Y. P. Guo, A. Guskov, S. Han, T. T. Han, T. Z. Han, X. Q. Hao, F. A. Harris, K. L. He, F. H. Heinsius, et al. (406 additional authors not shown)
aa r X i v : . [ h e p - e x ] J a n Observation of a resonant structure in e + e − → K + K − π π M. Ablikim , M. N. Achasov ,e , P. Adlarson , S. Ahmed , M. Albrecht , A. Amoroso A, C , Q. An , , Anita ,Y. Bai , O. Bakina , R. Baldini Ferroli A , I. Balossino A , Y. Ban ,m , K. Begzsuren , J. V. Bennett , N. Berger ,M. Bertani A , D. Bettoni A , F. Bianchi A, C , J Biernat , J. Bloms , I. Boyko , R. A. Briere , H. Cai , X. Cai , ,A. Calcaterra A , G. F. Cao , , N. Cao , , S. A. Cetin B , J. F. Chang , , W. L. Chang , , G. Chelkov ,c,d , D. Y. Chen ,G. Chen , H. S. Chen , , M. L. Chen , , S. J. Chen , X. R. Chen , Y. B. Chen , , W. Cheng C , G. Cibinetto A ,F. Cossio C , X. F. Cui , H. L. Dai , , J. P. Dai ,i , X. C. Dai , , A. Dbeyssi , D. Dedovich , Z. Y. Deng ,A. Denig , I. Denysenko , M. Destefanis A, C , F. De Mori A, C , Y. Ding , C. Dong , J. Dong , , L. Y. Dong , ,M. Y. Dong , , , S. X. Du , J. Fang , , S. S. Fang , , Y. Fang , R. Farinelli A, B , L. Fava B, C , F. Feldbauer ,G. Felici A , C. Q. Feng , , M. Fritsch , C. D. Fu , Y. Fu , X. L. Gao , , Y. Gao ,m , Y. Gao , Y. G. Gao ,I. Garzia A, B , E. M. Gersabeck , A. Gilman , K. Goetzen , L. Gong , W. X. Gong , , W. Gradl , M. Greco A, C ,L. M. Gu , M. H. Gu , , S. Gu , Y. T. Gu , C. Y Guan , , A. Q. Guo , L. B. Guo , R. P. Guo , Y. P. Guo ,Y. P. Guo ,j , A. Guskov , S. Han , T. T. Han , T. Z. Han ,j , X. Q. Hao , F. A. Harris , K. L. He , , F. H. Heinsius ,T. Held , Y. K. Heng , , , M. Himmelreich ,h , T. Holtmann , Y. R. Hou , Z. L. Hou , H. M. Hu , , J. F. Hu ,i ,T. Hu , , , Y. Hu , G. S. Huang , , L. Q. Huang , X. T. Huang , N. Huesken , T. Hussain , W. IkegamiAndersson , W. Imoehl , M. Irshad , , S. Jaeger , S. Janchiv ,l , Q. Ji , Q. P. Ji , X. B. Ji , , X. L. Ji , ,H. B. Jiang , X. S. Jiang , , , X. Y. Jiang , J. B. Jiao , Z. Jiao , D. P. Jin , , , S. Jin , Y. Jin , T. Johansson ,N. Kalantar-Nayestanaki , X. S. Kang , R. Kappert , M. Kavatsyuk , B. C. Ke , , I. K. Keshk , A. Khoukaz , P.Kiese , R. Kiuchi , R. Kliemt , L. Koch , O. B. Kolcu B,g , B. Kopf , M. Kuemmel , M. Kuessner , A. Kupsc ,M. G. Kurth , , W. K¨uhn , J. J. Lane , J. S. Lange , P. Larin , L. Lavezzi C , H. Leithoff , M. Lellmann , T. Lenz ,C. Li , C. H. Li , Cheng Li , , D. M. Li , F. Li , , G. Li , H. B. Li , , H. J. Li ,j , J. C. Li , J. L. Li , Ke Li ,L. K. Li , Lei Li , P. L. Li , , P. R. Li , S. Y. Li , W. D. Li , , W. G. Li , X. H. Li , , X. L. Li , X. N. Li , ,Z. B. Li , Z. Y. Li , H. Liang , , H. Liang , , Y. F. Liang , Y. T. Liang , L. Z. Liao , , J. Libby , C. X. Lin ,D. X. Lin , B. Liu ,i , B. J. Liu , C. X. Liu , D. Liu , , D. Y. Liu ,i , F. H. Liu , Fang Liu , Feng Liu , H. B. Liu ,H. M. Liu , , Huanhuan Liu , Huihui Liu , J. B. Liu , , J. Y. Liu , , K. Liu , K. Y. Liu , Ke Liu , L. Liu , ,L. Y. Liu , Q. Liu , S. B. Liu , , Shuai Liu , T. Liu , , X. Liu , X. Y. Liu , , Y. B. Liu , Z. A. Liu , , ,Z. Q. Liu , Y. F. Long ,m , X. C. Lou , , , H. J. Lu , J. D. Lu , , J. G. Lu , , X. L. Lu , Y. Lu , Y. P. Lu , ,C. L. Luo , M. X. Luo , P. W. Luo , T. Luo ,j , X. L. Luo , , S. Lusso C , X. R. Lyu , F. C. Ma , H. L. Ma , L. L.Ma , M. M. Ma , , Q. M. Ma , R. Q. Ma , , R. T. Ma , X. N. Ma , X. X. Ma , , X. Y. Ma , , Y. M. Ma ,F. E. Maas , M. Maggiora A, C , S. Maldaner , S. Malde , Q. A. Malik , A. Mangoni B , Y. J. Mao ,m , Z. P. Mao ,S. Marcello A, C , Z. X. Meng , J. G. Messchendorp , G. Mezzadri A , J. Min , , T. J. Min , R. E. Mitchell ,X. H. Mo , , , Y. J. Mo , C. Morales Morales , N. Yu. Muchnoi ,e , H. Muramatsu , S. Nakhoul ,h , Y. Nefedov ,F. Nerling ,h , I. B. Nikolaev ,e , Z. Ning , , S. Nisar ,k , S. L. Olsen , Q. Ouyang , , , S. Pacetti B , X. Pan , Y. Pan ,M. Papenbrock , A. Pathak , P. Patteri A , M. Pelizaeus , H. P. Peng , , K. Peters ,h , J. Pettersson , J. L. Ping ,R. G. Ping , , A. Pitka , R. Poling , V. Prasad , , H. Qi , , H. R. Qi , M. Qi , T. Y. Qi , S. Qian , , C. F. Qiao ,L. Q. Qin , X. P. Qin , X. S. Qin , Z. H. Qin , , J. F. Qiu , S. Q. Qu , K. H. Rashid , K. Ravindran , C. F. Redmer ,M. Richter , A. Rivetti C , V. Rodin , M. Rolo C , G. Rong , , Ch. Rosner , M. Rump , A. Sarantsev ,f , M. Savri´e B ,Y. Schelhaas , C. Schnier , K. Schoenning , D. C. Shan , W. Shan , X. Y. Shan , , M. Shao , , C. P. Shen ,P. X. Shen , X. Y. Shen , , H. Y. Sheng , H. C. Shi , , R. S. Shi , , X. Shi , , X. D Shi , , J. J. Song ,Q. Q. Song , , X. Y. Song , Y. X. Song ,m , S. Sosio A, C , C. Sowa , S. Spataro A, C , F. F. Sui , G. X. Sun ,J. F. Sun , L. Sun , S. S. Sun , , T. Sun , , W. Y. Sun , Y. J. Sun , , Y. K Sun , , Y. Z. Sun , Z. J. Sun , ,Z. T. Sun , Y. X. Tan , , C. J. Tang , G. Y. Tang , J. Tang , X. Tang , V. Thoren , B. Tsednee , I. Uman D ,B. Wang , B. L. Wang , C. W. Wang , D. Y. Wang ,m , H. P. Wang , , K. Wang , , L. L. Wang , L. S. Wang ,M. Wang , M. Z. Wang ,m , Meng Wang , , P. L. Wang , W. P. Wang , , X. Wang ,m , X. F. Wang , X. L. Wang ,j ,Y. Wang , , Y. Wang , Y. D. Wang , Y. F. Wang , , , Y. Q. Wang , Z. Wang , , Z. G. Wang , , Z. Y. Wang ,Ziyi Wang , Zongyuan Wang , , T. Weber , D. H. Wei , P. Weidenkaff , F. Weidner , H. W. Wen ,a , S. P. Wen ,D. J. White , U. Wiedner , G. Wilkinson , M. Wolke , L. Wollenberg , J. F. Wu , , L. H. Wu , L. J. Wu , , X. Wu ,j ,Z. Wu , , L. Xia , , H. Xiao ,j , S. Y. Xiao , Y. J. Xiao , , Z. J. Xiao , Y. G. Xie , , Y. H. Xie , T. Y. Xing , ,X. A. Xiong , , G. F. Xu , J. J. Xu , Q. J. Xu , W. Xu , , X. P. Xu , L. Yan A, C , L. Yan ,j , W. B. Yan , ,W. C. Yan , Xu Yan , H. J. Yang ,i , H. X. Yang , L. Yang , R. X. Yang , , S. L. Yang , , Y. H. Yang , Y. X. Yang ,Yifan Yang , , Zhi Yang , M. Ye , , M. H. Ye , J. H. Yin , Z. Y. You , B. X. Yu , , , C. X. Yu , G. Yu , ,J. S. Yu ,n , T. Yu , C. Z. Yuan , , W. Yuan A, C , X. Q. Yuan ,m , Y. Yuan , C. X. Yue , A. Yuncu B,b , A. A. Zafar ,Y. Zeng ,n , B. X. Zhang , B. Y. Zhang , , C. C. Zhang , D. H. Zhang , Guangyi Zhang , H. H. Zhang , H. Y. Zhang , ,J. L. Zhang , J. Q. Zhang , J. W. Zhang , , , J. Y. Zhang , J. Z. Zhang , , Jianyu Zhang , , Jiawei Zhang , , L. Zhang ,Lei Zhang , S. Zhang , S. F. Zhang , T. J. Zhang ,i , X. Y. Zhang , Y. Zhang , Y. H. Zhang , , Y. T. Zhang , ,Yan Zhang , , Yao Zhang , Yi Zhang ,j , Z. H. Zhang , Z. Y. Zhang , G. Zhao , J. Zhao , J. W. Zhao , , J. Y. Zhao , ,J. Z. Zhao , , Lei Zhao , , Ling Zhao , M. G. Zhao , Q. Zhao , S. J. Zhao , T. C. Zhao , Y. B. Zhao , , Z. G. Zhao , ,A. Zhemchugov ,c , B. Zheng , J. P. Zheng , , Y. Zheng ,m , Y. H. Zheng , B. Zhong , C. Zhong , L. Zhou , ,L. P. Zhou , , Q. Zhou , , X. Zhou , X. K. Zhou , X. R. Zhou , , A. N. Zhu , , J. Zhu , K. Zhu , K. J. Zhu , , ,S. H. Zhu , W. J. Zhu , X. L. Zhu , Y. C. Zhu , , Y. S. Zhu , , Z. A. Zhu , , J. Zhuang , , B. S. Zou , J. H. Zou (BESIII Collaboration) Institute of High Energy Physics, Beijing 100049, People’s Republic of China Beihang University, Beijing 100191, People’s Republic of China Beijing Institute of Petrochemical Technology, Beijing 102617, People’s Republic of China Bochum Ruhr-University, D-44780 Bochum, Germany Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA Central China Normal University, Wuhan 430079, People’s Republic of China China Center of Advanced Science and Technology, Beijing 100190, People’s Republic of China COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, 54000 Lahore, Pakistan Fudan University, Shanghai 200443, People’s Republic of China G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090, Russia GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany Guangxi Normal University, Guilin 541004, People’s Republic of China Guangxi University, Nanning 530004, People’s Republic of China Hangzhou Normal University, Hangzhou 310036, People’s Republic of China Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany Henan Normal University, Xinxiang 453007, People’s Republic of China Henan University of Science and Technology, Luoyang 471003, People’s Republic of China Huangshan College, Huangshan 245000, People’s Republic of China Hunan Normal University, Changsha 410081, People’s Republic of China Hunan University, Changsha 410082, People’s Republic of China Indian Institute of Technology Madras, Chennai 600036, India Indiana University, Bloomington, Indiana 47405, USA (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati,Italy; (B)INFN and University of Perugia, I-06100, Perugia, Italy (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of Ferrara, I-44122, Ferrara, Italy Institute of Modern Physics, Lanzhou 730000, People’s Republic of China Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330, Mongolia Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands Lanzhou University, Lanzhou 730000, People’s Republic of China Liaoning Normal University, Dalian 116029, People’s Republic of China Liaoning University, Shenyang 110036, People’s Republic of China Nanjing Normal University, Nanjing 210023, People’s Republic of China Nanjing University, Nanjing 210093, People’s Republic of China Nankai University, Tianjin 300071, People’s Republic of China Peking University, Beijing 100871, People’s Republic of China Qufu Normal University, Qufu 273165, People’s Republic of China Shandong Normal University, Jinan 250014, People’s Republic of China Shandong University, Jinan 250100, People’s Republic of China Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China Shanxi Normal University, Linfen 041004, People’s Republic of China Shanxi University, Taiyuan 030006, People’s Republic of China Sichuan University, Chengdu 610064, People’s Republic of China Soochow University, Suzhou 215006, People’s Republic of China Southeast University, Nanjing 211100, People’s Republic of China State Key Laboratory of Particle Detection and Electronics, Beijing 100049, Hefei 230026, People’s Republic of China Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China Tsinghua University, Beijing 100084, People’s Republic of China (A)Ankara University, 06100 Tandogan, Ankara, Turkey; (B)Istanbul Bilgi University, 34060 Eyup, Istanbul, Turkey;(C)Uludag University, 16059 Bursa, Turkey; (D)Near East University, Nicosia, North Cyprus, Mersin 10, Turkey University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China University of Hawaii, Honolulu, Hawaii 96822, USA University of Jinan, Jinan 250022, People’s Republic of China University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom University of Minnesota, Minneapolis, Minnesota 55455, USA University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany University of Oxford, Keble Rd, Oxford, UK OX13RH University of Science and Technology Liaoning, Anshan 114051, People’s Republic of China University of Science and Technology of China, Hefei 230026, People’s Republic of China University of South China, Hengyang 421001, People’s Republic of China University of the Punjab, Lahore-54590, Pakistan (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy Uppsala University, Box 516, SE-75120 Uppsala, Sweden Wuhan University, Wuhan 430072, People’s Republic of China Xinyang Normal University, Xinyang 464000, People’s Republic of China Zhejiang University, Hangzhou 310027, People’s Republic of China Zhengzhou University, Zhengzhou 450001, People’s Republic of China a Also at Ankara University,06100 Tandogan, Ankara, Turkey b Also at Bogazici University, 34342 Istanbul, Turkey c Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia d Also at the Functional Electronics Laboratory, Tomsk State University, Tomsk, 634050, Russia e Also at the Novosibirsk State University, Novosibirsk, 630090, Russia f Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia g Also at Istanbul Arel University, 34295 Istanbul, Turkey h Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany i Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education; Shanghai Key Laboratoryfor Particle Physics and Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240, People’s Republic of China j Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Instituteof Modern Physics, Fudan University, Shanghai 200443, People’s Republic of China k Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA l Currently at: Institute of Physics and Technology, Peace Ave.54B, Ulaanbaatar 13330, Mongolia m Also at State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, People’s Republic of China n School of Physics and Electronics, Hunan University, Changsha 410082, China (Dated: January 14, 2020)Using e + e − collision data samples with center-of-mass energies ranging from 2.000 to 2.644 GeV,collected by the BESIII detector at the BEPCII collider, and with a total integrated luminosityof 300 pb − , a partial-wave analysis is performed for the process e + e − → K + K − π π . The totalBorn cross sections for the process e + e − → K + K − π π , as well as the Born cross sections for thesubprocesses e + e − → φπ π , K + (1460) K − , K +1 (1400) K − , K +1 (1270) K − and K ∗ + (892) K ∗− (892),are measured versus the center-of-mass energy. The corresponding results for e + e − → K + K − π π and φπ π are consistent with those of BaBar and have much improved precision. By analyz-ing the cross sections for the four subprocesses, K + (1460) K − , K +1 (1400) K − , K +1 (1270) K − and K ∗ + (892) K ∗− (892), a structure with mass M = (2126.5 ± ± c and width Γ =(106.9 ± ± σ , although withvery limited significance in the subprocesses e + e − → K +1 (1270) K − and K ∗ + (892) K ∗− (892). Theresonant parameters of the observed structure suggest it can be identified with the φ (2170), thusthe results provide valuable input to the internal nature of the φ (2170). The vector meson state Y (2175), denoted as the φ (2170) by the Particle Data Group (PDG) [1], is cur-rently one of the most interesting particles in lighthadron spectroscopy. The φ (2170) was first observedby BaBar [2] and subsequently confirmed by severalother experiments [3–7]. The internal constituents ofthe φ (2170) are still unknown, which has stimulated ex-tensive theoretical discussions. Possible interpretationsof the φ (2170) include a conventional 3 S or 2 D s ¯ s state [8–11], an s ¯ sg hybrid [9, 12, 13], a tetraquarkstate [14–17], a Λ ¯Λ( S ) bound state [18–20], or a φKK resonance state [21], etc., but no interpretation has yetbeen established. Each of these theoretical models canaccommodate a resonant state with parameters similar tothose of the φ (2170), but they predict significantly differ-ent partial widths for individual decay modes, especiallythe K ( ∗ ) K ( ∗ ) decay modes, where the K ( ∗ ) is the groundor excited state of a K meson with different spin-parities.Consequently, studying the decay modes of the φ (2170),and precisely measuring their partial widths, plays a keyrole in determining the internal structure of the φ (2170).The BESII collaboration searched for the decay φ (2170) → K ∗ (892) ¯ K ∗ (892) via J/ψ → ηφ (2170) byusing 58 million J/ψ events [22]. No significant sig-nal was observed. The BaBar collaboration performedan analysis of e + e − → K + K − π + π − and K + K − π π using 454 fb − data via the initial state radiation(ISR) process [2]. Beside clearly observing the pro-cess e + e − → φππ , abundant K ∗ structures were ob-served in the Kπ ( π ) invariant mass spectrum, such asthe K ∗ (892) and K ∗ (1430), as well as the K (1270) and K (1400). It is worth noting that only about 1% ofthe e + e − → K + K − π + π − events were from the sub-process e + e − → K ∗ (892) ¯ K ∗ (892), while roughly 30%of the e + e − → K + K − π π events were from e + e − → K ∗ + (892) K ∗− (892). A comprehensive analysis, e.g. apartial-wave analysis (PWA), is desired to resolve thecontribution of individual components in these decays.Besides an excited φ state, the quark model also pre-dicts excited ρ and ω states in the 2 GeV/ c massrange [23]. Finding this set of excited vector mesonswould help establish the corresponding ρ , ω , and φ mesonfamilies and would set a baseline for theoretical models.Since these excited vector mesons can each decay into K ( ∗ ) K ( ∗ ) final states, analyzing the K ( ∗ ) K ( ∗ ) invariantmass spectra in e + e − annihilation becomes an effectivemeans to discover them.In this Letter, we present a PWA of the process e + e − → K + K − π π using data collected with the BE-SIII detector. The ten data samples used in this analysishave center-of-mass (c.m.) energies ranging from 2.000to 2.644 GeV and have a total integrated luminosity of300 pb − . The c.m. energy values and integrated lumi-nosities of each data set are presented in Table I in thesupplemental material [24]. Charge-conjugated processesare always included by default.Detailed descriptions of the design and performance ofthe BESIII detector can be found in Ref. [25]. A MonteCarlo (MC) simulation based on Geant4 [26], includingthe geometric description of the BESIII detector and itsresponse, is used to optimize the event selection crite-ria, estimate backgrounds, and determine the detectionefficiency. The signal MC samples are generated usingthe package
ConExc [27], which incorporates a higher-order ISR correction. Background samples of the pro-cesses e + e − → e + e − , µ + µ − and γγ are generated withthe Babayaga [28] generator, while e + e − → hadronsand two photon events are generated by the Luarlw [29]and
Bestwogam [30] generators, respectively.The selection criteria for charged tracks, particle iden-tification (PID), and photon candidates are the same asthose in Ref. [31].The process e + e − → K + K − π π results in the finalstate K + K − γγγγ . Thus, candidate events with only twooppositely-charged kaons and at least four photons areselected. To improve the kinematic resolution and sup-press background, a six-constraint (6C) kinematic fit im-posing energy-momentum conservation, as well as twoadditional π mass constraints, is carried out under thehypothesis e + e − → K + K − π π . The combination withminimum χ C is retained for further analysis. The can-didate events are required to satisfy χ C <
80. After theabove selection criteria, detailed studies indicate that thebackgrounds are negligible.Using the GPUPWA framework [32], a PWA is per-formed on the surviving candidate events to disen-tangle the intermediate processes present in e + e − → K + K − π π . The quasi two-body decay amplitudes inthe sequential decays are constructed using covariant ten-sor amplitudes [33]. The intermediate states are param-eterized with relativistic Breit-Wigner (BW) functions,except for the f (980), which is described with a Flatt´eformula [34]. The resonance parameters of the f (980)and the wide resonance σ in the fit are fixed to those inRef. [34] and Ref. [34, 35], respectively, and those of otherintermediate states are fixed to PDG values, or measuredin the analysis. To include the resolution for the nar-row φ (1020) resonance, a Gaussian function is convolvedwith the BW function, but this is not done for the otherresonances. The relative magnitudes and phases of theindividual intermediate processes are determined by per-forming an unbinned maximum likelihood fit using MI- NUIT [36].We start the fit procedure by including all possible in-termediate states in the PDG that conserve J PC , wherethese intermediate states can decay into K + K − , π π , K ± π , K + K − π or K ± π π final states. Then we ex-amine the statistical significance of the individual am-plitudes, and drop the ones with statistical significanceless than 5 σ . The process is repeated until no ampli-tude remains with a statistical significance less than 5 σ .After that, all the removed processes are reintroduced in-dividually to make sure that they are not needed in thefit. In the above approach, the statistical significance ofeach individual amplitude is determined by the changesin the negative log likelihood (NLL) value and the num-ber of free parameters in the fit with and without thecorresponding amplitude included.The above strategy is performed individually onthe data sets at √ s = 2 .
125 and 2.396 GeV,which have the largest luminosities among the tendata sets. The nominal solution for data at √ s = 2 .
125 GeV includes the two-body decayprocesses K + (1460) K − , K +1 (1270) K − , K +1 (1400) K − , K ∗ + (892) K ∗− (892), K ∗ +0 (1430) K ∗− (892), φ (1020) σ , φ (1020) f (980), φ (1020) f (1270) and ω (1420) π , aswell as the three-body decay processes K + K − σ , K + K − f (980) and K + K − f (1370). For the dataat √ s = 2 .
396 GeV, the additional intermedi-ate processes K ∗ +2 (1430) K ∗− (892), K ∗ + (892) K − π and φ (1020) f (1370) are included, but without the φ (1020) σ and φ (1020) f (1270) processes. An interesting decaymode K ∗ + (1410) K − , which is expected to have a sizeabledecay rate for a conventional 3 S s ¯ s state [9], is found tobe less than 3 σ in both data samples. In the above, thethree-body decays are treated as consecutive quasi two-body decays with a very broad resonance decaying into K + K − or K + π and modeled as a 1 − phase space distri-bution. The intermediate states K + (1460), K +1 (1270), K +1 (1400) decay into K ∗ + (892) π , and ω (1420) decaysinto K ∗± (892) K ∓ , followed by K ∗ + (892) → K + π . Thestate K ∗ +0 (1430) decays into K + π . The state φ (1020)decays into K + K − and σ , f (980), f (1270), f (1370)decay into π π . The masses and widths of the K (1460), K (1400), K (1270) and ω (1420) in the fit are deter-mined by scanning the likelihood value, and the resultsare consistent with the parameters in the PDG. Themasses and widths of other intermediate states are fixedto PDG values. The statistical significance of all inter-mediate processes are summarized in sections II and IIIof the supplemental material [24], respectively. The cor-responding comparison of invariant mass spectra and an-gular distributions between data and MC projections areshown in section IV of the supplemental material.For the other eight data samples, due to limited statis-tics, we do not perform the above optimization strategyto determine which intermediate processes to include. In-stead, we use the same intermediate processes as the datasets with nearby c.m. energy. The data sets with √ s =2.000, 2.100, 2.175, 2.200 and 2.232 GeV (referred to asgroup I data), use the same processes as √ s = 2 .
125 GeV,while the other three points (group II data) use the sameprocesses as √ s = 2 .
396 GeV.The total Born cross sections for e + e − → K + K − π π and the Born cross sections for the intermediate processesare obtained at each c.m. energy using: σ B = N sig L int | − Π | (1 + δ ) r B r ǫ , (1)where N sig is the corresponding signal yield, and is deter-mined by calculating the fraction according to the PWAresults for the individual intermediate process; L int isthe integrated luminosity; (1 + δ ) r is the ISR correc-tion factor obtained from a QED calculation [27, 37] andincorporating the input cross section from this analy-sis iteratively; | − Π | is the vacuum polarization factortaken from a QED calculation [38]; ǫ is the detectionefficiency obtained from a PWA-weighted MC sample;and B r is the product of branching ratios of the inter-mediate states as quoted in the PDG [1]. In the de-cay e + e − → K + (1460) K − , the branching fraction of K (1460) → K ∗ (892) π is included in the measured crosssection since it has never been measured.Two categories of systematic uncertainties are con-sidered in the measurement of the Born cross sections.The first category includes uncertainties associated withthe luminosity, track detection, PID, kinematic fit, ISRcorrection, and the branching fractions of intermediatestates. The uncertainty associated with the integratedluminosity is 1% at each energy point [39]. The uncer-tainty of the detection efficiency is 1% for each chargedtrack [40] and photon [41]. The PID efficiency uncer-tainty is 1.0% for each charged track [40]. The uncer-tainty related to the kinematic fit is estimated by correct-ing the helix parameters of the simulated charged tracksto match the resolution [42]. The uncertainty associatedwith the ISR correction factor is estimated to be the dif-ference of (1 + δ r ) ǫ between the last two iterations in thecross section measurement. The systematic uncertain-ties from the branching ratios of intermediate states inthe subsequent decays are taken from the PDG [1]. Thesecond category of uncertainties are from the PWA fitprocedure. Fits with alternative scenarios are performed,and the changes of signal yields are taken as systematicuncertainties. Uncertainties from the BW parameteriza-tion are estimated by replacing the constant-width BWwith the mass-dependent width. Uncertainties associatedwith the resonance parameters, which are taken from thePDG and fixed in the fit, are estimated by alternative fitssuperposing additional constraints on these resonance pa-rameters, where the superposed constraints follow Gaus-sian distributions with widths equal to their uncertain-ties. One thousand fits are performed, and the resultantstandard deviations of the signal yields are taken as sys-tematic uncertainties. Uncertainties associated with theadditional resonances are estimated by alternative fits in-cluding the components K ∗ (1410) K or the K ∗ (1430) K ∗ , which are most significant, but less than 5 σ . Uncertain-ties due to the barrier factor are estimated by varyingthe radius of the centrifugal barrier from 0.7 to 1.0 fm.To estimate the uncertainties on the detection efficiencyrelated to the fit parameters in the PWA, one hundredMC samples are generated with five hundred groups ofparameters of PWA amplitudes which is sampled from amulti-variable Gaussian function according to their meanvalues and their covariance error matrix from the nomi-nal fit. The standard deviations of the resultant detectionefficiencies are considered as the uncertainties.In the above procedure, the uncertainties associ-ated with the barrier factor, resonance parameteriza-tion and additional resonances are strongly affected bythe statistics. Thus, those uncertainties of data with √ s =2.125 GeV are assigned to the group I data, whilethose of data with √ s =2.396 GeV are assigned to thegroup II data. Assuming all sources of systematic uncer-tainties are independent, the total uncertainties are thequadratic sums of the individual values, shown in sectionV of the supplemental material [24], where the sources ofthe uncertainties tagged with ‘*’ are assumed to be 100%correlated among each energy points.The measured total Born cross sections for e + e − → K + K − π π and the Born cross sections for the sub-process e + e − → φπ π , summing over all the π π in-termediate processes and their interferences, are shownin Fig. 1. Good agreement is found with the previ-ous results from BaBar. In order to study the proper-ties of 1 −− states, the cross sections for the processes e + e − → K + (1460) K − , K +1 (1400) K − , K +1 (1270) K − and K ∗ + (892) K ∗− (892), referred to as the KK processes, areshown in Fig. 2. A clear peak between 2.1 and 2.2 GeVis present in the process e + e − → K + (1460) K − , and dipsare observed for the processes e + e − → K +1 (1400) K − and K +1 (1270) K − in almost the same energy region. Thismay be due to destructive interference between differentcomponents. No obvious structure or dip is present inthe process e + e − → K ∗ + (892) K ∗− (892). All the variousnumbers used in the cross section calculation are sum-marized in section I of the supplemental material [24]. (GeV)s ) ( nb ) π π - K + K → - e + ( e σ (a) BaBarBESIII (GeV)s ) ( nb ) π π φ → - e + ( e σ (b) BaBarBESIII
FIG. 1. The Born cross sections for (a) the process e + e − → K + K − π π and (b) the subprocess e + e − → φπ π . Thered squares are from this analysis; the blue dots are from theBaBar experiment. To further examine the structure, a binned χ fit,incorporating the correlated and uncorrelated uncer-tainties among different energy points, is performed to (GeV)s − − ) ( nb ) - ( ) K + - > K - e + ( e σ (a) (GeV)s − − ) ( nb ) - ( ) K + - > K - e + ( e σ (b) (GeV)s − − ) ( nb ) - ( ) K + - > K - e + ( e σ (c) (GeV)s ( )) ( nb ) * - ( ) K * + - > K - e + ( e σ (d) FIG. 2. Fit to the cross sections for e + e − to the final states(a) K + (1460) K − , (b) K +1 (1400) K − , (c) K +1 (1270) K − and(d) K ∗ + (892) K ∗− (892), where black dots with errors aredata, the black solid curves are the overall fit results, the redlong-dashed curves are from the intermediate state, the greenshort-dashed curves are from the continuum component, andthe blue dash-dotted curves are the interference contributionfor solution 1. the cross sections for the K + (1460) K − , K +1 (1400) K − , K +1 (1270) K − and K ∗ + (892) K ∗− (892) processes. Thefit probability density function (PDF) for the individualprocesses is the coherent sum of a continuum component f and a resonant component f : A = f + e iφ f , (2)where φ is the relative phase between the two compo-nents. By considering phase space Φ( √ s ), the energy-dependent cross section of the QED process, and therelative orbital angular momentum L in the two-bodydecay, the amplitude f is described as f = q L p Φ( √ s ) s n , (3)where q is the momentum of the daughter particle. Theresonant amplitude f is described with a BW function, f = M R √ s q π B r Γ e + e − R Γ R s − M R + iM R Γ R ( qq ) L s Φ( √ s )Φ( M R ) , (4)where M R is the mass of the structure, Γ R is the totalwidth, Γ e + e − R is its partial width to e + e − , B r is the decaybranching fraction to a given final state, and q is themomenta of the daughter particle in the rest frame ofthe parent particle ( M R ).A simultaneous fit, assuming the same structureamong the K + (1460) K − , K +1 (1400) K − , K +1 (1270) K − and K ∗ + (892) K ∗− (892) processes, is performed to themeasured cross sections, as shown in Fig. 2. In the fit, M R and Γ R are shared parameters between thefour processes and are floated, while n , the production B r Γ e + e − R , and the relative phase angle φ are floated andfinal state dependent. For e + e − → K +1 (1270) K − and K +1 (1400) K − , L = 0, while L = 1 for the other twomodes. The fit results have two solutions with equalfit quality, identical M R = (2126 . ± .
8) MeV/ c andΓ R = (106 . ± .
1) MeV, but different B r Γ e + e − R and φ forthe processes e + e − → K +1 (1400) K − and K +1 (1270) K − ,as summarized in Table I. The statistical significance ofthe structure is estimated with the change of χ (∆ χ )and the number of degrees of freedom (∆ndof) betweenthe scenarios with and without the structure includedin the fit. The overall statistical significance is 6.3 σ ,obtained with ∆ χ =63.8 and ∆ndof=10. The signif-icance of the resonant state for each KK process isalso estimated and summarized in Table I. The signif-icances of the resonant state in the processes e + e − → K + (1460) K − and K +1 (1400) K − are greater than 4.5 σ ,while no significant signal is found in the other two pro-cesses. We also estimate the upper limit at the 90% con-fidence level on the production B r Γ e + e − R to be 1.9 eVfor e + e − → K ∗ + (892) K ∗− (892) and 12.5(297.6) eV for e + e − → K +1 (1270) K − . TABLE I. A summary of fit results.
Channel B r Γ e + e − R (eV) φ (rad) Sig. ( σ ) K + (1460) K − ± ± K +1 (1400) K − solution 1 4.7 ± ± ± ± K +1 (1270) K − solution 1 7.6 ± ± ± ± K ∗ + (892) K ∗− (892) 0.04 ± ± The systematic uncertainties on the resonant param-eters come from the absolute c.m. energy measurement,the measured cross section, and the fit procedure. Theuncertainty of the c.m. energy from BEPCII is small,and is ignored in the determination of the parameters ofthe structure. The statistical and systematic uncertain-ties of the measured cross section are incorporated in thefit, thus no further uncertainty is necessary. The uncer-tainties associated with the fit procedure include thosefrom the fit range and signal model. The uncertaintyfrom the fit range is investigated by excluding the lastenergy point √ s = 2 .
644 GeV in the fit. The resultantchanges, 5.1 MeV/ c for mass and 9.1 MeV for width, aretaken as the systematic uncertainties. To assess the sys-tematic uncertainty associated with the signal model, analternative BW function with energy-dependent width isimplemented in the fit, and results in differences of 11.3MeV/ c and 26.5 MeV for mass and width, respectively,which are taken as the systematic uncertainties. Theoverall systematic uncertainties are the quadratic sum ofthe individual ones, 12.4 MeV/ c and 28.1 MeV for themass and width, respectively.In summary, a PWA of the process e + e − → K + K − π π is performed for ten data samples withc.m. energies from 2.000 to 2.644 GeV and with an inte-grated luminosity of 300 pb − . The Born cross sectionsfor e + e − → K + K − π π and φπ π are obtained andare consistent with those from the BaBar experiment.We also measure the cross sections for the processes e + e − → K + (1460) K − , K +1 (1400) K − , K +1 (1270) K − ,and K ∗ + (892) K ∗− (892), individually, and perform a si-multaneous fit on the obtained results. The fit results in astructure with mass M = (2126.5 ± ± c ,width Γ = (106.9 ± ± σ , where the uncertainties are statisticaland systematic, respectively. The structure is directlyproduced in e + e − collisions, thus has J P C = 1 −− . Thisstructure has a mass close to the masses of the vector par-ticles φ (2170), ρ (2150) and ω (2290) listed in the PDG [1].Its width is only consistent with the φ (21770) and is dif-ferent from the others by more than 3 σ .Assuming the observed structure is φ (2170), ourmeasurement implies that the φ (2170) has a siz-able partial width to K + (1460) K − , K +1 (1400) K − ,and K +1 (1270) K − , but a much smaller partial widthto K ∗ + (892) K ∗− (892) and K ∗ + (1410) K − . Accord-ing to Ref. [9], the 3 S s ¯ s state mainly decays to K ∗ + (892) K ∗− (892) and K ∗ + (1410) K − , but has a muchsmaller partial width to K +1 (1400) K − and K + (1460) K − .On the other hand, the 2 D s ¯ s state has an expectedpartial width to K +1 (1400) K − smaller than that to K ∗ + (1410) K − by a factor of 2-5 [9, 10]. A hybrid stateis expected to decay dominantly into K +1 (1270) K − and K +1 (1400) K − , while it should be highly suppressed in the modes K ∗ + (892) K ∗− (892) and K + (1460) K − [12]. Noneof the above theoretical expectations are in good agree-ment with our experimental results.The BESIII collaboration thanks the staff of BEPCIIand the IHEP computing center for their strong sup-port. This work is supported in part by NationalKey Basic Research Program of China under ContractNo. 2015CB856700; National Natural Science Founda-tion of China (NSFC) under Contracts Nos. 11625523,11635010, 11735014, 11822506, 11835012; the ChineseAcademy of Sciences (CAS) Large-Scale Scientific Facil-ity Program; Joint Large-Scale Scientific Facility Fundsof the NSFC and CAS under Contracts Nos. U1532257,U1532258, U1732263, U1832207; CAS Key Research Pro-gram of Frontier Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Pro-gram of CAS; INPAC and Shanghai Key Laboratory forParticle Physics and Cosmology; ERC under ContractNo. 758462; German Research Foundation DFG un-der Contracts Nos. Collaborative Research Center CRC1044, FOR 2359; Istituto Nazionale di Fisica Nucleare,Italy; Ministry of Development of Turkey under ContractNo. DPT2006K-120470; National Science and Technol-ogy fund; STFC (United Kingdom); The Knut and Al-ice Wallenberg Foundation (Sweden) under Contract No.2016.0157; The Royal Society, UK under Contracts Nos.DH140054, DH160214; The Swedish Research Coun-cil; U. S. Department of Energy under Contracts Nos.DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0012069;University of Groningen (RuG) and the Helmholtzzen-trum fuer Schwerionenforschung GmbH (GSI), Darm-stadt. [1] K. A. Olive et al. [Particle Data Group], Chin. Phys. C, , 090001 (2018).[2] B. Aubert et al. (BaBar Collaboration), Phys. Rev. D , 091103(R) (2006).[3] B. Aubert et al. (BaBar Collaboration), Phys. Rev. D , 012008 (2007).[4] C. P. Shen et al. (Belle Collaboration), Phys. Rev. D ,031101(R) (2009)[5] M. Ablikim et al. (BES Collaboration), Phys. Rev. Lett. , 102003 (2008).[6] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 052017 (2015).[7] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 012014 (2019).[8] T. Barnes, et al. , Phys. Rev. D , 49 (2007).[10] X. Wang et al. , Phys. Rev. D , 074024 (2012).[11] S. S. Afonin and I. V. Pusenkov, Phys. Rev. D , 094020(2014).[12] G. J. Ding and M. L. Yan, Phys. Lett. B , 390 (2007).[13] P. R. Page, E. S. Swanson, and A. P. Szczepaniak, Phys.Rev. D , 034016 (1999).[14] Z. G. Wang, Nucl. Phys. A , 106 (2007).[15] H. X. Chen et al. , Phys. Rev. D , 034012 (2008). [16] H. W. Ke and X. Q. Li, Phys. Rev. D , 036014 (2019).[17] N. V. Drenska, R. Faccini and A. D. Polosa, Phys. Lett.B , 160 (2008).[18] L. Zhao et al. , Phys. Rev. D , 054034 (2013).[19] C. Deng et al. , Phys. Rev. D , 074007 (2013).[20] Yubing Dong et al. , Phys. Rev. D , 074027 (2017).[21] A. Martinez Torres et al. , Phys. Rev. D , 074031(2008); S. Gomez-Avila, M. Napsuciale and E. Oset,Phys. Rev. D , 034018 (2009).[22] M. Ablikim et al. (BES Collaboration), Phys. Lett. B
27 (2010).[23] S. Godfrey and N. Isgur, Phys. Rev. D , 189 (1985).[24] See Supplemental Material at [URL to be inserted bypublisher] for a summary of the number of the numberof signal events, luminosity, cross section and systematicuncertainty at each energy point, the comparison of in-variant mass spectra and angular distribution betweendata and fit results at √ s =2.125 and 2.396 GeV.[25] M. Ablikim et al. (BESIII Collaboration), Nucl. Instrum.Meth. A , 3 (2010).[26] S. Agostinelli et al. ( Geant4
Collaboration), Nucl. In-strum. Meth. A , 250 (2003).[27] R. G. Ping, Chin. Phys. C , 083001 (2014).[28] G. Balossini et al. , Nucl. Phys. B758, 227 (2006). [29] B. Andersson and H. Hu, arXiv:hep-ph/9910285.[30] M. Ablikim, et al. (BESIII Collaboration), Chin. Phys.C , 063001 (2017).[31] M. Ablikim, et al. (BESIII Collaboration), Phys. Rev. D , 032003 (2011).[32] N. Berger, B. J. Liu, and J. K.Wang, J. Phys. Conf. Ser.219, 042031 (2010).[33] B. S. Zou and D. V. Bugg, Eur. Phys. J. A , 537 (2003).[34] M. Ablikim, et al. (BESII Collaboration), Phys. Lett. B , 149 (2004).[35] M. Ablikim, et al. (BESII Collaboration), Phys. Lett. B , 19 (2007). [36] F. James and M. Roos, Comput. Phys. Commun. 10, 343(1975).[37] E. A. Kuraev and V. S. Fadin, Sov. J. Nucl. Phys. ,466 (1985).[38] S. Actis et al ., Eur. Phys. J. C , 585 (2010).[39] M. Ablikim et al. (BESIII Collaboration), Chin. Phys. C , 063001 (2017).[40] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 032001 (2019).[41] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D , 052005 (2010).[42] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D87