Amplitude analysis and branching-fraction measurement of D^{+}_{s}\rightarrow K^{0}_{S}K^{-}π^{+}π^{+}
aa r X i v : . [ h e p - e x ] F e b Amplitude analysis and branching-fraction measurement of D + s → K S K − π + π + M. Ablikim , M. N. Achasov ,c , P. Adlarson , S. Ahmed , M. Albrecht , R. Aliberti , A. Amoroso A, C , M. R. An ,Q. An , , X. H. Bai , Y. Bai , O. Bakina , R. Baldini Ferroli A , I. Balossino A , Y. Ban ,k , K. Begzsuren ,N. Berger , M. Bertani A , D. Bettoni A , F. Bianchi A, C , J. Bloms , A. Bortone A, C , I. Boyko , R. A. Briere ,H. Cai , X. Cai , , A. Calcaterra A , G. F. Cao , , N. Cao , , S. A. Cetin A , J. F. Chang , , W. L. Chang , ,G. Chelkov ,b , D. Y. Chen , G. Chen , H. S. Chen , , M. L. Chen , , S. J. Chen , X. R. Chen , Y. B. Chen , ,Z. J Chen ,l , W. S. Cheng C , G. Cibinetto A , F. Cossio C , X. F. Cui , H. L. Dai , , X. C. Dai , , A. Dbeyssi ,R. E. de Boer , 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 , , , X. Dong , S. X. Du , Y. L. Fan , J. Fang , ,S. S. Fang , , Y. Fang , R. Farinelli A , L. Fava B, C , F. Feldbauer , G. Felici A , C. Q. Feng , , J. H. Feng ,M. Fritsch , C. D. Fu , Y. Gao , Y. Gao , , Y. Gao ,k , Y. G. Gao , I. Garzia A, B , P. T. Ge , C. Geng ,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 ,h , A. Guskov ,T. T. Han , W. Y. Han , X. Q. Hao , F. A. Harris , N H¨usken , , K. L. He , , F. H. Heinsius , C. H. Heinz ,T. Held , Y. K. Heng , , , C. Herold , M. Himmelreich ,f , T. Holtmann , Y. R. Hou , Z. L. Hou , H. M. Hu , ,J. F. Hu ,m , T. Hu , , , Y. Hu , G. S. Huang , , L. Q. Huang , X. T. Huang , Y. P. Huang , Z. Huang ,k ,T. Hussain , W. Ikegami Andersson , W. Imoehl , M. Irshad , , S. Jaeger , S. Janchiv ,j , Q. Ji , Q. P. Ji , X. B. Ji , ,X. L. Ji , , H. B. Jiang , X. S. Jiang , , , J. B. Jiao , Z. Jiao , 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 A,e , B. Kopf , M. Kuemmel , M. Kuessner , A. Kupsc ,M. G. Kurth , , W. K¨uhn , J. J. Lane , J. S. Lange , P. Larin , A. Lavania , L. Lavezzi A, C , Z. H. Lei , ,H. Leithoff , M. Lellmann , T. Lenz , C. Li , C. H. Li , Cheng Li , , D. M. Li , F. Li , , G. Li , H. Li , H. Li , ,H. B. Li , , H. J. Li ,h , J. L. Li , J. Q. Li , J. S. Li , Ke Li , L. K. Li , Lei Li , P. R. Li , S. Y. Li , W. D. Li , ,W. G. Li , X. H. Li , , X. L. Li , Z. Y. Li , H. Liang , , H. Liang , , H. Liang , Y. F. Liang , Y. T. Liang ,L. Z. Liao , , J. Libby , C. X. Lin , B. J. Liu , C. X. Liu , D. Liu , , F. H. Liu , Fang Liu , Feng Liu , H. B. Liu ,H. M. Liu , , Huanhuan Liu , Huihui Liu , J. B. Liu , , J. L. Liu , J. Y. Liu , , K. Liu , K. Y. Liu , Ke Liu ,L. Liu , , M. H. Liu ,h , P. L. Liu , Q. Liu , Q. Liu , S. B. Liu , , Shuai Liu , T. Liu , , W. M. Liu , , X. Liu ,Y. Liu , Y. B. Liu , Z. A. Liu , , , Z. Q. Liu , X. C. Lou , , , F. X. Lu , F. X. Lu , 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 ,h , 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. X. Ma , ,X. Y. Ma , , F. E. Maas , M. Maggiora A, C , S. Maldaner , S. Malde , A. Mangoni B , Y. J. Mao ,k , Z. P. Mao ,S. Marcello A, C , Z. X. Meng , J. G. Messchendorp , G. Mezzadri A , T. J. Min , R. E. Mitchell , X. H. Mo , , ,Y. J. Mo , N. Yu. Muchnoi ,c , H. Muramatsu , S. Nakhoul ,f , Y. Nefedov , F. Nerling ,f , I. B. Nikolaev ,c , Z. Ning , ,S. Nisar ,i , S. L. Olsen , Q. Ouyang , , , S. Pacetti B, C , X. Pan ,h , Y. Pan , A. Pathak , P. Patteri A , M. Pelizaeus ,H. P. Peng , , K. Peters ,f , J. Pettersson , J. L. Ping , R. G. Ping , , R. Poling , V. Prasad , , H. Qi , ,H. R. Qi , K. H. Qi , M. Qi , T. Y. Qi , T. Y. Qi , S. Qian , , W. B. Qian , Z. Qian , C. F. Qiao , L. Q. Qin ,X. P. Qin , X. S. Qin , Z. H. Qin , , J. F. Qiu , S. Q. Qu , K. Ravindran , C. F. Redmer , A. Rivetti C , V. Rodin ,M. Rolo C , G. Rong , , Ch. Rosner , M. Rump , H. S. Sang , A. Sarantsev ,d , Y. Schelhaas , C. Schnier ,K. Schoenning , M. Scodeggio A, B , D. C. Shan , W. Shan , X. Y. Shan , , J. F. Shangguan , M. Shao , ,C. P. Shen , P. X. Shen , X. Y. Shen , , H. C. Shi , , R. S. Shi , , X. Shi , , X. D Shi , , J. J. Song , W. M. Song , ,Y. X. Song ,k , S. Sosio A, C , S. Spataro A, C , K. X. Su , P. P. Su , F. F. Sui , G. X. Sun , H. K. Sun , J. F. Sun ,L. Sun , S. S. Sun , , T. Sun , , W. Y. Sun , W. Y. Sun , X Sun ,l , Y. J. Sun , , Y. K. Sun , , Y. Z. Sun ,Z. T. Sun , Y. H. Tan , Y. X. Tan , , C. J. Tang , G. Y. Tang , J. Tang , J. X. Teng , , V. Thoren , Y. T. Tian ,I. Uman B , B. Wang , C. W. Wang , D. Y. Wang ,k , H. J. Wang , H. P. Wang , , K. Wang , , L. L. Wang ,M. Wang , M. Z. Wang ,k , Meng Wang , , W. Wang , W. H. Wang , W. P. Wang , , X. Wang ,k , X. F. Wang ,X. L. Wang ,h , Y. Wang , , Y. Wang , Y. D. Wang , Y. F. Wang , , , Y. Q. Wang , Y. Y. Wang , Z. Wang , ,Z. Y. Wang , Ziyi Wang , Zongyuan Wang , , D. H. Wei , P. Weidenkaff , F. Weidner , 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 ,h , Z. Wu , ,L. Xia , , H. Xiao ,h , S. Y. Xiao , Z. J. Xiao , X. H. Xie ,k , Y. G. Xie , , Y. H. Xie , T. Y. Xing , , G. F. Xu ,Q. J. Xu , W. Xu , , X. P. Xu , Y. C. Xu , F. Yan ,h , L. Yan ,h , W. B. Yan , , W. C. Yan , Xu Yan , H. J. Yang ,g ,H. X. Yang , L. Yang , S. L. 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 ,l , T. Yu , C. Z. Yuan , , L. Yuan , X. Q. Yuan ,k , Y. Yuan ,Z. Y. Yuan , C. X. Yue , A. Yuncu A,a , A. A. Zafar , Y. Zeng ,l , B. X. Zhang , Guangyi Zhang , H. Zhang ,H. H. Zhang , H. H. Zhang , H. Y. Zhang , , J. J. Zhang , J. L. Zhang , J. Q. Zhang , J. W. Zhang , , ,J. Y. Zhang , J. Z. Zhang , , Jianyu Zhang , , Jiawei Zhang , , L. M. Zhang , L. Q. Zhang , Lei Zhang , S. Zhang ,S. F. Zhang , Shulei Zhang ,l , X. D. Zhang , X. Y. Zhang , Y. Zhang , Y. H. Zhang , , Y. T. Zhang , ,Yan Zhang , , Yao Zhang , Yi Zhang ,h , Z. H. Zhang , Z. Y. Zhang , G. Zhao , J. Zhao , J. Y. Zhao , , J. Z. Zhao , ,Lei Zhao , , Ling Zhao , M. G. Zhao , Q. Zhao , S. J. Zhao , Y. B. Zhao , , Y. X. Zhao , Z. G. Zhao , ,A. Zhemchugov ,b , B. Zheng , J. P. Zheng , , Y. Zheng ,k , Y. H. Zheng , B. Zhong , C. Zhong , 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 ,T. J. Zhu , W. J. Zhu ,h , W. J. Zhu , Y. C. Zhu , , Z. A. Zhu , , 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 INFN Laboratori Nazionali di Frascati , (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFNSezione di Perugia, I-06100, Perugia, Italy; (C)University of Perugia, I-06100, Perugia, Italy INFN Sezione di Ferrara, (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 Jilin University, Changchun 130012, People’s Republic of China 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 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 North China Electric Power University, Beijing 102206, 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 South China Normal University, Guangzhou 510006, 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 Suranaree University of Technology, University Avenue 111, Nakhon Ratchasima 30000, Thailand Tsinghua University, Beijing 100084, People’s Republic of China Turkish Accelerator Center Particle Factory Group, (A)Istanbul Bilgi University, 34060 Eyup, Istanbul, Turkey; (B)NearEast University, Nicosia, North Cyprus, Mersin 10, Turkey University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China University of Groningen, NL-9747 AA Groningen, The Netherlands 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 University of Turin and INFN, (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 Bogazici University, 34342 Istanbul, Turkey b Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia c Also at the Novosibirsk State University, Novosibirsk, 630090, Russia d Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia e Also at Istanbul Arel University, 34295 Istanbul, Turkey f Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany g 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 h Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of Modern Physics, FudanUniversity, Shanghai 200443, People’s Republic of China i Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA j Currently at: Institute of Physics and Technology, Peace Ave.54B, Ulaanbaatar 13330, Mongolia k Also at State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People’s Republic ofChina l School of Physics and Electronics, Hunan University, Changsha 410082, China m Also at Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China NormalUniversity, Guangzhou 510006, China
Using 6.32 fb − of e + e − collision data collected by the BESIII detector at the center-of-massenergies between 4.178 and 4.226 GeV, an amplitude analysis of the D + s → K S K − π + π + decays isperformed for the first time to determine the intermediate-resonant contributions. The dominantcomponent is the D + s → K ∗ (892) + K ∗ (892) decay with a fraction of (40 . ± . stat ± . sys )%. Ourresults of the amplitude analysis are used to obtain a more precise measurement of the branchingfraction of the D + s → K S K − π + π + decay, which is determined to be (1 . ± . stat ± . sys )%. I. INTRODUCTION
The decay D + s → K S K − π + π + is usually used as a“tag mode” for measurements related to the D + s me-son [1–5] due to its large branching fraction and lowbackground contamination. The inclusion of charge-conjugate states is implied throughout the paper. In 2013the CLEO Collaboration reported its branching fraction B ( D + s → K S K − π + π + ) to be (1 . ± . ± . − of e + e − collisions at a center-of-mass energy ( E cm ) of 4.17 GeV [6]. The measurementwas limited by the sample size and lack of knowledgeof the intermediate processes. In addition, the branch-ing fraction of D + s → K ∗ (892) + K ∗ (892) was deter-mined by the ARGUS Collaboration [7] more than twen-ty years ago, who claimed the contribution of D + s → K ∗ (892) + K ∗ (892) in the D + s → K S K − π + π + decays isalmost 100%. The ARGUS measurement suffers from lowstatistics and large uncertainties in the branching frac- tion of the reference decay D + s → φ (1020) π + . An ampli-tude analysis of the D + s → K S K − π + π + decays is neces-sary to investigate the resonant contributions, and there-by reduce the systematic uncertainties of its branchingfraction and for providing input to measurements whereamplitude information is essential.It is well known that two-body modes dominate D + s de-cays [8]. The majority of the observed two-body decayshave pseudoscalar-pseudoscalar or pseudoscalar-vectormesons in the final states. Among various kinds of D + s decay modes, vector-vector final states are of spe-cial interest. The ratios between different orbital an-gular momenta of the two vector mesons for the dom-inant quasi-two-body decay D + s → K ∗ (892) + K ∗ (892) provide valuable information on CP violation with T-violating triple-products [9]. In addition, several mesonswith J P = 0 − , + are reported in the mass region be-tween 1.2 and 1.6 GeV/ c and decay to the ( KKπ ) finalstate [10–13]. These are the η (1295), η (1405), η (1475), f (1285), f (1420) and f (1510), although many of thesestates are not well established.This paper presents the first amplitude analysis and animproved branching-fraction measurement of the D + s → K S K − π + π + decay with data samples corresponding to atotal integrated luminosity of 6.32 fb − collected by theBESIII detector at E cm between 4.178 and 4.226 GeV. II. DETECTOR AND DATA SETS
The detailed description of the BESIII detector can befound in Ref. [14]. It is a magnetic spectrometer locatedat the Beijing Electron Positron Collider (BEPCII) [15].The cylindrical core of the BESIII detector consists of ahelium-based multilayer drift chamber (MDC), a plasticscintillator time-of-flight system (TOF), and a CsI(Tl)electromagnetic calorimeter (EMC), which are all en-closed in a superconducting solenoidal magnet providinga 1.0 T magnetic field. The solenoid is supported by anoctagonal flux-return yoke with resistive plate countermuon-identifier modules interleaved with steel. The ac-ceptance of charged particles and photons is 93% overthe 4 π solid angle. The charged-particle momenta reso-lution at 1.0 GeV/ c is 0 . dE/dx ) resolution is 6% for the electrons from Bhabhascattering. The EMC measures photon energies with aresolution of 2 .
5% (5%) at 1 GeV in the barrel (end-cap)region. The time resolution of the TOF barrel part is68 ps, while that of the end cap part is 110 ps. The end-cap TOF was upgraded in 2015 with multi-gap resistiveplate chamber technology, providing a time resolution of60 ps [16, 17].The data samples used in this paper were accumulat-ed in the years 2013, 2016 and 2017 with E cm of 4.226,4.178 and 4.189 − e + e − annihilation is modeled with the genera-tor conexc [19], which includes the effects of the beamenergy spread and initial state radiation (ISR). The ISRproduction of vector charmonium states and the con-tinuum processes are incorporated in kkmc [20]. Theknown decay modes are generated using evtgen [21],which assumes the branching fractions reported by theParticle Data Group (PDG) [8]. The remaining unknowndecays from the charmonium states are generated with lundcharm [22]. The final state radiation from chargedtracks are simulated by the photos package [23]. TheGMC is used to estimate background and optimize selec-tion criteria.More than 10 million simulated events are generatedwith an uniform distribution in the phase space of the D + s → K S K − π + π + decay to perform the normalizationin the amplitude fit. Preliminary parameters of the am-plitude model are obtained from an initial fit to the da-ta. A signal Monte Carlo (SMC) sample is generatedaccording to the preliminary parameters and is used to validate the fit performance and to estimate the detectorefficiency. A final determination of the fit parameters isobtained by fitting the data using the SMC sample forthe normalization. III. EVENT SELECTIONS
The production of D ± s candidates is dominated by theprocess e + e − → D ∗ + s D − s , where the D ∗ + s meson de-cays to either γD + s or π D + s with branching fractionsof (93.5 ± ± D − s mesons is reconstructed first, with nine D − s prominent hadronic decay modes, as shown in Table I,and is referred to as the “single tag (ST)” candidates.The signal decay D + s → K S K − π + π + is reconstructed byselecting two π + , one K − and one K S candidates fromthe unused tracks in each ST event, and is referred to asthe sample of “double tag (DT)” candidates.All charged tracks reconstructed in the MDC mustsatisfy | cos θ | < .
93, where θ is the polar angle withrespect to the direction of the positron beam. Exceptfor K S daughters, they must originate from the inter-action point with a distance of closest approach lessthan 1 cm in the transverse plane and less than 10 cmalong the beam direction. The d E /d x information in theMDC and the time-of-fight information from the TOFare combined and used for particle identification (PID)by forming confidence levels CL K ( π ) for kaon (pion) hy-potheses. Kaon (pion) candidates are required to satisfy CL K ( π ) > CL π ( K ) .For the photon identification, it is required that eachelectromagnetic shower starts within 700 ns of the eventstart time and its energy is greater than 25 (50) MeV inthe barrel (end cap) with | cos θ | < .
80 ( | cos θ | ∈ [0.86,0.92]). The π and η candidates are reconstructed viadiphoton decays ( π /η → γγ ) with the invariant massof the γγ combination M γγ ∈ [0.115, 0.150] and [0.50,0.57] GeV/ c , respectively. The value of M γγ is con-strained to the π or η nominal mass [8] by a kinematicfit, and the χ of the kinematic fit must be less than 30.We reconstruct the η ′ → π + π − η candidates by requiring M π + π − η ∈ [0 . , . c .The K S candidates are selected by looping over allpairs of tracks with opposite charges, whose distances tothe interaction point along the beam direction are within20 cm. A primary vertex and a secondary vertex [24] arereconstructed and the decay length between the two isrequired to be greater than twice its uncertainty. Sincethe combinatorial background is low, this requirement isnot applied for the D − s → K S K − decay. The invariantmass M π + π − is required to be in the region [0.487, 0.511]GeV/ c . To prevent an event being retained by both the D − s → K S K − and D − s → K − π + π − selections, the valueof M π + π − is required to be outside of the mass range[0.487, 0.511] GeV/ c for the D − s → K − π + π − decay. IV. AMPLITUDE ANALYSISA. Selections for Amplitude Analysis
The tagged D − s candidates are constructed from the π + , K − , η , η ′ , K S and π mesons, while the signal D + s candidates are reconstructed from the K S , K − and two π + mesons. The requirements on the recoiling mass ofthe D + s , M rec , and the mass of the tagged D − s , M tag , aresummarized in Table I. The recoiling mass is calculatedas follows: M rec = r ( E cm − q | ~ p D + s | + m D + s ) − ~ p D + s . (1) Here, ~ p D + s is the three momentum of the D + s candidateand m D + s is its nominal mass [8]. TABLE I. The requirements of M rec for various energies and M tag for individual single-tagged modes. The K S , π ( η ) and η ′ mesons decay to π + π − , γγ and π + π − η final states, respec-tively. E cm (GeV) M rec (GeV/ c )4.178 [2.050, 2.180]4.189 [2.048, 2.190]4.199 [2.046, 2.200]4.209 [2.044, 2.210]4.219 [2.042, 2.220]4.226 [2.040, 2.220]Tag mode M tag (GeV/ c ) D − s → K S K − [1.948, 1.991] D − s → K + K − π − [1.950, 1.986] D − s → K S K − π [1.946, 1.987] D − s → K − π + π − [1.953, 1.983] D − s → π − η ′ [1.940, 1.996] D − s → K S K − π + π − [1.958, 1.980] D − s → K + K − π − π [1.947, 1.982] D − s → π + π − π − [1.952, 1.982] D − s → π − η [1.930, 2.000] Kinematic fits are performed of the process e + e − → D ∗± s D ∓ s → γD ± s D ∓ s with the photon assigned to eachcharm meson in turn, and the χ of the fit being used todecide between the D ∗ + s and D ∗− s hypotheses. The fitsinclude constraints from four-momentum conservation inthe e + e − system, and also constrain the invariant massesof K S , D ∗± s and tag-side D ± s candidates to their nominalmasses [8]. In order to ensure that all candidates fallwithin the kinematic boundary of the phase space, weperform a further kinematic fit in which the signal D ± s mass is constrained to its nominal value, and the updatedfour-momenta are used for the amplitude analysis.To suppress the background where the π − from thesignal decay and the π + from the tag modes are ex-changed, which fakes the signal and the same tag mode but with opposite charges, we perform kinematic fitswith D + s and D − s mass constraints for the two cas-es, and select the one with the smaller χ . To reducethe background coming from D → K − π + π + π − versus D → K S K + K − ( K S π + π − ), by exchanging π − from D and K S from D , faking the signal D + s → K S K − π + π + and the tag mode D − s → K + K − π − ( π + π − π − ), we re-ject events satisfying: | M ′ D − M PDG D | <
15 MeV /c , | M ′ D − M PDG D | <
15 MeV /c and | M ′ D − M ′ D | < | M ′ D + s − M ′ D − s | . Here, M PDG D is the nominal D mass [8], M ′ D , M ′ D , M ′ D + s and M ′ D − s are the invariant masses ofthe D → K − π + π + π − , D → K S K + K − ( K S π + π − ), D + s → K S K − π + π + , and D − s → K + K − π − ( π + π − π − )candidates, respectively.Figure 1 shows the invariant mass distributions of thesignal D + s , M sig , in data and the fit results. The signaldistribution is modeled with the simulated shape con-volved with a Gaussian function and the background isdescribed by a first-order Chebychev polynomial. The fit-ted yields for the signal are 744 ±
28, 415 ±
21 and 159 ± c , withpurities of (94.7 ± ± ± E cm = 4 . , . − . D + s mass region are retained for the amplitude anal-ysis. We compare the background yield and various dis-tributions of the events outside the signal region betweendata and GMC. The yield and distributions are found tobe consistent within the statistical uncertainties. Thebackground events in the signal region from GMC areused to estimate the background contributions in data. B. Likelihood Function
An unbinned-maximum-likelihood method is appliedto determine resonant contributions in the D + s → K S K − π + π + decays. The likelihood function is con-structed with a probability density function (PDF) of themomenta of the four daughter particles. The amplitudeof the n th intermediate state ( A n ) is A n = P n P n S n F n F n F n , (2)where the indices 1, 2 and 3 correspond to the two sub-sequent intermediate resonances and the D + s meson. S isthe spin factor constructed with the covariant tensor for-malism [25], F is the Blatt-Weisskopf barrier factor and P is the propagator of the intermediate resonance. Thetotal amplitude M is a coherent sum of the amplitudesof intermediate processes, M = X c n A n , (3)where c n = ρ n e iφ n are complex coefficients to be deter-mined from the fit to data. ) c (GeV/ sig M E v e n t s / . M e V / c ( a ) ) c (GeV/ sig M E v e n t s / . M e V / c ( b ) ) c (GeV/ sig M E v e n t s / . M e V / c ( c )FIG. 1. Fits to the M sig distributions of accepted candidates from the data samples taken at (a) E cm = 4.178, (b) 4.189 − The signal PDF f S ( p j ) is given by f S ( p j ) = ǫ ( p j ) | M ( p j ) | R ( p j ) R ǫ ( p j ) | M ( p j ) | R ( p j ) dp j , (4)where ǫ ( p j ) is the detection efficiency parameterized interms of the final four-momenta p j and j refers to the dif-ferent particles in the final states. R ( p j ) is the standardelement of the four-body phase space.The normalization is determined from the simulatedevents, Z ǫ ( p j ) | M ( p j ) | R ( p j ) dp j ≈ N sim N sim X k sim | M ( p k sim j ) | | M gen ( p k sim j ) | , (5) where k sim runs from 1 to N sim , the total number of sim-ulated events. M gen ( p j ) is the PDF used to generate thesimulated samples.The normalization takes into account the difference indetector efficiencies for PID and tracking between dataand simulation by assigning a weight to each simulatedevent γ ǫ ( p ) = Y i ǫ i, data ( p j ) ǫ i, sim ( p j ) , (6)where i denotes the four daughter particles. The normal-ization is then given by Z ǫ ( p j ) | M ( p j ) | R ( p j ) dp j ≈ N sim N sim X k sim γ ǫ ( p k sim j ) | M ( p k sim j ) | | M gen ( p k sim j ) | . (7) The total PDF f T ( p j ) is f T ( p j ) = w ǫ ( p j ) | M ( p j ) | R ( p j ) R ǫ ( p j ) | M ( p j ) | R ( p j ) dp j + (1 − w ) B ( p j ) R ( p j ) R B ( p j ) R ( p j ) dp j , (8)where w is the purity of the signal described by a constantparameter in the fit. We factorize out ǫ ( p j ) from f T ( p j )as ǫ ( p j ) is independent of the fit variables. Its contri-bution enters into the normalization and the background PDF. As a consequence, the combined PDF becomes f T ( p j ) = ǫ ( p j ) R ( p j )[ w | M ( p j ) | R ǫ ( p j ) | M ( p j ) | R ( p j ) dp j + (1 − w ) B ǫ ( p j ) R ǫ ( p j ) B ǫ ( p j ) R ( p j ) dp j ] , (9)where B ǫ ( p j ) = B ( p j ) /ǫ and the background PDF B ( p j )is parameterized using RooNDKeysPdf [26]. The nor-malization in the denominator of the background term iscalculated as Z ǫ ( p j ) B ǫ ( p j ) R ( p j ) dp j ≈ N sim N sim X k sim B ǫ ( p k sim j ) | M gen ( p k sim j ) | . (10) Finally the log-likelihood is written asln L = ln[ w | M ( p j ) | R ǫ ( p j ) | M ( p j ) | R ( p j ) dp j + (1 − w ) B ǫ ( p j ) R ǫ ( p j ) B ǫ ( p j ) R ( p j ) dp j ] , (11)and data samples collected at different E cm are fittedsimultaneously. C. Spin Factors
For the process a → bc , the four momenta of the parti-cles a , b and c are denoted as p a , p b and p c , respectively.The spin projection operators [25] are defined as P (1) µµ ′ ( a ) = − g µµ ′ + p a,µ p a,µ ′ p a ,P (2) µνµ ′ ν ′ ( a ) = 12 ( P (1) µµ ′ ( a ) P (1) νν ′ ( a ) + P (1) µν ′ ( a ) P (1) νµ ′ ( a )) − P (1) µν ( a ) P (1) µ ′ ν ′ ( a ) . (12)The pure orbital angular-momentum covariant tensorsare given by ˜ t (1) µ ( a ) = − P (1) µµ ′ ( a ) r µ ′ a , ˜ t (2) µν ( a ) = P (2) µνµ ′ ν ′ ( a ) r µ ′ a r ν ′ a , (13)where r a = p b − p c . The spin factors S ( p ) used in thispaper are constructed from the spin projection opera-tors and pure orbital angular-momentum covariant ten-sors and are listed in Table II. TABLE II. The spin factor S ( p ) for each decay chain. All op-erators, i.e. ˜ t , have the same definitions as in Ref. [25]. Scalar,pseudo-scalar, vector and axial-vector states are denoted by S , P , V and A , respectively. The [ S ], [ P ] and [ D ] denote theorbital angular-momentum quantum numbers L = 0, 1 and2, respectively.Decay chain S ( p ) D + s [ S ] → V V ˜ t (1) µ ( V ) ˜ t (1) µ ( V ) D + s [ P ] → V V ǫ µνλσ p µ ( D + s ) ˜ T (1) ν ( D + s ) × ˜ t (1) λ ( V ) ˜ t (1) σ ( V ) D + s [ D ] → V V ˜ T (2) µν ( D + s ) ˜ t (1) µ ( V ) ˜ t (1) ν ( V ) D + s → AP , A [ S ] → V P ˜ T (1) µ ( D + s ) P (1) µν ( A ) ˜ t (1) ν ( V ) D + s → AP , A → SP ˜ T (1) µ ( D + s )˜ t (1) µ ( A ) D + s → V S ˜ T (1) µ ( D + s ) ˜ t (1) µ ( V ) D + s → P P , P → V P p µ ( P )˜ t (1) µ ( V ) D + s → P P , P → SP D. Blatt-Weisskopf Barrier Factors
For the process a → bc , the Blatt-Weisskopf barrierfactor F L ( p j ) is parameterized as a function of the angu-lar momentum L and the momentum q of the daughter b or c in the rest system of a , F L ( q ) = z L X L ( q ) , (14)where z = qR . R is the effective radius of the barrier,which is fixed to 3.0 GeV − × ~ c for the intermediateresonances and 5.0 GeV − × ~ c for the D + s meson [27].The momentum-transfer squared is q = ( s a + s b − s c ) s a − s b , (15)where s a,b,c are the invariant-mass squared of particles a, b, c , respectively. The X L ( q ) factors are given by X L =0 ( q ) = 1 ,X L =1 ( q ) = r z + 1 ,X L =2 ( q ) = r z + 3 z + 9 . (16) E. Propagators
The propagators for the resonances K ∗ (892) + , K ∗ (892) , η (1295), η (1405), η (1475), f (1285), f (1420) and f (1510) are modeled by the relativistic Breit-Wigner function, which is given by P ( m ) = 1( m − s a ) − im Γ( m ) , Γ( m ) = Γ (cid:18) qq (cid:19) L +1 (cid:16) m m (cid:17) (cid:18) X L ( q ) X L ( q ) (cid:19) , (17)where m and Γ( m ) are the mass and width of the inter-mediate resonance, and q is the value of q when s a = m .The a (980) contribution is parameterized as theFlatt´e formula P a (980) = 1 M − s a − i ( g ηπ ρ ηπ ( s a ) + g KK ρ KK ( s a )) , (18) where ρ ηπ ( s a ) and ρ KK ( s a ) are the Lorentz-invariantphase-space factors defined as 2 q/ √ s a , and the couplingconstants g ηπ = 0 . ± .
004 GeV / c and g KK =(0 . ± . g ηπ [28].We use the same parameterization to describe the KπS -wave as Ref. [29], which is extracted from scatteringdata [30]. The model is built with a Breit-Wigner shapefor the K ∗ (1430) and an effective range parameteriza-tion for the non-resonant component, A ( m ) = F sin δ F e iδ F + R sin δ R e iδ R e i δ F , (19)with δ F = φ F + cot − (cid:20) aq + rq (cid:21) ,δ R = φ R + tan − (cid:20) M Γ( m Kπ ) M − m Kπ (cid:21) , where a and r are the scattering length and effective in-teraction length, respectively. The parameters F ( φ F )and R ( φ R ) are the magnitudes (phases) for the non-resonant term and the resonant contribution, respective-ly. The parameters M , F , φ F , R , φ R , a and r are fixed tothe results of the D → K S π + π − analysis by the B A B A R and Belle Collaborations [29]. F. Fit Fraction
The fit fraction (FF) for a quasi-two-body contribu-tion is independent of any normalization and phase con-ventions in the amplitude formalism, and hence providesa more useful measure of amplitude strengths than themagnitudes of each contribution. The definition of theFF for the n th contribution is FF n = R | c n A n ( p ) | R ( p ) dp R | P k c k A k ( p ) | R ( p ) dp ≈ N g, ph P l =1 | c n A n ( p l ) | N g, ph P l =1 | P k c k A k ( p l ) | , (20) where the integration is approximated by the sum of thesimulated events generated flatly over the phase spaceand without any efficiency effects included.To estimate the statistical uncertainties on the FFs,the calculation is repeated by randomly varying the fitparameters according to the error matrix. The resultingdistribution of each FF is fitted with a Gaussian func-tion, whose width gives the corresponding statistical un-certainty. G. Fit Results
In the fit, the magnitude ( ρ ) and phase ( φ ) of D + s → K ∗ (892) + K ∗ (892) with angular momentum L = 0 between K ∗ (892) + and K ∗ (892) is fixed to1 and 0, respectively, and the magnitudes and phas-es of the other contributions are kept floating. Themasses and widths of all resonances are fixed to thecorresponding PDG averages [8]. We consider pos-sible resonant contributions from a (980), K ∗ (892), K ∗ (1410), K ∗ (1430), K (1270), K (1400), η (1295), η (1405), η (1475), f (1285), f (1420), f (1510) and φ (1680) as well as non-resonant contributions. SU(3)flavor symmetry requires the magnitude and phases ofthe processes D + s → η (1475) π + , η (1475) → K ∗ (892) K S and D + s → η (1475) π + , η (1475) → K ∗ (892) + K − to bethe same [31]. Resonant or non-resonant contributionswith a significance of larger than four standard devia-tions are retained in the model, where the significanceis calculated from the change of the log-likelihood valuesand the corresponding degrees of freedom. Eleven am-plitude contributions are retained in the nominal fit, in-cluding a non-resonant component of K ∗ (892) + K − with L = 1 between K ∗ (892) + and K − . All the resonant andnon-resonant contributions and their φ , FFs and signif-icances are listed in Table III. The projections for thenine invariant-mass distributions are shown in Fig. 2.To validate the fit performance, 300 sets of SMC sam-ples with the same size as the data samples are generat-ed according to the nominal fit results in this analysis.Each sample is analyzed with the same method as for da-ta. The pull value is given by V pull = ( V fit − V input ) /σ fit ,where V input is the input value in the generator, V fit and σ fit is the output value and the corresponding statisti-cal uncertainty, respectively. The resulting pull distri-butions are fitted with Gaussian distributions. The fit-ted mean value of the pull distribution for the FF of D + s → η (1475) π + , η (1475) → ( K S π + ) S -wave K − deviatesfrom zero by more than 3.0 σ , we correct its FF accordingto the deviation and the uncertainty of the FF. H. Systematic Uncertainties
The systematic uncertainties for the amplitude analy-sis are studied in the following categories. • Kπ S -wave model. The fixed parameters of themodel are evaluated by varying the input valueswithin ± σ according to Ref. [29]. • Lineshape of a (980). The Flatt´e parametersare shifted by ± σ based on the values given inRef .[28]. • Effective barrier radius. The barrier radius are var-ied within ± − × ~ c for intermediate reso-nances and the D + s meson. • Masses and widths of the resonances considered.The masses and widths are shifted by ± σ basedon their values from the PDG [8]. • Background estimation. We shift the fractions ofthe signal in Eq. 9 according to the uncertainty as-sociated with the background estimation and takethe largest shift as the systematic uncertainty. • Experimental effects. To determine the systematicuncertainty due to tracking and PID efficiencies,we alter the fit by shifting the γ ǫ in Eq. 6 withinits uncertainty, and the change of the nominal fitresult is taken as the systematic uncertainty. • Neglected resonances. The intermediate processeswith statistical significance less than four standarddeviations are added one-by-one to the nominalcontributions. For each parameter, the maximumdifference with respect to the nominal fit result istaken as the corresponding systematic uncertainty. • Fit uncertainties. The fitted widths from the pulldistributions described in Sec. IV G are consistentwith 1.0 within 2 . σ . Therefore, the fit uncertain-ties are estimated properly and no systematic un-certainty is assigned from this source.All of the systematic uncertainties of the φ and FFs arelisted in Table IV. The total systematic uncertainties areobtained by adding the above systematic uncertainties inquadrature. V. BRANCHING-FRACTION MEASUREMENTA. Yields and Efficiencies
The selection criteria of the tagged D − s and signal D + s candidates are the same as in Sec. III, except for thefollowing requirements: (I) the requirement of the sec-ondary vertex fit for K S from the tag modes is removed,while that for the signal is retained; (II) a further re-quirement of p π ± /π > c is added to removethe soft π ± /π directly from D ∗± /D ∗ decays; (III) thetagged D − s candidates are reconstructed by looping overall their daughter tracks to form different combinations.If there are multiple candidates from the same event,the one with M rec closest to the D ∗± s mass is accepted;(IV) at least one of the D + s /D − s candidates must satisfy M rec > .
10 GeV/ c ; (V) the combination with average TABLE III. The φ , FFs and significances for different resonant contributions, labeled as I, II, III, · · · , XIII, respectively. Thefirst and second uncertainties are the statistical and systematic uncertainties, respectively. Here K ∗ (892) + , K ∗ (892) and a (980) − denote K ∗ (892) + → K S π + , K ∗ (892) → K − π + and a (980) − → K S K − , respectively, while K (892) ∗ K indicates K ∗ K S and K ∗ (892) + K − . The FF of IV (IIX) term is the sum of I, II and III (VIII and IX) terms after considering theinterference.Label Component φ FF(%) Significance ( σ )I D + s [ S ] → K ∗ (892) + K ∗ (892) ± ± > D + s [ P ] → K ∗ (892) + K ∗ (892) -1.61 ± ± ± ± D + s [ D ] → K ∗ (892) + K ∗ (892) -0.16 ± ± ± ± D + s → K ∗ (892) + K ∗ (892) ± ± D + s → K ∗ (892) + ( K − π + ) S − wave ± ± ± ± D + s → K ∗ (892) ( K S π + ) S − wave -1.57 ± ± ± ± D + s → η (1475) π + , η (1475) → a (980) − π + -1.95 ± ± ± ± D + s → η (1475) π + , η (1475) → K ∗ (892) K S ± ± ± ± D + s → η (1475) π + , η (1475) → K ∗ (892) + K − ± ± ± ± D + s → η (1475) π + , η (1475) → K ∗ (892) K ± ± D + s → η (1475) π + , η (1475) → ( K S π + ) S − wave K − ± ± ± ± D + s → f (1285) π + , f (1285) → a (980) − π + -0.89 ± ± ± ± D + s → ( K ∗ (892) + K − ) P π + ,( K ∗ (892) + K − ) P → K ∗ (892) + K − -1.07 ± ± ± ± mass M = [ M tag + M sig ] / D + s [8] is chosen among the multiple candidates.The ST yields ( N ST ) and DT yields ( N DT ) in data aredetermined by fitting the M tag distributions from differ-ent tag modes and M sig distributions, respectively. Ineach fit, the signal shape is modeled using the simulatedshape convolved with a Gaussian function, whose resolu-tion and mean are free parameters, and the backgroundis described with a second-order Chebychev polynomial.These fits give a total ST yield of N ST = 550496 ± M tag distributions at E cm = 4.178 GeV are shown inFig. 3 as an example. The total DT signal yield, N totDT , isdetermined to be 1332 ±
42, as shown in Fig. 4. The fitsto the M sig distribution for GMC are performed to es-timate the corresponding ST efficiencies ( ǫ ST ). The DTefficiencies ( ǫ DT ) are determined by GMC, in which ouramplitude analysis model is taken for the generation ofthe signal mode. B. Tagging Technique and Branching Fraction
The branching fraction for the signal mode is given by B sig = N totDT P i P j N ij ST · ǫ ij DT /ǫ ij ST , (21)where the indices i and j denote the i th tag mode and the j th center-of-mass energy point. The N ij ST and ǫ ij ST(DT) are the number of the ST candidates and the correspond-ing ST (DT) detection efficiency. Using Eq. 21 and the PDG value of the B ( K S → π + π − ) = (69.20 ± B ( D + s → K S K − π + π + ) = (1 . ± . , (22)where the uncertainty is statistical. C. SYSTEMATIC UNCERTAINTIES
The systematic uncertainties for the branching fractionmeasurement are studied in the following categories. • K ± and π ± tracking (PID) efficiencies. The track-ing (PID) efficiencies are studied using samples of e + e − → K + K − π + π − ( e + e − → K + K − K + K − , K + K − π + π − ( π ) and π + π − π + π − ( π )) events.The systematic uncertainties for K ± and π ± dueto tracking (PID) are estimated to be 0.8% and0.3% (0.8% and 0.5%), respectively. • K S reconstruction efficiency. The uncertainty forthe K S reconstruction efficiency is assigned as 1.5%per K S , obtained using control samples of J/ψ → K S K ± π ∓ and φK S K ± π ∓ decays. • Fit to the DT M sig distribution. The uncertaintyassociated with the modeling of the DT M sig distri-bution is studied with alternative models for signaland background. The uncertainties are estimat-ed by comparing with the fit results obtained usingthe signal and background shapes directly from thesimulated samples.0 ) c ) (GeV/ - K M(K E n t r i e s / M e V / c ( a ) ) c ) (GeV/ π M(K E n t r i e s / M e V / c ( b ) ) c ) (GeV/ π - M(K E n t r i e s / M e V / c ( c ) ) c ) (GeV/ π M(K E n t r i e s / M e V / c ( d ) ) c ) (GeV/ π - M(K E n t r i e s / M e V / c ( e ) ) c ) (GeV/ π - K M(K E n t r i e s / M e V / c ( f ) ) c ) (GeV/ π - K M(K E n t r i e s / M e V / c ( g ) ) c ) (GeV/ π π M( E n t r i e s / M e V / c ( h ) ) c ) (GeV/ π π - M(K E n t r i e s / M e V / c ( i )FIG. 2. The projections of (a) M K S K − , (b) M K S π +1 , (c) M K − π +2 , (d) M K S π +2 , (e) M K − π +1 , (f) M K S K − π +1 , (g) M K S K − π +2 , (h) M π − π +2 and (i) M K − π +1 π +2 for the nominal amplitude fit are shown from data samples at E cm between 4.178 and 4.226 GeV.The black points with error bars are data, the red histograms are the results of the nominal amplitude fit, the green shadedhistograms are the scaled GMC combinatorial background. For the identical pions, the one giving a lower K S π + invariant massis denoted as π +1 , the other is denoted as π +2 . • Fit to the ST M tag distribution. We changethe background shape from the second-orderChebychev polynomial to a third-order Chebychevpolynomial, causing a 0.18% relative change of thebranching fraction. The systematic uncertainty dueto the modeling of the signal distribution is deter-mined to be 0.16% by performing an alternative fitusing the shape directly obtained from the simu-lated sample. The quadratic sum of these terms,0.24%, is assigned as the systematic uncertainty. • Measurement method. The possible bias due to themeasurement method is estimated to be 0.3% bycomparing the measured branching fraction in theSMC, using the same method as in data analysis,to the value input in the SMC generation. • Statistics of simulated events. The uncertainty as-sociated with the limited statistics of GMC for thedetection efficiency is 0.3%. • Amplitude analysis model. The uncertainty from the amplitude analysis model is 0.6%, estimatedfrom the efficiency difference obtained by varyingthe fitted parameters c n in Eq. 3 according to theerror matrix.All the systematic uncertainties of the branching frac-tion measurement are listed in Table V. When added inquadrature they sum to a relative uncertainty of 3 . VI. CONCLUSION
Using 6.32 fb − of e + e − collision data collected bythe BESIII detector with center-of-mass energies between4.178 and 4.226 GeV, we report the first amplitude anal-ysis of the D + s → K S K − π + π + decays and an improvedmeasurement of the D + s → K S K − π + π + branching frac-tion. The model indicates that the quasi-two-body de-cay D + s → K ∗ (892) + K ∗ (892) is dominant, with a fit1 TABLE IV. Summary of systematic uncertainties on the φ and FFs from different sources, in units of the correspondingstatistical uncertainties: (I) Kπ S -wave model, (II) lineshape of a (980), (III) effective barrier radius, (IV) quoted masses andwidths, (V) background estimation, (VI) experimental effects, (VII) neglected resonances.Component SourceI II III IV V VI VII Total D + s [ S ] → K ∗ (892) + K ∗ (892) FF 0.18 0.12 0.41 0.43 0.09 0.04 1.55 1.67 D + s [ P ] → K ∗ (892) + K ∗ (892) φ D + s [ D ] → K ∗ (892) + K ∗ (892) φ D + s → K ∗ (892) + K ∗ (892) FF 0.18 0.00 0.24 0.52 0.00 0.00 1.55 1.66 D + s → K ∗ (892) + ( K − π + ) S -wave φ D + s → K ∗ (892) ( K S π + ) S -wave φ D + s → η (1475) π + , η (1475) → a (980) − π + φ D + s → η (1475) π + , η (1475) → K ∗ (892) K S φ D + s → η (1475) π + , η (1475) → K ∗ (892) + K − φ D + s → η (1475) π + , η (1475) → K ∗ (892) K FF 0.04 0.05 0.05 0.09 0.01 0.00 0.66 0.68 D + s → η (1475) π + , η (1475) → ( K S π + ) S -wave K − φ D + s → f (1285) π + , f (1285) → a (980) − π + φ D + s → ( K ∗ (892) + K − ) P π + ,( K ∗ (892) + K − ) P → K ∗ (892) + K − φ K S reconstruction efficiency 1.5DT M sig fit 1.7ST M tag fit 0.2Measurement method 0.3Statistics of simulated events 0.3Amplitude analysis model 0.6 B ( K S → π + π − ) [8] 0.1Total 3.3 fraction of (40 . ± . stat ± . sys )%. In addition, thereare significant contributions from f (1285), η (1475) and( K ∗ (892) + K − ) P in the mass spectrum of K S K − π + .The η (1475) meson decays to both K ∗ K and a (980) π final states, while the f (1285) meson decays only to a (980) π . The absolute branching fraction of the D + s → K S K − π + π + decay is determined to be (1 . ± . stat ± . sys )%, and the branching fractions for different com-ponents are listed in Table VI. The branching fraction ofthe quasi-two-body decay D + s → K ∗ (892) + K ∗ (892) iscalculated to be (5 . ± . stat ± . sys )%. Our mea-surements are consistent with the current world averages[8] but much more precise. ACKNOWLEDGMENTS
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 Research and Development Program of China un-der Contracts Nos. 2020YFA0406300, 2020YFA0406400;National Natural Science Foundation of China (NSFC)under Contracts Nos. 11625523, 11635010, 11735014,11822506, 11835012, 11935015, 11935016, 11935018,11961141012; the Chinese Academy of Sciences (CAS)2 π - K + K → -s D π - π + π → -s D π - π + K → -s D π - π - K + K → -s D - π + π - K S0 K → -s D π - K S0 K → -s D - K S0 K → -s D ’ η - π → -s D η - π → -s D ) c (GeV/ tag M ) × ) ( c E v e n t s / ( M e V / FIG. 3. Best fit results to the M tag distributions of the ST candidates from the data sample taken at E cm = 4.178 GeV. Thepoints with error bars are data. The red solid curves are the fit results. The blue dotted curves are the fitted backgroundshapes. The pair of pink arrows indicates the chosen signal regions. The green dotted curve in the D − s → K S K − ( D − s → π + η ′ )mode is the D − s → K S π − ( D − s → π + π − π − η ) component.TABLE VI. The branching fractions measured in this analysis and from PDG [8]. The K ∗ (892) + , K ∗ (892) and a (980) − denote K ∗ (892) + → K S π + , K ∗ (892) → K − π + and a (980) − → K S K − , respectively.Process BF(10 − )This analysis PDG D + s [ S ] → K ∗ (892) + K ∗ (892) ± ± D + s [ P ] → K ∗ (892) + K ∗ (892) ± ± D + s [ D ] → K ∗ (892) + K ∗ (892) ± ± D + s → K ∗ (892) + K ∗ (892) ± ± ± D + s → K ∗ (892) + ( K − π + ) S − wave ± ± D + s → K ∗ (892) ( K S π + ) S − wave ± ± D + s → η (1475) π + , η (1475) → a (980) − π + ± ± D + s → η (1475) π + , η (1475) → K ∗ (892) K S ± ± D + s → η (1475) π + , η (1475) → K ∗ (892) + K − ± ± D + s → η (1475) π + , η (1475) → K ∗ (892) K ± ± D + s → η (1475) π + , η (1475) → ( K S π + ) S − wave K − ± ± D + s → f (1285) π + , f (1285) → a (980) − π + ± ± D + s → ( K ∗ (892) + K − ) P π + ,( K ∗ (892) + K − ) P → K ∗ (892) + K − ± ± D + s → K S K − π + π + ± ± ± ) c (GeV/ sig M E v e n t s / . M e V / c FIG. 4. Best fit result to the M sig distributions of the DTcandidates from data samples at E cm between 4.178 and4.226 GeV. The points with error bars are data. The redsolid curve is the fit result. The blue dotted curve is thefitted background shape. Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC andCAS under Contracts Nos. U1732263, U1832107, U1832207, U2032104; CAS Key Research Program ofFrontier Sciences under Contracts Nos. 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