Measurement of the \tau lepton polarization and R(D^{*}) in the decay \bar{B}\rightarrow D^{*}\tau^{-}\bar{\nu}_{\tau}with one-prong hadronic \tau decays at Belle

Measurement of the lepton polarization and in the decay
with one-prong hadronic decays at Belle

S. Hirose Graduate School of Science, Nagoya University, Nagoya 464-8602    T. Iijima Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602 Graduate School of Science, Nagoya University, Nagoya 464-8602    I. Adachi High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193    K. Adamczyk H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342    H. Aihara Department of Physics, University of Tokyo, Tokyo 113-0033    S. Al Said Department of Physics, Faculty of Science, University of Tabuk, Tabuk 71451 Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah 21589    D. M. Asner Pacific Northwest National Laboratory, Richland, Washington 99352    H. Atmacan University of South Carolina, Columbia, South Carolina 29208    T. Aushev Moscow Institute of Physics and Technology, Moscow Region 141700    R. Ayad Department of Physics, Faculty of Science, University of Tabuk, Tabuk 71451    T. Aziz Tata Institute of Fundamental Research, Mumbai 400005    V. Babu Tata Institute of Fundamental Research, Mumbai 400005    I. Badhrees Department of Physics, Faculty of Science, University of Tabuk, Tabuk 71451 King Abdulaziz City for Science and Technology, Riyadh 11442    A. M. Bakich School of Physics, University of Sydney, New South Wales 2006    V. Bansal Pacific Northwest National Laboratory, Richland, Washington 99352    M. Berger Stefan Meyer Institute for Subatomic Physics, Vienna 1090    V. Bhardwaj Indian Institute of Science Education and Research Mohali, SAS Nagar, 140306    B. Bhuyan Indian Institute of Technology Guwahati, Assam 781039    J. Biswal J. Stefan Institute, 1000 Ljubljana    A. Bondar Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    A. Bozek H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342    M. Bračko University of Maribor, 2000 Maribor J. Stefan Institute, 1000 Ljubljana    T. E. Browder University of Hawaii, Honolulu, Hawaii 96822    D. Červenkov Faculty of Mathematics and Physics, Charles University, 121 16 Prague    M.-C. Chang Department of Physics, Fu Jen Catholic University, Taipei 24205    P. Chang Department of Physics, National Taiwan University, Taipei 10617    V. Chekelian Max-Planck-Institut für Physik, 80805 München    A. Chen National Central University, Chung-li 32054    B. G. Cheon Hanyang University, Seoul 133-791    K. Chilikin P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991 Moscow Physical Engineering Institute, Moscow 115409    K. Cho Korea Institute of Science and Technology Information, Daejeon 305-806    S.-K. Choi Gyeongsang National University, Chinju 660-701    Y. Choi Sungkyunkwan University, Suwon 440-746    S. Choudhury Indian Institute of Technology Hyderabad, Telangana 502285    D. Cinabro Wayne State University, Detroit, Michigan 48202    T. Czank Department of Physics, Tohoku University, Sendai 980-8578    N. Dash Indian Institute of Technology Bhubaneswar, Satya Nagar 751007    S. Di Carlo Wayne State University, Detroit, Michigan 48202    Z. Doležal Faculty of Mathematics and Physics, Charles University, 121 16 Prague    D. Dutta Tata Institute of Fundamental Research, Mumbai 400005    S. Eidelman Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    J. E. Fast Pacific Northwest National Laboratory, Richland, Washington 99352    T. Ferber Deutsches Elektronen–Synchrotron, 22607 Hamburg    B. G. Fulsom Pacific Northwest National Laboratory, Richland, Washington 99352    R. Garg Panjab University, Chandigarh 160014    V. Gaur Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    N. Gabyshev Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    A. Garmash Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    M. Gelb Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe    A. Giri Indian Institute of Technology Hyderabad, Telangana 502285    P. Goldenzweig Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe    B. Golob Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana J. Stefan Institute, 1000 Ljubljana    Y. Guan Indiana University, Bloomington, Indiana 47408 High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    E. Guido INFN - Sezione di Torino, 10125 Torino    J. Haba High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193    K. Hara High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    K. Hayasaka Niigata University, Niigata 950-2181    H. Hayashii Nara Women’s University, Nara 630-8506    M. T. Hedges University of Hawaii, Honolulu, Hawaii 96822    T. Higuchi Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583    W.-S. Hou Department of Physics, National Taiwan University, Taipei 10617    C.-L. Hsu School of Physics, University of Melbourne, Victoria 3010    K. Inami Graduate School of Science, Nagoya University, Nagoya 464-8602    G. Inguglia Deutsches Elektronen–Synchrotron, 22607 Hamburg    A. Ishikawa Department of Physics, Tohoku University, Sendai 980-8578    R. Itoh High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193    M. Iwasaki Osaka City University, Osaka 558-8585    I. Jaegle University of Florida, Gainesville, Florida 32611    H. B. Jeon Kyungpook National University, Daegu 702-701    Y. Jin Department of Physics, University of Tokyo, Tokyo 113-0033    T. Julius School of Physics, University of Melbourne, Victoria 3010    K. H. Kang Kyungpook National University, Daegu 702-701    G. Karyan Deutsches Elektronen–Synchrotron, 22607 Hamburg    T. Kawasaki Niigata University, Niigata 950-2181    D. Y. Kim Soongsil University, Seoul 156-743    J. B. Kim Korea University, Seoul 136-713    S. H. Kim Hanyang University, Seoul 133-791    Y. J. Kim Korea Institute of Science and Technology Information, Daejeon 305-806    K. Kinoshita University of Cincinnati, Cincinnati, Ohio 45221    P. Kodyš Faculty of Mathematics and Physics, Charles University, 121 16 Prague    S. Korpar University of Maribor, 2000 Maribor J. Stefan Institute, 1000 Ljubljana    D. Kotchetkov University of Hawaii, Honolulu, Hawaii 96822    P. Križan Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana J. Stefan Institute, 1000 Ljubljana    R. Kroeger University of Mississippi, University, Mississippi 38677    P. Krokovny Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    T. Kuhr Ludwig Maximilians University, 80539 Munich    R. Kulasiri Kennesaw State University, Kennesaw, Georgia 30144    R. Kumar Punjab Agricultural University, Ludhiana 141004    A. Kuzmin Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    Y.-J. Kwon Yonsei University, Seoul 120-749    J. S. Lange Justus-Liebig-Universität Gießen, 35392 Gießen    I. S. Lee Hanyang University, Seoul 133-791    L. K. Li Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049    Y. Li Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    L. Li Gioi Max-Planck-Institut für Physik, 80805 München    J. Libby Indian Institute of Technology Madras, Chennai 600036    D. Liventsev Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    M. Lubej J. Stefan Institute, 1000 Ljubljana    T. Matsuda University of Miyazaki, Miyazaki 889-2192    K. Miyabayashi Nara Women’s University, Nara 630-8506    H. Miyata Niigata University, Niigata 950-2181    G. B. Mohanty Tata Institute of Fundamental Research, Mumbai 400005    S. Mohanty Tata Institute of Fundamental Research, Mumbai 400005 Utkal University, Bhubaneswar 751004    H. K. Moon Korea University, Seoul 136-713    T. Mori Graduate School of Science, Nagoya University, Nagoya 464-8602    R. Mussa INFN - Sezione di Torino, 10125 Torino    K. R. Nakamura High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    M. Nakao High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193    T. Nanut J. Stefan Institute, 1000 Ljubljana    K. J. Nath Indian Institute of Technology Guwahati, Assam 781039    Z. Natkaniec H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342    M. Niiyama Kyoto University, Kyoto 606-8502    N. K. Nisar University of Pittsburgh, Pittsburgh, Pennsylvania 15260    S. Nishida High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193    S. Ogawa Toho University, Funabashi 274-8510    S. Okuno Kanagawa University, Yokohama 221-8686    H. Ono Nippon Dental University, Niigata 951-8580 Niigata University, Niigata 950-2181    B. Pal University of Cincinnati, Cincinnati, Ohio 45221    S. Pardi INFN - Sezione di Napoli, 80126 Napoli    C. W. Park Sungkyunkwan University, Suwon 440-746    H. Park Kyungpook National University, Daegu 702-701    S. Paul Department of Physics, Technische Universität München, 85748 Garching    R. Pestotnik J. Stefan Institute, 1000 Ljubljana    L. E. Piilonen Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    V. Popov Moscow Institute of Physics and Technology, Moscow Region 141700    M. Ritter Ludwig Maximilians University, 80539 Munich    A. Rostomyan Deutsches Elektronen–Synchrotron, 22607 Hamburg    M. Rozanska H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342    Y. Sakai High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193    M. Salehi University of Malaya, 50603 Kuala Lumpur Ludwig Maximilians University, 80539 Munich    S. Sandilya University of Cincinnati, Cincinnati, Ohio 45221    Y. Sato Graduate School of Science, Nagoya University, Nagoya 464-8602    O. Schneider École Polytechnique Fédérale de Lausanne (EPFL), Lausanne 1015    G. Schnell University of the Basque Country UPV/EHU, 48080 Bilbao IKERBASQUE, Basque Foundation for Science, 48013 Bilbao    C. Schwanda Institute of High Energy Physics, Vienna 1050    A. J. Schwartz University of Cincinnati, Cincinnati, Ohio 45221    Y. Seino Niigata University, Niigata 950-2181    K. Senyo Yamagata University, Yamagata 990-8560    M. E. Sevior School of Physics, University of Melbourne, Victoria 3010    V. Shebalin Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    T.-A. Shibata Tokyo Institute of Technology, Tokyo 152-8550    N. Shimizu Department of Physics, University of Tokyo, Tokyo 113-0033    J.-G. Shiu Department of Physics, National Taiwan University, Taipei 10617    F. Simon Max-Planck-Institut für Physik, 80805 München Excellence Cluster Universe, Technische Universität München, 85748 Garching    A. Sokolov Institute for High Energy Physics, Protvino 142281    E. Solovieva P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991 Moscow Institute of Physics and Technology, Moscow Region 141700    M. Starič J. Stefan Institute, 1000 Ljubljana    J. F. Strube Pacific Northwest National Laboratory, Richland, Washington 99352    J. Stypula H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342    M. Sumihama Gifu University, Gifu 501-1193    K. Sumisawa High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193    T. Sumiyoshi Tokyo Metropolitan University, Tokyo 192-0397    M. Takizawa Showa Pharmaceutical University, Tokyo 194-8543 J-PARC Branch, KEK Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 Theoretical Research Division, Nishina Center, RIKEN, Saitama 351-0198    U. Tamponi INFN - Sezione di Torino, 10125 Torino University of Torino, 10124 Torino    K. Tanida Advanced Science Research Center, Japan Atomic Energy Agency, Naka 319-1195    F. Tenchini School of Physics, University of Melbourne, Victoria 3010    K. Trabelsi High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193    M. Uchida Tokyo Institute of Technology, Tokyo 152-8550    T. Uglov P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991 Moscow Institute of Physics and Technology, Moscow Region 141700    S. Uno High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193    P. Urquijo School of Physics, University of Melbourne, Victoria 3010    C. Van Hulse University of the Basque Country UPV/EHU, 48080 Bilbao    G. Varner University of Hawaii, Honolulu, Hawaii 96822    K. E. Varvell School of Physics, University of Sydney, New South Wales 2006    V. Vorobyev Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    A. Vossen Indiana University, Bloomington, Indiana 47408    C. H. Wang National United University, Miao Li 36003    M.-Z. Wang Department of Physics, National Taiwan University, Taipei 10617    P. Wang Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049    X. L. Wang Pacific Northwest National Laboratory, Richland, Washington 99352 High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    S. Wehle Deutsches Elektronen–Synchrotron, 22607 Hamburg    E. Widmann Stefan Meyer Institute for Subatomic Physics, Vienna 1090    E. Won Korea University, Seoul 136-713    H. Yamamoto Department of Physics, Tohoku University, Sendai 980-8578    Y. Yamashita Nippon Dental University, Niigata 951-8580    H. Ye Deutsches Elektronen–Synchrotron, 22607 Hamburg    C. Z. Yuan Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049    Y. Yusa Niigata University, Niigata 950-2181    S. Zakharov P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991    Z. P. Zhang University of Science and Technology of China, Hefei 230026    V. Zhilich Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    V. Zhukova P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991 Moscow Physical Engineering Institute, Moscow 115409    V. Zhulanov Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090 Novosibirsk State University, Novosibirsk 630090    A. Zupanc Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana J. Stefan Institute, 1000 Ljubljana
Abstract

With the full data sample of pairs recorded by the Belle detector at the KEKB electron-positron collider, the decay is studied with the hadronic decays and . The polarization in two-body hadronic decays is measured, as well as the ratio of the branching fractions , where denotes an electron or a muon. Our results, and , are consistent with the theoretical predictions of the standard model. The polarization values of are excluded at the 90% confidence level.

pacs:
13.20.He, 14.40.Nd
preprint: Belle Preprint 2017-18 KEK Preprint 2017-26

The Belle Collaboration

I Introduction

Semileptonic decays to leptons (semitauonic decays) are theoretically well-studied processes within the standard model (SM) cite:Heiliger:1989 (); cite:Korner:1990 (); cite:Hwang:2000 (), where the decay process is represented by the tree-level diagram shown in Fig. 1. The lepton is more sensitive to new physics (NP) beyond the SM that couples strongly with mass. A prominent candidate is the two-Higgs-doublet model (2HDM) cite:2HDM:1989 (), where charged Higgs bosons appear. The contribution of the charged Higgs to the decay process  cite:CC () is suggested by many theoretical works (for example, Refs. cite:Grzadkowski:1992 (); cite:Tanaka:1995 (); cite:Soni:1997 (); cite:Itoh:2005 (); cite:Crivellin:2012 ()).

Figure 1: Feynman diagram of for the SM amplitude, where denotes or .

Experimentally, the decays have been studied by Belle cite:Belle:2007 (); cite:Belle:2010 (); cite:Belle:2015 (); cite:Belle:2016 (), BABAR cite:BaBar:2008 (); cite:BaBar:2012:letter (); cite:BaBar:2013:fullpaper () and LHCb cite:LHCb:2015 (). Most of these studies have measured ratios of branching fractions, defined as

(1)

The denominator is the average of for Belle and BABAR, and for LHCb. The ratio cancels numerous uncertainties common to the numerator and the denominator; these include the uncertainty in the Cabibbo-Kobayashi-Maskawa matrix element , many of the theoretical uncertainties on hadronic form factors (FFs), and experimental reconstruction effects. Recently, LHCb measured the mode using the three-prong decay  cite:LHCb:2017 (). To reduce the systematic uncertainty, is measured with the common final states between the numerator and the denominator, and is converted to by using the world-average values for and .

As of early 2016, the results from the three experiments cite:Belle:2015 (); cite:Belle:2016 (); cite:BaBar:2012:letter (); cite:BaBar:2013:fullpaper (); cite:LHCb:2015 () were and standard deviations (cite:HFLAV:2014 () away from the SM predictions of  cite:RD_FermiandMILC:2015 () or  cite:RD_HPQCD:2015 () and  cite:RDst:2012 (), respectively. The overall discrepancy with the SM was about 4. These deviations have been theoretically studied in the context of various NP models cite:RDst:2012 (); cite:Datta:2012 (); cite:Celis:2013 (); cite:Tanaka:2013 (); cite:Biancofiore:2013 (); cite:Dorsner:2013 (); cite:Sakaki:2013 (); cite:Hagiwara:2014 (); cite:Duraisamy:2014 (); cite:Sakaki:2015 (); cite:Freytsis:2015 (); cite:Li:2016 (); cite:Bhattacharya:2016 (); cite:Bardhan:2017 (); cite:Celis:2017 ().

In addition to , the polarizations of the lepton and the meson are sensitive to NP cite:Tanaka:1995 (); cite:Tanaka:2010 (); cite:RDst:2012 (); cite:Datta:2012 (); cite:Biancofiore:2013 (); cite:Tanaka:2013 (); cite:Sakaki:2013 (); cite:Duraisamy:2014 (); cite:Bhattacharya:2016 (); cite:Bardhan:2017 (). The lepton polarization is defined as

(2)

where denotes the decay rate of with a helicity of . The SM predicts  cite:Tanaka:2010 () and  cite:Tanaka:2013 (). For example, the type-II 2HDM allows to be between and for and between and for  cite:Tanaka:2013 (); cite:comment:2HDMII (), whereas a leptoquark model suggested in Ref. cite:Sakaki:2013 () with a leptoquark mass of 1 TeV allows to be between and . can be measured in two-body hadronic decays with the differential decay rate

(3)

where is the angle of the -daughter meson momentum with respect to the direction opposite the momentum of the system in the rest frame of . The parameter describes the sensitivity to for each -decay mode; in particular, for and for  cite:Hagiwara:1990 ().

In this paper, we describe details of the first measurement in the decay with the decays and reported in Ref. cite:Belle:2017 (). Our study includes an measurement independent of previous studies cite:Belle:2015 (); cite:Belle:2016 (); cite:BaBar:2012:letter (); cite:BaBar:2013:fullpaper (); cite:LHCb:2015 (), in which leptonic decays have been used.

Ii Experimental Apparatus

We use the full data sample containing pairs recorded with the Belle detector cite:Belle-detector:2002 () at the asymmetric-beam-energy collider KEKB cite:KEKB:2003 (). 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 (ECL) comprised of CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside the coil is instrumented to detect mesons and to identify muons (KLM). The detector is described in detail elsewhere cite:Belle-detector:2002 (). Two inner detector configurations were used. A 2.0 cm radius beampipe and a 3-layer SVD were used for the first sample of pairs, while a 1.5 cm radius beampipe, a 4-layer SVD and a small-cell inner drift chamber were used to record the remaining pairs cite:SVD2:2006 ().

Iii Monte Carlo Simulation

The Monte Carlo (MC) simulated events are used to establish the analysis criteria, study the background and estimate the signal reconstruction efficiency. Events with a pair are generated using EvtGen cite:EvtGen:2001 (), and the meson decays are reproduced based on branching fractions reported in Ref. cite:PDG:2016 (). The hadronization process of the meson decay with no experimentally-measured branching fraction is inclusively reproduced by Pythia cite:PYTHIA:2006 (). For continuum () events, hadronization of the initial quark pair is described by Pythia, and hadron decays are modeled by EvtGen. Final-state radiation from charged particles is added using Photos cite:PHOTOS:2016 (). Detector responses are reproduced by the Belle detector simulator based on Geantcite:GEANT:1984 (). The MC samples used in this analysis are described below.


The MC sample for the signal mode () is generated with hadronic FFs based on heavy quark effective theory (HQET). The following values of the hadronic FF parameters in the Caprini-Lellouch-Neubert scheme cite:Caprini:1998 (); cite:CLN:note () are used: , , and from the experimental world averages cite:HFLAV:2014 (), and with 10% uncertainty from the HQET estimation cite:RDst:2012 ().


The MC sample for the normalization mode () is generated based on HQET. Since the FF parameters used for the production of the normalization MC sample have been updated as described above, final-state kinematics are corrected to match the latest parameter values.

and


Semileptonic decays and , where denotes the excited charm meson states heavier than , comprise an important background category as they have a similar decay topology to the signal events. The MC sample for is generated based on the Isgur-Scora-Grinstein-Wise (ISGW) model cite:ISGW2:1995 (), and decay kinematics are corrected to match the Leibovich-Ligeti-Stewart-Wise (LLSW) model cite:LLSW:1998 (). The branching fractions for with , , and are taken from the world averages cite:HFLAV:2014 (). For the decays, in addition to experimentally-measured modes, we allow unmeasured final states consisting of a and one or two pions, a meson, or an meson based on quantum-number, phase-space and isospin considerations. The radially-excited modes are included so that the total branching fraction of becomes about 3%, which is expected from the difference between (where denotes all the possible charmed-meson states) and the sum of the exclusive branching fractions of . The MC sample is generated using the ISGW model. We take the branching fractions from the theoretical estimates of for each state cite:RDstst:2017 (). We use the average of the four approximations discussed in Ref cite:RDstst:2017 (). We do not consider or other semitauonic modes containing a charmed state heavier than as their small phase space suppresses the branching fractions.

Other background


The MC samples for other background processes, both events and continuum events, are generated based on the past experimental studies reported in Ref. cite:PDG:2016 (). Unmeasured decay channels are generated with Pythia through the inclusive hadronization process.

The MC sample sizes of the signal mode, the normalization mode, , , the background, and the process are 40, 10, 40, 400, 10, and 5 times larger, respectively, than the full Belle data sample.

Iv Event Reconstruction

iv.1 Reconstruction of the tag side

We conduct the analysis by first identifying events where one of the two mesons () is reconstructed in one of 1104 exclusive hadronic decays cite:Full-recon:2011 (). A hierarchical multivariate algorithm based on the NeuroBayes neural-network package is employed. More than 100 input variables are used to determine well-reconstructed candidates, including the difference between the energy of the reconstructed candidate and the beam energy in the center-of-mass (CM) frame , as well as the event shape variables for suppression of background. The quality of the candidate is synthesized in a single NeuroBayes output-variable classifier (). We require the beam-energy-constrained mass of the candidate , where is the reconstructed three-momentum in the CM frame, to be greater than 5.272 GeV and the value of to be between and . Throughout the paper, natural units with are used. We place a requirement on such that about 90% of true and about 30% of fake candidates are retained. If two or more candidates are retained in one event, we select the one with the highest .

Due to limited knowledge of hadronic decays, the branching fractions of the decay modes are not perfectly modeled in the MC simulation. It is therefore essential to calibrate the reconstruction efficiency (tagging efficiency) with control data samples. We determine a scale factor for each decay mode using events where the signal-side meson candidate () is reconstructed in modes. Further details of the calibration method are described in Ref. cite:Belle_Xulnu:2013 (). The ratio of measured to expected rates in each decay mode ranges from 0.2 to 1.4, depending on the decay mode, and is 0.72 on average. After the efficiency calibration, the tagging efficiencies are estimated to be about 0.20% for charged mesons and 0.15% for neutral mesons.

iv.2 Reconstruction of the signal side

We reconstruct the signal mode and the normalization mode using the particle candidates not used for reconstruction. The following decay modes are used for the daughter particles: , , , and for the candidate; and for the candidate; , , , , , , , , , , , , , , and for the candidate; and , and , respectively, for the light-meson candidates. A -daughter candidate or is combined with a candidate to form a candidate. For the normalization events, a charged lepton or is associated instead of or .

iv.2.1 Particle selection

First, daughter particles of and (, , , , and ) and charged leptons ( and ) are reconstructed. For reconstruction, we use different particle selections from those applied for the reconstruction described in Ref. cite:Full-recon:2011 ().

Charged particles are reconstructed using the SVD and the CDC. All tracks, except for -daughter candidates, are required to have  cm and  cm, where and are the impact parameters to the interaction point (IP) in the directions perpendicular and parallel, respectively, to the beam axis. Charged-particle types are identified by a likelihood ratio based on the responses of the sub-detector systems. Identification of and candidates is performed by combining measurements of specific ionization () in the CDC, the time of flight from the IP to the TOF counter and the photon yield in the ACC. For -daughter candidates, an additional proton veto is required in order to reduce background from baryonic decays such as . The ECL electromagnetic shower shape, track-to-cluster matching at the inner surface of the ECL, in the CDC, the photon yield in the ACC and the ratio of the cluster energy in the ECL to the track momentum measured with the SVD and the CDC are used to identify candidates cite:eID:2002 (). Muon candidates are selected based on their penetration range and transverse scattering in the KLM cite:muID:2002 (). To form candidates, we combine pairs of oppositely-charged tracks, treated as pions. Standard Belle selection criteria are applied cite:Belle_KsKsKs:2005 (): the reconstructed vertex must be detached from the IP, the momentum vector must point back to the IP, and the invariant mass must be within 30 MeV of the nominal mass cite:PDG:2016 (), which corresponds to about 8. (In this section, denotes the corresponding mass resolution.)

Photons are reconstructed using ECL clusters not matching to charged tracks. Photon energy thresholds of 50, 100 and 150 MeV are used in the barrel, forward-endcap and backward-endcap regions, respectively, of the ECL to reject low-energy background photons, such as those originating from the beams and hadronic interactions of particles with materials in the detector.

Neutral pions are reconstructed in the decay . For candidates from or decay, referred to as normal s, we impose the same photon energy thresholds described above. The candidate’s invariant mass must lie between 115 and 150 MeV, corresponding to about around the nominal mass cite:PDG:2016 (). In order to reduce the number of fake candidates, we apply the following candidate selection procedure. The candidates are sorted in descending order according to the energy of the most energetic daughter. If a given photon is the most energetic daughter of two or more candidates, they are sorted by the energy of the lower-energy daughter. We then retain the candidates whose daughter photons are not shared with a higher-ranked candidate. In this criterion, 76% of the correctly reconstructed candidates are selected while 54% of the fake candidates are removed. The retained candidates are used for and reconstruction described later.

For the soft from decay, we impose a relaxed photon energy threshold of 22 MeV in all ECL regions and the same requirement for the invariant mass of the two photons. Additionally, the energy asymmetry is required to be less than 0.6, where and are the energies of the high- and low-energy photon daughters in the laboratory frame. Here, we do not apply the normal- candidate selection procedure.

The candidate is formed from the combination of a and a . The candidate invariant mass must lie between 0.66 and 0.96 GeV.

iv.2.2 reconstruction

After reconstructing the light mesons, we reconstruct the candidates in 15 decay modes. The invariant mass requirements are optimized for each decay mode. For the modes used in forming candidates, the reconstructed invariant masses () are required to be within () of the nominal meson mass cite:PDG:2016 () for the high (low) signal-to-noise ratio (SNR) modes. For candidates, the requirements are loosened to and for the high- and low-SNR modes, respectively. The requirements for the candidates are for the high-SNR modes and for the low-SNR modes around the nominal meson mass cite:PDG:2016 (). Here, the high-SNR modes are , , , , , , , ; the low-SNR modes are all remaining modes. We reconstruct candidates by combining a candidate with a , , or soft . The candidates are selected based on the mass difference , where denotes the reconstructed invariant mass of the candidate. The , , , and candidates are required to have within , , and , respectively, of the nominal .

iv.2.3 selection

The candidates are formed by associating a -daughter meson (signal events) or a (normalization events) with a candidate. Allowed combinations are for , for , for and for , where or . We select one of the following meson combinations: , , and .

For the signal mode, if at least one possible candidate for the signal mode is found in an event, we calculate in the rest frame of the . Although this frame cannot be determined completely, equivalent kinematic information is obtained using the rest frame of the system. This frame is obtained by boosting the laboratory frame along with the three-momentum vector component of the momentum transfer

(4)

where denotes the four-momentum of the beam, , and , respectively. In this frame, the energy and the magnitude of the momentum of the lepton are determined only by as

(5)
(6)

where is the lepton mass. The cosine of the angle between the momenta of the lepton and its daughter meson is determined by

(7)

where and denote the energy and the momentum of the lepton (the daughter ) respectively, and is the mass of the daughter. Through a Lorentz transformation from the rest frame of the system to the rest frame, the following relation is obtained:

(8)

where is the -daughter momentum in the rest frame of , and and . Solving gives the value of . Events are required to lie in the physical region of , where 97% of the reconstructed signal events are retained. As shown in Fig. 2, there is a significant background peak near 1 in the sample due to the background. To reject this background, we only use the region in the fit to the sample.

Figure 2: Distribution of (MC) for the signal (red), (blue-hatched) and the other background (black-hatched) in the sample. The SM prediction on is assumed. All the signal selection requirements including and are applied.

Due to the kinematic constraint that must be greater than , almost no signal events exist with below 4 GeV. Therefore is required. The variable is the linear sum of the energy of ECL clusters not used in the event reconstruction. The ECL clusters satisfying the photon-energy requirement defined in the previous section are added to . Signal events ideally have equal to zero with a tail in the distribution from the beam background and split-off showers, separated from the main ECL cluster and reconstructed as photon candidates. We require to be less than 1.5 GeV.

For the normalization mode, we calculate the squared missing mass,

(9)

where denotes the four-momentum of the charged lepton and the other variables were defined earlier. The normalization events populate the region near because there is exactly one neutrino in an event. We require . We further require to be less than 1.5 GeV.

Finally, for both the signal and the normalization events, we require that there be no extra charged tracks with and , and normal candidates.

iv.3 Best candidate selection

After event reconstruction, the average number of retained candidates per event is about 1.09 for charged mesons and 1.03 for neutral mesons. In events where two or more candidates are reconstructed, 2.1 candidates are found on average. Multiple-candidate events mostly arise from more than one combination of a candidate with photons or soft pions. For the charged mode, about 2% of the events are reconstructed both in the and modes. Since the latter mode has a much higher branching fraction, we assign these events to the sample. The contribution of this type of multiple-candidate events is negligibly small in the neutral mode. We then select the most signal-like candidate as follows. For the events, we select the candidate with the most energetic photon associated with the . For the and events, we select the candidate with the soft that has an invariant mass nearest the nominal mass. For the events, we select one candidate at random since the multiple-candidate probability is only . After the candidate selection, roughly 2% of the retained events are reconstructed both in the and the samples. Since the MC study indicates that about 80% of such events originate from the decay, we assign these events to the sample.

iv.4 Sample composition

The reconstructed events are categorized in turn as below. Based on this categorization, we construct histogram probability density functions (PDFs) from the MC samples to perform a final fit.

Signal


Correctly reconstructed signal events that originate from events are categorized in this component. The yield is treated as a free parameter determined by and .

cross feed


Cross feed events, where the events are reconstructed as due to misreconstruction of one , or the events are reconstructed as by adding a random , comprise this component. Since these events originate from , they contribute to the determination. They are also used for the determination after the bias on is corrected by MC information.

Other cross feed


events with other decay modes also contribute to the signal sample. They originate mainly from with one or two missing , or with a low-momentum that does not reach the KLM. These two modes occupy about 80% of this component. The MC study shows that the cross feed events both from and have negligible impact on our measurement. In the fit, the yield of this category is determined by .


The decay contaminates the signal sample due to misassignment of as . We fix the yield in the signal sample from the fit to the distribution of the normalization sample.

and hadronic decays


The ( is also included in this category) and hadronic decays are the most uncertain component due to limited experimental knowledge. By missing a few particles such as mesons, the event topology resembles the signal event. We combine these decay modes into one component. The fractions of the decays and hadronic decays are about 10% and 90%, respectively, according to the MC study. Since it is difficult to estimate the yield of this component using MC simulation or to fix the yield using control data samples, we float the yield in the final fit. One exception is the collection of modes with two charm mesons such as and . Since the branching fractions of these modes have been studied experimentally, we fix their yield using the MC expectation after correction with the branching fractions based on Ref. cite:PDG:2016 ().

Continuum


Continuum events from the process provide a minor contribution at in the signal sample. We fix the yield using the MC expectation.

Fake


All events containing fake candidates are categorized in this component. This is the main background source in the charged meson sample. For the neutral sample, many candidates are reconstructed from the combination of a with a and therefore much more cleanly reconstructed than the other modes with or . The yield is determined from a comparison of the data and the MC sample in the sideband regions.

iv.5 Measurement Method of and

We use the following variables to measure yields of the signal and the normalization modes. For the normalization mode, is the most suitable variable due to its high purity. On the other hand, the shape of the distribution for the signal mode has a strong correlation with . To measure the signal yield, we use because it has a small correlation to and provides good discrimination between the signal and the background modes.

The value of is measured using the formula

(10)

where denotes the relevant branching fraction, and and ( and ) are the efficiencies (the observed yields) for the signal and the normalization modes, respectively. The indices and represent the decays ( or ) and the charges (charged or neutral ), respectively. Assuming isospin symmetry, we use .

Figure 3: Comparison of the distributions between the data (black circles) and the MC simulation (red rectangles) of the normalization mode. The area of the histograms are normalized to unity.

The value of is determined using the formula

(11)

where denotes the signal yield in the region and satisfies . This formula is obtained by calculating

(12)
(13)

The differential decay rate is given by Eq. (3). As with , we use the common parameters .

Due to detector efficiency effects, the measured polarization, , is biased from the true value of . To correct for this bias, we form a linear function that maps to using several MC sets with different . This function, denoted the correction function, is separately prepared for each sample since the detector bias depends on the given mode. We also make a correction function for the cross feed component to take into account the distortion of the distribution shape. In the correction, other kinematic distributions are assumed to be consistent with the SM predictions.

Figure 4: Comparisons between the data (black circles) and the MC simulation (red rectangles) in the sideband regions, where the distributions are normalized to unity. (a) distribution, (b) distribution for the mode, (c) distribution for the mode. All the corresponding channels are combined.

V Background Calibration and PDF Validation

To use the MC distributions as histogram PDFs, the MC simulation needs to be verified using calibration data samples. In this section, the calibration of the PDF shapes is discussed.

v.1 Signal PDF shape

To validate the shape of the signal component, we use the normalization mode as the control sample. It has similar properties to the signal component; there is no extra photon from the decay except for bremsstrahlung photons, and therefore the shape is mostly determined by the background photons. The normalization sample contains about 50 times more events than the expected signal yield. Figure 3 shows a comparison of between data and MC simulation. The pull of each bin is shown in the bottom panel; hereinafter, the pull in the th bin is defined as

(14)

where and denote the number of events and the statistical error, respectively, in the th bin of the data (MC) distribution. The fake yield is scaled based on the calibration discussed in the next section. Since the contribution from the other background components is negligibly small, it is fixed to the MC expectation. The shape in the MC sample agrees well with the data within statistical uncertainty.

v.2 Fake events

One of the most significant background components arises from fake candidates. The combinatorial fake background processes are difficult to model precisely in the MC simulation. The shapes for the data and the MC sample are compared using sideband regions of 50–500 MeV, 135–190 MeV, 135–190 MeV, and 140–500 MeV for , , , and , respectively; each excludes about around the peak. These sideband regions contain 5 to 50 times more events than the signal region. Figure 4 shows the comparison of the shapes. Although all the and modes are combined in these figures, the shape has been compared in 16 subsamples of modes, modes, modes, and the two regions. We find good agreement of the shape within the statistical uncertainty of these mass sideband data samples. We also check the distribution in the sideband region, as shown in Figs. 4 and 4. The distribution in the MC simulation also shows good agreement with the data within the statistical uncertainty.

In both the signal and the normalization samples, yield discrepancies of up to 20% are observed. The fake yields in the signal region of the MC simulation are scaled by the yield ratios of the data to the MC sample in the sideband regions.

v.3 and hadronic composition

Figure 5: Distribution of the for the sample. The solid histogram shows the MC distribution (the red-filled and the white components for the correct and fake candidates, respectively) and the black dots are the data distribution.
      decay mode