Angular analysis of B^{0}\rightarrow\phi K^{*} decays and search for CP violation at Belle

Angular analysis of decays and search for violation at Belle

M. Prim Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe    I. Adachi High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    H. Aihara Department of Physics, University of Tokyo, Tokyo 113-0033    D. M. Asner Pacific Northwest National Laboratory, Richland, Washington 99352    T. Aushev Institute for Theoretical and Experimental Physics, Moscow 117218    A. M. Bakich School of Physics, University of Sydney, NSW 2006    A. Bala Panjab University, Chandigarh 160014    B. Bhuyan Indian Institute of Technology Guwahati, Assam 781039    G. Bonvicini Wayne State University, Detroit, Michigan 48202    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    P. Chen Department of Physics, National Taiwan University, Taipei 10617    B. G. Cheon Hanyang University, Seoul 133-791    R. Chistov Institute for Theoretical and Experimental Physics, Moscow 117218    K. Cho Korea Institute of Science and Technology Information, Daejeon 305-806    V. Chobanova Max-Planck-Institut für Physik, 80805 München    Y. Choi Sungkyunkwan University, Suwon 440-746    D. Cinabro Wayne State University, Detroit, Michigan 48202    M. Danilov Institute for Theoretical and Experimental Physics, Moscow 117218 Moscow Physical Engineering Institute, Moscow 115409    Z. Doležal Faculty of Mathematics and Physics, Charles University, 121 16 Prague    Z. Drásal Faculty of Mathematics and Physics, Charles University, 121 16 Prague    D. Dutta Indian Institute of Technology Guwahati, Assam 781039    S. Eidelman Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090    H. Farhat Wayne State University, Detroit, Michigan 48202    M. Feindt Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe    T. Ferber Deutsches Elektronen–Synchrotron, 22607 Hamburg    A. Frey II. Physikalisches Institut, Georg-August-Universität Göttingen, 37073 Göttingen    V. Gaur Tata Institute of Fundamental Research, Mumbai 400005    S. Ganguly Wayne State University, Detroit, Michigan 48202    R. Gillard Wayne State University, Detroit, Michigan 48202    Y. M. Goh Hanyang University, Seoul 133-791    B. Golob Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana J. Stefan Institute, 1000 Ljubljana    H. Hayashii Nara Women’s University, Nara 630-8506    M. Heider Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe    Y. Hoshi Tohoku Gakuin University, Tagajo 985-8537    W.-S. Hou Department of Physics, National Taiwan University, Taipei 10617    Y. B. Hsiung Department of Physics, National Taiwan University, Taipei 10617    T. Iijima Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602 Graduate School of Science, Nagoya University, Nagoya 464-8602    K. Inami Graduate School of Science, Nagoya University, Nagoya 464-8602    A. Ishikawa Tohoku University, Sendai 980-8578    R. Itoh High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    I. Jaegle University of Hawaii, Honolulu, Hawaii 96822    T. Julius School of Physics, University of Melbourne, Victoria 3010    D. H. Kah Kyungpook National University, Daegu 702-701    H. Kawai Chiba University, Chiba 263-8522    T. Kawasaki Niigata University, Niigata 950-2181    H. Kichimi High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    C. Kiesling Max-Planck-Institut für Physik, 80805 München    D. Y. Kim Soongsil University, Seoul 156-743    H. O. Kim Kyungpook National University, Daegu 702-701    J. B. Kim Korea University, Seoul 136-713    J. H. Kim Korea Institute of Science and Technology Information, Daejeon 305-806    M. J. Kim Kyungpook National University, Daegu 702-701    Y. J. Kim Korea Institute of Science and Technology Information, Daejeon 305-806    K. Kinoshita University of Cincinnati, Cincinnati, Ohio 45221    J. Klucar J. Stefan Institute, 1000 Ljubljana    B. R. Ko Korea University, Seoul 136-713    P. Kodyš Faculty of Mathematics and Physics, Charles University, 121 16 Prague    P. Križan Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana J. Stefan Institute, 1000 Ljubljana    P. Krokovny Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090    B. Kronenbitter Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe    T. Kuhr Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe    T. Kumita Tokyo Metropolitan University, Tokyo 192-0397    Y.-J. Kwon Yonsei University, Seoul 120-749    J. S. Lange Justus-Liebig-Universität Gießen, 35392 Gießen    S.-H. Lee Korea University, Seoul 136-713    J. Li Seoul National University, Seoul 151-742    J. Libby Indian Institute of Technology Madras, Chennai 600036    P. Lukin Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090    D. Matvienko Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090    K. Miyabayashi Nara Women’s University, Nara 630-8506    H. Miyata Niigata University, Niigata 950-2181    R. Mizuk Institute for Theoretical and Experimental Physics, Moscow 117218 Moscow Physical Engineering Institute, Moscow 115409    G. B. Mohanty Tata Institute of Fundamental Research, Mumbai 400005    A. Moll Max-Planck-Institut für Physik, 80805 München Excellence Cluster Universe, Technische Universität München, 85748 Garching    N. Muramatsu Research Center for Electron Photon Science, Tohoku University, Sendai 980-8578    R. Mussa INFN - Sezione di Torino, 10125 Torino    I. Nakamura High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    E. Nakano Osaka City University, Osaka 558-8585    M. Nakao High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    M. Nayak Indian Institute of Technology Madras, Chennai 600036    E. Nedelkovska Max-Planck-Institut für Physik, 80805 München    C. Niebuhr Deutsches Elektronen–Synchrotron, 22607 Hamburg    N. K. Nisar Tata Institute of Fundamental Research, Mumbai 400005    S. Nishida High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    O. Nitoh Tokyo University of Agriculture and Technology, Tokyo 184-8588    Y. Onuki Department of Physics, University of Tokyo, Tokyo 113-0033    G. Pakhlova Institute for Theoretical and Experimental Physics, Moscow 117218    H. Park Kyungpook National University, Daegu 702-701    H. K. Park Kyungpook National University, Daegu 702-701    T. K. Pedlar Luther College, Decorah, Iowa 52101    R. Pestotnik J. Stefan Institute, 1000 Ljubljana    M. Petrič J. Stefan Institute, 1000 Ljubljana    L. E. Piilonen CNP, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    M. Ritter Max-Planck-Institut für Physik, 80805 München    M. Röhrken Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe    A. Rostomyan Deutsches Elektronen–Synchrotron, 22607 Hamburg    M. Rozanska H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342    H. Sahoo University of Hawaii, Honolulu, Hawaii 96822    T. Saito Tohoku University, Sendai 980-8578    Y. Sakai High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    S. Sandilya Tata Institute of Fundamental Research, Mumbai 400005    T. Sanuki Tohoku University, Sendai 980-8578    Y. Sato Tohoku University, Sendai 980-8578    V. Savinov University of Pittsburgh, Pittsburgh, Pennsylvania 15260    O. Schneider École Polytechnique Fédérale de Lausanne (EPFL), Lausanne 1015    G. Schnell University of the Basque Country UPV/EHU, 48080 Bilbao Ikerbasque, 48011 Bilbao    C. Schwanda Institute of High Energy Physics, Vienna 1050    D. Semmler Justus-Liebig-Universität Gießen, 35392 Gießen    K. Senyo Yamagata University, Yamagata 990-8560    M. E. Sevior School of Physics, University of Melbourne, Victoria 3010    M. Shapkin Institute for High Energy Physics, Protvino 142281    C. P. Shen Beihang University, Beijing 100191    T.-A. Shibata Tokyo Institute of Technology, Tokyo 152-8550    J.-G. Shiu Department of Physics, National Taiwan University, Taipei 10617    A. Sibidanov School of Physics, University of Sydney, NSW 2006    Y.-S. Sohn Yonsei University, Seoul 120-749    A. Sokolov Institute for High Energy Physics, Protvino 142281    E. Solovieva Institute for Theoretical and Experimental Physics, Moscow 117218    M. Starič J. Stefan Institute, 1000 Ljubljana    M. Steder Deutsches Elektronen–Synchrotron, 22607 Hamburg    T. Sumiyoshi Tokyo Metropolitan University, Tokyo 192-0397    U. Tamponi INFN - Sezione di Torino, 10125 Torino University of Torino, 10124 Torino    G. Tatishvili Pacific Northwest National Laboratory, Richland, Washington 99352    Y. Teramoto Osaka City University, Osaka 558-8585    K. Trabelsi High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    T. Tsuboyama High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    M. Uchida Tokyo Institute of Technology, Tokyo 152-8550    S. Uehara High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    T. Uglov Institute for Theoretical and Experimental Physics, Moscow 117218 Moscow Institute of Physics and Technology, Moscow Region 141700    Y. Unno Hanyang University, Seoul 133-791    S. Uno High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801    Y. Usov Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090    S. E. Vahsen University of Hawaii, Honolulu, Hawaii 96822    C. Van Hulse University of the Basque Country UPV/EHU, 48080 Bilbao    P. Vanhoefer Max-Planck-Institut für Physik, 80805 München    G. Varner University of Hawaii, Honolulu, Hawaii 96822    V. Vorobyev Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090    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 CNP, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    M. Watanabe Niigata University, Niigata 950-2181    Y. Watanabe Kanagawa University, Yokohama 221-8686    K. M. Williams CNP, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    E. Won Korea University, Seoul 136-713    Y. Yamashita Nippon Dental University, Niigata 951-8580    S. Yashchenko Deutsches Elektronen–Synchrotron, 22607 Hamburg    Z. P. Zhang University of Science and Technology of China, Hefei 230026    V. Zhilich Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090    A. Zupanc Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe
Abstract

We report the measurements of branching fractions and violation asymmetries in decays obtained in an angular analysis using the full data sample of pairs collected at the resonance with the Belle detector at the KEKB asymmetric-energy collider. We perform a partial wave analysis to distinguish among scalar [], vector [] and tensor [] components, and determine the corresponding branching fractions to be , and . We also measure the longitudinal polarization fraction in and decays to be and , respectively. The first quoted uncertainties are statistical and the second are systematic. In total, we measure 26 parameters related to branching fractions, polarization and violation in the system. No evidence for violation is found.

pacs:
13.25.Hw, 11.30.Er, 13.88.+e
preprint: Belle Preprint # 2013-16 KEK Preprint # 2013-26

The Belle Collaboration

I Introduction

In the standard model (SM) of electroweak interactions, the effect of violation is explained by a single irreducible phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix C (); KM (). So far, analyses HFAG () searching for violation have shown no significant deviation with respect to the SM predictions.

The CKM mechanism alone is not sufficient to explain the observed matter-antimatter asymmetry in the universe, and thus new sources of violation are necessary. Decays dominated by penguin (loop) transitions in the SM, such as , as shown in Fig. 1, are sensitive to such new contributions. New particles could appear in virtual loops, resulting in significant deviations from the SM expectations of negligible direct violation. Previous studies by Belle Belle_phiK () and BaBar BaBar_phiK () in did not find any evidence for violation. On the other hand, the longitudinal polarization fractions (Belle) and (BaBar) in this decay were found to deviate from a naive expectation based on the factorization approach Polarization (), which predicts a longitudinal polarization fraction close to unity. In contrast, BaBar measured the longitudinal polarization fraction in to be  BaBar_phiK (), consistent with the factorization prediction.

Figure 1: Penguin diagram of the decay .

In this paper, we present an improved analysis of the  CC () system using the full Belle data sample collected at the resonance. We perform a partial wave analysis to distinguish among the different states. Overall, 26 parameters related to branching fractions, polarization, interference effects and violation are measured.

The measurement of polarization in flavor specific decays can be used further to distinguish between -even and -odd fractions in the decay . This decay channel can also be used for a time-dependent measurement of the angle  alpha () of the CKM unitarity triangle in transitions.

Ii Analysis strategy

We perform a partial wave analysis of the system with and . We use the notation to indicate all possible contributions from scalar (S-wave, spin ), vector (P-wave, ) and tensor (D-wave, ) components from , and , respectively. We assume no further resonant contributions. The analysis region is limited to a invariant mass below  GeV, as the LASS model LASS (), used to parametrize the S-wave contribution, is not valid above this value. Furthermore, no significant contribution from states beyond  GeV is observed BaBar_highmass_states (). We use mass and angular distributions to distinguish among the three contributing channels , , and , and to determine the polarization in vector–vector and vector–tensor decays, as well as a number of parameters related to violation. We also determine the branching fraction for each of the three channels.

We first explain the parametrization of the angular distribution, which is followed by a description of the invariant-mass distribution. Finally, we derive the combined model of mass and angular distributions of partial waves used for the parameter extraction in a maximum likelihood fit.

ii.1 Angular distribution

The angular distribution in the system with and is described by the three helicity angles , , and , which are defined in the rest frame of the parent particles as illustrated in Fig. 2.

Figure 2: Definition of the three helicity angles given in the rest frame of the parent particles for the decay.

In general, due to the angular momentum conservation, the partial decay width for a two-body decay of a pseudoscalar meson into particles with spins and is given by

(1)

where are the spherical harmonics, the sum is over the helicity states , and is the complex weight of the corresponding helicity amplitude. The parameter takes all discrete values between and , with being the smaller of the two daughter particle spins and . As the is a vector meson, in this analysis, whereas for , for , and for . The partial decay width of each partial wave with spin is therefore

(2)

with being the complex weight of the corresponding helicity amplitude of the partial wave with spin .

The helicity basis is not a basis of eigenstates. Polarization measurements are commonly performed in the transversity basis of eigenstates with the transformation for two of the amplitudes. In this basis, the longitudinal polarization and the parallel polarization are even under transformation while the perpendicular component is -odd. Throughout this article, we use for and for related complex weights of the helicity and transversity amplitudes. Furthermore, depending on the context, we use either of the two bases with or . Where necessary, we explicitly state the basis used. We use polar coordinates to define the complex weights and apply the same implicit definition of the basis; e. g. would be the magnitude of the perpendicular D-wave component in the transversity basis.

ii.2 Mass distribution

To distinguish among different partial waves, we study their invariant-mass spectrum . To parametrize the lineshape of the P- and D-wave components as a function of the invariant mass , we use a relativistic spin-dependent Breit–Wigner (BW) amplitude  PDG ():

(3)

where we use the convention

(4)

For spin and , the mass-dependent widths are given by

(5)
(6)

where is the resonance width, the resonance mass, the momentum of a daughter particle in the rest frame of the resonance, this momentum evaluated at , and the interaction radius. This parametrization of the mass-dependent width uses the Blatt–Weisskopf penetration factors PDG ().

The S-wave component is parametrized using scattering results from the LASS experiment LASS (). It was found by LASS that the scattering is elastic up to about  GeV and thus can be parametrized as

(7)

where

(8)

representing a resonant contribution from while denoting a non-resonant contribution. The resonant part is defined as

(9)

where and are the resonance mass and width, and is given by

(10)

The non-resonant part is defined as

(11)

where is the scattering length and is the effective range.

The amplitude is obtained by multiplying the lineshape with the two-body phase space factor

(12)

The resonance parameters used in the analysis are given in Table 1.

Parameter
(MeV)
(MeV)
(GeV)
(GeV)
(GeV)
Table 1: Resonance parameters for S-, P-, and D-wave components. The parameters and for P- and D-wave are taken from Ref. PDG (), and interaction radii and S-wave parameters are taken from Ref. BaBar_phiK (), which includes updated values with respect to Ref. LASS ().

ii.3 Mass-angular distribution

We combine the mass distribution with the angular distribution to obtain the partial decay width

(13)

where is a phase space factor that takes into account the three-body kinematics in . As we expect no resonant charmless structure in the invariant-mass distribution, we assume a constant amplitude that can be computed for each value of following the section on kinematics in Ref. PDG () as

(14)

with () being the maximum (minimum) value of the Dalitz plot range of the invariant mass at a given value.

Parameter Definition
Table 2: Definitions of the 26 real parameters that are measured in the system. Three partial waves with spin are considered in the spectrum. The amplitude weights are defined in the text. The extra in the definition of and accounts for the sign flip of under transformation.

The matrix element squared is given by the coherent sum of the corresponding S-, P-, and D-wave amplitudes as

(15)

where we have omitted the explicit dependence of on for readability. Each partial wave for a given spin is parametrized as the product of the angular distribution from Eq. (2) and the mass distribution from Eq. (12). For the S-, P-, and D-wave, we obtain

(16)
(17)

and

(18)

respectively.

Overall, the seven complex helicity amplitudes contributing to these formulas can be parametrized by 14 real parameters (28 if and are measured independently).

We define the normalized partial decay width as

(19)

where [] is the matrix element for [], is depending on the charge of the primary charged kaon from the meson and is the overall normalization given by

(20)

By averaging the normalization over and , we can perform a simultaneous fit with a single reference amplitude of fixed magnitude, which defines the relative strengths of the amplitudes. If both final states are normalized independently, each with its own reference amplitude, and violation is observed, the interpretation of whether violation is in the reference amplitudes or all other amplitudes would be ambiguous.

Using these notations, we define the final set of parameters used in the analysis. For the matrix element , we define the weights as and, for , as . With defined as

(21)

and given by

(22)

where we use one -conserving and one -violating parameter per magnitude and phase. For only is possible, whereas, for and , three values and are allowed.

We choose as our reference phase, as the system is invariant under a global phase transformation. This effectively reduces the 28 parameters by one. Of the remaining 27 parameters, 26 can be measured in the system with . These 26 parameters can be used to define a more common set of parameters shown in Table 2, which are used in the review of polarization in decays in Ref. PDG (). For each partial wave , we define parameters such as the longitudinal (perpendicular) polarization fractions (), the relative phase of the parallel (perpendicular) amplitude () to the longitudinal amplitude, and strong phase difference between the partial waves and a number of parameters related to violation. The 27th parameter, , could only be measured in a time-dependent analysis of violation in decays that is beyond the scope of this analysis, so we fix . Furthermore, we fix as it has the largest relative magnitude among all amplitudes and choose it as our reference amplitude. Fixing does not decrease the number of free parameters as the absolute magnitude, defined by the signal yield, remains a free parameter in the fit. Overall, we are left with 26 real parameters to be determined.

In the previous analysis Belle_phiK (), a twofold phase ambiguity was observed in the decay of ; this is a fourfold ambiguity if and are measured independently, as the sets and solve all angular equations. Even the interference terms in are invariant under such transformation if we flip the sign of the strong phase . However, the mass dependence of is unique: it either increases or decreases with increasing invariant mass. We solve this ambiguity for and using Wigner’s causality principle Wigner (), which states that the phase of a resonance increases with increasing invariant mass.

From the measured weights, we can also calculate the triple-product correlations in , given in our previous measurement. The -odd quantities

(23)

from Ref. TripleProduct (); TripleProduct_note () and the corresponding asymmetries between and are sensitive to -odd violation.

Iii Event reconstruction

iii.1 Data sample and detector

We use the full Belle data sample, consisting of an integrated luminosity of containing pairs collected at the resonance at the KEKB asymmetric-energy (3.5 on 8 GeV) collider KEKB (). An additional data sample of integrated luminosity collected 60 MeV below the resonance, referred to as the off-resonance data, is utilized for background studies.

The Belle detector Belle () is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector, 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 composed of CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect mesons and to identify muons. Two inner detector configurations were used. A 2.0 cm beampipe and a 3-layer silicon vertex detector were used for the first sample of pairs, while a 1.5 cm beampipe, a 4-layer silicon detector and a small-cell inner drift chamber were used to record the remaining pairs svd2 ().

iii.2 Event reconstruction and selection

We reconstruct candidates in the decay mode with . The charged tracks are required to have a transverse (longitudinal) distance of closest approach to the interaction point (IP) of less than  cm. For particle identification (PID) of track candidates, specific energy loss measured in the CDC and information from the ACC and the TOF are combined using a likelihood-ratio approach. The selection requirement on the combined PID quantity has a kaon (pion) identification efficiency of with an associated pion (kaon) misidentification rate of for the track candidates not used as primary kaon from the meson. For a primary kaon from the meson candidate, the kaon identification efficiency is with an associated pion misidentification rate of . The invariant mass for candidates is required to be  GeV. The invariant mass must satisfy the criterion .

The selection of candidates is based on the beam-energy-constrained mass and the energy difference , where is the beam energy, and and are the momentum and energy of the candidates in the center-of-mass (CM) frame, respectively. Candidates with and are retained for further analysis. The range is used as the sideband, whereas is used as the nominal fit region.

In 17% of all signal events, more than one candidate passes the above selection; we select the candidate with the smallest for the hypothesis that all tracks form a common vertex within the IP region. This requirement selects the correct candidate with a probability of 64% according to Monte Carlo (MC) simulations.

The dominant background arises from continuum events, which are suppressed using a neural network (NN) implemented with the NeuroBayes package NeuroBayes (). In the NN, we combine , the polar angle of the candidate with respect to the beam direction in the CM frame, a likelihood constructed from 16 modified Fox–Wolfram moments SFW () and , the polar angle between the thrust axis of the candidate and the remaining tracks in the event. The NN assigns each candidate a value, , in the interval with being background (signal)-like. We require to reject 86% of the background while retaining 83% of the signal. Hereinafter, we refer to the continuum background, together with a 2% contribution from random combinations of tracks from events, as the combinatorial background.

The remaining background contribution arises from events and is due either to signal events in which we select a candidate with at least one track originating from the other [referred to as self-crossfeed (SCF)], or peaking background from decays. The SCF events are mainly due to partially reconstructed candidates, with a track from the other meson. Often, the pion momentum is low compared to the kaon momentum so that the direction of the system is dominated by the momentum. These combinations tend to peak in the region of high values. The peaking background originates from either with , which peaks sharply near in the distribution, or from events. We require to reject the peaking events completely as well as a majority of the SCF events. With respect to signal, about 5% of the events are due to SCF that will be discussed further in Sec. IV.4.

The reconstruction and selection procedures are established using MC events generated with the EvtGen program EvtGen () and a full detector simulation based on GEANT3 GEANT3 (). The PHOTOS package PHOTOS () is used to take into account final state radiation. The MC statistics for CKM-favored transitions and decays correspond to four times the data statistics. In addition, we use an MC sample of rare decays with 50 times the statistics of the data sample. We further use a very large sample of three-body phase space decays for our studies and several samples with different polarizations for cross-checks.

iii.3 Efficiency

We derive the four-dimensional efficiency function using MC samples of three-body phase space decays. It is found that the efficiency function can be parametrized by the product of one-dimensional projections . We model the efficiency as a function of with a second-order polynomial function. The efficiency as a function of is parametrized by a fourth-order polynomial function for and zero above. Both distributions are shown in Fig. 3. The efficiency as a function of and is found to be uniform.

Figure 3: Efficiency as a function of (a) and (b) . In (b) the dashed line indicates the region excluded from the analysis.

For a three-body phase-space decay, we obtain an averaged reconstruction efficiency of about within the analysis region. The reconstruction efficiency for a given partial wave depends on the observed angular distribution and can be obtained only after the polarization is measured. For the partial wave amplitudes with spin in Eqs. (16) to (18), we compute using

(24)

The numerator is the integral over the phase space with the efficiency included and is given by

(25)

where , , and are the nominal particle masses that limit the phase space. We omit the explicit dependencies of and for readability. The denominator of Eq. (24), , is given by the integral over the full phase space with a uniform efficiency

(26)

Iv Partial wave analysis

We use an unbinned extended maximum-likelihood (ML) fit to extract the 26 parameters related to polarization and violation defined in Eqs. (21) and (22), and denoted in the following. The log-likelihood function is given by

(27)

where is the total number of candidate events in the data set, is the number of contributions, is the expected number of events for the th contribution, is the probability density function (PDF) for the th contribution, is the nine-dimensional vector of observables for the th event, and denotes remaining parameters such as those related to PDF shapes.

We include three contributions in our fit model: the signal decay (), peaking background from decays (), and combinatorial background (). Each event is characterized by a nine-dimensional set of observables , with the beam-energy-constrained mass , the energy difference , the transformed continuum NN output , the invariant mass of the candidate , the invariant mass of the candidate , the three helicity angles , and , and the charge of the primary kaon from the meson, denoting the meson flavor. The transformed is used instead of as it has a Gaussian-like shape and can be described by an analytic parametrization.

iv.1 PDF parametrization

The PDF for a given contribution is constructed as a joint PDF of the distributions of the observables . With a few exceptions, explained below, we find no significant correlations among the fit observables. We use the method described in Ref. CAT () to check for linear and non-linear correlations among the observables using MC samples as well as sideband and off-resonance data for cross-checks.

The signal PDF for is modeled with a double Gaussian function for . The distribution is modeled with the sum of a Gaussian and two asymmetric Gaussian functions. In addition, to take into account a significant linear correlation between and for the signal, the mean of the distribution is parametrized by a linear function of . The distribution is parametrized by a sum of two asymmetric Gaussian functions. The candidate mass is modeled by a relativistic spin-dependent BW convolved with a Gaussian function to account for resolution effects; the BW parameters can be found in Table 3. For , the helicity angles and we refer to Eq. (19), which we multiply with the experimentally derived efficiency function to obtain the mass-angular signal PDF.

Parameter
(MeV)
(MeV)
(GeV)
(MeV)
(MeV)
Table 3: Parameters used for the resonance are taken from PDG (), except for , we make an assumption based on the values found in scattering. For , we use values from BES BES ().

The peaking background PDF for is constructed using the same parametrization as signal for , and . The distribution of the candidates is modelled by a Flatté function F76 (). The resonance parameters are given in Table 3. The distribution is parametrized by a relativistic spin-dependent BW for using the same parameters as the signal component. The angular distribution of this pseudoscalar to scalar–vector decay is uniform in and , and is proportional to ; we correct for detector acceptance effects. We use a distribution with equal probability for the two values of .

The combinatorial background PDF follows an empirically determined shape for the distribution, given by

(28)

where is a free parameter. This function was first introduced by the ARGUS Collaboration Argus (). The distribution is parametrized by a first-order polynomial function. The distribution is parametrized with a sum of two asymmetric Gaussians. To account for background that contains real candidates and a non-resonant component, the distribution is parametrized by the sum of resonant and non-resonant contributions. Similar to signal, the resonant contribution is parametrized with a relativistic spin-dependent BW convolved with the same resolution function. The non-resonant component is described by a threshold function as

(29)

where is the mass and a free parameter in the fit. The distribution is also parametrized by a sum of resonant and non-resonant components. The resonant component from is modelled with a relativistic spin-dependent BW using the same parameters as the signal component. The non-resonant contribution is parametrized by a fourth-order Chebyshev polynomial. We find a significant non-linear correlation between and in the non-resonant component of the combinatorial background. The resonant component in is uniform in , whereas the non-resonant contribution is parametrized by a fifth-order Chebyshev polynomial, where the parameters depend linearly on . The distribution is parametrized by a second-order Chebyshev polynomial and the distributions in and are uniform. The combinatorial background PDF is verified using off-resonance and sideband data. The contribution due to the combinatorial background from events, which is present in the sideband, has no significant effect on the shape parameters.

We use sideband data events to determine the free parameters of the combinatorial background PDF. Due to the presence of a clear peak in these events, we also determine the resolution (about 1 MeV) from this fit and use it for the signal model in the nominal fit region.

The , and distributions of the signal and peaking background components are cross-checked by fitting to a large-statistics control sample of events. In the control channel, we find excellent agreement between data and simulations for the distributions of and . We also confirm the linear correlation between