Angular analysis of decays and search for violation at Belle
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
The Belle Collaboration
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.
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.
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
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
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 ():
where we use the convention
For spin and , the mass-dependent widths are given by
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
representing a resonant contribution from while denoting a non-resonant contribution. The resonant part is defined as
where and are the resonance mass and width, and is given by
The non-resonant part is defined as
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
The resonance parameters used in the analysis are given in Table 1.
ii.3 Mass-angular distribution
We combine the mass distribution with the angular distribution to obtain the partial decay width
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
with () being the maximum (minimum) value of the Dalitz plot range of the invariant mass at a given value.
The matrix element squared is given by the coherent sum of the corresponding S-, P-, and D-wave amplitudes as
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
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
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
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
and given by
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.
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.
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.
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
The numerator is the integral over the phase space with the efficiency included and is given by
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
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
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.
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
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
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