Search for Direct Violation in ThreeBody Charmless Decay
Abstract
We report results on studies of violation in the threebody charmless decay . Evidence at the level for large direct violation in is found. This is the first evidence for violation in a charged meson decay. The analysis is performed using Dalitz analysis technique with a data sample that contains 386 million pairs collected near the resonance, with the Belle detector operating at the KEKB asymmetric energy collider.
pacs:
13.25.Hw, 11.30.Er, 14.40.NdThe Belle Collaboration
I Introduction
Decays of mesons to threebody charmless hadronic final states may provide new possibilities for violation searches. In contrast to decays to twobody final states where direct violation can only manifest itself as difference in decay rates for and mesons to charge conjugate final states, in threebody decays it can also be observed as a difference in relative phases between two quasitwobody channels. A necessary condition for observing direct violation in a twobody decay is a nontrivial strong phase difference between the conserving and violating amplitudes contributing to a particular final state. Although this condition (if satisfied) also enhances the sensitivity to violation in threebody decays, it is not required in general and direct violation in quasitwobody decays can also be observed with any strong phase difference via the interference with a nearby quasitwobody or nonresonant amplitude(s). Although direct violation has been observed in decays of neutral mesons dcpvK0 and recently in neutral meson decays dcpvB0 no violation in decays of charged mesons has been found to date. However, large direct violation is expected in some quasitwobody modes benekeneubert . Several other ideas to study violation utilizing decays to threebody final states have been proposed gronau ; b2hhhcp ; garmash2 ; hazumi .
First results on the amplitude analysis of the decay are described in Refs. khhdalitzbelle ; hhhdalitzbabar ; the first results on searches for direct violation from independent fits of the and samples are given in Ref. bellekppdcpv ; babarkppdcpv . The analysis of direct violation in the decay described in this paper is based on a simultaneous fit to and events. The results are obtained with a data sample of 357 fb containing 386 million pairs, collected with the Belle detector operating at the KEKB asymmetricenergy collider KEKB with a centerofmass (c.m.) energy at the resonance (onresonance data). The beam energies are 3.5 GeV for positrons and 8.0 GeV for electrons. For the study of the continuum background, we use data taken 60 MeV below the resonance (offresonance data).
Ii The Belle detector
The Belle detector Belle is a largesolidangle magnetic spectrometer based on a 1.5 T superconducting solenoid magnet. Charged particle tracking is provided by a silicon vertex detector and a 50layer central drift chamber (CDC) that surround the interaction point. Two inner detector configurations were used. A 2.0 cm beampipe and a 3layer silicon vertex detector was used for the first sample of 152 million pairs, while a 1.5 cm beampipe, a 4layer silicon detector and a smallcell inner drift chamber were used to record the remaining 234 million pairs Ushiroda . The charged particle acceptance covers laboratory polar angles between and , corresponding to about 92% of the total solid angle in the c.m. frame. The momentum resolution is determined from cosmic rays and events to be , where is the transverse momentum in GeV/.
Charged hadron identification is provided by measurements in the CDC, an array of 1188 aerogel Čerenkov counters (ACC), and a barrellike array of 128 timeofflight scintillation counters (TOF); information from the three subdetectors is combined to form a single likelihood ratio, which is then used in kaon and pion selection. At large momenta ( GeV/) only the ACC and CDC are used to separate charged pions and kaons since here the TOF provides no additional discrimination. Electromagnetic showering particles are detected in an array of 8736 CsI(Tl) crystals (ECL) that covers the same solid angle as the charged particle tracking system. The energy resolution for electromagnetic showers is , where is in GeV. Electron identification in Belle is based on a combination of measurements in the CDC, the response of the ACC, and the position, shape and total energy deposition (i.e., ) of the shower detected in the ECL. The electron identification efficiency is greater than 92% for tracks with GeV/ and the hadron misidentification probability is below 0.3%. The magnetic field is returned via an iron yoke that is instrumented to detect muons and mesons. We use a GEANTbased Monte Carlo (MC) simulation to model the response of the detector and determine its acceptance GEANT .
Iii Event Reconstruction
Charged tracks are selected with a set of track quality requirements based on the number of CDC hits and on the distances of closest approach to the interaction point. We also require that the track momenta transverse to the beam be greater than 0.1 GeV/ to reduce the low momentum combinatorial background. For charged kaon identification we impose a requirement on the particle identification variable which has 86% efficiency and a 7% fake rate from misidentified pions. Charged tracks that are positively identified as electrons or protons are excluded. Since the muon identification efficiency and fake rate vary significantly with the track momentum, we do not veto muons to avoid additional systematic errors.
We identify candidates using two variables: the energy difference and the beam constrained mass where the summation is over all particles from a candidate; and are their c.m. threemomenta and masses, respectively. The signal shape is fitted by a sum of two Gaussian functions with a common mean. In fits to the experimental data, we fix the width and the relative fraction of the second Gaussian function from MC simulation. The common mean of the two Gaussian functions and the width of the main Gaussian are floated. The shape for the background is parametrized by a linear function. The distribution for the signal events is parametrized by a single Gaussian function. The width is about 3 MeV/ and, in general, does not depend on the final state (unless photons are included in the reconstructed final state). The background shape is parametrized with an empirical function ArgusF , where and is a parameter.
Iv Background Suppression
The dominant background is due to ( and quarks) continuum events that have a crosssection about three times larger than that for the . This background is suppressed using variables that characterize the event topology. Since the two mesons produced from an decay are nearly at rest in the c.m. frame, their decay products are uncorrelated and the event tends to be spherical. In contrast, hadrons from continuum events tend to exhibit a twojet structure. We use , which is the angle between the thrust axis of the candidate and that of the rest of the event, to discriminate between the two cases. The distribution of is strongly peaked near for events and is nearly flat for decay while retaining 36% of the signal. A detailed description of the continuum suppression technique can be found in Ref. garmash2 and references therein. events. A Fisher discriminant is utilized for the further suppression of the continuum background. When combined, these two variables reject about 98% of the continuum background in the
Another background originates from other meson decays. We study the decays that proceed via penguin and tree transitions are not included in the generic MC sample and are generated separately. We find that the dominant final state is due to , decays. We veto MeV/. We also veto MeV/. To suppress the background due to misidentification, we also exclude candidates if the invariant mass of any pair of oppositely charged tracks from the candidate is consistent with the hypothesis within 25 MeV/ (), regardless of the particle identification information. Modes with in the final state contribute due to muonpion misidentification; the contribution from the submode is found to be negligible after the electron veto requirement. We exclude background by requiring MeV/ and MeV/, with a muon mass assignment used here for the pion candidates. Yet another small but clearly visible background is due to , decay with a complicated series of particle misidentifications; the charged kaon from the is misidentified as a pion, the is misidentified as a kaon and the as another pion. This background is excluded by applying a veto on the invariant mass of oppositely charged kaon and pion candidates: MeV/. The most significant background from charmless decays is found to originate from followed by . Another contribution comes from decay, where one of the two same charge pions is misidentified as a kaon. Finally, we consider a background from the decay . Although it does not directly contribute to the signal region, this background should be taken into account in order to correctly estimate the component of the background. signal by requiring , events by requiring and due to , related background to the generic events. Note that charmless hadronic related background using a large sample of MC generated
V Threebody Signal Yields
The distribution for candidates that pass all the selection requirements are shown in Fig. 1. In the fit to the distribution we fix the shape and normalization of the charmless PDG and known number of produced background component are free parameters. Results of the fit are shown in Fig. 1, where different components of the background are shown separately for comparison. There is a large increase in the level of GeV. This is mainly due to , decay. This decay mode produces the same final state as the studied process plus one extra pion that is not included in the energy difference calculation. The semileptonic decays , also contribute due to muonpion misidentification. The shape of these backgrounds is well described by MC simulation. Results of the fits are given in Table 1. related background in the region generic component we fix only the shape and let the normalization float. The slope and normalization of the events. For the background components from the measured branching fractions
Final state  Fraction of the  Signal Yield  

MeV  MeV  main Gaussian  (events)  (events)  (events)  
(fixed)  (fixed)  
(fixed)  (fixed)  
(fixed)  (fixed) 
For the analysis of quasitwobody intermediate states that contribute to the observed threebody signal, we define the signal and sideband regions as shown in Fig. 2. Defined in this way, the sidebands are equivalent to the following sidebands in terms of the threeparticle invariant mass and threeparticle momentum :
and
The signal region is defined as an ellipse around the and mean values:
The efficiency of the requirements that define the signal region is 0.927; the total number of events in the signal region is 7757. The relative fraction of signal events in the signal region is determined to be . There are 27855 events in the sideband region that is about seven times the estimated number of background events in the signal region.
Vi Amplitude Analysis
The amplitude analysis of threebody meson decay reported here is performed by means of an unbinned maximum likelihood fit. Details of the analysis technique are described in Ref. khhdalitzbelle . One of the important questions that arise in unbinned analysis is the estimation of the goodnessoffit. As the unbinned maximum likelihood fitting method does not provide a direct way to estimate the quality of the fit, we need a measure to assess how well any given fit represents the data. To do so the following procedure is applied. We first subdivide the entire Dalitz plot into 1 (GeV/)1 (GeV/)bins. If the number of events in the bin is smaller than it is combined with the adjacent bins until the number of events exceed the minimum required level. After completing this procedure, the entire Dalitz plot is divided into set of bins of varying size, and a variable for the multinomial distribution can be calculated as
(1) 
where is the number of events observed in th bin, and is the number of predicted events from the fit. For a large number of events this formulation becomes equivalent to the usual one. Since we are minimizing the unbinned likelihood function, our “” variable does not asymptotically follow a distribution but it is bounded by a variable with () degrees of freedom and a variable with () degrees of freedom, where is the number of fit parameters. Because it is bounded by two variables, it should be a useful statistic for comparing the relative goodness of fits for different models.
vi.1 Fitting the Background Shape
Before fitting the Dalitz plot for events in the signal region, we need to determine the distribution of background events. The background density function is determined from an unbinned likelihood fit to the events in the sidebands defined in Fig. 2. Figure 3(a) shows the Dalitz plot for sideband events.
We use the following empirical parameterization to describe the distribution of background events over the Dalitz plot
(2)  
where , and (), and are fit parameters; is a BreitWigner function. The first three terms in Eq. (2) are introduced to describe the background enhancement in the twoparticle low invariant mass regions. This enhancement originates mainly from continuum events. Due to the jetlike structure of this background, all three particles in a threebody combination have almost collinear momenta. Hence, the invariant mass of at least one pair of particles is in the low mass region. In addition, it is often the case that two high momentum particles are combined with a low momentum particle to form a candidate. In this case there are two pairs with low invariant masses and one pair with high invariant mass. This results in even stronger enhancement of the background in the corners of the Dalitz plot. This is taken into account by terms in Eq. (2). To account for the contribution from real and mesons, we introduce two more terms in Eq. (2), that are (noninterfering) squared BreitWigner amplitudes, with masses and widths fixed at world average values PDG .
The projections of the data and fits for the background events are shown in Fig. 4. The value of the fit is .
vi.2 Fitting the Signal
The Dalitz plot for events in the signal region is shown in Fig. 3(b). There are 7757 events in the signal region that satisfy all the selection requirements. As found in Ref. khhdalitzbelle the decay is well described by a matrix element that is a coherent sum of , , , , , quasitwobody channels and a nonresonant amplitude. The channel is added in order to describe an excess of signal events at GeV/. With current statistics, the contribution of is found to be significant, but not sufficient to fully explain the excess of signal events in this mass region. In this analysis we modify the model by adding two more quasitwobody channels: and and change the parameterization of the lineshape from a standard BreitWigner function to a coupled channel BreitWigner also known as Flatté parametrization Flatte . Although the branching fraction is only % PDG , the natural width is rather narrow. As a result a numerical factor of is introduced in the amplitude (relative to the amplitude) that compensates the smallness of the branching. As the independently measured branching fraction HFAG is comparable to that for , the interference between these two amplitudes might significantly distort the lineshape.
Finally, for violation studies the amplitude for each quasitwobody channel is modified to include two components: one that is independent of the sign of the charge and a component that changes sign with the charge of the meson. The resulting decay amplitude reads as
(3) 
with the nonresonant amplitude parametrized by an empirical function
(4) 
where , . Note that alternative parameterizations of the nonresonant amplitude possible khhdalitzbelle ; babarkppdcpv . The amplitudes and , relative phases and , mass, and of the , mass and width of the , and parameter of the nonresonant amplitude are fit parameters. With such a parameterization of the amplitude, the violating asymmetry for a particular quasitwobody channel can be calculated as
(5) 
To reduce the number of free fit parameters, we fit the data in two steps. First we fix all and fit the data assuming no violation. From this fit we determine the parameters of the ( GeV/, GeV/), ( GeV/, , ) and the parameter of the nonresonant amplitude (). We then fix these parameters and repeat the fit to data with and floating. In addition, we also assume no violation in and for the nonresonant amplitude. Possible effects of these assumptions are studied and considered in the final results as a part of the model uncertainty.
Channel  averaged  ,  ,  ,  Significance,  

fraction, %  degrees  degrees  %  
(fixed)  
(fixed)  
NonRes.  (fixed)  
The numerical values of the fit parameters are given in Table 2. The value of the fit is with fit parameters. Fit projections and the data are shown in Fig. 5. Figure 6 shows helicity angle distributions for several regions, where the helicity angle is defined as the angle between the direction of flight of the in the rest frame and the direction of candidate in the rest frame. Gaps visible in Fig. 6 are due to vetoes applied on invariant masses of twoparticle combinations. All plots shown in Figs. 5 and 6 demonstrate good agreement between data and the fit.
The statistical significance of the asymmetry quoted in Table 2 is calculated as , where and denote the maximum likelihood with nominal fit and with the asymmetry fixed at zero, respectively. The only channel where the statistical significance of the asymmetry exceeds the level is . Figures 7(a,b) show the invariant mass distributions for the mass region separately for and events. The effect is even more apparent when the distribution for the two helicity angle regions ( and ) shown in Fig. 7(cf) are compared.
Vii Systematic & Model Uncertainties
The dominant sources of systematic error are listed in Table 3. The systematic uncertainty in charged track reconstruction is estimated using partially reconstructed events and from comparison of the ratio of to events in data and MC. The uncertainty from the particle identification efficiency is estimated using pure samples of kaons and pions from decays, where the flavor is tagged using decays. The systematic uncertainty due to requirements on event shape variables is estimated from a comparison of the and distributions for signal MC events and shape parameterization by varying the parameters of the fitting function within their errors. The uncertainty in the background parameterization is estimated by varying the relative fraction of the background function within their errors. Reconstruction efficiency is determined using MC events distributed over phase space according to the matrix element corresponding to the best fit to data. The relevant systematic uncertainty is estimated to be at the level of one percent. Finally, to account for variations in reconstruction efficiency due to modifications in the detector setup and due to nonuniform datataking conditions (mainly beam related background conditions), we generate signal events with background events embedded. Background events are recorded with random triggers for each experiment. Signal MC events are generated for each experiment with statistics proportional to experimental data. The overall systematic uncertainty for the threebody branching fraction is . background component and the slope of the events in the data. We estimate the uncertainty due to the signal
Source  Error 

Charged track reconstruction  
PID  
Event Shape requirements  
Signal yield extraction  
Model  
MC statistics  
Luminosity measurement  
Total 
Source  

Rare background  
Detector asymmetry  
Signal yield extraction  
Total 
Note that in asymmetry calculation most of these systematic uncertainties cancel. The few remaining sources are listed in Table 4. Systematic uncertainty due to possible asymmetry in background from charmless decays is estimated by introducing an asymmetry equal to experimentally measured central value HFAG increased by one standard deviation to each charmless background component one by one and refitting the data. The possible bias due to intrinsic detector asymmetry in reconstruction of tracks of different charges is estimated using events in data.
To estimate the model dependent uncertainty in the branching fractions and asymmetries for individual quasitwobody intermediate states, we vary the default model and repeat the fit to data. Namely, we add one additional quasitwobody channel which is either , , or or remove or channel from the default model, use different assumptions on the spin of the state and use different parameterizations of the nonresonant amplitude. For estimation of the model uncertainty in charge asymmetries for individual quasitwobody channels in addition to model variations we fit the data with different assumptions on violation in different channels. Finally, we check the consistency of the results with those obtained from independent fits of and subsamples.
Viii Branching Fraction & Charge Asymmetry Results
In the preceding section we determined the relative fractions of various quasitwobody intermediate states in the threebody decay. To translate those numbers into absolute branching fractions, we first need to determine the branching fraction for the three body decay. To determine the reconstruction efficiency, we use MC simulation where events are distributed over the phase space according to the matrix elements obtained from the best fit to data. The corresponding reconstruction efficiency is %. Results of the branching fraction and violating asymmetry calculations are summarized in Table 5.
Mode  

%  
Charmless  
Nonresonant  
Ix Discussion & Conclusion
With a 357 fb data sample collected with the Belle detector, we made the first analysis of direct violation in the threebody charmless decay . Results on branching fraction and violating asymmetry calculations are summarized in Table 5. In all except the channel the measured asymmetry is below statistical significance. Evidence for large direct violation the decay is found in agreement with our results obtained with 253 fb bellekppdcpv and with results by BaBar babarkppdcpv . This is also in agreement with some theoretical predictions benekeneubert . The statistical significance of the asymmetry observed in is . Depending on the model used to fit the data the significance varies from to . If confirmed with a larger data sample, this would be the first observation of violation in the decay of a charged meson.
Acknowledgments
We thank the KEKB group for the excellent operation of the accelerator, the KEK cryogenics group for the efficient operation of the solenoid, and the KEK computer group and the National Institute of Informatics for valuable computing and SuperSINET network support. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology of Japan and the Japan Society for the Promotion of Science; the Australian Research Council and the Australian Department of Education, Science and Training; the National Science Foundation of China under contract No. 10175071; the Department of Science and Technology of India; the BK21 program of the Ministry of Education of Korea and the CHEP SRC program of the Korea Science and Engineering Foundation; the Polish State Committee for Scientific Research under contract No. 2P03B 01324; the Ministry of Science and Technology of the Russian Federation; the Ministry of Higher Education, Science and Technology of the Republic of Slovenia; the Swiss National Science Foundation; the National Science Council and the Ministry of Education of Taiwan; and the U.S. Department of Energy.
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(19)
Heavy Flavor Averaging Group, hepex/0505100, and updates available at
http://www.slac.stanford.edu/xorg/hfag/.