#

Measurement of the violation parameters in

decays

###### Abstract

We present a measurement of the charge-parity violating parameters in decays. The results are obtained from the final data sample containing pairs collected at the resonance with the Belle detector at the KEKB asymmetric-energy collider. We obtain the violation parameters

where and represent the direct and mixing-induced asymmetries in decays, respectively. Using an isospin analysis including results from other Belle measurements, we find is disfavored at the level, where is one of the three interior angles of the Cabibbo-Kobayashi-Maskawa unitarity triangle related to decays.

###### pacs:

11.30.Er, 12.15.Hh, 13.25.HwBelle Preprint 2013-11

KEK Preprint 2013-21

The Belle Collaboration

## I Introduction

Violation of the combined charge-parity symmetry ( violation) in the standard model (SM) arises from a single irreducible phase in the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix Cabibbo (); KM (). A main objective of the Belle experiment at KEK, Japan, is to over-constrain the unitarity triangle of the CKM matrix related to decays. This permits a precision test of the CKM mechanism for violation as well as the search for new physics (NP) effects. Mixing-induced violation in the sector has been clearly established by Belle jpsiks_Belle1 (); jpsiks_Belle2 () and BaBar jpsiks_BABAR1 (); jpsiks_BABAR2 () in the induced decay . There are many other modes that may provide additional information on various violating parameters.

Decays that proceed predominantly through the transition are sensitive to the interior angle of the unitarity triangle ^{1}^{1}1Another notation, , also exists in literature.. This paper describes a measurement of violation parameters in decays, whose dominant amplitudes are shown in Fig. 1. Belle, BaBar and LHCb have reported time-dependent asymmetries in related modes including pipi_Belle (); pipi_BaBar (); pipi_LHCb (), rhopi_Belle (); rhopi_BaBar (), rhorho_Belle (); rhorho_BaBar () and a1pi_Belle (); a1pi_BaBar ().

The decay of the can produce a pair in a coherent quantum-mechanical state, from which one meson () may be reconstructed in the decay mode. This decay mode does not determine whether the decayed as a or as a . The flavor of the other meson (), however, can be identified using information from the remaining charged particles and photons. This dictates the flavor of as it must be opposite that of the flavor at the time decays. The proper time interval between and , which decay at time and , respectively, is defined as measured in the frame. For the case of coherent pairs, the time-dependent decay rate for a eigenstate when possesses flavor , where has and has , is given by

(1) |

Here, is the lifetime and is the mass difference between the two mass eigenstates of the neutral meson. This time dependence assumes invariance, no violation in the mixing, and that the difference in decay rates between the two mass eigenstates is negligible. The parameter measures the direct violation, while is a measure of the amount of mixing-induced violation.

In the limit that only the dominant tree amplitude contributes, no flavor-dependent direct violation is expected and is . However, in the final state and other self-conjugate modes, the value of is shifted by an amount , due to the presence of additional penguin contributions that interfere with the dominant tree contribution (see Fig. 1). Thus, the observable mixing-induced parameter becomes .

Despite penguin contamination, it is still possible to determine in with an isospin analysis theory_su2 () by considering the set of decays into the three possible charge states for the pions. Here, the two pions in decays must have a total isospin of or , since . For the penguin contributions, only or is possible because the gluon is an isospin singlet carrying . However, is forbidden by Bose-Einstein statistics; thus, strong loop decays cannot contribute and hence decays only through the tree diagram in the limit of negligible electroweak penguins.

The complex and decay amplitudes obey the relations

(2) |

respectively, where the subscripts refer to the combination of the pion charges. The decay amplitudes can be represented as the triangles shown in Fig. 2. As is a pure tree mode, these triangles share the same base, , and can be determined from the difference between the two triangles. These triangles and can be fully determined from the branching fractions, , and , and the violation parameters, , and . This method has an eightfold discrete ambiguity in the determination of , which arises from the four triangle orientations about and the two solutions of in the measurement of .

Belle, BaBar and LHCb have reported measurements pipi_Belle (); pipi_BaBar (); pipi_LHCb (), summarized in Table 1, of the violation parameters reported here. The previous Belle measurements were based on a sample of 535 million pairs and are superseded by the analysis presented here.

Parameter | Belle | BaBar | LHCb |
---|---|---|---|

( pairs) | ( pairs) | (0.7 fb) | |

In Sec. II, we briefly describe the data set and Belle detector. We explain the selection criteria used to identify signal candidates and suppress backgrounds in Sec. III, followed by the fit method used to extract the signal component in Sec. IV. In Sec. V, the results of the fit are presented along with a discussion of the systematic uncertainties in Sec. VI. Finally, our conclusions are given in Sec. VII.

## Ii Data Set And Belle Detector

This measurement of the violation parameters in decays is based on the final data sample containing pairs collected with the Belle detector at the KEKB asymmetric-energy ( on ) collider KEKB (). At the resonance ( GeV), the Lorentz boost of the produced pairs is nearly along the direction, which is opposite the positron beam direction. We also use a data sample recorded at 60 MeV below the resonance, referred to as off-resonance data, for continuum (, where ) background studies.

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) comprising 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 (KLM). The detector is described in detail elsewhere Belle (). Two inner detector configurations were used. A 2.0-cm-radius beampipe and a three-layer silicon vertex detector (SVD1) were used for the first sample of pairs, while a 1.5-cm-radius beampipe, a four-layer silicon detector (SVD2) and a small-cell inner drift chamber were used to record the remaining pairs svd2 (). We use a GEANT-based Monte Carlo (MC) simulation to model the response of the detector and to determine its acceptance GEANT ().

## Iii Event Selection

The decay channel is reconstructed from two oppositely charged tracks. Charged tracks are identified using a loose requirement on the distance of closest approach with respect to the interaction point (IP) along the beam direction, , and in the transverse plane, . Additional SVD requirements of at least two hits and one hit ResFunc () are imposed on all charged tracks so that a good quality vertex of the reconstructed candidate can be determined. Using information obtained from the CDC, ACC and TOF, particle identification (PID) is determined from a likelihood ratio . Here, () is the likelihood that the particle is of type (). To suppress background due to electron misidentification, ECL information is used to veto particles consistent with the electron hypothesis. The PID ratios of the two charged tracks , are used in the fit model to discriminate among the three possible two-body channels: , and .

Reconstructed candidates are identified with two nearly uncorrelated kinematic variables: the beam-energy-constrained mass and the energy difference , where is the beam energy and () is the energy (momentum) of the meson, all evaluated in the center-of-mass system (CMS). The candidates that satisfy and are retained for further analysis.

The dominant background in the reconstruction of arises from continuum production. Since continuum events tend to be jetlike, in contrast to spherical decays, continuum background can be distinguished from signal using event-shape variables, which we combine into a Fisher discriminant nazi_stuff (). The training sample is taken from signal MC, while the training sample is from the off-resonance data sample. The Fisher discriminant is then constructed from the variables described in Ref. a1pi_Belle (). The variable providing the strongest discrimination against continuum is the cosine of the angle between the thrust direction (TB) and the thrust of the tag side (TO) . The thrust is defined as the vector that maximizes the sum of the longitudinal momenta of the particles. For a event, the pair is nearly at rest in the CMS, so the thrust axis of is uncorrelated with the thrust axis of . In a event, on the other hand, the decay products align along two nearly back-to-back jets, so the two thrust axes tend to be collinear. Before training, a loose requirement of is imposed that retains 90% of the signal while rejecting 50% of the continuum background. The range of the Fisher discriminant encompasses all signal and background events.

Backgrounds from charm () decays are found to be negligible and are thus not considered, while charmless () decays of the meson may contribute, though rarely in the same region of and where signal is present.

As the and are almost at rest in the CMS, the difference in decay time between the two candidates, , can be determined approximately from the displacement in between the final state decay vertices as

(3) |

The vertex of reconstructed candidates is determined from the charged daughters, with a further constraint coming from the known IP. The IP profile is smeared in the plane perpendicular to the axis to account for the finite flight length of the meson in that plane. To obtain the distribution, we reconstruct the tag side vertex from the tracks not used to reconstruct ResFunc (). Candidate events must satisfy the requirements and , where is the multitrack vertex goodness-of-fit, calculated in three-dimensional space without using the IP profile constraint jpsiks_Belle2 (). To avoid the necessity of also modeling the event-dependent observables that describe the resolution in the fit Punzi (), the vertex uncertainty is required to satisfy the loose criteria for multitrack vertices and for single-track vertices.

The flavor tagging procedure is described in Ref. Tagging (). The tagging information is represented by two parameters, the flavor and the flavor-tagging quality . The parameter is continuous and determined on an event-by-event basis with an algorithm trained on MC simulated events, ranging from zero for no flavor discrimination to unity for an unambiguous flavor assignment. To obtain a data-driven replacement for , we divide it into seven regions and determine a probability of mistagging for each region using high statistics control samples. Due to a nonzero probability of mistagging , the asymmetry in data is thus diluted by a factor instead of the MC-determined . The measure of the flavor tagging algorithm performance is the total effective tagging efficiency , rather than the raw tagging efficiency , as the statistical significance of the parameters is proportional to . These are determined from data to be and for the SVD1 and SVD2 data, respectively jpsiks_Belle2 ().

About 1% of events have more than one candidate. For these events, the candidate containing the two highest momentum tracks in the lab frame is selected.

Differences from the previous Belle analysis pipi_Belle () include an improved tracking algorithm that was applied to the SVD2 data sample and the inclusion of the event shape into the fit rather than the optimization of selection criteria for this variable. As the latter strategy results in a large increase of the continuum background level, a reduced fit region in and is chosen in order to reduce this background without significant loss of signal events. According to MC simulation, these changes increase the detection efficiency by 19% over the previous analysis at a cost of continuum levels rising 4.7 times higher in the signal region defined by the previous analysis.

## Iv Event Model

The violation parameters are extracted from a seven-dimensional unbinned extended maximum likelihood fit to , , , , and from a data sample divided into seven bins () in the flavor-tag quality and 2 SVD configurations . Seven categories are considered in the event model: signal, , and peaking backgrounds, continuum, charmless neutral and charged decays. For most categories, the linear correlations between fit variables are small, so the probability density function (PDF) for each category is taken as the product of individual PDFs for each variable: in each bin, unless stated otherwise.

### iv.1 Peaking models

The four peaking shapes, including the signal, are determined from reconstructed MC events. The PDFs for and are taken to be the sum of three Gaussian functions, where the two tail Gaussians are parametrized relative to the core, which incorporates calibration factors that correct for the difference between data and MC simulation. These factors calibrate the mean and width of the core Gaussian component. The PDF for is taken to be the sum of three Gaussians in each flavor-tag bin , where the shape parameters are identical for all peaking channels. Calibration factors that correct for the shape differences between data and MC are incorporated into the core mean and width. These factors for are determined directly in the fit, while for and , these factors are determined from a large-statistics control sample of decays. The shape is modeled with a two-dimensional histogram that has been corrected for the difference between data and MC in PID as determined from an independent study with inclusive decays. The PDF of and for is given by

(4) | |||||

which accounts for dilution from the probability of incorrect flavor tagging and the wrong tag difference between and , both of which are determined from flavor-specific control samples using the method described in Ref Tagging (). The physics parameters and are fixed to their respective current world averages PDG (). This PDF is convolved with the resolution function for neutral particles , as in Ref. jpsiks_Belle2 (). We consider the , distributions for the flavor-specific and peaking backgrounds separately with

(5) |

For the peaking background, the , PDF is taken to be the same as that for signal, but as has not yet been observed, the parameters are set to zero. To account for the outlier events not described by the resolution function, a broad Gaussian PDF is introduced for every category,

(6) |

### iv.2 Continuum model

The parametrization of the continuum model is based on the off-resonance data; however, all the shape parameters of , , and are floated in the fit. As continuum is the dominant component, extra care is taken to ensure that this background shape is understood as precisely as possible, incorporating correlations above 2%. The PDF for is an empirical ARGUS function ARGUS (), while is modeled by a linear fit in each flavor-tag bin with a slope parametrized by and , depending linearly on ,

(7) |

The shape is observed to shift depending on the PID region, so the PDF is a sum of two Gaussian functions in two PID regions, and ( or . A small correlation between the shape and flavor-tag is also observed due to the component of continuum. As an example, consider the case where two jets are produced in which one contains a and the other contains a . If a candidate is successfully reconstructed with the , it inhabits the flavor-specific sector of . Then the accompanying could then be used as part of the flavor-tagging routine, which leads to a preferred flavor tag of . This enhances the distribution in the region and depletes it in the region for . To account for this effect, we model with an effective asymmetry that modifies the two-dimensional PID histogram model , in each bin depending on the flavor tag,

(8) |

where

(9) | |||||

which we hereafter refer to as the “manta ray” function. The model,

(10) |

contains a lifetime and prompt component to account for the charmed and charmless contributions, respectively. It is convolved with a sum of two Gaussians,

(11) |

which uses the event-dependent error constructed from the estimated vertex resolution as a scale factor of the width parameters and .

### iv.3 model

The charmless background shape is determined from a large sample of MC events based on transitions that is further subdivided into neutral and charged samples. A sizeable correlation of 18% is found between and and is taken into account with a two-dimensional histogram. The PDF for is taken to be the sum of three Gaussians in each flavor-tag bin , similar to the peaking model. Here, we are able to fix the shape parameters from the peaking model except for the core mean and width. A similar correlation between the flavor tag and , similar to that in continuum, is also observed. Due to mixing in the neutral background, this effect is correlated with and . For the neutral background, the PDF is given by

and the charged background PDF is given by

(13) |

where are manta ray functions for each category and is the resolution function for charged events. As reconstructed background candidates may borrow a track from the tag side, the average lifetime tends to be smaller and is taken into account with the effective lifetime, .

### iv.4 Full model

The total likelihood for candidates in the fit region is

(14) |

which iterates over events, categories, flavor-tag bins and detector configurations. The fraction of events in each bin, for category , is denoted by . The fraction of signal events in each bin, , is calibrated with the control sample. Free parameters of the fit include the and yields, and . The individual and yields are parametrized in terms of their combined yield and the violating parameter , which are both free in the fit: . The remaining yields are fixed to and as determined from MC simulation. In addition, all shape parameters of the continuum model with the exception of the parameters are allowed to vary in the fit. In total, there are 116 free parameters in the fit: 10 for the peaking models, 104 for the continuum shape and 2 for the background.

To determine the component yields and violation parameters, in contrast to the previous Belle analysis pipi_Belle (), we fit all variables simultaneously. The previous analysis applied a two-step procedure where the event-dependent component probabilities were calculated from a fit without and . These were then used as input in a fit to and to set the fractions of each component to determine the parameters. Our procedure has the added benefit of further discrimination against continuum with the variable and makes the treatment of systematic uncertainties more straightforward, at a cost of analysis complexity and longer computational time. A pseudoexperiment study indicates a 10% improvement in statistical uncertainty of the parameters over the previous analysis method.

## V Results

From the fit to the data, the following violation parameters are obtained:

(15) |

where the first uncertainty is statistical and the second is the systematic error (Sec. VI). Signal-enhanced fit projections are shown in Figs. 3 and 4. The effects of neglecting the correlation between and in the peaking models can be seen there as the slight overestimation of signal; however, pseudoexperiments show that this choice does not bias the violation parameters. These results are the world’s most precise measurements of time-dependent violation parameters in . The statistical correlation coefficients between the violation parameters is . The peaking event yields including signal are , and , where the uncertainties are statistical only. From the yields obtained in the fit, the relative contributions of each component are found to be for , for , for continuum and for background. For the violating parameter , we obtain a value of , which is consistent with the latest Belle measurement hphm_Belle ().

Our results confirm violation in this channel as reported in previous measurements and other experiments pipi_Belle (); pipi_BaBar (); pipi_LHCb (), and the value for is in marginal agreement with the previous Belle measurement. As a test of the accuracy of the result, we perform a fit on the data set containing the first pairs, which corresponds to the integrated luminosity used in the previous analysis. We obtain which is in good agreement with the value shown in Table 1, considering the new tracking algorithm and the 19% increase in detection efficiency due to improved analysis strategy. In a separate fit to only the new data sample containing pairs, we obtain . Using a pseudoexperiment technique based on the fit result, we estimate the probability of a statistical fluctuation in the new data set causing the observed shift in central value of from our measurement with the first pairs to be 0.5%.

To test the validity of the resolution description, we perform a separate fit with a floating lifetime; the result for is consistent with the current world average PDG () within . As a further check of the resolution function and the parameters describing the probability of mistagging, we fit for the parameters of our control sample ; the results are consistent with the expected null asymmetry. Finally, we determine a possible fit bias from a MC study in which the peaking channels and backgrounds are obtained from GEANT-simulated events, and the continuum background is generated from our model of off-resonance data. The statistical errors observed in this study agree with those obtained from our fit to the data.

Using Eq. (2) and input from other Belle publications hphm_Belle (); pi0pi0_Belle (), an isospin analysis is performed to constrain the angle . A goodness-of-fit is constructed for the five amplitudes shown in Fig. 2, accounting for the correlations between our measured physics observables used as input. The is then converted into a value (CL) as shown in Fig. 5. The region is disfavored and the constraint on the shift in caused by the penguin contribution is at the level, including systematic uncertainties.

## Vi Systematic Uncertainties

Systematic errors from various sources are considered and estimated with independent internal studies and cross-checks. These are summarized in Table 2. Uncertainties affecting the vertex reconstruction include the IP profile, charged track selection based on track helix errors, helix parameter corrections, and vertex goodness-of-fit selection, bias and SVD misalignment. The fit model uncertainties including the fixed physics parameters and , parameters describing the difference between data and MC simulation, resolution function parameters, as well as the flavor-tagging performance parameters and , are varied by . The parametric and nonparametric shapes describing the background are varied within their uncertainties. For nonparameteric shapes (i.e., histograms), we vary the contents of the histogram bins by . The fit bias is determined from the difference between the generated and fitted physics parameters using pseudoexperiments. Finally, a large number of MC pseudoexperiments are generated and an ensemble test is performed to obtain possible systematic biases from interference on the tag side arising between the CKM-favored and doubly CKM-suppressed amplitudes in the final states used for flavor tagging tsi ().

Category | ||
---|---|---|

IP profile | 0.13 | 1.19 |

track selection | 0.30 | 0.33 |

Track helix errors | 0.00 | 0.01 |

selection | 0.01 | 0.03 |

Vertex quality selection | 0.37 | 0.23 |

bias | 0.50 | 0.40 |

Misalignment | 0.40 | 0.20 |

and | 0.12 | 0.09 |

Data/MC shape | 0.15 | 0.19 |

resolution function | 0.83 | 2.02 |

Flavor tagging | 0.40 | 0.31 |

Background Parametric shape | 0.15 | 0.28 |

Background Nonparametric shape | 0.37 | 0.57 |

Fit bias | 0.54 | 0.86 |

Tag-side interference | 3.18 | 0.17 |

Total | 3.48 | 2.68 |

## Vii Conclusion

We report an improved measurement of the violation parameters in decays, confirming violation in this channel as reported in previous measurements and other experiments pipi_Belle (); pipi_BaBar (); pipi_LHCb (). These results are based on the full Belle data sample after reprocessing with a new tracking algorithm and with an optimized analysis performed with a single simultaneous fit, and they supersede those of the previous Belle analysis pipi_Belle (). They are now the world’s most precise measurement of time-dependent violation parameters in , disfavoring the range , at the level.

## 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, the National Institute of Informatics, and the PNNL/EMSL computing group for valuable computing and SINET4 network support. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Research Council and the Australian Department of Industry, Innovation, Science and Research; Austrian Science Fund under Grant No. P 22742-N16; the National Natural Science Foundation of China under Contract No. 10575109, No. 10775142, No. 10875115 and No. 10825524; the Ministry of Education, Youth and Sports of the Czech Republic under Contract No. MSM0021620859; the Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft and the VolkswagenStiftung; the Department of Science and Technology of India; the Istituto Nazionale di Fisica Nucleare of Italy; the BK21 and WCU program of the Ministry of Education, Science and Technology; the National Research Foundation of Korea Grants No. 2010-0021174, No. 2011-0029457, No. 2012-0008143, No. 2012R1A1A2008330; the BRL program under NRF Grant No. KRF-2011-0020333; the GSDC of the Korea Institute of Science and Technology Information; the Polish Ministry of Science and Higher Education and the National Science Center; the Ministry of Education and Science of the Russian Federation and the Russian Federal Agency for Atomic Energy; the Slovenian Research Agency; the Basque Foundation for Science (IKERBASQUE) and the UPV/EHU under program UFI 11/55; the Swiss National Science Foundation; the National Science Council and the Ministry of Education of Taiwan; and the U.S. Department of Energy and the National Science Foundation. This work is supported by a Grant-in-Aid from MEXT for Science Research in a Priority Area (“New Development of Flavor Physics”) and from JSPS for Creative Scientific Research (“Evolution of Tau-lepton Physics”).

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