Study of decay at Belle
We present a study of the decay using a data sample collected with the Belle detector. The analysis is based on lepton-tagged signal events. The measured branching fraction is consistent with the world average value and has better accuracy. An analysis of the invariant mass spectrum reveals contributions from the as well as other states. For the first time the mass and width have been measured in decay: , . The mass is significantly different from the current world average value.
Pacs:13.30.Eg, 13.35.Dx, 13.66.Jn, 14.40.Ev, 14.60.Fg
BELLE Preprint 2007-28
KEK Preprint 2007-17
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and ,
lepton hadronic decays provide a laboratory for the study of low energy hadronic currents under very clean conditions. In these decays, the hadronic system is produced from the QCD vacuum via the charged weak current mediated by a boson. The decay amplitude can thus be factorized into a purely leptonic part including the and and a hadronic spectral function. Strangeness changing decays are suppressed by a factor of relative to Cabibbo-allowed modes. High-statistics measurements at factories provide excellent opportunities for studying the structure of the strange hadronic spectral functions in specific decay modes [1, 2, 3], the parameters of the intermediate states and the total strange hadronic spectral function .
The decay (unless specified otherwise, charge conjugate decays are implied throughout the paper) has the largest branching fraction of all Cabibbo-suppressed decays of the lepton. Early studies of this decay established that the main contribution to the invariant mass spectrum is from the meson [5, 6, 7]. Although scalar or tensor contributions are expected in theoretical models [8, 9] and not excluded experimentally [10, 11], the low statistics of previous investigations did not allow for a detailed study.
Here we report a precise measurement of the branching fraction for the decay as well as a study of its final state dynamics. This analysis is based on a data sample that contains 313 pairs, collected with the Belle detector at the KEKB energy-asymmetric (3.5 on 8 GeV) collider  operating at the resonance.
2 The Belle detector
The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter (ECL) comprised of CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside the coil is instrumented to detect mesons and to identify muons (KLM). Two inner detector configurations are used in this analysis. A beampipe with a radius of 2.0 cm and a 3-layer silicon vertex detector are used for the first sample of 124 pairs, while a 1.5 cm beampipe, a 4-layer silicon detector and a small-cell inner drift chamber are used to record the remaining 189 pairs . The detector is described in detail elsewhere .
3 Selection of events
We select events in which one decays to leptons, , while the other one decays via the hadronic channel , where denotes the hadronic system. Events where both ’s decay to leptons are used for normalization. This reduces systematic uncertainties substantially.
The selection process, which is optimized to suppress background while retaining a high efficiency for the decays under study, proceeds in two stages. The criteria of the first stage suppress beam background to a negligible level and reject most of the background from other physical processes. These criteria retain a % efficiency for events. We then select events having to tracks with a net charge less than or equal to one in absolute value. The extrapolation of each track to the interaction point (IP) is required to pass within in the transverse direction and in the longitudinal direction of the nominal collision point of the beams. Each track must have a transverse momentum in the center-of-mass (CM) frame larger than . At least one of the charged particles should have a transverse momentum higher than . The sum of the absolute values of the CM track momenta must be less than . The minimum opening angle for any pair of tracks is required to be larger than . The number of photons with a CM energy exceeding is required to be less than or equal to five. The total ECL energy deposition in the laboratory frame must be less than . The total energy of all photon candidates in the laboratory frame should satisfy . The missing four-momentum is calculated by subtracting the four-momentum of all charged tracks and photons from the beam four-momentum. The missing mass is required to satisfy . The polar angle of the missing momentum in the CM frame is required to be larger than or equal to and less than or equal to . The last two criteria are particularly effective in suppressing the backgrounds from radiative Bhabha, and two-photon processes.
At the second stage, two event classes are selected for further processing: a two-lepton sample and a lepton-hadron sample . To select electrons, a likelihood ratio requirement is applied, where the electron likelihood function and the non-electron function include information on the specific ionization () measurement by the CDC, the ratio of the cluster energy in the ECL to the track momentum measured in the CDC, the transverse ECL shower shape and the light yield in the ACC . The efficiency of this requirement for electrons is . To select muons, a likelihood ratio requirement is applied. It provides efficiency for muons. Each of the muon(), pion() and kaon() likelihood functions is evaluated from the information on the difference between the range calculated from the momentum of the particle and the range measured by KLM, and the of the KLM hits with respect to the extrapolated track . To separate pions from kaons, for each track we determine the pion () and kaon () likelihoods from the ACC response, the measurement in the CDC and the TOF flight-time measurement, and form a likelihood ratio . For pions we apply the requirement , which provides a pion identification efficiency of about , while keeping the pion fake rate at the level.
To evaluate the background and to calculate efficiencies, a Monte Carlo (MC) sample of pairs is produced with the KORALB/TAUOLA generators [17, 18]. The detector response is simulated by a GEANT3 based program .
3.1 Two-lepton events
For this class the and samples still contain contamination from radiative Bhabha and processes of about 50%, only events are used for normalization. To further suppress and charm backgrounds, we require the opening angle of the leptons to be larger than in the CM. As a result, we selected events of the type and events.
MC simulation indicates that there is an approximately contamination coming primarily from the two-photon process () and from events where the is misidentified as a lepton (). Contamination from other non- processes is found to be negligible (less than 0.1%). The numbers of and events after background subtraction are and , respectively. The detection efficiencies and their statistical errors are for and for events.
3.2 Lepton-hadron events
For this class we select events with only one lepton , one candidate and one charged pion . A meson is reconstructed from a pair of oppositely charged pions having invariant mass within MeV of the mass, which corresponds to a signal range. The pion momenta are then refitted with a common vertex constraint. The -distance between the two helices at the vertex position before the fit is required to be less than , where is defined as the direction opposite to the positron beam. The closest approach of at least one track to the IP in the plane must be larger than . The decay length of the candidate in the plane must satisfy . The -projection of the candidate decay length is required to be . The decay length must be larger than . The cosine of the azimuthal angle between the momentum vector and the decay vertex vector of the candidate is required to be larger than or equal to . The lepton- and lepton- opening angles are required to be larger than in the CM. events were selected for further analysis. Figure 1 shows a comparison of the MC and experimental distributions for the invariant mass of the candidate and the decay length.
Figure 1 (a) shows that MC mass resolution is slightly better than the experimental one resulting in a clear difference of the mass spectra in the region of the peak. However, the efficiency of the cut for the candidates is almost , hence the impact of this discrepancy on the detection efficiency is very small and is taken into account in the systematic uncertainty. In Fig. 1 (b) one can see a clear difference between the distributions in the region of small , where events of decay are located, however, in the region, where cm, populated mostly by true ’s the agreement is good. Figure 2 shows selected events on a plot of the decay length versus the invariant mass of the candidate.
The main background is from other decays: , , , . Using the branching fractions of these decays from Ref.  and detection efficiencies from MC simulation, the contamination from decays with a is calculated to be . decays contaminate the sample when a pair of oppositely charged pions is reconstructed as a fake . The invariant mass distribution of these fake ’s is flat in the region of the mass (see also Fig. 2). The number of background events is calculated from two sideband regions in the vs. plane, determined by the following criteria: and for the first region, and for the second one. These sidebands have the same area as the signal region. The fraction of signal events in the -sideband region is about , which is taken into account in the calculation of the MC signal detection efficiency. We observe a background of events in the signal region. In the sample there is a small contamination (of about for the -tagged and for the -tagged events) coming primarily from events, where the first pion was misidentified as a lepton. The non- background is found to be . After background subtraction signal events remain. Table 1 shows how they are distributed among the various tagging configurations.
4 branching fraction
The branching fraction is calculated according to the formula:
where , are the number and MC efficiency of the signal events, , are the number and MC efficiency of the two-lepton events, is the leptonic branching fraction taken from Ref. . Note that the tag-lepton () efficiency cancels in the ratio of the efficiencies, so the associated systematic uncertainty is reduced. The branching fractions calculated separately for each event configuration are given in Table 1, which also lists separately the averages for electrons and muons as well as the overall branching fraction.
|Lepton identification efficiency|
|Non- background subtraction|
|Pion identification efficiency|
Table 2 lists the different sources of systematic uncertainties for the branching fraction. The dominant contributions come from the detection efficiency and background subtraction. A systematic uncertainty in the detection efficiency receives contributions from the reconstruction of daughter pions (), the efficiency for fitting two pion tracks to a common vertex (), which was evaluated by varying the cut on the -distance between the two helices at the vertex position before the fit, and the efficiency of the selection criteria (), which was checked by varying cuts on the invariant mass . Systematic uncertainties arising from -background subtraction are , , and for the , , and modes, respectively. For the background from decay modes with a the uncertainties are determined by the corresponding uncertainties in their branching fractions taken from Ref. , except for the mode. Here we rely on the isospin relation and the CLEO result  to calculate the branching fraction . The uncertainty in the contamination by events is evaluated by varying the decay length cut.
The lepton detection efficiency is corrected using the two-photon data sample. An efficiency correction table is calculated in 70 bins on the plane of momentum vs. polar angle in the laboratory frame and then applied to the Monte Carlo efficiencies and . Hence, the uncertainty on the leptonic efficiency is determined by the statistics of the sample and the long-term stability, which is evaluated from the variation of the corrections calculated for time ordered subsamples of the experimental two-photon data. The pion identification efficiency in MC differs from that in data. In the signal sample, mesons provide a source of identified pions, which are used to calculate corrections to the MC efficiency. Therefore, the systematic uncertainty on the pion identification efficiency is determined by the statistical error of the correction, which is about . To calculate a signal MC sample is produced according to the model and the model dependence of is found to be negligible.
We also vary cuts on the pion momentum, the kaon momentum, and the total laboratory energy of photons () to check the stability of the branching fraction. The total systematic uncertainty of is obtained by adding all the contributions in quadrature. Our final result for the branching fraction is .
5 Analysis of the spectrum
where is the invariant mass squared and is the momentum in the rest frame:
The vector form factor is parameterized by the , and meson amplitudes:
where and are complex coefficients for the fractions of the and resonances, respectively. is a relativistic Breit-Wigner function:
where is the s-dependent total width of the resonance:
where if the system originates in the -wave state and is the resonance width at its peak.
The scalar form factor includes the and contributions, their fractions are described respectively by the complex constants and :
The experimental distribution is approximated in the mass range from 0.63 GeV/ to 1.78 GeV/ by a function calculated from the convolution of the spectrum given by Eq. (2) and the detector response function, which takes into account the efficiency and finite resolution of the detector. In all fits the mass and width as well as the total normalization are free parameters. Only the strengths (fractions) of the other ’s are free parameters, while their masses and widths are fixed at the world average values . In the approximation is chosen to be real, because is defined up to the common phase, which cancels in .
Figure 3 (a) and Table 3 show that the alone is not sufficient to describe the mass spectrum. To describe the enhancement near threshold, we introduce a amplitude, while for description of the distribution at higher invariant masses we try to include the , vector resonances (see Table 3) or the scalar (see Table 4). Figure 3 (b) demonstrates the good quality of the fit with the model. It can be seen from Tables 3, 4 that we cannot distinguish between the and models. The fit quality with the model (see the fourth column of Table 3) is worse than that of the and models.
|solution 1||solution 2|
|, - fixed||, - free|
It should be noted that the absolute value of a sum of two Breit-Wigner functions of mass () can have the same shape for two different sets of parameters. In the case of the model the relevant parameters are , and . This statement holds true when mass-independent widths are considered. If the width is mass-dependent, some difference in the spectra appears. If in the fit to the data the errors are large enough, we cannot distinguish these solutions by their values. For high statistics the two solutions can be distinguished by a test. While for the and models with a complicated vector form factor the values of are significantly different (due to the small () errors at the peak), in the case with a complicated scalar form factor different solutions result in similar values (see Table 4) due to the relatively low statistics in the region of the and peaks.
where is a real constant, is momentum in the rest frame (see Eq. (3)), and the phase is determined from the equation , where , are the model parameters. In this parameterization the non-resonant mechanism is given by the effective range term , while the resonant structure is described by the amplitude.
Table 5 shows the results of fits to the spectrum in models, where the non- mechanism is described by the LASS parameterization of the scalar form factor. In the first fit (see the second column of Table 5) and parameters were fixed at the LASS optimal values . In the second fit and were free parameters (see the third column of Table 5). The optimal values of and in our fit differ significantly from the values obtained by the LASS collaboration in experiments on scattering .
The model was considered as the default and was used to obtain the fraction in the final state, which was found to be . The error includes the model uncertainty, which was found by calculating this fraction in the fits with the other models mentioned above, as well as the uncertainty in the fit parameters. Finally we obtain .
6 Measurement of the mass and width
A fit to the invariant mass spectrum also provides a high precision measurement of the mass and width. We consider a fit with the model, which provides a good description of the data, as a reference, and use it to obtain the mass and width values. It can be seen from Table 3 that the statistical uncertainty is about for the mass and for the width. Two additional sources of uncertainty are studied: the effects of imperfect knowledge of the detector response function and model uncertainty.
The systematic uncertainty is studied with a MC sample by comparing the parameters implemented in the generator and its parameters after the full reconstruction procedure (the detector response function is determined from other statistically independent MC simulations of signal events). It is found to be for the mass and for the width.
The model uncertainty is investigated by fitting the mass spectrum with different models. The maximal difference from the reference value is considered as a model uncertainty. It is found to be for the mass and for the width.
As a result, the mass and width are and , where the first uncertainty is statistical, the second is systematic and the third is from the model.
The branching fraction of the decay has been measured using a data sample of collected with the Belle detector. Our result is:
Figure 4 (a) shows the results of various measurements of the branching fraction, along with the Particle Data Group (PDG) fit value ()  and our result. Our result is consistent with previous measurements, but is more precise.
The alone is not sufficient to describe the invariant mass spectrum. The best description is achieved in the and models. Future high precision studies of the invariant mass spectra in lepton decays with kaons combined with angular analysis, i.e. an application of the structure function formalism suggested in Ref. , will elucidate the nature of the scalar form factor. They will also check various theoretical models describing the scalar sector, e.g., the predictions of the resonance chiral theory  and the parameters of the resonance calculated from the Roy-Steiner representations in a model-independent way .
The product of and branching fractions is found to be:
also the mass and width are measured:
The values of the mass and width that we obtain are more precise than any of the existing measurements of these quantities listed in Ref.  and shown in Fig. 4 (b). While our determination of the width is compatible with most of the previous measurements within experimental errors, our mass value is systematically higher than those before and is in fact consistent with the world average value of the neutral mass, which is MeV/ . Note that all earlier mass measurements listed in Ref.  come from analysis of hadronic reactions and include the effects of final state interaction while our work presents a measurement based on decays, where the decay products of the are the only hadrons involved. It is also noteworthy that none of the previous measurements in Ref. , all of which were performed more than 20 years ago, present the systematic uncertainties for their measurements. Unfortunately, previous studies of the in lepton decays usually do not determine its parameters. The only published result we are aware of is that of ALEPH , which is consistent with ours. Its accuracy, however, is much worse and no systematic errors are presented, which precludes any detailed comparisons. A similar mass shift of was reported by CLEO , but no dedicated study of this effect was published. Future dedicated measurements of the parameters with high precision are necessary to clarify this discrepancy and shed light on the long standing issue of the electromagnetic mass difference between the charged and neutral [32, 33].
We are grateful to M. Jamin for interesting discussions. 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 Super-SINET 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 and the Knowledge Innovation Program of the Chinese Academy of Sciences under contract No. 10575109 and IHEP-U-503; the Department of Science and Technology of India; the BK21 program of the Ministry of Education of Korea, the CHEP SRC program and Basic Research program (grant No. R01-2005-000-10089-0) of the Korea Science and Engineering Foundation, and the Pure Basic Research Group program of the Korea Research Foundation; the Polish State Committee for Scientific Research; the Ministry of Education and Science of the Russian Federation and the Russian Federal Agency for Atomic Energy; the Slovenian Research Agency; the Swiss National Science Foundation; the National Science Council and the Ministry of Education of Taiwan; and the U.S. Department of Energy.
-  J. H. Kühn and E. Mirkes, Z. Phys. C 56 (1992) 661, Erratum-ibid. C 67 (1995) 364.
-  R. Decker, E. Mirkes, R. Sauer, Z. Wa̧s, Z. Phys. C 58 (1993) 445.
-  M. Finkemeier and E. Mirkes, Z. Phys. C 69 (1996) 243.
-  E. Gamiz et al., Phys. Rev. Lett. 94 (2005) 011803.
-  J. Dorfan et al. (MARK II Collaboration), Phys. Rev. Lett. 46 (1981) 215.
-  H. Albrecht et al. (ARGUS Collaboration), Z. Phys. C 41 (1988) 1.
-  M. Battle et al. (CLEO Collaboration), Phys. Rev. Lett. 73 (1994) 1079.
-  M. Finkemeier and E. Mirkes, Z. Phys. C 72 (1996) 619.
-  J.J. Godina Nava and G. Lopez Castro, Phys. Rev. D 52 (1995) 2850.
-  R. Barate et al. (ALEPH Collaboration), Eur. Phys. J. C 10 (1999) 1.
-  T.E. Coan et al. (CLEO Collaboration), Phys. Rev. D 53 (1996) 6037.
-  S. Kurokawa and E. Kikutani, Nucl. Instr. Meth. A 499 (2003) 1, and other papers included in this Volume.
-  Z. Natkaniec et al. (Belle SVD2 Group), Nucl. Instr. Meth. A 560 (2006) 1.
-  A. Abashian et al. (Belle Collaboration), Nucl. Instr. Meth. A 479 (2002) 117.
-  K. Hanagaki et al., Nucl. Instr. Meth. A 485 (2002) 490.
-  A. Abashian et al., Nucl. Instr. Meth. A 491 (2002) 69.
-  S. Jadach and Z. Wa̧s, Comp. Phys. Commun. 85 (1995) 453.
-  Z. Wa̧s, Nucl. Phys. Proc. Suppl. 98 (2001) 96.
-  R. Brun et al., GEANT 3.21, CERN Report No. DD/EE/84-1 (1984).
-  W.-M. Yao et al., J. Phys. G 33 (2006) 1.
-  R.A. Briere et al. (CLEO Collaboration), Phys. Rev. Lett. 90 (2003) 181802.
-  M. Ablikim et al. (BES Collaboration), Phys. Lett. B 633 (2006) 681.
-  D. Aston et al. (LASS Collaboration), Nucl. Phys. B 296 (1988) 493.
-  B. Aubert et al. (BaBar Collaboration), Phys. Rev. D 72 (2005) 072003.
-  M. Jamin, A. Pich and J. Portoles, Phys. Lett. B 640 (2006) 176.
-  S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C 48 (2006) 553.
-  G. Abbiendi et al. (OPAL Collaboration), Eur. Phys. J. C 13 (2000) 213.
-  R. Barate et al. (ALEPH Collaboration), Eur. Phys. J. 4 (1998) 29.
-  M. Acciarri et al. (L3 Collaboration), Phys. Lett. B 345 (1995) 93.
-  R. Barate et al. (ALEPH Collaboration), Eur. Phys. J. C 11 (1999) 599.
-  G. Bonvicini et al. (CLEO Collaboration), Phys. Rev. Lett. 88 (2002) 111803.
-  A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D 12 (1975) 147.
-  M. Aguilar-Benitez et al. (HBC Collaboration), Nucl. Phys. B 141 (1978) 101.