Measurement of the lepton polarization and in the decay
with one-prong hadronic decays at Belle
With the full data sample of pairs recorded by the Belle detector at the KEKB electron-positron collider, the decay is studied with the hadronic decays and . The polarization in two-body hadronic decays is measured, as well as the ratio of the branching fractions , where denotes an electron or a muon. Our results, and , are consistent with the theoretical predictions of the standard model. The polarization values of are excluded at the 90% confidence level.
The Belle Collaboration
Semileptonic decays to leptons (semitauonic decays) are theoretically well-studied processes within the standard model (SM) cite:Heiliger:1989 (); cite:Korner:1990 (); cite:Hwang:2000 (), where the decay process is represented by the tree-level diagram shown in Fig. 1. The lepton is more sensitive to new physics (NP) beyond the SM that couples strongly with mass. A prominent candidate is the two-Higgs-doublet model (2HDM) cite:2HDM:1989 (), where charged Higgs bosons appear. The contribution of the charged Higgs to the decay process cite:CC () is suggested by many theoretical works (for example, Refs. cite:Grzadkowski:1992 (); cite:Tanaka:1995 (); cite:Soni:1997 (); cite:Itoh:2005 (); cite:Crivellin:2012 ()).
Experimentally, the decays have been studied by Belle cite:Belle:2007 (); cite:Belle:2010 (); cite:Belle:2015 (); cite:Belle:2016 (), BABAR cite:BaBar:2008 (); cite:BaBar:2012:letter (); cite:BaBar:2013:fullpaper () and LHCb cite:LHCb:2015 (). Most of these studies have measured ratios of branching fractions, defined as
The denominator is the average of for Belle and BABAR, and for LHCb. The ratio cancels numerous uncertainties common to the numerator and the denominator; these include the uncertainty in the Cabibbo-Kobayashi-Maskawa matrix element , many of the theoretical uncertainties on hadronic form factors (FFs), and experimental reconstruction effects. Recently, LHCb measured the mode using the three-prong decay cite:LHCb:2017 (). To reduce the systematic uncertainty, is measured with the common final states between the numerator and the denominator, and is converted to by using the world-average values for and .
As of early 2016, the results from the three experiments cite:Belle:2015 (); cite:Belle:2016 (); cite:BaBar:2012:letter (); cite:BaBar:2013:fullpaper (); cite:LHCb:2015 () were and standard deviations () cite:HFLAV:2014 () away from the SM predictions of cite:RD_FermiandMILC:2015 () or cite:RD_HPQCD:2015 () and cite:RDst:2012 (), respectively. The overall discrepancy with the SM was about 4. These deviations have been theoretically studied in the context of various NP models cite:RDst:2012 (); cite:Datta:2012 (); cite:Celis:2013 (); cite:Tanaka:2013 (); cite:Biancofiore:2013 (); cite:Dorsner:2013 (); cite:Sakaki:2013 (); cite:Hagiwara:2014 (); cite:Duraisamy:2014 (); cite:Sakaki:2015 (); cite:Freytsis:2015 (); cite:Li:2016 (); cite:Bhattacharya:2016 (); cite:Bardhan:2017 (); cite:Celis:2017 ().
In addition to , the polarizations of the lepton and the meson are sensitive to NP cite:Tanaka:1995 (); cite:Tanaka:2010 (); cite:RDst:2012 (); cite:Datta:2012 (); cite:Biancofiore:2013 (); cite:Tanaka:2013 (); cite:Sakaki:2013 (); cite:Duraisamy:2014 (); cite:Bhattacharya:2016 (); cite:Bardhan:2017 (). The lepton polarization is defined as
where denotes the decay rate of with a helicity of . The SM predicts cite:Tanaka:2010 () and cite:Tanaka:2013 (). For example, the type-II 2HDM allows to be between and for and between and for cite:Tanaka:2013 (); cite:comment:2HDMII (), whereas a leptoquark model suggested in Ref. cite:Sakaki:2013 () with a leptoquark mass of 1 TeV allows to be between and . can be measured in two-body hadronic decays with the differential decay rate
where is the angle of the -daughter meson momentum with respect to the direction opposite the momentum of the system in the rest frame of . The parameter describes the sensitivity to for each -decay mode; in particular, for and for cite:Hagiwara:1990 ().
In this paper, we describe details of the first measurement in the decay with the decays and reported in Ref. cite:Belle:2017 (). Our study includes an measurement independent of previous studies cite:Belle:2015 (); cite:Belle:2016 (); cite:BaBar:2012:letter (); cite:BaBar:2013:fullpaper (); cite:LHCb:2015 (), in which leptonic decays have been used.
Ii Experimental Apparatus
We use the full data sample containing pairs recorded with the Belle detector cite:Belle-detector:2002 () at the asymmetric-beam-energy collider KEKB cite:KEKB:2003 (). 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). The detector is described in detail elsewhere cite:Belle-detector:2002 (). Two inner detector configurations were used. A 2.0 cm radius beampipe and a 3-layer SVD were used for the first sample of pairs, while a 1.5 cm radius beampipe, a 4-layer SVD and a small-cell inner drift chamber were used to record the remaining pairs cite:SVD2:2006 ().
Iii Monte Carlo Simulation
The Monte Carlo (MC) simulated events are used to establish the analysis criteria, study the background and estimate the signal reconstruction efficiency. Events with a pair are generated using EvtGen cite:EvtGen:2001 (), and the meson decays are reproduced based on branching fractions reported in Ref. cite:PDG:2016 (). The hadronization process of the meson decay with no experimentally-measured branching fraction is inclusively reproduced by Pythia cite:PYTHIA:2006 (). For continuum () events, hadronization of the initial quark pair is described by Pythia, and hadron decays are modeled by EvtGen. Final-state radiation from charged particles is added using Photos cite:PHOTOS:2016 (). Detector responses are reproduced by the Belle detector simulator based on Geant3 cite:GEANT:1984 (). The MC samples used in this analysis are described below.
The MC sample for the signal mode () is generated with hadronic FFs based on heavy quark effective theory (HQET). The following values of the hadronic FF parameters in the Caprini-Lellouch-Neubert scheme cite:Caprini:1998 (); cite:CLN:note () are used: , , and from the experimental world averages cite:HFLAV:2014 (), and with 10% uncertainty from the HQET estimation cite:RDst:2012 ().
The MC sample for the normalization mode () is generated based on HQET. Since the FF parameters used for the production of the normalization MC sample have been updated as described above, final-state kinematics are corrected to match the latest parameter values.
Semileptonic decays and , where denotes the excited charm meson states heavier than , comprise an important background category as they have a similar decay topology to the signal events. The MC sample for is generated based on the Isgur-Scora-Grinstein-Wise (ISGW) model cite:ISGW2:1995 (), and decay kinematics are corrected to match the Leibovich-Ligeti-Stewart-Wise (LLSW) model cite:LLSW:1998 (). The branching fractions for with , , and are taken from the world averages cite:HFLAV:2014 (). For the decays, in addition to experimentally-measured modes, we allow unmeasured final states consisting of a and one or two pions, a meson, or an meson based on quantum-number, phase-space and isospin considerations. The radially-excited modes are included so that the total branching fraction of becomes about 3%, which is expected from the difference between (where denotes all the possible charmed-meson states) and the sum of the exclusive branching fractions of . The MC sample is generated using the ISGW model. We take the branching fractions from the theoretical estimates of for each state cite:RDstst:2017 (). We use the average of the four approximations discussed in Ref cite:RDstst:2017 (). We do not consider or other semitauonic modes containing a charmed state heavier than as their small phase space suppresses the branching fractions.
- Other background
The MC samples for other background processes, both events and continuum events, are generated based on the past experimental studies reported in Ref. cite:PDG:2016 (). Unmeasured decay channels are generated with Pythia through the inclusive hadronization process.
The MC sample sizes of the signal mode, the normalization mode, , , the background, and the process are 40, 10, 40, 400, 10, and 5 times larger, respectively, than the full Belle data sample.
Iv Event Reconstruction
iv.1 Reconstruction of the tag side
We conduct the analysis by first identifying events where one of the two mesons () is reconstructed in one of 1104 exclusive hadronic decays cite:Full-recon:2011 (). A hierarchical multivariate algorithm based on the NeuroBayes neural-network package is employed. More than 100 input variables are used to determine well-reconstructed candidates, including the difference between the energy of the reconstructed candidate and the beam energy in the center-of-mass (CM) frame , as well as the event shape variables for suppression of background. The quality of the candidate is synthesized in a single NeuroBayes output-variable classifier (). We require the beam-energy-constrained mass of the candidate , where is the reconstructed three-momentum in the CM frame, to be greater than 5.272 GeV and the value of to be between and . Throughout the paper, natural units with are used. We place a requirement on such that about 90% of true and about 30% of fake candidates are retained. If two or more candidates are retained in one event, we select the one with the highest .
Due to limited knowledge of hadronic decays, the branching fractions of the decay modes are not perfectly modeled in the MC simulation. It is therefore essential to calibrate the reconstruction efficiency (tagging efficiency) with control data samples. We determine a scale factor for each decay mode using events where the signal-side meson candidate () is reconstructed in modes. Further details of the calibration method are described in Ref. cite:Belle_Xulnu:2013 (). The ratio of measured to expected rates in each decay mode ranges from 0.2 to 1.4, depending on the decay mode, and is 0.72 on average. After the efficiency calibration, the tagging efficiencies are estimated to be about 0.20% for charged mesons and 0.15% for neutral mesons.
iv.2 Reconstruction of the signal side
We reconstruct the signal mode and the normalization mode using the particle candidates not used for reconstruction. The following decay modes are used for the daughter particles: , , , and for the candidate; and for the candidate; , , , , , , , , , , , , , , and for the candidate; and , and , respectively, for the light-meson candidates. A -daughter candidate or is combined with a candidate to form a candidate. For the normalization events, a charged lepton or is associated instead of or .
iv.2.1 Particle selection
First, daughter particles of and (, , , , and ) and charged leptons ( and ) are reconstructed. For reconstruction, we use different particle selections from those applied for the reconstruction described in Ref. cite:Full-recon:2011 ().
Charged particles are reconstructed using the SVD and the CDC. All tracks, except for -daughter candidates, are required to have cm and cm, where and are the impact parameters to the interaction point (IP) in the directions perpendicular and parallel, respectively, to the beam axis. Charged-particle types are identified by a likelihood ratio based on the responses of the sub-detector systems. Identification of and candidates is performed by combining measurements of specific ionization () in the CDC, the time of flight from the IP to the TOF counter and the photon yield in the ACC. For -daughter candidates, an additional proton veto is required in order to reduce background from baryonic decays such as . The ECL electromagnetic shower shape, track-to-cluster matching at the inner surface of the ECL, in the CDC, the photon yield in the ACC and the ratio of the cluster energy in the ECL to the track momentum measured with the SVD and the CDC are used to identify candidates cite:eID:2002 (). Muon candidates are selected based on their penetration range and transverse scattering in the KLM cite:muID:2002 (). To form candidates, we combine pairs of oppositely-charged tracks, treated as pions. Standard Belle selection criteria are applied cite:Belle_KsKsKs:2005 (): the reconstructed vertex must be detached from the IP, the momentum vector must point back to the IP, and the invariant mass must be within 30 MeV of the nominal mass cite:PDG:2016 (), which corresponds to about 8. (In this section, denotes the corresponding mass resolution.)
Photons are reconstructed using ECL clusters not matching to charged tracks. Photon energy thresholds of 50, 100 and 150 MeV are used in the barrel, forward-endcap and backward-endcap regions, respectively, of the ECL to reject low-energy background photons, such as those originating from the beams and hadronic interactions of particles with materials in the detector.
Neutral pions are reconstructed in the decay . For candidates from or decay, referred to as normal s, we impose the same photon energy thresholds described above. The candidate’s invariant mass must lie between 115 and 150 MeV, corresponding to about around the nominal mass cite:PDG:2016 (). In order to reduce the number of fake candidates, we apply the following candidate selection procedure. The candidates are sorted in descending order according to the energy of the most energetic daughter. If a given photon is the most energetic daughter of two or more candidates, they are sorted by the energy of the lower-energy daughter. We then retain the candidates whose daughter photons are not shared with a higher-ranked candidate. In this criterion, 76% of the correctly reconstructed candidates are selected while 54% of the fake candidates are removed. The retained candidates are used for and reconstruction described later.
For the soft from decay, we impose a relaxed photon energy threshold of 22 MeV in all ECL regions and the same requirement for the invariant mass of the two photons. Additionally, the energy asymmetry is required to be less than 0.6, where and are the energies of the high- and low-energy photon daughters in the laboratory frame. Here, we do not apply the normal- candidate selection procedure.
The candidate is formed from the combination of a and a . The candidate invariant mass must lie between 0.66 and 0.96 GeV.
After reconstructing the light mesons, we reconstruct the candidates in 15 decay modes. The invariant mass requirements are optimized for each decay mode. For the modes used in forming candidates, the reconstructed invariant masses () are required to be within () of the nominal meson mass cite:PDG:2016 () for the high (low) signal-to-noise ratio (SNR) modes. For candidates, the requirements are loosened to and for the high- and low-SNR modes, respectively. The requirements for the candidates are for the high-SNR modes and for the low-SNR modes around the nominal meson mass cite:PDG:2016 (). Here, the high-SNR modes are , , , , , , , ; the low-SNR modes are all remaining modes. We reconstruct candidates by combining a candidate with a , , or soft . The candidates are selected based on the mass difference , where denotes the reconstructed invariant mass of the candidate. The , , , and candidates are required to have within , , and , respectively, of the nominal .
The candidates are formed by associating a -daughter meson (signal events) or a (normalization events) with a candidate. Allowed combinations are for , for , for and for , where or . We select one of the following meson combinations: , , and .
For the signal mode, if at least one possible candidate for the signal mode is found in an event, we calculate in the rest frame of the . Although this frame cannot be determined completely, equivalent kinematic information is obtained using the rest frame of the system. This frame is obtained by boosting the laboratory frame along with the three-momentum vector component of the momentum transfer
where denotes the four-momentum of the beam, , and , respectively. In this frame, the energy and the magnitude of the momentum of the lepton are determined only by as
where is the lepton mass. The cosine of the angle between the momenta of the lepton and its daughter meson is determined by
where and denote the energy and the momentum of the lepton (the daughter ) respectively, and is the mass of the daughter. Through a Lorentz transformation from the rest frame of the system to the rest frame, the following relation is obtained:
where is the -daughter momentum in the rest frame of , and and . Solving gives the value of . Events are required to lie in the physical region of , where 97% of the reconstructed signal events are retained. As shown in Fig. 2, there is a significant background peak near 1 in the sample due to the background. To reject this background, we only use the region in the fit to the sample.
Due to the kinematic constraint that must be greater than , almost no signal events exist with below 4 GeV. Therefore is required. The variable is the linear sum of the energy of ECL clusters not used in the event reconstruction. The ECL clusters satisfying the photon-energy requirement defined in the previous section are added to . Signal events ideally have equal to zero with a tail in the distribution from the beam background and split-off showers, separated from the main ECL cluster and reconstructed as photon candidates. We require to be less than 1.5 GeV.
For the normalization mode, we calculate the squared missing mass,
where denotes the four-momentum of the charged lepton and the other variables were defined earlier. The normalization events populate the region near because there is exactly one neutrino in an event. We require . We further require to be less than 1.5 GeV.
Finally, for both the signal and the normalization events, we require that there be no extra charged tracks with and , and normal candidates.
iv.3 Best candidate selection
After event reconstruction, the average number of retained candidates per event is about 1.09 for charged mesons and 1.03 for neutral mesons. In events where two or more candidates are reconstructed, 2.1 candidates are found on average. Multiple-candidate events mostly arise from more than one combination of a candidate with photons or soft pions. For the charged mode, about 2% of the events are reconstructed both in the and modes. Since the latter mode has a much higher branching fraction, we assign these events to the sample. The contribution of this type of multiple-candidate events is negligibly small in the neutral mode. We then select the most signal-like candidate as follows. For the events, we select the candidate with the most energetic photon associated with the . For the and events, we select the candidate with the soft that has an invariant mass nearest the nominal mass. For the events, we select one candidate at random since the multiple-candidate probability is only . After the candidate selection, roughly 2% of the retained events are reconstructed both in the and the samples. Since the MC study indicates that about 80% of such events originate from the decay, we assign these events to the sample.
iv.4 Sample composition
The reconstructed events are categorized in turn as below. Based on this categorization, we construct histogram probability density functions (PDFs) from the MC samples to perform a final fit.
Correctly reconstructed signal events that originate from events are categorized in this component. The yield is treated as a free parameter determined by and .
- cross feed
Cross feed events, where the events are reconstructed as due to misreconstruction of one , or the events are reconstructed as by adding a random , comprise this component. Since these events originate from , they contribute to the determination. They are also used for the determination after the bias on is corrected by MC information.
- Other cross feed
events with other decay modes also contribute to the signal sample. They originate mainly from with one or two missing , or with a low-momentum that does not reach the KLM. These two modes occupy about 80% of this component. The MC study shows that the cross feed events both from and have negligible impact on our measurement. In the fit, the yield of this category is determined by .
The decay contaminates the signal sample due to misassignment of as . We fix the yield in the signal sample from the fit to the distribution of the normalization sample.
- and hadronic decays
The ( is also included in this category) and hadronic decays are the most uncertain component due to limited experimental knowledge. By missing a few particles such as mesons, the event topology resembles the signal event. We combine these decay modes into one component. The fractions of the decays and hadronic decays are about 10% and 90%, respectively, according to the MC study. Since it is difficult to estimate the yield of this component using MC simulation or to fix the yield using control data samples, we float the yield in the final fit. One exception is the collection of modes with two charm mesons such as and . Since the branching fractions of these modes have been studied experimentally, we fix their yield using the MC expectation after correction with the branching fractions based on Ref. cite:PDG:2016 ().
Continuum events from the process provide a minor contribution at in the signal sample. We fix the yield using the MC expectation.
All events containing fake candidates are categorized in this component. This is the main background source in the charged meson sample. For the neutral sample, many candidates are reconstructed from the combination of a with a and therefore much more cleanly reconstructed than the other modes with or . The yield is determined from a comparison of the data and the MC sample in the sideband regions.
iv.5 Measurement Method of and
We use the following variables to measure yields of the signal and the normalization modes. For the normalization mode, is the most suitable variable due to its high purity. On the other hand, the shape of the distribution for the signal mode has a strong correlation with . To measure the signal yield, we use because it has a small correlation to and provides good discrimination between the signal and the background modes.
The value of is measured using the formula
where denotes the relevant branching fraction, and and ( and ) are the efficiencies (the observed yields) for the signal and the normalization modes, respectively. The indices and represent the decays ( or ) and the charges (charged or neutral ), respectively. Assuming isospin symmetry, we use .
The value of is determined using the formula
where denotes the signal yield in the region and satisfies . This formula is obtained by calculating
The differential decay rate is given by Eq. (3). As with , we use the common parameters .
Due to detector efficiency effects, the measured polarization, , is biased from the true value of . To correct for this bias, we form a linear function that maps to using several MC sets with different . This function, denoted the correction function, is separately prepared for each sample since the detector bias depends on the given mode. We also make a correction function for the cross feed component to take into account the distortion of the distribution shape. In the correction, other kinematic distributions are assumed to be consistent with the SM predictions.
V Background Calibration and PDF Validation
To use the MC distributions as histogram PDFs, the MC simulation needs to be verified using calibration data samples. In this section, the calibration of the PDF shapes is discussed.
v.1 Signal PDF shape
To validate the shape of the signal component, we use the normalization mode as the control sample. It has similar properties to the signal component; there is no extra photon from the decay except for bremsstrahlung photons, and therefore the shape is mostly determined by the background photons. The normalization sample contains about 50 times more events than the expected signal yield. Figure 3 shows a comparison of between data and MC simulation. The pull of each bin is shown in the bottom panel; hereinafter, the pull in the th bin is defined as
where and denote the number of events and the statistical error, respectively, in the th bin of the data (MC) distribution. The fake yield is scaled based on the calibration discussed in the next section. Since the contribution from the other background components is negligibly small, it is fixed to the MC expectation. The shape in the MC sample agrees well with the data within statistical uncertainty.
v.2 Fake events
One of the most significant background components arises from fake candidates. The combinatorial fake background processes are difficult to model precisely in the MC simulation. The shapes for the data and the MC sample are compared using sideband regions of 50–500 MeV, 135–190 MeV, 135–190 MeV, and 140–500 MeV for , , , and , respectively; each excludes about around the peak. These sideband regions contain 5 to 50 times more events than the signal region. Figure 4 shows the comparison of the shapes. Although all the and modes are combined in these figures, the shape has been compared in 16 subsamples of modes, modes, modes, and the two regions. We find good agreement of the shape within the statistical uncertainty of these mass sideband data samples. We also check the distribution in the sideband region, as shown in Figs. 4 and 4. The distribution in the MC simulation also shows good agreement with the data within the statistical uncertainty.
In both the signal and the normalization samples, yield discrepancies of up to 20% are observed. The fake yields in the signal region of the MC simulation are scaled by the yield ratios of the data to the MC sample in the sideband regions.
v.3 and hadronic composition