Measurement of Absolute Hadronic Branching Fractions of Mesons and Cross Sections at the
Using 281 pb of collisions recorded at the resonance with the CLEO-c detector at CESR, we determine absolute hadronic branching fractions of charged and neutral mesons using a double tag technique. Among measurements for three and six modes, we obtain reference branching fractions and , where the first uncertainty is statistical, the second is all systematic errors other than final state radiation (FSR), and the third is the systematic uncertainty due to FSR. We include FSR in these branching fractions by allowing for additional unobserved photons in the final state. Using an independent determination of the integrated luminosity, we also extract the cross sections and at a center of mass energy, MeV.
Measurements of absolute hadronic meson branching fractions play a central role in the study of the weak interaction because they serve to normalize many important meson and hence meson branching fractions. We present absolute measurements of the and branching fractions111Generally () will refer to either or ( or ), and specification of an explicit state and its decay daughters will imply a corresponding relationship for the and its daughters. for the Cabibbo favored decays , , , , , , , and , and for the Cabibbo suppressed decay . Two of these branching fractions, and , are particularly important because most and branching fractions are determined from ratios to one of these branching fractions Yao et al. (2006). As a result, almost all branching fractions in the weak decay of heavy quarks that involve or mesons are ultimately tied to one of these two branching fractions, called reference branching fractions in this paper. Furthermore, these reference branching fractions are used in many measurements of CKM matrix elements for and quark decay.
We previously reported results He et al. (2005) based on a subset of the data sample used in this analysis. The measurements presented here supersede those results.
We note that the Monte Carlo simulations used in calculating efficiencies in this analysis include final state radiation (FSR). Final state radiation reduces yields because candidates can fail the energy selection criteria (the limits described in Sec. IV) if the energies of the FSR photons are large enough. However, many branching fractions used in the Particle Data Group (PDG) averages Yao et al. (2006) do not take this effect into account. The selection criteria imposed in differing analyses correspond to differing maximum photon energies, and hence differing FSR effects on the observed yields and branching fractions. Had we not included FSR in our simulations, our quoted branching fractions would have been lower than we report; the difference is mode-dependent, ranging from 0.5% to 3%.
Ii Branching Fractions and Production Cross Sections
The data for these measurements were obtained in collisions at a center-of-mass energy GeV, near the peak of the resonance. At this energy, no additional hadrons accompany the and pairs that are produced. These unique final states provide a powerful tool for avoiding the most vexing problem in measuring absolute branching fractions at higher energies — the difficulty of accurately determining the number of mesons produced. Following a technique first introduced by the MARK III Collaboration Baltrusaitis et al. (1986); Adler et al. (1988), we select “single tag” (ST) events in which either a or is reconstructed without reference to the other particle, and “double tag” (DT) events in which both the and are reconstructed. Reconstruction of one particle as a ST serves to tag the event as either or . Absolute branching fractions for or decays can then be obtained from the fraction of ST events that are DT, without needing to know independently the integrated luminosity or the total number of events produced.
If violation is negligible, then the branching fractions and for and are equal. However, the efficiencies and for detection of these modes may be somewhat different since the cross sections for scattering of pions and kaons on the nuclei of the detector material depend on the charge of these particles. With the assumption that , the observed yields and of reconstructed and ST events will be
where is the number of events (either or events) produced in the experiment. The DT yield with (signal mode) and (tagging mode) will be
where is the efficiency for detecting DT events in modes and . Hence, the ratio of the DT yield () to the ST yield () provides an absolute measurement of the branching fraction ,
Due to the high segmentation and large solid angle of the CLEO-c detector and the low multiplicities of hadronic decays, . Hence, the ratio is insensitive to most systematic effects associated with the decay mode, and a signal branching fraction obtained using this procedure is nearly independent of the efficiency of the tagging mode. Of course, is sensitive to the signal mode efficiency (), whose uncertainties dominate the contribution to the systematic error from the efficiencies.
Finally, the number of pairs that were produced is given by
Since , the systematic error for is nearly independent of systematic uncertainties in the efficiencies.
Estimating errors and combining measurements using these expressions requires care because and are correlated (whether or not ) and measurements of using different tagging modes are also correlated. Although and branching fractions are statistically independent, systematic effects introduce significant correlations among them. Therefore, we utilize a fitting procedure Sun (2006) in which both charged and neutral meson yields are simultaneously fit to determine all of our charged and neutral branching fractions as well as the numbers of charged and neutral pairs that were produced (see Sec. IX). The input to the branching fraction fit includes both statistical and systematic uncertainties, as well as their correlations. We also perform corrections for backgrounds, efficiency, and crossfeed among modes directly in the fit, as the sizes of these adjustments depend on the fit parameters. Thus, all experimental measurements, such as yields, efficiencies, and background branching fractions, are treated in a consistent manner. As indicated above, we actually obtain and candidate yields separately in order to accommodate possible differences in efficiency, but we constrain charge conjugate branching fractions to be equal. However, we also search for violation by comparing yields for charge conjugate modes after subtraction of backgrounds and correction for efficiencies (see Sec. X).
We obtain the production cross sections for and by combining and , which are determined in the branching fraction fit, with a separate measurement of the integrated luminosity .
Iii The CLEO-c Detector
The CLEO-c detector is a modification of the CLEO III detector Kubota et al. (1992); Peterson et al. (2002); Artuso et al. (2005) in which the silicon-strip vertex detector has been replaced with a six-layer vertex drift chamber, whose wires are all at small stereo angles to the axis of the chamber cle (). These stereo angles allow hit reconstruction in the dimension parallel to the drift chamber axis. The charged particle tracking system, consisting of the vertex drift chamber and a 47-layer central drift chamber Peterson et al. (2002), operates in a 1.0 T magnetic field whose direction is along the drift chamber axis. The two drift chambers are coaxial, and the electron and positron beams collide at small angles to this common axis (see Appendix A). The root-mean-square (rms) momentum resolution achieved with the tracking system is approximately % at for tracks that traverse all layers of the drift chamber. Photons are detected in an electromagnetic calorimeter consisting of about 7800 CsI(Tl) crystals Kubota et al. (1992). The calorimeter attains an rms photon energy resolution of 2.2% at GeV and 5% at 100 MeV. The solid angle coverage for charged and neutral particles in the CLEO-c detector is 93% of .
We utilize two devices to obtain particle identification (PID) information to separate from : the central drift chamber, which provides measurements of ionization energy loss (), and a cylindrical ring-imaging Cherenkov (RICH) detector Artuso et al. (2005) surrounding the central drift chamber. The solid angle of the RICH detector is 80% of . As described in the next Section, for momenta below where separation is highly efficient and RICH separation is not, information is used alone. Above this threshold, and RICH information are combined if both are available. For momenta below (the entire momentum range of hadrons from decay at the ) the combined and RICH particle identification provides excellent separation of kaons and pions, as illustrated in Figs. 1 and 2.
Above there are modest decreases in and , the efficiencies for identifying pions and kaons, respectively; and modest increases in the probabilities, and of misidentifying a kaon as a pion or vice-versa, respectively. These efficiencies and misidentification probabilities are averaged over the whole solid angle of the tracking system. However, the RICH solid angle is about 86% of the solid angle of the tracking system, and within that solid angle, pion-kaon separation is excellent Artuso et al. (2005) above . Outside of the RICH acceptance, only information is available, and the lower PID efficiency from at high momentum leads to the modest decreases in performance observed in this high momentum region.
The response of the the CLEO-c detector was studied with a detailed GEANT-based gea () Monte Carlo (MC) simulation of particle trajectories generated by EvtGen Lange (2001), with final state radiation predicted by PHOTOS Barberio and Was (1994). Simulated events were reconstructed and selected for analysis with the reconstruction programs and selection criteria used for data.
The integrated luminosity (needed only to obtain production cross sections from and ) was measured using the QED processes , , and , achieving a relative systematic error of 1.0%, as described in Appendix C.
Iv Data Sample and Event Selection
In this analysis, we utilized a total integrated luminosity of pb of data collected at center of mass energies near GeV. The data were produced by the Cornell Electron Storage Ring (CESR), a symmetric collider, operating in a configuration cle () that includes twelve wiggler magnets222The first 56 pb of data were obtained in an earlier configuration of CESR with six wiggler magnets. to enhance synchrotron radiation damping at energies in the charm threshold region. The rms spread in with the twelve wiggler magnets is MeV.
In each event we reconstructed and candidates from combinations of final-state particles. Reconstruction begins with standardized requirements for , , , and candidates; these requirements are common to many CLEO-c analyses involving decays.
Charged tracks must be well-measured and satisfy track quality criteria, including the following requirements: the momentum of the track must be in the range ; the polar angle must be in the range ; and at least half of the layers traversed by the track must contain a reconstructed hit from that track. Track candidates must also be consistent with coming from the interaction region in three dimensions. The beams collided close to the origin of the coordinate system, but the collision point in the - plane (transverse to the axis of the drift chamber system) usually changed somewhat when CESR operating conditions changed significantly. Hence, we determined a separate average beam position for each data subset bounded by such changes. The period of validity for a given average beam position was as short as one run and as long as one hundred runs. (Most runs corresponded to a CESR fill and were typically between 40 and 60 minutes long.) For each track, we required that the distance of the track from the average beam position in the - plane must be less than 0.5 cm ( cm). Finally, we required that the track must pass within 5.0 cm of the origin in the direction ( cm). The requirements on and are approximately five times the standard deviation for the corresponding parameter.
We identified charged track candidates as pions or kaons using and RICH information. In the rare case that no useful information of either sort was available, we utilized the track as both a and a candidate. Otherwise, as described below, we either identified it as or , or rejected it if it was inconsistent with both hypotheses.
If information was available, we calculated and , where
from the measurements , the expected for pions and kaons of that momentum, and the measured resolution at that momentum. We rejected tracks as kaon candidates when was greater than 9, and similarly for pions. The difference was also calculated. If information was not available, this difference was set equal to 0.
We used RICH information if the track was within the RICH acceptance () and its momentum was above , which is far enough above the Cherenkov threshold for kaons that we expect good efficiency for kaons and pions. Furthermore, we required that valid RICH information was available for both pion and kaon hypotheses. We then rejected tracks as kaon candidates when the number of Cherenkov photons detected for the kaon hypothesis was less than three, and similarly for pions. When there were at least three photons for each hypothesis, we obtained a difference for the RICH, , from a likelihood ratio using the locations of Cherenkov photons and the track parameters Artuso et al. (2005). If RICH information was not available, we set this difference equal to 0.
The final particle identification requirement for a kaon (pion) candidate was that the track be more consistent with the kaon (pion) hypothesis than the pion (kaon) hypothesis. Specifically, we combined the and RICH differences in an overall difference, . Kaon candidates were required to have , and pion candidates were required to have . When , we utilized the track as both a and a candidate.
We formed neutral pion candidates from pairs of photons reconstructed in the calorimeter. The showers were required to pass photon quality requirements and to have energies greater than 30 MeV. An unconstrained mass was calculated from the energies and momenta of the two photons, under the assumption that the photons originated at the center of the detector. This mass was required to be within three standard deviations () of a nominal mass value that varied slightly with the total momentum of the candidate. The slight change in the nominal mass compensates for energy leakage in the calorimeter for energetic showers. The uncertainty on was calculated from the error matrices of the two photons; the values of were typically in the range 5 – 7 . We then performed a kinematic fit of the two photon candidates to the mass from the PDG Yao et al. (2006), and the resulting energy and momentum of the were used for further analysis.
We built candidates from pairs of intersecting opposite-charge tracks. These tracks were not subjected to the track quality or particle identification requirements described above. For each pair of tracks, we performed a constrained vertex fit and used the resulting track parameters to calculate the invariant mass, . We accepted the track pair as a candidate if the invariant mass was within of the mass from the PDG Yao et al. (2006). The resolution was . There is very little background under the peak in the distribution, so we did not impose requirements on track quality or particle identification of the daughters. Also, we did not impose other requirements commonly utilized in reconstructing candidates, e.g., requiring that the candidate come from the collision point. Imposition of any of these additional requirements would have necessitated evaluation of an another systematic uncertainty.
We formed and candidates in the three and six decay modes from combinations of , , and candidates selected using the requirements described above. Two variables reflecting energy and momentum conservation are used to identify valid candidates.
First, we calculated the energy difference, , where is the total measured energy of the particles in the candidate and is the mean value of the energies of the and beams. The value of was determined from accelerator parameters for each run. Candidates were rejected if they failed the requirements, given in Table 1, which were tailored for each individual decay mode. As mentioned in the Introduction, a candidate may be lost if FSR reduces below the lower limit set by the requirement. We include this effect in our MC simulations.
Second, we calculated the beam-constrained mass of the candidate by substituting the beam energy for the energy of the candidate, i.e.,
where is the measured total momentum of the particles in the candidate. Valid candidates produce a peak in at the mass. To obtain our yields, we fit the distribution for events with , as described in detail below.
For the ST analysis, if there was more than one candidate in a particular or decay mode, we chose the candidate with the smallest . Multiple candidates were very rare in some modes, including and , and more common in others. The largest multiple candidate rate occurred in , where approximately 18% of the events had more than one candidate.
In two-track events that were consistent with our requirements for decays, we imposed additional lepton veto requirements to eliminate , , and cosmic ray muon events. We eliminated the event if either the pion or kaon candidate track was consistent with being an electron or a muon, utilizing criteria described in Appendix B.5. A cosmic ray event where the muon has the same momentum as the kaon or pion in a decay at rest will peak in at the beam energy. Removing these events simplifies the description of the background shape in the fits. The events from and populate the distribution more uniformly. Since our DT modes all have at least four charged particles, the , , and cosmic ray muon event suppression requirements only affect the ST yields.
In the mode there is a background from Cabibbo suppressed decays to . To suppress this background, candidates are rejected if any pair of oppositely-charged pions (excluding those from the decay) falls within the range . This veto is applied for both ST and DT events.
To obtain a DT candidate, we applied the appropriate requirements from Table 1 to the candidate and the candidate in the DT mode. If there was more than one DT candidate with a given and decay mode, we chose the combination for which the average of and — i.e., — was closest to . This criterion selects the correct combination when an event contains multiple candidates due to mispartitioning. (Mispartitioning means that some tracks or s were assigned to the wrong candidate.) In studies of Monte Carlo events, we demonstrated that this procedure does not generate false peaks at the mass in the vs. distributions that are narrow enough or large enough to be confused with the DT signal.
V Generation and Study of Monte Carlo Events
We used Monte Carlo simulations to develop the procedures for measuring branching fractions and production cross sections, to understand the response of the CLEO-c detector, to determine parameters to use in fits for yields, to determine efficiencies for reconstructing particular and decay modes, and to estimate and understand possible backgrounds. In each case events were generated with the EvtGen program Lange (2001), and the response of the detector to the daughters of the decays was simulated with GEANT gea (). The EvtGen program includes simulation of initial-state-radiation (ISR) events, i.e., events in which the or the radiates a photon before the annihilation. The program PHOTOS Barberio and Was (1994) was used to simulate final state radiation — radiation of photons by the charged particles in the final state. We used PHOTOS version 2.15 and enabled the option of interference between radiation produced by the various charged particles. FSR causes a loss of efficiency due to energy lost to unreconstructed FSR photons; the largest effect is a 3% efficiency loss for the decay . We generated three types of Monte Carlo events:
generic Monte Carlo events, in which both the and the decay with branching fractions based on PDG 2004 Eidelman et al. (2004) averages, supplemented with estimates for modes not listed by the PDG,
single tag signal Monte Carlo events, in which either the or the always decays in one of the nine modes measured in this analysis while the or , respectively, decays generically, and
double tag signal Monte Carlo events, in which both the and the decay in particular modes.
We applied the same selection criteria for candidates and events when analyzing data and Monte Carlo events. We compared many distributions of particle kinematic quantities in data and Monte Carlo events to assess the accuracy and reliability of the modeling of the decay process (event generation) and Monte Carlo simulation of the detector response. The agreement between data and Monte Carlo events for both charged and neutral particles was excellent for almost all distributions of kinematic variables that we studied. The results of this analysis are not sensitive to the modest discrepancies that were observed in a few distributions. One exception is the resonant substructure in the multi-body final states studied in this analysis. The sensitivity of the analysis to the description of the multi-body substructure is discussed further in the section on systematic uncertainties.
Vi Determination of Efficiencies and Data Yields
We obtained yields in Monte Carlo events and data with unbinned likelihood fits to the distributions of (for single tags) and vs. (for double tags). We determined ST and DT efficiencies from the yields of signal Monte Carlo events. These efficiencies include the branching fractions for and decays. We corrected the MC efficiencies for modes involving daughters to be consistent with the updated value of in the PDG 2006 Yao et al. (2006) averages.
The functions and parameters used to model signals and non-peaking backgrounds in these fits are described in Sec. VI.1 and Appendix A. In Secs. VI.2 and VI.3 we discuss the fit procedures, the efficiencies, and the data yields for double and single tag events. Our procedure was to determine first the parameters describing the momentum resolution function in each mode by fitting double tag signal Monte Carlo events where the and decayed to charge conjugate final states. After determining these parameters, we used them in fitting all double and single tag modes in data and Monte Carlo events.
vi.1 Signal and Background Shapes and Parameters
Signal line shapes in the distributions depend on the beam energy spread, initial state radiation from the incident and , the resonance line shape, and momentum resolution. Appendix A describes the method used to combine these contributions to obtain the line shape function that we used to describe signals.
The distributions for and events have peaks at and , respectively, and radiative tails at higher masses due to ISR. The shapes of the peaks are due primarily to beam energy spread and momentum resolution. The radiative tails occur at because the momenta of mesons in events that have lost significant energy due to ISR are lower than the momenta of mesons in events without significant energy loss. Therefore, using in Eq. (6) to calculate leads to . As described in Appendix A, the shape of the radiative tail depends on the resonance line shape and the energy spectrum of the ISR photons.
For the fits to data, our resonance line shape description requires values of the mass and width ( and , respectively) and the Blatt-Weisskopf radius () (see Eqs. (15) and (16)). The resonance line shape primarily affects the distribution of the radiative tail at . Hence, our data cannot separate the effects of simultaneous changes to the mass, width, and Blatt-Weisskopf radius, and we require external input. The Particle Data Group Yao et al. (2006) reports three measurements of from MARK I Rapidis et al. (1977), DELCO Bacino et al. (1978), and MARK II Abrams et al. (1980), of MeV, MeV, and MeV, respectively. The PDG averages these to obtain MeV, and it also has a fit which gives MeV. Furthermore, there is a recent measurement from BES Ablikim et al. (2007) that gives a width of MeV. In addition to the width, BES also determines the mass of the to be . In our fits we adopted the BES values for the mass and width333For the width, we actually used the value MeV that appeared in a BES preprint before publication.. We take the Blatt-Weisskopf radius to be , which is favored by our data when and are fixed to the BES values. To assess the systematic uncertainties, we vary these parameters as discussed in Section VII.
We used a sum of three Gaussian functions to describe the momentum resolution of the detector,
Here, is the true momentum of the meson; is its reconstructed momentum; is the width of the core Gaussian; is the width of the second Gaussian; is the fraction of candidates that are smeared with the width of the second Gaussian; is the width of a third Gaussian; and is the fraction of candidates that are smeared with the width of the third Gaussian. All values of and determined from our fits (see below) are greater than 2, so the second Gaussian is significantly wider than the first and the third is significantly wider than the second.
Combinatorial backgrounds were described by a modified ARGUS function Albrecht et al. (1990)
where is the candidate mass (), is the endpoint given by the beam energy, and is a normalization constant. The modification of the original ARGUS function allows the power parameter, , to differ from the nominal value, . The parameters and were determined in each individual ST fit to data or MC simulations. Combinatorial backgrounds are very small in DT data, so for DT data and signal MC events, we fixed and used values of determined from much larger generic MC samples.
In DT fits, we must include a number of features in our fit function. Figure 3 shows the distribution of vs. for DT event candidates from data, and it illustrates the signal and background components in the - plane. The principal features of this two-dimensional distribution are the following.
There is an obvious signal peak in the region surrounding . The distribution of the signal candidates in this peak is influenced primarily by beam energy spread, and secondarily by the resonance shape and detector resolution. The signal also includes a tail due to initial state radiation along the vs. diagonal. This correlation is due to the fact that — neglecting measurement and reconstruction errors — the values of and calculated using the beam energy will both be too large by the same amount if energy was lost due to ISR.
There are horizontal and vertical bands centered at and , respectively. These bands contain DT candidates in which the () candidate was reconstructed correctly, but the () was not.
There is a diagonal band below the peak that continues through the signal region and the radiative tail. This band is populated by the following two sources of background.
There are “mispartitioned” candidates, in which all of the particles were found and reconstructed reasonably accurately, but one or more particles from the were interchanged with corresponding particles from the (e.g., s were interchanged between the and the ).
There are also continuum events in this band (i.e., annihilations into , , and quark pairs). Events fall into this band because all particles in the event were reconstructed and used to make the and the candidates, so the two candidates have equal momentum.
We accounted for the signal described in the first bullet with the DT signal line shape function given in Eq. (24). To account for the features in the second and third bullets, we included four different background terms in each fit:
Two background terms where one of the mesons is correctly reconstructed and the second is incorrectly reconstructed. These terms are described by a signal function of or for the correctly reconstructed or multiplied by an ARGUS function of or , for the or , respectively.
One ARGUS background shape in (defined above) for mispartitioned and continuum events, multiplied by a Gaussian in . The width of the Gaussian depends linearly on .
One background term represented by the product of an ARGUS function of and an ARGUS function of , to account for small combinatorial backgrounds.
The signal shape parameters describing the effects of detector resolution on the mass () distributions are determined by fits to DT signal Monte Carlo samples in which the and decay to charge conjugate final states. The four parameters controlling the two wide Gaussians in the resolution function are then fixed to these values in all other fits, and the core resolution and mass values are fixed in all other Monte Carlo fits. The DT Monte Carlo samples offer a significantly better signal to background ratio than the single tag samples, and there are insufficient statistics to determine these parameters well from data. Furthermore, double tag fits allow us to separate the effects of beam energy smearing and detector resolution. In single tag fits, the effects of detector resolution and beam energy smearing both broaden the distribution. In double tags, as indicated in Fig. 3, beam energy smearing moves the events along the diagonal line in a fully correlated way while the effects of detector resolution smear events isotropically, including perpendicular to this diagonal. The fitted momentum resolution parameters from Eq. (7) are given in Table 2.
vi.2 Double Tag Efficiencies and Data Yields
We determined double tag yields in data and Monte Carlo events from unbinned maximum likelihood fits to vs. distributions using the signal and background functions described in the previous Subsection. The efficiencies, yields from data, and peaking backgrounds (see Sec. VII) are given in Tables 3 and 4 for and events, respectively. Since the ARGUS backgrounds are small in signal MC, the errors in the efficiencies were estimated using binomial statistics.
|Double Tag Mode||Efficiency (%)||Data Yield||Background|
|Double Tag Mode||Efficiency (%)||Data Yield||Background|