Measurement of the form factors of the decay and determination of the CKM matrix element
Abstract
This article describes a determination of the CabibboKobayashiMaskawa matrix element from the decay using 711 fb of Belle data collected near the resonance. We simultaneously measure the product of the form factor normalization and the matrix element as well as the three parameters , and , which determine the form factors of this decay in the framework of the Heavy Quark Effective Theory. The results, based on about 120,000 reconstructed decays, are , , and . The branching fraction of is measured at the same time; we obtain a value of . The errors correspond to the statistical and systematic uncertainties. These results give the most precise determination of the form factor parameters and to date. In addition, a direct, modelindependent determination of the form factor shapes has been carried out.
pacs:
The Belle Collaboration
I Introduction
The study of the decay is important for several reasons. The total rate is proportional to the magnitude of the CabibboKobayashiMaskawa(CKM) matrix element Kobayashi:1973fv (); Cabibbo:1963yz () squared. Experimental investigation of the form factors of the decay can check theoretical models and possibly provide input to more detailed theoretical approaches. In addition, is a major background for charmless semileptonic decays, such as , or semileptonic decays with large missing energy, including . Precise knowledge of the form factors in the decay will thus help to reduce systematic uncertainties in these analyses.
This article is organized as follows: After introducing the theoretical framework for the study of decays in Section II, the experimental procedure is presented in detail in Section III. This is followed by a discussion of our results and the systematic uncertainties assuming the form factor parameterization of Caprini et al. Caprini:1997mu () in Section IV. Finally, a measurement of the form factor shapes is described in Section V.
This paper supersedes our previous result Abe:2001cs (), based on a subset of the data used in this analysis.
Ii Theoretical framework
ii.1 Kinematic variables
The decay ref:0 () proceeds through the treelevel transition shown in Fig. 1. Below we will follow the formulation proposed in reviews Neubert:1993mb (); Richman:1995wm (), where the kinematics of this process are fully characterized by four variables as discussed below.
The first is a function of the momenta of the and mesons, labeled and defined by
(1) 
where and are the masses of the and the mesons (5.279 and 2.010 GeV/, respectively Amsler:2008zzb ()), and are their fourmomenta, and . In the rest frame the expression for reduces to the Lorentz boost . The ranges of and are restricted by the kinematics of the decay, with corresponding to
(2) 
and to
(3) 
The point is also referred to as zero recoil.
The remaining three variables are the angles shown in Fig. 2:

, the angle between the direction of the lepton and the direction opposite the meson in the virtual rest frame;

, the angle between the direction of the meson and the direction opposite the meson in the rest frame;

, the angle between the plane formed by the decay and the plane formed by the decay, defined in the meson rest frame.
ii.2 Fourdimensional decay distribution
Three helicity amplitudes, labeled , , and , can be used to describe the Lorentz structure of the decay amplitude. These quantities correspond to the three polarization states of the , two transverse and one longitudinal. When neglecting the lepton mass, i.e., considering only electrons and muons, these amplitudes are expressed in terms of the three functions , , and as follows Neubert:1993mb ():
(4) 
where
(5)  
(6) 
with and . The functions and are defined in terms of the axial and vector form factors as,
(7) 
(8) 
By convention, the function is defined as
(9) 
The axial form factor dominates for . Furthermore, in the limit of infinite  and quark masses, a single form factor describes the decay, the socalled IsgurWise function Isgur:1989vq (); Isgur:1989ed ().
In terms of the three helicity amplitudes, the fully differential decay rate is given by
(10) 
with . Four onedimensional decay distributions can be obtained by integrating this decay rate over all but one of the four variables, , , , or . The differential decay rate as a function of is
(11) 
where
and is a known phase space factor,
A value of the form factor normalization is predicted by Heavy Quark Symmetry (HQS) Neubert:1993mb () in the infinite quarkmass limit. Lattice QCD can be utilized to calculate corrections to this limit. The most recent result obtained in unquenched lattice QCD is Bernard:2008dn ().
ii.3 Form factor parameterization
A parameterization of form factors , , and can be obtained using heavy quark effective theory (HQET). Perfect heavy quark symmetry implies that , i.e., the form factors and are identical for all values of and differ from only by a simple kinematic factor. Corrections to this approximation have been calculated in powers of and the strong coupling constant . Various parameterizations in powers of have been proposed. We adopt the following expressions derived by Caprini, Lellouch and Neubert Caprini:1997mu (),
(12)  
(13)  
(14) 
where . In addition to the form factor normalization , these expressions contain three free parameters, , , and . The values of these parameters cannot be calculated in a modelindependent manner. Instead, they have to be extracted by an analysis of experimental data.
Iii Experimental procedure
iii.1 Data sample and event selection
The data used in this analysis were taken with the Belle detector unknown:2000cg () at the KEKB asymmetricenergy collider Kurokawa:2001nw (). The Belle detector is a largesolidangle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50layer central drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrellike arrangement of timeofflight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a superconducting solenoid coil that provides a 1.5 T magnetic field. An iron fluxreturn located outside of the coil is instrumented to detect mesons and to identify muons (KLM). The detector is described in detail in Ref. unknown:2000cg (). Two inner detector configurations were used. A 2.0 cm beampipe and a 3layer silicon vertex detector were used for the first sample of million pairs, while a 1.5 cm beampipe, a 4layer silicon detector and a smallcell inner drift chamber were used to record the remaining million pairs svd2 ().
The data sample consists of 711 fb taken at the resonance, or about million events. Another 88 fb taken at 60 MeV below the resonance are used to estimate the non (continuum) background. The offresonance data is scaled by the integrated on to offresonance luminosity ratio corrected for the dependence of the cross section.
This data sample contains events recorded with two different detector setups as well as two different tracking algorithms and large differences in the input files used for Monte Carlo generation. To ensure that no systematic uncertainty appears due to inadequate consideration of these differences, we separate the data sample into four distinct sets labeled A (141 fb), B (274 fb), C (189 fb) and D (107 fb), where the number in parentheses indicates the integrated luminosity corresponding to the individual samples.
Monte Carlo generated samples of decays equivalent to about three times the integrated luminosity are used in this analysis. Monte Carlo simulated events are generated with the Evtgen program Lange:2001uf (), and full detector simulation based on GEANT Brun:1987ma () is applied. QED final state radiation in decays is added using the PHOTOS package Barberio:1993qi ().
Hadronic events are selected based on the charged track multiplicity and the visible energy in the calorimeter. The selection is described in detail elsewhere Abe:2001hj (). We also apply a requirement on the ratio of the second to the zeroth FoxWolfram moment Fox:1978vu (), , to reject continuum events.
iii.2 Event reconstruction
Charged tracks are required to originate from the interaction point by applying the following selections on the impact parameters in the and directions: cm and cm, respectively. In addition, we demand at least one associated hit in the SVD detector. For pion and kaon candidates, the Cherenkov light yield from the ACC, the timeofflight information from TOF, and from the CDC are required to be consistent with the appropriate mass hypothesis.
Neutral meson candidates are reconstructed in the decay channel. We fit the charged tracks to a common vertex and reject the candidate if the probability is below . The reconstructed mass is required to lie within MeV/ of the nominal mass of 1.865 GeV/ Amsler:2008zzb (), corresponding to about 2.5 times the experimental resolution measured from data.
The candidate is combined with an additional charged pion (oppositely charged with respect to the kaon candidate) to form a candidate. Due to the kinematics of the decay, the momentum of this pion does not exceed 350 MeV/. It is therefore referred to as the “slow” pion, . No impact parameter or SVD hit requirements are applied for . Again, a vertex fit is performed and the same vertex requirement is applied. The invariant mass difference between the and the candidates, , is required to be less than 165 MeV/. This selection is tightened after the background estimation described below. Additional continuum suppression is achieved by requiring that the momentum in the c.m. frame be below 2.45 GeV/.
Finally, the candidate is combined with an oppositely charged lepton (electron or muon). Electron candidates are identified using the ratio of the energy detected in the ECL to the track momentum, the ECL shower shape, position matching between track and ECL cluster, the energy loss in the CDC, and the response of the ACC counters. Muons are identified based on their penetration range and transverse scattering in the KLM detector. In the momentum region relevant to this analysis, charged leptons are identified with an efficiency of about 90% while the probability to misidentify a pion as an electron (muon) is 0.25% (1.4%) Hanagaki:2001fz (); Abashian:2002bd (). Lepton tracks have to be associated with at least one SVD hit. In the laboratory frame, the momentum of the electron (muon) is required to be greater than 0.30 GeV/ (0.60 GeV/). We also require the lepton momentum in the c.m. frame to be less than 2.4 GeV/ to reject continuum. More stringent lepton requirements are imposed later in the analysis.
For electron candidates we attempt bremsstrahlung recovery by searching for photons within a cone of degrees around the electron track. If such a photon is found, it is merged with the electron and the sum of the momenta is taken to be the lepton momentum.
iii.3 Background estimation
Because we do not reconstruct the other meson in the event, the momentum is a priori unknown. However, in the c.m. frame, one can show that the direction lies on a cone around the axis ref:2 () with an opening angle defined by:
(15) 
In this expression, is half of the c.m. energy and is . The quantities , and are calculated from the reconstructed system.
This cosine is also a powerful discriminator between signal and background: Signal events should lie in the interval , although – due to finite detector resolution – about 5% of the signal is reconstructed outside this interval. The background, on the other hand, does not have this restriction.
The signal lies predominantly in the region defined by 144 MeV/ 147 MeV/ and GeV/ ( GeV/) for electrons (muons). The region outside these thresholds can be used to estimate the background level.
We therefore perform a fit to the threedimensional , , ) distributions. The range between and is divided into 30 bins. The () range is divided into five (two) bins, with bin boundaries at MeV ( GeV/ for electrons and GeV/ for muons).
The background contained in the final sample has the following six components:

: background from decays with or and from nonresonant events, where the lepton has been correctly identified;

correlated background: background from processes other than decays in which the and the lepton originate from the same meson, e.g, , ;

uncorrelated background: the and the lepton come from different mesons and the lepton is not from a decay;

fake lepton: the charged lepton candidate is a misidentified hadron while the candidate may or may not be correctly reconstructed;

fake : the candidate is misreconstructed; the lepton candidate is identified correctly, but it is not from a decay;

continuum: background from processes.
To model the component, which consists of a total of four resonant (, , , ) and one nonresonant mode for both neutral and charged decays, we reweight the branching ratios of each subcomponent to match the values reported by the Particle Data Group Amsler:2008zzb (). For the resonant parts, only products of branching ratios are available and consequently we reweight these products. The shape of the momentum distributions is also reweighted in 22 bins of to match the predictions of the LLSW model Leibovich:1997em (); Urquijo:2006wd ().
All of the background components are modeled by MC simulation except for continuum events; these are modeled by offresonance data. For muon events, the shape of the fake lepton background is corrected by the ratio of the pion fake rate in the experimental data over the same quantity in the Monte Carlo, as measured using decays. The lepton identification efficiency is corrected by the ratio between experimental data and Monte Carlo in events Hanagaki:2001fz (); Abashian:2002bd (). The (, , ) distribution in the data is fitted using the TFractionFitter algorithm Barlow:1993dm () within ROOT Brun:1996ro (). The fit is done separately in each of eight subsamples defined by the experiment range and the lepton type. The results are given in Table 1. Figure 3 shows plots of the projections in for subsample B.
In all fits, the continuum normalization is fixed to the on to offresonance luminosity ratio, corrected for the dependence of the cross section. In general, the normalizations obtained by the fit agree well with the MC expectations except for the component and the fake component, which are overestimated in the MC. After the background determination only candidates satisfying the requirements , and GeV/ ( GeV/) for electrons (muons) are considered for further analysis.
A,  A,  B,  B,  
Num. Candidates  14802  14203  29217  26894 
Signal events 
11609 181  11139 190  23029 280  21002 258 
Signal ()  78.43 1.22  78.43 1.34  78.82 0.96  78.09 0.96 
() 
5.63 0.78  4.02 0.86  4.32 0.66  3.90 0.60 
Signal correlated () 
1.07 0.17  1.41 0.25  1.33 0.16  1.71 0.19 
Uncorrelated () 
7.24 0.35  6.01 0.40  7.19 0.31  6.31 0.29 
Fake () 
0.36 0.17  1.99 0.34  0.50 0.17  2.10 0.23 
Fake () 
2.59 0.12  2.81 0.13  3.07 0.11  2.96 0.10 
Continuum () 
4.68 0.54  5.32 0.59  4.77 0.38  4.93 0.40 
C,  C,  D,  D,  
Num. Candidates  22056  20428  15871  14719 
Signal events 
17301 240  15513 235  12365 189  11469 205 
Signal ()  78.44 1.09  75.94 1.15  77.91 1.19  77.92 1.39 
() 
5.15 0.71  5.22 0.71  4.54 0.72  4.67 0.86 
Signal correlated () 
1.56 0.27  2.07 0.37  2.01 0.26  2.73 0.43 
Uncorrelated () 
6.35 0.35  6.01 0.33  7.33 0.38  6.30 0.40 
Fake () 
0.75 0.18  2.26 0.28  0.30 0.19  1.68 0.38 
Fake () 
2.86 0.12  2.69 0.11  2.89 0.13  2.80 0.14 
Continuum () 
4.88 0.45  5.81 0.51  5.02 0.53  3.89 0.49 

iii.4 Kinematic variables
To calculate the four kinematic variables defined in Eq. 1 and Fig. 2 – , , and – which characterize the decay, we need to determine the rest frame. The direction is already known to lie on a cone around the axis with an opening angle in the c.m. frame, Equation (15). To initially determine the direction, we estimate the c.m. frame momentum vector of the nonsignal meson by summing the momenta of the remaining particles in the event ( ref:2 ()) and choose the direction on the cone that minimizes the difference to , Fig. 4.
To obtain , we exclude tracks passing far from the interaction point. The minimal requirements depend on the transverse momentum of the track, , and are set to cm ( cm, cm) or cm (50 cm, 20 cm) for a track MeV/ ( MeV/, MeV/). Track candidates that are compatible with a multiply reconstructed track generated by a lowmomentum particle spiraling in the central drift chamber are also checked for and only one of the multiple tracks is considered. Unmatched clusters in the barrel region must have an energy greater than 50 MeV. For clusters in the forward (backward) region, the threshold is at 100 MeV (150 MeV). We then compute (in the laboratory frame) by summing the 3momenta of the selected particles,
(16) 
where the index stands for all particles passing the conditions above, and transform this vector into the c.m. frame. Note that we do not introduce any mass assumption for the charged particles. The energy component of is determined by requiring to be .
With the rest frame reconstructed in this way, the resolutions in the kinematic variables are found to be about 0.025, 0.049, 0.050 and 13.5 for , , and , respectively.
Iv Analysis based on the parameterization of Caprini et al.
iv.1 Fit procedure
Our main goal is to extract the following quantities: the product of the form factor normalization and , (Eq. 11), and the three parameters , and that parameterize the form factors in the HQET framework (Eqs. 12–14). For this, we perform a binned fit to the , , and distributions over nearly the entire phase space. Instead of an unbinned fit, we fit the onedimensional projections of , , and . This avoids the difficulty of parameterizing the six background components and their correlations in four dimensions. In addition, the onedimensional projections have sufficient statistics in each bin. However, this approach introduces bintobin correlations that must be accounted for.
The distributions in , , and are divided into ten bins of equal width. The kinematically allowed values of are between and , but we restrict the fit range to values between and . In each subsample, there are thus 40 bins to be used in the fit. In the following, we label these bins with a single index , . The bins correspond to the bins of the distribution, to , to , and to the distribution.
The number of produced events in the bin , , is given by
(17) 
where is the number of mesons in the data sample, and are the and branching ratios into the final state under consideration Amsler:2008zzb (), is the lifetime Amsler:2008zzb (), and is the width obtained by integrating Eq. 10 in the kinematic variable corresponding to from the lower to the upper bin boundary (the other kinematic variables are integrated over their full range). This integration is numerical in the case of and analytic for the other variables. The expected number of events is related to as follows
(18) 
Here, is the probability that an event generated in the bin is reconstructed and passes all analysis cuts, and is the detector response matrix, i.e., it gives the probability that an event generated in the bin is observed in the bin . Both quantities are calculated using MC simulation. is the number of expected background events, estimated as described in Sect. III.3.
Next, we calculate the variance of . We consider the following contributions: the Poissonian uncertainty in ; fluctuations related to the efficiency, estimated by a binomial distribution with repetitions and known success probability ; a similar contribution related to using a multinomial distribution; and the uncertainty in the background contribution . This yields the following expression for ,
(19)  
The first term is the Poissonian uncertainty in . The second and third terms are the binomial and multinomial uncertainties, respectively, related to the finite real data size, where () is the total number of decays (the number of reconstructed decays) into the final state under consideration (, or ) in the real data. The quantities and are calculated from a finite signal MC sample ( and ); the corresponding uncertainties are estimated by the fourth and fifth terms. Finally, the last term is the background contribution , calculated as the sum of the different background component variances. For each background component defined in Section III.3 we estimate its contribution by linear error propagation of the results determined in Section III.3. For continuum, we estimate the error in the onresonance to offresonance luminosity ratio to be 1.0%. These variances give the diagonal elements of the covariance matrix .
In each subsample we calculate the offdiagonal elements of the covariance matrix as , where is the relative abundance of bin in the 2dimensional histogram obtained by plotting the kinematic variables against each other, is the relative number of entries in the 1dimensional distribution, and is the size of the sample. Covariances are calculated for the signal and the different background components in the MC samples, and added with appropriate normalizations.
The covariance matrix is inverted numerically within ROOT Brun:1996ro () and, labeling the electron and muon mode in each subsample with the index , functions are calculated,
(20) 
where is the number of events observed in bin in data sample . We sum these two functions in each subsample and minimize the global with MINUIT James:1975dr ().
We have tested this fit procedure using generic MC data samples. All results are consistent with expectations and show no indication of bias.
iv.2 Investigation of the efficiency of low momentum tracks
The tracking efficiency of the Belle experiment is reproduced well by MC simulations for tracks with momenta above 200 MeV/, which we refer to as “high momentum tracks”. However, a significant portion of the momentum spectrum of the slow pions emitted in the decay lies below this boundary. For low momenta, the effects of interactions with the detector material such as multiple scattering and energy loss become important and might lead to a deviation between data and MC in the reconstruction efficiency.
We use one half of the reconstructed sample to obtain corrections to the MC reconstruction efficiency in the low momentum range, measured using real data. The second half is used to perform the analysis with a statistically independent sample. The results of the background estimation shown in Table 1 are those obtained in the samples used for the analysis. Both of the samples contain about 120,000 signal events.
The sample used to investigate the efficiency of low momentum tracks is divided into a total of six bins in . The bin borders of the first five are 50 MeV, 100 MeV, 125 MeV, 150 MeV, 175 MeV and 200 MeV. The region beyond 200 MeV/ defines the sixth bin. By subtracting the background, we obtain an estimate of the signal in data and form the ratio with the signal in MC in each bin, .
The high momentum range is used as normalization, no efficiency correction is applied there. In the lower momentum bins we obtain the ratios , which are identical to the ratio of reconstruction efficiencies in the bins and the high momentum region, . We calculate this set of ratios for the electron and muon modes and form the weighted average, separately for each of the four subsamples. These values are applied as weights when filling the MC histograms to correct the reconstruction efficiency.
Most systematic uncertainties cancel out in the ratios . Only the uncertainties in the various background components give a small systematic contribution to the uncertainty.
This procedure assumes that the distribution of events in the spectrum is identical for data and MC. However, one of the aims of the analysis is to measure the form factor parameters that govern this distribution. Therefore, an iterative procedure is adopted: we calculate one set of corrections, apply them and perform the analysis to determine and the form factor parameters. We then calculate a new set of corrections using these results and repeat the analysis. The changes of the parameters during this iterative procedure are small and vanish after the third iteration. We assign an additional systematic uncertainty to our results based on the stability of the corrections against changes in the form factor parameters. As will be shown in Table 3, this is a negligibly small contribution.
Sample A  total sample  

1.248 0.102 0.022  1.285 0.114 0.028  1.259 0.076 0.019  

1.317 0.099 0.041  1.577 0.131 0.036  1.436 0.078 0.030 

0.804 0.076 0.017  0.768 0.093 0.020  0.795 0.058 0.015 

34.8 0.5 1.2  34.6 0.6 1.2  34.7 0.4 1.2 
/ n.d.f 
32.2 / 36.0  31.6 / 36.0  70.9 / 76.0 

0.651  0.676  0.643 
Sample B 
total sample  
1.169 0.079 0.011  1.167 0.088 0.016  1.168 0.059 0.011  

1.411 0.079 0.026  1.449 0.090 0.028  1.427 0.059 0.022 

0.902 0.054 0.011  0.859 0.061 0.013  0.882 0.041 0.010 

34.4 0.4 1.1  33.9 0.4 1.1  34.2 0.3 1.1 
/ n.d.f  22.7 / 36.0  36.5 / 36.0  60.7 / 76.0 

0.958  0.443  0.900 
Sample C 
total sample  
1.226 0.088 0.011  1.262 0.101 0.016  1.239 0.066 0.011  

1.363 0.086 0.026  1.480 0.107 0.033  1.411 0.066 0.023 

0.891 0.062 0.012  0.851 0.076 0.015  0.876 0.048 0.012 

34.4 0.5 1.1  33.9 0.5 1.1  34.2 0.3 1.1 
/ n.d.f 
38.6 / 36.0  38.2 / 36.0  81.4 / 76.0 

0.352  0.370  0.314 
Sample D  total sample  
1.321 0.102 0.019  1.174 0.106 0.020  1.247 0.073 0.014  

1.448 0.109 0.041  1.230 0.089 0.031  1.330 0.069 0.027 

0.791 0.081 0.019  0.931 0.071 0.015  0.864 0.053 0.014 

35.4 0.6 1.2  35.7 0.6 1.2  35.6 0.4 1.2 
/ n.d.f 
25.1 / 36.0  42.0 / 36.0  70.1 / 76.0 

0.913  0.226  0.669 

iv.3 Results of the fits and investigation of the systematic uncertainties in the subsamples
After applying all analysis cuts and subtracting backgrounds, a total of signal events are used for the analysis, divided into a total of four experimental subsamples as mentioned above. The result of the fit to these data is shown in Fig. 5 and Table 2. The per degree of freedom, , of all fits is good. Table 2 also gives the probabilities or Pvalues, .
To estimate the systematic uncertainties in these results, we consider contributions from the following sources: uncertainties in the background component normalizations, uncertainty in the MC tracking efficiency, errors in the world average of and as well as in the components Amsler:2008zzb (), uncertainties in the shape of the distribution of events based on the LLSW model Leibovich:1997em (), uncertainties in the lifetime Amsler:2008zzb (), and the uncertainties in the total number of mesons in the data sample.
To calculate these systematic uncertainties, we consider 300 pseudoexperiments in which one of 15 parameters is randomly varied, using a normal distribution. The entire analysis chain is repeated for every pseudoexperiment and new fit results are obtained, in total for 4500 variations. One standard deviation in the pseudoexperiment fit results for a given parameter is used as the systematic uncertainty in this parameter.
The parameters varied in the pseudoexperiments are as follows:

The corrections on the tracking efficiencies for low momentum tracks are varied within their respective uncertainties. To obtain the most conservative estimate, the uncertainties in different momentum bins are assumed to be fully correlated. Therefore, this component corresponds to a single parameter in the toy MC.

The lepton identification efficiencies are varied within their respective uncertainties Hanagaki:2001fz (); Abashian:2002bd ().

The normalization of the continuum background is not correlated with any of the other backgrounds, it is therefore varied individually within the uncertainty on the on to offresonance luminosity ratio, which is 1.0%.

Uncertainties in the composition of the D** component are accounted for by varying each of the components contributing to the background within the uncertainty reported by the Particle Data Group Amsler:2008zzb (). For the resonant modes, this is the uncertainty in the branching fraction products ; for the nonresonant mode, this is the uncertainty in .

In addition, the shape of the distributions of the components is varied according to the LLSW model Leibovich:1997em () and the uncertainties on the model parameters as determined in Ref. Urquijo:2006wd ().

The number of events is obtained from the product of the number of events in the sample with the branching fraction of to a pair. We vary the fraction within its uncertainty Amsler:2008zzb (). This affects both the overall normalization and the background distributions.
The uncertainties in the reconstruction of the high momentum tracks, the branching ratios and , the number of events in the sample, and the lifetime affect only , not the form factors. Therefore, their uncertainties are considered by analytical error propagation.
iv.4 Averaging the results of the subsamples
To obtain the average of the four subsamples, which have been measured independently, we use the algorithm applied by the Heavy Flavor Averaging Group TheHeavyFlavorAveragingGroup:2010qj () to obtain the world average for from semileptonic decays. This algorithm combines both the statistical and the systematic uncertainties. The correlations of some of these errors between different samples is considered. For example, the uncertainty on the branching fraction will lead to a fully correlated systematic uncertainty in each analysis.
The average is obtained with the MINUIT package James:1975dr () by using a minimization. Here, gives the total number of fit parameters, in our case . When calculating the average of measurements of the four fit parameters , a total of values are available as inputs, which we label as . In general this number can be labeled as . Each measurement corresponds to one of the parameters , which defines a primitive map . The statistical covariance matrix of each measurement is known, as well as the correlation between the samples. The latter are zero in our case. This information allows one to construct a dimensional covariance matrix containing the statistical uncertainties and to obtain the statistical part of the to be minimized: