Measurement of Moments of the Hadronic-Mass and -Energy Spectrum in Inclusive Semileptonic \kern 1.8pt\overline{\kern-1.8ptB}{}\rightarrow X_{c}\ell^{-}\overline{\nu} Decays

Measurement of Moments of the Hadronic-Mass and -Energy Spectrum in Inclusive Semileptonic Decays

Abstract

We present a measurement of moments of the inclusive hadronic-mass and -energy spectrum in semileptonic decays. This study is based on a sample of million decays recorded by the BABAR detector at the PEP-II -storage rings. We reconstruct the semileptonic decay by identifying a lepton in events tagged by a fully reconstructed hadronic decay of the second meson. We report preliminary results for the moments with and with and , with the mass of the hadronic system, its energy, and a constant of , for different minimal lepton momenta between and measured in the -meson rest frame. These are predicted in the framework of a Heavy Quark Expansion (HQE), which allows the extraction of the total semileptonic branching fraction, the CKM-matrix element , and the quark masses and , together with the dominant non-perturbative HQE parameters. We find as preliminary results and .

Submitted to the 2007 Europhysics Conference on High Energy Physics, Manchester, England.

pacs:
12.15.Ff, 12.15.Hh, 13.25.Hw, 13.30.Ce

BABAR-CONF-07/003

SLAC-PUB-12668

12

The BABAR Collaboration

I Introduction

Measurement of moments of the hadronic-mass Csorna et al. (2004); Aubert et al. (2004a); Acosta et al. (2005); Abdallah et al. (2006); Schwanda et al. (2007) and lepton-energy Aubert et al. (2004b); Abdallah et al. (2006); Abe et al. (2005) spectra in inclusive semileptonic decays  have been used to determine the non-perturbative QCD parameters describing these decays and the CKM matrix element .

Combined fits to these moments and moments of the photon-energy spectrum in decays Chen et al. (2001); Koppenburg et al. (2004); Aubert et al. (2005, 2006) in the context of Heavy Quark Expansions (HQE) of QCD have resulted in precise determinations of and , the mass of the quark. Specifically, they are reported to be and in the kinetic mass scheme Buchmüller and Flächer (2006) and and in the 1S scheme Bauer et al. (2004).

Lepton-energy moments are known with very good accuracy, but the precision of the hadronic-mass and photon-energy moments is limited by statistics. Therefore, we present here an updated measurement of the hadronic-mass moments with based on a larger dataset than previously used Aubert et al. (2004a). In addition we present measurements of the mixed hadron mass-energy moments with as proposed by Gambino and Uraltsev Gambino and Uraltsev (2004). All moments are presented for different cuts on the minimum energy of the charged lepton. The mixed moments use the mass  and the energy  of the  system in the meson rest frame of  decays,

(1)

with a constant , here fixed to be 0.65  as proposed in Gambino and Uraltsev (2004). They allow a more reliable extraction of the higher-order non-perturbative HQE parameters and thus they are expected to increase the precision on the extraction of and the quark masses and .

We perform a combined fit to the hadronic mass moments, measured moments of the lepton-energy spectrum, and moments of the photon energy spectrum in decays . The fit extracts values for , the quark masses and , the total semileptonic branching fraction , and the dominant non-perturbative HQE parameters. These are and , parameterizing effects at , and and , parameterizing effects at .

Ii BABAr Detector and Dataset

The analysis is based on data collected with the BABAR experiment Aubert et al. (2002) at the PEP-II asymmetric-energy storage rings PEP-II: An Asymmetric Factory. Conceptual Design Report (1993) at the Stanford Linear Accelerator Center between October 1999 and July 2004.

The BABAR tracking system used for charged particle and vertex reconstruction has two main components: a silicon vertex tracker (SVT) and a drift chamber (DCH), both operating within a 1.5-T magnetic field of a superconducting solenoid. The transverse momentum resolution is 0.47 % at 1. Photons are identified in an electromagnetic calorimeter (EMC) surrounding a detector of internally reflected Cherenkov light (DIRC), which associates Cherenkov photons with tracks for particle identification (PID). The energy of photons is measured with a resolution of 3 % at 1. Muon candidates are identified with the use of the instrumented flux return (IFR) of the solenoid. The detector covers the polar angle of in the center of mass (c.m.) frame.

The data sample consists of about 210.4 , corresponding to decays of . We use Monte Carlo (MC) simulated events to determine background distributions and to correct for detector acceptance and resolution effects. The simulation of the BABAR detector is realized with GEANT4 Agostinelli et al. (2003). Simulated  meson decays are generated using EvtGen Lange (2001). Final state radiation is modeled with PHOTOS Richter-Was (1993).

The simulations of decays use a parameterization of form factors for  Duboscq et al. (1996), and models for  Scora and Isgur (1995) and  Goity and Roberts (1995).

Iii Reconstruction of Semileptonic Decays

iii.1 Selection of Hadronic -Meson Decays

The analysis uses  events in which one of the mesons decays to hadrons and is fully reconstructed () and the semileptonic decay of the recoiling  meson () is identified by the presence of an electron or muon. While this approach results in a low overall event selection efficiency of only a few per mille, it allows for the determination of the momentum, charge, and flavor of the mesons. To obtain a large sample of mesons, many exclusive hadronic decays are reconstructed Aubert et al. (2004c). The kinematic consistency of these candidates is checked with two variables, the beam-energy-substituted mass and the energy difference . Here is the total energy in the c.m. frame, and denote the c.m. momentum and c.m. energy of the candidate. We require within three standard deviations, which range between 10 and 30 depending on the number of hadrons used for the reconstruction. For a given  decay mode, the purity is estimated as the signal fraction in events with . For events with one high-momentum lepton with  in the -meson rest frame, the purity is approximately 78 %.

iii.2 Selection of Semileptonic Decays

Semileptonic decays are identified by the presence of one and only one electron or muon above a minimum momentum measured in the rest frame of the . Electrons are selected with 94% average efficiency and a hadron misidentification rate in the order of 0.1%. Muons are identified with an efficiency ranging between 60% for momenta in the laboratory frame and 75% for momenta and a hadron misidentification rate between 1% for kaons and protons and 3% for pions. Efficiencies and misidentification rates are estimated from selected samples of electrons, muons, pions, and kaons. We impose the condition , where is the charge of the lepton and is the charge of the -quark of the . This condition is fulfilled for primary leptons, except for  events in which flavor mixing has occurred. We require the total observed charge of the event to be , allowing for a charge imbalance in events with low momentum tracks or photon conversions. In cases where only one charged track is present in the reconstructed  system, the total charge in the event is required to be equal to zero.

iii.3 Reconstruction of the Hadronic System

The hadronic system in the decay  is reconstructed from charged tracks and energy depositions in the calorimeter that are not associated with the or the charged lepton. Procedures are implemented to eliminate fake tracks, low-energy beam-generated photons, and energy depositions in the calorimeter originating from hadronic showers faking the presence of additional particles. Each track is assigned a specific particle type, either , , or , based on combined information from the different BABAR subdetectors. The four-momentum of the reconstructed hadronic system is calculated from the four-momenta of the reconstructed tracks , reconstructed using the mass of the identified particle type, and photons by . The hadronic mass is calculated from the reconstructed four-momenta as .

The four-momentum of the unmeasured neutrino is estimated from the missing four-momentum . Here, all four-momenta are measured in the laboratory frame. To ensure a well reconstructed hadronic system, we impose criteria on the missing energy, , the missing momentum, , and the difference of both quantities, . After having selected a well reconstructed and having imposed the selection criteria on , of signal decays and of background decays are retained.

Starting from a kinematically well defined initial state additional knowledge of the kinematics of the semileptonic final state is used in a kinematic fit to improve the overall resolution and reduce the bias of the measured values. The fit imposes four-momentum conservation, the equality of the masses of the two mesons, and constrains the mass of the neutrino, . The resulting average resolutions in and are and , respectively. The overall biases of the kinematically fitted hadronic system are found to be and , respectively. We require the fit to converge, thus ensuring that the constraints are fulfilled.

The background is composed of events (continuum background) and decays or in which the candidate is mistakenly reconstructed from particles coming from both mesons in the event (combinatorial background). Missing tracks and photons in the reconstructed hadronic system are not considered an additional source of background since they only affect its resolution. The effect of missing particles in the reconstruction is taken care of by further correction procedures. To quantify the amount of background in the signal region we perform a fit to the distribution of the candidates. We parametrize the background using an empirical threshold function Albrecht et al. (1987),

(2)

where , is the kinematic endpoint approximated by the mean c.m. energy, and is a free parameter defining the curvature of the function. The signal is parameterized with a modified Gaussian function Skwarnicki (1986) peaked at the -meson mass and corrected for radiation losses. The fit is performed separately for several bins in and to account for changing background contributions in different or regions, respectively. The background shape is determined in a signal-free region of the sideband, . Figure 1 shows the distribution for together with the fitted signal and background contributions.

Figure 1: spectrum of decays accompanied by a lepton with . The signal (solid line) and background (red dashed line) components of the fit are overlaid. The crossed area shows the background under the signal. The background control region in the sideband is indicated by the hatched area.

Residual background is estimated from MC simulations. It is composed of charmless semileptonic decays , hadrons misidentified as leptons, secondary leptons from semileptonic decays of , mesons or either in mixed events or produced in transitions, as well as leptons from decays of , and . The simulated background spectra are normalized to the number of events in data. We verify the normalization using an independent data control sample with inverted lepton charge correlation, .

Iv Hadronic Mass Moments

We present measurements of the moments , with , of the hadronic mass distribution in semileptonic meson decays . The moments are measured as functions of the lower limit on the lepton momentum, , between and calculated in the rest frame of the meson.

iv.1 Selected Event Sample

The selected event sample contains about background. For we find a total of signal events above a combinatorial and continuum background of events and residual background of events. For we find signal events above a background constituted of and combinatorial and residual events, respectively. Figure 2 shows the kinematically fitted distributions together with the extracted background shapes for and .

Figure 2: Kinematically fitted hadronic mass spectra for minimal lepton momenta (top) and (bottom) together with distributions of combinatorial background and background from non- decays (hatched area) as well as residual background (crossed area)

iv.2 Extraction of Moments

To extract unbiased moments , additional corrections have to be applied to correct for remaining effects that can distort the measured distribution. Contributing effects are the limited acceptance and resolution of the BABAR detector resulting in unmeasured particles and in misreconstructed energies and momenta of particles. Additionally measured particles not originating from the hadronic system and final state radiation of leptons contribute, too. We correct the kinematically fitted by applying correction factors on an event-by-event basis using the observed linear relationship between the moments of the measured mass and moments of the true underlying mass . Correction functions are constructed from MC simulations by calculating moments and in several bins of the true mass and fitting the observed dependence with a linear function.

Figure 3: Examples of calibration fucntions for in bins of , and . Shown are the extracted versus in bins of for (), (), and (). The results of fits of linear functions are overlaid as solid lines. A reference line with is superimposed (dashed line). There is only one calibration function with constructed but plotted for better comparableness in each bin.

Studies show that the bias of the measured is not constant over the whole phase space but depends on the resolution and total multiplicity of the reconstructed hadronic system, . Therefore, correction functions are derived in three bins of , three bins of , as well as in twelve bins of , each with a width of . Due to limited number of generated MC events, the binning in and is abandoned for . Overall we construct calibration functions for each order of moments. Figure 3 shows examples of correction functions for the moment in three bins of as well as in nine bins of and .

For each event the corrected mass is calculated by inverting the linear function,

(3)

with the offset and the slope of the correction function. Background contributions are subtracted by applying weight factors dependent on to each corrected hadronic mass, whereby each weight corresponds to the fraction of signal events expected in the corresponding part of the spectrum. This leads to the following expression used for the calculation of the moments:

(4)

The factors and are dependent on the order and minimal lepton momentum of the measured moment. They are determined in MC simulations and correct for small biases observed after the calibration. The factors account for the bias of the applied correction method and range between and . For we observe larger biases ranging between and for the lowest between and , respectively. The residual bias correction factor accounts for differences in selection efficiencies for different hadronic final states and QED radiation in the final state that is included in the measured hadron mass and distorts the measurement of the lepton’s momentum. The effect of radiative photons is estimated by employing PHOTOS. Our correction procedure results in moments which are free of photon radiation. The residual bias correction is estimated in MC simulations and typically ranges between and . For the moments and slighly higher correction factors are determined ranging between and as well as and , respectively.

This procedure is verified on a MC sample by applying the calibration to measured hadron masses of individual semileptonic decays, , , four resonant decays , and two non-resonant decays . Figure 4 shows the corrected moments and as functions of the true moments for minimal lepton momenta . The dashed line corresponds to . The calibration reproduces the true moments over the full mass range.

Figure 4: Calibrated () and uncorrected () moments (left) and (right) of individual hadronic modes for minimal lepton momenta . A reference line with is superimposed.
Figure 5: Measured hadronic mass moments with for different selection criteria on the minimal lepton momentum . The inner error bars correspond to the statistical uncertainties while the full error bars correspond to the total uncertainties. The moments are highly correlated.

iv.3 Systematic Studies

The principal systematic uncertainties are associated with the modeling of hadronic final states in semileptonic -meson decays, the bias of the calibration method, the subtraction of residual background contributions, the modeling of track and photon selection efficiencies, the identification of particles, as well as the stability of the results. The obtained results are summarized in Tables A.I and A.II for the measured moments with and selection criteria on the minimum lepton momentum ranging from to .

Modeling of Signal Decays

The uncertainty of the calibration method with respect to the chosen signal model is estimated by changing the composition of the simulated inclusive hadronic spectrum. The dependence on the simulation of high mass hadronic final states is estimated by constructing correction functions only from MC simulated hadronic events with hadronic masses , thereby removing the high mass tail of the simulated hadronic mass spectrum. The model dependence of the calibration method is found to contribute only little to the total systematic uncertainty. We estimate the model dependence of the residual bias correction by changing the composition of the inclusive hadronic spectrum, thereby omitting one or more decay modes. We associate a systematic uncertainty to the correction of the observed bias of the calibration method of half the size of the applied correction.

We study the effect of differences between data and MC in the multiplicity and distributions on the calibration method by changing the binning of the correction functions. The observed variation of the results is found to be covered by the statistical uncertainties of the calibration functions.

Background Subtraction

The branching fractions of background decays in the MC simulation are scaled to agree with current measurements Yao et al. (2006). The associated systematic uncertainty is estimated by varying these branching fractions within their uncertainties. At low , most of the studied background channels contribute to the systematic uncertainty, while at high , the systematic uncertainty is dominated by background from decays . Contributions from and decays are found to be negligible.

The uncertainty in the combinatorial background subtraction is estimated by varying the lower and upper limits of the sideband region in the distribution up and down by . The observed effect is found to be negligible.

Detector-Related Effects

We correct the MC simulation for differences to data in the selection efficiencies of charged tracks and photons, as well as identification efficiencies and misidentification rates of various particle types. The corrections are extracted from data and MC control samples.

The uncertainty of the photon selection efficiencies is found to be per photon independent of energy, polar angle and multiplicity. The systematic uncertainty in track finding efficiencies is estimated to be per track. We add in quadrature the statistical uncertainty of the control samples that depend on energy and polar angle of the track as well as the multiplicity of tracks in the reconstructed event. The misidentification of mesons as leptons is found to affect the overall normalization of the corresponding background spectra by .

While the latter two uncertainties give only small contributions to the total systematic uncertainty, the uncertainty associated with the selection efficiency of photons is found to be the main source of systematic uncertainties.

Stability of the Results

The stability of the results is tested by dividing the data into several independent subsamples: and , decays to electrons and muons, different run periods of roughly equal data-sample sizes, and two regions in the spectrum, and , characterized by different resolutions of the reconstructed hadronic system. No significant variations are observed.

The stability of the result under variation of the selection criteria on is tested by varying the applied cut between and . For most of the measured moments the observed variation is covered by other known systematic detector and MC simulation effects. In cases where the observed variation is not covered by those effects, we add an additional contribution to the systematic uncertainty of the measurement that compensates the observed difference .

Simulation of Radiation

We check the impact of low energetic photons by removing EMC neutral energy deposits with energies below from the reconstructed hadronic system. The effect on the measured moments is found to be negligible.

iv.4 Results

The measured hadronic mass moments with as functions of the minimal lepton momentum are depicted in Fig. 5. All measurements are correlated since they share subsets of selected events. Tables A.I and A.II summarize the numerical results. The statistical uncertainty consists of contributions from the data statistics and the statistics of the MC simulation used for the construction of the correction functions, for the subtraction of residual background, and the determination of the final bias correction. In most cases we find systematic uncertainties that exceed the statistical uncertainty by a factor of .

V Mixed Hadronic Mass- and Energy-Moments

The measurement of moments of the observable , a combination of the mass and energy of the inclusive  system, as defined in Eq. 1 , allow a more reliable extraction of the higher order HQE parameters , , , and . Thus a smaller uncertainty on the standard model parameters , , and could be achieved.

We present measurements of the moments , , and for different minimal lepton momenta between 0.8  and 1.9  calculated in the -meson rest frame. We calculate the central moments , , and the moments  and  as proposed in Gambino and Uraltsev (2004).

Due to the structure of the variable  as a difference of two measured values, its measured resolution and bias are larger than for the mass moments and the sensitivity to is increased wrt. to . The overall resolution of  after the kinematic fit for lepton momenta greater than 0.8  is measured to be 1.31  with a bias of -0.08 . We therefore introduce stronger requirements on the reconstruction quality of the event. We tighten the criteria on the neutrino observables. The variable is required to be between 0 and 0.3 . Due to the stronger requirements on the individual variables and have less influence on the resolution of the reconstrcuted hadronic system. Therefore, the cuts on the missing energy and the missing momentum in the event are loosened to and , respectively, as they do not yield significant improvement on the resolution of , and do not increase the ratio of signal to background events.

The final event sample contains about 22 % of background events. The background is composed of 12 % continuum and combinatorial background and 10 % decays of the signal meson other than the semileptonic decay . Combinatorial and continuum background is subtracted using the sideband of the distribution, as described above. The residual background events, containing a correctly reconstructed  meson, are subtracted using MC simulations. The dominant sources are pions misidentified as muons,  decays, and secondary semileptonic decays of and mesons.

The measured  spectra for cuts on the lepton momentum at  and are shown together with the backgound distributions in Fig. 6. We measure  () signal events for , respectively.

Figure 6: Spectra of  after the kinematic fit together with distributions of combinatorial background and background from non- decays (hatched area) as well as residual background (crossed area) for different minimal lepton momenta (a) and (b).

v.1 Extraction of Moments

To extract unbiased moments , effects that distort the  distribution need to be corrected. These are the limited detector acceptance, resulting in a loss of particles, the resolution of measured charged particle momenta and energy depositions in the EMC, as well as the radiation of final-state photons. These photons are included in the measured  system and thus lead to a modified energy and mass measurement of the inclusive system. In the case of radiation from the lepton, the lepton’s measured momentum is also lowered w.r.t. its initial momentum. The measured moments are corrected for the impact of these photons.

Figure 7: Examples of calibration curves for () in bins of , extracted separately for events  (a)-(c) and  (d)-(f). Shown are the extracted versus in bins of for (), (), and () integrated over multiplicity and bins. The results of fits of linear functions are overlaid as solid lines. Reference lines with are superimposed (dashed lines). Please note the logarithmic scales in (b), (c), (e), and (f).

As described before, we find linear relationships correcting the measured means to the true means described by first order polynomials. These functions vary with the measured lepton momentum, the measured , and the measured multiplicity of the inclusive system. The curves are therefore derived in three bins of and three bins of the multiplicity for each of the 12 lepton momentum bins of 100 . We also find differences for events containing an electron or a muon and therefore derive separate correction functions for these two classes of events. The measured  value is corrected on an event-by-event basis using the inverse of these functions:

(5)

Here and are the offset and the slope of the calibration function and differ for each order and for each of the abovementioned bins. Figure 7 shows calibration curves for the moments (), integrated over all multiplicity bins and bins in , for three different bins of . These calibration curves are extracted separately for events containing an electron or muon. Differences are mainly visible in the low momentum bin.

To verify this calibration procedure, we extract the moments of  of individual exclusive  modes on a MC sample and compare the calibrated moments to the true moments. The result of this study for the moments  is plotted in Fig. 8, confirming that the extraction method is able to reproduce the true moments. Small biases remaining after calibration are of the order of 1 % for and in the order of few percent for and and are corrected and treated in the systematic uncertainties.

Background contributions are subtracted applying  dependent weight factors on an event-by-event basis, leading to the following expression for the moments:

(6)

The bias correction factor depends on the minimal lepton momentum and the order of the extracted moments. It is derived on MC simulations and corrects for the small bias remaining after the calibration.

Figure 8: Result of the calibration verification procedure for different minimal lepton momenta (a) and (b). Moments  of exclusive modes on simulated events before calibration () and after calibration () plotted against the true moments for each mode. The dotted line shows the fit result to the calibrated moments, the resulting parameters are shown.

v.2 Systematic Studies

The main sources of systematic uncertainties have been identified as the simulation of the detector efficiency to detect neutral clusters. The corresponding effect from charged tracks is smaller but still contributes to the uncertainty on the moments. Their impact has been evaluated by randomly excluding neutral or charged candidates from the  system with a probability of 1.8 % for the neutral candidates and 0.8% for the charged tracks, corresponding to the systematic uncertainties of the efficiency extraction methods. For the tracks we add in quadrature the statistical uncertainties from the control samples to the 0.8 % systematic uncertainty. The uncertainty arising from the differences between data and MC in the distributions is evaluated by changing the selected region of to [0.0,0.2]  and [0.0, 0.4] . To evaluate the uncertainty due to the binning of the calibration curves in the multiplicity, we randomly increase the measured multiplicity used for the choice of the calibration curve by one with a probability of corresponding to observed differences between data and MC.

Smaller uncertainties arise from the unknown branching fractions of the background decay modes. Their branching fractions are scaled to agree with recent measurements Yao et al. (2006) and are varied within their uncertainties. The MC sample is corrected for differences in the identification efficiencies between data and MC for various particle types. The uncertainty on the background due to pions misidentified as muons is evaluated by changing the MC corrections within the statistical uncertainties of these data control samples. While the background shape does not vary, the amount decreases up to 8 %. For the estimate of the uncertainty due to particle identification, we propagate this variation into the extracted moments.

A similar variation procedure is applied for the branching fractions of the exclusive signal modes, varying them several times randomly within 10 % for the , 15 % for the , 50 % for the individual  modes and 75% for the non-resonant modes. The inclusive rate for the decays  is conserved by rescaling all other modes. In addition, all (non-resonant) modes are scaled in common, again randomly within 50%, keeping the inclusive decay rate  constant by rescaling the non-resonant () modes only. Experimental uncertainties on the signal branching fractions are fully covered by these variations Yao et al. (2006). This dependence of the extraction method results in changes of the calibration curve and bias correction, however the impact on the moments measured on data is small. We conservatively add half of the bias correction remaining after calibration to the uncertainty related to the extraction method.

The stability of the results has been tested by splitting the data sample into several independent subsamples: and , decays to electrons and muons, and different run periods of roughly equal data-sample sizes. No significant variations are observed.

Figure 9: Measured moments (a), (b), (c), and the central moments  with () and () (d), and  with () and () (e) for different cuts on the lepton momentum . The error bars indicate the statistical and the total errors, respectively. Please note the logarithmic scale on the -axis in plots (d) and (e). The moments are highly correlated.

v.3 Results

Figure 9 shows the results for the moments , , , and the central moments  and  for as a function of the  cut. The moments are highly correlated due to the overlapping data samples. The full numerical results and the statistical and the estimated systematic uncertainties can be found in Tables A.III - A.VII. A clear dependence on the lepton momentum selection criteria is observed for all moments, due to the varying contributions from higher mass final states with decreasing lepton momentum. Statistical uncertainties on the moments arise from the limited data sample, the width of the measured distribution , and limited statistics on the MC samples used for the extraction of background shapes, calibration curves, and bias correction. In most cases we obtain systematic uncertainties slightly exceeding the statistical uncertainty.

Vi Determination of and the quark masses and

At the parton level, the weak decay rate for can be calculated accurately; it is proportional to and depends on the quark masses, and . To relate measurements of the semileptonic -meson decay rate to , the parton-level calculations have to be corrected for effects of strong interactions. Heavy-Quark Expansions (HQEs) Voloshin and Shifman (1985); Chay et al. (1990); Bigi and Uraltsev (1992) have become a successful tool for calculating perturbative and non-perturbative QCD corrections Bigi et al. (1992, 1993); Blok et al. (1994); Manohar and Wise (1994); Gremm and Kapustin (1997) and for estimating their uncertainties.

In the kinetic-mass scheme Benson et al. (2003); Gambino and Uraltsev (2004); Benson et al. (2005); Aquila et al. (2005); Uraltsev (2005); Bigi et al. (2007), these expansions in and (the strong coupling constant) to order contain six parameters: the running kinetic masses of the and quarks, and , and four non-perturbative parameters. The parameter denotes the Wilson normalization scale that separates effects from long- and short-distance dynamics. The calculations are performed for Bigi et al. (1997). We determine these six parameters from a fit to the moments of the hadronic-mass and electron-energy Aubert et al. (2004b) distributions in semileptonic decays and moments of the photon-energy spectrum in decays Aubert et al. (2005, 2006).

In the kinetic-mass scheme the HQE to for the rate of semileptonic decays can be expressed as Benson et al. (2003)