Measurement and Interpretation of Moments in Inclusive Semileptonic Decays

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I. INTRODUCTION
The Standard Model of particle physics (SM) contains a large number of free parameters which can only be determined by experiment. Precision measurements of all of these parameters are essential for probing the validity range of the model by comparing many other precision measurements to SM calculations. Three of the parameters, the Cabibbo-Kobayashi-Maskawa (CKM) matrix element |V cb | [1,2] and the heavy quark masses m b and m c , can be related via Operator Product Expansions (OPE) to moments and rates of inclusive distributions in semileptonic B meson decays, B → X c ℓ − ν [3], and rare B-meson decays, B → X s γ, where X c and X s denote the hadronic systems with charm and strangeness in these final states, respectively. The quantities |V cb |, m b , m c , and nonperturbative parameters describing effects of the strong interaction can be determined from the measured rates and moments using expansions in 1/m b and the strong coupling constant α s with reliable uncertainty estimates.
While lepton-energy moments are known with good accuracy, the precision of the hadronic-mass and photonenergy moments is limited by statistics. Therefore, we present a new measurement of the hadronic-mass moments m k X with k = 1, . . . , 6 based on a larger dataset than previously used [5]. We also present the first measurement of the combined hadronic mass-and-energy mo- ‡ ‡ Also with Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France § § Also with Università di Sassari, Sassari, Italy ments n k X with k = 2, 4, 6 as proposed by Gambino and Uraltsev [20]. The combined moments n k X use the mass m X and the energy E X of the X c system in the B meson rest frame of B → X c ℓ − ν decays, with a constantΛ, here fixed to be 0.65 GeV as proposed in [20]. They are expected to allow a more reliable extraction of the higher-order nonperturbative HQE parameters and thus to increase the precision on the extraction of |V cb | and the quark masses m b and m c . All moments are determined for different values of the minimum energy of the charged lepton. We update our previous measurement of lepton-energy moments [9] using branching fraction measurements for background decays in [21] and improving the evaluation of systematic uncertainties.
Finally, we perform a combined fit to the hadronicmass moments, moments of the lepton-energy spectrum, and moments of the photon-energy spectrum in decays B → X s γ. The fit determines |V cb |, the quark masses m b and m c , the total semileptonic branching fraction B(B → X c ℓ − ν), and the dominant nonperturbative HQE parameters µ 2 π , µ 2 G , ρ 3 D , and ρ 3 LS . An alternative fit to the moments of n k X , of the lepton-energy, and of the photon energy in B → X s γ, leads to essentially the same results.

II. BABAR DETECTOR AND DATASETS
The work is based on data collected with the BABAR experiment [22] at the PEP-II asymmetric-energy e + e − storage rings [23] at the SLAC National Accelerator Laboratory.
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 GeV/c. 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 GeV. Muon candidates are identified with the use of the instrumented flux return (IFR) of the solenoid. The tracking system, EMC, and IFR cover the full azimuthal range and the polar-angle range 0.3 < θ < 2.7 rad in the laboratory frame, corresponding to a coverage of approximately 90% in the center-of-mass (c.m.) frame, where θ is the polar angle with respect to the electron direction. The DIRC fiducial volume corresponds to a c.m. frame coverage of about 84%.
The data sample for the hadronic moments measurements consists of about 210 fb −1 , corresponding to 232 × 10 6 decays Υ (4S) → BB. Our previous measurement of the lepton-energy moments, which is updated in this paper, was based on a data sample of about 51 × 10 6 Υ (4S) → BB decays. This corresponds to an integrated luminosity of 47 fb −1 on the Υ (4S) resonance. In addition, about 9 fb −1 of data recorded at an energy 40 MeV below the resonance (off-resonance) was used in the lepton-energy moments measurement for the subtraction of background not originating from the Υ (4S).
We use Monte Carlo (MC) simulated events to determine background distributions and to correct for detector acceptance and resolution effects. Simulated B-meson decays are generated using EvtGen [24]. The simulation of the BABAR detector is realized with GEANT4 [25] and final state radiation (FSR) is modeled using the PHOTOS code [26].
In the simulation of semileptonic decays B → X c ℓ − ν we use the branching fractions listed in Table I. For the dominant decay B → D * ℓ − ν we use a parameterization of form factors, based on heavy quark effective theory (HQET) [27][28][29]. Its differential rate is described by three helicity amplitudes which are expressed by the three parameters ρ 2 , R 1 , and R 2 . We choose the following values measured in [30]: R 1 = 1.18 ± 0.30 ± 0.12, R 2 = 0.71 ± 0.22 ± 0.07, and ρ 2 = 0.91 ± 0.15 ± 0.06. The quoted errors reflect statistical and systematic uncertainties. For decays B → Dℓ − ν and for decays to the higher-mass states D 1 , D ′ 1 , D * 0 , and D * 2 we use the ISGW2 model [31]. For the decays B → D ( * ) πℓν, we use the prescription by Goity and Roberts [32].

III. RECONSTRUCTION OF SEMILEPTONIC DECAYS FOR THE MEASUREMENT OF HADRONIC MOMENTS
The event selection and reconstruction for the hadronic-mass moments m k X and the combined massand-energy moments n k X are almost identical. As described in the corresponding sections IV and V, the only differences regard the requirements needed to ensure a good resolution in the observables of interest.
The analysis uses Υ (4S) → BB events in which one of the B mesons decays to hadrons and is fully reconstructed (B tag ), and the semileptonic decay of the recoiling B meson (B recoil ) 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 momentum, charge, and flavor of the B mesons.
The kinematic consistency of the B tag candidates is checked with two variables, the beam-energy-substituted mass m ES = s/4 − p 2 B and the energy difference ∆E = Semileptonic decays are identified by the presence of one and only one electron or muon above a minimum momentum p * ℓ,min measured in the rest frame of the B meson. If not stated otherwise, p * ℓ will denote in the following the lepton momentum measured in the B-meson rest frame. Electrons are identified by combining information from the EMC, the DCH, and the DIRC. They are re-quired to have a lab-frame momentum of p > 0.8 GeV/c and a polar angle in the range 0.41 < θ < 2.54 rad. In this range, electrons are selected with 94% average efficiency and a hadron misidentification rate of the order of 0.1%. Muon identification is mainly based on information obtained from the IFR. Muons are identified with an efficiency ranging between 60% for momenta p = 1 GeV/c in the laboratory frame and 75% for momenta p > 2 GeV/c. The misidentification rate ranges between 1% for kaons and protons and 3% for pions. Efficiencies and misidentification rates are estimated from control samples of electrons, muons, pions, and kaons. We impose the condition Q b Q ℓ < 0, where Q ℓ is the charge of the lepton and Q b is the charge of the b quark of the B tag . This condition is fulfilled for primary leptons originating directly from the B decay, except for B 0 B 0 events in which flavor mixing has occurred. We require the total observed charge of the event to be |Q tot | = |Q Btag + Q B recoil | ≤ 1, 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 X c system, the total charge in the event is required to be zero.

C. Reconstruction of the Hadronic System
The hadronic system X c in the decay B → X c ℓ − ν is reconstructed from charged tracks and energy deposits in the calorimeter that are not associated to the B tag or the charged lepton. We ignore tracks and energy deposits in the calorimeter which are compatible with the hypothesis of being reconstruction artifacts, low-energy beam-generated photons or calorimeter deposits originating from hadronic showers. Each track is assigned a specific particle type, either ( ) p , K ± , or π ± , based on combined information from the different BABAR subdetectors. Few events containing single protons are kept in the selection but removed later on in the background removal procedure. The four-momentum P Xc of the reconstructed hadronic system is obtained from the four-momenta of the reconstructed tracks P i,trk for the given mass assignment, and photons P i,γ by The hadronic mass is defined by m 2 X = P 2 Xc . The four-momentum of the unmeasured neutrino is calculated from the missing four-momentum P miss = P Υ (4S) − P Btag − P Xc − P ℓ . Here, all four-momenta are measured in the laboratory frame. To ensure a well reconstructed hadronic system, we impose criteria on the missing energy, E miss > 0.5 GeV, the missing momentum, p miss > 0.5 GeV/c, and the difference of both quantities, |E miss − cp miss | < 0.5 GeV.
We perform a kinematic fit exploiting the fact that B mesons are produced in a well-defined initial state e + e − → Υ (4S) → BB, to improve the resolution and reduce the bias on the reconstructed quantities. The fit imposes four-momentum conservation, the equality of the masses of the two B mesons, and constrains the mass of the neutrino, P 2 miss = 0. The resulting (original) average resolutions on the measurement of m X and n 2 X are 0.355 GeV/c 2 (0.425 GeV/c 2 ) and 1.05 GeV 2 (1.17 GeV 2 ), respectively. The average biases of m X and n 2 X after (before) the kinematic fit are found to be −0.096 GeV/c 2 (−0.254 GeV/c 2 ) and −0.11 GeV 2 (−0.37 GeV 2 ), respectively.
The background is composed of e + e − → qq (q = u, d, s, c) events (continuum background), Υ (4S) → B + B − or B 0 B 0 decays in which the B tag candidate is mistakenly reconstructed from particles coming from both B mesons in the event (combinatorial background), and non-signal decays of the recoiling B recoil meson (residual background). Signal events where the hadronic system is not fully reconstructed are not considered as an additional source of background. The effect of missing tracks and photons on the resolution of the kinematical quantities of interest is taken into account by applying the correction procedures described below.
To quantify the amount of continuum and combinatorial background in the m ES signal region we perform a fit to the m ES distribution of the B tag candidates. We parameterize the background using an empirical threshold function [39], where x = m ES /m ES,max , m ES,max = 5.289 GeV/c 2 is the kinematic endpoint approximated by the mean c.m. energy, and χ is a free parameter defining the curvature of the function. The signal is parameterized with a modified Gaussian function [40] peaked at the B-meson mass and corrected for radiation losses. The fit is performed separately for several bins in m X and n 2 X to account for changing background contributions. Figure 1 shows the m ES distribution for p * ℓ ≥ 0.8 GeV/c together with the fitted signal and background contributions. The shape of the continuum and combinatorial background as function of m X and n 2 X is determined in a signal-free region of the m ES sideband, 5.210 ≤ m ES ≤ 5.255 GeV/c 2 . Its overall size in the m ES signal region, m ES > 5.27 GeV/c 2 , is determined by rescaling with the relative background contributions in the signal and sideband regions as determined by the fit. Signal and sideband region are separated by 15 MeV/c 2 to avoid the leakage of signal events into the sideband region.
Residual background is estimated from MC simulations. It is composed of charmless semileptonic decays B → X u ℓ − ν, hadrons misidentified as leptons, secondary leptons from semileptonic decays of D ( * ) , D + s mesons or τ either in B 0 B 0 mixed events or produced in b → ccs transitions, as well as leptons from decays of J/ψ and ψ(2S). The branching fractions of the individual simulated background decays are scaled to agree with measurements [21,34,41,42]. The overall simulated background spectrum is normalized to the number of B tag events in data. We verify the normalization and the shape using an independent data control sample with inverted lepton charge correlation, Q b Q ℓ > 0.

IV. HADRONIC-MASS MOMENTS
We present measurements of the moments m k X , with k = 1, . . . 6, of the hadronic-mass distribution in semileptonic B-meson decays B → X c ℓ − ν. The moments are measured as functions of the lower limit on the lepton momentum p * ℓ,min between 0.8 GeV/c and 1.9 GeV/c, calculated in the rest frame of the B meson.

A. Selected Event Sample
We find 19, 212 events with p * ℓ ≥ 0.8 GeV/c, composed of 15, 085 ± 146 signal events above a combinatorial and continuum background of 2, 429 ± 43 events and residual background of 1, 696±19 events. Signal decays amount to 79% of the selected event sample. For p * ℓ ≥ 1.9 GeV/c, we find in total 2, 527 events composed of 2, 006 ± 53 signal events above a background of 271 ± 17 and 248 ± 7 combinatorial/continuum and residual events, respectively. Figure 2 shows the m X distributions after the kinematic fit together with the extracted background shapes for p * ℓ ≥ 0.8 GeV/c and p * ℓ ≥ 1.9 GeV/c.

B. Extraction of Moments
To extract unbiased moments m k X , we apply corrections to account for effects that distort the measured m X 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. In addition, there are contributions from measured particles not belonging to Hadronic-mass spectra after the kinematic fit for lepton momenta p * ℓ ≥ 0.8 GeV/c (top) and p * ℓ ≥ 1.9 GeV/c (bottom) together with distributions of combinatorial background and background from non-BB decays (red, hatched area) as well as residual background (blue, crossed area). The two background histograms are plotted on top of each other.
the hadronic system, especially photons originating from FSR of the primary leptons. These photons are included in the measured X c system and thus lead to a modified value of its mass; they also lower the momentum of the primary lepton. Both effects are included in our correction procedure.
We correct the kinematically-fitted value of m k X of each event by applying correction factors on an event-by-event basis using the observed linear relationship between the moments of the measured mass m k X,reco and the moments of the true mass m k X,true in MC spectra. The correction factors are determined from MC simulations by calculating moments m k X,reco and m k X,true in several bins of the true mass m X,true and fitting the observed dependence with a linear function, referred to as calibration function in the following.
We find that the bias of the measured moments m k X,reco is not constant over the whole phase space. Therefore, we derive the calibration functions in three bins of the particle multiplicity N Xc in the X c system, three bins of E miss − cp miss , as well as in twelve bins of p * ℓ , each with a width of 100 MeV/c. Due to the limited number of generated MC events, the binning in N Xc and E miss − cp miss is not used for p * ℓ,min ≥ 1.7 GeV/c. Overall we construct 84 calibration functions for each order of moments. The obtained calibration functions allow a  consistent extraction of moments for events containing an electron or a muon. Figure 3 shows examples of calibration functions for the moment m 2 X in three bins of p * ℓ as well as in nine bins of E miss − cp miss and N Xc .
For each data event i, the corrected mass m k X,calib,i is calculated by inverting the linear function, where A is the offset and B is the slope of the calibration function. Background contributions are removed by applying a weight factor w i to each corrected hadronic mass m k X,calib,i , where the weight is the expected fraction of signal events in the corresponding region of the m X,reco spectrum in Fig. 2. The expression used to cal-culate the moments is the following: with N ev the total number of selected events. The factors C cal and C true depend on the order k and the minimum lepton momentum p * ℓ,min of the measured moment. They are determined in MC simulations and correct for the residual small biases observed after the calibration. The factors C cal account for the bias of the applied correction method and are typically ranging between 1.01 and 1.06 for k = 1 . . . 5. Larger bias corrections C cal are observed for m 6 X ranging between the extremes 0.902 and 1.109. The residual bias-correction factor C true accounts for differences in selection efficiencies for different hadronic final states and FSR that is included in the measured hadron mass and distorts the measurement of the lepton's momentum. Our correction procedure results in moments which are within systematic uncertainties free of photon radiation. The correction C true is estimated in MC simulations and typically ranges between 0.994 and 1.007. For the moments m 5 X and m 6 X , slightly higher correction factors are determined, ranging between 0.990 and 1.014 for m 5 X and 0.986 and 1.024 for m 6 X . This correction procedure is verified on a MC sample by applying the calibration to measured hadron masses of individual semileptonic decays, B → Dℓ − ν, B → D * ℓ − ν, four resonant decays B → D * * ℓν, and two nonresonant decays B → D ( * ) πℓν. Figure 4 shows the corrected moments m 2 X and m 4 X as functions of the true moments for minimum lepton momenta p * ℓ ≥ 0.8 GeV/c. The dashed line corresponds to m k X,calib = m k X,true . The calibration reproduces the true moments over the full mass range.

C. Systematic Uncertainties and Tests
The main systematic uncertainties are associated with the modeling of hadronic final states in semileptonic Bmeson decays, the bias of the calibration method, the determination of residual background contributions, the modeling of track and photon selection efficiencies, and the identification of particles. The total systematic uncertainty is estimated by adding in quadrature five contributions, as described below. Tables A.I and A.II list the individual contributions to the systematic errors of the measured moments m k X with k = 1 . . . 6 for minimum lepton momenta ranging from 0.8 to 1.9 GeV/c.

MC Statistics
The effect of limited MC statistics on the extracted moments is evaluated using parameterized MC experiments. To study the effect on the calibration curves, the parameters of the fitted first-order polynomials are randomly varied within their uncertainties including correlations and new sets of moments are extracted. The overall uncertainty is determined by repeating this procedure 250 times and taking the r.m.s. of the distribution of the moments as the systematic uncertainty.
To estimate the effect of limited MC statistics in the residual background determination a similar method is applied by varying the parameters of the fit to the m ES distribution within their errors including correlations.

Simulation-Related Effects
We correct for differences between data and MC simulation 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 systematic uncertainties of the photon selection and track finding efficiencies are determined studying independent control samples. Their impact on the measured moments has been evaluated by randomly excluding neutral or charged candidates from the X c system with probabilities corresponding to the uncertainties of the efficiency extraction methods. The uncertainty of the photon selection efficiencies is found to be 1.8% per photon independent of energy, polar angle, and multiplicity. The uncertainty in track finding efficiencies consists of two parts. For each track, we add in quadrature 0.8% systematic uncertainty and 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 systematic uncertainty on the misidentification of π ± mesons as leptons is found to affect the overall normalization of the corresponding background spectra by 8%. The influence on the measured moments is estimated by varying the corresponding background within its uncertainty. The observed variation of moments is taken as a systematic uncertainty.
The impact of mismodeling FSR simulated with PHOTOS [26] is estimated by calculating moments from data using a set of calibration curves constructed from events simulated without FSR photons. The theoretical uncertainty associated with the calculations included in PHOTOS is conservativley assumed to be of the order of 20%. The systematic uncertainty connected to the mismodeling of FSR photons is therefore estimated to be 20% of the observed difference between the nominal moments and those from the MC simulation without FSR photons.
A significant fraction of the low-energy photons detected in the calorimeter are beam related. We check the impact of low-energy photons by removing EMC sig-nals with energies below 100 MeV from the reconstructed hadronic system. The effect on the measured moments is found to be negligible.
The stability of the result under variation of the selection criteria on E miss − cp miss is tested by varying the applied cut between |E miss − cp miss | < 0.2 GeV and |E miss − cp miss | < 1.4 GeV. For all measured moments, the observed variation is well covered by other known systematic detector and MC simulation effects. Therefore, no contribution is added to the systematic uncertainty.

Extraction Method
The systematic uncertainty of the calibration bias correction C cal is estimated to be (C cal − 1)/2.

Background Determination
The branching fractions of background decays in the MC simulation are scaled to agree with the current measurements [21,34,41,42]. The associated systematic uncertainty is estimated by varying these branching fractions within their uncertainties. At low p * ℓ,min , most of the studied background channels contribute to the systematic uncertainty equally, while at high p * ℓ,min , the systematic uncertainty is dominated by background from decays B → X u ℓ − ν. Contributions from J/ψ and ψ(2S) decays are found to be negligible.
The uncertainty in the combinatorial B tag background determination is estimated by varying the lower and upper limits of the sideband region in the m ES distribution up and down by 2.5 MeV/c 2 . The observed effect on all hadronic-mass moments is found to be negligible.

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 calibration functions only from MC simulated hadronic events with hadronic masses m X,true < 2.5 GeV/c 2 , thereby removing the high mass tail of the simulated hadronicmass spectrum. The model dependence of the calibration method is found to be a small contribution to the total systematic uncertainty. We estimate the model dependence of the residual bias correction C true by changing the composition of the inclusive hadronic spectrum, i.e. omitting one or more decay modes.
We study the effect of differences between data and MC simulation in the multiplicity and E miss − cp miss distributions on the calibration method by changing the binning of the calibration functions. The observed variation of the results are found to be covered by the statistical uncertainties of the calibration functions, and no contribution is added to the total systematic uncertainty.

Stability of the Results
The stability of the results is tested by dividing the data into several independent subsamples: B ± and B 0 , decays to electrons and muons, different run periods of roughly equal sample sizes, and two regions in the E miss − cp miss spectrum, −0.5 ≤ E miss − cp miss < 0 GeV and 0 ≤ E miss − cp miss < 0.5 GeV, characterized by different resolutions of the reconstructed hadronic system. No significant variations are observed.

D. Results
The measured hadronic-mass moments m k X after radiative correction with k = 1 . . . 6 as functions of the minimum lepton momentum p * ℓ,min are shown in Fig. 5. All measurements are correlated since they share subsets of selected events. Tables A.I and A.II summarize the numerical results. In most cases we find systematic uncertainties that exceed the statistical uncertainty by a factor of 2.5. The correlation matrix for the moments is given in the EPAPS document [43].

V. MOMENTS OF THE COMBINED MASS-AND-ENERGY SPECTRUM
The measurement of moments of the observable n 2 X , a combination of the mass and energy of the inclusive X c system, as defined in Eq. (1), is theoretically motivated and is expected to allow a more reliable extraction of the higher order HQE parameters µ 2 π and ρ 3 D [20]. We present measurements of the moments n 2 X , n 4 X , and n 6 X for different minimum lepton momenta between 0.8 GeV/c and 1.9 GeV/c in the B-meson rest frame.

A. Event Selection
Due to the structure of the variable n 2 X as a difference of two measured values, its measured resolution and bias are worse than for the mass moments. Also, the sensitivity to cuts on E miss − cp miss increases. The average resolution of n 2 X after the kinematic fit for lepton momenta greater than 0.8 GeV/c is measured to be 1.05 GeV 2 with a bias of -0.11 GeV 2 . We therefore introduce stronger requirements on the reconstruction quality of the event. We tighten the criteria on the neutrino observables by requiring E miss − cp miss to be between −0.2 and 0.3 GeV. Due to the stronger requirement, the individual variables E miss and p miss have less influence on the resolution of the reconstructed hadronic system. Therefore, the requirements on the missing energy and the missing momentum in the event are relaxed to E miss > 0 GeV and p miss > 0 GeV/c, respectively, as these requirements do not yield significant improvement on the resolution of n 2 X , and do not increase the ratio of signal to background events.
For p * ℓ ≥ 0.8 GeV/c and 1.9 GeV/c, there remain 10, 053 ± 142 and 1, 626 ± 52 signal events, respectively. Background events make up 22 % of the final event sample with p * ℓ ≥ 0.8 GeV/c. The background is composed of 12 % continuum and combinatorial background and 10 % decays of the signal B meson other than the semileptonic decay B → X c ℓ − ν. Combinatorial and continuum background is removed using the sideband of the m ES distribution, as described in section III C. The residual background events, containing a correctly reconstructed B tag meson, are removed using MC simulations. The dominant sources are pions misidentified as muons, B → X u ℓ − ν decays, and secondary semileptonic decays of D and D s mesons.
The measured n 2 X spectra for p * ℓ,min = 0.8 GeV/c and p * ℓ,min = 1.9 GeV/c are shown together with the background distributions in Fig. 6.

B. Extraction of Moments
The extraction of unbiased moments n k X from the measured n 2 X spectra follows a calibration procedure sim-ilar to the one used to extract the hadronic-mass moments as described in Section IV B. The linear calibration functions for k = 2, 4, 6 are derived from MC samples in three bins of E miss − cp miss and three bins of the X c -system multiplicity N Xc for each of the 12 lepton momentum bins of 100 MeV/c width. Because of differences in events containing electrons and muons, we also derive separate calibration functions for these two classes of events. Overall, we determine 216 linear calibration functions. The calibration again includes the effects of FSR photons which not only modify m X and p * ℓ , but also E X . We have verified that applying the calibration procedure on MC samples of individual exclusive B → X c ℓ − ν modes allows to reproduce the generated moments, as shown in Fig. 7. Small biases remaining after calibration are of the order of 1 % for n 2 X and of few percent for n 4 X and n 6 X . Background contributions are removed by applying n 2 X -dependent weight factors w i (n 2 X ) on an event-byevent basis, leading to the following expression for the determination of the moments:  The bias correction factors C(p * ℓ , k), depending on the minimum lepton momentum and the order of the extracted moments, are determined by MC simulations; they combine the two factors C cal and C true as described in Section IV B.

C. Systematic Uncertainties and Tests
We consider the same five sources of systematic uncertainties as for the mass moments described in Sections IV C 1 to IV C 5: MC statistics, simulation-related effects, extraction method, background determination, and modeling of signal decays. The individual contributions to the systematic error, listed in Table A.III, are estimated following procedures essentially identical to those described for the mass moments.
Because of the tighter cut on E miss − cp miss , the systematic uncertainty associated with this criterion is estimated in a different way. We first keep the lower limit fixed to the nominal value and vary the upper limit to 0.3 GeV/c to 0.25 GeV/c, 0.4 GeV/c, and 0.5 GeV/c. Then we fix the upper limit to its nominal value and vary the lower limit to −0.3 GeV/c and −0.1 GeV/c. The mean of the observed differences in the measured moments on data is taken as systematic uncertainty.
In the third study, we include the uncertainty from the binning of the calibration function in the multiplicity of the X c -system. For the choice of the calibration function, we randomly increase the measured multiplicity of the X c system by one with a probability of 5% corresponding to the observed difference between MC and data. The uncertainty in the bias-correction factor C(p * ℓ , k) is conservatively estimated as half of the applied correction.
Varying the branching fractions of the exclusive signal modes in the MC simulation has, in agreement with the mass-moment studies, a very small impact on the measured combined moments. Also, no significant variations of the results are observed when splitting the data sample into the same subsamples as for the mass moments. Figure 8 shows the results for the moments n 2 X , n 4 X , and n 6 X as a function of the minimum lepton momentum p * ℓ,min . The moments are highly correlated due to the overlapping data samples. The full numerical results and the statistical and the estimated systematic uncertainties are given in Table A.III. The systematic covariance matrix for the moments of different order and with different cuts on p * ℓ,min is built using statistical correlations. This correlation matrix for the moments is given in the EPAPS document [43].

D. Results
A clear dependence on the minimum lepton momentum is observed for all moments, due to the increasing contributions from higher-mass final states with decreasing lepton momentum. In most cases we obtain systematic uncertainties slightly exceeding the statistical uncertainty.

VI. MOMENTS OF THE ELECTRON-ENERGY SPECTRUM
Moments of the electron-energy spectrum for semileptonic decays B → X c e − ν averaged over charged and neutral B mesons have been measured in a data sample of 51 × 10 6 Υ (4S) → BB decays [9]. In the following, we present an overview of this analysis and update the results by using more recent measurements [21,41] of branching fractions of background processes.
In multi-hadron events as defined in [9], BB events are selected by requiring a semileptonic B decay with an identified electron (e tag ), with charge Q(e tag ) and a momentum 1.4 < p * e < 2.3 GeV/c, measured in the Υ (4S) rest frame. These events constitute a tagging sample used as normalization for the branching fraction. A second electron e sig , for which we require p * e > 0.5 GeV/c, is assigned either to the unlike-sign sample if the tagged sample contains an electron with Q(e tag ) = −Q(e sig ) or to the like-sign sample if Q(e tag ) = Q(e sig ). In events without B 0 B 0 mixing, primary electrons from semileptonic B decays belong to the unlike-sign sample while secondary electrons contribute to the like-sign sample. Secondary electrons originating from the same B as the e tag are removed from the unlike-sign sample by the requirement cos α * > 1.0 − p * e c/ GeV and cos α * > −0.2, where α * is the angle between the two electrons in the Υ (4S) rest frame. Corrections for the small residual background of unlike-sign pairs originating from the same B fulfilling this requirement are described in [9]. Additional background corrections for electrons from J/ψ → e + e − decays, continuum events, photon conversions, π 0 → e + e − γ Dalitz decays, and misidentified hadrons are also described in [9]. Figure 9 shows the electron-momentum spectra together with the contributions of the backgrounds. Further backgrounds arise from decays of τ leptons, charmed mesons produced in b → ccs decays, and J/ψ or ψ(2S) → e + e − decays with only one detected electron. We also need to correct for cases where the tagged electron does not originate from a semileptonic B decay. These backgrounds are irreducible. Their contributions to the three samples -single electrons, like-sign, and unlike-sign pairs -are estimated from MC simulations, using the ISGW2 model [31] to describe semileptonic D and D s -meson decays. As an important update to the results in [9], the branching fractions of these backgrounds are recalculated to match the recent measurements [21].  [9] and estimated backgrounds (histograms) for electron candidates in (a) the unlike-sign sample, and (b) the like-sign sample. The background spectra are updated wrt. the previous publication with more recent branching-fraction measurements [21,41].
After the like-and unlike-sign samples have been corrected for electron identification efficiency, these irreducible background spectra are subtracted. To account for B 0 B 0 mixing, we determine the number of primary electrons in the i-th p * -bin from the like-sign and unlikesign pairs as where χ 0 = 0.1878 ± 0.0024 [21] is the B 0 B 0 mixing parameter, f 0 = B(Υ (4S) → B 0 B 0 ) = 0.491 ± 0.007 [21], [21]. The parameter ǫ i α * is the efficiency of the additional requirement for the unlike-sign sample as defined in Eq. (7). The spectrum obtained from Eq. (8) is corrected for the effects of bremsstrahlung in the detector material using MC simulation. Figure 10 shows the resulting spectrum of primary electrons.
We determine the partial branching fraction as ( i N i b→c,u )/(N tag ǫ evt ǫ cuts ), where i runs over all bins with E e > E 0 . For the background-corrected number N tag of tag electrons we find N tag = (3617±4±22)×10 3 , where the uncertainties are statistical and systematic, respectively. The parameter ǫ evt = (98.9 ± 0.5)% refers to the relative efficiency for selecting two-electron events compared to events with a single e tag , and ǫ cuts = (82.8 ± 0.3)% is the acceptance for the signal electron for E 0 = 0.6 GeV. The result is B(B → Xeν(γ), E e > 0.6 GeV) = (10.30 ± 0.06 ± 0.21)%, where the errors correspond to the statistical and systematic uncertainties, respectively. In the B-meson rest frame, we define 0) for n = 2, 3 and the partial branching fraction B(E 0 ) = τ B R 0 (E 0 , 0), where τ B is the average lifetime of charged and neutral B mesons. The calculation of the moments is done as in [9] and includes corrections for charmless semileptonic decays, the movement of the B mesons in the c.m. frame, biases due to the event selection criteria, and binning effects. The spectra and moments presented are those of B → X c eν(γ) decays with any number of photons. Since current theoretical predictions on the lepton-energy moments do not incorporate photon emission, we also present a second set of of moments with corrections for the impact of QED radiation using the PHOTOS code [26]. Figure 11 shows the moments of B → X c eν(γ) decays as a function of E 0 , and Table II lists the main systematic errors for E 0 = 0.6 and 1.5 GeV. The complete listing of all moments and the full correlation matrix, with and without PHOTOS corrections can be found in [43].

VII. DETERMINATION OF |V cb | AND THE QUARK MASSES m b AND mc
At the parton level, the weak decay rate for b → cℓν can be calculated accurately; it is proportional to |V cb | 2 and depends on the quark masses m b and m c . To relate measurements of the semileptonic B-meson decay rate to |V cb |, the parton-level calculations have to be corrected for effects of strong interactions. Heavy Quark Expansions (HQEs) [44][45][46] have become a successful tool for calculating perturbative and nonperturbative QCD corrections [47][48][49][50][51] and for estimating their uncertainties.  In the kinetic-mass scheme [11,20,[52][53][54][55], these expansions in 1/m b and the strong coupling constant α s (m b ) to order O(1/m 3 b ) contain six parameters: the running kinetic masses of the b and c quarks, m b (µ) and m c (µ), and four nonperturbative parameters. The parameter µ denotes the Wilson factorization scale that separates effects from long-and short-distance dynamics. The calculations are performed for µ = 1 GeV [56]. It has been shown that the expressions for the moments have only a small scale dependence [17]. We determine these six parameters and |V cb | from fits to moments of the hadronic-mass, combined mass-and-energy, and electron-energy distributions in semileptonic B decays B → X c ℓ − ν and moments of the photon-energy spectrum in decays B → X s γ [14][15][16].
The performed fit uses a linearized expression for the dependence of |V cb | on the values of heavy-quark parameters, expanded around a priori estimates of these parameters [11]: Here m b and m c are in GeV/c 2 and all other parameters of the expansion are in GeV k ; τ B refers to the average lifetime of B mesons produced at the Υ (4S), measured in picoseconds. HQEs in terms of the same heavy-quark parameters are available for hadronic-mass, combined mass-and-energy, electron-energy, and photonenergy moments. Predictions for those moments are obtained from an analytical calculation [57]. We use these calculations to determine |V cb |, the total semileptonic branching fraction B(B → X c ℓ − ν), the quark masses m b and m c , as well as the heavy-quark parameters µ 2 π , µ 2 G , ρ 3 D , and ρ 3 LS , from a simultaneous χ 2 fit to the measured moments and partial branching fractions, all as functions of the minimum lepton momentum p * ℓ,min and minimum photon energy E γ,min .

A. Extraction Formalism
The fit method designed to extract the HQE parameters from the measured moments has been reported previously [17,58]. It is based on a χ 2 minimization, The vectors M exp and M HQE contain the measured moments and the corresponding moments calculated by theory, respectively. Furthermore, the expression in Eq. (11) contains the total covariance matrix C tot = C exp + C HQE defined as the sum of the experimental C exp and theoretical C HQE covariance matrices (see Section VII C). The total semileptonic branching fraction B(B → X c ℓ − ν) is extracted in the fit by extrapolating the measured partial branching fractions B p * ℓ,min (B → X c ℓ − ν) with p * ℓ ≥ p * ℓ,min to the full lepton energy spectrum. Using HQE predictions of the relative decay fraction R p * ℓ,min = p * ℓ,min dΓSL dp * ℓ dp * ℓ 0 dΓSL dp * ℓ dp * ℓ , the total branching fraction can be introduced as a free parameter in the fit. It is given by Using Eqs. (10) and (11) together with the measured average B-meson lifetime τ B and the total branching fraction, allows the calculation of |V cb |: Thereby, |V cb | is introduced as an additional free parameter to the fit. To propagate the uncertainty on τ B properly into the extracted result for |V cb |, τ B is added as an additional measurement to the vectors of measured and predicted quantities, M exp and M HQE . The nonperturbative parameters µ 2 G and ρ 3 LS have been estimated from the B-B * mass splitting and heavyquark sum rules to be µ 2 G = (0.35 ± 0.07) GeV 2 and ρ 3 LS = (−0.15 ± 0.10) GeV 3 [17], respectively. Both parameters are restricted in the fit by imposing Gaussian error constraints.

B. Experimental Input
The combined fit is performed on a subset of available moment measurements with correlations below 95% to ensure the invertibility of the covariance matrix. Since the omitted measurements are characterized by high correlations to other measurements considered in the fit, they do not contribute significant additional information, and the overall sensitivity of the results is not affected. Choosing a different subset of moments gives consistent results. We perform two fits to the following set of measured moments, thereby including either the hadronicmass moments or the moments of the combined massand-energy spectrum: • Hadronic-mass moments are used as presented in this paper. We select the following subset for the fit: m 2 X for p * ℓ ≥ 0.9, 1.1, 1.3, 1.5 GeV/c, m 4 X for p * ℓ ≥ 0.8, 1.0, 1.2, 1.4 GeV/c, and m 6 X for p * ℓ ≥ 0.9, 1.1, 1.3, 1.5 GeV/c.

C. Theoretical Uncertainties
As discussed in [17] and specified in [20], the following theoretical uncertainties are taken into account: The uncertainty related to the uncalculated perturbative corrections to the Wilson coefficients of nonperturbative operators are estimated by varying the corresponding parameters µ 2 π and µ 2 G by 20% and ρ 3 D and ρ 3 LS by 30% around their expected values. Uncertainties for the per- For the extracted value of |V cb | an additional error of 1.4% is added for the uncertainty in the expansion of the semileptonic rate Γ SL [11,55]. It accounts for remaining uncertainties in the perturbative corrections to the leading operator, uncalculated perturbative corrections to the chromomagnetic and Darwin operator, higher order power corrections, and possible nonperturbative effects in the operators with charm fields. This uncertainty is not included in the theoretical covariance matrix C HQE but is listed separately as a theoretical uncertainty on |V cb |.
For the predicted photon-energy moments E n γ , additional uncertainties are taken into account. As outlined in [52], uncertainties of 30% of the applied bias correction to the photon-energy moments and half the difference in the moments derived from two different distributionfunction ansätze have to be considered. Both contributions are added linearly [17].
The theoretical covariance matrix C HQE is constructed by assuming fully correlated theoretical uncertainties for a given moment with different lepton-momentum or photon-energy cutoffs and assuming uncorrelated theoretical uncertainties for moments of different orders and types. The additional uncertainties considered for the photon-energy moments are assumed to be uncorrelated for different moments and photon-energy cutoffs.

D. Results
In the following, the results of the two fits, one including the measurement of hadronic-mass moments and the other including the measured moments of the combined mass-and-energy spectrum instead, are discussed.
We use a parameterized MC simulation to separate fit parameter uncertainties into experimental and theoretical contributions. The simulation uses a set of expected moments randomly varied with either C tot or C exp . Fits to these moments allow for the determination of the expected total and experimental uncertainties, respectively. The final experimental and theoretical uncertainties are calculated from the final total uncertainties by means of their simulated relative expected fractions.

Combined Fit Including Hadronic-Mass Moments
A comparison of the fit including hadronic-mass moments with the measured moments is shown in Fig. 12. The moments m X and m 3 X as well as the combined mass-and-energy moments are not included in the fit and thus provide an unbiased comparison with the fitted HQE prediction. We find an overall good agreement, also indicated by χ 2 = 10.9 for 28 degrees of freedom. Results for the SM and HQE parameters extracted from the fit are summarized in Table III. We find |V cb | = (42.05 ± 0.45 ± 0.70) × 10 −3 , B(B → X c e −ν ) = (10.64 ± 0.17 ± 0.06)%, m b = (4.549 ± 0.031 ± 0.038) GeV/c 2 , and m c = (1.077 ± 0.041 ± 0.062) GeV/c 2 , where the errors correspond to experimental and theoretical uncer-tainties, respectively. The fitted quark masses have a large correlation of 95% resulting in a more precise determination of the quark mass difference, m b − m c = (3.472 ± 0.032) GeV/c 2 , where the error is the total uncertainty. We translate the quark masses which were extracted in the kinetic scheme into the MS scheme using calculations up to O(α 2 s ) accuracy [11]. The translation yields m b (m b ) = (4.186 ± 0.044 ± 0.015) GeV/c 2 and m c (m c ) = (1.196 ± 0.059 ± 0.050) GeV/c 2 , where the first uncertainty is a translation of the uncertainty obtained in the kinetic scheme and the second corresponds to an estimate for the uncertainty of the transformation itself. Figure 13 shows a comparison of the measured moments and the fit including the measured combined massand-energy moments. We find an overall good agreement with χ 2 = 8.2 for 28 degrees of freedom. The fit yields predictions of the hadronic-mass moments that are in good agreement with the measurement. Numerical results of the fit are summarized in Table IV

Comparison of Results
Comparing the result of the fit that includes moments of the n 2 X distribution with that including hadronic-mass moments instead, we find good agreement of all fit parameters and their uncertainties. The differences between the fit values are 0.2 σ for |V cb |, 0.3 σ for m b , and 0.3 σ for m c . The uncertainties of all fit parameters in both fits agree within 8%. Figure 14 shows ∆χ 2 = 1 contours of both fits in the (m b , |V cb |) and (m b , µ 2 π ) planes. We find an almost identical precision for the fit values of |V cb |, m b , and µ 2 π . In the Figure, we also show the results of two fits with reduced sets of input measurements. To illustrate the influence of the photon-energy measurements, a fit with only hadronic-mass and lepton-energy moments is performed. For further comparison we also perform a fit with only hadronic-mass moments and partial branching fractions.
The fits with reduced experimental input show a significantly reduced accuracy of the extracted parameters.
As our primary results we choose the values extracted from the fit with hadronic-mass moments since this fit has been used extensively before. Its results are in good agreement with earlier determinations [17,59], but their uncertainties are slightly larger because of the restrictions to BABAR data only.
The use of combined mass-and-energy moments n 2 X does not lead to a more precise determination of the fundamental physics parameters |V cb |, m b , and m c . However, the agreement of both fits confirms that higherorder corrections, which are needed for the expansion of the hadronic-mass moments but not for the n 2 X moments, have been estimated correctly. A significant change in the uncertainties of the SM and HQE parameters would have indicated a too naive treatment of the corrections for the mass moments [57]. Consequently, the presented results have increased the confidence into the validity of error estimates that have to be made for a reliable determination of m b , m c , and |V cb |.

VIII. ACKNOWLEDGMENTS
We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the US Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council       IV: Results of the simultaneous fit to moments of the combined mass-and-energy spectrum, electron-energy moments, and photon-energy moments. For |V cb | we account for an additional theoretical uncertainty of 1.4% from the uncertainty in the expansion of ΓSL. Correlation coefficients for all parameters are summarized below the central values.  A.I: Results for the moments m k X with k = 1 . . . 3 for different minimum lepton momenta p * ℓ,min with statistical and systematic uncertainties. The systematic uncertainties are grouped in five categories having related sources: MC statistics contains the statistical uncertainties of the calibration curves and of the residual background. Simulation related is the sum of uncertainties due to neutral and charged reconstruction efficiency differences in data and MC, particle identification, and mismodeling of final state radiation. The category extraction method contains the conservative estimate of half of the bias correction. The category background sums all contributions from the variation of the residual background components. The category signal model sums the impact of the variation of the signal decay branching fractions. Minimum lepton momenta are given in GeV/c. Moments and uncertainties are given in ( GeV/c 2 ) k .  Results for the moments m k X with k = 4 . . . 6 for different minimum lepton momenta p * ℓ,min with statistical and systematic uncertainties. The systematic uncertainties are grouped in five categories having related sources: MC statistics contains the statistical uncertainties of the calibration curves and of the residual background. Simulation related is the sum of uncertainties due to neutral and charged reconstruction efficiency differences in data and MC, particle identification, and mismodeling of final state radiation. The category extraction method contains the conservative estimate of half of the bias correction. The category background sums all contributions from the variation of the residual background components. The category signal model sums the impact of the variation of the signal decay branching fractions. moment measurements. Minimum lepton momenta are given in GeV/c. Moments and uncertainties are given in ( GeV/c 2 ) k .  A.III: Results for n k X for k = 2, 4, 6 for all minimum lepton momentum values p * ℓ,min . The statistical uncertainty contains the uncertainty arising from the limited data sample and an additional statistical uncertainty arising from the determination of the combinatorial background. The systematic uncertainties are grouped in five categories having related sources: MC statistics contains the statistical uncertainties of the calibration curves and of the residual background. Simulation related is the sum of neutral and charged reconstruction efficiency differences in data and MC, Emiss − cpmiss differences, mismodeling of final state radiation, and PID impact. The category extraction method contains the conservative estimate of half of the bias correction and the impact of the calibration curve binning. The category background sums all contributions from the variation of the residual background components. The category signal model sums the impact of the variation of the signal decay branching fractions.