Heavy Quark Parameters and Vcb from Spectral Moments in Semileptonic B Decays

We extract the heavy quark masses and non-perturbative parameters from the Delphi preliminary measurements of the first three moments of the charged lepton energy and hadronic mass distributions in semileptonic B decays, using a multi-parameter fit. We adopt two formalisms, one of which does not rely on a 1/mc expansion and makes use of running quark masses. The data are consistent and the level of accuracy of the experimental inputs largely determines the present sensitivity. The results allow to improve on the uncertainty in the extraction of Vcb.


Introduction
The Operator Product Expansion (OPE) represents a foundation for extracting the |V ub | and |V cb | elements of the CKM mixing matrix from inclusive semileptonic (s.l.) B decays. In this framework, the decay width is expressed in terms of quark masses, and non-perturbative effects are described by expectation values of heavy quark operators, some of which are presently poorly known. The experimental accuracy already achieved, and that expected from the large data sets recorded by the B-factories, makes the ensuing theory uncertainty a serious limitation. Extracting the heavy quark masses and the non-perturbative parameters, arising from the 1/m 2 b and 1/m 3 b corrections, directly from the data has therefore become a key issue. There have already been |V cb | determinations from the first moment of distributions in s.l. and b → X s γ decays, and the 1/m 3 b corrections, estimated from parameter ranges, have been found to represent an important source of uncertainty [1]. These ranges, based on dimensional arguments, are affected by a considerable degree of arbitrariness.
In order to circumvent these problems, we introduce in this Letter a multi-parameter fit to determine the relevant 1/m 2 b and 1/m 3 b parameters, together with the heavy quark masses, from the first three moments of the leptonic energy and hadronic mass spectra in s.l. B decays. Results are based on preliminary data obtained by the Delphi Collaboration. Moments are measured without cuts on the lepton energy in the B rest frame. We consider two formalisms, one of which is new and relies on fewer theoretical assumptions. The use of higher moments guarantees a sensitivity to these parameters and the simultaneous use of the hadronic and leptonic spectra ensures that a larger number of parameters can be kept free in the fit. We discuss the results both in terms of the extraction of the parameters and the implications for |V cb |, and as a consistency check of the underlying theoretical assumptions.

Extracting Non-Perturbative Parameters
The moments of the hadronic and leptonic spectra in s.l. B decays have recently been measured by several experiments [1][2][3][4][5]. We consider here moments of the charged lepton energy distribution defined as and moments of the distribution of M X , the invariant hadronic mass, whereM D = 1.973 GeV is the spin averaged D meson mass and no cut on the charged lepton energy is assumed.
The theoretical framework to interpret these data has long been known and it is based on the OPE. Different implementations exist, depending on the way the quark masses are treated. For instance, the m b and m c masses can be taken as independent parameters or subject to a constraint on m b −m c , imposed from the measured B ( * ) and D ( * ) meson masses. The second choice introduces a 1/m c expansion. Another option concerns the normalization scheme used for quark masses and nonperturbative parameters. One approach is to use short-distance masses, such as the low-scale running masses. Alternatively, the pole mass scheme can be used.
The OPE expresses lepton moments through quark masses as a double expansion in α s and 1/m b : It is possible to re-express the heavy quark masses, m Q , in the above equations, in terms of the meson masses, M H Q , through the relation [6]: The use of these expressions introduces an explicit dependence of the non-local correlators contributing to ρ 3 nl . In the notation of [7], ρ 3 nl corresponds to linear combinations of T 1−4 . Here, we employ the following two formalisms. The first one is based on the kinetic running masses, m Q (µ), and non-perturbative parameters, introduced in [8]. No charm mass expansion is assumed. The second formalism employs quark pole masses and the B ( * ) and D ( * ) meson mass relations. It represents a useful reference, as it has been already adopted in several studies.
Contributions through O(α 2 s β 0 ) [9,10] and O(1/m 3 b ) [11][12][13][14] to the moments are available. Depending on the formulation adopted, the number of parameters involved at this order ranges from six to nine. Some of these parameters, like m b and λ 2 ≃ µ 2 G /3, are relatively well known. Others, notably those which appear at O(1/m 3 b ), are virtually unknown.

2.1
The m b (µ), m c (µ) and µ 2 π (µ) Formalism The running kinetic quark masses m b (µ) and m c (µ) are considered here as two independent parameters. Apart from µ 2 π (µ) and µ 2 G (µ), defined as expectation values in the actual B meson, there are two 1/m 3 b parameters, ρ 3 D and ρ 3 LS . The effect of ρ 3 LS turns out to be numerically small. In Eqs. (3) and (4) the mass ratio r is given by (m c (µ)/m b (µ)) 2 , and the b quark mass is understood as m b (µ). The Table 1: Numerical values of the coefficients in Eq.(3) evaluated at r=0.06 and m b (1 GeV) = 4.6 GeV.  Table 2: Numerical values of the coefficients in Eq.(4) evaluated at r=0.06 and m b (1 GeV) = 4.6 GeV.  Table 1, for the central values of m b (1 GeV) = 4.6 GeV and r ≃ 0.06 obtained in our fit. Although we quote only the leading-order perturbative coefficients, we also include second-order BLM corrections in the analysis. Detailed expressions for the coefficients will be presented elsewhere.
In the case of hadronic moments, we discard in Eq. (4) coefficients b nl , c nl with l > 1, and d nl , The value of the other coefficients, at r = 0.06, are listed in Table 2. Here we consider only O(α s ) corrections and evaluate them using α s = 0.3.

TheΛ and λ 1 Formalism
This widely used scheme results from the combination of the OPE with the HQET. Following the notation of Ref. [14], the moments are expressed in the following general form: where k = n and k = 2n for leptonic and hadronic moments, respectively, while a 0 = 0 for hadronic moments.M B = 5.3135 GeV is the spin-averaged B meson mass. The second order BLM corrections 1 are expressed in terms of β 0 = 11 − 2/3n f , where we take n f = 3. The coefficients a i , b i , c i , d i are given in Table 3 for the first three leptonic, M 1,2,3 (E ℓ ), and hadronic moments, M 1,2,3 (M X ). In the Table 3: Numerical values of the coefficients in Eq. (6).
leptonic case the a 1 coefficients agree with ref. [12], while the coefficients of the first two hadronic moments agree with [13,14]. Details of the derivation will be presented elsewhere.
The non-perturbative parameters in Eq.(6) are related to those in Section 2.1 by the following relations, valid up to O(α s ): Perturbative corrections introduce a significant numerical difference between the parameters in the two schemes. At µ = 1 GeV: A well known problem of this formalism is the instability of the perturbative series, due to the use of the pole quark masses. Large higher order corrections are however expected to cancel in the relation between physical observables, as long as all observables involved in the analysis are computed at the same order in α s [6,15]. We also note that, as a consequence of the HQET mass relations for the mesons, the intrinsic expansion parameter in Eq.(6) is 1/M D , rather than 1/M B . The convergence of this expansion has been questioned, in view of indications [16,17] that the matrix elements T i of some non-local operators could be larger than that expected from dimensional estimates.

Fits and Results
This analysis is based on the preliminary Delphi measurements [3,4] of the first three moments of the hadronic mass and charged lepton energy, summarised in Table 4. Owing to the large boost of B hadrons in Z 0 → bb events, the acceptance of the analyses can be extended down to the start of the lepton energy spectrum, making their theoretical interpretation more direct. The results correspond to B 0 d and B − u mesons decays only. We perform a χ 2 fit to these six moments, using the two theoretical frameworks discussed above.
In the fit we also impose some additional constraints derived from independent determinations.
In the kinetic mass scheme, we fit the full set of parameters: m b (1 GeV), m c (1 GeV), µ 2 π , ρ 3 D and ρ 3 LS . We impose µ 2 G =(0.35±0.05) GeV 2 [16] and ρ 3 LS =(-0.15±0.15) GeV 3 . Two mass constraints have also been applied: m b (1 GeV)=(4.57±0.10) GeV [18], and, to be conservative, m c (1 GeV)=(1.05±0.30) GeV. The most stringent is that on m b (1 GeV). It must be noted that this constraint is largely equivalent to that derived from the first moment of the photon energy spectrum in b → sγ in other studies [2]. Results are obtained for α s (m b ) = 0.22 ± 0.01 and are shown in Table 5. In order to study the effect of the bounds on m b,c introduced, the fit has been repeated unconstrained. Results are consistent, although the accuracy on the masses degrades. In particular we find m b (1 GeV)=(4.61±0.15) GeV. It is interesting to observe that the mass constraints applied are of the scale of the fit sensitivity. Also, the central values of the heavy quark masses are in good agreement with independent determinations [18,19].
In the alternative approach based on pole masses, the fit extractsΛ, λ 1 , λ 2 , ρ 1 and ρ 2 . We fix T i = 0 and impose two constraints from M B * − M B and M D * − M D which effectively reduce by two the number of free parameters. The results are given in Table 6.
Projections of the constraints from the six moments in the m b -µ 2 π and m b -ρ 3 D planes are shown in Fig. 1 and those in theΛ-λ 1 andΛ-ρ 1 planes in Fig. 2. The χ 2 /n.d.f. of the fits is 0.96 and 0.35 in the two formulations. Since the contributions proportional to ρ 3 LS in the moment expressions are numerically suppressed, the fit is only marginally sensitive to its size and the result is determined by the constraint applied. By removing this, the fit would give ρ 3 LS =(-1.0±0.7) GeV 3 . In contrast, the value of the leading 1/m 3 b correction (parameterised by ρ 3 D or ρ 1 ) can be determined with satisfactory accuracy and its range agrees with theoretical expectations [16].
Systematic uncertainties due to ranges of residual parameters which have been fixed and missing terms in the expansions have been estimated. For the running mass formalism we propagate the  uncertainty on α s and evaluate the effect of removing the BLM corrections from the lepton moments.
In this scheme that is a small effect and higher order perturbative corrections are expected to be under control. Dimensional estimates suggest that 1/m 4 b effects do not exceed the present experimental resolution. Other systematic uncertainties will be addressed in a dedicated publication. the fit. It is therefore possible to test a posteriori the consistency of the meson mass expansion by comparing theΛ values obtained in the two cases. We findΛ(B) −Λ(D) = −0.086±0.092. This is also a test of the size of the non-local terms.
In both approaches, the OPE predictions for the six moments, computed with the available precision, have a common intersection in the multi-parameter space and the quality of the fit is good. Within the present experimental accuracy, we therefore do not see the need to introduce higher order terms to establish agreement with the data. In particular, the first leptonic and hadronic moments are highly correlated and depend on nearly the same combination of heavy quark masses. Fixing this from M 1 (M X ), one finds M 1 (E ℓ ) = 1.377 GeV which agrees well with the measured value of (1.383±0.015) GeV. This provides a non-trivial consistency check of the OPE. The overall agreement represents both a test of the theory and suggests constraints on the size of the 1/m 4 b terms and of other missing corrections. Similarly, the observed agreement strongly supports the validity of quark-hadron duality in the B decay shape variables.
At present the achieved experimental resolution matches the available theoretical accuracy. With more precise data soon becoming available, it is important to improve the latter, particularly for higher hadronic moments. One way to improve the convergence of the heavy quark expansion could be to employ different kinematic variables. We propose to consider N 2 X = M 2 X − 2ΛE X , where M X and E X are the hadronic mass and energy andΛ a fixed mass parameter. ChoosingΛ near

Implications for |V cb |
The value of |V cb | obtained from the total s.l. decay width depends on the OPE parameters extracted above. We discuss now the implications of our results for |V cb |, using the input parameters given in Table 7. The uncertainties on the BR(b → Xℓ − ν) have been increased compared to ref. [22] for not using the heavy quark forward-backward asymmetries in the LEP global electroweak fit and to account for the ±15% uncertainty on the equality of s.l. partial width of b baryons and mesons. The inclusive s.l. decay width has been calculated through second order in perturbative QCD. Second order BLM corrections were obtained in [23], all-order BLM terms are available from [24], whereas second-order non-BLM corrections have been estimated in [25]. Non-perturbative corrections start at order O(1/m 2 b ) [11] and O(1/m 3 b ) corrections have also been calculated [7]. Electroweak corrections have also been taken into account [26].
The determination of |V cb | and the contributions of the various parameters in the kinetic mass scheme is described in [27]. An approximate formula which displays the dependence on the different parameters is: A detailed discussion of the theoretical uncertainties on |V cb | goes beyond the scope of this paper. Here we focus on the uncertainty arising from the heavy quark masses and non-perturbative parameters determined in the fit. It is evaluated using the full fit error matrix which leads to ±1.5%.
There is an additional uncertainty coming from the limited accuracy of the theoretical expressions which have been used. We take the range m b /2 < µ ′ < m b for the scale µ ′ at which α s is evaluated and find a ±1% effect 2 . In summary, we obtain: where the first uncertainty reflects the accuracy on the s.l. width determination.
The expression for the inclusive b s.l. width in the pole mass scheme is known to the same order. The fit results have been used to obtain: |V cb | = 0.0413 × 1 ± 0.016| meas ± 0.017| f it ± 0.006| nl ± 0.021| pert .
Again, the first two uncertainties correspond to the decay width measurement and to the fitted parameters, respectively. The third uncertainty refers to the T i=1,4 parameters which have been varied within the range (0 ± (0.5) 3 ) GeV 3 . The uncertainty from the truncation of the perturbative QCD series is again estimated by varying the scale at which α s is evaluated between m b /2 and 2m b .
Here the perturbative uncertainty is larger and reflects the slower convergence of the perturbative series when the pole mass scheme is employed.

Conclusions
The values of the heavy quark masses have been determined, together with the leading 1/m 2 b and 1/m 3 b parameters, from a fit to the first three moments of the charged lepton energy and hadronic mass spectra in s.l. decays, measured in a preliminary analysis of the Delphi data. The absence of a charged lepton energy cut in the analysis makes the heavy quark expansion more reliable and allows us to include higher moments. We have adopted two different formalisms: one based on lowenergy running quark masses, which does not rely on a 1/m c expansion, and the other on pole quark masses. The constraints from the six moments agree well and the size of the dominant 1/m 3 b term has been found to be compatible with theoretical estimates. The fit is largely insensitive to non-local correlators and to the spin-orbital operator.
Propagating the ranges of the OPE parameters to the determination of |V cb | reduces the theoretical uncertainty due to the 1/m 3 b corrections below 2%. Furthermore, the use of a fit changes the nature of these uncertainties and partly removes the arbitrariness arising from estimates based on parameters ranges.