The q 2 moments in inclusive semileptonic B decays

: We compute the first moments of the q 2 distribution in inclusive semileptonic B decays as functions of the lower cut on q 2 , confirming a number of results given in the literature and adding the O ( α 2 s β 0 ) BLM contributions. We then include the q 2 -moments recently measured by Belle and Belle II in a global fit to the moments. The new data are compatible with the other measurements and slightly decrease the uncertainty on the nonperturbative parameters and on | V cb | . Our updated value is | V cb | = (41 . 97 ± 0 . 48) × 10 − 3 .


Introduction
The measurement of the first few moments of the kinematic distributions in inclusive semileptonic B decays has been instrumental in determining the CKM matrix element |V cb | from these decays.For over 20 years the moments of the lepton energy and hadronic invariant mass distributions measured by CLEO, DELPHI, CDF, BaBar and Belle [1][2][3][4][5][6][7] have been employed in global fits [8][9][10][11][12][13] to extract |V cb |.The fits are based on the Operator Product Expansion (OPE) which governs these inclusive decays, and use the moments to get information on the nonperturbative parameters of the OPE, namely the matrix elements of the local operators.Thanks to a sustained effort in computing higher order effects, which culminated in the O(α 3 s ) calculation of the semileptonic width [14], the inclusive determination of |V cb | has currently a 1.2% uncertainty [12].
The moments of the lepton invariant mass (q 2 ) distribution have been recently proposed as a basis to extract |V cb | with a reduced set of higher dimensional parameters [15].Although a couple of q 2 moments had been measured by CLEO long ago, they were subject to a lower cut on the lepton energy and therefore unsuitable to the method proposed in [15], which relies on Reparametrization Invariance (RPI) [16,17] relations and requires instead a lower cut on q 2 .This has led to new and precise measurements of the first few q 2 moments by the Belle and Belle II collaborations [18,19], which were then employed in [20], together with additional information, to extract |V cb | with the RPI method of Ref. [15].The purpose of this paper is twofold: first, we want to revisit the OPE calculation of the q 2 moments, checking and extending the results that can be found in the literature; second, we want to verify the compatibility of these new measurements with previous results for leptonic and hadronic moments, and to study their impact on the global OPE fit and the determination of |V cb |.
As we plan to make use of all the available inclusive measurements, we perform our calculation of the q 2 moments in the kinetic scheme framework introduced in [21] and later adopted and developed in [10][11][12] and do not employ the RPI approach of [15].We compute the first three moments of the q 2 spectrum subject to a lower cut on q 2 , including all available higher order effects.In particular we compute power corrections up to O(1/m 3 b ) starting from the structure functions given in [22][23][24].The O(1/m 4 b ), O(1/m 5 b ), and O(1/(m 2 c m 3 b )) effects depend on a number of additional nonperturbative parameters and are available for the leptonic and hadronic moments [17,25], but would require a dedicated calculation which we postpone to a future publication.We also include the O(α s ) and O(α 2 s β 0 ) perturbative corrections starting from the triple differential distribution given in [26,27] and [26], respectively.Finally, we include all the O(α s /m 2 b ) and the O(α s ρ 3 D /m 3 b ) and O(α s ρ 3  LS /m 3 b ) contribution using Refs.[28,29] and [30] and RPI, respectively.Additional O(α s /m 3 b ) corrections might be expected from four-quark operators [31].We find perfect agreement with the results given in [14,15,30], but the O(α 2 s β 0 ) perturbative corrections to the q 2 moments are presented here for the first time.They are expected to provide the bulk of the two-loop corrections.
As they do not depend on µ 2 π , the q 2 moments can play an important role in a global fit to inclusive data.Indeed, fits based only on the leptonic and hadronic moments find a high correlation between the matrix elements µ 2 π and ρ 3 D , 0.73 in the default fit of [12].As they probe a new direction in the parameter space, the inclusion of the q 2 moments in the fit can improve the determination of the OPE parameters and decrease the uncertainty of the inclusive determination of |V cb |.Moreover, the Belle and Belle II measurements [18,19] are the first measurements of inclusive semileptonic B decays since 2009, and it is important to test their compatibility with older measurements of leptonic and hadronic moments.Indeed, the value of ρ 3 D (1 GeV) obtained in [12], 0.185±0.031GeV 3 , is in strong tension with the one extracted in [20] in a fit including only terms up to O(1/m 3 b ), 0.03 ± 0.02 GeV 3 , suggesting that the q 2 moments may be incompatible with previous measurements.Even though the fits of Refs.[12,20] employ slightly different conventions for the OPE parameters, and therefore the two above values of ρ 3 D should not be compared directly, and even though the results for |V cb | seem unaffected by this discrepancy, the situation needs to be clarified.This paper is organized as follows.In Sec. 2 we describe our calculation of the q 2 moments and compare our results with the Belle and Belle II measurements.In Sec. 3 we discuss the inclusion of these measurements in the global fit to inclusive semileptonic data performed in [12].Sec. 4 contains our conclusions.

The calculation
The kinematics of inclusive semileptonic B decays is described by three independent variables.We work in the B meson rest frame, with momentum p B = m B v, with v = (1, 0, 0, 0).We will denote with p X the momentum of the hadronic state and with q the momentum of the lepton-neutrino pair.Momentum conservation implies where p b and p h are the momenta of the b quark and the partonic inclusive final state, respectively.We choose as independent kinematical variables the dilepton invariant mass q 2 , the charged lepton energy v • p ℓ = E ℓ , and the charm off-shellness u defined as We will mostly work with the dimensionless quantities In the absence of cuts on the lepton energy, the kinematical boundaries on these variables are ) ) where q0 = (1 + q2 − ρ − û)/2.We now assume1 p b = m b v and introduce The hadronic invariant mass and the leptonic invariant mass q 2 can be re-expressed in terms of partonic quantities using (2.1) (2.9) It follows that the moments of q 2 and M 2 X can be expressed in terms of the same building blocks, namely the moments of the partonic energy and invariant mass.Their expressions are related by the replacement Λ ↔ −m b .This simple kinematic relation allows for useful checks.
Following [28] and assuming massless leptons, we write the triple differential width as where ), the W i are structure functions which encode the hadronic physics and are functions of q 2 and u, but not of E ℓ .The structure functions admit an OPE and can be written, in the on-shell scheme, as where β 0 = 11 − 2 3 n f and we have retained only the known contributions.Throughout this work the coupling constant α s ≡ α ) is evaluated at the generic scale µ s , and indicated explicitly in the terms where it makes a difference.The nonperturbative parameters µ 2 π , µ 2 G , ρ 3 D , ρ 3 LS are matrix elements of local operators in the physical B meson states (not taking the m b → ∞ limit), see [24,29].Furthermore, unless otherwise specified, the MS scale of µ 2 G , ρ 3 D and ρ 3 LS is set to m b .Our goal is to compute the q2 spectrum from Eqs. (2.10, 2.11).Integrating over the lepton energy gives where q0 is to be seen as the function of û and q2 given below (2.6).We notice that the function W 3 drops out when integrating over the whole lepton energy range.
For the integration in û we need the explicit expressions for the structure functions, which can be found in [22-24, 28, 29, 32], except for W ) contribution to the q 2 spectrum is given in [30] without recourse to the structure functions.
We write the result of integrating (2.12) in û as where S (p,n) are functions of q2 with p labelling the term in the heavy mass expansion and n the perturbative order in α s .Whenever the W (p,n) i are available, they can be computed from (q 2 , û) .
(2.14) The results are conveniently expressed in terms of The leading order result is in agreement with [15,30].The O(1/m 2 b ) power corrections at leading order in α s are S (π,0) (q 2 ) = −S (0,0) (q 2 ) , (2.17) in agreement with RPI and [15,30], respectively.We compute the leading order expressions for the ρ 3 LS and ρ 3 D terms from W i (q 2 , û) = Im[T i ]/(2π) of [24], or equivalently from the results of [32] . The results for the q 2 spectrum are where S (D,0) agrees with −C (0) ρ D /(24π 2 ) from [30].Unlike S (0,0) and S (G,0) , S (D,0) has a non-integrable singularity at the endpoint q2 which originates in the kinematic square root in Eq. (2.12).The derivatives of this nonanalytic term emerge from the integration of the δ (n) that appear in the W i and are more and more singular at the endpoint for increasing n.Notice that the singularity appears only in the q 2 spectrum, and not in the total rate or in the moments of the q 2 spectrum, which can be computed from the double differential distribution (2.12) using a different order of integration.It is therefore possible to introduce a regularisation.Ref. [30] employs dimensional regularisation, but other choices would lead to equivalent results.The regularised spectrum is then obtained by replacing the singular term with a plus distribution and a Dirac δ: with ξ = q2 /q 2 max .Here the plus distribution is defined in the usual way, with f (ξ) an arbitrary test function.
Let us now turn to the O(α s ) contributions to (2.13).Starting from the O(α s ) contributions to the form factors W i given in [26,28], we find excellent numerical agreement with [30], namely where C NLO,F 0 is given in Eq. ( 50) of [30].For the O(α s µ2 π ) term we have verified numerically that the formulas for W (π,1) i in [28] lead to the relation S (π,1) (q 2 ) = −S (0,1) (q 2 ) . (2.24) We proceed in the same way for the O(α s µ 2 G ) term using W (G,1) i from [29].However, the non-perturbative parameters are defined differently in [30] and [29], as the former includes the perturbative Wilson coefficient of the chromomagnetic operator in the definition of µ 2 G , where Taking this into account we have verified that where µ G are given in [30].Finally, the O(α s ρ3 D ) terms have only recently been computed in [30] and we employ their results where we have again taken into account that 2 ρ 3 D (µ) with the Darwin operator Wilson coefficient in the HQET Lagrangian given by As previously stated the O(α s ρ 3 LS ) corrections in the on-shell scheme can be inferred from reparametrization invariance 3 [31] S (LS,1) (q 2 ) = −S (G,1) (q 2 ) . (2.31) In Figure 1 we plot the q2 spectrum as a function of q 2 , normalized to the standard prefactor Γ 0 .The spectrum is shown up to q 2 = 12.2 GeV and one can see the rise of the endpoint singularity related to the O(ρ 3 D ) contribution at q 2 max = (m b −m c ) 2 = 12.27 GeV 2 .We note that the perturbative corrections to the spectrum are sizeable, but they largely cancel in the normalised moments, where the power corrections will be more important as they significantly change the spectrum at high q 2 .)) respectively in black, blue and green.We used the inputs

The moments
We define the q 2 moments as and the normalized moments as . (2.33) In the above ratios we always re-expand the moments in α s and 1/m b .We study the first three central moments, for different values of q 2 cut .We start with the results in the on-shell scheme.After expanding in α s and in 1/m b and neglecting higher order terms, each moment Q i can be expressed as where the µ 2 π terms have dropped out.Again, the renormalization scale of µ 2 G , ρ 3 D and ρ 3  LS is set equal to m b .Table 1 presents the results of the various contributions in (2.35)    1: Values of the first three central moments of the q 2 spectrum in the on-shell scheme for two values of q 2 cut .All quantities are in GeV to the appropriate power.
for two representative values of q 2 cut , using m b = 4.8 GeV and ρ = 0.073.The leading power perturbative corrections are invariably small, while the power corrections are sizeable, especially for the higher moments.

BLM corrections
The O(α 2 s β 0 ) corrections to the hadronic structure functions and therefore to the triple differential distribution have been computed in Ref. [26], which gives explicit numerical results for the leptonic and hadronic form factors, as well as a FORTRAN code implementing the calculation.This code performs a multidimensional numerical integration over the gluon mass, the variable u, two Feynman parameters and q 2 , and can compute directly the BLM corrections to the q 2 moments.However, due to very large cancellations between the different terms, reaching a good precision in the correction to Q(2,0) i is a daunting task, especially for i = 2, 3 (the same problem affects the lepton energy moments).We have therefore first computed with high precision S (0,2) (q 2 ), namely the BLM correction to the spectrum (2.13), from which we then compute the corrections to the moments at different values of q 2 cut by a simple one dimensional integration.This procedure allows us to reach a much higher numerical accuracy, especially for Q(2,0) is obtained by a mix of analytical and numerical integrations at 60 values of q 2 in the whole allowed range.The set of points is then interpolated with a cubic polynomial of coefficients c i , reproducing S (0,2) (q 2 ) with excellent accuracy.To include the ρ dependence, we repeated this procedure for 6 different values of ρ ∈ [0.035, 0.087].The coefficients of the cubic polynomial are then turned into c i (ρ) by another cubic fit to the 6 points in ρ.This allows us to have an interpolated formula for S (0,2) (q 2 , ρ), and the quality of the interpolation is checked by comparing the moments with the direct numerical integration.In the case of Q(2,0) 2,3 the uncertainty is much smaller than the one we have combining the results of numerical integration for the linear moments M i .In the supplemental material we provide approximate formulas for Q(2,0) i for values of ρ in the above range.

Change to the kinetic scheme and results
We are now ready to translate our predictions to the kinetic scheme [33].Since we do not have the complete O(α 2 s ) contributions to the q 2 -moments, we perform the change of scheme including only the O(α 2 s β 0 ) terms, given in [34].Notice that the three-loop relations between on-shell and kinetic scheme parameters have been computed in [35].Denoting by m b (µ k ) the kinetic mass at the cutoff scale µ k , the on-shell b mass is given by its µ k → 0 limit, and similarly for µ 2 π and ρ 3 D .The relevant relations are with where as stated above we neglect the term proportional to C A in [ Λ(µ k )] pert , since it is not enhanced by β 0 .We will choose as our default value For what concerns the charm mass, we adopt the MS scheme at a scale µ c , choosing as a reference µ c = 2 GeV.The well-known relation between the pole and the MS charm mass [36] is Before we present numerical results in the kinetic scheme we recall that since the kinetic scheme employs the hard cutoff µ k in the b quark reference frame, it breaks Lorentz invariance and RPI. 4 For instance, in the kinetic scheme the two terms in the combination that multiplies ) corrections due to the different power of m b in the denominator.Therefore in the kinetic scheme we write the analogue of (2.35) 4 After a change of scheme for m b and the HQE parameters, RPI still implies definite relations between the Wilson coefficients of the HQE parameters.However, these relations change order by order, unlike in the native HQET (on-shell) derivation; this is what we mean by breaking RPI.However, if one employs RPI to reduce the number of HQE parameters as in Refs.[15,20], the perturbative scheme must preserve the RPI identities otherwise the reduction of parameters may be violated by perturbative corrections.as where . Table 2 reports the coefficients of this expansion for q 2 cut = 2 and 8 GeV 2 using µ k = 1 GeV and the central values m b (1 GeV) = 4.573 GeV and m c (2 GeV) = 1.092GeV from the default fit of [12].
In Figure 2 we show our results for the first three central q 2 -moments as functions of the minimum value of q 2 , q 2 cut .For the quark masses and OPE parameters we have employed the central values and scale settings of the default fit in [12].The results are compared with the Belle and Belle II measurements.We observe that the perturbative and power corrections to the first moment are small, except for very high q 2 cut .On the other hand, the power corrections to the second and third central moments are quite sizeable, as expected.We also observe that our calculation of the variance Q 2 , which is a positive definite quantity, becomes negative for values of q 2 cut slightly larger than 8 GeV 2 .This is suggestive of relevant higher order effects in the high q 2 tail that are not included in our calculation.In our analysis of the next section we will not consider experimental data with q 2 cut > 8 GeV 2 .In Fig. 2 we observe a good agreement between our predictions for the first q 2 -moment and the Belle and Belle II measurements.However, the Belle measurements of Q 2 and Q 3 are consistently above our predictions.In order to understand whether there is a serious discrepancy, we show in Figure 3 our predictions together with our estimate of the theory uncertainty and with the parametric uncertainty from the default fit of [12].To estimate the theory uncertainty we employ the same method as [10,12], combining in quadrature  the variations in Q i obtained varying the parameters by fixed amounts.The resulting uncertainties tend to be relatively conservative and improve significantly the agreement with the Belle and Belle II data.We note that the predictions of Q 2,3 are particularly sensitive to ρ 3 D , and we can definitely expect the experimental data for Q 2,3 to have an important impact in the determination of ρ 3 D through a global fit.While we will investigate this in the next section, we can already determine the size of ρ 3 D preferred by the central values of the Belle and Belle II Q 2,3 measurements, assuming all other inputs are unchanged.In the case q 2 cut = 6 GeV 2 , the Belle and Belle II central values of Q 3 prefer ρ 3 D ≈ 0.12 GeV 3 and 0.19 GeV 3 , respectively, with an experimental uncertainty of around 0.03 GeV 3 .Different values of q 2 cut lead to roughly similar results, with lower values of ρ 3 D preferred (with larger experimental uncertainty) at lower q 2 cut .Similarly, for q 2 cut = 6 GeV 2 , the Belle and Belle II central values of Q 2 prefer ρ 3 D ≈ 0.11 GeV 3 and 0.16 GeV 3 , respectively, with an experimental uncertainty between 0.020 and 0.025 GeV 3 .In summary, even considering the theory uncertainty of our predictions, the Belle data for Q 2,3 appear in tension with the results of the fit of [12], but they are also in tension with the Belle II results: for instance Q 3 measured at q 2 cut = 6 GeV 2 by Belle and Belle II is 0.18(35) GeV 6 and 1.16(38) GeV 6 , respectively (a ∼ 2σ tension).It is also worth mentioning that even the low range of ρ 3 D favoured by the Belle q 2 -moments data is quite far from the results of the fit without higher power corrections in [20].
The above considerations on Q 2,3 depend significantly on the inclusion of the BLM corrections in our predictions.Indeed, we see in Fig. 2 that they shift Q 2,3 up by an amount similar to the difference between then Belle and Belle II results.This is directly related to the large BLM contribution to the perturbative definition of ρ 3 D (µ), cfr.Eq. (2.36).We also observe a sizeable residual dependence on the scale µ s at which α s is evaluated (in Figs. 2 and 3 we employ µ s = m b /2), which is however within our theory uncertainty estimate.
The O(α 2 s ) non-BLM corrections to the q 2 moments are only available for q 2 cut = 0 [37].Their size, relative to the BLM ones, depends on the scheme and the scales employed for the masses.It is useful to recall the case of the leptonic and hadronic moments: when both the b and c masses are in the on-shell or kinetic scheme the two-loop non-BLM corrections to the leptonic and hadronic moments are dominated by the BLM ones [38].However, the non-BLM corrections become larger when m c is expressed in the MS scheme, especially for µ c = 3 GeV.This is due to the large mass anomalous dimension, which enhances non-BLM logarithms in the calculation.As a consequence, the perturbative series tend to converge more slowly at µ c = 3 GeV and one has larger non-BLM contributions.This is the reason why we prefer a lower scale µ c ∼ 2 GeV, for which the BLM corrections still provide the bulk of the complete O(α 2 s ) corrections to leptonic and hadronic moments [38].Comparing our BLM calculation of the q 2 moments at q 2 cut = 0 with the calculation of Ref. [37] we find that in the on-shell scheme the complete O(α 2 s ) corrections to the first two central q 2 moments agree with the BLM result within 30% or better.The non-BLM correction to the third central moment are 45% of the BLM ones.In the case of the kinetic scheme with MS c quark mass, Ref. [37] provides results only for µ c = 3 GeV, Eq. ( 33), in which case the non-BLM corrections are in general larger than the BLM ones.We have also compared the results given in Eq. (33) at O(α 2 s ) with our BLM calculation using µ c = 2 GeV and µ s = m b /2 (the default values used in the next Section) and find good numerical agreement and deviations between 20% and 47% of the BLM corrections.This suggests both the usefulness of the BLM computation (for small µ c ) and the need for a complete O(α 2 s ) calculation with generic cuts on q 2 .

A global fit
In this section we present fits to the semileptonic moments that include the q 2 moments measured by Belle and Belle II.As a first step, we repeat the fit to hadronic and leptonic moments of Ref. [12] using exactly the same default inputs and settings, but including the new data.This will allow us to evaluate the impact of the new data on the global fit.Later on we will update some of the inputs, introduce a few improvements, and provide our final results.

Impact of the q 2 moments
In order to proceed with the first step, we need to briefly review the strategy for the correlation among theoretical uncertainties adopted in [12] and generalise it to the case of the q 2 moments.Indeed, the treatment of central q 2 -moments data measured with different lower cuts on q 2 presents problems analogous to those discussed in [10] for the hadronic and leptonic moments measured with different lower cuts on the lepton energy.The extent to which the theoretical uncertainties of different observables are correlated is a subtle issue.In general, such correlations are neglected but there are cases where this is unreasonable.For instance, linear moments of different orders are certainly highly correlated, and the same holds for moments measured at nearby values of q 2 cut .In the default approach employed in [12] (option D of [10]) the theoretical uncertainties to different central moments are considered uncorrelated and the correlation between the same moment at different cuts on the lepton energy are modeled in such a way that the correlation is highest for adjacent cuts and low for distant cuts, and is lower when the cuts get closer to the endpoint, where one expects higher order corrections to be more important.To extend this approach to the q 2 -moments, we introduce a factor which represents the correlation between moments measured at q 2 cuts differing by 0.5 GeV 2 .The correlation between moments measured at q 2 cuts further away is given by the product of ξ computed at all intermediate values of q 2 cut spaced by 0.5 GeV 2 .The parameters q2 and ∆ q are chosen in such a way that the correlation between far-away cuts and between nearby cuts close to the endpoint becomes small.Our default values are q2 = 9 GeV 2 and ∆ q = 1.4 GeV 2 and we will discuss later the effect of changing these values.As an illustration, the correlation between the theoretical errors of a generic moment with cuts at 2 GeV 2 and 6 GeV 2 is given by 7 k=0 ξ(2.25 GeV 2 + 0.5k) = 0.855, while for cuts at 2 GeV 2 and 8 GeV 2 the correlation decreases to 0.49.
We compare the results of the default fit of [12] with fits including the Belle and Belle II data in Table 3.Because of the large number of highly correlated data points and in analogy with the leptonic and hadronic moments, we make a selection of the q 2 -moments data: from the Belle II dataset we choose the first three moments at five values of the lower cut q 2 cut = {1.5, 3.0, 4.5, 6.0, 7.5} GeV 2 .Similarly, in the Belle dataset we select the moments with q 2 cut = {3.0,4.5, 6.0, 7.5} GeV 2 .We have checked that the fits are very stable with respect to the choice of the subset of cuts to be included.We use the correlations between Belle and Belle II data that were employed in [20]. 5We see in Table 3 that there is excellent agreement among the various fits, with a small downward shift of µ 2 π and ρ 3 D (and consequently of V cb ) with respect to the results of [12].The uncertainty on ρ 3 D is reduced significantly, but this reflects in only a small reduction of the final uncertainty on |V cb | from 5.1×10 −4 to 4.8×10 −4 .This is mostly due to the relevance of the theoretical uncertainties.The analogue of Fig. 3 with the parameters resulting from the fit including Belle and Belle and µ c = 2 GeV.All parameters are in GeV at the appropriate power and all, except m c , in the kinetic scheme at µ k = 1 GeV.The first row shows the central values and the second row the uncertainties.The first case corresponds to the default fit of [12].II data is presented in Fig. 4. We observe a clear reduction of the parametric uncertainty, mostly due to the improved determination of ρ 3 D .We have performed a number of other fits, changing the scales and selecting different subsets of data.In particular, we study the dependence on the model of theoretical correlations by varying ∆ q in between 0.7 and 3 GeV 2 .The results of the global fits including both Belle and Belle II data are shown in Fig. 5: they depend very little on the choice for ∆ q .As can be seen from (3.1) the value of q2 controls the region in q 2 cut where the correlation between adjacent measurements starts to decrease because of fast growing higher order effects.Values of q2 lower than 9 GeV 2 would lead to ξ(q 2 cut ) similar to those obtained with large ∆ q , while values of q2 higher than 9 GeV 2 appear unjustified.
The results of fits with various subsets of data are shown in Fig. 6.The fits with only hadronic moments and only q 2 -moments also include the measurements of the branching fraction at different values of the cut on E ℓ in order to determine |V cb |.However, even including the branching fraction measurements, the only q 2 -moments fit is unable to meaningfully constrain µ 2 π , as these moments are insensitive to this parameter.As a consequence, the result for |V cb | also suffers from a large uncertainty, as can be seen in Fig. 6.We also show the results of fits performed setting µ s = m b and µ c = 3 GeV.While there are differences, especially in the values of µ 2 π and ρ 3 D , all fits are consistent within uncertainties.In order to test the importance of the inclusion of data measured with different values of the cuts on E ℓ and q 2 , we also perform a fit using only a single cut for each leptonic, hadronic or q 2 -moment.We select q 2 cut = 5.5 GeV 2 , and the cut on E ℓ closest to 1 GeV, depending on the experiment (we exclude Delphi's results, obtained without a cut on E ℓ ), and use the Belle and BaBar measurements of the branching fraction at the lowest value E cut = 0.6 GeV.This fit does not depend at all on the modelling of the theoretical correlations in the q 2 -moments discussed above and only minimally on the modelling of the correlations in leptonic and hadronic moments.The results are perfectly compatible with those in Table 3 and only slightly less precise: the final result is |V cb | = 41.80(53) 10 −3 .We conclude that, with the inclusion of q 2 -moments, using moments at different cuts adds little to the global fit.

Update of the semileptonic fit
Thus far we have used the same inputs adopted in the default analysis of [12].In order to provide our final results we update the lattice QCD constraints on the b and c quark masses using the latest FLAG review [39].The new FLAG N f = 2 + 1 + 1 heavy quark where we have indicated the number of active quark flavours, which has to be taken into account in the conversion to the kinetic scheme.Converting m  (11) GeV and then using the three loop results of [14,35] (scheme B) we obtain the kinetic mass of the b quark m b (1 GeV) = 4.562 (18) GeV . (3.3) Concerning the charm mass, we observe that the latest FLAG average has a larger uncertainty than in 2021, due to tensions between different determinations.Our default input is m c (2 GeV) = 1.094 (11) GeV, obtained evolving m c in (3.2) from 3 to 2 GeV.For α s (M Z ) we use the PDG value 0.1179(9) [40] and we keep the same constraints µ 2 G (m b ) = 0.35(7)GeV 2 and ρ 3 LS = −0.15(10)GeV 3 employed in [12].The QED corrections to the leptonic moments have been recently computed in Ref. [41], where small but non-negligible differences have been found with respect to the BaBar estimate based on PHOTOS.We have investigated the importance of these differences in the context of the global fit.Let us illustrate our procedure with the example of the branching fraction measured for E ℓ > E cut , R(E cut ).BaBar has measured [1,7] a photon inclusive branching fraction, R incl (E cut ) and estimated the leading logarithmic soft-photon QED contribution ∆R(E cut ) using PHOTOS [42].The QED-subtracted branching ratio that we want to compare with our QCD-only theoretical predictions is therefore The QED contribution ∆R(E cut ) have been computed in [41] including the complete virtual contributions, electroweak effects, and leading logarithmic power suppressed terms.In order to employ this new calculation in place of the PHOTOS estimate we write where ∆R new is the result of the calculation of [41] and R new QCD is a more precise value for the QED-subtracted branching fraction.Hence we can express the correction in the form of a multiplicative factor ζ QED From Ref. [41] we obtain the values which we use to correct BaBar results for the branching fraction.The first number is particularly important because the branching fraction at the lowest E cut drives the determination of the inclusive semileptonic branching fraction in the fits and because its sizeable −0.8% shift may affect the |V cb | determination in a visible way.We do not change the uncertainties and correlations given by BaBar.We have proceeded in the same way with the leptonic moments measured by BaBar, and found that the changes in the fit are minimal.The Belle measurement of the leptonic moments [6] subtracts the QED effects in a way similar to what done by BaBar, but their paper does not provide the size of the subtraction.We are therefore unable to improve the QED treatment on the Belle data.Overall, the modified treatment of QED corrections leads to a −0.23% change in |V cb |.
Finally, we briefly comment on the calculation of O(α 3 s ) contributions to the semileptonic moments at zero cuts performed in [37].We have checked that the O(α 3 s ) contributions in the kinetic scheme given in [37] are generally well within our estimate of the theoretical uncertainty.The only exception appears to be the third hadronic central moment, where our ∼ 15% uncertainty falls short of an O(α 3 s ) contribution exceeding 25%.We therefore increase the theoretical uncertainty of the third hadronic moments for the values of E cut where it is lower than 30%.This affects mostly the third hadronic moment measured by Delphi [4], which has an experimental uncertainty of about 20% and favours a low ρ 3 D , and results in an increase of ∼ 0.008 GeV 3 of the central value of ρ 3 D in the fit.Our final results are summarised in Table 4, where we present a global fit to hadronic, leptonic and q 2 -moments that employs the updated heavy quark masses, an enlarged theory uncertainty for the third hadronic moment, and includes, for the BaBar measurements, the QED effects computed in [41].The changes with respect to the global fit (last row) of Table 3 are minor and mostly concern the determination of the branching fraction and a −0.1% shift of |V cb |.In Fig. 7 we show the regions of ∆χ 2 < 1 in the 2D planes (µ 2 π , ρ 3 D ) and (ρ 3 D , |V cb |), for the sets of data B-F of Fig. 6 after the various updates discussed in this section.

Summary and outlook
The recent measurements of the q 2 -moments by Belle and Belle II [18,19] has opened new opportunities for the study of inclusive semileptonic B decays.In this paper we have presented the results of a new calculation of the moments of the q 2 spectrum in inclusive semileptonic B decays that includes contributions up to O(α 2 s β 0 ) and O(α s Λ 3 QCD /m 3 b ).In particular, we have reproduced many of the results presented in Refs.[15,30] and computed for the first time the BLM corrections O(α 2 s β 0 ) to the q 2 -moments.If we employ the results of the default fit of [12] as inputs, our predictions for the central moments of the q 2 spectrum are in excellent agreement with Belle II data [19], while there is a mild tension with Belle data [18] in the case of the second and third central moments.As a matter of fact, the Belle and Belle II for those moments differ by about 2σ.
The inclusion of the q 2 -moments in the global fit confirms the above picture.The q 2moments lower slightly the value of ρ 3 D (m b ) by half a σ and that of |V cb | by a fraction of a σ, decreasing the final uncertainty on them from 0.031 to 0.018 GeV 3 and from 0.51×10 −3 to 0.48 ×10 −3 , respectively.Because of its correlation with ρ 3 D , the determination of µ 2 π also benefit from the new data, with the uncertainty going down from 0.056 to 0.042 GeV 2 .We have also included the results of the new calculation of QED and electroweak effects on the lepton energy spectrum and moments [41].Applying them to the BaBar data only, they lower the values of the branching fraction and of |V cb | by about 0.23%.Our final result for |V cb |, obtained updating the input charm and bottom masses and increasing the uncertainty on the hadronic moments, is This is still in tension with most estimates based on the Belle and BaBar measurements of exclusive decay B → D * ℓν [43][44][45][46][47][48][49], but agrees well with the very recent Belle and Belle II results [50,51] and with analyses of B → Dℓν [52,53].Interestingly, we also find that a global fit to moments measured at a single cut on E ℓ and q 2 , which minimally depends on the correlations among theory errors, gives very similar results.This corroborates our study of the dependence on the modelling of theory correlations.Further improvements of the inclusive determination of |V cb | may come from new and more precise measurements of the leptonic and hadronic moments at Belle II, which could also measure the Forward-Backward asymmetry and related observables for the first time, bringing a new sensitivity to µ 2 G to the fits [54,55].The new measurements should be able to improve the treatment of QED corrections using the results of [41].It will be useful to investigate the higher power contributions of O(Λ 4 QCD /m 4 b , Λ 5 QCD /(m 2 c m 3 b ), Λ 5 QCD /m 5 b ) in the q 2 -moments, in analogy to what has been done in [25] for the hadronic and leptonic moments.As far as perturbative corrections are concerned, a complete O(α 2 s ) calculation of the q 2 -moments at arbitrary q 2 cut is feasible and necessary.The poor convergence of the perturbative series for the third hadronic moments observed at O(α 3 s ) in [37] should also be investigated.In the longer term, we expect lattice calculation of the inclusive semileptonic B decays [56][57][58] to validate and complement the OPE calculations.

Figure 2 :
Figure 2: Comparison of the first three central moments in the kinetic scheme between theoretical prediction and experimental data from Belle [18] (red dots) and Belle II [19] (red squares).The various curves represent calculations including all terms at leading power in m b (LP), up to O(1/m 2 b ) (NLP), up to O(1/m 3 b ) (NNLP), and up to O(α 0 s , α 1 s , α 2 s β 0 ) (LO, NLO, BLM).

Figure 3 :
Figure 3: Results for the first three central moments including the theory uncertainty bands(green) and the parametric uncertainty from the fit[12] results (blue).The combined errors are not shown.

Figure 4 :
Figure 4: Results for the central moments including the theory uncertainty bands (green) and the parametric uncertainty from the results of the fit performed in this paper (blue).The combined errors are not shown.

Figure 5 :
Figure 5: Results of global fits performed using different values of ∆ q in (3.1).In the bottom-right panel we show the dependence of ξ on q 2cut for different values of ∆ q .

Figure 6 :
Figure 6: Fit results for different data sets (A-F), different choice of µ s (G) and of the MS scale for the charm mass (H).The fit F corresponds to the last row of Table3.

Table 2 :
Table of numerical values for the expansion coefficients of the central moments in the kinetic scheme using µ k = 1 GeV, m b (1 GeV) = 4.573 GeV, and m c (2 GeV) = 1.092GeV.All quantities are in GeV to the appropriate power.

Table 3 :
Global fit results with and without the q 2 moments from Belle/Belle II for µ s = m b /2

Table 4 :
Results of the updated fit in our default scenario (µ c = 2 GeV, µ s = m b /2).All parameters are in GeV at the appropriate power and all, except m c , in the kinetic scheme at µ k = 1 GeV.The first and second rows give central values and uncertainties, the correlation matrix follows.χ 2 min = 40.4and χ 2 min /dof = 0.546.