Taming the higher power corrections in semileptonic B decays

We study the effect of dimension 7 and 8 operators on inclusive semileptonic B decays and the extraction of |Vcb|. Using moments of semileptonic B decay spectra and information based on the Lowest-Lying State saturation Approximation (LLSA) we perform a global fit of the nonperturbative parameters of the heavy quark expansion including for the first time the O(1/mb^{4,5}) contributions. Higher power corrections appear to have a very small effect on the extraction of |Vcb|, independently of the weight we attribute to the LLSA.


I. INTRODUCTION
The results of the B Factories and LHC place stringent constraints on new physics in the flavour sector. Only small deviations from the SM are allowed, and their detection represents an experimental and theoretical challenge. In the next few years a wealth of new experimental results will come from Belle-II and from the high-luminosity phase of LHC.
In this context, the precise determination of the parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix remains a high priority, as it is instrumental to constraining new physics models and to setting bounds on the scale of new effective interactions. However, the determination of the CKM element V cb , which plays a special role in tests of the CKM unitarity and in FCNC transitions, is plagued by a long-standing ∼3σ tension between the analyses based on inclusive and exclusive decays. This is unlikely to signal new physics [1] and calls for a thorough investigation of all possible sources of theoretical uncertainty.
The determination of |V cb | from inclusive semileptonic B decays is based on an Operator Product Expansion (OPE) [2][3][4][5] which allows us to parameterize all of the non-perturbative physics in terms of the expectation values of local operators in the B-meson to be extracted from experimental data. Since the contribution of higher dimensional operators is suppressed by powers of the heavy quark mass, only the operators of low dimension are expected to be relevant. Current fits of inclusive semileptonic B decays [6] use experimental data on the moments of kinematic distributions to constrain the power corrections up to 1/m 3 b terms, corresponding to dimension ≤ 6 operators, and neglect higher power corrections altogether.
While present data appear to be well described by these fits, investigations of higher power corrections are mandatory to test the convergence of the heavy quark mass expansion. Moreover, the OPE does not lead to an expansion of inclusive observables in inverse powers of m b but also contains terms of O(1/m n b 1/m k c ), with odd n ≥ 3 and even k ≥ 2, sometimes dubbed intrinsic charm (IC) contributions [7][8][9], which alter the actual power counting since . Higher power corrections have been studied in [10,11], where nine new operators of dimension 7 and eighteen new operators of dimension 8 have been identified and their Wilson coefficients computed at the tree-level. A rough estimate of the matrix elements of these 27 new operators is given by the Lowest-Lying State Approximation (LLSA) [11,12], which assumes that the lowest lying heavy meson states saturate a sum-rule for the insertion of a heavy meson state sum. The energy and is expected to be valid within 50-100% [12].
In this Letter, after briefly reviewing the structure of the 1/m 4,5 b corrections computed in [11], we study their inclusion in the fit of Ref. [6] and discuss how the results depend on the uncertainty associated to the LLSA.

II. POWER CORRECTIONS AND MATRIX ELEMENTS
Our analysis is based on the calculation of higher power corrections of [11], which is performed at leading order in α s . The inclusive observables considered below (width, moments of kinematic distributions) can be calculated by an appropriate (weighted) phase-space integral of the differential decay width where all the soft hadronic information is contained in the hadronic tensor W µν = − 1 π Im T µν . The hadronic tensor is the imaginary part of the forward matrix element of a time-ordered product of weak currents. The charm quark in this forward matrix element propagates in a background field. We expand the background field propagator S BGF , with momentum The coefficients A  [13], and at order 1/m 4,5 b [11]. At the lowest non-trivial order, corresponding to dimension 5 operators, the non-perturbative parameters are given by b ) corrections in the fit to the semileptonic moments on which the inclusive determination of |V cb | is based. We will use the LLSA ansatz, proposed in [11] and made more systematic in [12], to constrain the 27 new parameters.
The goal of LLSA is to estimate expectation values of local operators of the form where Γ is a Dirac matrix. Splitting the chain of covariant derivatives into two shorter ones labeled by A k 1 and C n k and inserting a full set of intermediate states between them one finds in the heavy quark limit [11,12] where |H n are hadronic states with the appropriate quantum numbers. The LLSA assumes that the sum of intermediate states is saturated by the lowest-lying state that can contribute, i.e. either the ground-state multiplet B, B * or the first excited states with = 1. Indeed, the matrix elements involving time derivatives like B|biD j iD k 0 iD l b|B are saturated by P -wave intermediate states, with parity opposite to that of the ground state. Including these states in the sum leads to extra powers of the P -wave excitation energy, = M P − M B . While there exist separate contributions coming from the spin 1 2 , 3 2 light degrees of freedom, we In the following we use the notation of [11], according to which the nine matrix elements involved coincide with those identified in [12], even though different notations are adopted.
It is useful to redefine the 1/m 4 b parameters to account for combinatorial factors. In practice, we expand the (anti-)commutators and count the number of terms after expunging those which are of higher order in 1/m b due to the equations of motion. We then expect the parameters to have a natural scale of O(Λ n QCD ), with n the dimension of the corresponding operator, as is also the case for the parameters in Eq. (3). The rescaled parameters are No such redefinition is necessary for the 1/m 5 b parameters, as they were already defined in this way. The LLSA expressions for the m i , r i are reported in Table I.

III. INCLUSIVE OBSERVABLES
The OPE allows us to express sufficiently inclusive observables as a double series in α s and Λ QCD /m b . In fact, the non-perturbative corrections to the semileptonic differential rate . Perturbative corrections are known up to NNLO [14][15][16][17] and the mixed O(α s µ 2 π,G /m 2 b ) corrections [18][19][20] have also been calculated. The expansion requires knowledge of the expectation values of local operators in the B-meson. These non-pertubative parameters can be determined from measurements of the normalized moments of the lepton energy and invariant hadronic mass distributions in inclusive B → X c ν decays, M 2n where E is the lepton energy, m 2 X the invariant hadronic mass squared and E cut an experimental lower cut on the lepton energy applied by the experiments. The cut dependence of the moments provides additional information on the OPE parameters we are fitting. For moments with n > 1, it is convenient to employ central moments, computed relative to We also have information on the lepton energy cut dependence of the inclusive width, which can be studied introducing R * = Γ E >Ecut /Γ tot . The information on the non-perturbative parameters obtained from a fit to these observables enables us to then extract |V cb | from the total semileptonic width [6,[21][22][23][24].
All analyses have so far considered only the minimal set of four matrix elements which contributions have never been included, although a rough estimate of their importance has been given in [11]. From the results of that paper we have computed all the O(1/m 4,5 b ) corrections to the first three hadronic and leptonic moments and to R * ; we will now employ these expressions in the global fit to determine

IV. THE FIT
We upgrade the fit strategy introduced in [24] in the kinetic scheme, and use as a baseline the default parameters and settings most recently employed in [6]. In particular, we use the same experimental data; the full list of available measurements [25][26][27][28][29][30][31] and the leptonic energy cuts employed in the fit is given in Table 1 of Ref. [24]. We also employ the MS scheme for the charm mass and use the constraints m c (3GeV) = 0.986(13)GeV [32], µ 2 G (m b ) = 0.35(7)GeV 2 , ρ 3 LS = −0.15(10)GeV 3 . The inclusion of higher power corrections allows us to slightly decrease the theoretical errors, which are estimated using the method of Ref. [24], i.e. varying the HQE parameters by fixed amounts in the calculation of an observable. Here we use the same settings as in [6], except for the variation in ρ 3 D,LS , which we decrease from 30% to 22%, to take into account the

V. RESULTS
We report the results of the default fit in Table II. In Fig. 1 we compare the µ 2 π,G , ρ 3 and comparing it to the BR in Table II divided by τ B , we get |V cb |. The value of |V cb | is remarkably close to that obtained in [6] and the quality of the fit is very good, χ 2 /dof = 0.46, but somewhat higher than in [6].
To verify the stability of the fit with respect to the choices we made for the LLSA uncertainty, we varied this uncertainty by a multiplicative factor ξ. The results are shown in Fig. 2: |V cb | changes very little. Of course, increasing the uncertainty on the higherorder matrix elements too much is equivalent to ignoring the LLSA completely, which would be unwise. We can therefore estimate the uncertainty related to the assumptions on the LLSA error by varying ξ between 0.7 and 1.3, obtaining the relative variations on the main parameters We will include this uncertainty in the final error on |V cb |. We also vary over the range 0.4 ± 0.1GeV to gauge the related uncertainty. The dependence of the parameters on the choice of excitation energy can be seen in Fig. 3, and the resulting relative uncertainties are which are mostly negligible. After the implementation of various higher order effects the inclusive determination of V cb appears robust. Further improvements may come from the calculation of O(α s /m 3 b ) and O(α 3 s ) effects, from lattice QCD determination of some of the non-perturbative parameters, and from new [35] and more precise measurements at Belle-II.