Higher order corrections to inclusive semileptonic B decays

We describe the computation of the O ( α S ) corrections to the Wilson coe ﬃ cients of the kinetic and chromomagnetic operators in inclusive semileptonic B decays. We can thus evaluate the complete correction at order α S Λ 2 QCD / m 2 b to the semileptonic width and the ﬁrst two leptonic moments.


Introduction
The study of semileptonic B decays (first initiated in [1,2]) has its foundation on an Operator Product Expansion together with an heavy quark expansion. The OPE expresses the differential decay rate (and consequently the total width, leptonic and hadronic moments) as a double series in α S and Λ QCD /m b . The leading term reproduces the free quark decay, while subsequent corrections take into account interactions inside the meson and further reduce the theoretical uncertainty. Currently several corrections are known: up to O(α 2 S ) in the free decay [3,4,5], while the nonperturbative terms already calculated include O(Λ 2 QCD /m 2 b ) [2], O(Λ 3 QCD /m 3 b ) [6] and a first investigation of O((Λ QCD /m b ) 4,5 ) [7]. The terms in the heavy mass expansion are expressed as B-meson matrix elements of local operators, growing in dimension: they are customarily parametrized as μ 2 π and μ 2 G at order . Regarding the perturbative corrections to these paramenters, there has been a first investigation, about μ 2 π alone, in reference [8] and then the complete calculation for both μ 2 π and μ 2 G has been carried out in the two papers relevant to this proceeding, [9] and [10]. Quite recently the coefficient of μ 2 G for the total width has been calculated again in the limit m c → 0 [11], yield-ing results compatible with our own. The method we used to perform the calculation can be traced back to [12], where it was employed for the simpler process B → X s γ, then it has been further developed in [9] for the coefficient of μ 2 π and finally extended to the full calculation of μ 2 G in [10].

The inclusive semileptonic decay
The triple differential decay distribution for an inclusive semileptonic decay is where p t = p B −p e −p ν −p X and labels e,ν and X represent the final lepton, neutrino and hadronic state X C ; q is the transferred momentum to the leptonic couple and H w the weak Hamiltonian The first step to simplify eq. (1) is to express it as the product of an hadronic tensor W αβ and a leptonic tensor L αβ , which is possible since leptons do not interact strongly (and we work at leading order in the ew interactions): the leptonic tensor is fairly simple L αβ = 2 p α e p β ν + p α ν p β e − p e · p ν g αβ −i ρβσα p e ρ p ν σ (4) while the hadronic tensor still contains unknown matrix elements with J α L the usual left-handed weak current (described in eq. (2)). To further manipulate W αβ we can relate it to the discontinuity of a time-ordered product across a cut. So by defining (6) we obtain the relation Up to this point we are still handling tensors with two indices, in order to simplify things a bit in this regard we can express the whole tensor W αβ in terms of structure functions W i : where v is the four-velocity of theB meson: There would be two additional functions W 4 and W 5 factoring structures containing q, but since we are dealing with massless leptons, the leptonic tensor yields zero when contracted with q: and so only W 1 , W 2 and W 3 contribute in the end. Eq. (7) thus becomes

The operator product expansion
Having related W i to a product of currents (eq. (6) and eq. (10)), we can now employ an operator product expansion and express the latter as the sum of local operators: in order to do so we consider an initial b quark which is slightly off-shell (being inside aB meson), p b = m b v + k with k of the order of Λ QCD , and expand in k up to O(k 2 ). If we then use QCD perturbation theory to calculate the coefficients of the O i operators, we obtain a double series in α S and Λ QCD /m b : where n is the dimension of the operator. It has to be noted that we could have followed the same steps with the addition of an extrernal soft gluon and we would have expected a result similar to eq. (12), but possibly with different coefficients and operators. This consideration will play an important role in the determination of the coefficient of μ 2 G .

Heavy quark effective theory
Even after having found all operators in eq. (12) with the respective coefficients, we must still calculate their matrix elements betweenB meson states. This could be done in QCD, like the rest of the calculation so far, but it is convenient to switch to HQET, an effective theory specifically tailored around the idea of heavy quarks being very close to their mass shell. In practice what we do is substituting the old quark field in terms of the new one and then take advantage of all the properties that hold true inside HQET in order to simplify the result and minimize the number of operators we have to deal with. The equation of motion is of particular importance it implies that no operator appears at O(Λ QCD /m b ) which cannot be written in terms of higher dimension operators. So in the end we have one operator at dimension three, describing the decay of a free quark, which is better to express in QCD: and then two operators at dimension five, the kinetic operator and the chromomagnetic operator: As becomes apparent from eq. (17), the chromomagnetic operator contains a gluon field, so its cofficient can be calculated only from the current product emitting an external gluon. The operators are parametrized as follows: having neglected higher order power corrections and introducing dimension d = 4 − 2 .
When expressing the final amplitude what really matters, rather than the operators themselves, are their matrix elements evaluated betweenB states, so we can now rewrite eq. (12) as

One loop calculation
At tree level the use of HQET is sufficient to solve most of the problems, but carring out the calculation at one loop (contributing diagrams shown in figure 1) presents many more challenges: namely lenghtier expressions, loop integrals, and divergencies to be cancelled through renormalization. The latter deserve some deeper discussion, in order to get rid of all the poles we need to be precise when matching HQET with QCD. We are equating After having taken into account all the counterterms, both the lhs and the rhs become ultraviolet free, but still retain infrared divergencies: since QCD and HQET reproduce the same infrared behaviour, these will cancel out and so the one-loop Wilson coefficient c i, j,1 {α} (what we wanted to calculate in the first place) will be finite. The relevant counterterms are: where ξ is the Feynman gauge and bare quantities are defined as follows:

Results
The calculation we have briefly sketched has been carried out in ref. [9] and [10], respectively for the oneloop coefficient of μ 2 π and μ 2 G . The full analytic result for the structure functions W i can be found in the Appendix of each paper. Numerical result are not as lenghty and can be discussed here. Considering on-shell quark masses of m b = 4.6GeV and m c = 1.15GeV we obtain a total width of The parameter μ 2 G is renormalized at the scale μ = m b . Assuming α S = 0.25, the one-loop correction increases the μ 2 G coefficient by 7%. Then there are the first and second central leptonic moments here the relative NLO correction to the μ 2 G coefficient are bigger than in the width, amounting to +28% and +23%. All the above can also be calculated inside the kinetic scheme (see [13,14]) with cutoff μ kin = 1GeV and they yield: here μ 2 G coefficients always increase with the addition of the NLO corrections, respectively by +15%, +20% and +20%.

Conclusions
We have calculated the O(α S ) corrections to the Wilson coefficients of the kinetic and chromomagnetic operators in inclusive semileptonic B decays. The complete O(α S Λ 2 QCD /m 2 b ) contribution to the width is just a few per mill, but the corrections to the first two leptonic moments are comparable to the experimental errors. In order to estimate their effect on |V cb |, these corrections have to be included in the global fit to the moments, which will be the subject of a future publication.