Perturbative corrections to power suppressed effects in $\bar B\to X_u\ell\nu$

We compute the $O(\alpha_s)$ corrections to the Wilson coefficients of the dimension five operators in inclusive semileptonic $B$ decays in the limit of a massless final quark. Our calculation agrees with reparameterization invariance and with previous results for the total width and places constraints on the shape functions that enter those decays.


Introduction
Despite a significant experimental effort at the B factories, the current status of the determination of the CKM matrix element V ub is far from satisfactory. The magnitude of V ub is determined from semileptonic B decays without charm and in the inclusive case stringent phase-space cuts must be employed to suppress the dominant B → X c ν background. The modern description of these inclusive decays is based on a non-local Operator Product Expansion (OPE) [1,2], where nonperturbative shape functions (SFs) play the role of parton distribution functions of the b quark inside the B meson. Among the theoretical frameworks that incorporate this formalism, BLNP [3], GGOU [4], and DGE [5] are currently employed by the Heavy Flavour Averaging Group (HFLAV) [6]. The latest average values of |V ub | in these three frameworks, |V ub | BLNP = 4.44(26) × 10 −3 , |V ub | GGOU = 4.32(18) × 10 −3 , |V ub | DGE = 3.99(14) × 10 −3 , do not agree well with each other. Moreover, the values obtained from different experimental analyses are not always compatible within their stated theoretical and experimental uncertainties. The latest endpoint analysis by BaBar [7], in particular, shows a strong dependence on the model used to simulate the signal and leads to sharply different results in BLNP and GGOU. This is the most precise analysis to date; in GGOU and DGE it favours a lower |V ub | and it is therefore in better agreement with |V ub | B→π ν av = 3.70(16) × 10 −3 , (1.1) the value extracted from B → π ν data together with lattice QCD determinations of the relevant form factor [6]. It is also worth mentioning that a preliminary tagged analysis based on the full Belle data set [8] indicates a better agreement both among theoretical frameworks and with Eq. (1.1). The large statistics available at Belle II should help clarify the matter in various ways, see [9]. In particular, it should be possible to calibrate and validate the different frameworks directly on data, especially on differential distributions which are sensitive to the SFs. The SIMBA [10,11] and NNVub [12] methods both aim at a model-independent parametrisation of the relevant SFs and are well posed to analyse the future Belle II data in an efficient way.
In view of these interesting prospects, various improvements are necessary on the theoretical side, among which the inclusion of O(α 2 s ) corrections not enhanced by β 0 [13] and of O(α s /m 2 b ) effects that modify the OPE constraints on the SFs. The latter corrections have been computed at the level of form factors (and therefore of the triple differential distribution) for the inclusive decays to charm [14,15], see also [16,17], but due to the intricate interplay of soft and collinear singularities the limit of m c → 0 is far from trivial, especially since in the case at hand the infrared singularities are power-like. One possibility is to repeat the calculation setting m c = 0 from the start, but we will show instead that the m c → 0 limit can be taken in a conceptually simple manner, reproducing the expected pattern of collinear and soft-collinear singularities, as well as a few existing results.
Our method consists in systematically disentangling all singularities that emerge in the m c → 0 limit at the level of the form factors W i ; since the phase space integrals of the form factors are infrared safe, one can reorganise them in such a way to remove the mass singularities completely. In this way we obtain analytic results for both O(α s µ 2 π /m 2 b ) and O(α s µ 2 G /m 2 b ) corrections to the form factors and therefore to the triple differential distribution. Our results for the O(α s µ 2 π /m 2 b ) corrections satisfy the reparametrization invariance relations obtained in [18], while the O(α s µ 2 G /m 2 b ) corrections reproduce the shift in the total width computed at m c = 0 in Ref. [17]. We also use our results to compute the O(α s ) corrections to the q 0 -moments of the individual form factors, which place crucial constraints on the SFs.
The outline of this paper is as follows. In section 2 we introduce our notation and review the known O(α s ) corrections to the triple differential rate in the charmed case. Section 3 gives an elementary illustration of our method, taking the limit m c → 0 of the O(α s ) corrections and recovering the known results. In section 4 we apply the method to the O(α s Λ 2 /m 2 b ) corrections, with all analytic results given in the Appendix. In section 5 we check that our results for the O(α s µ 2 π /m 2 b ) satisfy the reparametrization invariance relations. Section 6 is devoted to a few applications: we compute the total decay rate, the q 2 spectrum, and the first moments of the form factors. Finally, section 7 summarises our findings.

Notation and O(α s ) corrections
We will consider the decay of a B meson of four-momentum p B = M B v into a lepton pair with momentum q and a hadronic final state with momentum p = p B − q. Let us first assume that the hadronic final state contains a charm quark with mass m c and express the b-quark decay kinematics in terms of the dimensionless quantities where p = m b v is the momentum of the b quark and the physical range is given by We will also employ the energy of the hadronic system normalized to the b mass 3) The case of tree-level kinematics corresponds toû = 0; we indicate the corresponding energy of the hadronic final state as The normalized total leptonic energy iŝ We also introduce a threshold factor In the case of tree-level kinematics, the threshold factor becomes λ 0 = 4(E 2 0 − ρ). It is convenient to introduce a short-hand notation for the square root of λ: The differential B → X ν decay rate is proportional to the product of a leptonic and a hadronic rank-2 tensors, where the hadronic tensor W µν describes all the QCD dynamics in the decay. It is customary to decompose W µν into form factors, whereq µ = q µ /m b , v µ is the four-velocity of the B meson, and the W i are functions ofq 2 andq 0 , or equivalently ofq 2 andû. In the limit of massless leptons only W 1,2,3 contribute to the decay rate and one has whereû + , defined in (2.2), represents the kinematic boundary onû, andÊ = E /m b is the normalized charged lepton energy. Thanks to the OPE, the structure functions can be expanded in series of α s and Λ QCD /m b . There is no term linear in Λ QCD /m b and therefore (2.10) where we have neglected terms of higher order in the expansion parameters. µ 2 π and µ 2 G are the B-meson matrix elements of the only gauge-invariant dimension 5 operators that can be formed from the b quark and gluon fields [19,20]. In the Standard Model the leading order coefficients are given by The tree-level nonperturbative coefficients W (π,0) i and W (G,0) i [20] are given in compact form in [14,15]. The leading perturbative corrections to the free quark decay have been computed in [21] and refs. therein. They read where S i = S + ∆ i and 13) and the functions R i are given in Eqs. (2.32-2.34) of Ref. [21]. 1 The integrals I 1 , I 1,0 , I 2,0 , and I 4,0 are given in Eqs. (A.6-8) of [14] and the plus distribution is defined by its action on a generic test function f (û): 2 (2.14)

The massless limit
We now take the limit m c → 0, i.e. ρ → 0, of the O(α s ) corrections to the form factors, W i . Of course, collinear divergences emerge in this way, leading to ln ρ and ln 2 ρ in W i , which however are compensated upon integration overû, as collinear logs arise from the phase space integration as well. As the phase space integrals of W i are infrared safe, one can therefore reorganise the expressions for W i in order to remove completely the mass singularities. In practice it is sufficient to consider the integral where f (û) is a generic test function. Let us first consider the limit for ρ → 0 of the coefficient of the δ(û), the function S given in (2.13). The integrals I k,0 admit the simple expansions and we therefore have We now consider the real emission contributions given by R i . Their structure is where r i , s i , t i are functions ofq 2 andû that are regular in the limitû, ρ → 0. Clearly, the collinear singularities atû = 0 are regulated by ρ. To expose them, let us start with the second term in (3.6) and observe that for a test function f (û) and therefore in the second term of (3.6) and in the last term in the universal part of (2.12) we can safely make the replacement and take the limit of s i for ρ → 0. This extracts one of the singularities we were looking for. Let us now turn to the first term in (3.6). In the limit ρ → 0 the coefficient of r we can therefore use the replacement and take the limit of r (2) i for ρ → 0. A linear combination of the two above replacement rules deals with the coefficient of r Let us now consider the plus distribution in (2.12), and in particular the part involving I 1 . Here the singularity is hidden in the integral I 1 and in itsû → 0 limit. They are given by where t and t 0 have been introduced in (2.7). Let us first focus on which is a function ofû, ρ, andq 2 and is non-analytic atû = ρ = 0. Indeed, expanding (3.11) for w = 1 −q 2 û, ρ we find that its leading singularity is The difference of (3.11) and (3.12) is however regular in the limit ρ → 0, and we can split (3.11) into a singular and a regular piece, Denoting by I 1 the limit of I 1 for ρ → 0, the function B is given by which has only a logarithmic (integrable) singularity inû and can be considered regular for our purposes. We can now use the definition of the plus distribution with a test function f (û) and reorganize the integral as follows: Keeping in mind that we can drop all O(ρ) terms, the first term in the last line is the sum of the integrals of the first two terms on the rhs of (3.13). We can simplify the second term by using (3.13) again, and obtain several terms, among which a logarithmic plus distribution, which signals the appearance of the soft-collinear divergence. Finally, in the last term we can use the ρ → 0 expansion of I 1,0 given in (3.2). The result is 3 Notice that the integrals of B(q 2 ,û) in the first and second term of the second line of (3.15) cancel each other.
We are now in the position to take the limit for ρ → 0 of the whole W (1) i . Collecting all terms we verify that the mass singularities cancel completely and obtain, with w = 1 −q 2 , and the functions R (1) i are given by with m c = 0 performed in Ref. [22].
The method employed in the previous section can be readily extended to take the m c → 0 limit of the O(α s Λ 2 /m 2 b ) results obtained in Refs. [14,15]. The main difference is that perturbative corrections to power suppressed effects induce power-like divergences, including collinear power divergences in the m c → 0. On the other hand, the most complicated features of these singularities are determined by the same integral I 1 that we have encountered in the previous section, as the calculations of the O(α s ) and O(α s Λ 2 /m 2 b ) corrections are based on the same building blocks (master integrals). The divergences in the corrections related to the kinetic operator and proportional to µ 2 π are stronger than in those proportional to µ 2 G . It is therefore instructive to start reviewing the structure of the O(α s µ 2 π /m 2 b ) contributions for finite charm mass: where the generalized plus distributions are defined by can be written as where p (j) i , q i , r i , s i , t i are also functions ofq 2 andû that are regular in the limitû, ρ → 0. Notice that the expressions for the R (π) i given in [14] have a different form, as they also contain powers ofû in the denominators. This is because Ref. [14] reduces the coefficients of the plus distributions by Taylor expanding them aroundû = 0, namely employs and similar identities which simplify the coefficients of the plus distributions. However, in Ref. [14] such identities have been applied for finite ρ. The non-analyticity of I 1 at ρ =û = 0 implies that the limit ρ → 0 should be taken before simplifying the coefficients of the plus distributions. We have therefore used the results of the calculation [14] before the final simplifications. Working in the same way as we did after (3.6) and using the definition (4.2) of the generalized plus distributions, we can isolate the divergences in R (π) i . For instance, let us consider where f (û) is again a generic test function. Subtraction of the divergent parts leads to where the last two integrals can be solved and expanded in ρ, while the first has no mass singularity and after setting ρ = 0 corresponds to the action of [1/(û 2 )] + on f (û). We therefore find the replacement rule and proceeding in a similar way we also find where the power divergences in ρ have become apparent. These rules together with (3.7) allow us to isolate the singularities of R (π) i in the limit of vanishing ρ. Like in the case studied in the previous section, the coefficients of the plus distributions contain the integral I 1 and one has to disentangle the collinear singularities starting from the definition of the plus distributions.
As a preliminary step in that direction let us consider the action of a third-order plusdistribution on the product of I 1 and a generic test-function f (û). It can be rearranged in the following way where I 1,1 and I 1,2 indicate the first and second derivatives of I 1 with respect toû evaluated atû = 0. If we now denote by P (n) I 1 the Taylor expansion of I 1 aroundû = 0 through order u n−1 , we see that the structures are regular atû = 0 for finite ρ and determine the form of the resulting distributions.
In analogy with what we did in Eq. (3.13) they can be expressed in terms of a divergent piece with power singularities inû and a residual finite (or integrable-divergent) function where D 1 (q 2 ,û, ρ) = ln(ρ/(û + ρ))/ûw and B 1 (q 2 ,û) = B(q 2 ,û), following the notation of Eq. (3.13). The integrals of the divergent pieces converge for ρ = 0 and can be expanded in powers of ρ. The relevant ones are given by 14) Let us now return to (4.10) and consider the last term on the rhs. We can rewrite I 1 − I 1,0 using (3.13) as because the rest of the integral is regular atû = 0. The first two terms correspond to plus distributions, and using also the ρ expansion of I 1,0 (3.2) we arrive at where the arguments (q 2 ,û) of the B i are understood. We can then expand B 1 in powers ofû, as reported in (3.14), 1û + ..., (4.19) and notice that the higher orders in theû expansion of B 2,3 have to be related to those of B 1 , see (4.12). In particular, one finds so that the second line of (4.18) becomes . Combining Eqs. (4.18) and (4.21) we see that in the massless limit I 1 [ 1 u 3 ] + can be expressed in terms of various distributions, with coefficients that contain divergences as strong as 1/ρ 2 . We recall that similar lower order plus distributions can be reduced using (for n ≥ 1)û 1 It is also worth noting that the coefficients d i , e i , f i in (4.1) contain inverse powers ofû + ρ, which may generate additional divergences. However, combining algebraic manipulations like ρ (û + ρ) 4 with (4.22), one can remove any such inverse power from the coefficients of the plus distributions.
We are finally ready to take the massless limit for all the terms in (4.1). As expected all power and logarithmic divergences in ρ cancel out in the form factors W For what concerns the O(α s ) corrections to the coefficients of the chromomagnetic matrix element, namely W (G,1) i , they can be computed from the results of Ref. [15] using the same procedure we have followed for W (π,1) i . The results are also given in the Appendix.

Reparametrization Invariance relations
Reparametrization Invariance (RI) [23,24] connects different orders in the heavy quark expansion. This in general implies relations among the coefficients of a number of operators, see e.g. [25], but we are interested only in the way RI links the coefficient of the kinetic operator to the coefficient of the leading, dimension 3 operator. In the total rate this corresponds to a rescaling factor 1−µ 2 π /2m 2 b on the leading power result, which corresponds to the relativistic dilation factor of the lifetime of a moving quark and applies at any order in perturbation theory. The relations for differential distributions have been studied by Manohar who has derived RI relations [18] directly at the level of the structure functions W i . They are valid to all orders in perturbation theory and give the coefficient of the O(α s µ 2 π /m 2 b ) corrections in terms of the O(α s ) coefficient and its derivatives: These relations have been verified in [14] for decays to charm. Here we verify them in the massless case as well. To this purpose we need the first two derivatives of the plus distributions in Eq. (3.18). They can be re-expressed in terms of the higher order plus distributions introduced in Eq. (4.2) and of delta functions:

Applications
The results for W (π,1) i and W (G,1) i in the massless case can be employed in Eq. (2.10) to compute the O(α s Λ 2 /m 2 b ) corrections to the total rate and to the moments of various differential distributions in B → X u ν. We first compute the total rate in the pole mass scheme and find where Γ 0 = G 2 F |V ub | 2 m 5 b /192π 3 is the lowest order result, and the O(α s ) contributions are a standard result, see [22]. As already discussed, the O(α s µ 2 π /m 2 b ) corrections are dictated by RI. The non-trivial O(α s µ 2 G /m 2 b ) correction to the total width is sizeable and amounts to almost a quarter of the O(µ 2 G /m 2 b ) correction, but comes with a sign opposite to the O(α s µ 2 π /m 2 b ) correction and tends to cancel it. Using α s = 0.22, m b = 4.55GeV, µ 2 π = 0.43GeV 2 and µ 2 G (m b ) = 0.35GeV 2 , the total shift induced by O(α s Λ 2 /m 2 b ) contributions amounts to -0.4%. Our result for the O(α s µ 2 G /m 2 b ) correction to the total width agrees with Ref. [17], where the O(α s µ 2 G /m 2 b ) correction to the total width and to a few q 2 moments has been computed in an expansion in m c /m b , and the limit m c → 0 can be read from the first term in the expansion.
We have also computed theq 2 distribution. It is displayed in Fig. 1, using the same inputs as above. One observes that the total correction is very small over the wholeq 2 range, except close to the endpoint, which is a region dominated by soft dynamics.
As explained in the Introduction, the rate subject to experimental cuts is determined by shape functions (SFs) that satisfy OPE constraints. Indeed, the corrections we have computed in this paper have an important effect on these constraints, which are related to theq 0 -moments of the form factors W i . In the GGOU framework of Ref. [4], a q 2dependent SF is associated to each form factor W i , which is in turn described by the convolution formula Here W pert i represents the purely perturbative part of the structure functions in the kinetic scheme, and the structure function W i depends on a hard cutoff µ =μm b ∼ 1GeV that is meant to separate perturbative and non-perturbative contributions. While the SFs F i describe all nonperturbative physics, theq 0 -moments (or equivalentlyû-moments) of (6.2) must match their OPE prediction, which can be shown to place constraints on the SFs moments, κ n F i (κ,q 2 ,μ)dκ. This matching has been performed at the tree-level in [4] but the O(α s Λ 2 /m 2 b ) calculation of this paper permits to extend it at O(α s ). In the following we compute the first threeq 0 -moments up to O(α s Λ 2 /m 2 b ) for fixedq 2 , leaving a detailed discussion of the constraints on the SFs to a future publication, which will also deal with the phenomenological consequences.
Let us consider the central moments of the power suppressed contributions only atq 2 = 0. On the other hand, we note that the physical range (6.5) becomes narrower for largerq 2 and vanishes at the maximal value,q 2 = 1. In order to include in the integration most of the nonperturbative part of the spectral function, we therefore consider a larger range. 4 This will be important for placing meaningful constraints on the SFs in the GGOU framework [4], where there is a q 2 -dependent SF associated to each form factor W i , and the J (n,i) i,X are the building blocks necessary to achieve that. The tree-level expressions J    W (X,1) i , respectively. In the Appendix we provide analytic results for J (n,1) i,π (0) and J (n,1) i,G (0). Let us also introduce In Fig. 2 we compare the moments J  3 . We observed that if we compute the zeroth moments in the physical range (6.5), the impact of O(α s Λ 2 /m 2 b ) corrections is much larger, with the exception of the smallest values ofq 2 . The reason why this does not imply large O(α s Λ 2 /m 2 b ) corrections to the total width and the q 2 spectrum has to do with the prefactors of W i in the differential width. For what concerns the higher moments, the O(α s Λ 2 /m 2 b ) corrections are generally moderate, but significant in a few cases, as a consequence of cancellations occurring at the tree level.

Summary
We have presented an analytic calculation of the O(α s ) corrections to the Wilson coefficient of the kinetic and chromomagnetic operators in inclusive semileptonic decays without charm. Our results agree with reparametrization invariance relations and with a previous result on the total width. We find small corrections to the total rate and to the q 2 spectrum, generally below 1% and more significant corrections to some of the moments of the form factors. Our results place constraints on the SFs that describe B → X u ν decays, and in particular allow for a determination of the perturbative corrections to their moments. This may prove useful in view of the higher precision expected at Belle II.
In the above expressions the coefficients of the derivatives of δ(û) have been reduced using integration by parts identities like as well as identities such as (4.4) and (4.22).
The analogous results for the coefficients of the matrix element of the chromomagnetic operator are where w