Revisiting radiative leptonic $B$ decay

In this paper, we summarize the existing methods of solving the evolution equation of the leading-twist $B$-meson LCDA. Then in the Mellin space, we derive a factorization formula with next-to-leading-logarithmic (NLL) resummation for the form factors $F_{A,V}$ in the $B \to \gamma \ell\nu$ decay at leading power in $\Lambda/m_b$. Furthermore, we investigate the power suppressed local contributions, the factorizable non-local contributions which are suppressed by $1/E_\gamma$ and $1/m_b$, and the soft contributions to the form factors. In the numerical analysis, employing the two-loop-level hard function and jet function we find that both the resummation effect and the power corrections can sizably decrease the form factors. Finally, the integrated branching ratios are also calculated which could be compared with the future experimental data.


Introduction
The radiative leptonic decay of B meson is of interest as it is the most important channel to extract the parameters of the B-meson light-cone distribution amplitudes (LCDAs) and to test the factorization theorem when the emitted photon is energetic. A precision study of this mode is also helpful in decreasing the background to the purely leptonic decay process B − → − ν which is important to determine the CKM matrix element V ub . The radiative leptonic B → γ ν decay amplitude is defined by the QCD matrix element In the rest frame of the B meson with momentum p B = m B v , it is convenient to introduce two light-cone vectors n µ andn µ with the definitions At leading order in QED and considering the constraints from Ward identity, the amplitude can be parameterized as [1,2] A where the contribution due to final-state radiation has been taken into account by the redefinition of the axial form factorF A (E γ ). At leading power in Λ/m b the QCD factorization formula for the B → γ form factors F A,V has been derived [3,4], and was confirmed under the framework of soft-collinear effective theory (SCET) [28,29]. The form factors F A,V (E γ ) can be factorized into a convolution of the hard function, jet function and B-meson LCDA. The hard function arises from the matching between heavy-to-light current in QCD and SCET I operators, and it has been calculated up to two-loop level [1]. The jet function can be obtained from the matching between SCET I and SCET II [28], and the next-tonext-to-leading-order (NNLO) correction is recently obtained [7]. The matrix elements of SCET II operators are actually the definition of B-meson LCDA. All the ingredients in the factorization formula depend on the factorization scale, and the radiative corrections will lead to large logarithmic terms which need to be resummed. For the hard function, the three-loop anomalous dimension is known [8][9][10][11], and the two-loop-level anomalous dimensions both for the B-meson LCDA and the jet function are recently calculated [7,12]. Therefore, the sufficient condition for a complete NLL resummation is ready, and it was firstly derived in [13] by performing a Laplace transformation to the B-meson LCDA.
Although the factorization formula of B → γ ν decay is well established at leading power, the power corrections are important for finite bottom-quark mass. The subleading-power corrections suppressed by a factor of O(Λ/m b ) were considered at tree level [1], where a symmetry-preserving form factor ξ(E γ ) was introduced to parameterize the non-local power corrections. The soft contribution from the endpoint region of the momentum of the light quark inside the B meson was firstly studied using dispersion relation and quark-hadron duality in [14], and the QCD correction to the soft contribution at one loop has been computed in [2], in addition to the contribution from threeparticle LCDAs. A comprehensive study on the local and non-local power suppressed contributions, the soft contribution and the higher-twist contribution to the B → γ ν decay was presented in [15]. The contribution from the hadronic structure of the photon which can be defined by the matrix elements of power-suppressed SCET operators was studied in [16,17]. Besides, based on transversemomentum-dependent factorization, the power corrections to B → γ ν decay was investigated in [18]. All these studies indicate that the power-suppressed contribution is sizable, and should not be neglected in the determination of the first inverse moment of the B-meson LCDA.
In this paper, we will make improvements from two aspects. The first one is to derive the scaleindependent leading-power factorization formula at the NLL level in the Mellin space, and the second one is to perform the phenomenological analysis after combining the NLL leading-power result with the power corrections. This paper is organized as follows: In the next section we will review the evolution of the leading-twist B-meson LCDA. In the third section we derive the scale-independent factorization formula of the B → γ form factors and discuss the power-suppressed contributions. What follows is the phenomenological analysis. Concluding discussions are presented in the last section.

The evolution of the B-meson LCDA
The B-meson LCDA is one of the most important ingredients of the QCD factorization formula for exclusive B decays. The two-particle LCDAs of the B meson in the heavy-quark effective theory (HQET) can be obtained from the coordinate-space matrix elements [19] 0|q The LCDAsΦ ± (t, z 2 ) in curly brackets can be expanded around z 2 = 0. In the limit z 2 → 0, t → τ = n · z/2, the B-meson LCDAs in the momentum space are defined through the Fourier transformation At leading power only φ + (ω) is relevant in the B → γ ν decay, and the evolution equation of φ + (ω) is the well-known Lange-Neubert equation [20]: with µ the renormalization scale. At one-loop level, the anomalous dimensions are written by with the "plus" function defined as In the position space, the evolution equation of B-meson LCDA takes the form [21] d d ln µΦ where at one-loop level No matter in the momentum space or in the position space, the evolution equation of the B-meson LCDA is integro-differential equation. It is difficult to be solved directly, one has to simplify it by an integral transformation. So far there exist the following treatments.
• Performing the Fourier transformation with respect to ln(ω/µ) (or Mellin transform ω N −1 for N = iθ) then the evolution kernel of ϕ + (θ, µ) is obtained as [22] ϕ Through the inverse Fourier transformation, we arrive at the solution to the evolution equation in the momentum space which is written by where ω < = min(ω, ω ), ω > = max(ω, ω ) and the functions V and g take the form • Performing the Mellin transformation to the evolution equation in the position space [21,23] In the Mellin space the evolution equation takes a simple form where ϑ(j) = 0 at one-loop level. The solution in the Mellin space can be obtained directlỹ • It was found that if the B-meson LCDA is transformed into the so-called "dual" space, the evolution kernel will be diagonalized [24]. The LCDA in the dual space can be obtained by which satisfies an ordinary differential equation Then it is simple to write down the solution The different method mentioned above is equivalent, and the LCDAs φ + (ω), ϕ + (θ),Φ + (t),φ + (j), ρ + (ω ) are the different expression of an identical objective. Because the momentum space and the position space are related through standard Fourier transformation, we are able to derivẽ andφ Then we haveφ At one-loop level, the most convenient method is to work in the dual space, since the Bessel function is the eigenfunction of Lange-Neubert kernel, which is confirmed in [25,26]. The Lange-Neubert kernel can be expressed as a logarithm of the generator of special conformal transformations along the light cone. When the eigenfunction of the generator is transformed to the momentum space, it is nothing but the Bessel function in Eq. (19). The two-loop-level anomalous dimension of the B-meson LCDA is firstly calculated in the coordinate space in [12], and it is simpler to be expressed in the dual space where sη + (s) = ρ + (1/s) and a = α s /(4π). This equation is also transformed into the momentum space in [7], results in the two-loop-level Lange-Neubert equation. The advantage of solving the evolution equation at two-loop level in the dual space does not hold as the two-loop evolution kernel is not diagonal in this space. On the contrary, the elegant form of the evolution equation (Eq. (16)) in the Mellin space maintains. Thus Eq. (18) is still the solution of the evolution equation up to two-loop level, with [23] ϑ(j) = aϑ (1) In a recent paper [13], an alternative approach to solve the evolution equation at two-loop level is pointed out. The essential idea of this approach is to perform Laplace transformation to the B-meson LCDA,φ whereω is a fixed reference scale, which can be used to eliminate the logarithmic moment σ 1 in the factorization formula of B → γν . Then one could derive with the definition whereγ + (1, x; α s ) starts from two-loop level and the specific expression can be seen in [13]. After the Laplace transformation, the solution to the evolution equation reads The normalization N (µ s , µ) depends on the factorization scale through where the quantity a γ , a Γ and S(µ s , µ) are given as We note that the LCDA ϕ + (j) is related toφ + (η) through

B → γ form factors
At leading power in Λ/m b , the QCD factorization formula for the B → γ form factors is written as At one-loop level, the hard function and jet function are given below [27][28][29] with r = 2E γ /m b . The result of two-loop level hard function and jet function can be found in [1,7]. As the hard function and jet function contain large logarithmic terms, it is important to perform the resummation to improve the convergence of the perturbative series. The first complete NLL resummation is given in [13], where the jet function J(L p , µ j ) ≡ J ⊥ (−p 2 , µ j ) = J ⊥ (n · pω, µ j ) with L p = ln(p 2 /µ 2 j ). Now we derive the scale independent factorization formula at NLL level in the Mellin space. The evolution of the B-meson LCDA has been known in the Mellin space, then we need to perform the Mellin transformation to the jet function, while it is not well-defined. Alternately, we follow the method in [13] to replace the first argument of J(L p , µ j ) by a derivative operator, i.e.
Taking advantage of Eq. (22), we have Employing the evolution function of hard function and B-meson LCDA, we obtain where U 1 (E γ , µ h , µ) and U 2 (E γ , µ h , µ) are evolution factor of hard function and B-meson decay constant in the HQET respectively and the specific expression can be found in the appendix of [1,2]. The parameter j depends on the factorization scale through The resummed factorization formula in the Mellin space seems more compact than Eq. (37). Now we turn to the power suppressed contributions. At leading power, F V and F A are equal due to the left-handedness of the weak interaction current and helicity-conservation of the quark-gluon interaction in the high-energy limit, while this relation might be broken by the power corrections.
In [15], the power suppressed contribution is separated into the symmetry-preserving part ξ and symmetry-breaking part ∆ξ, i.e.
When the power suppressed contribution is from region x 2 ∼ 1/Λ 2 , where x denotes the separation between the quark-photon vertex and the weak current, it is called the soft contribution. The soft contributions with higher twist B-meson LCDA are considered in [15]. While they are highly suppressed and numerically very small, thus we neglect them in this study. Then the symmetry breaking part only contains the local contribution and it can be written by The symmetry preserving part can be divided into three part, i.e. ξ(E γ ) = ξ 1 , and the explicit expressions for the first two parts are where φ 3 (ω 1 , ω 2 ) and φ 4 (ω 1 , ω 2 ) are the three particle twist-3 and twist-4 B-meson LCDA respectively. φ − t3 (ω) are the "genuine" twist-three contribution to the LCDA φ − (ω) [15]. The soft contribution with QCD corrections is obtained as with K(µ) the factor relating the QCD decay constant of B-meson to the HQET one, M 2 and s 0 are Borel mass and threshold parameter respectively. The evolution kernel U (E γ , µ h , µ h , µ) = U 1 (E γ , µ h , µ)/U 2 (E γ , µ h , µ). The effective LCDA ρ + eff (ω , µ) takes the form To obtain this function, one must generalize the photon momentum from p 2 = 0 to −p 2 = 0, then calculate the generalized hard-collinear function at this Euclidean region, perform the dispersion treatment to the convolution of hard-collinear function with B-meson LCDA and take the limit p 2 → 0. The soft contribution actually include the hadronic effect of the photon, it must overlap with the contribution of photon LCDA, which will be investigated in future work.

Phenomenological analysis
The fundamental nonperturbative inputs entering the factorization formula of B → γν decay include the two-particle and three-particle B-meson distribution amplitudes up to the twist-four accuracy, the decay constant of the B-meson and the parameters appears in the dispersion approach. In the numerical analysis we will employ the following three-parameter model for leading twist B-meson LCDA [15] φ where U (α, γ, x) is the confluent hypergeometric function of the second kind. In dual space, this model has a simpler expression where 1 F 1 (α, β, z) is the confluent hypergeometric function of the first kind. In the leading power factorization formula, only the first inverse moment and the logarithmic moments enter the factorization formula, which are defined by For the three-parameter model, the first inverse moment and the first two logarithmic moments are obtained as If the parameter α = β, the three-parameter model is nothing but the familiar exponential model [19] φ which will be set to be our default model. To estimate the error from the models, we will let α − β vary at the region −0.5 < α − β < 0.5, then we employ two models with α = 2.0, β = 1.5 and , whose determination has been discussed extensively in the context of exclusive B-meson decays (see [30][31][32][33] for more discussions). Here we will employ λ B (1 GeV) = 0.35 ± 0.05 GeV which is consistent with the calculations of the semileptonic B → π form factors with B-meson LCDAs in the framework of light-cone sum rules [30]. The figure of leading twist B-meson LCDA with three-parameter model is plotted in Fig. 1. In the factorization formula (31), a new parameterω is introduced to eliminate σ 1 , it can be determined once the parameters α, β and λ B are given. In addition, utilizing the three-parameter B-meson LCDA model, the logarithmic moments σ 2,3,4 are also determined. All the parameters are collected in Table 1.  Table 1: Numerical values of the nonperturbative parameters entering the leading twist LCDA of the B meson. Here the energy scale is µ s = 1 GeV and λ B is fixed at 0.35 GeV.
The higher-twist LCDAs must incorporate the correct low-momentum behaviour and satisfy the equations of motion. All the suggested models can be obtained as particular cases of a more general ansatz here the function f (ω) obeys the following normalization condition then one can derive the following results Take advantage of the above results, the NLP contribution with 1/E γ and 1/m b corrections can be obtained as To highlight the influence of the power suppressed contributions, we present the numerical result of the form factors with different contributions, where the photon energy E γ is fixed at 2.2 GeV, and λ B and σ n is set to be free parameters. The leading power result in the first parenthesis is borrowed from [13], and the power suppressed contributions include the symmetry breaking local term, the symmetry preserving 1/E γ , 1/m b term as well as soft contribution. It is obvious that power corrections are sizeable, and more important than the σ n terms. Therefore, the power suppressed contribution must play an important role in the determination of λ B . For a more detailed study on the subleading power corrections, we leave for a future work.
To test the effect of the large logarithm resummation, we plot the E γ dependence of the leading power form factor F A/V,LP in Fig. 2. For the leading logarithmic resummation, we employ tree-level hard function and jet function, one-loop level anomalous dimension and two-loop cusp anomalous dimension. For the NLL resummation case, we follow the convention in [13]. In [13], the contribution from NNLO hard function and jet function are also considered. Strictly speaking, to resum the logarithmic terms in the NNLO jet function, we need three-loop anomalous dimension of B-meson LCDA, which has not been obtained now. While it is not important phenomenologically as the hard-collinear scale µ j is actually close to the soft scale µ s . From Fig. 2 we can see that the NLL resummation effect significantly decreases the LL result, and the NNLO result is about 5% smaller compared with the NLL result. The form factors including both LP contribution and NLP corrections are plotted in Fig. 3, where for LP contribution we adopt the same result as in [13]. Compared with the LP result in Fig. 2, the power corrections significantly decrease the form factors, and the symmetry breaking effect from the NLP local contribution is also sizable. The uncertainties are denoted by the band in Fig. 3. To obtain the uncertainties, we considered various sources, including the decay constant f B = 0.192±0.0043 GeV, the first inverse moment λ B = 0.35±0.05 GeV, the hard scale µ h and hardcollinear scale µ j (the same with [13]), the models of leading twist B-meson LCDA in Tab inversely proportional to it. Having the form factors at hand, the differential decay width is expressed as To guarantee the reliability of our calculation, we cut the photon energy at E γ > 1.5 GeV. We integrate over differential decay width at the interval [1.5 GeV, m B /2], then multiply it by the lifetime of B meson to obtain the branching ratio Br(E γ > 1.5 GeV). If we fix λ B = 0.35 GeV, the branching ratio reads Br(B → γν ) = 0.40 +0.14 −0.24 × 10 −6 , where the uncertainties come from the same source as the form factors (except for λ B ). The dependence of branching ratio on λ B is presented in Fig. 4, where the parameter λ B varies between the interval [0.3 GeV, 0.4 GeV]. We can see that the large uncertainties prevent us form the precise determination of the parameter λ B , thus it is important to reduce the uncertainty of the parameters, especially that from B meson LCDA, and to obtain more precision prediction of the power suppressed contribution.

Summary
The radiative leptonic decay mode B → γν is interesting both theoretically and experimentally. It plays an irreplaceable role in the determination of parameters of the B-meson LCDA. The factorizationscale dependence of the B-meson LCDA is governed by the LN evolution equation, which is an integro-differential equation and not easily solved. We summarized the existing method of solving the LN evolution equation, both for the one-loop and the two-loop anomalous dimensions. Then we derive a factorization formula with NLL resummation for the form factors appears at leading power in the Mellin space, which is equivalent to the one obtained in [13] but written in a more compact form. The power corrections to the B → γν are sizeable, and many efforts have been put into investigating the NLP contributions. In this paper we include the power suppressed local contributions, the factorizable non-local contributions which are suppressed by 1/E γ and 1/m b , and the soft contributions.
In the numerical analysis, we found that the NLL-resummation effect significantly decreases the leading-power form factors, and the NNLO correction brings about an additional 5% reduction. The NLP contributions are combined with leading-power NNLO contributions, and also manifestly decrease the form factors. We also calculated the integrated branching fractions of the B → γν decay. The large uncertainty from various sources makes it hard to determine the parameters λ B and other logarithmic moments accurately. In future work, we will consider the NLP corrections more systematically in the framework of the SCET and hope that it can help to reduce the theoretical uncertainty.