Factorization of Radiative Leptonic Decays of $B^-$ and $D^-$ Mesons Including the Soft Photon Region

In this work, we study the radiative leptonic decays of $B^-$ and $D^-$ mesons using factorization approach. Factorization is proved to be valid explicitly at 1-loop level at any order of $O(\Lambda_{\rm QCD}\left/m_Q\right.)$. We consider the contribution in the soft photon region that $E_{\gamma} \sim \left. \Lambda^2_{\rm QCD} /\right. m_Q$. The numerical results shows that, the soft photon region is very important for both the $B$ and $D$ mesons. The branching ratios of $B\to \gamma e\nu_e$ is $5.21\times 10^{-6}$, which is about $3$ times of the result obtained by only considering the hard photon region $E_{\gamma}\sim m_Q$. And for the case of $D\to \gamma e\nu_e$, the result of the branching ratio is $1.92\times 10^{-5}$.


I. INTRODUCTION
The study of the heavy meson decays is an important subject for understanding the high energy physics and standard model [1]. The rare decays of the pseudoscalar meson also provide a sensitive probe for new physics [2]. In recent years, both experimental and theoretical studies have been improved greatly [3]. However, due to the quark confinement, one cannot probe the quarks directly in the experiments. The effect of hadronic boundstate have to be considered theoretically when treating hadronic decays. The hadronic bound-state effect is certainly highly none-perturbative in QCD. However the limitation in understanding and controlling the non-perturbative effects in QCD is so far still a problem.
Varies theoretical methods on how to deal with the non-perturbative effects have been developed. An important approach known as factorization [4,5] has been greatly developed to study the decay of the hadrons [6]. In Ref. [7][8][9], the factorization is constructed using the soft-collinear effective theory (SCET) [10]. The idea of factorization is to absorb the infrared (IR) behavior into the wave-function of hadron, so the matrix element can be written as the convolution of wave-function and a hard kernel The wave-function should be determined by non-perturbative methods.
The radiative leptonic decay of heavy mesons provides a good opportunity to study the factorization approach, where strong interaction is involved only in one hadronic external state. Many works has been done on this decay mode using factorization [9,[11][12][13]. In these studies, the heavy quark is treated in the heavy quark effect theory (HQET) [14].
In Ref. [15], the factorization of radiative leptonic decay of B meson is revisited with the next-to-leading logarithmic (NLL) contribution and O(Λ QCD /m Q ) contribution at tree level considered. In Ref. [16] factorization is proved to order O(Λ QCD / m Q ). In those works, the energy of photon in the hard region is treated where E γ ∼ m Q . One cannot obtain the valid results in the region of soft photon because O(Λ QCD /E γ ) 2 contribution is neglected.
The soft photon region is very important in the radiative decay both theoretically and numerically. Consider the tree level result for example, if the Λ QCD /E γ contribution in the decay amplitude is preserved, the branching ratios will be increased by where ∆E γ is the cut on photon energy to regulate the soft photon. Using Λ QCD = 0.2 GeV, and ∆E γ can be as small as ∆E γ ∼ 0.01 GeV, the result is possible to increase quite a lot no matter how heavy the meson is. However, this region is still absence in the previous works.
In this work, factorization of the radiative leptonic decays of heavy mesons is proved to be valid explicitly at 1-loop level at any order of O(Λ QCD /m Q ), where the soft photon region as E γ ∼ Λ 2 QCD / m Q is also considered. The numerical results show that, the soft photon region is very important for both B and D mesons. The branching ratios of B → γeν e can be increased to 5.21 × 10 −6 , which is about 3 times of the result we obtained in Ref. [16], where only hard photon region is seriously treated. For the case of D → γeν e , the result is 1.92 × 10 −5 , which is close to Ref. [16].
The remainder of the paper is organized as follows. In Sec. II, we present the factorization at tree level and discuss the kinematic of the radiative decay and the wave-function. In Sec. III, the 1-loop corrections of the wave-function are discussed. The factorization at 1-loop order is presented in Sec. IV. In Sec. V, we obtain the result in the soft photon region and hard photon region separately, and briefly discuss the resummation of the large logarithms. The numerical results are presented in Sec. VI. Finally Sec. VII is a summary.

II. TREE LEVEL RESULT AND KINEMATICS
The heavy pseudoscalar meson, B or D meson is constituted with a quark and an antiquark, where one is heavy and the other is light. To study the factorization, we consider the state of a free quark and an anti-quark at first. The wave-function of the state can be defined as Φ(k q , k Q ) = d 4 xd 4 y exp(ik q · x) exp(ik Q · y) 0|q(x) [x, y] Q(y)|qQ (3) where [x, y] denotes the Wilson line [17]. And the matrix element is defined as The prove of factorization is to prove that, the matrix element can be written as the convolution of a wave-function Φ and a hard-scattering kernel T , where T is infrared(IR) finite and independent of the external state.
We start with the matrix elements at tree level. The Feynman diagrams of the radiative leptonic decay at tree level can be shown as Fig. 1. The contribution of Fig. 1.d is suppressed (pq)P µ L Q(p Q ) l/ ε * γ i( / p γ + / p l + m l ) 2 (p γ ·p l ) P Lµν (5) where pq and p Q are the momenta of the anti-quarkq and quark Q, respectively. p γ , p l and p ν are the momenta of photon, lepton and neutrino, ε γ denotes the polarization vector of photon, and P µ L is defined as γ µ (1 − γ 5 ). Lorentz invariant definition of the wave-function in coordinate space is where S and s are spin labels ofq and Q, respectively. Then we find And then the matrix element is with Using Eqs. (7-9), we find there are a few different valid ways to define the tree level hard-scattering kernel, which fulfill the requirement of the factorization. For example, the one we used in Ref. [16] is and another valid definition is: We find those two different T (0) 's will lead to different results. Especially for the case that emitting a soft photon where E γ is small, using the definition in Eq. (10), there will be IR divergence which can not be canceled, and the factorization will fail. However, if we use the definition of Eq. (11), we find that, at 1-loop order, all IR divergence can be absorbed into the wave-function to all orders of Λ QCD / m Q . We will briefly discuss it in Sec. III. To study the decay including the soft photon region, we choose the definition in Eq. (11).
The amplitude A c leads to another hard scattering kernel T (0) c . We find The kinematic feature of the soft photon region and hard photon region is very different.
We work in the rest-frame of the meson, and we choose the frame such that the direction of the photon momentum is on the opposite z axis, so the momentum of the photon can be written as p γ = (E γ , 0, 0, −E γ ), with 0 ≤ E γ ≤ m Q /2. To consider the contribution when the photon is soft, we consider the region of the energy of photon E γ separately. In the soft photon region, we assume E γ ∼ Λ 2 QCD /m Q , while in the hard photon region, we assume E γ ∼ m Q .
When E γ ∼ m Q , only part of F (0) a is at the leading order of heavy quark expansion The others are at the order O (Λ QCD /m Q ). Gauge invariance is kept order by order. When is at the leading order, and With the help of u Q , the F can be written as is found to be at sub-leading order because the polarization vector ε dose not have the 0-component.
Different from the hard photon case, in the soft photon region, gauge invariance is not kept at the leading order explicitly. In fact, part of F (0) c becomes leading order. We find that gauge invariance is kept explicitly to the sub-leading order. So, for E γ is small, we should consider the sub-leading order contribution.

III. THE CORRECTION OF WAVE-FUNCTION AT O(α s ) ORDER
The expansion of the decay amplitude can be written as [12] At the 1-loop level, we find The 1-loop corrections of wave-function come from the QCD interaction and the Wilson-Line. The later can be written as the path-ordered exponential [12,17] [ The corrections are shown in Fig. 2.
We use Φ (1) q to represent the correction with the gluon from the Wilson Line connected to the light quark external leg. So the correction in Fig. 2.a can be written as [16] Φ (1)  where C F is (N 2 − 1)/2N = 4/3 for SU(N = 3) gauge group, i and j are the matrix index in spinner space, so Similar to Φ q , the correction in Fig. 2 and Φ (1) We use Φ Wfc to denote the correction shown in Fig. 2.c. We find and Φ (1) When the wave functions at 1-loop order convolute with the tree level hard function T 0 , the momenta in T 0 , k Q and kq will not be on the mass-shell because of the momentum of the gluon that connects with the quark line flowing into it. The δ functions in Eqs. (19), extq and Φ (1) extQ . We find that, they have the same forms as the free particle 1-loop QCD box correction and external leg corrections.

IV. 1-LOOP FACTORIZATION
The hard-scattering kernel at 1-loop, denoted as T (1) can be calculated by using the 1loop QCD corrections of free quarks F (1) , and the 1-loop corrections of the wave function For simplicity, we define For convenience, we will establish our results as Φ (0) ⊗ T (1) instead of T (1) , because the delta function in Φ (0) will just replace the transmission momentum k Q and kq in T (1) with the momentums of the quarks p Q and pq.
In the calculation, the collinear IR divergences are regulated by assigning a small mass m q to the light quark. The soft IR divergences will not appear explicitly in the calculations.
We use the dimensional regulator (DR) in D = 4 − ǫ dimension and MS scheme [18] to regulate the ultraviolet (UV) divergences, and N UV is defined as We use the renormalization scale equal to the factorization scale.
A. corrections of F respectively. For simplicity, we define q = p γ − pq.
The F and we find The corresponding correction of the wave-function to the F a , which can be written as Using the method given in Appendix A, we find So we can obtain The correction F In Ref. [12,16], the correction of wave-function corresponding to F (1)Wfc a is found to be 0.
However, using the T (0) a in Eq. (11), we find The result is given in Eq. (B7) correspondingly we find The correction corresponding to F a , which can be written as We calculate Weak vertex correction F a , and finally we get where B 0 , C i and C ij are scalar Passarino-Veltman functions(Pa-Ve function) [19] and defined as and C 1 , C 2 and C ij are defined in C µ and C µν with with C µ (r 2 10 , r 2 12 , r 2 20 ; m 2 0 , m 2 1 , m 2 2 ) = In the expression above, the function C's are short for And a 1 in Eq. (38) is defined as: There is no IR Div caused by gluon in those integrals. For convenience, we will give the result of C 0 , C ij , B 0 and a 1 up to the sub-leading order after expansion in the next section.
When taking the renormalization scale equal to the factorization scale, the corrections of the external legs and the box diagram are equal to the relevant corrections to the wavefunction, then we can obtain are shown in Fig. 4 and Fig. 2. The respectively. For simplicity, we define Q = p Q − p γ .
There are collinear IR divergences in F b , which can be written as is given in Eq. (A33). We find the collinear IR divergences are canceled to all orders, and the result is where the C i 's and C ij 's are short for the Pa-Ve functions C i (0, x − y + w − z, x; 0, 0, x) and is larger then leading order. This behavior is new for the soft photon case. We will discuss it in the next section.
The corresponding correction of the wave function is Φ Then the kernel can be obtained by using Eq. (25).
The corresponding correction of wave function is The result of the momentum-integration is where the C 0 and C µ are short for Pa-Ve functions C 0 (x − y, 0, x; 0, x, x) and C µ (x − y, 0, x; 0, x, x), and and The kernel can be obtained by using Eq. (25).
The case of T a is similar, we find After calculating Φ c , the hard kernels we get are similar to Eqs. (44) and Eq. (59). The results are And because the momentum of quarks appear together as so we find is not only free from the infrared divergence, but also free from the hard dynamics corresponding to external state. Although there may contain some hard dynamics in the wave function, the wave function is process-independent, i.e., it is universal. In treating meson decays, the wave function of meson should be obtained from nonperturbative method.
Also, we didn't assume the energy of photon E γ to be large. So we can extend the valid region of factorization in Refs. [12,16] to E γ ∼ Λ 2 QCD / m Q . In such region, we can investigate the soft photon IR Div in radiative decay.
In the next section, we will investigate both the hard photon region and the soft photon region, and the result will be established up to the sub-leading order.

V. EXPANSION AND LARGE LOGARITHMS
There are large logarithms and double large logarithms in Φ (0) ⊗ T (1) . Those large logarithms may break the α s perturbation. To obtain the reliable result of the decay amplitude, those large logarithms should be resummed [20]. We concentrate on these large logarithms at the leading order of Λ QCD /m Q , because those higher order terms are not large and will not affect the α s expansion.
We concentrate on the contribution of the hard scattering kernel to the amplitude. The amplitude can be obtained by replacing the wave-function with the one obtained in Ref. [21].
The matrix element can be written as is the notation for Dirac spinner of the meson. The functions T (0) and T (1) are coefficients which are scalars in spinner space multiplied by some simple Dirac structures. So it is natural to write them in the form of products of coefficients and operators. After convolution, F µ can be written as with p q = ( m 2 q + k 2 , − k) and p Q = ( m 2 Q + k 2 , k) denote the on-shell momenta of the light anti-quark and the heavy quark in the bound state.
After the convolution, the Dirac structures can be simplified. The matrix element can be decomposed as [12,22] To calculate the form factors F A and F V in Eq. (67), we introduce 6 operators Then we can concentrate on those C n O n terms. The large logarithms are related to the power counting and expansion, and the power counting and expansion are different in soft and hard photon region, so we discuss them separately.

A. the hard photon region
When E γ ∼ m Q , keeping only the leading order and the sub-leading order terms, we find (70) where d 1 is defined in Eq. (53). Except for C 6 , the other C j i 's denote the coefficients for O i 's, and the products are at O(Λ QCD /m Q ) j order. It is now clear that the coefficient with large logarithm at order O(1) is C 0 1 . And the counter term of C 0 1 at the leading order is which leads to the group function (the anomalous dimension of the operator O 1 at the order The effective field theory (EFT) is often used to resume the large logarithms [10,12]. In general, the coefficient can be written as with H known as the hard-function, and J known as the jet function. Inspired by the idea of resummation using an EFT, we find we can separate the coefficient as a product of coefficients at different scale, as with H(µ) is the hard-function and J(µ) = J 1 (µ) × J 2 (µ) is the jet function. We find the with and We do not need the explicit form of H(µ) and J 1 (µ). We can assume H(µ) is at scale m Q , Compare them with C 0 1 , we find J 2 can be defined as Then the resummed C 0 1 can be written as where C 0 1r denotes the resummed C 0 1 . There is no longer large logarithms in J 2 (µ) when which means that, if the resummed result of the coefficient C 0 1r is expanded to the leading order at scale µ = √ z, it will go back to the coefficient C 0 1 as expected. We also notice that, when E γ becomes smaller, for example, when E γ ∼ m Q Λ QCD , C 0 1r (µ) can still correctly resume all the large logarithms in the leading order in C 0 1 (µ) because all the remaining log (x/y) terms will go to sub-leading order.

B. the soft photon region
In soft photon region that E γ ∼ Λ 2 QCD /m Q , we find Then we concentrate on the resummation of the large logarithms in C 0 2 . The anomalous dimension for O 2 at leading order is found to be Similar as the hard photon case, we assume and the scales are Similar to Eq. (82), we separate the anomalous dimension as The RGE evaluation of h(µ) is found to be and the result of j 1 (µ) is and j 2 can be obtained as where one can see that the scale of j 2 is at xz/y ∼ m Q Λ QCD . At this scale, all the logarithm terms are small.
and then evaluate h(µ) and j 1 (µ) to xz/y , considering √ w ≥ xz/y ∼ m Q Λ QCD , we The remaining γ j 2 can reproduce the log (µ 2 y/xz) term in j 2 (µ). Notice that, the scale , which implies that the relevant scale is at We will use this scale in the numerical calculation.

VI. NUMERICAL RESULTS
The contribution of Fig. 1.c depends on not only E γ but also on p ν and p l . For simplicity, we treat this term separately, then the form factors are defined as When E γ ∼ m Q , up to sub-leading order, we find When E γ ∼ Λ 2 QCD /m Q , up to sub-leading order, we find In the numerical calculation, the values of parameters we take are [21] m Ref. [24], the form factors are fitted as The fitted result of the tree level form factors is shown in Fig. 8, which can be written as In the calculation, we treat the 3-momentum of the light-quark k as small quantity, i.e. The result that F A will go from positive to negative when E γ become small can also been found in Eq. (2.9) of Ref. [15], because the signs of the 1/E 2 γ term and 1/E γ terms are To calculate the decay width, we need the result of |F A |, |F B | and Re(F A ). The fit of the form factors of B meson at one-loop level is shown in Fig. 9, and the fitted results can be written as On the other hand, F c can be related to the decay constant [21,24] by Using the fitted results of F A , F V and F c , and using the Cabibbo -Kobayashi -Maskawa (CKM) matrix elements [25] and the decay constant f B [21] as following we can obtain the branching ratios.
There are IR divergences in the radiative leptonic decays in the case that the photon is soft or the photon is collinear with the emitted lepton. Theoretically this IR divergences can be canceled by adding the decay rate of the radiative leptonic decay with the pure leptonic decay rate with one-loop virtual photon correction [26]. The radiative leptonic decay can not be distinguished from the pure leptonic decay in experiment when the photon energy is smaller than the experimental resolution to the photon energy. So the decay rate of the radiative leptonic decay depend on the experimental resolution to the photon energy which is denoted by ∆E γ .
Using ∆E γ = 10MeV [27], we obtain the branch ratios of B → γeν e . We also calculate The dependence of the branching ratios at 1-loop level on the resolution of photon energy is shown in Fig. 10 and Table I. The search for the radiative leptonic decay B + → ℓ + ν ℓ γ was performed by BABAR Collaboration in 2009 [28] and Belle Collaboration in 2015 [29].
The upper limit for the branching ratio of the radiative leptonic decay was presented. The upper limit given by BABAR collaboration is BR(B → eν e γ) < 17 × 10 −6 for the photon threshold energy E γ > 20MeV [28], while the result of Belle Collaboration is BR(B → eν e γ) < 6.1 × 10 −6 for the photon threshold E γ > 1GeV, and BR(B → eν e γ) < 9.3 × 10 −6 for the photon threshold E γ > 400MeV [29]. Compared with the experimental upper limit, our results given in Fig. 10 and Table I are consistent with the experimental data.   the numerical results with E γ > m Q / 2 − Λ QCD , and in the soft photon region, we calculate The fit of the form factors of D → γeν e at 1-loop level are shown in Fig. 11, and the fitted results can be written as Using [21,25] f D = 0.205 GeV −1 , V cd = 0.225 (120) We obtain the branching ratios of D → γeν e , and the branching ratios given with the dependence on ∆E γ are shown in Table. II.  We find that, for D mesons, the soft photon region is also very important. Compared with the tree level result of Ref. [16] where the contribution of order Λ QCD / E γ is neglected, the branching ratio is increased about 2 times. However, the enhancement is not so large compared to the B meson case. We find that it is because of the contribution of C 3 O 3 in Eq. (111). C 3 O 3 is a Λ 2 QCD /E γ m Q order contribution, while the contribution of C 2 O 2 is at the order of Λ QCD /E γ . For B meson, the mass of b quark is very large, so the contribution of C 3 O 3 compared to C 2 O 2 is small. However, for D meson, with the charge of the quark, we find As a result, the Λ QCD /E γ contribution could be canceled a lot by C 3 O 3 which is relatively a Λ QCD /m Q order contribution.

VII. SUMMARY
In this paper, we study the factorization of the radiative leptonic decays of B − and D − mesons. Compared with the work in Ref. [16], the factorization is extended to include the O(Λ QCD /E γ ) contributions. The factorization is proved at 1-loop order, and the IR divergences are found to be absorbed into the wave-function at any order of Λ QCD /m Q explicitly. The hard kernel at order O(α s ) is obtained. We list the hard kernel at order O(α s Λ QCD /m Q ) explicitly in both the soft photon region and the hard photon region. We use the wave function obtained in Ref. [21] to calculate the numerical results. The branching ratio of B − → γeν is found to be at the order of 5.21 × 10 −6 , which is a little larger then the previous works [11,24,[30][31][32]. The results for the D mesons radiative leptonic decay in the previous works are in the range from 10 −3 to 10 −6 , our results is about 1.92 × 10 −5 .
We also find that the contribution of the soft photon region is very important especially for B meson, the branching ratio is increased to be about 3 times of the result in Ref. [16], which is obtained using factorization and including Λ QCD / m Q contributions but treating E γ ∼ m Q . This result implies the importance of O(Λ QCD /E γ ) contributions, which increases the branching ratios at the soft photon region no matter how large the heavy quark mass is. For D → γeν e , we find the O(Λ QCD /E γ ) contribution is also very important, but not as important as the B meson decay.
Q ⊗ T which are canceled by can be canceled to all orders, which has been proved in Ref. [16]. On the other hand, the IR Div in F becomes sub-leading order when E γ is small, which is new, and can not be canceled if we use T (0) a in Ref. [16]. So the calculation of Φ as an example of how those corrections of wave-function are calculated. In the calculation we use Integral by part(IBP) method [33].
The tree-level amplitude T Then the convolution of the wave-function and the hard amplitude can be written as With the help ofvq, we find And I 1 can be treated as The second term in Eq. A5 can be worked out directly by integrate with l first, and then integrate with α. The result is free from IR Div, for convenience, we define From the integral we find f 1 is free from IR Div. The other terms will be calculated with I 2 together later. Part of I 2 can be worked out, and we define When integrate with α, the divergences in 1 l 2 and 1 l+2l·pq in Eq. (A7) can cancel each other, and we find From the integral we find f 2 is free from IR Div. The other part of I 2 can be written as where C 0 is the Pa-Ve function C 0 (−z, x − y + w − z, x; 0, 0, x). We will use C denote C(−z, x − y + w − z, x; 0, 0, x) in this section. We find We can perform the Feynman parameter directly on the last term, so that The first term is We add an m 2 q term in the numerator and integrate with α, so, it is And the second term is The first term can be integrated with α directly, and the second term is just f 1 , so it is Finally, we find The I 3 term can be worked out together with I 4 , which we defined as the sum of I 3 and I 4 is Using the IBP relation and we find With the help ofvq, it is The α integral can be worked out by the substitution l → αl And now we can integrate with α, and the result is where C µ is the Pa-Ve function. And then we find The first term will vanish. Then we can calculate I 3 by using 2I 3 = (I 3 + I 4 ) + (I 3 − I 4 ), so Using the results we define So that, combine all those terms together we have Now we are going to show the IR divergence cancellation. F can be written as with the help ofvq, they are Finally, we can obtain We know that, there are IR divergences in both, C 0 and C 1 . However, they are canceled and will not show up in T , and the cancellation can be shown without using any order expansion over the heavy quark mass. Therefor, at 1-loop order, the cancellation of IR Div can be established to all orders over the heavy quark mass expansion.
Finally, we find The correction can be written as At first, we adjust the region of α and β, first, we make the substitution α → α − β, then we change the order of α and β, so After integrating over the parameter β, we can obtain I = I 1 + I 2 the integrating over α in the first term can be integrated directly, and using IBP relation we can obtain The integrating over α in the first term in Eq. B13 can be integrated directly, and the second term can be written as Then using IBP relation we obtain the first term is just i 1 , the second term is just i 2 , and the third term can be transformed to i 1 use IBP relation so we find And then we define The calculation of i 4 is simpler then i 2 , we have the first term is i 3 , and also use IBP relation, we find the second term can be transformed to i 3 , as there is no UV Div in i 3 , so use D = 4 we find the α can be integrated now, and finally, we find and the last term is And now we can write I 1 as And then is the integral with I 2 , after Feynman parameter, and neglect the terms with even times of α because they will vanish, and then we find m Q α 4 (l 2 − ∆ α 2 ) 3 ∆ = X 2 Q 2 − XQ 2 + Xm 2 Q (B35) Now we can integrate α directly, so Finally, we find