Next-to-Leading-logarithm threshold resummation for exclusive $B$ meson decays

We extend the threshold resummation of the large logarithms $\ln x$ which appear in factorization formulas for exclusive $B$ meson decays, $x$ being a spectator momentum fraction, to the next-to-leading-logarithm (NLL) accuracy. It is shown that the NLL resummation effect provides suppression in the end-point region with $x\sim 0$ stronger than the leading-logarithm (LL) one, and thus improves perturbative analyses of the above processes. We revisit the $B\to K\pi$ decays under the NLL resummation, and find that it induces 20-25\% variation of the direct CP asymmetries compared to those from the LL resummation. Our way to avoid the Landau singularity in the inverse Mellin transformation causes little theoretical uncertainty.


I. INTRODUCTION
Factorization theorems have been one of the major theoretical approaches to exclusive B meson decays, in which a decay process is factorized into a convolution of a hard kernel with hadron distribution amplitudes. A crucial issue on the application of factorization theorems to a key ingredient of these decays, B meson transition form factors, is the end-point singularity, which appears as a spectator parton carries a vanishing momentum fraction x → 0 [1][2][3]. Because of this end-point singularity, a B meson transition form factor is treated as a nonperturbative input in the QCD-improved factorization approach [4] based on the collinear factorization theorem. In the soft-collinear effective theory the end-point singularity can be removed by the zero-bin subtraction [5], so that a B meson transition form factor becomes factorizable in the collinear factorization. It was argued that a parton transverse momentum k T is not negligible, when the end-point region is important. The perturbative QCD approach based on the k T factorization theorem was then proposed [6][7][8], in which the end-point singularity is regularized by a parton k T , and a B meson transition form factor also becomes factorizable [9].
An alternative removal of the end-point singularity in the framework of the collinear factorization has been suggested in [10]. When the end-point region dominates, the double logarithms α s ln 2 x from radiative corrections [2,11] should be organized to all orders to improve perturbative expansion. The first systematic study was done in [10], where these double logarithms were factorized from exclusive B meson decays into a universal jet function, and resummed up to the leading-logarithm (LL) accuracy. It was then shown that the resultant jet function vanishes quickly at x → 0, and suppresses the end-point singularities in the B → π form factors. The threshold resummation effect on more complicated two-body hadronic B meson decays, which involve the annihilation and nonfactorizable amplitudes in addition to the factorizable one proportional to a transition form factor, was analysed in [12] and implemented in the PQCD approach widely afterwards.
In this paper we will extend the LL threshold resummation performed in [10] to the next-to-leading-logarithm (NLL) accuracy. To accomplish this task, we calculate the jet function stated above at one loop to identify the complete large logarithms, solve an evolution equation for the jet function in the Mellin space to get all-order summation of the logarithms, match the all-order summation to the one-loop result to determine the initial condition of the jet function, and follow the best fit method in [9] to obtain the threshold resummaiton in the momentum fraction x space. It will be demonstrated that the NLL jet function exhibits suppression at the end point x ∼ 0 stronger than the LL one. Because the threshold resummation modifies hard decay kernels by including partial higher order contributions, hadron distribution amplitudes, such as the uncertain B meson distribution amplitude, should be adjusted accordingly to maintain B meson transition form factors [13]. Therefore, we compare the LL and NLL resummation effects by investigating their impacts on the CP asymmetries in the B → Kπ decays, which are less sensitive to choices of hadron distribution amplitudes. It is found that the replacement of the LL jet function by the NLL one in the PQCD factorization formulas causes about 20-25% variation of the B → Kπ direct CP asymmetries, an effect which needs to be taken into account for precision calculations of B meson decays.
In Sec. II we compute the one-loop jet function, derive its evolution equation and the corresponding evolution kernels in the Mellin space, and solve the evolution equation to attain the NLL threshold resummation. The best fit method is employed to transform the jet function from the Mellin space back to the momentum fraction space in Sec. III. It is verified that the Landau singularity in the inverse Mellin transformation can be avoided, and the theoretical uncertainty is under control in the above method. The impacts of the LL and NLL resummations on the CP asymmetries in the B → Kπ decays are then examined and compared. Section IV contains the conclusion and outlook.

A. One-loop Quark Diagrams
The definition of the jet function J(x) in terms of a quark field and its associated Wilson link, which is constructed from the factorization of the radiative decay B(P 1 ) → γ(P 2 )lν, is referred to [10]. The Wilson link runs in the direction n, that contains the arbitrary components n + and n − . The quark momentum has been parametrized as P 2 − k, where the photon momentum P 2 is in the minus direction and the momentum k = (xP + 1 , 0, 0 T ) of the light quark in the B meson is in the plus direction. That is, this quark is slightly off-shell by (P 2 − k) 2 = −2xP + 1 P − 2 ≡ −xQ 2 . The leading-order (LO) jet function has been chosen as J (0) (x) = 1. The one-loop vertex correction in Fig. 1 is written as where C F = 4/3 is a color factor, and µ is the renormalization scale. The projector v / + v / − /4, with the light-like vectors v + = (1, 0, 0 T ) and v − = (0, 1, 0 T ), arises from the factorization of the jet function [10]. A straightforward evaluation gives for n 2 > 0, with the n-dependent factor ξ 2 ≡ 4(P 2 · n) 2 /n 2 and the Euler constant γ E . Note that the jet function depends on the Lorentz invariants (P 2 − k) · n ≈ P 2 · n in the small x limit and n 2 , and that the Feynman rules associated with the Wilson link shows a scale invariance in n. These facts explain why the two vectors P 2 − k and n appear via the ratio ξ 2 in Eq. (2). The self-energy correction in Fig. 1 is expressed as We then have the O(α s ) jet function which is independent of µ, ie., ultraviolet finite. We apply the Mellin transformation from the momentum fraction x space to the moment N spaceJ It implies that the transformed jet functionJ(N) at large N collects the contribution mainly from the small x region. The Mellin transformation of the LO jet function, J (0) (N) = 1/N, is trivial. The Mellin transformation of Eq. (4) yields, in terms of the variableN ≡ N exp(γ E ), in the large N limit up to corrections down by powers of 1/N.

B. Evolution Equation for J
As indicated by the above one-loop calculation, the important logarithms in the jet function depend on the factor ξ 2 ≡ 4ν 2 P −2 2 . To resum these logarithms, we construct the evolution equation for the jet function [14,15] The derivative respect to n α applies to the Feynman rules of the Wilson link, generating with the special vertexn The technique of varying Wilson links has been applied to the resummation of various types of logarithms, such as the rapidity logaritms in the B meson wave function [16], and the joint logarithms in the pion wave function [17]. The Ward identity for the summation over the special vertices leads to the factorization of the soft function K and the hard function G from the derivative of the jet function [14,15] Figure 2 depicts the factorization of the soft function K at O(α s ), which contains two pieces K 1 and K 2 . The former is written as with the momentum P J = P 2 − k, where the gluon mass m 2 , serving as an infrared regulator, will approach to zero eventually. Choosing the additive counterterm we have The loop momentum l flows through the jet function for K 2 , so they appear in a convolution Performing the contour integration over l − for −(1 − x)P + 1 < l + < 0 followed by the integration over the transverse momentum l T , and employing the variable change u = x − l + /P + 1 , we arrive at with the infrared regulator M 2 ≡ ν 2 m 2 /P +2 1 . The Mellin transformation of the first integral in Eq. (15) gives where the order of the integrations over x and u has been exchanged in the first term, the variable change x = u(1 − t) has been applied, and the upper bound 1 − x of the integration variable t in the second term has been approximated by 1. This approximation holds up to an infrared finite constant, which will be compensated by matching later. The further approximation The above result is subject to corrections down by a power of 1/N. We also apply the Mellin transformation to the second integral in Eq. (15): The sum of Eqs. (17) and (18) yields the Mellin transformation of Eq. (15), which is then combined with Eq. (13) into It is seen that the infrared regulator m 2 has disappeared in the above combination. The first diagram for the O(α s ) hard function G in Fig. 3 contributes We have dropped the small momentum k, to which the hard function is not sensitive. Instead, the infrared regulator m 2 is introduced, whose dependence will be removed by the subtraction below. Note that the above expression is free of an ultraviolet divergence. The final result for G in Fig. 3 is with the additive counterterm δG = −δK, where the subtraction K 1 avoids double counting of the soft contribution.

C. Solution to Evolution Equation
The functions K and G involve only single soft and ultraviolet logarithms, respectively, so they can be treated by renormalization-group (RG) methods: where the anomalous dimension λ K = µdδK/dµ is given up to two loops by [18] with C A = 3 being a color factor and n f the number of quark flavors. We allow the scale µ to evolve to P + 1 /N in K and to 2ν 2 P − 2 / √ e in G, obtaining the RG solution of K + G, Substituting the above evolution kernel into Eq. (10), we solve for the jet functioñ with the variable changes µ = wP + 1 andν 2 = √ eP + 1 y/(2P − 2 ). The initial conditionJ in (N) for the jet function is determined via matching: we expand Eq. (26) to O(α s ) for a fixed coupling constant, and compare it with Eq. (6) to get If an order-unity constant C is introduced into the exponent, the initial condition will be modified accordingly: For a heavy-to-light transition at maximal recoil, we have P + 1 = P − 2 = m B / √ 2, m B being the B meson mass. Choosing the factor ν 2 = 1/2, ie., ξ 2 = Q 2 , and neglecting the running of the coupling constant, we derive the jet function in the Mellin spacẽ This is the improvement of the threshold resummation with a fixed coupling constant to the NLL accuracy. The inverse Mellin transformation brings the jet function back to the momentum fraction space, with the coefficient In the above formula c is an arbitrary real constant larger than the real parts of all the poles of the integrand, the variable change N = exp(t + iπ) (N = exp(t − iπ)) has been adopted for the piece of contour above (below) the branch cut in Fig. 3 of [10], and the further variable change t + γ E − 1/2 → t has been made. It is found that Eq. (30) exhibits the features similar to those of the LL jet fucntion [10]: it vanishes as x → 0 and x → 1, and it is normalized to unity up to corrections of O(α s ). Next we take into account the running effect of the coupling constant by inserting α s (µ) = 4π/[β 0 ln(µ 2 /Λ 2 )] into Eq. (26), with β 0 = 11 − 2n f /3 and the QCD scale Λ ≡ Λ QCD , and arrive at for ν 2 = 1/2. Note that the above expression vanishes asN approaches to the Landau pole,N → P + 1 /Λ, namely, as N ≈ 8.4 for m B = 5.28 GeV and Λ = 0.25 GeV [13].

III. NUMERICAL RESULTS
In this section we examine the NLL threshold resummation effect on various CP asymmetries in the B → Kπ decays. The first step is to convert the jet function in the Mellin space to the momentum fraction space, which is usually done through the inverse Mellin transformation defined by the first line of Eq. (30). Due to the existence of the Landau pole, an extrapolation of Eq. (32) in the large N region is necessary for avoiding this singularity, which then introduces theoretical uncertainty. On the other hand, it has been observed [19] that the threshold resummation effect is mainly governed by the behavior of Eq. (32) at intermediate N for currently accessible energy scales. Therefore, we will employ the best fit method proposed in [9], instead of the inverse Mellin transformation, for the aforementioned conversion: the Mellin transformation of a parametrized jet function is fit to Eq. (32) in the intermediate N region.
We parametrize the jet function in the momentum fraction space as which is motivated by the feature of Eq. (30), ie., vanishing as x → 0 and x → 1. The prefactor has been introduced to obey the normalization dxJ(x) =J(1) = 1. It implies that we have chosen the initial condition at LO,J in (N) = 1/N, since we intend to focus on effects from the resummaton. To be consistent, the B → Kπ factorization formulas with the LO hard kernels will be adopted for the numerical study below. We mention that Eq. (32) is roughly, but not exactly, equal to unity as N = 1 even with the LOJ in (N).   The best fits to Eq. (32) for n f = 4 in the ranges from N = 1 to 3, N = 1 to 4, and N = 1 to 5 produce the curves displayed in Fig. 4, which exhibit good agreement with Eq. (32) at intermediate N, and start to deviate from Eq. (32) as N > 6. We take the jet function from the N = 1-4 fit with the parameters α = 0.58 and β = 0.47 to generate our results, and those from the N = 1-3 fit (α = 0.43 and β = 0.33) and from the N = 1-5 fit (α = 0.76 and β = 0.67) to estimate the theoretical uncertainty. The similarity among the three fitted jet functions guarantees that the uncertainty from avoiding the Landau singularity is not severe in our best fit method. Compared to the LL jet function [9], the NLL one provides stronger suppression at the end points of x, with which particles involved in the hard decay kernels tend to be more off-shell, and the perturbative analysis of the B → Kπ decays is expected to be more reliable.
For the Cabibbo-Kobayashi-Maskawa matrix elements, we take the Wolfenstein parametrization with the values A = 0.836 ± 0.015, λ = 0.22453 ± 0.00044,ρ = 0.122 +0.018 −0.017 andη = 0.355 +0.012 −0.011 [20]. The hadronic inputs, including meson masses and decay constants, meson distribution amplitudes, chiral scales m 0 associated with the pion and kaon twist-3 distribution amplitudes, and the QCD scale Λ QCD are the same as in [13]. The factorization formulas for the relevant B → Kπ decay amplitudes with the LO hard kernels are also referred to [13]. The outcomes for the CP asymmetries under the LL and NLL threshold resummations are listed in Table I, in which the values in the column labelled by LL well reproduce the corresponding ones in [13]. It is found that the NLL effect enhances the direct CP asymmetry A CP (K 0 π − ) by 25%, and decreases the other three direct CP asymmetries by 20-25% relative to the LL results. The mixing-induced CP asymmetry S K 0 π 0 , increasing by only 3%, is less sensitive to the replacement of the jet function. It is understandable, because this observable is supposed to be close to sin(2φ 1 ), φ 1 being the weak phase, in penguin-dominated modes like B → Kπ. The comparison of the column labelled by NLL (N = 1-4) with those labelled by NLL (N = 1-3) and NLL (N = 1-5) indicates that the theoretical uncertainty is under control: except A CP (K 0 π − ), whose uncertainty amounts up to 20%, the other CP asymmetries change by lower than 10%.

IV. CONCLUSION
In this paper we have improved the LL threshold resummation for exclusive B meson decays to the NLL accuracy. The recipe contains the computation of the one-loop jet function factorized out of decay amplitudes, the derivation of the evolution kernels, the matching of the resummation formula to the one-loop jet function, and the inclusion of the running effect of the coupling constant. It has been observed that the NLL threshold resummation suppresses the end-point region with x ∼ 0 more strongly than the LL one. Since we focused on the resummaton effect, we did not take into account the NLO piece in the initial condition of the jet function. For consistency, we worked on the PQCD factorization formulas for the B → Kπ decays with the LO hard kernels. It has been explained that the different LL and NLL threshold resummation effects can be compared unambiguously through the investigation of the CP asymmetries. We have shown that the replacement of the LL jet function by the NLL one causes about 20-25% variation of the B → Kπ direct CP asymmetries, which is not negligible for precision analyses for B meson decays. On the contrary, the mixing-induced CP asymmetry almost remains untouched under the above replacement. Moreover, the theoretical uncertainty from the inverse Mellin transformation of the threshold resummation is under control.
The implementation of the NLL threshold resummation derived here in the PQCD approach to exclusive B meson decays is nontrivial, and demands more efforts. As pointed out in the Introduction, the threshold resummation modifies hard decay kernels by including partial higher order contributions, so hadron distribution amplitudes should be adjusted accordingly. In principle, it is more appropriate to execute this task in a global study of many two-body hadronic B meson decay modes. A global fit to available data based on the PQCD approach with the NLL threshold resummation will be attempted in near future.