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

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 $x\to 0$ SHB ; ASY ; BF . Because of this end-point singularity, a $B$ meson transition form factor is treated as a nonperturbative input in the QCD-improved factorization approach BBNS based on the collinear factorization theorem. In the soft-collinear effective theory the end-point singularity can be removed by the zero-bin subtraction Manohar:2006nz , 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 LY1 ; KLS ; LUY , in which the end-point singularity is regularized by a parton $k_{T}$, and a $B$ meson transition form factor also becomes factorizable TLS .

An alternative removal of the end-point singularity in the framework of the collinear factorization has been suggested in Li:2001ay . When the end-point region dominates, the double logarithms $\alpha_{s}\ln^{2}x$ from radiative corrections ASY ; KPY should be organized to all orders to improve perturbative expansion. The first systematic study was done in Li:2001ay , 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 $x\to 0$, and suppresses the end-point singularities in the $B\to\pi$ 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 Li:2002mi and implemented in the PQCD approach widely afterwards.

In this paper we will extend the LL threshold resummation performed in Li:2001ay 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 TLS 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 $x\sim 0$ 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 Li:2005kt . Therefore, we compare the LL and NLL resummation effects by investigating their impacts on the CP asymmetries in the $B\to K\pi$ 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 $B\to K\pi$ 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 $B\to K\pi$ decays are then examined and compared. Section IV contains the conclusion and outlook.

In this section we examine the NLL threshold resummation effect on various CP asymmetries in the $B\to K\pi$ 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 LL2 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 TLS , 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 $x\to 0$ and $x\to 1$. The prefactor has been introduced to obey the normalization $\int dxJ(x)=\tilde{J}(1)=1$. It implies that we have chosen the initial condition at LO, $\tilde{J}_{in}(N)=1/N$, since we intend to focus on effects from the resummaton. To be consistent, the $B\to K\pi$ 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 LO $\tilde{J}_{in}(N)$. The equality can be made exact by choosing $\nu^{2}=\exp(1/2-\gamma_{E})/2\approx 0.47$, quite close to $\nu^{2}=1/2$ taken in this work. The Mellin transformation of Eq. (33) is then fit to Eq. (32) in the intermediate $N$ region, and its deviation from Eq. (32) at large $N$ is regarded as an extrapolation to avoid the Landau singularity.

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 $\alpha=0.58$ and $\beta=0.47$ to generate our results, and those from the $N=1$-3 fit ($\alpha=0.43$ and $\beta=0.33$) and from the $N=1$-5 fit ($\alpha=0.76$ and $\beta=0.67$) 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 $J(x)\propto[x(1-x)]^{0.3}$ TLS , 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 $B\to K\pi$ decays is expected to be more reliable.

For the Cabibbo-Kobayashi-Maskawa matrix elements, we take the Wolfenstein parametrization with the values $A=0.836\pm 0.015$, $\lambda=0.22453\pm 0.00044$, $\bar{\rho}=0.122^{+0.018}_{-0.017}$ and $\bar{\eta}=0.355^{+0.012}_{-0.011}$ Tanabashi:2018oca . 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 $\Lambda_{\rm QCD}$ are the same as in Li:2005kt . The factorization formulas for the relevant $B\to K\pi$ decay amplitudes with the LO hard kernels are also referred to Li:2005kt . The outcomes for the CP asymmetries under the LL and NLL threshold resummations are listed in Table 1, in which the values in the column labelled by LL well reproduce the corresponding ones in Li:2005kt . It is found that the NLL effect enhances the direct CP asymmetry $A_{CP}(K^{0}\pi^{-})$ by 25%, and decreases the other three direct CP asymmetries by 20-25% relative to the LL results. The mixing-induced CP asymmetry $S_{K^{0}\pi^{0}}$, 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 $\sin(2\phi_{1})$, $\phi_{1}$ being the weak phase, in penguin-dominated modes like $B\to K\pi$. 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 $A_{CP}(K^{0}\pi^{-})$, whose uncertainty amounts up to 20%, the other CP asymmetries change by lower than 10%.

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 $x\sim 0$ 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 $B\to K\pi$ 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 $B\to K\pi$ 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.

Acknowledgement

We thank S. Mishima for useful discussions. This work was supported in part by MOST of R.O.C. under Grant No. MOST-107-2119-M-001-035-MY3, and by NSFC under Grant No.11347030.