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

Abstract

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.