Next-to-leading-order corrections to $B \to \pi$ form factors in $k_T$ factorization

TL;DR

This paper computes next-to-leading-order corrections to $B o \pi$ form factors within the $k_T$ factorization framework, demonstrating infrared finiteness and confirming the universality of the pion wave function.

Contribution

It provides a detailed calculation of NLO corrections to $B o \pi$ form factors in $k_T$ factorization, confirming the cancellation of infrared divergences and the universality of the pion wave function.

Findings

Infrared logarithms cancel between diagrams.

NLO corrections are up to 30% at large recoil.

Hard kernel remains infrared finite, validating $k_T$ factorization.

Abstract

We calculate next-to-leading-order (NLO) corrections to the $B\to\pi$ transition form factors at leading twist in the $k_T$ factorization theorem. Light partons off-shell by $k_T^2$ are considered in the quark diagrams, in the effective diagrams for the $B$ meson wave function defined with the effective heavy-quark field, and in the effective diagrams for the pion wave function. It is explicitly demonstrated that the infrared logarithms $\ln k_T^2$ cancel between the above sets of diagrams, as deriving the $k_T$-dependent NLO hard kernel from their difference. The infrared finiteness of the hard kernel confirms the application of the $k_T$ factorization theorem to $B$ meson semileptonic decays. The NLO pion wave function is identical to those constructed from the pion transition and electromagnetic form factors, consistent with its universality. Choosing the renormalization and factorization scales lower than the $B$ meson mass, the NLO corrections are under control: they amount only up to 30% of the form factors at large recoil of the pion, when varying models for the meson wave functions.