The next-to-leading order corrections to $B \to \rho$ transition in the $k_T$ factorization

TL;DR

This paper calculates the next-to-leading order corrections to the $B o ho$ transition using $k_T$ factorization, demonstrating divergence cancellations and factorization of amplitudes into wave functions and hard kernels.

Contribution

It extends the NLO factorization analysis of $B o ho$ transitions from collinear to $k_T$ factorization, detailing divergence cancellations and wave function factorization.

Findings

Soft divergences cancel at the quark level.

Collinear divergences absorbed into NLO wave functions.

Full NLO amplitudes factorize into wave functions and finite hard kernels.

Abstract

In this paper, we investigate the factorization hypothesis step by step for the exclusive process $B \to \rho$ at next-to-leading order (NLO) with the collinear factorization approach, and then we extend our results to the $k_T$ factorization frame. We show that the soft divergence from the specific NLO diagrams will cancel each other at the quark level, while the remaining collinear divergence can be absorbed into the NLO wave functions. The full NLO amplitudes can be factorized into two parts: the NLO $B$ and $\rho$ meson wave functions containing the collinear divergence and the leading order (LO) finite hard kernels. We give the general form of the nonlocal hadron matrix for the NLO $B$ and $\rho$ meson wave functions and all results of factorization for different twists' combinations.