The perturbative QCD factorization of $\rho \gamma^{\star} \to \rho$

TL;DR

This paper proves the validity of QCD factorization at NLO for the exclusive process 92 2, extending it to k_T factorization, and demonstrates the cancellation of soft divergences with the absorption of collinear divergences into meson wave functions.

Contribution

It provides a detailed proof of NLO factorization for 92 2 in both collinear and k_T frameworks, including the derivation of polarized NLO 92 wave functions.

Findings

Soft divergences cancel at the quark level.

Collinear divergences absorbed into NLO meson wave functions.

Full NLO amplitudes expressed as convolution of wave functions and finite hard kernels.

Abstract

In this paper we firstly demonstrate step by step that the factorization hypothesis is valid at the next-to-leading order (NLO) for the exclusive process $\rho \gamma^{\star} \to \rho$ by employing the collinear factorization approach, and then extend this proof to the case of the $k_T$ factorization by taking into account the transversal momentum of the light external quark (anti-quark) lines in the $\rho$ meson. At the NLO level, we then show that the soft divergences from different sub-diagrams will be canceled each other in the quark level, while the remaining collinear divergences can be absorbed into the NLO meson wave functions. The full NLO amplitudes can therefore be factorized as the convolution of the NLO wave functions $ \Phi^{(1)}_{\rho}$ and the infrared-finite leading order (LO) hard kernels $G^0_{X,IJ,kl}$ in the $k_T$ factorization. We also write down the polarized NLO $\rho$ meson wave functions in the form of nonlocal hadron matrix elements with the gauge factor integral path deviating from the light cone. These NLO $\rho$ meson wave functions can be used to calculate the NLO hard corrections to some relevant exclusive processes, such as $B \to \rho$ transition.