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

In this paper, we firstly varify that the factorization hypothesis is valid for the exclusive process $\rho \gamma^{\star} \to \pi$ at the next-to-leading order (NLO) with the collinear factorization approach, and then extend this proof to the case of the $k_T$ factorization approach. We particularly show that at the NLO level, the soft divergences in the full quark level calculation could be canceled completely as for the $\pi \gamma^{\star} \to \pi$ process where only the pseudoscalar $\pi$ meson involved, and the remaining collinear divergences can be absorbed into the NLO hadron wave functions. The full amplitudes can be factorized as the convolution of the NLO wave functions and the infrared-finite hard kernels with these factorization approaches. We also write out the NLO meson distribution amplitudes in the form of nonlocal matrix elements.