  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.

High Energy Physics - Phenomenology;High Energy Physics - Experiment