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

In this paper we first 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 $k_{\mathrm{T}}$ factorization by taking into account the transversal momentum of the light external quark (antiquark) lines in the $\rho$ meson. At the NLO level, we then show that the soft divergences from different subdiagrams will cancel 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 ${\mathrm{\Phi }}_{\rho }^{(1)}$ and the infrared-finite leading order (LO) hard kernels $G_{X,IJ,kl}^0$ in the $k_{\mathrm{T}}$ 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.

Figure 1

The leading-order quark diagrams for the $\rho {\gamma }^{\star }\to \rho$ form factor, with the virtual photon vertices, denoted by the symbol $•$, setting at four different positions.

Figure 2

The Feynman diagrams which provide the NLO QCD corrections to Fig. 1 with an additional gluon (blue curves) emitted from the quark (antiquark) lines of the initial $\rho$ meson.

Figure 3

$\mathcal{O}({\alpha }_s)$ effective diagrams for the initial longitudinal $\rho$ meson wave function. The vertical double line denotes the Wilson line along the light cone, whose Feynman rule is $v_{\rho }/(v·l)$ as described in Eqs. (36), (50), (51), (52), (53).

Figure 4

Analogous to Fig. 2, but with the additional gluon (blue curves) emitted from the quark (antiquark) lines of the final state $\rho$ meson.

Figure 5

The deviation of the integral (Wilson link) from the light corn by $\mathbf{b}$ in the coordinate space for the two-parton meson wave function.