QCD factorizations in gamma* gamma* -> rho rho

Abstract

We calculate the lowest order QCD amplitude, i.e. the quark exchange contribution, to the forward production amplitude of a pair of longitudinally polarized $\rho$ mesons in the scattering of two virtual photons ${\gamma }^{\ast }(Q_1){\gamma }^{\ast }(Q_2)\to {\rho }_L^0{\rho }_L^0$. We show that the scattering amplitude simultaneously factorizes in two quite different ways: the part with transverse photons is described by the QCD factorization formula involving the generalized distribution amplitude of two final $\rho$ mesons, whereas the part with longitudinally polarized photons takes the QCD factorized form with the ${\gamma }_L^{\ast }\to {\rho }_L^0$ transition distribution amplitude. Perturbative expressions for these, in general, non-perturbative functions are obtained in terms of the $\rho -$meson distribution amplitude.