QCD factorizations in gamma* gamma* -> rho rho

We calculate the lowest order QCD amplitude, i.e. the quark exchange contribution, to the forward production amplitude of a pair of longitudinally polarized $ρ$ mesons in the scattering of two virtual photons $γ^*(Q_1) γ^*(Q_2) \to ρ^0_L ρ^0_L$. 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 $ρ$ mesons, whereas the part with longitudinally polarized photons takes the QCD factorized form with the $γ^*_L \to ρ^0_L$ transition distribution amplitude. Perturbative expressions for these, in general, non-perturbative functions are obtained in terms of the $ρ-$meson distribution amplitude.

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