Next-to-next-to-leading order ${\cal O}(α^2α_s^2)$ results for top quark pair production in photon--photon collisions: The loop-by-loop contribution

We calculate the so--called loop-by-loop contributions to the next-to-next-to-leading order ${\cal O}(\alpha^2\alpha_s^2)$ radiative QCD corrections for the production of heavy quark pairs in the collisions of unpolarized on--shell photons. In particular, we present analytical results for the squared matrix elements that correspond to the product of the one--loop amplitudes. All results of the perturbative calculation are given in the dimensional regularization scheme. These results represent the Abelian part of the corresponding gluon--induced next-to-next-to-leading order cross section for heavy quark pair hadroproduction.