Application of FIRE for the calculation of photon matrix elements

The next-to-next-to-leading order (the order ααs2) corrections to the first moment of the polarized virtual photon structure function g1γ (x,Q2, P2) are studied in perturbative QCD for the kinematical region Λ2 ≪ P2 ≪ Q2, where −Q2 (−P2) is the mass square of the probe (target) photon and Λ is the QCD scale parameter. In order to evaluate the two-loop Feynman diagrams for the photon matrix element of the gluon operator, I apply the recently developed algorithm FIRE which reduces a complicated sum of scalar Feynman integrals to a linear combination of a few master integrals. The details of the calculation are presented.