Decorated-box-diagram contributions to Bhabha scattering. Pt. 1

We evaluate, in the light-energy limit, s>>vertical stroke tvertical stroke >>m2>>λ2, the sum of amplitudes corresponding to a class of Feynman diagrams describing two-loop virtual photonic corrections to Bhabha scattering. The diagrams considered are box and crossed-box diagrams with an extra photon decorating one of the fermion lines. The mathematical method employed is that of Mellin transforms. In the eikonal approximation, this sum of two-loop amplitudes has previously been evaluated, and found to be equal to the sum of the box and crossed-box amplitudes, multiplied by the electric form factor of the electron. We obtain a similar factorization, but with the form factor replaced by another expression involving the logarithms log(λ2/m2) and log(λ2/vertical stroke tvertical stroke ). (orig.)