Master integrals with one massive propagator for the two-loop electroweak form factor

Abstract

We compute the master integrals containing one massive propagator entering the two-loop electroweak form factor, i.e. the process f fbar --> X, where f fbar is an on-shell massless fermion pair and X is a singlet particle under SU(2)L x U(1)Y, such as a virtual gluon or an hypothetical Z'. The method used is that of the differential equation in the evolution variable x = -s/m^2, where s is the c.m. energy squared and m is the mass of the W or Z bosons (assumed to be degenerate). The 1/\epsilon poles and the finite parts are computed exactly in terms of one-dimensional harmonic polylogarithms of the variable x, H(w;x), with \epsilon=2-D/2 and D the space-time dimension. We present large-momentum expansions of the master integrals, i.e. expansions for |s| >> m^2, which are relevant for the study of infrared properties of the Standard Model. We also derive small-momentum expansions of the master integrals, i.e. expansions in the region |s| << m^2, related to the threshold behaviour of the form factor (soft probe). Comparison with previous results in the literature is performed finding complete agreement.