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

We compute the master integrals containing one massive propagator entering the two-loop electroweak form factor, i.e., the process ff̄→X, where ff̄ is an on-shell massless fermion pair and X is a singlet particle under SU(2)L×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/m2, 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/ϵ poles and the finite parts are computed exactly in terms of one-dimensional harmonic polylogarithms of the variable x, H(w→;x), with ϵ=2−D/2 and D the space–time dimension. We present large-momentum expansions of the master integrals, i.e., expansions for |s|⪢m2, 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|⪡m2, related to the threshold behaviour of the form factor (soft probe). Comparison with previous results in the literature is performed finding complete agreement.