Master integrals with 2 and 3 massive propagators for the 2 loop electroweak form-factor - planar case

We compute the master integrals containing 2 and 3 massive propagators entering the planar amplitudes of the 2-loop electroweak form factor. By this me mean the process ff¯?X, where ff¯ is an on-shell massless fermion pair and X is a singlet under the electroweak gauge group SU(2L)×U(1Y). This work is a continuation of our previous evaluation of master integrals containing at most 1 massive propagator. The masses of the W, Z and Higgs bosons are assumed to be degenerate. The 1/? poles and the finite parts are computed exactly in terms of a new class of 1-dimensional harmonic polylogarithms of the variable x=-s/m2, with ?=2-D/2, D the space-time dimension and s the center-of-mass energy squared. Since thresholds and pseudothresholds in s=±4m2 do appear in addition to the old ones in s=0,±m2, an extension of the basis function set involving complex constants and radicals is introduced, together with a set of recursion equations to reduce integrals with semi-integer powers. It is shown that the basic properties of the harmonic polylogarithms are maintained by the generalization. We derive small-momentum expansions |s|?m2 of all the 6-denominator amplitudes. Comparison with previous results in the literature is performed finding complete agreement.