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

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We compute the master integrals containing 2 and 3 massive propagators entering the planar amplitudes of the 2-loop electroweak form factor. The masses of the $W$, $Z$ and Higgs bosons are assumed to be degenerate. This work is a continuation of our previous evaluation of master integrals containing at most 1 massive propagator. The 1/\epsilon poles and the finite parts are computed exactly in terms of a {\it new} class of 1-dimensional harmonic polylogarithms of the variable x, with \epsilon=2-D/2 and D the pace-time dimension. Since thresholds and pseudothresholds in s=\pm 4m^2 do appear in addition to the old ones in s=0,\pm m^2, 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| << m^2 of all the 6-denominator amplitudes. We also present large momentum expansions |s| >> m^2 of all the 6-denominator amplitudes which can be represented in terms of ordinary harmonic polylogarithms. Comparison with previous results in the literature is performed finding complete agreement.