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 f (f) over bar -->X, where f (f) over bar is an on-shell massless fermion pair and X is a singlet under the electroweak gauge group SU(2)(L) x U(2)(gamma). 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/epsilon poles and the finite parts are computed exactly in terms of a new class of 1-dimensional harmonic polylogarithms of the variable x=-s/m(2), with epsilon=2-D/2,D the space-time dimension and s the center-of-mass energy squared. Since thresholds and pseudothresholds in s=+/-4m(2) do appear in addition to the old ones in s=0, +/-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\much less than m(2) of all the 6-denominator amplitudes. Comparison with previous results in the literature is performed finding complete agreement. (C) 2004 Elsevier B.V. All rights reserved.