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

Abstract 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 ¯ → X , where f f ¯ is an on-shell massless fermion pair and X is a singlet under the electroweak gauge group SU ( 2 ) L × U ( 1 ) Y . 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 / e 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 e = 2 − D / 2 , D the space–time dimension and s the center-of-mass energy squared. Since thresholds and pseudothresholds in s = ± 4 m 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 | ≪ m 2 of all the 6-denominator amplitudes. Comparison with previous results in the literature is performed finding complete agreement.