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

TL;DR

This work delivers a complete analytic evaluation of master integrals for planar two-loop electroweak form-factor amplitudes with 2 and 3 massive propagators under degenerate masses, by extending the harmonic polylogarithm framework to include complex constants and radicals. The authors employ a differential-equation approach to compute 25 master integrals and express results in generalized harmonic polylogarithms (GHPLs) with a carefully expanded basis to capture thresholds at s=0, m^2, 4m^2 and pseudothresholds. They provide explicit ε-expansions for the master integrals, along with small- and large-momentum expansions, and validate their results against existing literature. The work also details reducible diagrams and offers comprehensive appendices with 1-loop and factorizable master integrals, underpinning precision calculations in electroweak two-loop corrections and guiding future extensions to crossed-ladder topologies and complete x → 1/x closure. Overall, the study significantly advances analytic techniques for multi-mass two-loop integrals and their representation in advanced polylogarithmic frameworks, with implications for high-precision EW phenomenology.

Abstract

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/εpoles and the finite parts are computed exactly in terms of a {\it new} class of 1-dimensional harmonic polylogarithms of the variable x, with ε=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.

Figures (6)

• Figure 1: Vertex-correction diagrams with 2 and 3 massive propagators. The topologies related to these diagrams are real 6-denominator topologies (see text). The graphical conventions are the same as in our previous paper UgoRo.
• Figure 3: The set of 7 independent 6-denominator diagrams, with 2 and 3 massive propagators.
• Figure 4: The set of 17 independent 5-denominator diagrams, with 2 and 3 massive propagators.
• Figure 5: The set of 11 independent 4-denominator diagrams, with 2 and 3 massive denominators.
• Figure 6: The set of 4 independent 3-denominator diagrams together with the single 2-denominator diagram, with 2 and 3 massive propagators.