Master integrals with one massive propagator for the two-loop electroweak form factor

TL;DR

This work computes the master integrals with a single massive propagator entering the two-loop electroweak form factor by reducing full two-loop vertex diagrams to a finite master basis via IBP and two scalar-amplitude schemes. The master integrals are solved using differential equations in the evolution variable $x = -s/m^2$, with poles and finite parts expressed exactly in terms of one-dimensional harmonic polylogarithms $H(ullet;x)$, enabling precise large- and small-momentum expansions. The authors provide explicit results for topologies with up to five denominators and all six-denominator planar and non-planar cases, including detailed large-$|s|$ and small-$|s|$ expansions and cross-checks with existing literature. The approach yields a compact, functionally rich representation suitable for infrared analyses in the Standard Model and offers a methodological foundation for more complex multi-mass cases, while highlighting potential needs for new special functions beyond one-mass propagator diagrams.

Abstract

We compute the master integrals containing one massive propagator entering the two-loop electroweak form factor, i.e. the process f fbar --> X, where f fbar is an on-shell massless fermion pair and X is a singlet particle under SU(2)L x U(1)Y, such as a virtual gluon or an hypothetical Z'. The method used is that of the differential equation in the evolution variable x = -s/m^2, where s is the c.m. energy squared and m is the mass of the W or Z bosons (assumed to be degenerate). The 1/εpoles and the finite parts are computed exactly in terms of one-dimensional harmonic polylogarithms of the variable x, H(w;x), with ε=2-D/2 and D the space-time dimension. We present large-momentum expansions of the master integrals, i.e. expansions for |s| >> m^2, which are relevant for the study of infrared properties of the Standard Model. We also derive small-momentum expansions of the master integrals, i.e. expansions in the region |s| << m^2, related to the threshold behaviour of the form factor (soft probe). Comparison with previous results in the literature is performed finding complete agreement.

Figures (9)

• Figure 1: two-loop vertex diagrams with up to one massive propagator. They involve the ladder, crossed-ladder and vertex-insertion topologies and are real six-denominator topologies (see text).
• Figure 2: two-loop vertex diagrams with up to one massive propagator involving a self-energy correction to the basic one-loop vertices. The related topologies are five and four denominator topologies (see text).
• Figure 3: Double box auxiliary diagram for the case of the vertex diagram in Fig. \ref{['fig1']} (b). The dashed line represents the auxiliary denominator $P_{7}$.
• Figure 4: The set of 9 independent 6-denominator topologies.
• Figure 5: The set of 22 independent 5-denominator topologies.