Two-Loop Master Integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering

TL;DR

This work delivers the full set of master integrals required for two-loop mixed QCD–EW virtual corrections to Drell–Yan production with massless external states, under the W/Z degeneracy approximation. By reducing to 49 MIs and casting their differential equations in a canonical form via the Magnus expansion, the authors obtain a clean ε-dependent system whose solutions are expressed as Chen’s iterated integrals and, where possible, in Goncharov polylogarithms (GPLs). Boundary conditions are fixed through limiting kinematics or regularity at pseudo-thresholds, and the final ε-expansions (up to weight four) are provided in both analytic GPL/Chen forms and numerical representations. The one- and two-mass four-point integrals emerge as novel results, with cross-checks against SecDec validating the results, and a mixed Chen–Goncharov representation enabling efficient numerical evaluation across the kinematic space. This work thus furnishes essential ingredients for precise, differential predictions of Drell–Yan observables at the two-loop mixed QCD–EW level, relevant for high-precision SM tests and new-physics backgrounds at the LHC.

Abstract

We present the calculation of the master integrals needed for the two-loop QCDxEW corrections to $ q + \bar{q} \to l^- + l^+$ and $ q + \bar{q}' \to l^- + \overlineν \, , $ for massless external particles. We treat W and Z bosons as degenerate in mass. We identify three types of diagrams, according to the presence of massive internal lines: the no-mass type, the one-mass type, and the two-mass type, where all massive propagators, when occurring, contain the same mass value. We find a basis of 49 master integrals and evaluate them with the method of the differential equations. The Magnus exponential is employed to choose a set of master integrals that obeys a canonical system of differential equations. Boundary conditions are found either by matching the solutions onto simpler integrals in special kinematic configurations, or by requiring the regularity of the solution at pseudo-thresholds. The canonical master integrals are finally given as Taylor series around d=4 space-time dimensions, up to order four, with coefficients given in terms of iterated integrals, respectively up to weight four.

Figures (6)

• Figure 1: One-loop topologies. Thin lines represent massless external particles and propagators, while thick lines represent massive propagators.
• Figure 2: Two-loop topologies. Thin lines represent massless external particles and propagators, while thick lines represent massive propagators.
• Figure 3: One-loop one-mass MIs $\mathcal{T}_{1,\ldots,5}$. Thin lines represent massless external particles and propagators; thick lines stand for massive propagators; an horizontal (vertical) dashed external line represents an off-shell leg with squared momentum equal to $s$ ($t$); dots indicate squared propagators.
• Figure 4: Two-loop one-mass MIs $\mathcal{T}_{1,\ldots,31}$. The conventions are as in figure \ref{['Fig:1M1LMIs']}.
• Figure 5: One-loop two-mass MIs $\mathcal{T}_{1,\ldots,6}$. The conventions are as in figure \ref{['Fig:1M1LMIs']}.