Two-Loop Integrals for Planar Five-Point One-Mass Processes

TL;DR

The authors tackle the challenge of planar two-loop five-point integrals with one external mass by constructing a canonical differential-equation framework and a pure master-basis, then solving these equations analytically via generalized power series along one-dimensional contours. They derive the symbol alphabet from cut differential equations, uncover a remarkably compact set of letters, and demonstrate that the extended Steinmann relations hold to all orders in ε. The results enable precise evaluation across all kinematic regions and provide high-precision benchmarks and phase-space plots relevant for NNLO W+jets phenomenology at the LHC. The work also outlines a practical semi-numerical workflow, including finite-field techniques and boundary-condition strategies, with potential extensions to non-planar topologies and public implementations.

Abstract

We present the computation of a full set of planar five-point two-loop master integrals with one external mass. These integrals are an important ingredient for two-loop scattering amplitudes for two-jet-associated W-boson production at leading color in QCD. We provide a set of pure integrals together with differential equations in canonical form. We obtain analytic differential equations efficiently from numerical samples over finite fields, fitting an ansatz built from symbol letters. The symbol alphabet itself is constructed from cut differential equations and we find that it can be written in a remarkably compact form. We comment on the analytic properties of the integrals and confirm the extended Steinmann relations, which govern the double discontinuities of Feynman integrals, to all orders in $ε$. We solve the differential equations in terms of generalized power series on single-parameter contours in the space of Mandelstam invariants. This form of the solution trivializes the analytic continuation and the integrals can be evaluated in all kinematic regions with arbitrary numerical precision.

Figures (27)

• Figure 1: Two-loop five-point one-mass topologies. The thick external line with label 1 denotes the massive external leg.
• Figure 2: Propagator structures of two-loop five-point master integrals.
• Figure 3: Topologies with more masters than in the limit $p_1^2\to0$.
• Figure 4: Triangle-triangle topology with five master integrals. The scalar integral, normalized as in eq. \ref{['eq:normBadSlashed']} is the only integral that depends on letters $\{W_{35}, W_{36}, W_{39}\}$ at weight 4.
• Figure 5: Triangle-triangle topology with two master integrals. The scalar integral, normalized by $\epsilon^4(s_{34}+s_{45})$, is the only integral that depends on $W_{21}$ at weight 4. Its image under $(2\leftrightarrow 5,3\leftrightarrow 4)$ is the only integral that depends on $W_{20}$ at weight 4.