Cutting massless four-loop propagators

TL;DR

This paper computes previously unknown master integrals for specific 4-loop massless propagator cuts, enhancing the understanding of unitarity cuts in quantum field theory through analytical and numerical methods.

Contribution

It introduces new calculations of 3-particle and 4-particle cut integrals at 4 loops, using dimensional recurrence relations and direct integration, with results verified numerically and analytically.

Findings

Analytic series in epsilon with multiple zeta values up to weight 12.

Dimensional recurrence matrices and numerical tools provided.

Verification through numerical checks and consistency with known integrals.

Abstract

Among the unitarity cuts of 4-loop massless propagators two kinds are currently fully known: the 2-particle cuts with 3 loops corresponding to form-factors, and the 5-particle phase-space integrals. In this paper we calculate master integrals for the remaining ones: 3-particle cuts with 2 loops, and 4-particle cuts with 1 loop. The 4-particle cuts are calculated by solving dimensional recurrence relations. The 3-particle cuts are integrated directly using 1->3 amplitudes with 2 loops, which we also re-derive here up to transcendentality weight 7. The results are verified both numerically, and by showing consistency with previously known integrals using Cutkosky rules. We provide the analytic results in the space-time dimension 4-2{\epsilon} as series in {\epsilon} with coefficients being multiple zeta values up to weight 12. In the ancillary files we also provide dimensional recurrence matrices and SummerTime files suitable for numerical evaluation of the series in arbitrary dimensions with any precision.