Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity

TL;DR

The paper computes planar two-loop five-gluon amplitudes in the leading-color limit using a variant of numerical unitarity extended to finite-field arithmetic, enabling exact extraction of integral coefficients and a complete reduction to a minimal master-integral basis. It introduces unitarity-compatible IBP identities solved via syzygy equations and employs momentum-twistor parameterizations to keep all quantities field-valued, avoiding complex numbers. The implementation combines on-shell cut equations, finite-field rational reconstruction, and analytic master integrals to produce high-precision results for all four independent helicity configurations, with validation against the universal IR/UV pole structure. This approach offers a scalable path to analytic forms via functional reconstruction and generalizes to other multi-loop, multi-leg amplitudes, including non-planar cases.

Abstract

We present a calculation of the planar two-loop five-gluon amplitudes. The amplitudes are obtained in a variant of the generalized unitarity approach suitable for numerical computations, which we extend for use with finite field arithmetics. Employing a new method for the generation of unitarity-compatible integration-by-parts identities, all helicity amplitudes are reduced to a linear combination of master integrals for the first time. The approach allows us to compute exact values for the integral coefficients at rational phase-space points. All required master integrals are known analytically, and we obtain arbitrary-precision values for the amplitudes.

Figures (3)

• Figure 1: Displayed are the conventions for assigning propagators in a two-loop diagram.
• Figure 2: Hierarchy of propagator structures for two-loop five-point gluon scattering. Only topologically inequivalent structures are shown.
• Figure 3: Propagator structures with master integrals