Bootstrapping pentagon functions

TL;DR

This paper extends the space of pentagon functions to non-planar cases, classifies relevant functions up to weight four, and uses bootstrap methods with Mellin-Barnes insights to evaluate complex five-particle integrals.

Contribution

It introduces a non-planar pentagon function space and a bootstrap approach for two-loop five-particle scattering amplitudes.

Findings

Classified non-planar pentagon functions up to weight four.

Constrained symbol entries using branch cut information.

Successfully evaluated symbols of two non-trivial five-particle integrals.

Abstract

In PRL 116 (2016) no.6, 062001, the space of planar pentagon functions that describes all two-loop on-shell five-particle scattering amplitudes was introduced. In the present paper we present a natural extension of this space to non-planar pentagon functions. This provides the basis for our pentagon bootstrap program. We classify the relevant functions up to weight four, which is relevant for two-loop scattering amplitudes. We constrain the first entry of the symbol of the functions using information on branch cuts. Drawing on an analogy from the planar case, we introduce a conjectural second-entry condition on the symbol. We then show that the information on the function space, when complemented with some additional insights, can be used to efficiently bootstrap individual Feynman integrals. The extra information is read off of Mellin-Barnes representations of the integrals, either by evaluating simple asymptotic limits, or by taking discontinuities in the kinematic variables. We use this method to evaluate the symbols of two non-trivial non-planar five-particle integrals, up to and including the finite part.