Elliptic Functions and Maximal Unitarity

TL;DR

This paper explores the algebraic geometry of loop integrals in quantum field theory, focusing on integrals with irrational components, and introduces methods to compute their coefficients using elliptic functions.

Contribution

It introduces a novel analysis of algebraic varieties with irrational components in maximal unitarity cuts and derives projectors using elliptic functions for two-loop integrals.

Findings

Derived unique projectors for master integrals using multivariate residues.

Analyzed algebraic varieties with irrational components in loop integrals.

Generated integration-by-parts identities analytically from elliptic functions.

Abstract

Scattering amplitudes at loop level can be reduced to a basis of linearly independent Feynman integrals. The integral coefficients are extracted from generalized unitarity cuts which define algebraic varieties. The topology of an algebraic variety characterizes the difficulty of applying maximal cuts. In this work, we analyze a novel class of integrals whose maximal cuts give rise to an algebraic variety with irrational irreducible components. As a phenomenologically relevant example we examine the two-loop planar double-box contribution with internal massive lines. We derive unique projectors for all four master integrals in terms of multivariate residues along with Weierstrass' elliptic functions. We also show how to generate the leading-topology part of otherwise infeasible integration-by-parts identities analytically from exact meromorphic differential forms.