Elliptic Functions and Maximal Unitarity

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.

Figures (2)

• Figure 1: The planar double box.
• Figure 2: The distribution of and relations among the eight poles $(z_1,\dots,z_8)$ in the underlying lattice of the elliptic functions. It can be seen that $z_{2i} = z_{2i-1}+z_2$ for $i = 1,2,3,4$ and $z_{i+2} = z_i+\omega_1+\omega_2$ for $i = 1,2,5,6$.