Two-loop Integral Reduction from Elliptic and Hyperelliptic Curves

TL;DR

This paper introduces a novel algebraic geometry-based algorithm for reducing two-loop Feynman integrals by relating IBP relations to exact meromorphic one-forms on algebraic curves, avoiding explicit integral calculations.

Contribution

It presents a new efficient method for integral reduction in two-loop diagrams using algebraic geometry, bypassing complex integral evaluations.

Findings

Successfully applied to complex two-loop diagrams with massive legs

No explicit elliptic or hyperelliptic integral computations required

Demonstrates efficiency and effectiveness of the algebraic geometry approach

Abstract

We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves. Since it is easy to find such exact meromorphic one-forms from algebraic geometry, this idea provides a new highly efficient algorithm for integral reduction. We demonstrate the power of this method via several complicated two-loop diagrams with internal massive legs. No explicit elliptic or hyperelliptic integral computation is needed for our method.