The Landau Bootstrap

TL;DR

This paper introduces a bootstrap method for Feynman integrals based on their singularities, providing a new way to determine their values without relying on non-perturbative constraints.

Contribution

The authors develop a singularity-based bootstrap approach for Feynman integrals, extending existing constraints to cases with equal or zero mass particles.

Findings

Successfully bootstrapped the four-point double box integral with a massive loop.

Extended singularity constraints to integrals with equal or vanishing masses.

Demonstrated the method's effectiveness on multiple examples.

Abstract

We advocate a strategy of bootstrapping Feynman integrals from just knowledge of their singular behavior. This approach is complementary to other bootstrap programs, which exploit non-perturbative constraints such as unitarity, or amplitude-level constraints such as gauge invariance. We begin by studying where a Feynman integral can become singular, and the behavior it exhibits near these singularities. We then characterize the space of functions that we expect the integral to evaluate to, in order to formulate an appropriate ansatz. Finally, we derive constraints on where each singularity can appear in this ansatz, and use information about the expansion of the integral around singular points in order to determine the value of all remaining free coefficients. Throughout, we highlight how constraints that have previously only been derived for integrals with generic masses can be extended to integrals involving particles of equal or vanishing mass. We illustrate the effectiveness of this approach by bootstrapping a number of examples, including the four-point double box with a massive internal loop.