Conformal Symmetry and Feynman Integrals

TL;DR

This paper investigates how hidden singularities can break conformal symmetry in finite loop integrals, leading to differential equations whose solutions are explored through a bootstrap approach, exemplified by the 6D penta-box integral.

Contribution

It introduces a novel method to handle conformal symmetry breaking in loop integrals using differential equations and bootstrap techniques, applied to a 6D penta-box integral.

Findings

Derived linear second-order differential equations for loop integrals.

Identified the non-homogeneous part of the equations from anomaly contact terms.

Successfully bootstrapped the 6D penta-box integral solution.

Abstract

Singularities hidden in the collinear region around an external massless leg may lead to conformal symmetry breaking in otherwise conformally invariant finite loop momentum integrals. For an $\ell$-loop integral, this mechanism leads to a set of linear $2$nd-order differential equations with a non-homogeneous part. The latter, due to the contact nature of the anomaly in momentum space, is determined by $(\ell-1)$-loop information. Solving such differential equations in general is an open problem. In the case of 5-particle amplitudes up to two loops, the function space is known, and we can thus follow a bootstrap approach to write down the solution. As a first application of this method, we bootstrap the 6D penta-box integral.