Feynman integrals and critical modular $L$-values

TL;DR

This paper proves Broadhurst's conjecture that a specific Feynman integral can be expressed in terms of a critical modular L-value, confirming a long-standing prediction in mathematical physics and number theory.

Contribution

The paper establishes the validity of Broadhurst's conjecture and extends similar identities to other polynomials in the family, advancing understanding of Feynman integrals and modular forms.

Findings

Broadhurst's conjecture is proven true.

Feynman integrals are expressed in terms of modular L-values.

Identities for other polynomials in the family are established.

Abstract

Broadhurst conjectured that the Feynman integral associated to the polynomial corresponding to $t=1$ in the one-parameter family $(1+x_1+x_2+x_3)(1+x_1^{-1}+x_2^{-1}+x_3^{-1})-t$ is expressible in terms of $L(f,2),$ where $f$ is a cusp form of weight $3$ and level $15$. Bloch, Kerr and Vanhove have recently proved that the conjecture holds up to a rational factor. In this paper, we prove that Broadhurst's conjecture is true. Similar identities involving Feynman integrals associated to other polynomials in the same family are also established.