Proof of the zig-zag conjecture

TL;DR

This paper proves the long-standing zig-zag conjecture in quantum field theory by explicitly constructing single-valued multiple polylogarithms, confirming the amplitudes of zig-zag graphs relate to Riemann zeta values.

Contribution

It provides a proof of the zig-zag conjecture by constructing a family of single-valued multiple polylogarithms, establishing the amplitudes of zig-zag graphs in quantum field theory.

Findings

Proves the zig-zag conjecture in quantum field theory.

Identifies zig-zag graphs as the only primitive graphs with known amplitudes.

Connects graph amplitudes to Riemann zeta values.

Abstract

A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the amplitudes of the zig-zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms. The zig-zag graphs therefore provide the only infinite family of primitive graphs in $\phi^4_4$ theory (in fact, in any renormalisable quantum field theory in four dimensions) whose amplitudes are now known.