Proof of the zig-zag conjecture

TL;DR

The paper proves the Broadhurst-Kreimer zig-zag conjecture by constructing two tailored families of single-valued multiple polylogarithms, built from Hoffman-type multiple zeta values, and controlling their monodromy on the punctured Riemann sphere. It advances the parametric-integration approach by exploiting a precise decomposition of noncommutative generating series into odd and even zeta components, guided by Zagier's evaluations, to produce exact period formulas for the zig-zag graphs. The result establishes that zig-zag amplitudes in φ^4_4 are rational multiples of a single odd zeta value, confirming the conjecture and highlighting the zig-zag family as a primitive in four-dimensional renormalisable quantum field theories, with implications for motivic coaction and Galois symmetry of Feynman periods.

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 $φ^4_4$ theory (in fact, in any renormalisable quantum field theory in four dimensions) whose amplitudes are now known.

Theorems & Definitions (43)

• Theorem 1.1
• Corollary 1.2
• Definition 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Remark 1.7
• Lemma 2.1
• proof
• Theorem 2.2