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arXiv · 1208.1890

Proof of the zig-zag conjecture

Francis Brown·Oliver Schnetz
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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.

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Francis Brown, Oliver Schnetz. 2012-08-09. Proof of the zig-zag conjecture. https://arxiv.org/abs/1208.1890

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