Proof of the zig-zag conjecture

2012/08/09 by Francis Brown, Brown, Francis, Oliver Schnetz +1 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1208.1890

openalex publication_date 2012/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ϕ⁴₄ theory (in fact, in any renormalisable quantum field theory in four dimensions) whose amplitudes are now known.

Citations

• Framings for graph hypersurfaces
• Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird

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• The Non-Planar Four-Point Integrand and Konishi Dimension in N=4 Super Yang-Mills Theory at Five Loops
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