Single-valued multiple polylogarithms and a proof of the zig–zag conjecture

A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the periods 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 which correspond to multiple zeta values ζ(2,…,2,3,2,…2)ζ(2,…,2,3,2,…2) and using the method of graphical functions. The zig–zag graphs are the only infinite family of primitive graphs in ϕ44 theory (in fact, in any renormalisable quantum field theory in four dimensions) whose periods are now known.