Conformal Triangles and Zig-Zag Diagrams

TL;DR

This paper introduces an integral representation for zig-zag Feynman diagrams in fishnet field theory, enabling exact evaluations and proving a conjecture related to multi-loop diagrams impacting the beta-function in 4D phi^4 theory.

Contribution

It provides a new integral representation for zig-zag diagrams and proves the Broadhurst-Kreimer conjecture on their values in multi-loop cases.

Findings

Exact evaluation of zig-zag diagrams in special cases

Proof of Broadhurst-Kreimer conjecture for multi-loop diagrams

Contribution to understanding the beta-function in 4D phi^4 theory

Abstract

A convenient integral representation for zig-zag four-point and two-point planar Feynman diagrams relevant to the bi-scalar D-dimensional fishnet field theory is obtained. This representation gives a possibility to evaluate exactly diagrams of the zig-zag series in special cases. In particular, we give a fairly simple proof of the Broadhurst-Kreimer conjecture about the values of zig-zag multi-loop two-point diagrams which make a significant contribution to the renormalization group beta-function in 4-dimensional phi^4 theory.