Integral representation for three-loop banana graph

TL;DR

This paper introduces a new integral representation for the three-loop banana graph in quantum field theory, enabling analytical and numerical calculations across different dimensions and thresholds, and facilitating the computation of more complex integrals.

Contribution

It develops a novel integral representation for the three-loop banana integral, generalizable to any order in epsilon expansion and applicable to complex three-loop diagrams.

Findings

Representation valid in 2-2ε dimensions

Applicable for numerical evaluation above and below threshold

Facilitates computation of complex three-loop integrals

Abstract

It has recently been shown that two-loop kite-type diagrams can be computed analytically in terms of iterated integrals with algebraic kernels. This result was obtained using a new integral representation for two-loop sunset subgraphs. In this paper, we have developed a similar representation for a three-loop banana integral in $d = 2-2\varepsilon$ dimensions. This answer can be generalized up to any given order in the $\varepsilon$-expansion and can be calculated numerically both below and above the threshold. We also demonstrate how this result can be used to compute more complex three-loop integrals containing the three-loop banana as a subgraph.