The criterion of irreducibility of multi-loop Feynman integrals

TL;DR

The paper introduces a sufficient irreducibility criterion for multi-loop Feynman integrals within the IBP framework by seeking a separating solution to the recurrence relations. It constructs an auxiliary integral representation $s(\underline{n})=\int dx_1...dx_N\, \frac{1}{x_1^{n_1}\cdots x_N^{n_N}} g(x_i)$ and imposes $R(I^-,I^+)s(\underline{n})=0$ via $R(\partial_i,x_i)g(x_i)=0$, with $g(x_i)$ built from polynomials of non-integer degree. The criterion is applied to a master 3-loop massless non-planar integral, where a suitable contour and Taylor-expansion argument yield $s(1,1,1,1,0,1,1,1)=1$, proving irreducibility of $B(1,1,1,1,0,1,1,1)$. Overall, the work links the algebraic structure of IBP recurrences to concrete irreducibility tests and outlines a path to verify irreducibility up to at least four loops.

Abstract

The integration by parts recurrence relations allow to reduce some Feynman integrals to more simple ones (with some lines missing). Nevertheless the possibility of such reduction for the given particular integral was unclear. The recently proposed technique for studying the recurrence relations as by-product provides with simple criterion of the irreducibility.