Feynman integrals and hyperlogarithms

TL;DR

This paper develops recursive formulas for massless Feynman integrals, introduces hyperlogarithm algorithms, and proves that certain multi-loop integrals can be expressed in terms of multiple polylogarithms, enabling effective computation.

Contribution

It provides a new method to express massless 3- and 4-point Feynman integrals as linear combinations of convergent integrals and demonstrates their representation via multiple polylogarithms at all orders.

Findings

Feynman integrals can be written as linear combinations of convergent integrals.

Massless 3- and 4-point graphs' integrals are expressible in terms of multiple polylogarithms.

Explicit computation of a non-multiple zeta value counterterm in massless phi^4 theory.

Abstract

We study Feynman integrals in the representation with Schwinger parameters and derive recursive integral formulas for massless 3- and 4-point functions. Properties of analytic (including dimensional) regularization are summarized and we prove that in the Euclidean region, each Feynman integral can be written as a linear combination of convergent Feynman integrals. This means that one can choose a basis of convergent master integrals and need not evaluate any divergent Feynman graph directly. Secondly we give a self-contained account of hyperlogarithms and explain in detail the algorithms needed for their application to the evaluation of multivariate integrals. We define a new method to track singularities of such integrals and present a computer program that implements the integration method. As our main result, we prove the existence of infinite families of massless 3- and 4-point graphs (including the ladder box graphs with arbitrary loop number and their minors) whose Feynman integrals can be expressed in terms of multiple polylogarithms, to all orders in the epsilon-expansion. These integrals can be computed effectively with the presented program. We include interesting examples of explicit results for Feynman integrals with up to 6 loops. In particular we present the first exactly computed counterterm in massless phi^4 theory which is not a multiple zeta value, but a linear combination of multiple polylogarithms at primitive sixth roots of unity (and divided by $\sqrt{3}$). To this end we derive a parity result on the reducibility of the real- and imaginary parts of such numbers into products and terms of lower depth.