Generalised ladders and single-valued polylogarithms

TL;DR

This paper introduces a new class of loop integrals generalizing ladder series, solved using single-valued polylogarithmic functions, with explicit recursive constructions and simplified symbol extraction, connecting to vacuum diagrams like wheels and zigzags.

Contribution

It develops a recursive method to explicitly construct generalized ladder integrals using differential equations and single-valued polylogarithms, extending known ladder series.

Findings

Explicit formula for the simplest generalized ladder integral

Recursive construction method for the class of integrals

Connection to vacuum diagrams including wheels and zigzags

Abstract

We introduce and solve an infinite class of loop integrals which generalises the well-known ladder series. The integrals are described in terms of single-valued polylogarithmic functions which satisfy certain differential equations. The combination of the differential equations and single-valued behaviour allow us to explicitly construct the polylogarithms recursively. For this class of integrals the symbol may be read off from the integrand in a particularly simple way. We give an explicit formula for the simplest generalisation of the ladder series. We also relate the generalised ladder integrals to a class of vacuum diagrams which includes both the wheels and the zigzags.