Generalised ladders and single-valued polylogarithms

TL;DR

The paper develops a comprehensive framework to evaluate an infinite class of loop integrals that generalise ladder diagrams, by expressing the results as single-valued polylogarithms that satisfy a hierarchy of differential equations. A key insight is that the symbol can be read off directly from the integrand using shuffle Hopf algebra, while single-valuedness, implemented recursively, fixes the remaining ambiguities and multi-zeta terms. The author provides explicit all-loop constructions for the depth-3 non-ladder family, connects the top-part to Knizhnik-Zamolodchikov equations, and extends the approach to conformal four-point and two-point limits, yielding new results for vacuum graphs including wheels and zigzags. These methods illuminate the transcendental structure of high-loop diagrams and furnish a practical path to analytic and numerical evaluations of complex conformal integrals.

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.

Figures (7)

• Figure 1: The four-point ladder integrals with the associated four-point and three-point dual diagrams.
• Figure 2: The generalised ladder integrals defined in dual coordinate space.
• Figure 3: Reading off the word $w$ associated to a generalised ladder integral. To simplify the picture the propagators between the integration vertices and the external points have been shortened to stubs.
• Figure 4: The depth 3 family of generalised ladders.
• Figure 5: Conformal four-point integrals which generalise the three-point ladders introduced in section \ref{['sect-genlad']}. Taking the limit $x_3 \rightarrow \infty$ yields generalised ladders with both three-point and four-point integration vertices.