Generalised ladders and single-valued polylogs

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.

Quantum gravity from conformal field theory

“…The particular class of single-valued polylogarithms of interest here are linear combinations of polylogarithms constructed on the singularities (or 'letters') {x, 1 − x,x, 1 −x} such that they are single-valued whenx is taken to be the complex conjugate of x. They are constructed in general in [50] and appear in many contexts to discuss the perturbative contributions to the correlation functions p 1 p 2 p 3 p 4 [40,42] as well as in multi-Regge kinematics of scattering amplitudes [51,52] and Feynman integral calculations [53,54]. Since our result for the double discontinuity F…”

Integral representations combining ladders and crossed-ladders

“…[40]. Mathematically, the completely massless case is special (in D = 4), since here the "proper" (non-crossed) ladder graphs become conformally invariant, and possess closed-form expressions in terms of polylogarithms for any N [35,[41][42][43][44][45][46].…”

Leading singularities and off-shell conformal integrals

“…[63] it was shown that infinite classes of generalised ladder integrals can be expressed in terms of single-valued combinations of HPLs. Single-valued analogues of HPLs were studied in detail in ref.…”

“…As we will demonstrate in the following sections, this integral obeys a second-order differential equation whose solution is uniquely specified by imposing single-valued behaviour, similar to the generalised ladders considered in ref. [63]. The four-loop contribution to the stress-tensor four-point function in N = 4 SYM contains some integrals that do not obviously obey any such differential equations, and with the effort presented here we also wanted to learn to what extent the two-step procedure of deriving symbols and subsequently uplifting them to functions can be repeated for those cases.…”

“…[63], slightly simpler, but very similar, equations were analysed for a class of generalised ladder integrals. The analysis of ref.…”