Thermal one-point functions and single-valued polylogarithms

TL;DR

This paper reveals that thermal one-point functions in certain free quantum field theories are expressed through single-valued polylogarithms, linking complex analysis, conformal field theory, and Feynman diagram representations.

Contribution

It establishes a novel connection between thermal one-point functions, single-valued polylogarithms, and deformations of generalized free CFTs, expanding understanding of thermal correlators in quantum field theories.

Findings

Thermal one-point functions are given by single-valued polylogarithms.

The polylogarithm rank relates to the spacetime dimension d.

The results generalize known conformal ladder graph representations.

Abstract

I point out that the thermal one-point functions of a pair of relevant operators in massive free QFTs, in odd dimensions and in the presence of an imaginary chemical potential for a U(1) global charge, are given by certain classes of single-valued polylogarithms. This result is verified by a direct calculation using the thermal OPE. The complex argument of the polylogarithms parametrize a two-dimensional subspace of relevant deformations of generalised free CFTs, while the rank of the polylogarithms is related to the dimension d. This may be compared with the well-known representation of single-valued polylogarithms as multiloop Feynman amplitudes. As an example, the thermal one-point function of the U(1) charge in d-dimensions generalises the thermal average of the twist operator in a pair of harmonic oscillators and is given by the well-known conformal ladder graphs in four dimensions.