The analytic bootstrap at finite temperature

TL;DR

This paper introduces universal formulas for thermal two-point functions of scalar operators using their analytic structure, satisfying bootstrap conditions and applicable to both weakly and strongly-coupled theories.

Contribution

It develops a dispersion relation-based approach to compute thermal correlators that satisfy bootstrap constraints, with explicit tests in various theories including the 3d Ising model.

Findings

Derived a dispersion relation fixing correlators up to a constant.

Constructed a thermal two-point function formula summing over images.

Confirmed agreement with Monte Carlo simulations in the 3d Ising model.

Abstract

We propose new universal formulae for thermal two-point functions of scalar operators based on their analytic structure, constructed to manifestly satisfy all the bootstrap conditions. We derive a dispersion relation in the complexified time plane, which fixes the correlator up to an additive constant and theory-dependent dynamical information. At non-zero spatial separation we introduce a formula for the thermal two-point function obtained by summing over images of the dispersion relation result obtained in the OPE regime. This construction satisfies all thermal bootstrap conditions, with the exception of clustering at infinite distance, which must be verified on a case-by-case basis. We test our results both in weakly and strongly-coupled theories. In particular, we show that the asymptotic behavior for the heavy sector proposed in~\cite{Marchetto:2023xap} and its correction can be explicitly derived from the dispersion relation. We combine analytical and numerical results to compute the thermal two-point function of the energy operator in the $3d$ Ising model and find agreement with Monte Carlo simulations.