Moduli Spaces in CFT: Bootstrap Equation in a Perturbative Example

TL;DR

This paper explores bootstrap constraints in conformal field theories with spontaneous symmetry breaking, using a perturbative example to analyze the spectrum and form factors in moduli spaces.

Contribution

It introduces a bootstrap equation involving new data like asymptotic spectrum and form factors, and applies it to a perturbative model to verify its implications.

Findings

Bootstrap equation converges well in the perturbative model

Explicit relations among theory data are verified

Spectrum and form factors satisfy bootstrap constraints

Abstract

Conformal field theories that exhibit spontaneous breaking of conformal symmetry (a moduli space of vacua) must satisfy a set of bootstrap constraints, involving the usual data (scaling dimensions and OPE coefficients) as well as new data such as the spectrum of asymptotic states in the broken vacuum and form factors. The simplest bootstrap equation arises by expanding a two-point function of local operators in two channels, at short distance using the OPE and at large distance using the EFT in the broken vacuum. We illustrate this equation in what is arguably the simplest perturbative model that exhibits conformal symmetry breaking, namely the real $ABC$ model in $d = 4 -\epsilon$ dimensions. We investigate the convergence properties of the bootstrap equation and check explicitly many of the non-trivial relations that it imposes on theory data.