Conformal Bootstrap At Large Charge

TL;DR

This paper explores the structure of conformal field theories with global symmetries at large charge, classifying solutions to crossing equations and connecting them to effective field theories of Goldstone modes.

Contribution

It introduces a classification of large charge solutions in CFTs based on Regge trajectories and their EFT descriptions, including a unique solution for a single trajectory.

Findings

Finite Regge trajectory contributions at large charge

Unique solution for one Regge trajectory as Goldstone EFT

Polynomial roots encode multi-trajectory solutions

Abstract

We consider unitary CFTs with continuous global symmetries in $d>2$. We consider a state created by the lightest operator of large charge $Q \gg 1$ and analyze the correlator of two light charged operators in this state. We assume that the correlator admits a well-defined large $Q$ expansion and, relatedly, that the macroscopic (thermodynamic) limit of the correlator exists. We find that the crossing equations admit a consistent truncation, where only a finite number $N$ of Regge trajectories contribute to the correlator at leading nontrivial order. We classify all such truncated solutions to the crossing. For one Regge trajectory $N=1$, the solution is unique and given by the effective field theory of a Goldstone mode. For two or more Regge trajectories $N \geq 2$, the solutions are encoded in roots of a certain degree $N$ polynomial. Some of the solutions admit a simple weakly coupled EFT description, whereas others do not. In the weakly coupled case, each Regge trajectory corresponds to a field in the effective Lagrangian.