A note on the large charge expansion in 4d CFT

TL;DR

This paper investigates universal large charge expansions in 4d conformal field theories with $U(1)$ symmetry, revealing a new $Q^0 ext{log}Q$ correction to operator dimensions and analyzing subleading OPE corrections.

Contribution

It identifies a universal $Q^0 ext{log}Q$ correction in 4d CFTs and computes the first subleading OPE correction for small charge insertions, extending previous 3d and 4d analyses.

Findings

Discovery of a universal $Q^0 ext{log}Q$ correction in 4d CFT operator dimensions.

Calculation of the first subleading correction to OPE coefficients for small charge insertions.

Contrast between 3d and 4d quantum fluctuation effects in large charge expansions.

Abstract

In this letter, we discuss certain universal predictions of the large charge expansion in conformal field theories with $U(1)$ symmetry, mainly focusing on four-dimensional theories. We show that, while in three dimensions quantum fluctuations are responsible for the existence of a theory-independent $Q^0$ term in the scaling dimension $\Delta_Q$ of the lightest operator with fixed charge $Q\gg 1$, in four dimensions the same mechanism provides a universal $Q^0\log Q$ correction to $\Delta_Q$. Previous works discussing four-dimensional theories failed in identifying this term. We also compute the first subleading correction to the OPE coefficient corresponding to the insertion of an arbitrary primary operator with small charge $q\ll Q$ in between the minimal energy states with charge $Q$ and $Q+q$, both in three and four dimensions. This contribution does not depend on the operator insertion and, similarly to the quantum effects in $\Delta_Q$, in four dimensions it scales logarithmically with $Q$.