A note on the large charge expansion in 4d CFT

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 Q0 term in the scaling dimension ΔQ of the lightest operator with fixed charge Q≫1, in four dimensions the same mechanism provides a universal Q0log⁡Q correction to ΔQ. 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≪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 ΔQ, in four dimensions it scales logarithmically with Q.