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 Q(0) term in the scaling dimension Delta(Q) of the lightest operator with fixed charge Q >> 1, in four dimensions the same mechanism provides a universal Q(0) log Q correction to Delta(Q). Previous works discussing fourdimensional 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 Delta(Q), in four dimensions it scales logarithmically with Q. (C) 2020 The Author. Published by Elsevier B.V.