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 three dimensions quantum fluctuations are responsible for existence a theory-independent $Q^0$ term scaling dimension $\Delta_Q$ lightest operator fixed $Q\gg 1$, four same mechanism provides $Q^0\log Q$ correction to $\Delta_Q$. Previous works discussing failed identifying term. also compute first subleading OPE coefficient corresponding insertion an arbitrary primary small $q\ll between minimal energy states $Q$ and $Q+q$, both dimensions. This contribution does not depend and, similarly effects $\Delta_Q$, it scales logarithmically $Q$.