Note on non-vacuum conformal family contributions to R\'enyi entropy in two-dimensional CFT

Abstract

We calculate the contributions of a general non-vacuum conformal family to R\'enyi entropy in two-dimensional conformal field theory (CFT). The primary operator of the conformal family can be either non-chiral or chiral, and we denote its scaling dimension by $\Delta$. For the case of two short intervals on complex plane, we expand the R\'enyi mutual information by the cross ratio $x$ to order $x^{2\Delta+2}$. For the case of one interval on torus with the temperature being low, we expand the R\'enyi entropy by $q=\exp(-2\pi\beta/L)$, with $\beta$ being the inverse temperature and $L$ being the spatial period, to order $q^{\Delta+2}$. To make the result meaningful, we require that the scaling dimension $\Delta$ cannot be too small. For two intervals on complex plane we need $\Delta>1$, and for one interval on torus we need $\Delta>2$. We work in small Newton constant limit in gravity side and so large central charge limit in CFT side, and find matches of gravity and CFT results.