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

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 $\mathrm{\Delta }$. For the case of two short intervals on complex plane, we expand the R'enyi mutual information by the cross ratio $x$ to order $x^{2\mathrm{\Delta }+2}$. For the case of one interval on torus with the temperature being low, we expand the R'enyi entropy by $q=\exp (-2\pi \beta \mathrm{/}L)$, with $\beta$ being the inverse temperature and $L$ being the spatial period, to order $q^{\mathrm{\Delta }+2}$. To make the result meaningful, we require that the scaling dimension $\mathrm{\Delta }$ cannot be too small. For two intervals on complex plane we need $\mathrm{\Delta }>1$, and for one interval on torus we need $\mathrm{\Delta }>2$. 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.