On short interval expansion of R\'enyi entropy

R\'enyi entanglement entropy provides a new window to study the AdS/CFT correspondence. In this paper we consider the short interval expansion of R\'enyi entanglement entropy in two-dimensional conformal field theory. This amounts to do the operator product expansion of the twist operators. We focus on the vacuum Verma module and consider the quasiprimary operators constructed from the stress tensors. After obtaining the expansion coefficients of the twist operators to level 6 in vacuum Verma module, we compute the leading contributions to the R\'enyi entropy, to order 6 in the short interval expansion. In the case of one short interval on cylinder, we reproduce the first several leading contributions to the R\'enyi entropy. In the case of two short disjoint intervals with a small cross ratio $x$, we obtain not only the classical and 1-loop quantum contributions to the R\'enyi entropy to order $x^6$, both of which are in perfect match with the ones found in gravity, but also the leading $1/c$ contributions, which corresponds to 2-loop corrections in the bulk.