Five dimensional $O(N)$-symmetric CFTs from conformal bootstrap

We investigate the conformal bootstrap approach to $O(N)$ symmetric CFTs in five dimension with particular emphasis on the lower bound on the current central charge. The bound has a local minimum for all $N>1$, and in the large $N$ limit we propose that the minimum is saturated by the critical $O(N)$ vector model at the UV fixed point, the existence of which has been recently argued by Fei, Giombi, and Klebanov. The location of the minimum is generically different from the minimum of the lower bound of the energy-momentum tensor central charge when it exists for smaller $N$. To better understand the situation, we examine the lower bounds of the current central charge of $O(N)$ symmetric CFTs in three dimension to compare. We find the similar agreement in the large $N$ limit but the discrepancy for smaller $N$ with the other sectors of the conformal bootstrap.