Recent programs on conformal bootstrap suggest an empirical relationship
between the existence of non-trivial conformal field theories and non-trivial
features such as a kink in the unitarity bound of conformal dimensions in the
conformal bootstrap equations. We report the existene of non-trivial kinks in
the unitarity bound of scalar operators in the adjoint representation of the
$SU(N)$ symmetric conformal field theories. They have interesting properties
(1) the kinks exist in $d<6$ dimensions (2) the location of kinks are when the
unitarity bound hits the space-time dimension $d$ (3) there exists a "conformal
window" of $N<N_*$, where $N_* \sim 15 $ in $d=4$ and $N_* \sim 20$ in $d=5$.