Bootstrapping O(N) vector models in 4 < d < 6

We use the conformal bootstrap to study conformal field theories with O(N) global symmetry in d = 5 and d = 5.95 space-time dimensions that have a scalar operator phi(i) transforming as an O(N) vector. The crossing symmetry of the four-point function of this O(N) vector operator, along with unitarity assumptions, determines constraints on the scaling dimensions of conformal primary operators in the phi(i) x phi(j) operator product expansion-Imposing a lower bound on the second smallest scaling dimension of such an O(N)-singlet conformal primary, and varying the scaling dimension of the lowest one, we obtain an allowed region that exhibits a kink located very close to the interacting O(N)-symmetric conformal field theory conjectured to exist recently by Fei, Giombi, and Klebanov. Under reasonable assumptions on the dimension of the second lowest O(N) singlet in the phi(i) x phi(j) operator product expansion, we observe that this kink disappears in d = 5 for small enough N, suggesting that in this case an interacting O(N) conformal field theory may cease to exist for N below a certain critical value.