First-order and pseudo-first-order transition in the high dimensional $O(N)\otimes O(M)$ model

Using the renormalization group approach, we consider the $O(N)\otimes O(M)$ model in four and more dimensions. We find that independently on $N$ and $M$, for $N\geq M\geq 2$, a transition can be of both the first and second order. In $d>4$, we also cannot exclude a pseudo-first-order behavior. As specific physically interesting cases, we consider the lattice version of the $O(2)\otimes O(2)$, $O(3)\otimes O(2)$ and $O(3)\otimes O(3)$ sigma models on a four dimensional hypercubic lattice. In all these cases, we find a distinct first-order transition.