Bifundamental Multiscalar Fixed Points in $d=3-\epsilon$

We study fixed-points of scalar fields that transform in the bifundamental representation of $O(N)\times O(M)$ in $3-\epsilon$ dimensions, generalizing the classic tricritical sextic vector model. In the limit where $N$ is large but $M$ is finite, we determine the complete beta function to order $1/N$ for arbitrary $M$. We find a rich collection of large-$N$ fixed-points in $d=3$, as well as fixed-points in $d=3-\epsilon$, that can be studied to all orders in the parameter $\hat{\epsilon}=N\epsilon$. With the goal of defining a large-$N$ nonsupersymmetric conformal field theory dominated by a web of planar diagrams, we also study fixed-points in the ``bifundamental'' large-$N$ limit, in which $M$ and $N$ are both large, but the ratio $M/N$ is held fixed. We find a unique infrared fixed-point in $d=3-\epsilon$, which we determine to order $\epsilon^2$. When $M/N \ll 1$, we also find an ultraviolet fixed-point in $d=3$ and $d=3-\epsilon$ that merges with the infrared fixed-point at $\epsilon \sim O(M/N)$. We expect at least one of two candidate fixed-points in integer dimensions -- the infrared fixed-point in $d=2$ and the ultraviolet fixed-point in $d=3$ -- to survive for finite values of $M/N$.