Tensor $O(N)$ model near six dimensions: fixed points and conformal windows from four loops

In search of non-trivial field theories in high dimensions, we study further the tensor representation of the $O(N)$-symmetric $\phi^4$ field theory introduced by Herbut and Janssen (Phys. Rev. D. 93, 085005 (2016)), by using four-loop perturbation theory in two cubic interaction coupling constants near six dimensions. For infinitesimal values of the parameter $\epsilon=(6-d)/2$ we find infrared-stable fixed point with two relevant quadratic operators for $N$ within the conformal windows $1<N<2.653$ and $2.999<N<4$, and compute critical exponents at this fixed point to the order $\epsilon^4$. Taking the four-loop beta-functions at their face value we determine the higher-order corrections to the edges of the above conformal windows at finite $\epsilon$, to find both intervals to shrink to zero above $\epsilon\approx 0.15$. The disappearance of the conformal windows with the increase of $\epsilon$ is due to the collision of the Wilson-Fisher $\mathcal{O}(\epsilon)$ infrared fixed point with the $\mathcal{O}(1)$ mixed-stable fixed point that appears at two and persists at higher loops. The latter may be understood as a Banks-Zaks type fixed point that becomes weakly coupled near the right edge of either conformal window. The consequences and issues raised by such an evolution of the flow with dimension are discussed. It is also shown both within the perturbation theory and exactly that the tensor representation at $N=3$ and right at the $\mathcal{O}(\epsilon)$ infrared-stable fixed point exhibits an emergent $U(3)$ symmetry. A role of this enlarged symmetry in possible protection of the infrared fixed point at $N=3$ is noted.