Sextic tensor model in rank 3 at next-to-leading order

Abstract

Abstract We compute the four-loop beta functions of short and long-range multi-scalar models with general sextic interactions and complex fields. We then specialize the beta functions to a U(N)3 symmetry and study the renormalization group at next-to-leading order in N and small ϵ. In the short-range case, ϵ is the deviation from the critical dimension while it is the deviation from the critical scaling of the free propagator in the long-range case. This allows us to find the 1/N corrections to the rank-3 sextic tensor model of [1]. In the short-range case, we still find a non-trivial real IR stable fixed point, with a diagonalizable stability matrix. All couplings, except for the so-called wheel coupling, have terms of order ϵ0 at leading and next-to-leading order, which makes this fixed point different from the other melonic fixed points found in quartic models. In the long-range case, the corrections to the fixed point are instead not perturbative in ϵ and hence unreliable; we thus find no precursor of the large-N fixed point.