RG stability of integrable fishnet models

Abstract

We address the question of perturbative consistency in the scalar fishnet models presented by Caetano, Gürdoğan and Kazakov [1, 2]. We argue that their 3-dimensional ϕ6 fishnet model becomes perturbatively stable under renormalization in the large N limit, in contrast to what happens in their 4-dimensional ϕ4 fishnet model, in which double trace terms are known to be generated by the RG flow. We point out that there is a direct way to modify this second theory that protects it from such corrections. Additionally, we observe that the 6-dimensional ϕ3 Lagrangian that spans an hexagonal integrable scalar fishnet is consistent at the perturbative level as well. The nontriviality and simplicity of this last model is illustrated by computing the anomalous dimensions of its tr ϕiϕj operators to all perturbative orders.