RG stability of integrable fishnet models

TL;DR

The paper analyzes perturbative consistency and RG stability of scalar fishnet models arising from gamma-deformations, focusing on planar large-$N$ dynamics. It demonstrates that the 3D $\\phi^{6}$ fishnet is perturbatively stable, while the 4D $\\phi^{4}$ model generically generates double-trace terms, though a simple refinement can protect it; it also presents a perturbatively consistent 6D $\\phi^{3}$ hexagonal fishnet with all-orders anomalous-dimension calculations for single-trace operators. In the hexagonal model, anomalous dimensions for $\\mathrm{tr}(\\phi_i\\phi_j)$ are obtained via a closed integral equation, yielding real branches within a finite $\\alpha$-range and explicit algebraic forms for some cases, with perturbative convergence shown up to high orders. The work connects these fishnet constructions to Zamolodchikov integrability, mapping diagrammatics to 1+1 spin chains and offering insights into finite-size corrections and potential nonperturbative completions, while proposing flavor-based refinements as a route to robust planar sectors.

Abstract

We address the question of perturbative consistency in the scalar fishnet models presented by Caetano, Gurdogan and Kazakov\cite{Gurdogan:2015csr, Caetano:2016ydc}. We argue that their 3-dimensional $φ^{6}$ fishnet model becomes perturbatively stable under renormalization in the large $N$ limit, in contrast to what happens in their 4-dimensional $φ^{4}$ 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 $φ^{3}$ 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 $\text{tr}φ_i φ_j$ operators to all perturbative orders.

Figures (14)

• Figure 1: Anomalous dimension of $\hbox{tr}\phi_i^2$ (black) and $\hbox{tr}\phi_i\phi_{j\neq i}$ (red) in the hexagonal fishnet model as a function of the only meaningful combination of the couplings, $\alpha$.
• Figure 2: The three types of fishnets that have been proven integrable for, respectively, $4$, $3$ and $6$ dimensional target spaces happen to coincide with the three regular tilings of flat 2-dimensional space.
• Figure 3: The tree structure for square fishnet models (left) can be depicted as a square fishnet(right). Horizontal lines represent one field species of field in the theory, vertical lines represent the other, we can consistently consider them all oriented rightwards and upwards. The nodes of the tree we have marked with the same symbol are represented by the same fishnet vertex. Wrapping once a tile of the fishnet corresponds to jumping from a branch of the tree to the consecutive one.
• Figure 4: The hexagon diagram in the 3 dimensional fishnet model and the path on the lattice that produces it. External legs could be contracted with each other or with other tiles. Different fields in the diagram are painted with different colors. In the lattice, the type of filed is uniquely determined by the direction of the line.
• Figure 5: An example on how to transform a path in a lattice into a diagram. Points coloured with the same color are contracted afterwards. In the lattice, horizontal lines are $\phi_1$, vertical lines are $\phi_2$.