Chiral limit of $$ \mathcal{N} $$ = 4 SYM and ABJM and integrable Feynman graphs

Abstract

We consider a special double scaling limit, recently introduced by two of the authors, combining weak coupling and large imaginary twist, for the γ-twisted $$ \mathcal{N} $$ N = 4 SYM theory. We also establish the analogous limit for ABJM theory. The resulting non-gauge chiral 4D and 3D theories of interacting scalars and fermions are integrable in the planar limit. In spite of the breakdown of conformality by double-trace interactions, most of the correlators for local operators of these theories are conformal, with non-trivial anomalous dimensions defined by specific, integrable Feynman diagrams. We discuss the details of this diagrammatics. We construct the doubly-scaled asymptotic Bethe ansatz (ABA) equations for multi-magnon states in these theories. Each entry of the mixing matrix of local conformal operators in the simplest of these theories — the bi-scalar model in 4D and tri-scalar model in 3D — is given by a single Feynman diagram at any given loop order. The related diagrams are in principle computable, up to a few scheme dependent constants, by integrability methods (quantum spectral curve or ABA). These constants should be fixed from direct computations of a few simplest graphs. This integrability-based method is advocated to be able to provide information about some high loop order graphs which are hardly computable by other known methods. We exemplify our approach with specific five-loop graphs.

RG stability of integrable fishnet models

“…In this section we generalize the set of Lagrangians presented by Caetano, Gürdogan and Kazakov [1,2] and discuss under which circumstances the resulting construction is stable under the RG flow, and therefore protected from the appearance of double trace operators.…”

“…As a result, the 3 dimensional theory introduced by [2] is conformal, at least perturbatively, in the planar limit.…”

“…On the one hand, some fishnet models emerge as gamma deformations of N = 4 SYM and ABJM, and therefore, at least these cases are expected to share the same integrability properties, despite the fishnet case presents some structural differences in relation to the undeformed case. 4 See [2] for a complete discussion on this integrability approach. On the other hand, as we mentioned in the introduction, Zamolodchikov proved in [10] that in scalar regular fishnet models the planar diagrams satisfy a Yang-Baxter equation in terms of their position/momentum propagators, which opens the perspective of applying integrability techniques of noncompact representations of orthogonal groups (see [27] and references therein, for instance).…”

“…In this context an unexpected observation was made [1,2]: a particular limit of γ-deformation that was dubbed the "double scaling limit" in [1] gives rise to scalar matrix models with classically marginal couplings and a surprisingly simple diagrammatic structure. More precisely, in the large N limit of these models the only nontrivial dependence on the coupling for most quantities arises at the perturbative level from fishnet Feynman JHEP06(2017)012 diagrams, i.e., arrangements of propagators in the shape of a regular lattice, which is made of triangular and square cells in the cases of ABJM and N = 4 SYM respectively.…”

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Abstract:We address the question of perturbative consistency in the scalar fishnet models presented by Caetano, Gürdogan and Kazakov [1,2]. 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.

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