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} = 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.

Highlights

  • The examples of solvable, or integrable QFTs in more than two dimensions are very rare. only two such theories with non-trivial integrable dynamics are known:1 fourdimensional N = 4 SYM theory and three-dimensional ABJM model in the planar, or’t Hooft limit [1]

  • The related diagrams are in principle computable, up to a few scheme dependent constants, by integrability methods

  • This paper is organised as follows: in the first two sections we will introduce the χFT4 and χFT3 theories emerging from double scaling limits of γ-deformed N = 4 SYM and for ABJM theories, repectively, and describe their planar diagrammatics for the two-point functions of single-trace operators

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Summary

Introduction

The examples of solvable, or integrable QFTs in more than two dimensions are very rare. The γ-deformed N = 4 SYM and ABJM theories are not conformal in a strict sense, even in the planar limit [17, 18] The reason for this is the presence of double-trace interactions of the type Tr(χ1χ2)Tr(χ3χ4) that renormalise the Lagrangian already at one-loop order and that the beta functions for the couplings of these interactions are not zero (they were computed at one loop in [19]). This paper is organised as follows: in the first two sections we will introduce the χFT4 and χFT3 theories emerging from double scaling limits of γ-deformed N = 4 SYM and for ABJM theories, repectively, and describe their planar diagrammatics for the two-point functions of single-trace operators. In the following we will explore the diagrammatics and integrability of the uniformly double scaled theories presented above but an analogous analysis applies to this other class of models as well

Planar diagrammatics of χFT4 and the breakdown of conformality
Planar diagrams for correlation functions of χFT4
Planar diagrams for correlation function of χFT3
Asymptotic Bethe ansatz for the spectrum of χFT4 and χFT3
Multi-magnon states
Dispersion relation
Computing multi-loop graphs from the ABA spectrum
The dilatation operator
Two-point functions in dimensional regularization
Two magnons at four loops
Predictions at five loops
Wrapping effects in the bi-scalar chiral model
Conclusions
Strongly twisted β-deformed ABJM Lagrangian
E Feynman integrals

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