Chiral limit of N = 4 SYM and ABJM and integrable Feynman graphs

TL;DR

<3-5 sentence high-level summary> We address the problem of understanding solvable quantum field theories beyond two dimensions by constructing chiral field theories χFTs as double-scaling limits of γ-twisted N=4 SYM and ABJM. The core approach combines planar integrability (via doubly scaled asymptotic Bethe Ansatz) with a drastically simplified Feynman diagrammatics (globe, wheel, and spider-web graphs) to determine anomalous dimensions and operator mixing. The key contributions include explicit DS-ABA equations for χFT_4 and χFT_3, a detailed mapping between operator mixing and single Feynman diagrams up to high loops, and a framework to fix higher-loop integrals using spectral data, including five-loop examples. This work opens pathways to exact high-loop information in non-conformal, chirally oriented theories and provides a testing ground for QSC-based methods in a controlled DS setting.

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.

Figures (13)

• Figure 1: Feynman rules for the bi-scalar theory. Double lines represent the colour dependence of the fields. There are two types of propagators: solid lines correspond to $\phi^1$ and dashed lines to $\phi^2$ fields. The arrows besides the double lines indicate the flavour flow of complex scalars. The only interaction vertex is a particular quartic one and its orientation implies a sense of chirality for the graphs it enters. The vertex with opposite orientation (chirality) is absent.
• Figure 2: The one-loop planar Feynman graphs which could contribute to the renormalization of the single coupling $\xi$ (on the left picture) or generate the mass (on the right picture). But on each graph, only one of two vertices is present in perturbation theory and the other vertex, indicated with red, has a wrong ordering of the fields, so that these diagrams do not contribute. This argument can be generalized to any loop order. Therefore the mass is not generated and the coupling $\xi$ is not running in the planar limit.
• Figure 3: Non-planar diagrams that generate couplings of the form ${\rm Tr}(\phi^1\phi_2^\dagger)\,{\rm Tr}(\phi^2\phi_1^\dagger)$ and $\bigl[{\rm Tr}(\phi^1\phi_1^\dagger)\bigr]^2$.
• Figure 4: Loop corrections to the two-point function of the BMN vacuum operator in the bi-scalar theory. It is clearly seen that in the bulk these graphs have the regular "fishnet" structure.
• Figure 5: Amputated graphs for pair correlators of BMN-vacuum -- wheel graphs.