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On a CFT limit of planar $γ_i$-deformed $\mathcal{N}=4$ SYM theory

Christoph Sieg, Matthias Wilhelm

TL;DR

This work shows that the proposed planar limit of the gamma_i-deformed N=4 SYM theory is not conformal unless double-trace quartic couplings are included. By adding these couplings and their counterterms, the authors derive one-loop beta-functions and analyze two-loop planar anomalous dimensions, revealing complex conformal fixed points for certain couplings. The study demonstrates that planar multi-point functions are also sensitive to the double-trace sector, and proposes integrability-based tests at the conformal points. The results clarify the role of double-trace interactions in achieving a consistent, potentially conformal limit and open avenues for exploring integrability in this simplified framework.

Abstract

We show that an integrable four-dimensional non-unitary field theory that was recently proposed as a certain limit of the $γ_i$-deformed $\mathcal{N}=4$ SYM theory is incomplete and not conformal -- not even in the planar limit. We complete this theory by double-trace couplings and find conformal one-loop fix-points when admitting respective complex coupling constants. These couplings must not be neglected in the planar limit, as they can contribute to planar multi-point functions. Based on our results for certain two-loop planar anomalous dimensions, we propose tests of integrability.

On a CFT limit of planar $γ_i$-deformed $\mathcal{N}=4$ SYM theory

TL;DR

This work shows that the proposed planar limit of the gamma_i-deformed N=4 SYM theory is not conformal unless double-trace quartic couplings are included. By adding these couplings and their counterterms, the authors derive one-loop beta-functions and analyze two-loop planar anomalous dimensions, revealing complex conformal fixed points for certain couplings. The study demonstrates that planar multi-point functions are also sensitive to the double-trace sector, and proposes integrability-based tests at the conformal points. The results clarify the role of double-trace interactions in achieving a consistent, potentially conformal limit and open avenues for exploring integrability in this simplified framework.

Abstract

We show that an integrable four-dimensional non-unitary field theory that was recently proposed as a certain limit of the -deformed SYM theory is incomplete and not conformal -- not even in the planar limit. We complete this theory by double-trace couplings and find conformal one-loop fix-points when admitting respective complex coupling constants. These couplings must not be neglected in the planar limit, as they can contribute to planar multi-point functions. Based on our results for certain two-loop planar anomalous dimensions, we propose tests of integrability.

Paper Structure

This paper contains 7 sections, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The proposed theory
  3. Running double-trace couplings
  4. Effect on the planar spectrum
  5. Effect on planar multi-point correlation functions
  6. Conclusions
  7. Acknowledgment

Figures (1)

  • Figure 1: The planar four-point function of the operators \ref{['forpointfunction']} receives a UV divergent contribution associated with a one-loop vertex correction. The divergence is canceled by the second diagram that contains the counter term of the respective double-trace coupling \ref{['doubletracecouplings']}. The four-point function is shown in double-line notation and the operators are depicted in red.