Abstract
We show that an integrable four-dimensional non-unitary field theory that was recently proposed as a certain limit of the $\gamma_i$-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.
Highlights
In a recent paper [1], a certain limit is applied to the γi-deformation of N = 4 SYM theory [2], and the authors claim that the resulting non-unitary theory is conformal in the planar limit where the number N of colors is sent to infinity.In this letter, we point out that the Lagrangian given in [1] is incomplete and does not define a CFT – not even in the planar limit
We show that an integrable four-dimensional non-unitary field theory that was recently proposed as a certain limit of the γi-deformed N = 4 SYM theory is incomplete and not conformal – not even in the planar limit
We show this explicitly by determining to two-loop order the planar anomalous dimensions of several single-trace operators composed of two scalar fields
Summary
We complete the theory by the missing couplings that are required for renormalizability. They are quartic and have double-trace color structures and non-trivial β-functions [9]. Planar multi-point correlation functions for infinitely many composite single-trace operators depend on these couplings and are sensitive to the β-functions. We show this explicitly by determining to two-loop order the planar anomalous dimensions of several single-trace operators composed of two scalar fields. We give an example for a planar four-point correlation function of further operators that depends on one of the doubletrace couplings. Allowing for complex coupling constants in this model, whose single-trace part is already nonunitary, we find (one-loop) fix-points
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