Strongly $\gamma$-Deformed $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory

We demonstrate by explicit multiloop calculation that $\gamma$-deformed planar $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory, supplemented with a set of double-trace counterterms, has two nontrivial fixed points in the recently proposed double scaling limit, combining vanishing ’t Hooft coupling and large imaginary deformation parameter. We provide evidence that, at the fixed points, the theory is described by an integrable nonunitary four-dimensional conformal field theory. We find a closed expression for the four-point correlation function of the simplest protected operators and use it to compute the exact conformal data of operators with arbitrary Lorentz spin. We conjecture that both conformal symmetry and integrability should survive in $\gamma$-deformed planar $\mathcal{N}=4$ SYM theory for arbitrary values of the deformation parameters.

Figure 1

Feynman diagrams contributing to two-point correlation functions $G_2$, $G_3$ (left) and $G_1$ (right) in the planar limit. Interaction vertices in the left diagram describe either the single-trace coupling ${\xi }^2$ or the double-trace coupling ${\alpha }_i^2$ (with $i=2$, 3) depending on the choice of $G_i$. The right diagram consists of the chain of scalar loops joined together through the double-trace coupling ${\alpha }_1^2$. Each internal scalar loop is built using the single-trace coupling ${\xi }^2$.