Bi-scalar integrable CFT at any dimension

TL;DR

This paper extends the 4D bi-scalar CFT to a D-dimensional setting, establishing a conformal, planar-integrable theory built from two complex adjoint scalars with nonlocal kinetic terms and a chiral quartic interaction, supplemented by D-dimensional double-trace counterterms. The isotropic case ($\omega=D/4$) yields two complex-conjugate fixed points for $\alpha_1^2(\xi)$, where the theory becomes a non-unitary CFT and certain bilinear operators are protected in the planar limit. Integrability is made explicit via a graph-building operator tied to an $SO(1,D+1)$ conformal spin chain, with the transfer matrix providing a conserved charge and generating fishnet cylinder graphs. The authors compute an exact four-point function of two-scalar operators, obtaining a conformal partial-wave expansion with pole structure at $h_{\Delta,S}=\xi^4$, from which they extract exact operator dimensions and OPE data for exchanged operators across general $D$, including explicit weak-coupling expansions and the distinction between even and odd dimensions. The results pave the way for further exploration of higher-point functions, non-planar effects, and potential string duals in this broad D-dimensional integrable CFT framework.

Abstract

We propose a $D$-dimensional generalization of $4D$ bi-scalar conformal quantum field theory recently introduced by Gürdogan and one of the authors as a strong-twist double scaling limit of $γ$-deformed $\mathcal{N}=4$ SYM theory. Similarly to the $4D$ case, this D-dimensional CFT is also dominated by "fishnet" Feynman graphs and is integrable in the planar limit. The dynamics of these graphs is described by the integrable conformal $SO(D+1,1)$ spin chain. In $2D$ it is the analogue of L. Lipatov's $SL(2,\mathbb{C})$ spin chain for the Regge limit of $QCD$, but with the spins $s=1/4$ instead of $s=0$. Generalizing recent $4D$ results of Grabner, Gromov, Korchemsky and one of the authors to any $D$ we compute exactly, at any coupling, a four point correlation function, dominated by the simplest fishnet graphs of cylindric topology, and extract from it exact dimensions of R-charge 2 operators with any spin and some of their OPE structure constants.

Figures (4)

• Figure 1: Loop expansion of $\langle{\text{tr}}(\phi_1^2)(x){\text{tr}}(\phi_1^2)^\dagger(0) \rangle$ planar graphs up to 2-loops.
• Figure 2: Graphical representation of the transfer matrix as a convolution of R-kernels according to formulas \ref{['transfer_matrix']}and \ref{['Rmat']}. Black dots are integration points and the weights of propagators are written in the second and third R-kernel.
• Figure 3: Graphical representation of the kernel of the graph-building operator for generic $D$ and $\omega$. It is otained by setting $u=-\frac{D}{4}$ in the transfer matrix \ref{['transfer_matrix']} presented on Fig. \ref{['t_matrix']}, so that $x_{jj'+1~}$--type type propagators disappear while $x_{j'+1}-y_{j}$ -type propagators are replaced by $\delta^{(D)}(x_{j'+1}-y_{j})$ factors. After that, integration over the points $y_{j}$ is equivalent to setting $y_{j}=x_{j'+1}$.
• Figure 4: General fishnet graphs up to $\alpha^2_1$ order in the expansion of four point function \ref{['G4-int']}.