Integrability of Conformal Fishnet Theory

TL;DR

We address the problem of determining exact scaling dimensions of wheel-type operators in planar bi-scalar chiFT_4, where wheel graphs control the perturbative expansion. We develop an integrability-based framework by identifying a conformal SU(2,2) spin chain underlying the fishnet graphs, and relate it to the Quantum Spectral Curve (QSC) of N=4 SYM. Using QSC and spin-chain methods, we derive a fourth-order Baxter equation with model-dependent constants and impose quantization conditions to fix Δ(ξ) at arbitrary coupling; for J=3 we illustrate with explicit m^2 = -ξ^6 constraints and a numerical procedure. In addition to weak-coupling expansions to high orders, we derive strong-coupling spectra from finite-gap/semi-classical quantization, exhibiting two branches corresponding to different bare dimensions, with Bohr-Sommerfeld type formulas. Our results establish a concrete bridge between fishnet integrability and AdS/CFT-inspired integrability, suggesting broader applicability to other χFT theories and to noncompact spin chains.

Abstract

We study integrability of fishnet-type Feynman graphs arising in planar four-dimensional bi-scalar chiral theory recently proposed in arXiv:1512.06704 as a special double scaling limit of gamma-deformed $\mathcal{N}=4$ SYM theory. We show that the transfer matrix "building" the fishnet graphs emerges from the $R-$matrix of non-compact conformal $SU(2,2)$ Heisenberg spin chain with spins belonging to principal series representations of the four-dimensional conformal group. We demonstrate explicitly a relationship between this integrable spin chain and the Quantum Spectral Curve (QSC) of $\mathcal{N}=4$ SYM. Using QSC and spin chain methods, we construct Baxter equation for $Q-$functions of the conformal spin chain needed for computation of the anomalous dimensions of operators of the type $\text{tr}(φ_1^J)$ where $φ_1$ is one of the two scalars of the theory. For $J=3$ we derive from QSC a quantization condition that fixes the relevant solution of Baxter equation. The scaling dimensions of the operators only receive contributions from wheel-like graphs. We develop integrability techniques to compute the divergent part of these graphs and use it to present the weak coupling expansion of dimensions to very high orders. Then we apply our exact equations to calculate the anomalous dimensions with $J=3$ to practically unlimited precision at any coupling. These equations also describe an infinite tower of local conformal operators all carrying the same charge $J=3$. The method should be applicable for any $J$ and, in principle, to any local operators of bi-scalar theory. We show that at strong coupling the scaling dimensions can be derived from semiclassical quantization of finite gap solutions describing an integrable system of noncompact $SU(2,2)$ spins. This bears similarities with the classical strings arising in the strongly coupled limit of $\mathcal{N}=4$ SYM.

Figures (14)

• Figure 1: The "wheel" Feynman graph with $M$ frames defining the order $\xi^{2M{J}}$ of the perturbative expansion of anomalous dimension of the BMN vacuum operator $\text{Tr}\phi_1^{J}$ ($M=4\,,\,\, {J}=10$ on this picture).
• Figure 2: Numerical results for the scaling dimension of the operator ${\text{tr}}(\phi_1^3)$ as a function of the coupling $\xi^3$. We observe a "phase transition" at $\xi^3\simeq 0.21$ where the scaling dimension takes the value $\Delta=2$ and becomes imaginary. This point defines the radius of convergency of the weak coupling expansion. The second branch, starting from $\Delta(0)=1$, arises due to the symmetry of the Baxter equation (\ref{['Baxter24']}) under $\Delta\to 4-\Delta$.
• Figure 3: The "wheel" Feynman graph corresponding to $O(\xi^{24})$ term in the weak coupling expansion \ref{['D3exp']} of anomalous dimension of the operator $\text{Tr}( \phi_1^3)$.
• Figure 4: A "wheel" Feynman graph with $M$ frames and ${J}$ spokes whose external legs have been joined at the point $x$ ($M=5$ and ${J}=9$ in this picture).
• Figure 5: Diagrammatic representation of the $R-$operator (\ref{['R']}). Solid line with index $\alpha$ stands for $1/(x^2)^\alpha$. The values of indices depend on the spectral parameter and are given by $\alpha_+=-1-u$, $\alpha_-=1-u$ and $\beta=2+u$.