Evolution equations beyond one loop from conformal symmetry

TL;DR

The work develops a framework in which exact conformal invariance at the critical point in non-integer dimensions constrains the evolution of light-ray operators in integer dimensions through deformed collinear $sl(2)$ generators. By separating evolution kernels into $sl(2)$-invariant and non-invariant parts and using Ward identities to fix the non-invariant pieces from lower-loop $S_+$ data, the authors reconstruct high-order kernels from anomalous dimensions. They explicitly demonstrate the method by obtaining the three-loop evolution kernels for twist-two operators in the $O(n)$ $ obreak ext{phi}^4$ theory and the two-loop kernels in the $ obreak ext{su}(n)$ matrix $ obreak ext{phi}^3$ theory, highlighting a nontrivial two-loop correction to $S_+$ in the former. The approach promises a practical path toward high-order results in gauge theories such as QCD by leveraging conformal constraints and avoids the full computation of forward kernels in many cases.

Abstract

We study implications of exact conformal invariance of scalar quantum field theories at the critical point in non-integer dimensions for the evolution kernels of the light-ray operators in physical (integer) dimensions. We demonstrate that all constraints due the conformal symmetry are encoded in the form of the generators of the collinear sl(2) subgroup. Two of them, S_- and S_0, can be fixed at all loops in terms of the evolution kernel, while the generator of special conformal transformations, S_+, receives nontrivial corrections which can be calculated order by order in perturbation theory. Provided that the generator S_+ is known at the k-1 loop order, one can fix the evolution kernel in physical dimension to the k-loop accuracy up to terms that are invariant with respect to the tree-level generators. The invariant parts can easily be restored from the anomalous dimensions. The method is illustrated on two examples: The O(n)-symmetric phi^4 theory in d=4 to the three-loop accuracy, and the su(n) matrix phi^3 theory in d=6 to the two-loop accuracy. We expect that the same technique can be used in gauge theories e.g. in QCD.

Figures (3)

• Figure 1: One-loop diagrams for the 1PI Green function $\Gamma_2(\underline{z},\underline{p})$ in the $\varphi^4$-theory (left) and $\varphi^3$-theory (right). The boxes denote the insertion of the light-ray operator $[\mathcal{O}(z_1,z_2)]$.
• Figure 2: A diagrammatic representation for the 1PI Green function $\delta\Gamma_2(\underline{z},\underline{p})$ (\ref{['DeltaSC']}) to the two-loop accuracy. The black boxes stand for the light-ray operator insertion, the filled (black) circles denote the usual $\varphi^4$ interaction vertex and the open circles correspond to an insertion of $\delta S_R$.
• Figure 3: Renormalization of light-ray operators $\mathcal{O}_j(z_1,z_2)$ (filled square) in the $\varphi^3$ theory to the two-loop accuracy.