Evolution equations beyond one loop from conformal symmetry

TL;DR

This paper explores how conformal symmetry constrains the evolution equations of light-ray operators in scalar quantum field theories beyond one loop, providing a systematic method to determine higher-order corrections.

Contribution

It introduces a method to derive evolution kernels at higher loops using conformal symmetry constraints, applicable to scalar theories and potentially gauge theories like QCD.

Findings

Derived three-loop evolution kernels for phi^4 theory in 4D.

Calculated two-loop evolution kernels for phi^3 theory in 6D.

Demonstrated the method's applicability to different scalar models.

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.