From Operator Product Expansion to Anomalous Dimensions

TL;DR

This paper introduces a novel method for calculating anomalous dimensions and renormalization functions using operator product expansion and large momentum expansion, applicable to scalar and other quantum field theories.

Contribution

It develops a general framework for computing anomalous dimensions via Wilson coefficients' ultraviolet finiteness, extending to various models including scalar and Yukawa theories.

Findings

Computed anomalous dimensions of $\,\phi^Q$ operator up to four loops.

Validated the method against known large N results.

Extended the approach to non-scalar models like Gross-Neveu-Yukawa.

Abstract

We propose a new method for computing the renormalization functions, which is based on the ideas of operator product expansion and large momentum expansion. In this method, the renormalization $Z$-factors are determined by the ultraviolet finiteness of Wilson coefficients in the dimensional regularization scheme. The ultraviolet divergence is extracted solely from two-point integrals at the large momentum limit. We develop this method in scalar field theories and establish a general framework for computing anomalous dimensions of fields, mass, couplings and composite operators. In particular, it is applied to the 6-dimensional cubic scalar theory and the 4-dimensional quartic scalar theory. We demonstrate this method by computing the anomalous dimension of the $\phi^Q$ operator in cubic theory up to four loops for arbitrary $Q$, which is in agreement with the known result in the large $N$ limit. The idea of computing anomalous dimensions from the operator production expansion is general and can be extended beyond scalar theories. This is demonstrated through examples of the Gross-Neveu-Yukawa model with generic operators.