From operator product expansion to anomalous dimensions

Abstract

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 ϕ 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.

Keywords

• Renormalization and Regularization
• Renormalization Group
• Scattering Amplitudes