Non-diagrammatic calculation of QCD one-loop beta-function based on the renormalization group equation

TL;DR

This paper presents a diagram-free method to derive the one-loop beta-function in QCD using the background field method, higher covariant derivatives, and the Fujikawa approach, emphasizing gauge invariance and anomalies.

Contribution

It introduces a novel, diagramless calculation of the QCD one-loop beta-function based on the renormalization group and anomaly analysis, avoiding Feynman diagrams.

Findings

Demonstrates the anomalous origin of the one-loop beta-function in QCD.

Shows the effectiveness of background field method with higher covariant derivatives.

Provides a simple, gauge-invariant derivation of the beta-function.

Abstract

Using the higher covariant derivatives regularization of gauge theories in the framework of the background field method, supplemented with one-loop Pauli-Villars regulator fields, we obtain a version of the renormalization group equation for the regulator fields, whose vacuum energy depends on the background gauge field. It is evaluated using an anomalous Ward-Takahashi identity, which is related to the rescaling anomaly of the auxiliary fields, obtained by the Fujikawa approach. In this way the anomalous origin of the one-loop beta-function in QCD is clearly shown in terms of scaling of effective Lagrangians without the use of any Feynman diagram. The simplicity of the method is due to the preservation of the background and quantum gauge invariance in any step of the calculation.