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

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 β-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. Introduction In an interesting paper Polchinski demonstrated that Wilson’s decimation method [1] applied to continuum field theory is sufficient to provide proof of perturbative renormalization [2]. To obtain the exact renormalization group equation (ERGE) he used an obvious identity, which consists in setting the integral functional of a proper total derivative to zero, and the independence of the partition function on the scale. Such an idea corresponds to a reparameterization of the partition function since the total derivative emerges from a field redefinition [3, 4]. This derivation of ERGE has to be modified when external gauge fields are present. In such a case, as they could be field dependent, we can not discard the singular terms that appear when developing the total derivative before having studied their possible physical relevance. In fact, as was shown in ref. [6] in the context of N = 1 supersymmetric Yang-Mills theory, they assume the meaning of a vacuum energy and are responsible for the exact one-loop running of the holomorphic gauge coupling [7]. Here the analysis in [6, 8] is extended to non-supersymmetric gauge theories. In those papers the gauge invariant regularization proposed by Arkani-Hamed and Murayama [9] is