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

Using the higher covariant derivatives and gauge invariant Pauli-Villars regularization of gauge theories in the framework of the background field method, we obtain a version of the exact 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 gauge invariance in any step of the calculation. Introduction In an interesting article Polchinski demonstrated that Wilson’s decimation method [1] applied to continuum field theory is as much as we need to give a proof of perturbative renormalization [2]. To obtain the exact renormalization group equation (ERGE) he used an obvious identity, which consists in setting to zero the integral functional of a proper total derivative, and the independence of the partition function on the scale. Such idea corresponds to a reparametrization of the partition function since the total derivative emerges from a field redefinition, as shown for instance in refs. [3]. This derivation of ERGE has to be modified when there are present external gauge fields. In this case, as they could be field dependent, we can not discard the singular terms that appear developing the total derivative before having studied their possible physical relevance. In fact, as was shown in ref. [4] in the context of N = 1 supersymmetric Yang-Mills theory, they assume the meaning of a vacuum energy and are responsible for the Shifman-Vainshtein relation of exact β-function [5]. Here we extend the analysis in [4, 6] to non-supersymmetric gauge theories. In these papers is used the gauge invariant regularization proposed by Arkani-Hamed and Murayama [7], which consists in giving a big mass to the extra fields1 of a finite theory with extended supersymmetry. In conventional gauge theories we have obviously to resort to a different regularization. The attractive properties of gauge invariance, non-perturbative meaning and applicability to chiral With extra field we mean a field of the finite theory which does not appear in the theory we are regulating.