Non-diagrammatic calculation of QCD one-loop beta-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 Pa uli-Villars regulator fields, we obtain a version of the renormalization gr oup equation for the regulator fields, whose vacuum energy depends on the b ackground gauge field. It is evaluated using an anomalous Ward-Takahashi id entity, 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 effe ctive 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.