Renormalization aspects of the Yang-Mills theory with a cutoff

The paper discusses renormalization aspects of the quantum four-dimensional Yang-Mills theory with a cutoff regularization in the coordinate representation. The background field method is used to formulate a generating functional, and the regularization is introduced through quasi-local probabilistic averaging. Two main types of regularization are proposed: strong deformation, which consists in averaging fluctuation fields, and weak deformation, which is a covariant generalization of the first case with respect to gauge transformations of the background field. We study singular contributions for the first two quantum corrections in this paper and compare them in detail with the case of dimensional regularization. The consistency of the action and the equation of motion after introducing the regularization and making a renormalization procedure is analyzed. New counter-vertices are studied, in particular their locality properties and dependence on the regularization parameter.