The heat kernel method in renormalization theory on the example of scalar models and Yang-Mills theory

The heat kernel method is widely used in various fields of theoretical and mathematical physics. We consider the application of this approach to the theory of renormalization. As a regularization scheme we use the cutoff regularization in the coordinate representation. First, as a trial model, we consider the scalar cubic theory, which, on the one hand, simple in calculations, but at the same time reflects the main ideas of applying the heat kernel method. We consider the two-loop renormalization of this model. Then the two-loop renormalization of the four-dimensional quantum Yang-Mills theory will be presented. Also, we discuss the background field formalism in the framework of the renormalization theory, and consider several ideas for modifying the regularization scheme.