Regularization of propagators with background field and their logarithms in $4$-dimensions

We provide different attempts to regularize simultaneously background field dependent propagators and traces of their logarithms for quantum field models in $4$-dimensional euclidian space-time. As was shown in the literature, infinities in the trace of the logarithm and in higher order loop diagramms are of different nature and require different approaches in regularization. We argue that the trace of the loagarithm itself is a finite (w.r.t regularization parameter) quantity. While the correspondent divergence in the effective action arises from the measure of the functional integral imposed by some Ward-like identities.