Finiteness of N =4 super-Yang-Mills effective action in terms of dressed N =1 superfields

We argue in favor of the independence on any scale, ultraviolet or infrared, in kernels of the effective action expressed in terms of dressed ${\cal N} =1$ superfields for the case of ${\cal N} =4$ super-Yang--Mills theory. Under ``scale independence '' of the effective action of dressed mean superfields we mean its `` finiteness in the off-shell limit of removing all the regularizations.'' This off-shell limit is scale independent because no scale remains inside these kernels after removing the regularizations. We use two types of regularization: regularization by dimensional reduction and regularization by higher derivatives in its supersymmetric form. Based on the Slavnov-Taylor identity we show that dressed fields of matter and of vector multiplets can be introduced to express the effective action in terms of them. Kernels of the effective action expressed in terms of such dressed effective fields do not depend on the ultraviolet scale. In the case of dimensional reduction, by using the developed technique we show how the problem of inconsistency of the dimensional reduction can be solved. Using Piguet and Sibold formalism, we indicate that the dependence on the infrared scale disappears off shell in both the regularizations.