Exact $$\beta $$-functions for $$\mathcal{N}=1$$ supersymmetric theories finite in the lowest loops

Abstract

AbstractWe consider a one-loop finite $$\mathcal{N}=1$$ N = 1 supersymmetric theory in such a renormalization scheme that the first L contributions to the gauge $$\beta $$ β -function and the first $$(L-1)$$ ( L - 1 ) contributions to the anomalous dimension of the matter superfields and to the Yukawa $$\beta $$ β -function vanish. It is demonstrated that in this case the NSVZ equation and the exact equation for the Yukawa $$\beta $$ β -function in the first nontrivial order are valid for an arbitrary renormalization prescription respecting the above assumption. This implies that under this assumption the $$(L+1)$$ ( L + 1 ) -loop contribution to the gauge $$\beta $$ β -function and the L-loop contribution to the Yukawa $$\beta $$ β -function are always expressed in terms of the L-loop contribution to the anomalous dimension of the matter superfields. This statement generalizes the result of Grisaru, Milewski, and Zanon that for a theory finite in L loops the $$(L+1)$$ ( L + 1 ) -loop contribution to the $$\beta $$ β -function also vanishes. In particular, it gives a simple explanation why their result is valid although the NSVZ equation does not hold in an arbitrary subtraction scheme.