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

Abstract

We consider a one-loop finite ${\cal N}=1$ supersymmetric theory in such a renormalization scheme that the first $L$ contributions to the gauge $β$-function and the first $(L-1)$ contributions to the anomalous dimension of the matter superfields and to the Yukawa $β$-function vanish. It is demonstrated that in this case the NSVZ equation and the exact equation for the Yukawa $β$-function in the first nontrivial order are valid for an arbitrary renormalization prescription respecting the above assumption. This implies that under this assumption the $(L+1)$-loop contribution to the gauge $β$-function and the $L$-loop contribution to the Yukawa $β$-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 $(L+1)$-loop contribution to the $β$-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.