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

TL;DR

This paper derives exact expressions for the beta functions in certain finite ${ m N}=1$ supersymmetric theories, showing how higher-loop contributions relate to lower-loop anomalous dimensions under specific renormalization schemes.

Contribution

It generalizes previous results by demonstrating that in a specific scheme, higher-loop beta functions are determined by lower-loop anomalous dimensions, extending the understanding of finiteness in supersymmetric theories.

Findings

Exact beta functions are derived for theories finite in the lowest loops.

The NSVZ equation holds under a broad class of renormalization schemes.

Higher-loop contributions are expressed in terms of lower-loop anomalous dimensions.

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 $\beta$-function and the first $(L-1)$ contributions to the anomalous dimension of the matter superfields and to the Yukawa $\beta$-function vanish. It is demonstrated that in this case the NSVZ equation and the exact equation for the Yukawa $\beta$-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 $\beta$-function and the $L$-loop contribution to the Yukawa $\beta$-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 $\beta$-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.