New form of the exact NSVZ $\beta$-function: the three-loop verification for terms containing Yukawa couplings

TL;DR

This paper verifies a new form of the NSVZ $eta$-function at three loops for terms with Yukawa couplings in ${ m extbf{N=1}}$ supersymmetric gauge theories, confirming its scheme independence when using higher covariant derivative regularization.

Contribution

It demonstrates the validity of the new NSVZ $eta$-function form at three loops for Yukawa terms, independent of the subtraction scheme, using higher covariant derivative regularization.

Findings

The new NSVZ relation holds at three loops for Yukawa terms.

The relation is scheme-independent when defined with bare couplings.

The NSVZ scheme is achieved with a specific subtraction scheme in the higher covariant derivative regularization.

Abstract

We investigate a recently proposed new form of the exact NSVZ $\beta$-function, which relates the $\beta$-function to the anomalous dimensions of the quantum gauge superfield, of the Faddeev--Popov ghosts, and of the chiral matter superfields. Namely, for the general renormalizable ${\cal N}=1$ supersymmetric gauge theory, regularized by higher covariant derivatives, the sum of all three-loop contributions to the $\beta$-function containing the Yukawa couplings is compared with the corresponding two-loop contributions to the anomalous dimensions of the quantum superfields. It is demonstrated that for the considered terms both new and original forms of the NSVZ relation are valid independently of the subtraction scheme if the renormalization group functions are defined in terms of the bare couplings. This result is obtained from the equality relating the loop integrals, which, in turn, follows from the factorization of the integrals for the $\beta$-function into integrals of double total derivatives. For the renormalization group functions defined in terms of the renormalized couplings we verify that the NSVZ scheme is obtained with the higher covariant derivative regularization supplemented by the subtraction scheme in which only powers of $\ln\Lambda/\mu$ are included into the renormalization constants.