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

Abstract

We investigate a recently proposed new form of the exact NSVZ β-function, which relates the β-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 N = 1 supersymmetric gauge theory, regularized by higher covariant derivatives, the sum of all three-loop contributions to the β-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 β-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 Λ/µ are included into the renormalization constants.

Supersymmetry, quantum corrections, and the higher derivative regularization

“…The above results have been verified in the three-loop approximation for terms containing the Yukawa couplings [43,44]. For the higher derivative regulators F(x) = 1 + x n ; R(x) = 1 + x m RGFs defined in terms of the renormalized couplings are…”

“…Because the NSVZ relation in the form (12) has the same graphical interpretation as in the Abelian case, it is reasonable to suggest that as in the Abelian case it is obtained with the higher covariant derivative regularization for RGFs defined in terms of the bare couplings independently of the renormalization prescription. This proposal has been confirmed by explicit calculations of the terms containing the Yukawa couplings in the three-loop β-function and in the two-loop anomalous dimensions of the quantum superfields [43,44]. For the other terms it has been verified in the previous order [45].…”

“…The calculations made in [43,44] allow to verify the terms in Eqs. (12) and 18which contain anomalous dimensions of the quantum gauge superfield and of the matter superfield.…”

Three-loop contribution of the Faddeev–Popov ghosts to the $$\beta $$-function of $$\mathcal{N}=1$$ supersymmetric gauge theories and the NSVZ relation

“…1 However, the derivation of the NSVZ relation in the non-Abelian case turns out to be more complicated. Nevertheless, the calculations made with the higher covariant derivative regularization in the lowest orders (see, e.g., [60][61][62][63][64]) reveal the same features as in the Abelian case. In particular, they demonstrate that all integrals for the β-function defined in terms of the bare couplings are integrals of double total derivatives.…”

Two-loop renormalization of the Faddeev-Popov ghosts in $$ \mathcal{N}=1 $$ supersymmetric gauge theories regularized by higher derivatives

“…It is this form that follows from the perturbative calculations, see, e.g., the calculations in refs. [28,35]. As in the Abelian case, the NSVZ relation follows from the factorization of loop integrals into integrals of total [36] and double total [37] derivatives in the case of using the HD regularization.…”