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

Abstract

For the general renormalizable N = 1 supersymmetric gauge theory we investigate renormalization of the Faddeev-Popov ghosts using the higher covariant derivative regularization. First, we find the two-loop anomalous dimension defined in terms of the bare coupling constant in the general ξ-gauge. It is demonstrated that for doing this calculation one should take into account that the quantum gauge superfield is renormalized in a nonlinear way. Next, we obtain the two-loop anomalous dimension of the Faddeev-Popov ghosts defined in terms of the renormalized coupling constant and examine its dependence on the subtraction scheme.

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

“…Another calculation made in Ref. [69] explicitly demonstrates that the renormalization group equations are satisfied only if this nonlinear renormalization is taken into account. That is why constructing the generating functional one should replace the gauge superfield V by a nonlinear function…”

“…Then, we extract terms which correspond to the cuts of internal ghost lines and compare them with the two-loop ghost anomalous dimension calculated in Ref. [69].…”

Gauge dependence of the one-loop divergences in 6D, N=(1,0) abelian theory

“…For example, for N = 1 supersymmetric gauge theories in the one-loop approximation ghosts are not renormalized in the Feynman gauge, while divergences appear for ξ = 1 [35]. For calculations in higher orders, the knowledge of gauge dependence in the lower-order approximations is also essential, see, e.g., [36]. These are the reasons why a vast literature is devoted to calculations in non-minimal gauges.…”

Supersymmetry, quantum corrections, and the higher derivative regularization

“…However, a nontrivial check can be obtain only by comparing the two-loop ghost anomalous dimension and the three-loop β-function, because only stating from this approximation the scheme dependence becomes essential. The two-loop anomalous dimension of the Faddeev-Popov ghosts has been calculated in [46]. Making this calculation it is very important to take into account that the quantum gauge superfield is renormalized in a non-linear way [32,33].…”

“…This equation should be taken into account together with the one-loop renormalization of the coupling constant and of the gauge parameter. Then the Faddeev-Popov ghost anomalous dimension defined in terms of the bare couplings for the higher derivative regulators R(x) = K(x) = 1 + x m ; F(x) = 1 + x n is given by the scheme-independent expression [46] γ where a ≡ M/Λ; a ϕ ≡ M ϕ /Λ are ratios of the Pauli-Villars masses to the parameter Λ. Note that writing Eq.…”