We consider a general non-Abelian renormalizable \( \mathcal{N} \) = 1 supersymmetric gauge theory, regularized by higher covariant derivatives without breaking the BRST invariance, and calculate one-loop divergences for a general form of higher derivative regulator and of the gauge fixing term. It is demonstrated that the momentum integrals giving the one-loop β-function are integrals of double total derivatives independently of a particular choice of the higher derivative term. Evaluating them we reproduce the well-known result for the one-loop β-function. Also we find that the three-point ghost vertices with a single line of the quantum gauge superfield are not renormalized in the considered approximation.

The perturbative all-loop derivation of the NSVZ $$\beta $$β -function for $${{\mathcal {N}}}=1$$N = 1 supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the $$\beta $$β -function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the $$\beta $$β -function and these anomalous dimensions for the renormalization group functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD + MSL prescription.

The effective diagram technique based on the Schwinger-Dyson equations is constructed for \( \mathcal{N} \) = 1 SQED with N_f flavors, regularized by higher derivatives. Using these effective diagrams, it is possible to derive the exact NSVZ relation between the β-function and the anomalous dimension of the matter superfields exactly in all loops, if the renormalization group functions are defined in terms of the bare coupling constant. In particular, we verify that all integrals which give the β-function defined in terms of the bare coupling constant are integrals of double total derivatives and prove some identities relating Green functions.

Recently the exact NSVZ $\beta$-function was rewritten in the form of a relation between the $\beta$-function and the anomalous dimensions of the quantum gauge superfield, of the Faddeev--Popov ghosts, and of the matter superfields. It was also suggested that this form of the NSVZ equation follows from an underlying equation relating two-point Green functions of the theory. Here we demonstrate that this relation is satisfied at the two-loop level for the non-Abelian ${\cal N}=1$ supersymmetric gauge theories in the case of using the simplest (BRST non-invariant) version of the higher covariant derivative regularization. Consequently, the integrals giving the two-loop $\beta$-function can be reduced to the one-loop integrals giving the anomalous dimensions of the quantum gauge superfield, of the Faddeev--Popov ghosts, and of the matter superfields.Comment: 12 pages, 3 figures, the version accepted for publication in Physics Letters

We discuss why the Slavnov higher covariant derivative regularization appeared to be an excellent instrument for investigating quantum corrections in supersymmetric gauge theories. For example, it allows demonstrating that the β-function in these theories is given by integrals of double total derivatives and to construct the Novikov-Shifman-Vainshtein-Zakharov (NSVZ) renormalization prescription in all loops. It was also used to derive the non-renormalization theorem for the triple gauge-ghost vertices. With the help of this theorem the exact NSVZ β-function was rewritten in a new form, which revealed its perturbative origin. Moreover, in the case of using the higher covariant derivative regularization, it is possible to construct a method for obtaining the β-function of N = 1 supersymmetric gauge theories, which simplifies the calculations to a great extent. This method is illustrated by an explicit two-loop calculation made in the general ξ-gauge.