[en] A method that may underlie an attempt at proving the previously proposed new identity for Green's functions is described for N = 1 supersymmetric massless electrodynamics regularized by higher derivatives. With the aid of this method, it is shown that some contributions to the identity in question do indeed vanish.

[en] We describe some calculations, which demonstrate that the β-function in N = 1 supersymmetric theories, regularized by higher covariant derivatives, is given by integrals of double total derivatives. This allows to calculate one of the loop integrals analytically and obtain the NSVZ β-function in the considered approximation. For the N = 1 supersymmetric electrodynamics the factorization of integrals into integrals of double total derivatives can be proved in all orders using a special technique. As a result, the exact NSVZ β-function is obtained by explicit summation of Feynman diagrams without a redefinition of the coupling constant.

[en] Using the higher covariant derivative regularization we calculate a two-loop β-function for the N = 1 supersymmetric Yang-Mills theory with the matter superfields, containing the cubic superpotential. It is found that all integrals, defining this function, are integrals of total derivatives.

[en] Most calculations of quantum corrections in supersymmetric theories are made with the dimensional reduction, which is a modification of the dimensional regularization. However, it is well known that the dimensional reduction is not self-consistent. A consistent regularization, which does not break the supersymmetry, is the higher covariant derivative regularization. However, the integrals obtained with this regularization can not be usually calculated analytically. We discuss application of this regularization to the calculations in supersymmetric theories. In particular, it is demonstrated that integrals defining the β-function are possibly integrals of total derivatives. This feature allows to explain the origin of the exact NSVZ β-function, relating the β-function with the anomalous dimensions of the matter superfields. However, integrals for the anomalous dimension should be calculated numerically.

[en] We investigate the NSVZ equation in $\mathcal{N}=1$ supersymmetric gauge theories, which relates the $\mathrm{\beta <!--\; ??\; -->}$ ‑function to the anomalous dimension of the matter superfields. In particular, we argue that it is closely connected with the non-renormalization theorem for the three-point gauge-ghost vertices (in which one external leg corresponds to the quantum gauge superfield). Using finiteness of these vertices the exact NSVZ $\mathrm{\beta <!--\; ??\; -->}$ -function can be equivalently presented in the form of a relation between the $\mathrm{\beta <!--\; ??\; -->}$ -function and the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This equation allows explaining how the NSVZ equation appears in the perturbation theory for theories regularized by higher covariant derivatives and constructing the NSVZ scheme in all orders in the case of using this regularization. The results are verified by an explicit three-loop calculation of the terms quartic in the Yukawa couplings.

[en] The paper contains analysis of the one-loop effective action for affine-metric gravity of the Hilbert–Einstein type with the cosmological term. We discuss different approaches to the calculation of the effective action, which depends on two independent variables, namely, the metric tensor and the affine connection. In the one-loop approximation we explain how the effective action can be obtained, if, at the first step of the calculation, the metric tensor is integrated out. It is demonstrated that the result is the same as in the case when one starts by integrating out the connection. (paper)

[en] We verify the identity which relates the two-point Green’s functions of $\mathcal{N}=1$ SQED with N_f flavors, regularized by higher derivatives, by explicit calculations in the three-loop approximation. This identity explains why, in the limit of the vanishing external momentum, the two-point Green’s function of the gauge superfield is given by integrals of double total derivatives in the momentum space. It also makes it possible to exactly derive the NSVZ β-function in all loops if the renormalization group functions are defined in terms of the bare coupling constant. In order to verify the considered identity, we use it to construct integrals giving the three-loop β-function starting from the two-point Green’s functions of matter superfields in the two-loop approximation. Then we demonstrate that the results for these integrals coincide with the sums of the corresponding three-loop supergraphs.

[en] The three-loop Adler D-function for $\mathcal{N}=1$ SQCD in the $\stackrel{¯<!--\; ??\; -->}{\mathrm{D}\mathrm{R}}$ scheme is calculated starting from the three-loop result recently obtained with the higher covariant derivative regularization. For this purpose, for the theory regularized by higher derivatives we find a subtraction scheme in which the Green functions coincide with the ones obtained with the dimensional reduction and the modified minimal subtraction prescription for the renormalization of the SQCD coupling constant and of the matter superfields. Also we calculate the D-function in the $\stackrel{¯<!--\; ??\; -->}{\mathrm{D}\mathrm{R}}$ scheme for all renormalization constants (including the one for the electromagnetic coupling constant which appears due to the SQCD corrections). It is shown that the results do not satisfy the NSVZ-like equation relating the D-function to the anomalous dimension of the matter superfields. However, the NSVZ-like scheme can be constructed with the help of a properly tuned finite renormalization. It is also demonstrated that the three-loop D-function defined in terms of the bare couplings with the dimensional reduction does not satisfy the NSVZ-like equation for an arbitrary renormalization prescription. We also investigate a possibility to present the results in the form of the β-expansion and the scheme dependence of this expansion.