We consider a general non-Abelian renormalizable 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.

At the three-loop level we analyze, how the NSVZ relation appears for N = 1 SQED regularized by the dimensional reduction. This is done by the method analogous to the one which was earlier used for the theories regularized by higher derivatives. Within the dimensional technique, the loop integrals cannot be written as integrals of double total derivatives. However, similar structures can be written in the considered approximation and are taken as a starting point. Then we demonstrate that, unlike the higher derivative regularization, the NSVZ relation is not valid for the renormalization group functions defined in terms of the bare coupling constant. However, for the renormalization group functions defined in terms of the renormalized coupling constant, it is possible to impose boundary conditions to the renormalization constants giving the NSVZ scheme in the three-loop order. They are similar to the all-loop ones defining the NSVZ scheme obtained with the higher derivative regularization, but are more complicated. The NSVZ schemes constructed with the dimensional reduction and with the higher derivative regularization are related by a finite renormalization in the considered approximation.

In the case of using the higher derivative regularization for N = 1 SQED with N f flavors the loop integrals giving the β-function are integrals of double total derivatives in the momentum space. This feature allows to reduce one of the loop integrals to an integral of the δ-function and to derive the NSVZ relation for the renormalization group functions defined in terms of the bare coupling constant. In this paper we consider N = 1 SQED with N f flavors regularized by the dimensional reduction in the DR-scheme. Evaluating the scheme-dependent three-loop contribution to the β-function proportional to (N f ) 2 we find the structures analogous to integrals of the δ-singularities. After adding the scheme-independent terms proportional to (N f ) 1 we obtain the known result for the three-loop β-function.

The Earth's density distribution can be approximately considered piecewise continuous at the scale of two-flavor oscillations of typical solar neutrinos, such as the beryllium-7 and boron-8 neutrinos. This quite general assumption appears to be enough to analytically calculate the day-night asymmetry factor for such neutrinos. Using the explicit time averaging procedure, we show that, within the leading-order approximation, this factor is determined by the electron density within about one oscillation length under the detector, namely, in the Earth's crust (and upper mantle for high-energy neutrinos). We also evaluate the effect of the inner Earth's structure on the observed asymmetry and show that it is suppressed and mainly comes from the neutrinos observed near the winter and summer solstices. As a result, we arrive at the strict interval constraint on the asymmetry, which is valid within quite a wide class of Earth models.

The three-loop Adler D-function for N = 1 SQCD in the DR 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 DR 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.