We find the three-loop contribution to the βfunction of N = 1 supersymmetric gauge theories regularized by higher covariant derivatives produced by the supergraphs containing loops of the Faddeev-Popov ghosts. This is done using a recently proposed algorithm, which essentially simplifies such multiloop calculations. The result is presented in the form of an integral of double total derivatives in the momentum space. The considered contribution to the β-function is compared with the two-loop anomalous dimension of the Faddeev-Popov ghosts. This allows verifying the validity of the NSVZ equation written as a relation between the β-function and the anomalous dimensions of the quantum superfields. It is demonstrated that in the considered approximation the NSVZ equation is satisfied for the renormalization group functions defined in terms of the bare couplings. The necessity of the nonlinear renormalization for the quantum gauge superfield is also confirmed.

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.

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For $$ \mathcal{N} $$ N = 1 SQED with Nf flavors regularized by higher derivatives in the general ξ-gauge we calculate the three-loop anomalous dimension of the matter superfields defined in terms of the bare coupling constant and demonstrate its gauge independence. After this the four-loop β-function defined in terms of the bare coupling constant is obtained with the help of the NSVZ equation, which is valid for these renormalization group functions in all loops. Next, we calculate the three-loop anomalous dimension and the four-loop β-function defined in terms of the renormalized coupling constant for an arbitrary subtraction scheme supplementing the higher derivative regularization. Then we construct a renormalization prescription for which the results coincide with the ones in the $$ \overline{\mathrm{DR}} $$ DR ¯ -scheme and describe all NSVZ schemes in the considered approximation. Also we demonstrate the existence of a subtraction scheme in which the anomalous dimension does not depend on Nf, while the β-function contains only terms of the first order in Nf. This scheme is obtained with the help of a finite renormalization compatible with a structure of quantum corrections and is NSVZ. The existence of such an NSVZ scheme is also proved in all loops.

We verify a recently proposed method for obtaining a β-function of N = 1 supersymmetric gauge theories regularized by higher derivatives by an explicit calculation. According to this method, a β-function can be found by calculating specially modified vacuum supergraphs instead of a much larger number of the two-point superdiagrams. The result is produced in the form of a certain integral of double total derivatives with respect to the loop momenta. Here we compare the results obtained for the three-loop β-function of N = 1 SQED in the general ξ-gauge with the help of this method and with the help of the standard calculation. Their coincidence confirms the correctness of the new method and the general argumentation used for its derivation. Also we verify that in the considered approximation the NSVZ relation is valid for the renormalization group functions defined in terms of the bare coupling constant and for the ones defined in terms of the renormalized coupling constant in the HD+MSL scheme, both its sides being gauge-independent.1 Note that with dimensional reduction the integrals for the β-function are not integrals of total derivatives [36]. Moreover, the results of Ref. [37] indicate that RGFs defined in terms of the bare couplings do not satisfy the NSVZ equation for theories regularized by dimensional reduction.2 In this scheme the theory is regularized by higher derivatives, and divergences are removed with the help of minimal subtractions of logarithms [39,40]. Also the HD+MSL scheme can be constructed by imposing certain boundary conditions on renormalization constants [41].