The β-function of $$ \mathcal{N} $$ = 1 supersymmetric gauge theories regularized by higher covariant derivatives as an integral of double total derivatives

Abstract

For a general N = 1 supersymmetric gauge theory regularized by higher covariant derivatives we prove in all orders that the β-function defined in terms of the bare couplings is given by integrals of double total derivatives with respect to loop momenta. With the help of the technique used for this proof it is possible to construct a method for obtaining these loop integrals, which essentially simplifies the calculations. As an illustration of this method, we find the expression for the three-loop contribution to the β-function containing the Yukawa couplings and compare it with the result of the standard calculations made earlier. Also we briefly discuss, how the structure of the loop integrals for the β-function considered in this paper can be used for the all-loop perturbative derivation of the NSVZ relation in the non-Abelian case.1 In the Landau gauge ξ → 0 a similar statement was known earlier for the usual (non-supersymmetric) Yang-Mills theory [15] and for N = 1 SYM formulated in terms of the component fields [16]. In the former case this statement was explicitly verified by the four-loop calculation in Ref. [17].

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

“…According to Ref. [65], it is possible to construct integrals of double total derivatives contributing to the β-function (defined in terms of the bare couplings in the case of using the higher covariant derivative regularization) with the help of a special algorithm, which essentially simplifies the calculations. This occurs, because this algorithm reduces the calculation of superdiagrams with two external lines of the background gauge superfield to the evaluation of certain superdiagrams without external lines.…”

“…The former superfields cancel one-loop divergences coming from the gauge and ghost loops, while the latter ones cancel one-loop divergences introduced by the matter loop. The actions for the Pauli-Villars superfields and the explicit expression for the generating functional can be found in [65]. It is important that the masses of the Pauli-Villars superfields should be proportional to the parameter in the higher derivative term,…”

“…The first and second steps have been done in Refs. [2] and [65], respectively. However, the singular contributions have not yet been summed.…”

“…To calculate the three-loop contributions to the β-function coming from the considered class of supergraphs, we will use the algorithm proposed in Ref. [65]. It essentially simplifies the calculations and produces the result in the form of integrals of double total derivatives.…”

“…The algorithm for constructing integrals of double total derivatives proposed in Ref. [65] is described in Sect. 3.…”