Abstract
For a general mathcal{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.
Highlights
Well-known statement that the superpotential in N = 1 supersymmetric theories is not renormalized [13]
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
The calculations of quantum corrections made in the DR-scheme in refs. [28,29,30,31,32] demonstrate that the NSVZ relation is not valid for this renormalization prescription
Summary
(We assume that the representation RPV is chosen in such a way that this condition can be satisfied. It is possible to use the adjoint representation.) To obtain a regularized theory with a single dimensionful parameter, it is necessary to require that the Pauli-Villars masses Mφ and M should be proportional to the parameter Λ, Mφ = aφΛ; M = aΛ. It is important that we consider a regularization for which aφ and a do not depend on couplings. The effective action is standardly defined as the Legendre transform of the generating functional W = −i ln Z for connected Green functions, Γ[V , V, φi, c, c] = W − Ssources. Where the sources should be expressed in terms of (super)fields from the equations δW δJA.
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