Conformal Invariant Supersymmetric Theories in Four Dimensions: A Practical Application of BRS Cohomology

The antighost equation in N = 1 super-Yang-Mills theories

“…This theorem is stated in general terms and proven in [6] as Theorem A.1. Its main consequence is Corollary A.2 of [6], which m a y be paraphrased as follows.…”

“…The latter property w as shown in [6] as a consequence of the nonrenormalization theorem for the chiral vertices [7]. But, since the latter theorem requires exact supersymmetry and exact superspace Feynman rules, and since supersymmetry is broken in some way in most cases of interest, an alternative for the niteness of K 3 , generalizing the one given in [5] for the nonsupersymmetric case, will be given now using the antighost equation (3.6).…”

“…Let the in nitesimal linear transformation 6 A i =ie i j A j ; A i = i A j e j i ; (4.1) acting on the matter chiral super elds { the numbers e i j being the elements of an hermitian matrix which commutes with the gauge group generators T a { be a symmetry of the theory up to a possible soft breaking due to the mass terms. Such a symmetry is renormalizable [4] and may be expressed by a W ard identity where we h a v e i n troduced the chiral superconnection ' = e D e : The main statement of the nonrenormalization theorem is that the anomaly coe cient r in (4.3) is exactly given by its one-loop approximation.…”

Constraints on finite soft supersymmetry-breaking terms

“…According to the finiteness theorem of ref. [17], the theory is then finite 2 to all orders in perturbation theory, if the one-loop anomalous dimensions γ (1) j i given in (A.2) vanish, i.e., if…”

“…Therefore, there exist, in principle, various finite models for a given matter content. However, during the early studies [14,15], the theorem [17] that guarantees all-order finiteness and requires the existence of power series solution to any finite order in perturbation theory was not known.…”