Finiteness in N=1 SYM Theories

I present a criterion for all-order finiteness in N =1 SYM theories. Three applications are given; they yield all-order finite N=1 SYM models with global symmetries of the superpotential. 1. INTRODUCTION The aim of this paper is to present applications of the criterion for all-order finiteness in N = 1 SYM of [ 1]. All-order finiteness is here meant in the sense of exact vanishing of the perturbative β-functions. The all-order finiteness criterion is an exact result, with hypotheses operating exclusively at the one-loop level. It is based on the structure of the supercurrent anomaly multiplet, which relates the conformal anomalies to the axial ones. The axial anomalies being non-renormalized, they are given by their one-loop values. Vanishing of the latter is guaranteed by the hypothesis that the one-loop gauge β-function, as well as the one-loop anomalous dimensions, vanish (these two conditions are known to yield one-loop finiteness [ 2]). A further hypothesis on the unicity of the solution to the conditions of vanishing one-loop Yukawa β-functions comes from imposing, as a consistency requirement, that reduction of the couplings be verified. Therefore, the all-order finiteness result is the one-loop result supplemented by a consistency requirement for higher orders [hyp. (iv) below]. We shall first review one-loop finiteness (section 2). In section 3, we state the all-order result. Section 4 presents three applications. For related approaches to all-order finiteness in N = 1 SYM, see [ 3, 4, 5].