One - loop renormalisation of N = 12 supersymmetric gauge theory in the adjoint representation

1 2 supersymmetric gauge theory coupled to chiral matter in the adjoint representation, and investigate the one-loop 1 2 supersymmetric theories (i.e. theories defined on non-anticommutative super-space) have recently attracted much attention[1]–[4]. Such theories are non-hermitian and only have half the supersymmetry of the corresponding N = 1 theory. These theories are not power-counting renormalisable 1 but it has been argued[7]–[10] that they are in fact nevertheless renormalisable, in other words only a finite number of additional terms need to be added to the lagrangian to absorb divergences to all orders. In previous work we have confirmed this renormalisability at the one-loop level. In particular we have shown that although divergent gauge non-invariant terms are generated at the one-loop level, they can be removed by divergent field redefinitions leading to a renormalisable theory in which N = 1 2 supersymmetry is preserved at the one-loop level in both the pure gauge case[11] and in the case of chiral matter in the fundamental representation[12]. On the other hand, the authors of Ref. [13] obtained the one loop effective action for pure N = 1 2 supersymmetry using a superfield formalism. Although they found divergent contributions which broke supergauge invariance, their final result was gauge-invariant without the need for any redefinition. In subsequent work[14] it was shown that the N = 1 2 superfield action requires modification to ensure renormalisability, which is consistent with our findings in the component formulation[12]. It was pointed out in Ref. [4] that an N = 1 2 supersymmetric theory can also be constructed with matter in the adjoint representation. Our purpose here is to repeat the analysis of Ref. [12] for the adjoint case, then proceed to consider the addition of superpotential terms, which will turn out to be a non-trivial task. The adjoint action of Ref. [4] was written for the gauge group U (N). As we noted in Refs. [11], [12], at the quantum level the U (N) gauge invariance cannot be retained. In the case of chiral matter in the fundamental representation we were obliged to consider a modified theory with the gauge group SU (N) ⊗ U (1). In the adjoint case with a trilinear superpotential, it will turn out that the matter fields must also be in a representation of SU (N) ⊗ U (1). However, for simplicity of exposition we shall start by considering the adjoint case without a superpotential, …