One - loop renormalisation of massive N = 12 supersymmetric gauge theory

We construct the general N = 1 2 supersymmetric gauge theory coupled to massive chiral matter, and show that it is renormalisable at one loop. There has recently been much interest in theories defined on non-anticommutative su-perspace [1]–[4]. Such theories are non-hermitian and turn out to have only half the super-symmetry of the corresponding ordinary supersymmetric theory–hence the term " N = 1 2 supersymmetry ". These theories are not power-counting renormalisable 1 but it has been argued[7]–[10] that they are in fact nevertheless renormalisable, in the sense that only a finite number of additional terms need to be added to the lagrangian to absorb divergences to all orders. This is primarily because although the theory contains operators of dimension five and higher, they are not accompanied by their hermitian conjugates which would be required to generate divergent diagrams. This argument does not of course guarantee that the precise form of the lagrangian will be preserved by renormalisation; nor does the N = 1 2 supersymmetry, since some terms in the lagrangian are inert under this symmetry. Moreover, the argument also requires (in the gauged case) the assumption of gauge invari-ance to rule out some classes of divergent structure. In previous work we have shown that although divergent gauge non-invariant terms are generated, they can be removed by a divergent field redefinition[11]; and that in the case of N = 1 2 supersymmetry with chiral matter[12] the joint requirements of renormalisability and N = 1 2 supersymmetry impose the choice of gauge group SU (N) ⊗ U (1) (rather than U (N) or SU (N)). In Ref. [12] there was no superpotential for the chiral matter. In the present work we show that in general the only superpotential terms which can be added consist of mass terms for the chiral and antichiral fields (linking the fundamental and antifundamental representations). 1 2 supersymmetric SU (N) ⊗ U (1) gauge theory coupled to 1 See Refs. [5][6] for other discussions of the ultraviolet properties of these theories.