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We discuss the non-anticommutative (N=1/2) supersymmetric U(1) gauge theoryin four dimensions, including a superpotential. We perform the one-looprenormalisation of the model, including the complete set of terms necessary forrenormalisability, showing in detail how the eliminated and uneliminated formsof the theory lead to equivalent results.

The non-anticommutative supersymmetric U(1) gauge theory