SU(2) x U(1) nonanticommutative N = 2 supersymmetric gauge theory

We derive the master function governing the component action of the four-dimensional nonanticommutative (NAC) and fully N = 2 supersymmetric gauge field theory with a nonsimple gauge group U(2) = SU(2) x U(1). The new NAC master function is a nontrivial generalization of the known master functions in the NAC, N = 2 supersymmetric gauge theories with the U(1) and SU(2) gauge groups. We use a Lorentz-singlet NAC-deformation parameter and an N = 2 supersymmetric star (Moyal) product, which do not break any of the fundamental symmetries of the undeformed N = 2 gauge theory. The scalar potential in the NAC-deformed theory is calculated. We also propose the non-Abelian BPS-type equations in the case of the NAC-deformed N = 2 gauge theory with the SU(2) gauge group, and comment on the SU(3) case too. The NAC-deformed field theories can be thought of as the effective (nonperturbative) N = 2 gauge field theories in a certain (scalar only) N = 2 supergravity background.