The effective hyper-K"ahler potential in the N=2 supersymmetric QCD

The effective low-energy hyper-K"ahler potential for a massive N=2 matter in the N=2 super-QCD is investigated. The N=2 extended supersymmetry severely restricts that N=2 matter self-couplings so that their exact form can be fixed by a few parameters, which is apparent in the N=2 harmonic superspace. In the N=2 QED with a single matter hypermultiplet, the one-loop perturbative calculations lead to the Taub-NUT hyper-K"ahler metric in the massive case, and a free metric in the massless case. It is remarkable that the naive non-renormalization `theorem' does not apply. There exists a manifestly N=2 supersymmetric duality transformation converting the low-energy effective action for the N=2 QED hypermultiplet into a sum of the quadratic and the improved (non-polynomial) actions for an N=2 tensor multiplet. The duality transformation also gives a simple connection between the low-energy effective action in the N=2 harmonic superspace and the component results.