Towards the Complete N=2 Superfield Born-Infeld Action with Partially Broken N=4 Supersymmetry

We propose a systematic way of constructing N = 2, d = 4 superfield Born-Infeld action with a second nonlinearly realized N = 2 supersymmetry. The latter, together with the manifest N = 2 supersymmetry, form a central-charge extended N = 4, d = 4 supersymmetry. We embed the Goldstone-Maxwell N = 2 multiplet into an infinite-dimensional off-shell supermultiplet of this N = 4 supersymmetry and impose an infinite set of covariant constraints which eliminate all extra N = 2 superfields through the Goldstone-Maxwell one. The Born-Infeld superfield Lagrangian density is one of these composite superfields. The constraints can be solved by iterations to any order in the fields. We present the sought N = 2 Born-Infeld action up to the 10th order. It encompasses the action found earlier by Kuzenko and Theisen to the 8th order from a self-duality requirement. This is a strong indication that the complete N = 2 Born-Infeld action with partially broken N = 4 supersymmetry is also self-dual.