Dual superfield approach to supersymmetric mechanics with spin variables

We consider a reducible $\mathcal{N}=4$, $d=1$ multiplet described by a real superfield as a coupling of the mirror $(\mathbf{1},\mathbf{4},\mathbf{3})$ and ordinary $(\mathbf{3},\mathbf{4},\mathbf{1})$ multiplets. Employing this so-called “long multiplet”, we construct a coupled system of dynamical and semi-dynamical multiplets. We show that the corresponding {\it on-shell} model reproduces the model of Fedoruk, Ivanov and Lechtenfeld presented in 2012. Furthermore, there is a hidden supersymmetry acting on the long multiplet that extends the full world-line supersymmetry to SU(2$\mathrm{\mid }$2). In other words, the $\mathcal{N}=4$ long multiplet can be derived from an irreducible SU(2$\mathrm{\mid }$2) multiplet.