$\mathcal{N}=8$ invariant interaction of dynamical and semi-dynamical $\mathcal{N}=4$ multiplets

We present a new model of $\mathcal{N}=8$ mechanics with semi-dynamic supermultiplets. The model is constructed as an interaction of $\mathcal{N}=4$ supermultiplets which carry an implicit $\mathcal{N}=4$ supersymmetry. The initial field content consists of three dynamical $(\mathbf{1},\mathbf{4},\mathbf{3})$ multiplets: one bosonic and two fermionic. To ensure implicit $\mathcal{N}=4$ supersymmetry, we introduce the superfields describing three semi-dynamical $(\mathbf{4},\mathbf{4},\mathbf{0})$ multiplets: one fermionic and two bosonic. To avoid the second-order Lagrangian for fermions from the fermionic $(\mathbf{1},\mathbf{4},\mathbf{3})$ multiplets, the conversion of their velocities into new auxiliary fields is carried out. After conversion, these multiplets turn into semi-dynamical mirror $(\mathbf{4},\mathbf{4},\mathbf{0})$ multiplets without non-canonical terms in the $\mathcal{N}=8$ Lagrangian at the component level. The final $\mathcal{N}=8$ multiplet content is $(\mathbf{1},\mathbf{8},\mathbf{7})\oplus (\mathbf{8},\mathbf{8},\mathbf{0})$. As a first step to the ultimate $\mathcal{N}=4$ superfield formulation of the model, we remind a natural description of the standard and mirror $(\mathbf{4},\mathbf{4},\mathbf{0})$ multiplets in the framework of $\mathcal{N}=4,d=1$ biharmonic superspace.