Enhancement of Supersymmetry via Renormalization Group Flow and the Superconformal Index

We find a four-dimensional $\mathcal{N}=1$ gauge theory which flows to the minimal interacting $\mathcal{N}=2$ superconformal field theory, the Argyres-Douglas theory, in the infrared up to the extra free chiral multiplets. The gauge theory is obtained from a certain $\mathcal{N}=1$ preserving deformation of the $\mathcal{N}=2$ $SU(2)$ gauge theory with four fundamental hypermultiplets. From this description, we compute the full superconformal index and find agreements with the known results in special limits.