${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace

A superfield formalism for the minimal ${\mathbb{Z}}_2^2$-graded version of supersymmetry is developed. This is done by using the recently introduced definition of integration on the minimal ${\mathbb{Z}}_2^2$-superspace. It is shown that one may construct ${\mathbb{Z}}_2^2$-supersymmetric action by the procedure similar to the standard supersymmetry. However, the Lagrangian obtained has very general interaction terms, which give rise to a ${\mathbb{Z}}_2^2$-graded extension of many known theories defined in two-dimensional spacetime. As an illustration, we will give a ${\mathbb{Z}}_2^2$-extension of the sine-Gordon model different from the one already discussed in the literature.