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We investigate superfield formulation of the minimal$\mathbb{Z}_2^2$-supersymmetry. It is shown that the integrability on$\mathbb{Z}_2^2$-superspace guarantees the invariance of action. Then wepresent two superfields which carry distinct irreducible representations of the$\mathbb{Z}_2^2$-supersymmetry algebra. One of them gives integrable Lagrangianand the other does not. We also show that integrable superfields with different$\mathbb{Z}_2^2$-degree also carry irreducible representations and they giveinvariant actions. To perform this analysis, the representation theory of theminimal $\mathbb{Z}_2^2$-supersymmetry algebra is studied in some detail.

Comments of $\mathbb{Z}_2^2$-supersymmetry in superfield formalism