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We discuss the role of a class of higher dimensional operators in 4D N=1supersymmetric effective theories. The Lagrangian in such theories is anexpansion in momenta below the scale of "new physics" ($\Lambda$) and containsthe effective operators generated by integrating out the "heavy states" above$\Lambda$ present in the UV complete theory. We go beyond the "traditional"leading order in this momentum expansion (in $\partial/\Lambda$). Keepingmanifest supersymmetry and using superfield {\it constraints} we show that thecorresponding higher dimensional (derivative) operators in the sectors ofchiral, linear and vector superfields of a Lagrangian can be "unfolded" intosecond-order operators. The "unfolded" formulation has only polynomialinteractions and additional massive superfields, some of which are ghost-likeif the effective operators were {\it quadratic} in fields. Using thisformulation, the UV theory emerges naturally and fixes the (otherwise unknown)coefficient and sign of the initial (higher derivative) operators. Integratingthe massive fields of the "unfolded" formulation generates an effective theorywith only polynomial effective interactions relevant for phenomenology. We alsoprovide several examples of "unfolding" of theories with higher derivative {\itinteractions} in the gauge or matter sectors that are actually ghost-free. Wethen illustrate how our method can be applied even when including {\it allorders} in the momentum expansion, by using an infinite set of superfieldconstraints and an iterative procedure, with similar results.

Effective operators in SUSY, superfield constraints and searches for a UV completion