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

We discuss the role of a class of higher dimensional operators in 4D N=1 super-symmetric effective theories. The Lagrangian in such theories is an expansion in momenta below the scale of new physics (I >) and contains the effective operators generated by integrating out the heavy states above I > present in the UV complete theory. We go beyond the traditional leading order in this momentum expansion (in a,/I >). Keeping manifest supersymmetry and using superfield constraints we show that the corresponding higher dimensional (derivative) operators in the sectors of chiral, linear and vector superfields of a Lagrangian can be unfolded into second-order operators. The unfolded formulation has only polynomial interactions and additional massive superfields, some of which are ghost-like if the effective operators were quadratic in fields. Using this formulation, the UV theory emerges naturally and fixes the (otherwise unknown) coefficient and sign of the initial (higher derivative) operators. Integrating the massive fields of the unfolded formulation generates an effective theory with only polynomial effective interactions relevant for phenomenology. We also provide several examples of unfolding of theories with higher derivative interactions in the gauge or matter sectors that are actually ghost-free. We then illustrate how our method can be applied even when including all orders in the momentum expansion, by using an infinite set of superfield constraints and an iterative procedure, with similar results.