Comments on the nilpotent constraint of the goldstino superfield

Abstract

Superfield constraints were often used in the past, in particular to describe the Akulov-Volkov action of the goldstino by a superfield formulation with $L=(Φ^\dagger Φ)_D + [(fΦ)_F + h.c.]$ endowed with the nilpotent constraint $Φ^2=0$ for the goldstino superfield ($Φ$). Inspired by this, such constraint is often used to define the goldstino superfield even in the presence of additional superfields, for example in models of "nilpotent inflation". In this review we show that the nilpotent property is not valid in general, under the assumption of a microscopic (ultraviolet) description of the theory with linear supermultiplets. Sometimes only weaker versions of the nilpotent relation are true such as $Φ^3=0$ or $Φ^4=0$ ($Φ^2\not=0$) in the infrared (far below the UV scale) under the further requirement of decoupling all additional scalars (coupling to sgoldstino), something not always possible (e.g. if light scalars exist). In such cases the weaker nilpotent property is not specific to the goldstino superfield anymore. We review the restrictions for the Kahler curvature tensor and superpotential $W$ under which $Φ^2=0$ remains true in infrared, assuming linear supermultiplets in the microscopic description. One can reverse the arguments to demand that the nilpotent condition, initially an infrared property, be extended even in the presence of additional superfields, but this may question the nature of supersymmetry breaking or the existence of a perturbative ultraviolet completion with linear supermultiplets.