Minimal massive completions of partially broken higher-spin models

Abstract

The partially broken Fronsdal (PBF) model describes massless particles of arbitrary integer spin $s$ with a gauge symmetry smaller than Fronsdal's, and differs from it only by the coefficient $c$ of the term $(\p\cdot h')^2$. We ask whether it admits a genuinely new massive completion. Adding a mass term and the most general local auxiliary sector with at most two derivatives, and demanding that all lower-spin sectors be non-dynamical, we show that for $s=3$ a single auxiliary scalar is never enough, while two scalars give a solution unique up to field redefinitions: one scalar is algebraic and its elimination exactly undoes the deformation, while the other is the auxiliary scalar of the Aragone--Deser--Yang (ADY) model. The same mechanism, with an algebraic vector, appears for $s=4$ and extends to arbitrary spin, and we conjecture that the resulting auxiliary content is minimal. In this minimal completion $c$ does not survive the mass: the correlation functions of $h$ coincide with those of the ADY model. For the Skvortsov--Vasiliev (SV) model, which belongs to the same family, we construct a massive version in which the Weyl-like symmetry survives the mass, and show that its auxiliary sector is fixed by the absence of lower-spin modes, within a general ansatz for $s=3$ and within a restricted one for $s=4$. For the partially broken SV model we obtain instead a massive completion which requires no auxiliary field beyond those of the massive SV model, and in which $c$ survives the mass through the same contact term as in the massless theory.