Goldstones with Extended Shift Symmetries

We consider scalar field theories invariant under extended shift symmetries consisting of higher order polynomials in the spacetime coordinates. These generalize ordinary shift symmetries and the linear shift symmetries of the galileons. We find Wess-Zumino Lagrangians which transform up to total derivatives under these symmetries, and which possess fewer derivatives per field and lower order equations of motion than the strictly invariant terms. In the non-relativistic context, where the extended shifts are purely spatial, these theories may describe multi-critical Goldstone bosons. In the relativistic case, where the shifts involve the full spacetime coordinate, these theories generally propagate extra ghostly degrees of freedom.