Goldstones with Extended Shift Symmetries

TL;DR

The work extends the concept of shift symmetries to polynomial, spatially and temporally extended forms for a scalar field, and develops a systematic coset construction to identify both strictly invariant and Wess–Zumino (WZ) Lagrangians under these symmetries.A key result is that WZ terms exist with fewer derivatives per field than strictly invariant terms, such as $\mathcal{L}_1\sim\phi$, $\mathcal{L}_2\sim(\nabla^2\phi)^2$, and higher-order cubic and beyond terms, all yielding lower-order or controlled equations of motion (e.g., up to fourth order) while preserving the symmetry up to a total derivative.In the non-relativistic limit these constructions describe multi-critical Goldstone bosons with dispersion relations containing higher powers of momentum (e.g., $\omega^2(\mathbf{k})\sim a_2\mathbf{k}^2+a_4\mathbf{k}^4+\cdots$), whereas relativistic realizations generally introduce ghostly degrees of freedom due to higher-derivative invariant terms.The paper also demonstrates that for traceless extended symmetries, the standard two-derivative kinetic term can itself arise as a WZ term, and discusses potential implications for coupling to gravity and condensed-matter systems.

Abstract

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.