Abstract
We initiate the classification of nonrelativistic effective field theories (EFTs) for Nambu-Goldstone (NG) bosons, possessing a set of redundant, coordinate-dependent symmetries. Similarly to the relativistic case, such EFTs are natural candidates for “exceptional” theories, whose scattering amplitudes feature an enhanced soft limit, that is, scale with a higher power of momentum at long wavelengths than expected based on the mere presence of Adler’s zero. The starting point of our framework is the assumption of invariance under spacetime translations and spatial rotations. The setup is nevertheless general enough to accommodate a variety of nontrivial kinematical algebras, including the Poincaré, Galilei (or Bargmann) and Carroll algebras. Our main result is an explicit construction of the nonrelativistic versions of two infinite classes of exceptional theories: the multi-Galileon and the multi-flavor Dirac-Born-Infeld (DBI) theories. In both cases, we uncover novel Wess-Zumino terms, not present in their relativistic counterparts, realizing nontrivially the shift symmetries acting on the NG fields. We demonstrate how the symmetries of the Galileon and DBI theories can be made compatible with a nonrelativistic, quadratic dispersion relation of (some of) the NG modes.
Highlights
effective field theories (EFTs) for NG bosons is well-understood in both relativistic [1,2,3] and nonrelativistic [4,5,6] systems
We have initiated the classification of exceptional EFTs living in spacetimes with a non-Lorentzian kinematical algebra
We first mapped the landscape of possible symmetry Lie algebras, obtained by augmenting the algebra of spacetime translations and spatial rotations with additional scalar and vector generators
Summary
We will answer the purely mathematical question as to what symmetry structures can be obtained by augmenting the Lie algebra of spacetime translations and spatial rotations with additional, scalar or vector, generators. Are completely fixed by the symmetric matrix gAB, required to be an invariant tensor under the representation ti of the algebra of Qis. Note that the first two lines of eq (2.5) give a precise definition of the QAs and QABs. The only commutators that remain to be determined are those of Qi with KμA and with the QA, QAB subsets of scalars. The Lie algebra of all the generators, Jμν, Pμ, KμA, Qi, is uniquely determined by the subalgebra of scalar generators Qi, defined by the structure constants fikj, its affine representation Ti, and the corresponding rank-2 symmetric invariant tensor gAB. In order to describe physical systems with invariance under time translations, the Lie algebra must possess the corresponding generator: the Hamiltonian H This can be included among the Qis; we need not specify its commutators separately. This assumption is justified for any d > 2, as explained in detail in appendix B
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