Nonrelativistic effective field theories with enhanced symmetries and soft behavior

Abstract: We systematically explore the landscape of nonrelativistic effective field theories with a local $S$-matrix and enhanced symmetries and soft behavior. The exploration is carried out using both conventional quantum field theory methods based on symmetry arguments, and recently developed on-shell recursion relations. We show that, in contrary to relativistic theories, enhancement of the soft limit of scattering amplitudes in nonrelativistic theories is generally not a byproduct of symmetry alone, but requires additional low-energy data. Sufficient conditions for enhanced scattering amplitudes can be derived by combining symmetries and dispersion relations of the scattered particles. This has direct consequences for the infrared dynamics that different types of nonrelativistic Nambu-Goldstone bosons can exhibit. We then use a bottom-up soft bootstrap approach to narrow down the landscape of nonrelativistic effective field theories that possess a consistent low-energy $S$-matrix. We recover two exceptional theories of a complex Schr\"odinger-type scalar, namely the $\mathbb{C} P^1$ nonlinear sigma model and the Schr\"odinger-Dirac-Born-Infeld theory. Moreover, we use soft recursion to prove a no-go theorem ruling out the existence of other exceptional Schr\"odinger-type theories. We also prove that all exceptional theories of a single real scalar with a linear dispersion relation are necessarily Lorentz-invariant. Soft recursion allows us to obtain some further general bounds on the landscape of nonrelativistic effective theories with enhanced soft limits. Finally, we present a novel theory of a complex scalar with a technically natural quartic dispersion relation. Altogether, our work represents the first step of a program to extend the developments in the study of scattering amplitudes to theories without Lorentz invariance.