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 Schrödinger-type scalar, namely the ℂP1 nonlinear sigma model and the Schrödinger-Dirac-Born-Infeld theory. Moreover, we use soft recursion to prove a no-go theorem ruling out the existence of other exceptional Schrödinger-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.
Highlights
Effective field theory (EFT) is a general framework that encodes the dynamics of the degrees of freedom present in a physical system below a given energy scale
EFT is powerful for physical systems with an ordered ground state, where the low-energy dynamics is dominated by Nambu-Goldstone (NG) modes of the symmetry spontaneously broken by the order parameter
We review the results of a previous classification of nonrelativistic EFTs with enhanced symmetries [36], which provides a basis for the discussion of scattering amplitudes in the rest of the paper
Summary
Effective field theory (EFT) is a general framework that encodes the dynamics of the degrees of freedom present in a physical system below a given energy scale. This happens as a rule when different global symmetries generate locally indistinguishable fluctuations of the order parameter [10–13] While such redundant symmetries impose nonlinear constraints on the low-energy effective action similar to any other spontaneously broken symmetry, it is not obvious what they imply for actual physical observables. Within a generic EFT for a scalar field, imposing the condition that the scattering amplitudes in the soft limit have the Adler zero property (σ = 1) will constrain the Wilson coefficients of the many operators one can add to the effective Lagrangian. The role of such constraints is to ensure cancellations between various contributions to the S-matrix, which leads to the Adler zero in the soft limit. In order to help the reader orient in the text, we give a brief overview of the contents and main results of the individual sections
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