A complete Lorentz-to-Galileo dictionary for direct Dark Matter detection

We determine the most general non-relativistic theory of DM-nucleon scattering complying with the sole requirement of Lorentz invariance, for spin-0 and spin-1/2 DM. To do so, we first classify a comprehensive list of amplitude terms encompassing the most general Lorentz-covariant 2-to-2 DM-nucleon scattering amplitude. We then match each term to a Galilean-invariant operator at leading-order in the non-relativistic expansion, for both elastic and inelastic (endothermic and exothermic) scattering. Our complete Lorentz-to-Galileo mapping can be used to promptly determine the non-relativistic DM-nucleon interaction and the associated nuclear form factor for any given Lorentz-invariant DM model. It applies to both renormalizable and non-renormalizable theories (such as effective field theories at all orders), at any order of a perturbative expansion. We use our results to prove that, at leading order, Lorentz invariance does not impose restrictions on the set of 16 Galilean-invariant operators commonly used to parametrize the non-relativistic DM-nucleon interaction. We also predict the lowest effective-operator dimension at which the non-relativistic operators appear in the effective field theory of a singlet DM particle.