  Yukawa-type interactions between heavy Dirac fermions and a scalar field are
a common ingredient in various extensions of the Standard Model. Despite of
that, the non-relativistic limit of the scalar Yukawa theory has not yet been
studied in full generality in a rigorous and model-independent way. In this
paper we intend to fill this gap by initiating a series of investigations that
make use of modern effective field theory (EFT) techniques. In particular, we
aim at constructing suitable non-relativistic and potential non-relativistic
EFTs of Yukawa interactions (denoted as NRY and pNRY respectively) in close
analogy to the well known and phenomenologically successful non-relativistic
QCD (NRQCD) and potential non-relativistic QCD (pNRQCD). The phenomenological
motivation for our study lies in the possibility to explain the existing
cosmological observations by introducing heavy fermionic dark matter particles
that interact with each other by exchanging a light scalar mediator. A
systematic study of this compelling scenario in the framework of
non-relativistic EFTs (NREFTs) constitutes the main novelty of our approach as
compared to the existing studies.
