Yukawa Bound States of a Large Number of Fermions

We consider the bound state problem for a field theory that contains a Dirac fermion $\chi$ that Yukawa couples to a (light) scalar field $\phi$. We are interested in bound states with a large number $N$ of $\chi$ particles. A Fermi gas model is used to numerically determine the dependence of the radius $R$ of these bound states on $N$ and also the dependence of the binding energy on $N$. Since scalar interactions with relativistic $\chi$'s are suppressed two regimes emerge. For modest values of $N$ the state is composed of non-relativistic $\chi$ particles. In this regime as $N$ increases $R$ decreases. Eventually the core region becomes relativistic and the size of the state starts to increase as $N$ increases. As a result, for fixed Yukawa coupling and $\chi$ mass, there is a minimum sized state that occurs roughly at the value of $N$ where the core region first becomes relativistic. We also compute an elastic scattering form factor that can be relevant for direct detection if the dark matter is composed of such $\chi$ particles.