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.
