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.