Yukawa Bound States of a Large Number of Fermions

Abstract

We consider the bound state problem for a field theory that contains a Dirac fermion x that Yukawa couples to a (light) scalar field ø. We are interested in bound states with a large number N of x 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 x's are suppressed two regimes emerge. For modest values of N the state is composed of non-relativistic x 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 x mass, there is a minimum sized state that occurs roughly at the value of N where the core region first becomes relativistic. As an application to dark matter, our analysis offers the possibility of having a supermassive thermal DM candidate. We also compute an elastic scattering form factor that can be relevant for direct detection if the dark matter is composed of such x particles.