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 $χ$ that Yukawa couples to a (light) scalar field $ϕ$. We are interested in bound states with a large number $N$ of $χ$ 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 $χ$'s are suppressed two regimes emerge. For modest values of $N$ the state is composed of non-relativistic $χ$ 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 $χ$ 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 $χ$ particles.

14 pages, 7 figures