Stability of Fermi ball against deformation from spherical shape

The stability of a Fermi ball (F-ball), which is a kind of non-topological soliton accompanying the breakdown of the approximate $Z_2$ symmetry, is investigated in three situations: the case it is electrically neutral, the case it is electrically charged and unscreened, and the case it is electrically charged and screened. We argue only the third case is physically meaningful since the neutral F-ball is unstable and the case of the unscreened charged one is observationally excluded when it has a sizable contribution to CDM. We found that the energy scale of the breakdown of the approximate $Z_2$ symmetry $v$ should satisfy $v