Gauged Q ball in a piecewise parabolic potential
  Q ball solutions are considered within the theory of a complex scalar field with a gauged   U(1) symmetry and a parabolic-type potential. In the thin-walled limit, we show explicitly that there is a maximum size for these objects because of the repulsive Coulomb force. The size of Q ball will increase with the decrease of local minimum of the potential. And when the two minima degenerate, the energy stored within the surface of the Q ball becomes significant.   Furthermore, we find an analytic expression for gauged Q ball, which is beyond the conventional thin-walled limit. 
Xin-zhou Li, Jian-gang Hao, Dao-jun Liu and Guang Chen