In supersymmetric generalizations of the standard model, all stable $Q$-balls are associated with some flat directions. We show that, if the flat direction has both the baryon number and the lepton number, the scalar field inside the $Q$-ball can deviate slightly from the flat direction in the ground state. We identify the true ground states of such nontopological solitons, including the electrically neutral and electrically charged $Q$-balls.

• Received 9 September 2008

DOI:https://doi.org/10.1103/PhysRevD.78.075014