Perturbation Analysis of Deformed Q-Balls and Primordial Magnetic Field
  We study the excited states of the Q-balls by performing stationary perturbation on the spherical Q-balls. We find the exact solution of the stationary perturbation of the global Q-ball with thin wall approximation. For local Q-balls we solve the equations of motion for the perturbative part approximately by using expansion about the coupling constant. Furthermore we comment on the magnetic field generated by the excited Q-balls during the phase transition precipitated by solitosyhthesis and give an implication into cosmology. 
Tetsuya Shiromizu, Tomoko Uesugi and Mayumi Aoki