The calculation of Dirac sea effects on finite solitons

The effects of polarization of the Dirac sea on finite solitons in a simple theory in which fermions interact with a single scalar field are studied. The mass shift for a given background scalar field is computed numerically and compared to approximations arising from expansions in inverse powers of the effective fermion mass and in powers of derivatives of the background scalar field. The conditions under which such approximations succeed are discussed. When such approximations work one can derive local equations of motion for the soliton fields which include the effects of polarizing the Dirac sea. These new equations are studied and energy minimization is used to explore the effects of the Dirac sea on the structure of the soliton. Calculations for a typical Friedberg-Lee soliton are presented, and it is shown that, while the approximations do not work well for fields employed to model the quark structure of nucleons, they do provide an upper bound for the mass of the soliton. A scalar field typical of those used to model ¹⁶O in quantum hadrodynamics is also studied, and it is shown that, when the effective potential is supplemented by the next term occurring in a derivative expansion, the renormalized shift in the energy of the Dirac sea is well approximated.