Quantum Corrections to Deep Bags ∗

Nontopological solitons, or “bags,” can arise when fermions acquire their mass through a Yukawa coupling to some scalar field. Bags have played an important role in models of baryons, nuclei, and more recently, in the idea that a Higgs condensate may form around a very heavy top quark. It has been claimed that deep bags, which correspond to tightly-bound states of fermions, will form when the Yukawa coupling is strong. Quantum corrections, however, are significant in this regime. We examine the effects of these quantum corrections on the formation of nontopological solitons in an exactly solvable large-N model. We find that quantum bags differ dramatically from those of the classical theory. In particular, for large Yukawa coupling, the bags remain shallow and the fermions weakly bound. ∗ Supported by NSF grant PHY-90-96198 and by the Alfred P. Sloan Foundation. Presented at the XXVI International Conference on High Energy Physics, August, 1992. Nontopological solitons, or “bags,” can arise when fermions acquire their mass through a Yukawa coupling to a scalar field. Bags have been used to describe bound states of fermions. In particle physics, the SLAC bag played an early and important role in describing the confinement of quarks inside hadrons. In nuclear physics, nontopological bags have been successfully used to model the binding of nucleons within nuclei. More recently, bags have also been discussed in conjunction with the phenomenology of the heavy top-Higgs system. Nontopological solitons are coherent states in which the expectation value of a scalar field is reduced from its vacuum value by the presence of a fermion field. The solitons carry fermion number because the fermion is energetically bound to the bag. They are stable because they have lower energy than any other configuration with the same quantum numbers. The solitons form because the energy gained by decreasing the fermion mass is greater than the energy lost through the potential and gradient terms in the scalar-field Hamiltonian. At the classical level, as the Yukawa coupling g gets large, the fermions become tightly bound inside deep bags, whose energy and radius are independent of g. In the full quantum theory, however, quantum corrections can be very important, especially in the nonperturbative regime of large g. One must check to see whether bags still form. There are two types of quantum fluctuations to consider: those in the scalar field, which can destroy the coherent state, and those in the fermion field, which can collapse the bag. We examine bag formation in a consistent quantum field theory. We consider a theory with N Dirac fermions ψ coupled to a real scalar field φ. We solve the quantum theory to leading order in the large-N expansion for any value of the Yukawa coupling g. We find that the full quantum theory supports nontopological bags. The bags correspond to bound states of N fermions, with a binding energy of less than about 5%. The quantum bags differ significantly from those in the classical theory, where the binding energy approaches 100% for large g. The quantum corrections invalidate the classical picture of tightly-bound fermions inside deep bags. We first present our model. The Lagrangian density is L0 = 1 2 (∂μφ0) 2 − λ0 8N (φ0 − u0N) + N