Cloudy bag model: Convergent perturbation expansion for the nucleon

The concept of chiral symmetry has been of great importance in elementary-particle physics for many years. ' ' In the context of massless quarks and quantum chromodynamics it is, of course, an exact symmetry, and should survive the proof of confinement in some way. It is not surprising, therefore, that immediately after the presentation of the original MIT bag model4 Chodos and Thorn' attempted to repair its obvious violation of chiral symmetry. Their method relied on introducing massless, elementary o and m fields which coupled to the quarks, only at the bag surface, in such a way as to restore exact chiral symmetry. In the past year or so, interest in this problem has been dramatically revived. For a longer review of these developments we refer to the discussion of Ref. 6, but a few brief comments will be useful here. Brown and collaborators have argued that the w field is actually a crucial aspect of the confinement process for the nucleon. ' That is, chiral symmetry should be manifest in the Wigner-Weyl mode inside (no pions), and in the Nambu-Goldstone mode outside (the pion is the Goldstone boson). ' In their purely classical model this external pion exerts a large pressure on the bag, resulting in a confinement volume of a few tenths of a fermi for the nucleon. They argue further that such a picture (the "little bag") would be more consistent with classical nuclear physics. On the other hand, Jaffe' and others' have developed classical models of a bag surrounded by a pion field which merely acts as a small perturbation on the usual MIT ground state. Once again the pion field appears as a Goldstone boson, excluded from the interior of the bag. Recent work by Johnson has also shown the importance of collective qq excitations in the volume about the MIT bag." However, the phenomenological replacement of such excitations by a pion field has not yet been clarified. At the same time as these developments using a classical pion field were taking place, the TRIUMF-University of Washington group has constructed a quantized version of the theory —the "cloudy bag model" (CBM)."" In order to avoid technical problems, the CBM (like that of Chodos and Thorn) allows the pion field to penetrate the bag. By working only to lowest order in the pion field, which is assumed to be small, it was possible to derive a Hamiltonian (see Sec. II) describing an interacting-system of (bare) nucleons, deltas, and pions. In Ref. 12, hereafter CBM-1, this Hamiltonian was used to settle the longstanding problem of the nature of the (3, 3) resonance. In CBM-2 (Ref. I3) this model has been used, with considerable success, to calculate pionic corrections to the MIT model of the nucleon (g„,magnetic moment, and charge radii). In the CBM work the contribution from the pion field inside the bag was rather small, and could be justified as a crude approximation to the effectof virtual qq pairs inside the bag. Indeed it is just this point which was made recently by DeTar. ' " His work provides some formal link between the theory of Jaffe' with the pion excluded from the bag and the CBM Hamiltonian, which he also used in a calculation of nucleon properties. There is a great deal of interesting physics in these developments, but for our present purpose we note only that all groups, except Stony Brook, rely on a perturbative treatment of pionic effects. This perturbative treatment has two aspects. First, the exponential coupling at the bag surface