Variation after Spin-Isospin Projection in the Skyrme Model

Abstract

2) In this model, the nucleon emerges as a topological soliton of the nonlinear chiral field. The hedgehog ansatz is in use as an intrinsic baryon state, which breaks the spin-isospin symmetry. Hence, a suitable spin-isospin projection is needed to construct physical baryons out of the hedgehog state. Most commonly used is the collective coordinate method (CCM) of Adkins et al.,3) which corresponds to rotation of hedgehog rigid rotor in isospin space. Recently, another method of spin-isospin projection was proposed by Hosaka and Toki based on the generator coordinate method (GeM). in the chiral bag plus Skyrmion hybrid model (CSH).4) In this method, whose powerfulness has been demonstrated in nuclear physics for the case of deformed nuclei and BCS states, the intrinsic state is considered as a mixed symmetry state, out of which states with desired quantum number are projected out. S ) They demonstrated that the nucleon and the delta get much smaller masses from those of CCM, which ought to be larger than the hedgehog mass. For example, MN=1l80 MeV, M.d=1400 MeV in GCM, while CCM provides MN=1570 MeV and M.d=1810 MeV in the limit of the zero bag radius (R-tO). In addition, the axial coupling gA was found gA=1.33 in GCM as compared to gA=O.89 in CCM. Those values are now close to the experimental values. Note that the above values are obtained from the Skyrme Lagrangian with 1,=93 MeV and e=4.5. 6 ) This GCM projection method suggests another development. Up to now, the variation of the hedgehog function is performed on the intrinsic state with mixed symmetry (variation before projection). We may reverse the order of calculations and perform variation after projection so that we minimize the energy of the state with good symmetry. Hence, each baryon has a different hedgehog function. In this paper we would like to report the numerical results of the variation after spin-isospin projection of the hedgehog state for nucleon, delta and higher spin-isospin baryons in the Skyrmion limit of the CSH (R-tO). We follow precisely the formulation of Hosaka and Toki,4) which is sketched briefly for clarity of notations. Since the Skyrme model is defined in the three sphere S3 in the four dimensional Euclidean space E 4, we can write the Lagrangian as