Analysis of a Classical Chiral Bag Model

We study a classical chiral bag model with a Mexican hat-type potential for the self-coupling of the pion fields. We assume a static spherical bag of radius R, the hedgehog ansatz for the chiral fields and that the quarks are all in the lowest lying s state. We have considered three classes of models, the cloudy or pantopionic bags, the little or exopionic bags and the endopionic bags, where the pions are allowed all through space, only outside the bag and only inside the bag respectively. In all cases, the quarks are confined in the interior. The Euler-Lagrange equations give coupled second order differential equations and boundary conditions at the surface of the bag for the meson fields. Finiteness of the solutions and the usual sigma-model prescriptions provide the additional conditions needed to determine uniquely the system. The latter has already been solved for extreme values of the potential term coupling constant (lamda). Here we develop a method to obtain solutions for arbitrary values of (lamda). Our method consists in reducing the two coupled second order differential equations to first order ones in such a way as to implement the proper conditions at the origin and at infinity, but without further loss of generality. These two equations are expansions of the first derivative of both pion fields in powers of each variable that enters the equations of motion. By imposing the proper conditions at the boundary of the bag we determine initial conditions with which we start the numerical integration and directly obtain solutions. Using these solutions, we calculate the bag radius R, the bag constant B and the total ground state energy E for wide ranges of the two free parameters of the theory, namely the coupling constant (lamda) and the quark frequency (OMEGA). We focus our study on the endopionic bags, the least known class, and compare our results with the familiar ones of the other classes.