Covariant confinement model for the calculation of radial excitations of the pion

We describe the mixing of q (q) over bar pseudoscalar states with longitudi nal q (q) over bar axial-vector states, making use of a relativistic quark model that includes a model of confinement. (In the absence of the confinem ent model, our model reduces to the Nambu-Jona-Lasinio model.) In addition to the pion, we find J(P) = 0(-) states at 1.18, 1.36, 1.47, 1.63, and 1.68 GeV. The first two of these states are in the region of the pi(1300) that is assigned a mass of 1300+/- 100 MeV and a width of 200-600 MeV in the dat a tables. We provide values of the coupled-channel q (q) over bar T matrix, as well as the mixing angle, which is energy-dependent in bur analysis. In addition, we describe a model of confinement for longitudinal axial-vector q (q) over bar states that is used in the calculation of vacuum polarizati on diagrams. (That analysis supplements our previous study of confinement i n the case of pseudoscalar mesons.) We show that our confinement model may be made covariant. We use the covariant model to calculate the decay of the various states, pi, to the pi+rho and pi+sigma channels at one-loop order. At one-loop order, it is found that only the nodeless state at 1.18 GeV an d the state at 1.36 GeV have significant widths for pi'-->pi+sigma. These s tates have somewhat larger widths for the decay pi'-->pi+rho, leading to Ga mma(tot)=0.368 GeV for the state at 1.18 GeV and 0.150 GeV for the state at 1.36 GeV: We note that the state Ills GeV is a mixed pseudoscalar-axial-ve ctor state, while the state at 1.36 GeV is the pi(2S) state to a good appro ximation, since it has a very small admixture of axial-vector components. T here is information concerning the decay pi' -->pi'+(pi+pi)(L=0) that is ex tracted from experimental data for three-body final states. Our (nodeless) state at 1.18 GeV has the correct energy and width to fit that data. Howeve r, our widths for pi'-->pi+(pi+pi)(L=1) are larger than those for pi'-->pi(pi+pi)(L=0) That suggests that final-state interactions are probably quite important in understanding the branching ratios for pi' decays to States o f three pions. Our results also suggest that, if we were to study the pi(13 00), and include final-state interactions, it is necessary to include both the 1.18 GeV and the 1.36 GeV states in the analysis. (On the other hand, s ince the 1.36 GeV state is a 2S state, it may be only weakly excited in the reactions used to generate final states of three pions.) [S0556-2813(99)05 802-1].