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

Abstract

We describe the mixing of $q\overline{q}$ pseudoscalar states with longitudinal $q\overline{q}$ axial-vector states, making use of a relativistic quark model that includes a model of confinement. (In the absence of the confinement model, our model reduces to the Nambu--Jona-Lasinio model.) In addition to the pion, we find ${J}^{P}{=0}^{\ensuremath{-}}$ 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 \ensuremath{\pi}(1300) that is assigned a mass of $1300\ifmmode\pm\else\textpm\fi{}100\mathrm{MeV}$ and a width of 200--600 MeV in the data tables. We provide values of the coupled-channel $q\overline{q}$ T matrix, as well as the mixing angle, which is energy-dependent in our analysis. In addition, we describe a model of confinement for longitudinal axial-vector $q\overline{q}$ states that is used in the calculation of vacuum polarization diagrams. (That analysis supplements our previous study of confinement in 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, ${\ensuremath{\pi}}^{\ensuremath{'}},$ to the $\ensuremath{\pi}+\ensuremath{\rho}$ and $\ensuremath{\pi}+\ensuremath{\sigma}$ channels at one-loop order. At one-loop order, it is found that only the nodeless state at 1.18 GeV and the state at 1.36 GeV have significant widths for ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\sigma}.$ These states have somewhat larger widths for the decay ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\rho},$ leading to ${\ensuremath{\Gamma}}_{\mathrm{tot}}=0.368\mathrm{GeV}$ for the state at 1.18 GeV and 0.150 GeV for the state at 1.36 GeV. We note that the state 1.18 GeV is a mixed pseudoscalar--axial-vector state, while the state at 1.36 GeV is the $\ensuremath{\pi}(2S)$ state to a good approximation, since it has a very small admixture of axial-vector components. There is information concerning the decay ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+(\ensuremath{\pi}+\ensuremath{\pi}{)}_{L=0}$ that is extracted 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. However, our widths for ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+(\ensuremath{\pi}+\ensuremath{\pi}{)}_{L=1}$ are larger than those for ${\ensuremath{\pi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\pi}+(\ensuremath{\pi}+\ensuremath{\pi}{)}_{L=0}.$ That suggests that final-state interactions are probably quite important in understanding the branching ratios for ${\ensuremath{\pi}}^{\ensuremath{'}}$ decays to states of three pions. Our results also suggest that, if we were to study the \ensuremath{\pi}(1300), 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, since 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.)