The pion and an improved static bag model

Abstract

Quark-model calculations involve an extended static object localized in space. We introduce new methods, involving momentum-space wave packets, which account for this localization. These methods have little effect on heavy states, whose sizes are large compared to their Compton size 1/m, but are very important for light particles such as the pion. In this treatment the pion's mass is naturally very small, and, in order to connect with a spontaneously broken chiral symmetry, we require that ${m}_{\ensuremath{\pi}}$ vanish when the light quarks are massless. Expanding about this limit (and also readjusting the fit to other hadrons), we obtain ${m}_{q}=\frac{({m}_{u}+{m}_{d})}{2}=33$ MeV. We calculate ${F}_{\ensuremath{\pi}}\ensuremath{\approx}145$ MeV (using a normalization such that ${F}_{\ensuremath{\pi}}{|}_{\mathrm{exp}}=93$ MeV), $\frac{{F}_{K}}{{F}_{\ensuremath{\pi}}}\ensuremath{\approx}1$, and various corrections to static properties of baryons. In addition we explore the relationship of our methods with chiral perturbation theory, deriving the formula ${{m}_{\ensuremath{\pi}}}^{2}=({m}_{u}+{m}_{d})〈\ensuremath{\pi}(p)|\overline{q}(0)q(0)|\ensuremath{\pi}(p)〉$ in the appropriate approximation and commenting on the quark mass obtained from the nucleon's $\ensuremath{\sigma}$ term. Finally we discuss the bag model's use of the scalar density $\overline{q}q$ as an order parameter describing the separation of the spontaneously broken vacuum phase from the perturbative vacuum of the bag's interior.