The pion and an improved static bag model

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 $m_{\pi }$ vanish when the light quarks are massless. Expanding about this limit (and also readjusting the fit to other hadrons), we obtain $m_q=\frac{(m_u+m_d)}{2}=33\mathrm{}$ MeV. We calculate $F_{\pi }\approx 145$ MeV (using a normalization such that $F_{\pi }{|}_{\exp }=93\mathrm{}$ MeV), $\frac{F_K}{F_{\pi }}\approx 1$, and various corrections to static properties of baryons. In addition we explore the relationship of our methods with chiral perturbation theory, deriving the formula ${m_{\pi }}^2=(m_u+m_d)〈\pi \mathrm{}\left(p\right)\left|\overline{q}\mathrm{}\right(0\left)q\right(0\left)\mathrm{}\right|\pi \mathrm{}\left(p\right)\mathrm{}〉$ in the appropriate approximation and commenting on the quark mass obtained from the nucleon's $\sigma$ 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.