The pion and an improved static bag model

TLDR: In this article, momentum-space wave packets are used to approximate the mass of the pion in the bag model, and the formula m/sub q/=(m-italic/sub u/+m/sub d/)/2=33 MeV is derived.

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/sub ..pi../ vanish when the light quarks are massless. Expanding about this limit (and also readjusting the fit to other hadrons), we obtain m/sub q/=(m-italic/sub u/+m/sub d/)/2=33 MeV. We calculate F/sub ..pi../ approx. = 145 MeV (using a normalization such that F/sub ..pi../ vertical-bar /sub exp/=93 MeV), F/sub K//F/sub ..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/sub ..pi../ /sup 2/=(m-italic/sub u/+m/sub d/) 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 q-barq as an order parameter describing themore » separation of the spontaneously broken vacuum phase from the perturbative vacuum of the bag's interior.« less