Quark model of light mesons with dynamically broken chiral symmetry

Abstract

We study the meson spectrum in a model with a confining Lorentz-vector---and hence chiral-invariant---interaction between massless quark fields. As shown in a previous work, chiral invariance is spontaneously broken. In the case of the harmonic oscillator, as the Fourier transform of the potential is the Laplacian of a \ensuremath{\delta} function, the Bethe-Salpeter (BS) equation---a system of linear integral equations in general---splits into a system of differential equations that we solve in the broken vacuum. Without appealing to any spin-spin interaction, we find, besides the massless pseudoscalar, a vector meson at the right scale and an excited pion and two vectors in the 1--2-GeV region. Moreover, we find a large L-S splitting with the expected ordering for a vector interaction. We study in detail the BS wave function for the pion in motion, necessary to compute axial-vector-current matrix elements, and recover well known relations of current algebra. We compute ${f}_{\ensuremath{\pi}}$ and find on general grounds that ${f}_{\ensuremath{\pi}\mathcal{'}}$=0 in the chiral limit, where \ensuremath{\pi}' is any radially excited pion. The pion satisfies the expected dispersion law for a Goldstone boson, \ensuremath{\omega}(p)\ensuremath{\rightarrow}cp (p\ensuremath{\rightarrow}0). .AE