Dynamical generation of the gauged SU(2) linear sigma model

Abstract

The fermion and meson sectors of the quark-level SU(2) linear sigma model are dynamically generated from a meson-quark Lagrangian, with the quark (q) and meson ($\sigma,\pi$) fields all treated as elementary, having neither bare masses nor expectation values. In the chiral limit, the masses are predicted to be $m_q = f_\pi g, m_\pi = 0, m_\sigma = 2m_q$, and we also find that the quark-meson coupling $g = 2\pi/sqrt{N_c}$, the three-meson coupling is $g' = m_\sigma^2/2f_\pi = 2gm_q$ and the four-meson coupling is $\lambda = 2g^2 = g'/f_\pi$, where $f_\pi \simeq 90$ MeV is the pion decay constant and $N_c = 3$ is the color number. By gauging this model one can generate the couplings to the vector mesons $\rho$ and $A_1$, including the quark-vector coupling constant $g_\rho = 2\pi, g_{\rho\pi\pi}, g_{A_1\rho\pi}$ and the masses $M_\rho\sim 700$ MeV, $m_{A_1}\simeq \sqrt{3}m_\rho$; of course the vector and axial vector currents remain conserved throughout.