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

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 $g=2\pi \mathrm{/}sqrt{N}_c$, the three-meson coupling is $g’ = m_\sigma^2/2f_\pi = 2gm_q$andthefour-mesoncouplingis$\lambda = 2g^2 = g’/f_\pi$,where$f_\pi \simeq 90$MeVisthepiondecayconstantand$N_c = 3$ 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.