Constituent quark-based linear $\sigma$ model (L$\sigma$M) quark and scalar mesons, vector meson dominance

After describing the SU(2) linear sigma model (L$\sigma$M), we dynamically generate it using the B.W. Lee null tadpole sum (characterizing the true vacuum) together with the dimensional regularization lemma. Next we generate the chiral-limiting (CL) nonstrange and strange constituent quark masses ${\hat m}=325.7$MeV;$m_s=486$ MeV away from the CL. Finally, we study vector meson dominance (VMD) and the pion, kaon charge radii and the loop-order $\rho \to \pi \gamma$, ${\pi }^0\to \gamma \gamma$ amplitudes in the quark model. Lastly, we verify this procedure using tree-order VMD graphs.