The quark-level linear σ model

This review of the quark-level linear σ model (QLLσM) is based upon the dynamical realization of the pseudoscalar and scalar mesons as a linear representation of SU(2) x SU(2) chiral symmetry, with the symmetry weakly broken by current quark masses. In its simplest SU(2) incarnation, with two non-strange quark flavors and three colors, this nonperturbative theory, which can be selfconsistently bootstrapped in loop order, is shown to accurately reproduce a host of low-energy observables with only one parameter, namely the pion decay constant 𝑓_π. Extending the scheme to SU(3) by including the strange quark, equally good results are obtained for many strong, electromagnetic, and weak processes just with two extra constants, viz. 𝑓_Κ and ⟨π|H_weak|Κ⟩. Links are made with the vector-meson-dominance model, the BCS theory of superconductivity, and chiral-symmetry restoration at high temperature. Finally, these ideas are cautiously generalized to the electroweak sector, including the W, Z, and Higgs bosons, and also to CP violation.