Lie-algebra approach to symmetry breaking

Abstract

A formal Lie-algebra approach to symmetry breaking is studied in an attempt to reduce the arbitrariness of Lagrangian (Hamiltonian) models which include several free parameters and/or ad hoc symmetry groups. From Lie algebra it is shown that the unbroken Lagrangian vacuum symmetry can be identified from a linear function of integers which are Cartan matrix elements. In broken symmetry if the breaking operators form an algebra then the breaking symmetry (or symmetries) can be identified from linear functions of integers characteristic of the breaking symmetries. The results are applied to the Dirac Hamiltonian of a sum of flavored fermions and colored bosons in the absence of dynamical symmetry breaking. In the partially reduced quadratic Hamiltonian the breaking-operator functions are shown to consist of terms of order ${g}^{2}$, $g$, and ${g}^{0}$ in the color coupling constants and identified with strong (boson-boson), medium strong (boson-fermion), and fine-structure (fermion-fermion) interactions. The breaking operators include a boson helicity operator in addition to the familiar fermion helicity and "spin-orbit" terms. Within the broken vacuum defined by the conventional formalism, the field divergence yields a gauge which is a linear function of Cartan matrix integers and which specifies the vacuum symmetry. We find that the vacuum symmetry is chiral SU(3)\ifmmode\times\else\texttimes\fi{}SU(3) and the axial-vector-current divergence gives a PCAC (partially conserved axialvector current)-like function of the Cartan matrix integers which reduces to PCAC for SU(2)\ifmmode\times\else\texttimes\fi{}SU(2) breaking. For the mass spectra of the nonets ${J}^{P}={0}^{\ensuremath{-}},\frac{1}{{2}^{+}},{1}^{\ensuremath{-}}$ the integer runs through the sequence 3,0, - 1, - 2, which indicates that the breaking subgroups are the simple Lie groups. Exact axial-vector-current conservation indicates a breaking sum rule which generates octet enhancement. Finally, the second-order breaking terms are obtained from the second-order spin tensor sum of the completely reduced quartic Hamiltonian. The breaking terms include the "anomalous" $^{*}F_{\ensuremath{\mu}\ensuremath{\nu}}{F}_{\ensuremath{\mu}\ensuremath{\nu}}$ term found by Schwinger, as well as fermion and boson helicity-breaking terms. Nonvanishing of the axial-vector-current divergence indicates the presence of solitons or, for electromagnetic coupling, of magnetic monopoles as the sources of strong fields.