Nonlinear Realizations of Chiral Symmetry

1968/02/25 by Steven Weinberg · 1 voice · 27 citations
Physics and Astronomy · #Algebra over a field #Black Holes and Theoretical Physics #Chiral anomaly #Chiral symmetry breaking #Coupling constant #Covariant transformation #Current algebra #Fermion #Mathematical physics #Meson #Particle physics #Particle physics theoretical and experimental studies #Physics #Pion #Pure mathematics #Quantum Chromodynamics and Particle Interactions #Quantum mechanics #Scattering #Symmetry breaking

paper · doi:10.1103/physrev.166.1568

openalex publication_date 1968/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

We explore possible realizations of chiral symmetry, based on isotopic multiplets of fields whose transformation rules involve only isotopic-spin matrices and the pion field. The transformation rules are unique, up to possible redefinitions of the pion field. Chiral-invariant Lagrangians can be constructed by forming isotopic-spin-conserving functions of a covariant pion derivative, plus other fields and their covariant derivatives. The resulting models are essentially equivalent to those that have been derived by treating chirality as an ordinary linear symmetry broken by the vacuum, except that we do not have to commit ourselves as to the grouping of hadrons into chiral multiplets; as a result, the unrenormalized value of \fracg_Ag_V need not be unity. We classify the possible choices of the chiral-symmetry-breaking term in the Lagrangian according to their chiral transformation properties, and give the values of the pion-pion scattering lengths for each choice. If the symmetry-breaking term has the simplest possible transformation properties, then the scattering lengths are those previously derived from current algebra. An alternative method of constructing chiral-invariant Lagrangians, using \ensuremathρ mesons to form covariant derivatives, is also presented. In this formalism, \ensuremathρ dominance is automatic, and the current-algebra result from the \ensuremathρ-meson coupling constant arises from the independent assumption that \ensuremathρ mesons couple universally to pions and other particles. Including \ensuremathρ mesons in the Lagrangian has no effect on the \ensuremathπ\ensuremath-\ensuremathπ scattering lengths, because chiral invariance requires that we also include direct pion self-couplings which cancel the \ensuremathρ-exchange diagrams for pion energies near threshold.

Citations

• Theory of strong interactions
• Broken Symmetries and the Masses of Gauge Bosons
• Global Conservation Laws and Massless Particles
• Broken Symmetry and the Mass of Gauge Vector Mesons
• On the symmetries of strong and weak interactions
• The axial vector current in beta decay
• Conservation of Isotopic Spin and Isotopic Gauge Invariance

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• Nucleon-nucleon scattering with perturbative pions: The uncoupled P-wave channels
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• On the general structure of gauged Wess-Zumino-Witten terms
• Coupled-channel study of crypto-exotic baryons with charm
• Dynamically Generated Hidden-Charm Baryon Resonances
• Effective chiral theory of nucleon–nucleon scattering
• Baryon chiral perturbation theory A.D. 1994
• Fermions on the worldsheet of effective strings via coset construction
• (1+1)-dimensional baryons from the SU(N) color-flavor transformation
• Electromagnetic couplings of the chiral perturbation theory Lagrangian from the perturbative chiral quark model
• Short-range correlations in isospin symmetric and asymmetric nuclear matter: A microscopic perspective
• Dyson–Schwinger Equations: A Tool for Hadron Physics
• Cutoff regulators in chiral nuclear effective field theory
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• Leading logarithms in the anomalous sector of two-flavour QCD
• Dual interpretations of pion clouds in relativistic heavy ion collisions
• Theory of Incommensurate Magnetic Correlations Across the Insulator-Superconductor Transition of UnderdopedLa2−xSrxCuO4
• String dynamics, spontaneous breaking of supersymmetry, and dual scalar field theory
• Universality of theρMeson Coupling in Effective Field Theory
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Discussions

• You misread your text: would lead back to our linear sigma model definitely does not mean it would lead to (28.12) with the scalar fields $\tilde \sigma$ and $\tilde \pi$ (representing the real and im [stackexchange, -1 points]