Realization Of Chiral Symmetry Research Articles

Algebraic realizations of chiral symmetry and the Veneziano model

First we prove that the chiral multiplets are infinite in the Veneziano model. Next we present the cancellation scheme of Brout, Englert and Truffin in the language of algebraic realizations of chiral symmetry. The criticisms of Riazuddin, Fayyazuddin and Ahmad of realizations of chiral symmetry will be rejected.

Multiple Production of Massless Pions as a Consistency Test in the Algebraic Realization of Chiral Symmetry

Multiple Production of Massless Pions as a Consistency Test in the Algebraic Realization of Chiral Symmetry

Superconvergence and Algebraic Realizations of Chiral Symmetry

The consequences of assuming that there is no isospin $T=2$ part in the commutator [${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{Q}}_{5}^{a},{\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{Q}}_{5}^{b}$] of time derivatives of the axial charges ${Q}_{5}^{a}$ and ${Q}_{5}^{b}$ have been investigated. The superconvergence condition derived from this assumption provides constraints on Weinberg's mass matrix. The mixing angles introduced by Weinberg for the $\ensuremath{\pi}\ensuremath{-}{A}_{1}\ensuremath{-}\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}$ system of mesons and for the nucleon resonances belonging to the (1, \textonehalf{}) and (\textonehalf{}, 0) representations of the chiral group are now determined. When only $\ensuremath{\rho}$-wave pion interactions are considered, it is shown that the new constraints on the mass matrix lead to mass degeneracy for the symmetric representations of $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(4)$. The constraints on the spin spectrum derivable from the superconvergence conditions are briefly mentioned.

Pion Production and the Algebraic Realization of Chiral Symmetry

Pion Production and the Algebraic Realization of Chiral Symmetry

Realization of chiral symmetry in the meson states through SU (4) ⊗ O(3)

We propose a saturation scheme for the equations [X a(h), X b(h)] = i ϵ abc T c , [X a(h), [X b(h), m 2]] = 1 3 δ ab[X c(h), [X c(h), m 2]] , within the lower meson states, which belong to the 15 L = 0.1 representations of SU (4) ⊗ O(3). With the introduction of two mixing parameters we are able to derive the chiral couplings of these particles in reasonable agreement with experiment. In particular the helicity properties of A 1 and B decays come out correctly.

SU(3)Symmetry and Algebraic Realizations of Chiral Symmetry

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Spin Spectrum Derivable from Algebraic Realizations of Chiral Symmetry

Superconvergence relations are known to obey algebraic equations involving the spin and the mass. States of the same helicity fall into reducible representations of $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$. When the helicity is \ifmmode\pm\else\textpm\fi{}\textonehalf{}, the two representations are related both by parity operation and by spin raising or lowering operators in the helicity space. The combination of the two yields sufficient constraints to solve for a number of properties of low-lying resonances, with results in good agreement with experiment.

On the non-local realization of chiral symmetry

It is shown that there exists a non-local, non-hermitian, effective Lagrangian giving in the tree approximation a description of scattering phenomena correct at all energies. Because such a Lagrangian cannot be invariant under the ordinary local, non-linear realization of chiral symmetry a non-local realization of chiral SU 2 × SU 2 is introduced and its consequences are studied.

Algebraic Realization of Chiral Symmetry and Veneziano Model

It is shown that the algebraic realization of chiral symmetry in which one assigns the mesons to a representation of the chiral algebra is not consistent with the Veneziano model.Received 6 June 1969DOI:https://doi.org/10.1103/PhysRevLett.23.103©1969 American Physical Society

Realization of Chiral Symmetry in a Curved Isospin Space

The nonlinear realizations of the chiral group SU(2)⊗SU(2) are studied from a geometric point of view. The three-dimensional nonlinear realization, associated with the pion field, is considered as a group of coordinate transformations in a three-dimensional isospin space of constant curvature, leaving invariant the line element. Spinor realizations in general coordinates are constructed by combined coordinate-spin-space transformations in analogy to Pauli's method for spinors in general relativity. The description of vector mesons and possible chiral-invariant Lagrangians, which yield the various nonlinear models in specific frames of general coordinates, are discussed.

Algebraic Realizations of Chiral Symmetry

The sum of tree graphs for forward pion scattering, generated by any chiral-invariant Lagrangian, is required to grow no faster at high energies than the actual scattering amplitude. In consequence, algebraic restrictions must be imposed on the axial-vector coupling matrix X and the mass matrix ${m}^{2}$: For each helicity, X must, together with the isospin T, form a representation of $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$, and ${m}^{2}$ must behave with respect to commutation with T and X as the sum of a chiral scalar and the fourth component of a chiral four-vector. If it is further assumed that the contribution of tree graphs to inelastic forward pion scattering vanishes at high energy, the two parts of the mass matrix must commute; this fixes various mixing angles, and leads to predictions like ${m}_{\ensuremath{\sigma}}={m}_{\ensuremath{\rho}}$, ${m}_{{A}_{1}}=\sqrt{2}{m}_{\ensuremath{\rho}}$, ${\ensuremath{\Gamma}}_{\ensuremath{\rho}}=135$ MeV, etc. If all pion transitions involved only $p$-wave pions, then X would form part of the algebra of $\mathrm{SU}(4)$, and the mass matrix would behave as the sum of a 1- and a 20-dimensional representation of $\mathrm{SU}(4)$; if $s$-wave transitions are allowed, then the algebra must be enlarged to at least $\mathrm{SO}(7)$.

Note on Nonlinear Realizations of Chiral Symmetry

Note on Nonlinear Realizations of Chiral Symmetry

Nonlinear Realizations of Chiral Symmetry

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 $\frac{{g}_{A}}{{g}_{V}}$ 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 $\ensuremath{\rho}$ mesons to form covariant derivatives, is also presented. In this formalism, $\ensuremath{\rho}$ dominance is automatic, and the current-algebra result from the $\ensuremath{\rho}$-meson coupling constant arises from the independent assumption that $\ensuremath{\rho}$ mesons couple universally to pions and other particles. Including $\ensuremath{\rho}$ mesons in the Lagrangian has no effect on the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths, because chiral invariance requires that we also include direct pion self-couplings which cancel the $\ensuremath{\rho}$-exchange diagrams for pion energies near threshold.