Algebraic Realizations of Chiral Symmetry

Abstract

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 $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(2\right)\mathrm{}$, 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 $m_{\sigma }=m_{\rho }$, $m_{A_1}=\sqrt{2}{m}_{\rho }$, ${\Gamma }_{\rho }=135\mathrm{}$ MeV, etc. If all pion transitions involved only $p$-wave pions, then X would form part of the algebra of $\mathrm{SU}\left(4\right)\mathrm{}$, and the mass matrix would behave as the sum of a 1- and a 20-dimensional representation of $\mathrm{SU}\left(4\right)\mathrm{}$; if $s$-wave transitions are allowed, then the algebra must be enlarged to at least $\mathrm{SO}\left(7\right)\mathrm{}$.