Superconvergence and Algebraic Realizations of Chiral Symmetry

Abstract

The consequences of assuming that there is no isospin $T=2\mathrm{}$ part in the commutator [${\dot{Q}}_5^a,{\ddot{Q}}_5^b$] 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 $\pi -A_1-\rho -\sigma$ system of mesons and for the nucleon resonances belonging to the (1, ½) and (½, 0) representations of the chiral group are now determined. When only $\rho$-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 $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(4\right)\mathrm{}$. The constraints on the spin spectrum derivable from the superconvergence conditions are briefly mentioned.