Harmonic-oscillator pattern arising from an algebraic approach to chiral symmetry

Abstract

In a previous work we proposed a saturation scheme for theSU3⊗SU3 chiral algebra within an infinite set ofSU6⊗O3 states. A mixing operator was introduced which transforms the axial charges of theSU6 solution into the physical ones,Qi5. Here we saturate the Weinberg equation for the (mass)2 operator [Q5+, [Q5+,m2]]=0, between meson states, in a perturbative approach. The generatorZ of the mixing operators, for which formerly only the transformation properties underSU6×O3 were given, is now completely established asZ=(W×M)z, whereW is theW-spin operator andM is the co-ordinate of the three-dimensional harmonic oscillator. In a perturbative expansion of the (mass)2 operator, the lowest term consists of two pieces, the harmonic-oscillator energy and a spin-orbit coupling of the form (−1)L+1(L·S+1/2). The resulting (mass)2 consists of families of equispaced linearly rising trajectories.