Pionic decays and saturation of current-algebra sum rules in a nonrelativistic expansion of the quark shell model

Abstract

Pionic decays of hadrons are calculated using a PCAC (partial conservation of axial-vector current) prescription and a quark shell model with quarks bound by a central potential, described by the Dirac equation. The Dirac Hamiltonian and operators are expanded in $\frac{v}{c}$, the internal quark velocity. Then, one finds an exact saturation of the current-algebra sum rules as defined in the SU(2) \ensuremath{\bigotimes} SU(2) symmetry of Gilman-Harari and Weinberg up to order $\frac{{v}^{4}}{{c}^{4}}$. The saturation is obtained without need of exotics, with the usual excitations of the ground state. The relation with the $P=\ensuremath{\infty}$ approach is clarified. The corrections found with respect to previous quark models in $L=2$ decays are discussed. They do not solve the problem of $\mathrm{SU}{(6)}_{W}$ coupling signs. Finally, the whole Weinberg scheme of linear SU(2) \ensuremath{\bigotimes} SU(2) symmetry is completed by the expression of the chiral-breaking part of the mass operator ${{m}_{4}}^{2}$.