Saturation of Chiral SU(2) ⊗ SU(2) Algebras,A1ρπandρ′Couplings

The interrelation between the two different types of sum rules, the chiral SU(2) ⊗ SU(2) spectral-function sum rules and the ones based on the saturation of the chiral SU(2) ⊗ SU(2) charge algebras (which also includes the time derivative of axial-vector charge) especially studied by Gilman and Harari (G-H), is investigated through the charge-charge density algebras and under the same single-particle approximation. It is shown that, if the $\rho$ meson is the only $I=1\mathrm{}$ vector meson, (i) the G-H saturation is justified for the case of helicity $\lambda =0\mathrm{}$, (ii) the whole set of sum rules is entirely consistent with each other, including the second-spectral-function sum rules, and (iii) ${m_A}^2\simeq 2{m_{\rho }}^2$ thus follows to the extent that the KSRF (Kawarabayashi-Suzuki-Riaduddin-Fayyazuddin) relation is experimentally satisfied. The modification due to the addition of the newly discovered ${\rho }^{\prime }$ meson to the scheme is studied. It is found that the ${\rho }^{\prime }$ cannot play a dominant role and its relevant couplings, ${\rho }^{\prime }\pi \pi$ and ${\rho }^{\prime }-\gamma$, are restricted to small values. Typically, we obtain $\Gamma ({\rho }^{\prime }\to \pi \pi )\simeq 50-80$ MeV and ${\left(\frac{G_{\rho }}{G_{\rho }}\right)}^2\simeq 0.2$, if the ${\rho }^{\prime }$ is treated as the triplet $D$ states of the quark model and no more vector mesons are introduced. Some remarks are also made about the $A_1\rho \pi$ coupling.