Sum Rules Involving theρandA1Mesons and theA1→ρ+πDecay

We derive sum rules involving the $\pi$, $\rho$, and $A_1$ mesons by using a chiral $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(2\right)\mathrm{}$ charge-charge density algebra and the hypothesis of partially conserved axial-vector current. We invoke the mass-shell extrapolation ${m_{\pi }}^2\to 0$ but not ${m_{\rho }}^2\to 0$. We derive both the $\rho$ dominance for the pion electromagnetic form factor, $G_{{\rho }^0{\pi }^{+}{\pi }^{-}}{G}_{\rho }=\sqrt{2}{m_{\rho }}^2$, and the first chiral spectral-function sum rule, $\frac{{G_{\rho }}^2}{{m_{\rho }}^2}-\frac{{G_A}^2}{{m_A}^2}={F_{\pi }}^2$. However, the second one, $G_A=G_{\rho }$, does not follow without further assumptions. Nevertheless, we note that from the above two sum rules, if $\Gamma \left(\rho \right)\simeq 130$ MeV, the relation $m_A\simeq \sqrt{2}{m}_{\rho }$ leads to $G_A\simeq G_{\rho }$ or vice versa. The decay $A_1\to \rho +\pi$ is discussed. Contrary to the soft-pion calculation, our approach is consistent with having a smaller $\Gamma (A_1\to \rho +\pi )$, around 100 MeV. Comparison with the results of other approaches is made.