A solution to theρ−πpuzzle: Spontaneously broken symmetries of the quark model

This article proposes a solution to the long-standing $\rho -\pi$ puzzle: How can the $\rho$ and $\pi$ be members of a quark model U(6) 36 and the $\pi$ be a Nambu-Goldstone boson satisfying partial conservation of the axial-vector current (PCAC)? Our solution to the puzzle requires a revision of conventional concepts regarding the vector mesons $\rho$, $\omega$, $K^{*}$, and $\phi$. Just as the $\pi$ is a Goldstone state, a collective excitation of the Nambu-Jona-Lasinio type, transforming as a member of the (3,¯3) + (¯3,3) representation of the chiral SU(3) × SU(3) group, so also the $\rho$ transforms like (3,¯3) + (¯3,3) and is also a collective state, a "dormant" Goldstone boson that is a true Goldstone boson in the static chiral U(6) × U(6) limit. The static chiral U(6) × U(6) is to be spontaneously broken to static U(6) in the vacuum. Relativisitc effects provide for U(6) breaking and a massive $\rho$. This viewpoint has many consequences. Vector-meson dominance is a consequence of spontaneously broken chiral symmetry—the mechanism that couples the axial-vector current to the $\pi$ couples the vector current to the $\rho$. The transition rate is calculated as $\gamma _{\rho }^{}{}_{}{}^{-1\mathrm{}}=\frac{f_{\pi }}{m_{\rho }}$ in rough agreement with experiment. This picture requires soft $\rho \text{'}\mathrm{s}$ to decouple; but this requirement is not in conflict with any experimental features of the vector mesons. The chiral partner of the $\rho$ is not the $A_1$ but the $B\left(1235\right)\mathrm{}$. The experimental absence of the $A_1$ is no longer a theoretical embarrassment in this scheme. As the analog of PCAC for the pion we establish a tensor-field identity for the $\rho$ meson in which the $\rho$ is interpreted as a dormant Goldstone state. The decays $\delta \to \eta +\pi$ $B\to \omega +\pi$, $\epsilon \to 2\pi$ are estimated and are found to be in agreement with the observed rates. A static U(6) × U(6) generalization of the $\Sigma$ model is presented with the $\pi$, $\rho$, $\sigma$, $B$ in the (6,¯6) + (¯6,6) representation. The $\rho$ emerges as a dormant Goldstone boson in this model. Symmetry breaking in the model leads to the remarkable relation $m_{\rho }^{}{}_{}{}^2-m_{\pi }^{}{}_{}{}^2=m_B^{}{}_{}{}^2-m_{\delta }^{}{}_{}{}^2$, satisfied within 0.5%. Others' efforts towards an integration of PCAC with the quark model, particularly in the context of the Melosh transformation, are discussed.