A Solution to the rho-pi Puzzle: Spontaneously Broken Symmetries of the Quark Model

This article proposes a solution to the long-standing $\ensuremath{\rho}\ensuremath{-}\ensuremath{\pi}$ puzzle: How can the $\ensuremath{\rho}$ and $\ensuremath{\pi}$ be members of a quark model U(6) 36 and the $\ensuremath{\pi}$ 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 $\ensuremath{\rho}$, $\ensuremath{\omega}$, ${K}^{*}$, and $\ensuremath{\varphi}$. Just as the $\ensuremath{\pi}$ is a Goldstone state, a collective excitation of the Nambu-Jona-Lasinio type, transforming as a member of the (3,\ifmmode\bar\else\textasciimacron\fi{}3) + (\ifmmode\bar\else\textasciimacron\fi{}3,3) representation of the chiral SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) group, so also the $\ensuremath{\rho}$ transforms like (3,\ifmmode\bar\else\textasciimacron\fi{}3) + (\ifmmode\bar\else\textasciimacron\fi{}3,3) and is also a collective state, a "dormant" Goldstone boson that is a true Goldstone boson in the static chiral U(6) \ifmmode\times\else\texttimes\fi{} U(6) limit. The static chiral U(6) \ifmmode\times\else\texttimes\fi{} U(6) is to be spontaneously broken to static U(6) in the vacuum. Relativisitc effects provide for U(6) breaking and a massive $\ensuremath{\rho}$. 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 $\ensuremath{\pi}$ couples the vector current to the $\ensuremath{\rho}$. The transition rate is calculated as $\ensuremath{\gamma}_{\ensuremath{\rho}}^{}{}_{}{}^{\ensuremath{-}1}=\frac{{f}_{\ensuremath{\pi}}}{{m}_{\ensuremath{\rho}}}$ in rough agreement with experiment. This picture requires soft $\ensuremath{\rho}'\mathrm{s}$ to decouple; but this requirement is not in conflict with any experimental features of the vector mesons. The chiral partner of the $\ensuremath{\rho}$ is not the ${A}_{1}$ but the $B(1235)$. 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 $\ensuremath{\rho}$ meson in which the $\ensuremath{\rho}$ is interpreted as a dormant Goldstone state. The decays $\ensuremath{\delta}\ensuremath{\rightarrow}\ensuremath{\eta}+\ensuremath{\pi}$ $B\ensuremath{\rightarrow}\ensuremath{\omega}+\ensuremath{\pi}$, $\ensuremath{\epsilon}\ensuremath{\rightarrow}2\ensuremath{\pi}$ are estimated and are found to be in agreement with the observed rates. A static U(6) \ifmmode\times\else\texttimes\fi{} U(6) generalization of the $\ensuremath{\Sigma}$ model is presented with the $\ensuremath{\pi}$, $\ensuremath{\rho}$, $\ensuremath{\sigma}$, $B$ in the (6,\ifmmode\bar\else\textasciimacron\fi{}6) + (\ifmmode\bar\else\textasciimacron\fi{}6,6) representation. The $\ensuremath{\rho}$ emerges as a dormant Goldstone boson in this model. Symmetry breaking in the model leads to the remarkable relation $m_{\ensuremath{\rho}}^{}{}_{}{}^{2}\ensuremath{-}m_{\ensuremath{\pi}}^{}{}_{}{}^{2}=m_{B}^{}{}_{}{}^{2}\ensuremath{-}m_{\ensuremath{\delta}}^{}{}_{}{}^{2}$, 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.