Proving Rho Meson be a Dynamical Gauge Boson of Hidden Local Symmetry

The rho meson has long been successfully identified with a dynamical gauge boson of the Hidden Local Symmetry (HLS) $H_{\rm local}$ in the nonlinear sigma model $G/H$ gauge equivalent to the model having the symmetry $G_{\rm global}\times H_{\rm local}$, with $G= [SU(2)_L \times SU(2)_R]\simeq O(4), H=SU(2)_{V}\simeq O(3)$, however under a hitherto unproven assumption that its kinetic term is dynamically generated, together with an ad hoc choice of the auxiliary field parameter "$a=2$". We prove this assumption, thereby solving the long-standing mystery: The rho meson kinetic term is generated simply by the large $N$ limit of the Grassmannian model $G/H=O(N)/[O(N-3)\times O(3)] $ gauge equivalent to $O(N)_{\rm global}\times [O(N-3)\times O(3)]_{\rm local}$, extrapolated to $N=4$, $O(4)_{\rm global}\times O(3)_{\rm local}$, with all the phenomenologically successful "$a=2$ results" i.e., $\rho$-universality, KSRF relation and the Vector Meson Dominance, realized {\it independently of the parameter "$a$".} This in turn establishes validity of the large $N$ dynamics at quantitative level directly by the experiments. The relevant cutoff reads $\Lambda \simeq 4 \pi F_\pi$ for $N=4$, which is regarded as a matching scale of the HLS as a "magnetic dual" to QCD. Skyrmion is stabilized by such a dynamically generated rho meson without recourse to the underlying QCD, further signal of the duality. The unbroken phase with massless rho meson may be realized as a novel chiral restored hadronic phase in hot/dense QCD.