Dynamical Gauge Boson of Hidden Local Symmetry within the Standard Model

The Standard Model (SM) Higgs Lagrangian is straightforwardly rewritten into the {\it scale-invariant} nonlinear sigma model $G/H=[SU(2)_L \times SU(2)_R]/SU(2)_{V}\simeq O(4)/O(3)$, with the (approximate) scale symmetry realized nonlinearly by the (pseudo) dilaton ($=$ SM Higgs). It is further gauge equivalent to that having the symmetry $O(4)_{\rm global}\times O(3)_{\rm local}$, with $O(3)_{\rm local}$ being the Hidden Local Symmetry (HLS). In the large $N$ limit of the scale-invariant version of the Grassmannian model $G/H=O(N)/[O(N-3)\times O(3)] $ $\simeq O(N)_{\rm global}\times [O(N-3)\times O(3)]_{\rm local}$, identical to the SM for $N\rightarrow 4$, we show that the kinetic term of the HLS gauge bosons ("SM rho") $\rho_\mu$ of the $O(3)_{\rm local}\simeq [SU(2)_V]_{\rm local}$ are dynamically generated by the nonperturbative dynamics of the SM itself. The dynamical SM rho stabilizes the skyrmion ("SM skyrmion") $X_s$ as a dark matter candidate within the SM: The mass $M_{X_s} ={\cal O}(10\, {\rm GeV})$ consistent with the direct search experiments implies the induced HLS gauge coupling $g_{_{\rm HLS}}={\cal O}(10^3)$, which realizes the relic abundance, $\Omega_{X_s} h^2 ={\cal O}(0.1)$. If instead $g_{_{\rm HLS}}\lesssim 3.5$ ($M_\rho \lesssim 1.2 $ TeV), the SM rho could be detected with "narrow width" $\lesssim 100 \,{\rm GeV}$ at LHC, having all the "$a=2$ results" of the generic HLS Lagrangian ${\cal L}_A+ a {\cal L}_V$, i.e., $\rho$-universality, KSRF relations and the vector meson dominance, independently of "$a$". There exists the second order phase phase transition to the unbroken phase having massless $\rho_\mu$ and massive $\pi$ (no longer NG bosons), both becoming massless free particles just on the transition point (scale-invariant ultraviolet fixed point).The results readily apply to the 2-flavored QCD as well.