Dynamical Gauge Boson of Hidden Local Symmetry within the Standard Model

Abstract

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 $O(4{)}_{\mathrm{global}}\times O(3{)}_{\mathrm{local}}$, with $O(3{)}_{\mathrm{local}}$ being the Hidden Local Symmetry (HLS). In the large $N$ limit of the scale-invariant version of the Grassmannian model $G/H=O(N)/[O(N-3)\times O(3)]$ $\simeq O(N{)}_{\mathrm{global}}\times [O(N-3)\times O(3){]}_{\mathrm{local}}$, identical to the SM for $N\to 4$, we show that the kinetic term of the HLS gauge bosons ("SM rho") ${\rho }_{\mu }$ of the $O(3{)}_{\mathrm{local}}\simeq [SU(2{)}_V{]}_{\mathrm{local}}$ 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 $M_{X_s}=\mathcal{O}(10\mathrm{GeV})$ consistent with the direct search experiments implies the induced HLS gauge coupling $g_{{}_{\mathrm{HLS}}}=\mathcal{O}(10^3)$, which realizes the relic abundance, ${\Omega }_{X_s}{h}^2=\mathcal{O}(0.1)$. If instead $g_{{}_{\mathrm{HLS}}}\lesssim 3.5$ ($M_{\rho }\lesssim 1.2$ TeV), the SM rho could be detected with "narrow width" $\lesssim 100\mathrm{GeV}$ at LHC, having all the "$a=2$ results" of the generic HLS Lagrangian ${\mathcal{L}}_A+a{\mathcal{L}}_V$, 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.

Comments: 52 pages, for readability preliminary formal discussions moved to appendix, some phenomenological comments added, typos corrected