pi+ pi0 Mass Difference in the Hidden Local Symmetry: A Dynamical Origin of Little Higgs

We calculate pi+ pi0 mass difference $\Delta m_\pi^2 \equiv m_{\pi^+}^2 -m_{\pi^0}^2$ in the Hidden Local Symmetry (HLS) model, based on the Wilsonian matching and Wilsonian renormalization-group equations. Even without a1 meson the result agrees well with the experiment in sharp contrast to the conventional approach where the a1 meson plays a crucial role. For large Nf QCD, there arises a large hierarchy between $\Delta m_\pi^2$ and the pi decay constant $F_\pi^2$, $\Delta m_\pi^2/F_\pi^2 \ll 1$, near the critical point where the chiral symmetry gets restored as the vector manifestation and the HLS model becomes a little Higgs model with two sites and two links, with the dynamically generated gauge coupling of the composite rho becoming vanishingly small.