$E\ll \Lambda_\chi$ where $\Lambda_\chi\approx
4\pi f_\pi\sim 1$ GeV, the only relevant degrees of freedom are the pions, that is, Goldstone bosons, emerging from the spontaneous breaking of chiral symmetry $SU(N_f)_L\times SU(N_f)_R$ to $SU(N_f)_{L+R}$. The dynamics involving the pion field, $U=exp(2i\pi/F_\pi)$ [^3] is encoded in a chiral Lagrangian expanded in derivatives, the leading term of which is given by low-energy theorems. Now we make the most obvious and “trivial" observation that one can always write the $U$ field in a product form if one is willing to introduce a redundant field. Define the L/R chiral fields with the redundant field $\sigma$ \_[L,R]{}=e\^[i/F\_]{}e\^[i/F\_]{}\[chifields\] $U$ can then be rewritten as U=\_L\^\_R. In this form, we unearth a local symmetry \_[L,R]{}h(x)\_[L,R]{} with $h(x)\in SU(N_f)_V$. As it stands, we have not done anything new, so there is no new physics here. However if we elevate the local symmetry to a local gauge symmetry by introducing a gauge field – which we will denote by $V_\mu \in SU(N_f)_V$ – and endow it with a kinetic energy term, then it becomes quite a different story. First of all, this procedure allows one to go up systematically in some expansion scheme (such as chiral expansion) in energy from 