5 which transforms as Note that, in the HLS, the entire symmetry G$_{global}$ × H$_{local}$ is spontaneously broken to its subgroup H$_{global}$ = [ Sp (2 N$_{f}$ )]$_{global}$, so that the NG bosons σ = σ $^{α}$Sα are absorbed into the HLS gauge bosons. The basic quantities to construct the HLS Lagrangian are the following two Maurer-Cartan 1-forms: where the covariant derivatives are read from the transformation properties in Eqs. (2.10) and (2.13) as with V$_{ν}$ = V α ν S α and G$_{ν}$ = G A ν T A being the H$_{local}$ gauge bosons and the external gauge bosons corresponding to the chiral symmetry (see Appendix A), respectively. These gauge bosons transform as It should be noticed that in the HLS formalism we can introduce the “vector” bosons V$_{ν}$ and the external chiral gauge bosons G$_{ν}$ independently. Since the covariantized 1-forms ˆ α$_{⊥}$$_{ν}$ and ˆ α$_{‖}$$_{ν}$ in Eqs. (2.14) and (2.15) transform homogeneously: we have the following two invariants: We introduce the kinetic term of the “vector” bosons: ξ$_{R}$ → hξ$_{R}$ ( ¯ Σ g T ¯ Σ $^{†}$) . (2.13) ˆ α$_{⊥}$$_{ν}$ = [( D$_{ν}$ξ$_{R}$ ) ξ † R − ( D$_{ν}$ξ$_{L}$ ) ξ † $_{L}$] / (2 i ) , (2.14) ˆ α$_{‖}$$_{ν}$ = [( D$_{ν}$ξ$_{R}$ ) ξ † R + ( D$_{ν}$ξ$_{L}$ ) ξ † $_{L}$] / (2 i ) , (2.15) D$_{ν}$ξ$_{L}$ = ∂$_{ν}$ξ$_{L}$ − iV$_{ν}$ξ$_{L}$ + iξ$_{L}$G$_{ν}$, (2.16) D$_{ν}$ξ$_{R}$ = ∂$_{ν}$ξ$_{R}$ − iV$_{ν}$ξ$_{R}$ + iξ$_{R}$ ( ¯ Σ G T ν ¯ Σ) , (2.17) V$_{ν}$ → V ′ ν = hV$_{ν}$h † + ih ( ∂$_{ν}$h $^{†}$) , (2.18) G$_{ν}$ → G ′ ν = gG$_{ν}$g † + ig ( ∂$_{ν}$g $^{†}$) . (2.19) ˆ α ν ⊥ , ‖ → h ˆ α ν ⊥ , $_{‖}$( x ) h $^{†}$, (2.20) f 2 $_{π}$tr[ˆ α$_{⊥}$$_{ν}$ ˆ α ν $_{⊥}$] , f 2 $_{σ}$tr[ˆ α$_{‖}$$_{ν}$ ˆ α ν $_{‖}$] . (2.21) − 1 2 g $_{2}$tr[ V$_{νρ}$V $^{νρ}$] , (2.22)