4 where with the decay constant f$_{π}$ . #1 Transformation property of ξ ( π ) under the chiral symmetry is given by From this together with the relations in Eq. (2.1), we see that Σ transforms linearly under the chiral symmetry as The effective Lagrangian including the NG bosons should be invariant under the global SU (2 N$_{f}$ ) group and under the Lorentz transformation, which is given by [3] Next, we include the “vector” boson fields #2 into the Lagrangian based on the hidden local symmetry (HLS) [9–11]. We decompose the field Σ as where ξ$_{L}$ is given by with σ being the NG bosons associated with the spontaneous breaking of the HLS and f$_{σ}$ the corresponding decay constant. The transformation property of ξ$_{L}$ is given by where For constructing the HLS Lagrangian, it is convenient to introduce the field ξ$_{R}$ by ξ ( π ) = e $^{iπ/f$_{π}$}$, π = π $^{a}$X$^{a}$, (2.4) ξ ( π ) → gξ ( π ) h $^{†}$( π, g ) , ( h ∈ Sp (2 N$_{f}$ ) , g ∈ SU (2 N$_{f}$ )) . (2.5) Σ → g Σ g $^{T}$. (2.6) L = f 2 π 4 tr[( ∂$_{ν}$ Σ)( ∂ $^{ν}$Σ $^{†}$)] . (2.7) Σ = ξ † $_{L}$¯ Σ ξ T $_{L}$, (2.8) ξ$_{L}$ = ξ ( σ ) ξ $^{†}$( π ) , ( ξ ( σ ) = e $^{iσ/f$_{σ}$}$, σ = σ $^{α}$S$^{α}$) , (2.9) ξ$_{L}$ → hξ$_{L}$g $^{†}$, (2.10) h ∈ [ Sp (2 N$_{f}$ )]$_{local}$ , g ∈ [ SU (2 N$_{f}$ )]$_{global}$ . (2.11) ξ$_{R}$ = ¯ Σ( ξ † $_{L}$( x )) T ¯ Σ † = ξ ( σ ) ξ ( π ) , (2.12) #1 Note that the NG bosons consist of the mesons with J P = 0 − and the (anti-) baryons with J P = 0 $^{+}$. #2 Here the “vector” boson fields imply the meson fields with J P = 1 − and the (anti-) baryon fields with J P = 1 $^{+}$.