20 as q$_{L}$ under the SU (2)$_{color}$ symmetry as well as under the Lorentz symmetry. Then, by introducing the field Ψ as the kinetic term in Eq.(A.8) is rewritten as This is invariant under the SU (2 N$_{f}$ ) transformation of Ψ given as Similarly, using the field Ψ, we rewrite L ext-scalar and L ext-vector in Eqs. (A.4) and (A.5) as where χ and G µ are external fields of 2 N$_{f}$ × 2 N$_{f}$ matrices defined as Transformation properties of the external fields under SU (2 N$_{f}$ ) are given by In this appendix, we show the explicit representation of the generators of SU (4). We con- sider the form of the generators following Ref. [8] for convenience. They can be represented Appendix B: Explicit realization of the SU (4) generators Ψ =       q 1 L q 2 L . q N$_{f}$ L σ$_{2}$τ$_{2}$q ∗ R, 1 σ$_{2}$τ$_{2}$q ∗ R, 2 . σ$_{2}$τ$_{2}$q ∗ R,N$_{f}$      $^{,}$ (A.10) ∫ d $^{4}$x i ¯ ψγ$_{ν}$D $^{ν}$ψ = ∫ d $^{4}$x i Ψ $^{†}$σ$_{ν}$ D $^{ν}$Ψ . (A.11) Ψ → g Ψ , ( g ∈ SU (2 N$_{f}$ )) . (A.12) L ext-scalar = 1 2 Ψ $^{T}$σ$_{2}$ τ$_{2}$χ Ψ + (h.c) , L ext-vector = Ψ $^{†}$σ$_{µ}$ G $^{µ}$Ψ , (A.13) χ ≡   Q − i R − ( S − i P ) T S − i R ( Q + i R ) †   , (A.14) G µ ≡   V µ + A µ ( B µ + D $^{µ}$) † B µ + D µ − ( V µ − A $^{µ}$)T   . (A.15) G$_{µ}$ → gG$_{µ}$g † + ig ( ∂$_{µ}$g $^{†}$) , χ → g $^{∗}$χg $^{†}$. (A.16)