22 where G$_{µν}$ is the filed strength of the external chiral gauge fields defined as From Eq. (C.1) together with other building blocks in TABLE I, a complete list of the O ( p $^{4}$) Lagrangian invariant under the C and P transformation is constructed as where G$_{µν}$ ≡ ∂$_{µ}$G$_{ν}$ − ∂$_{ν}$G$_{µ}$ − i [ G$_{µ}$, G$_{ν}$ ] . (C.2) L (4) = L (4) y + L (4) w + L (4) $_{z}$, (C.3) L (4) y = y$_{1}$ tr[ˆ α$_{⊥}$$_{µ}$ ˆ α µ $_{⊥}$ˆ α$_{⊥}$$_{ν}$ ˆ α ν $_{⊥}$] + y$_{2}$ tr[ˆ α$_{⊥}$$_{µ}$ ˆ α$_{⊥}$$_{ν}$ ˆ α µ $_{⊥}$ˆ α ν $_{⊥}$] + y$_{3}$ tr[ˆ α$_{‖}$$_{µ}$ ˆ α µ ‖ ˆ α$_{‖}$$_{ν}$ ˆ α ν $_{‖}$] + y$_{4}$ tr[ˆ α$_{‖}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ˆ α µ ‖ ˆ α ν $_{‖}$] + y$_{5}$ tr[ˆ α$_{⊥}$$_{µ}$ ˆ α µ $_{⊥}$ˆ α$_{‖}$$_{ν}$ ˆ α ν $_{‖}$] + y$_{6}$ tr[ˆ α$_{⊥}$$_{µ}$ ˆ α$_{⊥}$$_{ν}$ ˆ α µ ‖ ˆ α ν $_{‖}$] + y$_{7}$ tr[ˆ α$_{⊥}$$_{µ}$ ˆ α$_{⊥}$$_{ν}$ ˆ α ν $_{‖}$ˆ α µ $_{‖}$] + y$_{8}$ { tr[ˆ α$_{⊥}$$_{µ}$ ˆ α µ ‖ ˆ α$_{⊥}$$_{ν}$ ˆ α ν $_{‖}$] + tr[ˆ α$_{⊥}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ˆ α ν $_{⊥}$ˆ α µ $_{‖}$] } + y$_{9}$ tr[ˆ α$_{⊥}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ˆ α µ $_{⊥}$ˆ α ν $_{‖}$] + y$_{10}$ (tr[ˆ α$_{⊥}$$_{µ}$ ˆ α µ $_{⊥}$]) 2 + y$_{11}$ tr[ˆ α$_{⊥}$$_{µ}$ ˆ α$_{⊥}$$_{ν}$ ]tr[ˆ α µ $_{⊥}$ˆ α ν $_{⊥}$] + y$_{12}$ (tr[ˆ α$_{‖}$$_{µ}$ ˆ α µ $_{‖}$]) 2 + y$_{13}$ tr[ˆ α$_{‖}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ]tr[ˆ α µ ‖ ˆ α ν $_{‖}$] + y$_{14}$ tr[ˆ α$_{⊥}$$_{µ}$ ˆ α µ $_{⊥}$]tr[ˆ α$_{‖}$$_{ν}$ ˆ α ν $_{‖}$] + y$_{15}$ tr[ˆ α$_{⊥}$$_{µ}$ ˆ α$_{⊥}$$_{ν}$ ]tr[ˆ α µ ‖ ˆ α ν $_{‖}$] , (C.4) L (4) w = w$_{1}$ f 2 χ f 2 π tr[ˆ α$_{⊥}$$_{µ}$ ˆ α µ $_{⊥}$( ˆ χ + ˆ χ $^{†}$)] + w$_{2}$ f 2 χ f 2 π tr[ˆ α$_{⊥}$$_{µ}$ ˆ α µ $_{⊥}$]tr[ ˆ χ + ˆ χ $^{†}$] + w$_{3}$ f 2 χ f 2 π tr[ˆ α$_{‖}$$_{µ}$ ˆ α µ $_{‖}$( ˆ χ + ˆ χ $^{†}$)] + w$_{4}$ f 2 χ f 2 π tr[ˆ α$_{‖}$$_{µ}$ ˆ α µ $_{‖}$]tr[ ˆ χ + ˆ χ $^{†}$] + w$_{5}$ f 2 χ f 2 π tr[(ˆ α µ ‖ ˆ α$_{⊥}$$_{µ}$ − ˆ α$_{⊥}$$_{µ}$ ˆ α µ $_{‖}$)( ˆ χ − ˆ χ $^{†}$)] + w$_{6}$ f 2 χ f 2 π tr[( ˆ χ + ˆ χ $^{†}$) $^{2}$] + w$_{7}$ f 2 χ f 2 π (tr[ ˆ χ + ˆ χ $^{†}$]) 2 + w$_{8}$ f 2 χ f 2 π tr[( ˆ χ − ˆ χ $^{†}$) $^{2}$] + w$_{9}$ f 2 χ f 2 π (tr[ ˆ χ − ˆ χ $^{†}$]) $^{2}$, (C.5) L (4) z = z$_{1}$ tr[ ˆ V µν ˆ V $^{µν}$] + z$_{2}$ tr[ ˆ A µν ˆ A $^{µν}$] + z$_{3}$ tr[ ˆ V $_{µν}$V $^{µν}$] + iz$_{4}$ tr[ V$_{µν}$ ˆ α µ $_{⊥}$ˆ α ν $_{⊥}$] + iz$_{5}$ tr[ V$_{µν}$ ˆ α µ ‖ ˆ α ν $_{‖}$] + iz$_{6}$ tr[ ˆ V µν ˆ α µ $_{⊥}$ˆ α ν $_{⊥}$] + iz$_{7}$ tr[ ˆ V µν ˆ α µ ‖ ˆ α ν $_{‖}$] − iz$_{8}$ tr[ ˆ A $_{µν}$(ˆ α µ $_{⊥}$ˆ α ν ‖ + ˆ α µ ‖ ˆ α ν $_{⊥}$)] . (C.6) [1] See e.g. S. Muroya, A. Nakamura, C. Nonaka and T. Takaishi, Prog. Theor. Phys. 110 , 615 (2003).