content_image/353221.jpg content_image/353220.jpg content_image/353222.jpg content_image/353223.jpg content_image/353224.jpg content_image/353225.jpg content_image/353226.jpg
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 

<img src='content_image/353221.jpg'>

<img src='content_image/353220.jpg'>

<img src='content_image/353222.jpg'>

Similarly, using the field Ψ, we rewrite L ext-scalar and L ext-vector in Eqs. (A.4) and (A.5) as 

<img src='content_image/353223.jpg'>

where χ and G µ are external fields of 2 N$_{f}$ × 2 N$_{f}$ matrices defined as 

<img src='content_image/353224.jpg'>

<img src='content_image/353225.jpg'>

Transformation properties of the external fields under SU (2 N$_{f}$ ) are given by 

## Appendix B: Explicit realization of the SU (4) generators 

<img src='content_image/353226.jpg'>

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 

