content_image/350733.jpg content_image/350734.jpg content_image/350735.jpg
The present analysis is valid when the “axial-vector” bosons (mesons with J P = 1 + and baryons with J P = 1 $^{−}$) are heavy. When the “axial-vector” bosons are light, we need to include these states. This can be done in the framework of the generalized HLS [10, 19]. 

Using this formalism, we may investigate the phase structure in the range of µ$_{B}$ wider than that studied in the present analysis. We hope to obtain a clue to understand the real-life QCD with three colors at finite baryon density through these analyses. 

## Acknowledgments 

This work is supported in part by Global COE Program “Quest for Fundamental Princi- ples in the Universe” of Nagoya University (G07). The work of M.H. is supported in part by the JSPS Grant-in-Aid for Scientific Research (c) 20540262 and Grant-in-Aid for Scientific Research on Innovative Areas (No. 2104) “Quest on New Hadrons with Variety of Flavors” from MEXT. The work of C.N. is supported in part by the Mitsubishi Foundation. 

## Appendix A: QCD Lagrangian with external source fields 

In this appendix, we give the QCD Lagrangian with the external source fields. 

We start with the ordinary QCD Lagrangian with N$_{f}$ massless quarks: 

where 

<img src='content_image/350733.jpg'>

<img src='content_image/350734.jpg'>

with G$_{ν}$ and g$_{s}$ being the gluon field matrix and the gauge coupling constant. Note that the gluon field matrix is expressed as G$_{ν}$ = G a ν τ$_{a}$ 2 where τ$_{a}$ is the Pauli matrix of SU (2)$_{color}$ defined as 

<img src='content_image/350735.jpg'>

