FIG. 4: µ$_{B}$ -dependences of the “vector” boson masses. The curve on the left figure is for the degenerate ρ and ω mesons and those on the right figure are for baryonic and anti-baryonic “vector” bosons. We use M$_{V}$ /M$_{π}$ = 5 / 4, C$_{1}$ = 0, C$_{2}$ = 1 and g = 1 to make the curves. coefficients C$_{1}$ and C$_{2}$ are not determined by the symmetry structure. To study the effect of O ( p $^{4}$) corrections we refer to the mass spectra obtained by the lattice simulation in Ref. [7], which shows that the masses of ρ and ω mesons are degenerate, and that both of them are stable against the change of the chemical potential for µ$_{B}$ < M$_{π}$ and decrease for µ$_{B}$ > M$_{π}$ . From Eqs. (4.6) and (4.7) the degeneracy of m$_{ρ}$ and m$_{ω}$ is realized for C$_{1}$ = 0, and both m$_{ρ}$ and m$_{ω}$ decrease for C$_{2}$ > 0. From Eq. (4.5) together with X $^{4}$X4 = 1, the choice of C$_{1}$ = 0 and C$_{2}$ > 0 implies that the O ( p $^{4}$) terms provide a negative contribution to all the “vector” boson masses equally, and that V$_{B}$$_{+}$ and V$_{B}$$_{−}$ do not mix with each other, even though the baryon number U (1)$_{B}$ symmetry is spontaneously broken. On the other hand, the choice of C$_{1}$ = 0 and C$_{2}$ < 0 implies that the O ( p $^{4}$) terms provide a positive contribution to all the “vector” boson masses equally (see FIG. 3). The effect of nonzero C$_{1}$ produces the mass difference between m$_{ρ}$ and m$_{ω}$ , and this difference is linked to the mixing strength between V$_{B}$$_{+}$ and V$_{B}$$_{−}$ as in Eq. (4.9). This relation is obtained from the symmetry breaking pattern and the assumption that all the bosons other than π and V are heavy enough to be neglected in the Lagrangian. #5 Thus, a violation of Eq. (4.9), when only π and V are light degrees of freedom, may signal a new phase transition. We consider the case with M$_{V}$ /M$_{π}$ = 5 / 4, i.e., relatively heavy π s, which corresponds to the one in the lattice analysis [7]. As an example, we plot the µ$_{B}$ -dependence of the “vector” #5 We expect that the contribution of higher order terms such as O ( p $^{6}$) is small enough.