11 FIG. 2: The masses of the “vector” bosons in unit of M$_{π}$ as a function of µ$_{B}$/M$_{π}$ . We use M$_{V}$ /M$_{π}$ = 4 and g = 1 to make the plot. From this, we obtain the masses as where we assumed M$_{V}$ > M$_{π}$ , so that M$_{V}$ > µ$_{B}$ cos θ . Let us consider the µ$_{B}$ -dependences of the masses of the “vector” bosons. Equation (3.16) indicates that the ordinary vector mesons with J P = 1 − do not change their masses at all. On the other hand, from Eq. (3.19) together with Eq. (3.10), we find that the mass of the baryon ( V$_{B}$$_{+}$ ) with J P = 1 + decreases for µ$_{B}$ < M$_{π}$ and turns to increase for µ$_{B}$ > M$_{π}$ , and the mass of anti-baryon ( V$_{B}$$_{−}$ ) with J P = 1 + shows the opposite behavior. This indicates that the phase transition can be observed by seeing the masses of baryons with J P = 1 $^{+}$. We stress that, for any value of µ$_{B}$ , the mass eigenstates are given by V$_{B}$$_{+}$ and V$_{B}$$_{−}$ , which are nothing but the eigenstates of the baryon number; V$_{B}$$_{+}$ carries B = +1 and V$_{B}$$_{−}$ does B = − 1. In other words, the baryon with J P = 1 + does not mix with the anti-baryon having the same spin and parity, even though the baryon number U (1)$_{B}$ symmetry is spontaneously broken for µ$_{B}$ > M$_{π}$ . Note that this feature holds only at the leading order: The mixing will generally appear when we include the higher order terms (see the next section). In FIG. 2, we plot the masses of “vector” bosons described in Eqs. (3.16) and (3.19) for M$_{V}$ /M$_{π}$ = 4 as an example. In this figure, we can see the µ$_{B}$ -dependence of the masses of the “vector” bosons explained above. m$_{V}$$_{B}$$_{+}$ = M$_{V}$ − µ$_{B}$ cos θ, m$_{V}$$_{B}$$_{−}$ = M$_{V}$ + µ$_{B}$ cos θ, (3.19)