FIG. 3: µ$_{B}$ -dependences of the “vector” boson masses. The curves on the left figure are 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$_{π}$ = 4, C$_{1}$ = 0, C$_{2}$ = ± 4 and g = 1 to make the curves. Dashed (blue) curves stand for C$_{2}$ = 4. Solid (red) curves stand for C$_{2}$ = − 4. where C$_{1}$ and C$_{2}$ are certain linear combinations of ¯ y$_{1}$ , ¯ y$_{2}$ and ¯ y$_{3}$ . Applying the same procedure as that in section 3 with the correction in Eq. (4.5), we obtain the masses of ρ meson and ω meson as The quadratic term for ( V$_{B}$$_{+}$, V$_{B}$$_{−}$ ) is obtained as The masses of “vector” bosons with J P = 1 + are obtained by diagonalizing the mass matrix in Eq. (4.8). We can see that the mixing between V$_{B}$$_{+}$ and V$_{B}$$_{−}$ is related with the mass difference between m$_{ρ}$ and m$_{ω}$ as Equations (4.6), (4.7) and (4.8) show that the corrections to the masses from O ( p $^{4}$) terms include the factor of sin θ , which is zero for µ$_{B}$ < M$_{π}$ . Then, the O ( p $^{4}$) corrections appear only for µ$_{B}$ > M$_{π}$ , where the U (1)$_{B}$ is spontaneously broken. It should be noticed that two m$_{ρ}$ = √ M 2 V + C$_{1}$ − C$_{2}$ 4 ( gµ$_{B}$ sin θ ) $^{2}$, (4.6) m$_{ω}$ = √ M 2 V − C$_{1}$ + C$_{2}$ 4 ( gµ$_{B}$ sin θ ) $^{2}$. (4.7) ( ˆ V † $_{B$_{+}$}$ˆ V † $_{$_{B$_{−}$}$)}$   ( E + µ$_{B}$ cos θ ) 2 − M 2 V + C$_{2}$ $_{16}$( gµ$_{B}$ sin θ ) 2 C$_{1}$ $_{16}$( gµ$_{B}$ sin θ ) 2 C$_{1}$ $_{16}$( gµ$_{B}$ sin θ ) 2 ( E − µ$_{B}$ cos θ ) 2 − M 2 V + C$_{2}$ $_{16}$( gµ$_{B}$ sin θ ) 2       ˆ V$_{B}$$_{+}$ ˆ V$_{B}$$_{−}$    . (4.8) m 2 ρ − m 2 ω = C$_{1}$ 2 ( gµ$_{B}$ sin θ ) $^{2}$. (4.9)