10 We define the masses of “vector” bosons as the energies determined from the zero mo- mentum limit of the dispersion relation. We shift the “vector” boson fields as and retain quadratic terms only in Eq. (2.30). Performing the Fourier transformation and taking the low-momentum limit yields where f αβγ is a structure constant of Sp (4), M$_{V}$ is the mass of “vector” boson at µ$_{B}$ = 0; and ˆ V is defined as Using the basis of Sp (4) generators shown in Appendix B, one finds that the matrix Γ for the inverse propagator at zero momentum limit is block diagonal with four 1 × 1 terms and three 2 × 2 blocks. The four diagonal 1 × 1 terms are composed of ˆ V $^{1}$, ˆ V $^{2}$, ˆ V 3 corresponding to the ρ meson and ˆ V 4 corresponding to the ω meson. The masses of these states are obtained as Three 2 × 2 blocks for ( ˆ V $^{5}$, ˆ V $^{6}$), ( ˆ V $^{7}$, ˆ V $^{8}$) and ( ˆ V $^{9}$, ˆ V $^{10}$) are identical because of the “isospin” symmetry. The 2 × 2 block for ˆ V 5 and ˆ V 6 is obtained as This matrix is diagonalized by the fields V$_{B}$$_{+}$ and V$_{B}$$_{−}$ defined in Table II as V$_{ν}$ ( x ) → V$_{ν}$ ( x ) + ˜ V$_{ν}$, (3.12) ∫ d $^{4}$x [ − 1 2 g $_{2}$tr( V$_{νρ}$V $^{νρ}$) + f 2 $_{σ}$tr( V$_{ν}$V $^{ν}$) ] → 1 2 g 2 ˆ V α j ( − E ) [ E $^{2}$δ αβ − 2 if $^{αβγ}$( ˜ V$_{0}$ ) $^{γ}$E + f $^{αγθ}$f $^{βδγ}$( ˜ V$_{0}$ ) $^{γ}$( ˜ V$_{0}$ ) δ − M 2 $_{V}$δ αβ ] ˆ V β j ( E ) ≡ ˆ V α j Γ αβ ˆ V β j , (3.13) M$_{V}$ ≡ gf$_{σ}$, (3.14) V α ν ( x ) = ∫ d $^{4}$k (2 π ) 4 ˆ V α ν ( k ) e − $^{ikx}$. (3.15) ( ˆ V 5 † j ˆ V 6 † j )   E 2 − M 2 V + µ 2 B cos $^{2}$θ 2 iµ$_{B}$E cos θ − 2 iµ$_{B}$E cos θ E 2 − M 2 V + µ 2 B cos $^{2}$θ     ˆ V 5 j ˆ V 6 j   . (3.17) ( ˆ V † B + ,j ˆ V † B$_{−}$$_{,j}$)   ( E + µ$_{B}$ cos θ ) 2 − M 2 V 0 0 ( E − µ$_{B}$ cos θ ) 2 − M 2 V     ˆ V$_{B}$$_{+}$$_{,j}$ ˆ V$_{B}$$_{−}$$_{,j}$   . (3.18) m$_{ρ,ω}$ = M$_{V}$ . (3.16)