content_image/330428.jpg content_image/330429.jpg content_image/330431.jpg content_image/330430.jpg content_image/330434.jpg content_image/330432.jpg content_image/330433.jpg
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 

<img src='content_image/330428.jpg'>

and retain quadratic terms only in Eq. (2.30). Performing the Fourier transformation and taking the low-momentum limit yields 

<img src='content_image/330429.jpg'>

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 

<img src='content_image/330431.jpg'>

<img src='content_image/330430.jpg'>

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 

<img src='content_image/330434.jpg'>

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 

<img src='content_image/330432.jpg'>

This matrix is diagonalized by the fields V$_{B}$$_{+}$ and V$_{B}$$_{−}$ defined in Table II as 

<img src='content_image/330433.jpg'>

