content_image/338209.jpg content_image/338210.jpg content_image/338211.jpg
We study the vacuum structure of the model in the case of N$_{f}$ = 2 using the leading order Lagrangian, and show that the flavor-singlet “ ω ”-meson carrying J P = 1 − has a vacuum expectation value in the time component. As a result, the phase structure is the same as the one determined by including only the pseudo-NG bosons [3]: For µ$_{B}$ > M$_{π}$ , a baryonic pseudo-NG boson ( J P = 0 + state) condenses, which causes the spontaneous breaking of the baryon number symmetry, U (1)$_{B}$. We show that the mass of the anti-baryon (baryon) with J P = 1 + increases (decreases) for µ$_{B}$ < M$_{π}$ and turns to decrease (increase) for µ$_{B}$ > M$_{π}$ . These behaviors signal the phase transition of U (1)$_{B}$ breaking. The effect of higher order terms is shown to make ρ and ω meson masses decrease for µ$_{B}$ > M$_{π}$ consistently with lattice data [6, 7]. Furthermore, the mass difference between ρ and ω mesons is proportional to the mixing strength between the diquark baryon with J P = 1 + and the anti-baryon. 

The paper is organized as follows. In section 2 we construct the chiral Lagrangian based on the HLS. The vacuum structure and the µ$_{B}$ -dependences of the masses are studied at the leading order in section 3. In section 4 we show the effects of higher order terms to the masses and mixings. Section 5 is devoted to a summary and discussions. Several intricate calculations and useful formulas are summarized in Appendices A-C. 

## 2. HLS MODEL IN TWO-COLOR QCD 

Let us construct a low energy effective Lagrangian including NG bosons associated with the spontaneous chiral symmetry breaking SU (2 N$_{f}$ ) → Sp (2 N$_{f}$ ), following Ref. [3]. 

In the following we divide the hermitian generators, { T $^{A}$} of SU (2 N$_{f}$ ) normalized as tr[ T $^{A}$T $^{B}$] = δ $^{AB}$/ 2, into two classes: The generators of Sp (2 N$_{f}$ ) denoted by { S $^{α}$} with α = 1 , ..., 2 N 2 $_{f}$+ N$_{f}$ ; and the remaining generators of SU (2 N$_{f}$ ) by { X $^{a}$} with a = 1 , ..., 2 N 2 f − N$_{f}$ − 1. These generators satisfy the relations 

where ¯ Σ is a 2 N$_{f}$ × 2 N$_{f}$ matrix satisfying following properties: 

The chiral symmetry breaking gives 2 N 2 f − N$_{f}$ − 1 NG bosons π which are encoded in the 2 N$_{f}$ × 2 N$_{f}$ matrix as 

<img src='content_image/338209.jpg'>

<img src='content_image/338210.jpg'>

<img src='content_image/338211.jpg'>

