content_image/371303.jpg content_image/371304.jpg content_image/371305.jpg content_image/371306.jpg content_image/371308.jpg content_image/371307.jpg content_image/371310.jpg content_image/371309.jpg content_image/371311.jpg
where 

<img src='content_image/371303.jpg'>

with the decay constant f$_{π}$ . #1 Transformation property of ξ ( π ) under the chiral symmetry is given by 

<img src='content_image/371304.jpg'>

From this together with the relations in Eq. (2.1), we see that Σ transforms linearly under the chiral symmetry as 

<img src='content_image/371305.jpg'>

The effective Lagrangian including the NG bosons should be invariant under the global SU (2 N$_{f}$ ) group and under the Lorentz transformation, which is given by [3] 

<img src='content_image/371306.jpg'>

Next, we include the “vector” boson fields #2 into the Lagrangian based on the hidden local symmetry (HLS) [9–11]. We decompose the field Σ as 

where ξ$_{L}$ is given by 

<img src='content_image/371308.jpg'>

<img src='content_image/371307.jpg'>

with σ being the NG bosons associated with the spontaneous breaking of the HLS and f$_{σ}$ the corresponding decay constant. The transformation property of ξ$_{L}$ is given by 

where 

<img src='content_image/371310.jpg'>

<img src='content_image/371309.jpg'>

For constructing the HLS Lagrangian, it is convenient to introduce the field ξ$_{R}$ by 

<img src='content_image/371311.jpg'>

