content_image/350906.jpg content_image/350907.jpg content_image/350908.jpg content_image/350909.jpg content_image/350910.jpg content_image/350911.jpg content_image/350912.jpg content_image/350913.jpg content_image/350914.jpg content_image/350915.jpg
which transforms as 

<img src='content_image/350906.jpg'>

Note that, in the HLS, the entire symmetry G$_{global}$ × H$_{local}$ is spontaneously broken to its subgroup H$_{global}$ = [ Sp (2 N$_{f}$ )]$_{global}$, so that the NG bosons σ = σ $^{α}$Sα are absorbed into the HLS gauge bosons. The basic quantities to construct the HLS Lagrangian are the following two Maurer-Cartan 1-forms: 

<img src='content_image/350907.jpg'>

<img src='content_image/350908.jpg'>

where the covariant derivatives are read from the transformation properties in Eqs. (2.10) and (2.13) as 

<img src='content_image/350909.jpg'>

<img src='content_image/350910.jpg'>

with V$_{ν}$ = V α ν S α and G$_{ν}$ = G A ν T A being the H$_{local}$ gauge bosons and the external gauge bosons corresponding to the chiral symmetry (see Appendix A), respectively. These gauge bosons transform as 

<img src='content_image/350911.jpg'>

<img src='content_image/350912.jpg'>

It should be noticed that in the HLS formalism we can introduce the “vector” bosons V$_{ν}$ and the external chiral gauge bosons G$_{ν}$ independently. Since the covariantized 1-forms ˆ α$_{⊥}$$_{ν}$ and ˆ α$_{‖}$$_{ν}$ in Eqs. (2.14) and (2.15) transform homogeneously: 

we have the following two invariants: 

We introduce the kinetic term of the “vector” bosons: 

<img src='content_image/350913.jpg'>

<img src='content_image/350914.jpg'>

<img src='content_image/350915.jpg'>

