content_image/346530.jpg content_image/346531.jpg content_image/346532.jpg content_image/346533.jpg content_image/346534.jpg content_image/346535.jpg
where g is the HLS gauge coupling constant and V$_{νρ}$ is the field strength defined by 

<img src='content_image/346530.jpg'>

In addition, we include the external scalar and pseudoscalar source fields χ given in Eq. (A.14) into the Lagrangian. Note that the vacuum expectation value (VEV) of χ gives the explicit chiral symmetry breaking due to the current quark masses as 

<img src='content_image/346531.jpg'>

The χ transforms under the chiral symmetry as χ → g $^{∗}$χg$^{†}$. (see Eq. (A.16)) Since ˆ α$_{‖}$$_{ν}$ as well as ˆ α$_{⊥}$$_{ν}$ transforms homogeneously under the HLS, it is convenient to convert χ into a field ˆ χ as 

<img src='content_image/346532.jpg'>

where G is a parameter carrying the mass dimension one. ˆ χ in Eq. (2.25) transforms homogeneously under the HLS (see Eqs. (2.10) and (2.13) for the transformation properties of ξ$_{L,R}$ ): 

For convenience, we summarize the transformation properties of the building blocks under parity ( P ), charge-conjugation ( C ), and HLS in TABLE I. In this table, Σ$_{c}$ is the 2 N$_{f}$ × 2 N$_{f}$ matrix defined as 

which determines the direction of the chiral condensate, i.e. how to embed the Sp (2 N$_{f}$ ) into SU (2 N$_{f}$ ). Ω is the 2 N$_{f}$ × 2 N$_{f}$ matrix given as 

<img src='content_image/346533.jpg'>

<img src='content_image/346534.jpg'>

<img src='content_image/346535.jpg'>

