6 where g is the HLS gauge coupling constant and V$_{νρ}$ is the field strength defined by 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 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 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 V$_{νρ}$ ≡ ∂$_{ν}$V$_{ρ}$ − ∂$_{ρ}$V$_{ν}$ − i [ V$_{ν}$, V$_{ρ}$ ] . (2.23) 〈 χ 〉 =      − m$_{q}$$_{1}$ 0$_{(}$$_{N}$$_{f}$$_{×}$$_{N}$$_{f}$$_{)}$ . . . − m$_{q}$$_{N}$$_{f}$ m$_{q}$$_{1}$ . . . 0$_{(}$$_{N}$$_{f}$$_{×}$$_{N}$$_{f}$$_{)}$ m$_{q}$$_{N}$$_{f}$     $^{.}$ (2.24) ˆ χ = 2 Gξ$_{R}$ ¯ Σ χξ † $_{L}$, (2.25) ˆ χ → h ˆ χh $^{†}$. (2.26) Σ$_{c}$ ≡   0 1$_{N}$ − 1$_{N}$ 0   , (2.27) Ω ≡   0 1$_{N}$ 1$_{N}$ 0   . (2.28)