12 4. EFFECT OF HIGHER ORDER TERMS In this section, we consider the effects of higher order terms in the hidden local symmetry (HLS). In QCD with three colors it is known that, thanks to the gauge invariance of the HLS, we can perform the systematic derivative expansion with including vector mesons in addition to the pseudo Nambu-Goldstone bosons when the masses of vector mesons are lighter than the chiral symmetry breaking scale (the chiral perturbation theory with the HLS [11, 15–17]). We adopt the same counting rule in the present case. Generally, there are 32 terms in the O ( p $^{4}$) HLS Lagrangian [11, 17]. We present a complete list of O ( p $^{4}$) Lagrangian in Appendix C. Here we include only the terms which do not alter the vacuum structure given in Eq. (3.7), neglecting the effect of current quark masses at O ( p $^{4}$). In this case we have only three combinations which give corrections to the “vector” boson masses: where ¯ y$_{1}$ , ¯ y$_{2}$ and ¯ y$_{3}$ are coefficients not determined by the HLS. We consider the case of N$_{f}$ = 2 in the following analysis. From the vacuum expectation values (VEVs) of ξ ( π ) and V$_{ν}$ given in Eqs. (3.8) and (3.11), VEVs of ˆ α$_{‖}$$_{µ}$ and ˆ α$_{⊥}$$_{µ}$ are determined as where X 4 is the fourth component of the broken generators given in Eq. (B.6). Substituting the VEV 〈 ˆ α$_{⊥}$$_{µ}$ 〉 into Eqs. (4.1)-(4.3) we obtain the correction to the masses of “vector” bosons as L (4) 1 = ¯ y$_{1}$ tr[ ˆ α$_{⊥}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ˆ α µ $_{⊥}$ˆ α ν ‖ − ˆ α$_{⊥}$$_{µ}$ ˆ α µ $_{⊥}$ˆ α$_{‖}$$_{ν}$ ˆ α ν ‖ + ˆ α$_{⊥}$$_{µ}$ ˆ α$_{⊥}$$_{ν}$ ˆ α µ ‖ ˆ α ν ‖ − 1 2 (ˆ α$_{⊥}$$_{µ}$ ˆ α µ ‖ ˆ α$_{⊥}$$_{ν}$ ˆ α ν ‖ + ˆ α$_{⊥}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ˆ α ν $_{⊥}$ˆ α µ $_{‖}$) ] , (4.1) L (4) 2 = ¯ y$_{2}$ tr[ ˆ α$_{⊥}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ˆ α µ $_{⊥}$ˆ α ν ‖ − ˆ α$_{⊥}$$_{µ}$ ˆ α$_{⊥}$$_{ν}$ ˆ α ν $_{‖}$ˆ α µ ‖ + ˆ α$_{⊥}$$_{µ}$ ˆ α$_{⊥}$$_{ν}$ ˆ α µ $_{‖}$ˆ α ν ‖ − 1 2 (ˆ α$_{⊥}$$_{µ}$ ˆ α µ ‖ ˆ α$_{⊥}$$_{ν}$ ˆ α ν ‖ + ˆ α$_{⊥}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ˆ α ν $_{⊥}$ˆ α µ $_{‖}$) ] , (4.2) L (4) 3 = ¯ y$_{3}$ tr[ ˆ α$_{⊥}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ˆ α µ $_{⊥}$ˆ α ν ‖ − ˆ α$_{⊥}$$_{µ}$ ˆ α µ $_{⊥}$ˆ α$_{‖}$$_{ν}$ ˆ α ν ‖ + ˆ α$_{⊥}$$_{µ}$ ˆ α$_{⊥}$$_{ν}$ ˆ α ν $_{‖}$ˆ α µ ‖ − 1 2 (ˆ α$_{⊥}$$_{µ}$ ˆ α µ ‖ ˆ α$_{⊥}$$_{ν}$ ˆ α ν ‖ + ˆ α$_{⊥}$$_{µ}$ ˆ α$_{‖}$$_{ν}$ ˆ α ν $_{⊥}$ˆ α µ $_{‖}$) ] , (4.3) 〈 ˆ α$_{‖}$$_{µ}$ 〉 = 0 , 〈 ˆ α$_{⊥}$$_{µ}$ 〉 = √$_{2}$$_{µ}$$_{B}$$_{δ}$$_{µ}$$_{0}$ sin θX $^{4}$, (4.4) L (4) 1 + L (4) 2 + L (4) 3 O ( V 2 j ) → 2 µ 2 B sin 2 θ [ C$_{1}$ tr( X $^{4}$V$_{j}$ X $^{4}$V $^{j}$) − C$_{2}$ tr( X $^{4}$X$^{4}$V$_{j}$ V $^{j}$)] , (4.5)