21 S α = 1 $_{2}$√$_{2}$   A B B † − A T   , X a = 1 $_{2}$√$_{2}$   C D D † C T   , (B.1) as where A is hermitian, C is hermitian and traceless, B T = B and D T = − D . The { S } are also Sp (4) generators since they obey Eq. (2.1). We define For α = 1 , 2 , 3, we have the standard Pauli matrices, while for α = 4 we define τ 4 = 1 . These are simply the generators for SU (2) × U (1). For α = 5 , ..., 10 and The five broken generators { X } are where τ a are the standard Pauli matrices. For a = 4 , 5 and The generators are normalized as follows: In this appendix, we present a complete list of the O ( p $^{4}$) HLS Lagrangian for general N$_{f}$ and N$_{C}$ = 2, following Refs. [11, 17]. For the construction, we need the building blocks ˆ V µν = 1 2 [ ξ$_{R}$ ¯ Σ G T $_{µν}$¯ Σ ξ † R + ξ$_{L}$G$_{µν}$ξ † $_{L}$] , ˆ A µν = 1 2 [ ξ$_{R}$ ¯ Σ G T $_{µν}$¯ Σ ξ † R − ξ$_{L}$G$_{µν}$ξ † $_{L}$] , (C.1) tr( S $^{α}$S$^{β}$) = 1 2 δ $^{αβ}$, tr( X $^{a}$X$^{b}$) = 1 2 δ $^{ab}$, tr( S $^{α}$X$^{a}$) = 0 . (B.8) D 4 = τ $^{2}$, D5 = iτ $^{2}$. (B.7) X a = 1 $_{2}$√$_{2}$   0 D a ( D $^{a}$) † 0   , ( a = 4 , 5) , (B.6) X a = 1 $_{2}$√$_{2}$   τ a 0 0 ( τ $^{a}$)T   , ( a = 1 , 2 , 3) , (B.5) B 5 = 1$_{2}$ , B 6 = i 1$_{2}$ , B 7 = τ $^{3}$, B8 = iτ $^{3}$, B9 = τ $^{1}$, B10 = iτ $^{1}$. (B.4) S α = 1 $_{2}$√$_{2}$   0 B α ( B $^{α}$) † 0   , ( α = 5 , ..., 10) , (B.3) S α = 1 $_{2}$√$_{2}$   τ α 0 0 − ( τ $^{α}$)T   , ( α = 1 , 2 , 3 , 4 ) . (B.2) Appendix C: O ( p $^{4}$) HLS Lagrangian