content_image/370835.jpg content_image/370852.jpg content_image/370851.jpg content_image/370850.jpg content_image/370849.jpg content_image/370848.jpg content_image/370847.jpg content_image/370846.jpg content_image/370845.jpg
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 

<img src='content_image/370835.jpg'>

<img src='content_image/370852.jpg'>

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 

<img src='content_image/370851.jpg'>

<img src='content_image/370850.jpg'>

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: 

<img src='content_image/370849.jpg'>

<img src='content_image/370848.jpg'>

<img src='content_image/370847.jpg'>

<img src='content_image/370846.jpg'>

## Appendix C: O ( p $^{4}$) HLS Lagrangian 

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 

<img src='content_image/370845.jpg'>

