In the exponential representation after spontaneous symmetry breaking of the $SU(2) \times SU(2)$ by a VEV, $ v $, the non-linear sigma model can be written,

\begin{align} {\cal L} & = \frac{1}{2} \left[ ( \partial ^\mu S ) ^2 - 2 \mu ^2 S ^2 \right] + \frac{ ( v + S ) ^2 }{ 4} \mbox{Tr} \left( \partial _\mu \Sigma \partial ^\mu \Sigma ^\dagger \right) - \lambda v S ^3 - \frac{ \lambda }{ 4} S ^4 \\ & + \bar{\psi} i \partial_\mu \gamma^\mu \psi - g ( v + S ) \left( \bar{\psi} _L \Sigma \psi _R + \bar{\psi} _R \Sigma \psi _L \right) \end{align}

where $S$ and $\Sigma$ are related to the linear representation of $\sigma$ and ${\vec \pi}$ by,

\begin{equation}

\pi = \sigma + i {\vec \tau} \cdot {\vec \pi} = ( v + S ) \Sigma \quad , \quad \Sigma = \exp \left( \frac{ i {\vec \tau} \cdot {\vec \pi} ' }{ v } \right)

\end{equation}

Spontaneous symmetry breaking lead to three Goldstone bosons, $\vec \pi $. These are typically identified as the pions. But where did this identification come from? We started with 4 unknown scalar fields and an $SU(2)\times SU(2)$ flavor symmetry. Why would we expect the Goldstone bosons of this broken symmetry give rise to composite particles made of the fermions?