I need to understand how starting from the free Lagrangian $$ \mathscr{L} = \bar{q}(i \not\partial - \hat{m})q $$
and based on the chiral angle associated with the pion field and the quark field rotated by the chiral transformation $$ q'= \exp\left[i\dfrac{\pi \cdot \tau\gamma^5}{2 f_\pi} \right]q $$
arrive at $$ \mathscr{L} = \bar{q}(i \not\partial - M)q - \dfrac{1}{2f_\pi}\partial_\pi \pi\cdot\bar{q}\gamma^5 \tau q - i \dfrac{\hat{m}}{f_\pi}\pi \pi\cdot\bar{q}\gamma^5 \tau q + \dfrac{\hat{m}}{2 f_\pi^2}\pi^2\bar{q}q $$
where$$ \mathscr{L} = \bar{q}(i \not\partial - m)q - \dfrac{1}{2f_\pi}\partial_\mu \pi\cdot\bar{q}\gamma^\mu\gamma^5 \tau q - i \dfrac{ {m}}{f_\pi} \pi\cdot\bar{q}\gamma^5 \tau q + \dfrac{ {m}}{2 f_\pi^2}\pi^2\bar{q}q $$ where the quark field operators $q'$ are written as $q$.
