# Pion decay as a point-particle

The $$\pi^-$$ meson is a composite particle of $$\bar{u}d$$ quarks, but for many practical purposes it can be treated as a point particle with an effective interaction. The vertex responsible for the $$\pi^-(p)\rightarrow e^-(q_1)+\bar{\nu}_e(q_2)$$ can be written as: $$(-i)\sqrt{2}G_FV_{ud}f_{\pi}\gamma^\mu\gamma_Lp_\mu$$

I want to write the amplitude of this process but I don't know what to do with the pion. So far I've written: $$iM=\bar{u}(q1)[(-i)\sqrt{2}G_FV_{ud}f_{\pi}\gamma^\mu\gamma_Lp_\mu ]v(q2)$$ But I still have a pion meson entering the vertex, so how do I take that into account? Does the above expression suffice?

On the face of it, your expression looks fine, and the pion momentum is the only usable trace of the annihilated pion. So you must evaluate the pion momentum you wrote. In your conventions, you have conservation of momentum, so $$p = q_1-q_2$$, and you must proceed to apply the equations of motion on your spinors.
Hint: the one of the neutrino will collapse and disappear; while the one on the electron will lead to $$~~/\!\!\! q_1 \to m_e$$, so then the spinor bilinear in your expression will reduce to $$m_e \bar u \gamma_L v$$.