Scattering amplitude with a change in basis of fields Suppose I know the Feynman rules for the scattering process $\pi^j \pi^k \rightarrow \pi^l \pi^m$ where $j,k,l,m$ can be $1, 2$ or $3$. Define the charged pion fields as $\pi^\pm=\frac{1}{\sqrt{2}}(\pi^1 \pm i \pi^2)$ and neutral pion field as $\pi^0=\pi^3$. I would like to derive the scattering amplitudes for processes like $\pi^+ \pi^- \rightarrow \pi^+ \pi^-$ from my knowledge of scattering amplitude of $\pi^j \pi^k \rightarrow \pi^l \pi^m$. How should I proceed?
I suppose it can be done by clever change of indices in Feynman rules, but I am unable to see how exactly. 
 A: I assume your Lagrangian might have a term of the form $\bar{N}\vec{\pi}.\vec{\tau}\gamma^5N$, or something with $\vec{\pi}.\vec{\tau}$. One method is to expand the following and see how $\pi^+$ and $\pi^-$ come into your Lagrangian,
\begin{align*}
\vec{\pi}.\vec{\tau} &= \pi^1\sigma^1 + \pi^2\sigma^2 +\pi^3\sigma^3 \\
        &= \begin{pmatrix}
         0 && \pi^1 \\
         \pi^1 && 0
         \end{pmatrix}
         + 
         \begin{pmatrix}
          0 && -i \pi^2 \\
          i \pi^2 && 0 
         \end{pmatrix}
         + 
         \begin{pmatrix}
          \pi^3 && 0 \\
          0 && -\pi^3
         \end{pmatrix} \\
        &= \begin{pmatrix}
         \pi^3 && \pi^1 - i \pi^2 \\
         \pi^1 + i\pi^2 && -\pi^3
         \end{pmatrix} \\
        &= \begin{pmatrix}
         \pi^0 && \sqrt{2}\pi^- \\
         \sqrt{2}\pi^+ && -\pi^0
         \end{pmatrix}.
\end{align*}
Then expand your Lagrangian in these fields, 
\begin{align*}
    \bar{N} \vec{\pi}.\vec{\tau}\gamma^5 N &= \begin{pmatrix} \bar{p} && \bar{n} \end{pmatrix}
         \begin{pmatrix}
         \pi^0 && \sqrt{2}\pi^- \\
         \sqrt{2}\pi^+ && -\pi^0
         \end{pmatrix}\gamma^5
         \begin{pmatrix}
          p \\ n 
         \end{pmatrix}\\
&= \bar{p}\pi^0\gamma^5 p + \sqrt{2} \bar{p}\pi^-\gamma^5 n + \sqrt{2} \bar{n} \pi^+ \gamma^5 p - \bar{n} \pi^0 \gamma^5 n
\end{align*}
After expanding this out you can read off your Feynman rules with these new fields. 
