How to compute scattering amplitude $\gamma\pi^+\to\pi^+\pi^0$ I wish to find the amplitude for process $\gamma\pi^+\to\pi^+\pi^0$ at low energies. I am familiar with the basic concepts and techniques of QFT but have never dealt with the scattering processes including hadrons. In this case, I do not have a clue where to start. I am basically asking for a reference where this process is computed and/or what are the key points one needs to have in mind thinking about the reaction: appropriate effective Lagrangian, certain selection rules etc.
 A: The standard chiral lagrangian has a $\pi\to-\pi$ symmetry that forbids processes with an odd number of Goldstone bosons. This is not a symmetry of QCD, however, and these reactions are described by the Wess-Zumino term. The Wess-Zumino term is not a local lagrangian in 4-d, but it can be written as a local lagrangian on a 5-d manifold which has 4-d Minkowski space as a boundary. This is discussed in many text books, see, for example, Section VII-5 of Donoghue, Golowich and Holstein. 
Note that $\gamma\pi^+\to \pi^+\pi^0$ is not a process that can be easily observed (there are no pion targets), but you can observe related processes like 
radiative rho meson decay $\rho\to \pi\gamma$ (the rho is a $\pi\pi$ resonance), and $\gamma p\to n\pi^+\pi^0$.  
A: A small clarification to @Thomas 's impeccable complete answer, for  readers impatient with the mechanics of the WZW effective actions. The crucial point is that G-parity is violated by electromagnetism, since isospin is. 
Essentially, the relevant effective interaction term you tease out of WZW is proportional to the parity-conserving 
$$
\frac{e}{24\pi^2 f^3}\int d^4x ~ A_\mu \epsilon^{\mu\nu\kappa\lambda} \operatorname {Tr} (\partial_\nu \mathbf { \pi} \partial_\kappa \mathbf {\pi} \partial_\lambda \mathbf {\pi}  )\propto \int d^4x ~ A_\mu \epsilon^{\mu\nu\kappa\lambda}   \partial_\nu   \pi^0 \partial_\kappa \pi^+ \partial_\lambda  \pi^-  ~.
$$
