How would I write the effective Lagrangian in the lowest order perturbation theory for the (standard model tree-level prohibited) process $$\pi^0\rightarrow 2\gamma?$$ My first guess would be something like $$L=L_{\mbox{free scalar}}+L_{\mbox{free photons}}+g\phi A^\mu A_\mu.$$ Where the free scalar field has mass $m=m_\pi$ and the free photons have ~$F_{\mu\nu}F^{\mu\nu}$. I don't think the above effective lagrangian respects gauge invariance, so I instead write $$L=L_{\mbox{free scalar}}+L_{\mbox{free photons}}+g\phi F^{\mu\nu} F_{\mu\nu},$$ where $g$ has mass dimension $-1$. Is this the correct approach to effective Lagrangians? If this is correct, then would my Feynman rules be vertices with $(-ig)$ and external photon polarizations $\epsilon^*$ analogous to QED?
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