Why is $\pi^0 \rightarrow \gamma \rightarrow e^-e^+$ forbidden but $\pi^+\rightarrow W^+ \rightarrow e^+ \nu_e$ allowed? I am slightly confused why the decay $\pi^0 \rightarrow \gamma \rightarrow e^-e^+$ is forbidden. A naive guess would say that the intemediate photon $\gamma$ has a spin 1, the initial pion has a spin $0$ therefore this violates spin conservation. However, on the same reasoning $\pi^+\rightarrow W^+ \rightarrow e^+ \nu_e$ would be forbidden, but it's not. Therefore I cannot see why we cannot have  $\pi^0 \rightarrow \gamma \rightarrow e^-e^+$  yet still have  $\pi^+\rightarrow W^+ \rightarrow e^+ \nu_e$. Please can someone explain? (p.s. all intermediate particles should be taken as virtual). 
 A: The QCD and QED themselves conserve parity. The conclusion of this statement is that all corresponding effective vertices must conserve the parity. The only coupling of $\pi^{0}$ to $\gamma$ conserving the parity is
$$
L_{\pi^{0}} \simeq \frac{\pi^{0}}{\Lambda}\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta},
$$ 
which doesn't allow your decay process $\pi^{0} \to \gamma^{*} \to e^{-}e^{+}$. To understand this, note that the pion fields are pseudo-scalars, while the photon field is the vector field, while the Levi-Civita tensor $\epsilon^{\mu\nu\alpha\beta}$ is the pseudo-tensor. 
However, it is possible to construct the effective vertex allowing the decay $\pi^{0}\to e^{+}e^{-}$ through $Z$-boson, i.e., through weak interactions. The reason is that they directly violate the parity. Therefore, it is possible to construct phenomenological parity-violating low-dimensional effective interaction vertex 
$$
L_{\pi^{0}}' \simeq \Lambda'\partial^{\mu}\pi^{0}Z_{\mu},
$$
allowing the decay process $\pi^{0} \to Z^{*} \to e^{+}e^{-}$. 
By the same reason, it is easy to construct parity violating vertex 
$$
L_{\pi^{+}} = \tilde{\Lambda}\partial^{\mu}\pi^{+}W^{-}_{\mu} + \text{ h.c.},
$$ 
allowing your decay process $\pi^{+}\to W^{+*} \to l^{+}\nu_{l}$.
A: your question could be generalized: pion decays to odd number of photon is forbidden. That's a conclusion of Furry's theorem: even number of photons is forbidden to odd numbers of photon
Here the only assumption is that strong interaction and EM process charge conjugation symmetry.
As for the second process, because charge conjugation is no longer a symmetry in weak interaction, the decay process is allowed.
