Why is the decay of a neutral pion $\to$ electron-positron loop suppressed? To my understanding the decay of a neutral pion into an electron-positron pair can only happen by the electromagnetic force and the mediation of two virtual photons in a triangle-diagram, so it is loop-suppressed.
What I'm failing to understand is: What is forbidding the direct decay into an electron-positron pair rather than 2 gamma rays? And why is the weak decay forbidden?
I assume I'm missing a conservation law, the question is which one?
 A: In the pion reference frame the two outgoing leptons are very boosted, hence helicity and chirality almost coincide. The angular momentum conservation forces them to have opposite spins, since the pion spin is zero. Therefore, they will have the same helicity, which is highly suppressed in this kinematic regime, because of the vector nature of the QED interactions (see for example Thomson, Modern Particle Physics, chapter 6).
Just my two cents.
A: The weak decay is not forbidden.  You can have $q\overline q \to ZZ \to e^+e^-$ using a loop like in the EM decay.  You need a loop to conserve angular momentum, because a single $Z$ or $\gamma$ has spin $1$ while the $\pi^0$ has spin $0$. If an EM decay is possible, it is so much faster that it will dominate. You can see this in the overall $\pi^0$ decay rate compared to the charged pion decay rate.  The neutral pion decays $10^9$ times faster because it is an EM decay.  The charged pions have to decay weakly as there is nothing they can decay into electromagnetically.
A: By comparing it to the $\pi^- \rightarrow l^- + \bar{\nu_{l} }$ decay, where the lepton and neutrino have the same helicity but is also helicity suppressed since the lepton is massive and contain contributions from left- and right-handed chirality. The antineutrino has right-handed chirality so it participates in the weak exchange.
The process $\pi^0 \rightarrow e^+e^-$ decay is doubly weak-suppressed since both the decay electron and positron have the same right-handed helicity. Only right-handed chiral antiparticles and left-handed chiral particles participate in the weak interaction.
The electron and positron are not massless therefore this right-handed helicity also contains contributions from both left- and right-handed chirality. The neutral pion can decay electromagnetically via loops with a branching ratio of $10^{-9}$.

