The neutral pion
In \(e^+e^-\) collisions one can observe collisions of almost real photons emitted by the electrons and positrons at small angles to the beam axis. In such collisions the photons can produce a neutral pion. The cross section of this reaction is proportional to the decay width \(\Gamma(\pi^0\to\gamma\gamma)\) and provides a way to measure it.
(i) Compute the cross section of the reaction \(\gamma+\gamma\to\pi^0\) in QED: \[\sigma(\gamma+\gamma\to\pi^0) = \frac{8\pi^2\,\Gamma(\pi^0\to\gamma\gamma)}{m_\pi}\,\delta\bigl((p+k)^2-m_\pi^2\bigr),\] where \(p,k\) are the 4-momenta of the photons (Low’s formula; the photons are unpolarized).
(ii) Check that the matrix element of the reaction \(\gamma+\gamma\to\pi^0\) is nonzero only when the helicities of the photons are equal (for linear polarizations, when they are mutually perpendicular).
(iii) Check that in the decay \(\pi^0\to\gamma\gamma\) the combination \((\boldsymbol{E}_1+\boldsymbol{E}_2)\cdot (\boldsymbol{B}_1+\boldsymbol{B}_2)\) is nonzero only for the same photon polarizations.