QED Feynman Diagram: e+e- to mu+mu-, |M|^2 and the Cross Section

A tree-level QED (quantum electrodynamics) playground for $e^+e^- \to \mu^+\mu^-$. Panel A draws the $s$-channel Feynman diagram and reads off the Feynman-rule bookkeeping: each vertex carries a factor $e$, so a $V$-vertex amplitude scales as $\alpha^{V/2}$ and $|M|^2$ as $\alpha^V$; a one-loop variant inserts a vacuum-polarisation bubble (four vertices, $\alpha^4$, suppressed by $\alpha^2$ relative to the tree). Panel B plots the total cross section $\sigma(\sqrt{s})$ in nanobarns on log-log axes: it is exactly zero below $\sqrt{s} = 2 m_\mu$, rises through the muon-pair threshold to a peak, then falls as a straight power-law line ($\sigma \sim 1/s$). Panel C shows the Mandelstam invariants with the exact identity $s + t + u = 2 m_e^2 + 2 m_\mu^2$ and the differential cross section $d\sigma/d\Omega$, which is even in $\cos\theta$ (forward-backward symmetric) and tends to $1 + \cos^2\theta$ ultrarelativistically, becoming isotropic near threshold. The Mandelstam identity $s + t + u = 2 m_e^2 + 2 m_\mu^2$ holds, the cross section turns on sharply at the muon-pair threshold and falls as $1/s$, and the angular distribution is forward-backward symmetric, tending to $1 + \cos^2\theta$ ultrarelativistically.