= Solution
Take the <quantum electrodynamics> interaction density to be $\mathcal L_{\mathrm{int}}=-e\bar\psi\gamma^\mu\psi A_\mu$. The momentum-space <Feynman-gauge photon propagator>, with Lorentz indices $\mu,\nu$ and momentum $k$, is
$$
\boxed{D_{\mu\nu}(k)=\frac{-i\eta_{\mu\nu}}{k^2+i\epsilon}.}
$$
The electron-photon vertex from the <QED Feynman rules> is
$$
\boxed{-ie\gamma^\mu,}
$$
with its spinor indices connecting the incident and outgoing fermion lines. The muon-photon vertex has the same coupling. The <Feynman i-epsilon prescription> specifies the pole boundary condition; using a different overall charge-sign convention changes both vertices consistently and leaves the squared amplitude unchanged.
For the <electron-positron annihilation into a muon pair>, the exchanged <photon> has $k=p+q$ and $k^2=s$. Up to the conventional overall phase, the <scattering amplitude> is
$$
\mathcal M=\frac{e^2}{s}\,[\bar v_r(q)\gamma^\mu u_s(p)]\,[\bar u_{s'}(p')\gamma_\mu v_{r'}(q')].
$$
Back to article page
