= Three-photon fermion Ward identity
{title2=$\mathcal A(\epsilon_\ell\to q_\ell)=0$}
For tree <Electron>-<Positron> annihilation into three <photons>, all six orders along the <fermion> line must be summed. Write $D(r)=\not r-m$ and $S(r)=D(r)^{-1}=(\not r+m)/(r^2-m^2)$. Contracting an internal photon with its momentum gives $S(R)\not q S(r)=S(R)-S(r)$ when $q=r-R$. Contracting an end photon removes its adjacent <Dirac propagator> by the external <Dirac equation>, with a minus sign at the electron end and a plus sign at the positron end. For either fixed order of the two uncontracted photons, the three insertion positions therefore telescope to zero. Summing their two orders proves the <Ward identity>. The mass terms in both remaining <Dirac propagators> are essential when $m\ne0$. The result expresses the invariance of the <scattering amplitude> under $\epsilon_\ell\mapsto\epsilon_\ell+cq_\ell$.
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