= Solution
Write an amplitude with an external massless vector of momentum $k$ as
$$
\mathcal M=\epsilon_\mu(k)\mathcal M^\mu(k,\ldots).
$$
Changing the polarization representative by $\epsilon_\mu\mapsto\epsilon_\mu+c,k_\mu$ is a <gauge transformation>. The physical amplitude must be unchanged, so the <Ward identity> is
$$
\boxed{k_\mu\mathcal M^\mu=0}.
$$
Thus a longitudinal pure-gauge polarization decouples from the S-matrix. This on-shell statement is the amplitude counterpart of the off-shell <Ward-Takahashi identity>.
Back to article page
