= Double-longitudinal Ward identity for gluon fusion
{title2=$k_1^\mu k_2^\nu\mathcal M_{\mu\nu}=0$}
For massless $gg\to q\bar q$, define $i\mathcal M$ by the Feynman-rule product with <quark> vertex $igT^a\gamma^\mu$. The ordered <quark> exchanges contribute $-g^2\bar u\not k_1[T^a,T^b]v$ after double contraction. Three-gluon exchange contributes $ig^2f^{abc}\bar u\not k_1T^cv$, so the sum vanishes by the <Lie algebra>. This does not assert off-shell transversality of each open index; the full <Slavnov-Taylor identity> includes <Faddeev-Popov ghost> terms.
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