  A recent letter Cai et al. [2107.14548] within a phenomenological dark matter
framework with a massive graviton in the external state indicated a divergence
with increasing centre-of-momentum energy arising from the longitudinal
polarizations of the graviton. In this letter we point out that in processes
such as graviton-photon production from matter annihilation, $f\bar{f} \to
G\gamma$, no such anomalous divergences occur at tree-level. This then applies
to other tree-level amplitudes related by crossing symmetry such as $\gamma f
\to Gf$, $Gf \to {\gamma}f$, ${\gamma}f \to Gf$, $f \to fG{\gamma}$ and so on.
We show this by explicitly computing the relevant tree-level diagrams, where we
find that delicate cancellations ensure that all anomalously growing terms are
well-regulated. Effectively at tree-level this is consistent with the operation
of a Ward identity associated with the external photon for such amplitudes. The
same tree-level results apply if the photon is replaced by a gluon. These
results are important for cosmological models of dark matter within the
framework of extra dimensions.
