Is it sufficient to consider the uncharged dark matter and a $\mathbb{Z}_2$ symmetry to prevent its decay into photons?

Question

Higgs boson, though electrically neutral, can decay to produce a pair of photons i.e., have electromagnetic interactions. See the answer here. Therefore, it appears that making dark matter electrically uncharged is not enough to prevent its decay into photons. For example, @marmot points out that axions can decay into photons. However, dark matter models usually also involve $\mathbb{Z}_2$ symmetries to prevent the decay of dark matter particles. Then how is it possible that the dark matter, like the axion, can still decay?

Answer 1

Both Higgs and (hypothetically) axions do not decay directly into photons, but indirectly through "virtual loops" involving other particles.

(Have a look also here.)

So, yes, if something doesn't carry electric charge then it cannot directly "touch" photons :)