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The $Z\to \gamma \gamma$ and $H\to \gamma \gamma \gamma$ decays are strictlyforbidden in the Standard Model, but they can be induced by theories thatviolate Lorentz symmetry or the CPT theorem. By assuming that a nonvanishing$Z\gamma \gamma$ vertex is induced in some context of new physics, and byanalyzing the reaction $H\to \gamma Z^*\to \gamma \gamma \gamma$ in the $Z$resonance, we obtain an estimation for the branching ratio of the $H\to \gamma\gamma \gamma$ decay. Specifically, it is found that $BR(H\to \gamma \gamma\gamma)\lesssim BR(H\to \gamma Z)BR_{Exp}(Z\to \gamma \gamma)$, which is of theorder of $10^{-7}$.

Implications of a nonvanishing $Z \gamma \gamma$ vertex on the $H\to \gamma \gamma \gamma$ decay