The $Z\to \gamma \gamma$ and $H\to \gamma \gamma \gamma$ decays are strictly forbidden in the Standard Model, but they can be induced by theories that violate Lorentz symmetry or the CPT theorem. By assuming that a nonvanishing $Z\gamma \gamma$ vertex is induced in some context of new physics, and by analyzing 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 the order of $10^{-7}$.

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