Decays Ztoγγ and Zto gg in the Standard Model Extension

The $Z\to \gamma \gamma$ and $Z\to gg$ decays are studied in the context of the renormalizable version of the Standard Model Extension. The $CPT$-odd $\bar{\psi}\gamma_5 \gamma^\mu b_\mu\psi$ bilinear interaction, which involves the constant background field $b_\alpha$ and which has been a subject of interest in literature, is considered. It is shown that the $Z\to \gamma \gamma$ and $Z\to gg$ decays, which are strictly zero in the standard model, can be generated radiatively at the one-loop level. It is found that these decays are gauge invariant and free of ultraviolet divergences, and that the corresponding decay widths only depend on the spatial component of the background field $b$.