Rare Exclusive Decays of the Z-boson into S-wave Quarkonia within the Bethe-Salpeter Formalism

This paper investigates the rare decays of the Z boson into S-wave quarkonia within the Bethe-Salpeter formalism. Both the production of double S-wave quarkonia and radiative decays into single S-wave quarkonium are analyzed. For double quarkonia production, i.e., $Z\to VV$ and $Z\to VP$ (where $V$ and $P$ represent vector and pseudoscalar quarkonia, respectively), we consider the leading order contribution from both QCD as well as electromagnetic transition via virtual photon (QED) amplitudes. The heavy quark limit is adopted to simplify calculations. Additionally, we have introduced another possible leading order channel for Z-boson decays to double S-wave quarkonia, where the Z boson decays into bottomonium plus charmonium via QED amplitude. Such processes may include $Z\to J/\psi+\Upsilon(1S)$, $Z\to\eta_{b}+J/\psi$ and $Z\to\eta_{c}+\Upsilon(1S)$. Moreover, we have also studied radiative Z boson decays to S-wave quarkonium, namely $Z\to X(Q\bar{Q})\gamma$, where $X(Q\bar{Q})=J/\psi$, $\Upsilon$, $\eta_{c}$, and $\eta_{b}$. Interestingly, for double charmonium and radiative charmonium production, our results are larger than the NRQCD finding, while for the bottomonium case, our findings are comparatively smaller. This shows that charmonium is a relativistic particle, while bottomonium is a non-relativistic particle.