On the decays $B \to K^{(*)} + $ leptonium

We determine the rates of the $B$ meson decays into a $K^{(*)}$ and an $\ell^+ \ell^-$ bound state, the leptonium, where $\ell = e,\mu,\tau$. The two spin states of the leptonium, the spin singlet and the spin triplet, couple to the axial current and to the vector current, respectively, thus probing different helicity structures of the underlying $b\to s \ell^+\ell^-$ effective Hamiltonian. Since ortho- and para-leptonia have different decay modes, a distinction between the two is relatively easy and these decays may become a cross check for the results of lepton-flavour-violation searches obtained with free leptons. We find that some of the decays involving muon and tau have a branching ratio of the order of $10^{-13}$ and they may become accessible at the LHCb with 50 fb$^{-1}$ of integrated luminosity. In addition, since the tau-pair threshold lies right between the $J/\psi$ and the $\psi(2S)$ resonances, we estimate the charm-loop contribution to the decays $B \to K^{(*)} +$tauonium.