Theoretical analysis of $B^0 \to \phi \ell^+ \ell^-$ decay

Abstract

In the Standard Model (SM), the $b \to s\, (d)$ flavor-changing neutral currents, being loop-induced, are standard experimental channels for testing the SM precisely and searching for possible physics beyond the SM. Pure annihilation $B$-meson decays originating by these currents are of significant interest as they are extremely suppressed in the SM and New Physics effects can increase substantially their decay widths. Typical examples of these annihilation processes are radiative and semileptonic decays with $\rho^0$-, $\omega$-, and $\phi$-production like $B_s^0 \to \rho^0\, (\omega)\, \gamma$, $B_s^0 \to \rho^0\, (\omega)\, \ell^+ \ell^-$, $B^0 \to \phi\, \gamma$, and $B^0 \to \phi\, \ell^+ \ell^-$, where $\ell = e,\, \mu$ is the charged lepton. At the beginning of 2022, the LHCb Collaboration presented the upper limit on the $B^0 \to \phi \mu^+ \mu^-$ decay branching fraction ${\cal B}_{\rm exp} (B^0 \to \phi \mu^+ \mu^-) < 3.2 \times 10^{-9}$, and it is important to have a precise SM prediction for this decay. Here, we present theoretical predictions for $B^0 \to \phi\, \ell^+ \ell^-$ branching fraction in the lepton-pair invariant mass range $1~{\rm GeV}^2 < q^2 < 8~{\rm GeV}^2$, so far without taking into account $\omega - \phi$ mixing effect. The main goal is to study a dependence of the branching fraction on the choice of the $B$-meson distribution amplitude model. Theoretical prediction for the total branching fraction ${\cal B}_{\rm th} (B^0 \to \phi \ell^+ \ell^-) \sim 10^{-12}$, being an order of magnitude estimate, is far below the LHCb experimental limit.