  We present first theoretical predictions for various observables related to
the $\bar B_d \to \mu^+ \mu^- e^+ e^-$ decay within the framework of the
Standard Model. Our study encompasses the branching ratios, differential
distributions, and forward-backward leptonic asymmetries. To obtain these
predictions, we account for several contributions: resonance contribution from
$\rho(770)$ and significant contribution $\omega(782)$; contributions from four
charmonium resonances: $\psi(3770)$, $\psi(4040)$, $\psi(4160)$, and
$\psi(4415)$; additional contributions from the "tails" of $J/\psi$ and
$\psi(2S)$ resonances; non-resonant contributions arising from virtual photon
emission by a $b$-quark of $B_d$ meson, bremsstrahlung, and weak annihilation.
In our calculations we employ the model of vector meson dominance (VMD) to
assess the resonance contributions accurately. We provide the predictions for
the branching ratio of the $\bar B_d \to \mu^+ \mu^- e^+ e^-$ decay both with
and without the resonant contribution from $\omega(782)$.
