Observation of $B^0_s$-$\bar{B}^0_s$ mixing and measurement of mixing frequencies using semileptonic B decays

The $B^0_s$ and $B^0$ mixing frequencies, $\Delta m_s$ and $\Delta m_d$, are measured using a data sample corresponding to an integrated luminosity of 1.0 fb$^{-1}$ collected by the LHCb experiment in $pp$ collisions at a centre of mass energy of 7 TeV during 2011. Around 1.8x10$^{6}$ candidate events are selected of the type $B^0_{(s)} \to D^-_{(s)} \mu^+$(+anything), where about half are from peaking and combinatorial backgrounds. To determine the $B$ decay times, a correction is required for the momentum carried by missing particles, which is performed using a simulation-based statistical method. Associated production of muons or mesons allows us to tag the initial-state flavour and so to resolve oscillations due to mixing. We obtain \begin{align} &\Delta m_s = ( 17.93 \pm 0.22\,\textrm{(stat)} \pm 0.15\,\textrm{(syst)}) \,\textrm{ps}^{-1} \nonumber ,\\ &\Delta m_d = ( 0.503 \pm 0.011\,\textrm{(stat)} \pm 0.013\,\textrm{(syst)}) \,\textrm{ps}^{-1}\nonumber . \end{align} The hypothesis of no oscillations is rejected by the equivalent of 5.8 standard deviations for $B^0_s$ and 13.0 standard deviations for $B^0$. This is the first observation of $B^0_s$ mixing to be made using only semileptonic decays.