Measurement of $B0\bar B0$ Oscillation Frequency and Calibration of Flavour Tagging with $B \rightarrow D\ell X$ Decays

Using $p\bar p$ collision data at $\sqrt{s} = 1.96$~TeV collected with the CDF\,II experiment at the Fermilab Tevatron collider, we measure the oscillation frequency $\Delta m_d$ of $B0 \bar B0$ mixing using partially reconstructed semileptonic decays $B \rightarrow D\ell X$. The charge of the lepton in the final state identifies the flavour of the $B0$~meson at its decay. The flavour of the $B0$ meson at production is inferred by an opposite side lepton or jet charge tag. This study also serves as calibration of the flavour tagging algorithms used in a search for $B0_s$~mixing using semileptonic $B0_s \rightarrow D^+_s\ell X$ decays.