Measurement of prompt charm production and the decay $D^{0}\rightarrow K^{+}\pi^{-}\pi^{+}\pi^{-}$ at LHCb

Prompt charm production in $pp$ collisions at a centre-of-mass energy of 7 TeV is measured with the decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$, using an integrated luminosity of 15.0 nb$ ^{-1}$ collected by the LHCb detector at CERN. The measurement is carried out in two-dimensional bins of $D^{0}$ rapidity and transverse momentum in the region $2.0 < y < 4.5$ and $0 < p_{\mathrm{T}} < 8\,\mathrm{GeV}/c$. The results are found to have excellent agreement with those measured using the decay $D^{0}\rightarrow K^{-}\pi^{+}$. With 1.0 fb $^{-1}$ of integrated luminosity, the time dependent ratio of $D^{0}\rightarrow K^{+}\pi^{-}\pi^{+}\pi^{-}$ to $D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$ decays is used to determine the ratio of doubly Cabibbo suppressed to Cabibbo favoured decay rates \begin{eqnarray*} r_{D}^{2}=\left(0.341\pm0.017\pm0.006\right)\% \end{eqnarray*} where the first uncertainty is statistical and the second is systematic. This is a previously unmeasured quantity, and can be used to measure the branching fraction of the doubly Cabibbo suppressed decay $D^{0}\rightarrow K^{+}\pi^{-}\pi^{+}\pi^{-}$ \begin{eqnarray*} \mathcal{B}\left(D^{0}\rightarrow K^{+}\pi^{-}\pi^{+}\pi^{-}\right) \mathrm{\, via \,\, DCS } = \left(2.75\pm0.14\pm0.05\pm0.07\right)\times10^{-4} \end{eqnarray*} where the final uncertainty is from the branching fraction of the Cabibbo favoured decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{-}\pi^{+}$. In addition, the no-mixing hypothesis is excluded with a probability that corresponds to 2.8 standard deviations, and the time-integrated mixing rate is measured as $R_{M}=\left( 0.04 \pm 0.88 \right)\times 10^{-4}$.