Mass distributions for all selected signal candidates. Left, the $K^+K^-\pi^+$ invariant mass, where the known mass of the $D^+_s$ has been subtracted. Right, the $D\mu$ normalized mass as defined in Eq. 2. Neutral candidates are those of the form $D^{\mp}\mu^\pm$, while double-charged candidates are those of the form $D^{\pm}\mu^\pm$. The double-charged candidates arise from several background sources, most of which are also present in the neutral sample. In the left plot, the neutral sample exhibits much larger $D$ mass peaks, indicative of the large $B$ signal component.

Input to obtain the $k$-factor correction from the fully-simulated $B^0_s$ sample. For each event the ratio of reconstructed to generated momentum, $p_{\textrm{rec}}/p_{\textrm{sim}}$ is plotted against the normalized $D\mu$ mass ($n$ in Eq. 2). The curve shows a fourth-order polynomial resulting from a fit to the mean of the distribution (in bins of $n$).

Illustration of the decay time resolution obtained from a fully simulated $B^0$ signal sample. The left plots demonstrate the Gaussian fits (solid lines) using the full LHCb simulated data (filled), to determine the decay time resolution. Each measured (reconstructed and corrected) time, $t'$, is compared to the corresponding simulated decay time, $t$. The results are shown for several bins of $t'$. The dependence on decay time of the mean (bias, $\mu$) and width (standard deviation, $\sigma$) can be fitted with a quadratic or cubic function of either $t$ or $t'$. The right hand plot shows a quadratic fit to the widths.

Distribution of measured $K^+K^-\pi^+$ mass, where the known mass of the $D^+_s$ has been subtracted. Black points show the data, and the various lines overlay the result of the fit. The small step at $-50$ MeV$c^{-2}$ is the result of differences in tagging efficiency for the $B^0_s$ and $B^0$ hypotheses.

Measured $B$ decay-time distribution, overlaid with projections of the fit, for background-only regions. Top left: a region between the two signal peaks, $-80$ to $-20$ MeV$c^{-2}$ (with respect to the known mass of the $D_s^+$), showing only low decay times. Top right: a region to the right of the signal peaks $20$ to $100$ MeV$c^{-2}$, showing only low decay times. Bottom row: the same on an extended decay-time scale and logarithmic. The legend is the same as in Fig. 4.

Measured $B$ decay-time distribution, overlaid with projections of the fit, for signal regions. Top left: for odd-tags, high-$n$ and a region of $\pm 20$ MeV$c^{-2}$ around the $D^+_s$ mass peak, showing only low decay times, where $B^0_s$ oscillations can be clearly seen. Top right: for odd-tags and all $n$ for a region of $\pm 20$ MeV$c^{-2}$ around the $D^+$ mass peak, showing only low decay times. Bottom row: for both tags and all $n$ for regions of $\pm 20$ MeV$c^{-2}$ around the $D^+_s$ (left) and $D^+$ (right) mass peaks. The legend is the same as in Fig. 4.

Tagged (mixing) asymmetry, $(N_+-N_-)/(N_++N_-)$, as a function of $B$ decay time. The left plot shows the asymmetry for events for a region of $\pm 20$ MeV$c^{-2}$ around the $D^+_s$ mass peak, and the right plot shows the corresponding asymmetry around the $D^+$ mass peak. The black points show the data and the curves are projections of the fitted PDF. On the left plot the fast oscillations of $B^0_s$ are gradually washed out by the increasingly poor decay-time resolution.

Result of using Fourier transforms to search for the $\Delta m_s$-peak. The image on the left is constructed from bins of the $K^+K^-\pi^+$ mass which are 25 MeV$c^{-2}$ in width, analysed in steps of 5 MeV$c^{-2}$ such that a smooth image is produced. The colour scale (blue-green-yellow-red) is an arbitrary linear representation of the signal intensity; dark blue is used for zero and below. The vertical dashed line is drawn at $18.0$ ps$^{-1}$. The apparent double-peak structure is an artifact of this image. On the right a slice around the $D^+_s$ mass region shows only the peak as used to measure the central value and rms width.

Animated gif made out of all figures.

A selection of fitted parameter values, for which statistical uncertainties only are given. The $B^0_s$ signal fraction includes contributions from any detached $D^+_s$ production. When the omitted fractions (of combinatorial background components) are included, the total fraction sums to unity within each $n$ region separately.

Sources of systematic uncertainty on $\Delta m_s$ and $\Delta m_d$. "Simulation" implies a combination of full LHCb simulation and pseudo-experiment studies.

See CDS-Supplementary.pdf and 6.supplementary.tex Supplementary material for LHCb-PAPER-2013-036. Figures are contained in the figs subdirectory, named after their appearance in the document. PNG, SVG, PDF and EPS are supplied where possible. For the one animated figure, PDN, GIF, and individual PNGs are given.