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

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## Abstract

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 \Delta m_s = (17.93 \pm 0.22 (stat) \pm 0.15 (syst)) ps^{-1}, \Delta m_d = (0.503 \pm 0.011 (stat) \pm 0.013 (syst)) ps^{-1}. 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.

## Figures and captions

 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. Fig1.pdf [65 KiB] HiDef png [156 KiB] Thumbnail [56 KiB] *.C file 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 $n$). Fig2.pdf [66 KiB] HiDef png [1 MiB] Thumbnail [183 KiB] *.C file 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. Fig3.pdf [74 KiB] HiDef png [294 KiB] Thumbnail [84 KiB] *.C file 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. Fig4.pdf [121 KiB] HiDef png [340 KiB] Thumbnail [118 KiB] *.C file 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. Fig5a-[..].pdf [100 KiB] HiDef png [368 KiB] Thumbnail [137 KiB] *.C file Fig5b-[..].pdf [99 KiB] HiDef png [370 KiB] Thumbnail [138 KiB] *.C file Fig5c-[..].pdf [132 KiB] HiDef png [403 KiB] Thumbnail [132 KiB] *.C file Fig5d-[..].pdf [128 KiB] HiDef png [369 KiB] Thumbnail [125 KiB] *.C file 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. Fig6a-[..].pdf [98 KiB] HiDef png [422 KiB] Thumbnail [147 KiB] *.C file Fig6b-[..].pdf [90 KiB] HiDef png [441 KiB] Thumbnail [148 KiB] *.C file Fig6c-[..].pdf [122 KiB] HiDef png [340 KiB] Thumbnail [117 KiB] *.C file Fig6d-[..].pdf [126 KiB] HiDef png [406 KiB] Thumbnail [132 KiB] *.C file 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. Fig7a-left.pdf [77 KiB] HiDef png [293 KiB] Thumbnail [125 KiB] *.C file Fig7b-[..].pdf [65 KiB] HiDef png [207 KiB] Thumbnail [93 KiB] *.C file 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. Fig8a-left.pdf [214 KiB] HiDef png [1 MiB] Thumbnail [190 KiB] *.C file Fig8b-[..].pdf [26 KiB] HiDef png [71 KiB] Thumbnail [14 KiB] *.C file Animated gif made out of all figures. PAPER-2013-036.gif Thumbnail

## Tables and captions

 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. Table_1.pdf [46 KiB] HiDef png [117 KiB] Thumbnail [16 KiB] tex code 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. Table_1.pdf [46 KiB] HiDef png [117 KiB] Thumbnail [16 KiB] tex code Sources of systematic uncertainty on $\Delta m_s$ and $\Delta m_d$. "Simulation" implies a combination of full LHCb simulation and pseudo-experiment studies. Table_2.pdf [54 KiB] HiDef png [76 KiB] Thumbnail [11 KiB] tex code 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. Table_1.pdf [46 KiB] HiDef png [117 KiB] Thumbnail [16 KiB] tex code Sources of systematic uncertainty on $\Delta m_s$ and $\Delta m_d$. "Simulation" implies a combination of full LHCb simulation and pseudo-experiment studies. Table_2.pdf [54 KiB] HiDef png [76 KiB] Thumbnail [11 KiB] tex code

## Supplementary Material [file]

 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. CDS-Su[..].pdf [1 MiB] HiDef png [147 KiB] Thumbnail [30 KiB] *C file Fig10a[..].pdf [94 KiB] HiDef png [295 KiB] Thumbnail [108 KiB] *C file Fig10b[..].pdf [95 KiB] HiDef png [308 KiB] Thumbnail [111 KiB] *C file Fig11a[..].pdf [91 KiB] HiDef png [282 KiB] Thumbnail [103 KiB] *C file Fig11b[..].pdf [90 KiB] HiDef png [271 KiB] Thumbnail [99 KiB] *C file Fig11c[..].pdf [91 KiB] HiDef png [272 KiB] Thumbnail [101 KiB] *C file Fig12-[..].pdf [67 KiB] HiDef png [297 KiB] Thumbnail [87 KiB] *C file Fig13-[..].pdf [311 KiB] HiDef png [2 MiB] Thumbnail [224 KiB] *C file Fig8a-[..].pdf [33 KiB] HiDef png [36 KiB] Thumbnail [7 KiB] *C file Fig8b-[..].pdf [33 KiB] HiDef png [35 KiB] Thumbnail [8 KiB] *C file

Created on 09 December 2018.Citation count from INSPIRE on 18 December 2018.