Measurement of CP asymmetry in B 0 s → D ∓ s K ± decays

: We report on measurements of the time-dependent CP violating observables in B 0 s → D ∓ s K ± decays using a dataset corresponding to 1 . 0 fb − 1 of pp collisions recorded with the LHCb detector. We ﬁnd the CP violating observables C f = 0 . 53 ± 0 . 25 ± 0 . 04, A ∆Γ f S ¯ f = − 0 . 36 ± 0 . 34 ± 0 . 08, where the uncertainties are statistical and systematic, respec-tively. Using these observables together with a recent measurement of the B 0 s mixing phase − 2 β s leads to the ﬁrst extraction of the CKM angle γ from B 0 s → D ∓ s K ± decays, ﬁnding γ = (115 +28 − 43 ) ◦ modulo 180 ◦ at 68% CL, where the error contains both statistical and systematic uncertainties.


Introduction
A key characteristic of the Standard Model (SM) is that CP violation originates from a single phase in the CKM quark-mixing matrix [1,2]. In the SM the CKM matrix is unitary, leading to the condition V ud V * ub + V cd V * cb + V td V tb = 0, where V ij are the CKM matrix elements. This relation is represented as a triangle in the complex plane, with angles α, β and γ, and an area proportional to the amount of CP violation in the quark sector of the SM [3][4][5]. The angle γ ≡ arg(−V ud V * ub /V cd V * cb ) is the least well-known angle of the CKM angles. Its current best determination was obtained by LHCb from a combination of measurements concerning B + , B 0 and B 0 s decays to final states with a D (s) meson and one or more light mesons [6]. Decay-time-dependent analyses of tree-level B 0 (s) → D ∓ (s) h ± (h = π, K) decays 1 are sensitive to the angle γ through CP violation in the interference of mixing and decay amplitudes [7][8][9][10]. A comparison between the value of the CKM angle γ obtained from tree-level processes, with the measurements of γ and other unitary triangle parameters in loop-level processes, provides a powerful consistency check of the SM picture of CP violation.
Due to the interference between mixing and decay amplitudes, the physical CPviolating parameters in these decays are functions of a combination of the angle γ and the relevant mixing phase, namely γ + 2β (β ≡ arg(−V cd V * cb /V td V * tb )) in the B 0 and γ − 2β s (β s ≡ arg(−V ts V * tb /V cs V * cb )) in the B 0 s system. Measurements of these physical quantities can therefore be interpreted in terms of the angles γ or β (s) by using independent determinations of the other parameter as input. Such measurements have been performed by both the BaBar [11,12] and Belle [13,14] collaborations using B 0 → D ( * )∓ π ± decays. In these decays, the ratios between the interfering b → u and b → c amplitudes are small, r D ( * ) π = |A(B 0 → D ( * )− π + )/A(B 0 → D ( * )+ π − )| ≈ 0.02, which limits the sensitivity to the CKM angle γ [15].
The leading-order Feynman diagrams contributing to the interference of decay and mixing in B 0 s → D ∓ s K ± decays are shown in Fig. 1. In contrast to B 0 → D ( * )∓ π ± decays, here both the B 0 s → D − s K + (b → csū) and the B 0 s → D + s K − (b → ucs) decay amplitudes are of O(λ 3 ), where λ ≈ 0.23 [16,17] is the sine of the Cabibbo angle, and the ratio of the amplitudes of the interfering diagrams is approximately |V * ub V cs /V cb V * us | ≈ 0.4. Moreover, the sizeable decay-width difference in the B 0 s system, ∆Γ s [18], allows the determination of γ − 2β s from the sinusoidal and hyperbolic terms of the decay-time evolution (see Eqs. 1 and 2) up to a two-fold ambiguity.
This paper presents an updated measurement with respect to Ref.
[19] of the CPviolating parameters and of γ − 2β s in B 0 s → D ∓ s K ± decays using a data set corresponding to an integrated luminosity of 1

Decay rate equations and CP violation parameters
The time-dependent-decay rates of the initially produced flavour eigenstates |B 0 s (t = 0) and |B 0 s (t = 0) are given by where λ f ≡ (q/p)(A f /A f ) and A f (A f ) is the amplitude of a B 0 s (B 0 s ) decay to the final state f , Γ s corresponds to the average B 0 s decay width, while ∆Γ s indicates the decaywidth difference between the light, |B L , and heavy, |B H , B 0 s mass eigenstates, defined as Γ B L − Γ B H and ∆m s is the mixing frequency in the B 0 s system defined as m B H − m B L . The complex coefficients p and q relate the B 0 s meson mass eigenstates, to the flavour eigenstates, where with |p| 2 + |q| 2 = 1. Equations similar to 1 and 2 can be written for the decays to the The CP -asymmetry parameters are given by The equality C f = −C f results from |q/p| = 1 and |λ f | = |1/λ f |, i.e. assuming no CP violation in either the mixing, in agreement with current measurements [20], or in the decay amplitude, which is justified as only a single amplitude contributes to each initial to final state transition. The CP parameters are related to the magnitude of the amplitude ratio r DsK ≡ |λ , and the weak-phase difference γ − 2β s by the following equations

Analysis strategy
The analysis strategy consists of a two-stage procedure. After the event selection, an unbinned extended maximum likelihood fit, referred to as the multivariate fit, is performed to separate signal B 0 s → D ∓ s K ± candidates from background contributions. The multivariate fit uses the B 0 s and D − s invariant masses and the log-likelihood difference between the pion and kaon hypotheses, L(K/π), for the K ± candidate. Using information from this fit, signal weights for each candidate are obtained using the sPlot technique [21]. At the second stage, the CP violation parameters are measured from a fit to the weighted decay-time distribution, referred to as the sFit [22] procedure, where the initial flavour of the B 0 s candidate is inferred by means of several flavour-tagging algorithms optimised using data and simulation samples. The full procedure is validated using the flavour-specific B 0 s → D − s π + decay, yielding approximately 16 times more signal than B 0 s → D ∓ s K ± decays. Precise determination of the decay-time resolution model and of the decay-time acceptance, as well as the calibration of the flavour-tagging algorithms, are obtained from B 0 s → D − s π + decays and subsequently used in the sFit procedure to the B 0 s → D ∓ s K ± candidates. The analysis strategy largely follows that described in Ref. [19]. Most of the inputs are updated, in particular the candidate selection, the flavour tagging calibration and the decay-time resolution are optimised on the current data and simulation samples. A more refined estimate of the systematic uncertainties is also performed. After a brief description of the LHCb detector in Sec. 2, the event selection is reported in Sec. 3. The relevant inputs for the multivariate fit and its results for B 0 s → D ∓ s K ± and B 0 s → D − s π + decays are outlined in Secs. 4. The flavour-tagging parameters and the decay-time resolution model are described in Secs. 5 and 6, respectively. The decay-time acceptance is reported in Sec. 7 followed by the results of the sFit procedure applied to B 0 s → D ∓ s K ± candidates in Sec. 8. The evaluation of the systematic uncertainties and the interpretation for the CKM angle γ are summarised in Secs. 9 and 10, respectively. Conclusions are drawn in Sec. 11.

Detector and software
The LHCb detector [23,24] is a single-arm forward spectrometer covering the pseudorapidity range 2 < η < 5, designed for the study of particles containing b or c quarks. The detector includes a high-precision tracking system consisting of a siliconstrip vertex detector surrounding the pp interaction region [25], a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm, and three stations of silicon-strip detectors and straw drift tubes [26] placed downstream of the magnet. The polarity of the dipole magnet is reversed periodically throughout data taking to control systematic effects. The tracking system provides a measurement of momentum, p, of charged particles with a relative uncertainty that varies from 0.5% at low momentum to 1.0% at 200 GeV/c. The minimum distance of a track to a primary vertex (PV), the impact parameter (IP), is measured with a resolution of (15 + 29/p T ) µm, where p T is the component of the momentum transverse to the beam, in GeV/c. Particle identification (PID) of charged hadrons is achieved using information from two ring-imaging Cherenkov detectors [27].
The online event selection is performed by a trigger [28], which consists of a hardware stage, based on information from the calorimeters and muon systems, followed by a software stage, which applies a full event reconstruction. At the hardware trigger stage, events are required to have a muon with high p T or a hadron, photon or electron with high transverse energy in the calorimeters. For hadrons, the transverse energy threshold is 3.5 GeV. The software trigger requires a two-, three-or four-track secondary vertex with a significant displacement from any primary pp interaction vertex. At least one charged particle must have a transverse momentum p T > 1.6 GeV/c and be inconsistent with originating from any PV. A multivariate algorithm [29] is used for the identification of secondary vertices consistent with the decay of a b hadron.
In the simulation, pp collisions are generated using Pythia [30,31] with a specific LHCb configuration [32]. Decays of hadronic particles are described by EvtGen [33], in which final-state radiation is generated using Photos [34]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [35] as described in Ref. [36].

Candidate selection
First, D − s → K − K + π − , D − s → K − π + π − , and D − s → π − π + π − candidates are formed from reconstructed charged particles. These D − s candidates are subsequently combined with a fourth particle, referred to as the "companion", to form B 0 s → D ∓ s K ± or B 0 s → D − s π + candidates, depending on the PID information of the companion particle. The decay-time resolution is improved by performing a kinematic fit [37] in which the B 0 s candidate is assigned to a PV for which it has the smallest impact parameter χ 2 , defined as the difference in the χ 2 of the vertex fit for a given PV reconstructed with and without the considered particle. Similarly, the B 0 s invariant mass resolution is improved by constraining the D − s invariant mass to its world-average value. A selection of reconstructed candidates is made using a similar multivariate secondaryvertex algorithm as that applied at the trigger level, but with offline-quality reconstruction [29]. Combinatorial background is further suppressed by a gradient boosted decision tree (BDTG) algorithm [38,39], which is trained on B 0 s → D − s π + data. Only the D − s → K − K + π − final state selected with additional PID requirements is considered in order to enrich the training sample with signal candidates. Since all channels in this analysis have similar kinematics, and no PID information is used as input to the BDTG, the resulting BDTG performs equally well on the other D − s decay modes. The optimal working point is chosen to maximise the significance of the B 0 s → D ∓ s K ± signal.
In addition, the B 0 s and D − s candidates are required to have a measured mass within [5300, 5800] MeV/c 2 and [1930,2015] MeV/c 2 , respectively.
Finally, a combination of PID information and kinematic vetoes is used to distinguish the different D − s final states from each other (D − s → K − π + π − , D − s → π − π + π − and D − s → K − K + π − , the latter being subdivided into D − s → φπ − , D − s → K * (892) 0 K − and D − s → (KKπ) nonres ) and from cross-feed backgrounds such as The selection structure and most criteria are identical to those used in Ref. [19]; the specific values of certain PID selection requirements were updated to perform optimally with the latest event reconstruction algorithms. Less than 1% of the events passing the selection requirements contain more than one signal candidate. All candidates are used in the analysis.
The signal and background probability density functions (PDFs) for the multivariate fit are obtained using a mixture of data-driven approaches and simulation. The simulated events are corrected for differences in the transverse momentum and event occupancy distributions between simulation and data, as well as for the kinematics-dependent efficiency of the PID selection requirements. The shape of the B 0 s invariant mass distribution for signal candidates is modelled using the sum of two Crystal Ball functions with a common mean [40]. This choice of functions provides a good description of the main peak as well as the radiative tail and reconstruction effects. The signal PDFs are determined separately for the B 0 s → D ∓ s K ± and B 0 s → D − s π + decays from simulation, taking into account different D − s final states. The shapes are fixed in the nominal fit with two exceptions. The common mean of the Crystal Ball functions is left free for both B 0 s → D − s π + and B 0 s → D ∓ s K ± , compensating for differences in the mass reconstruction between simulation and data. A scale factor accounting for data-simulation differences in the signal width is left free in the B 0 s → D − s π + fit and is subsequently fixed to its measured value in the fit to the B 0 s → D ∓ s K ± sample. The functional form of the combinatorial background is taken from the B 0 s invariant mass sideband (above 5800 MeV/c 2 ), with all parameters left free to vary in the multivariate fit. It is parametrised separately for each D − s mode either by an exponential function or by the sum of an exponential function and a constant offset. The shapes of the fully or partially reconstructed backgrounds are fixed from simulated events, corrected to reproduce the PID efficiency and kinematics in data, using a nonparametric kernel estimation method (KEYS) [41]. An exception is background due to B 0 mesons decaying to the same final state as signal, which is parametrised by the signal PDF shifted by the known B 0 -B 0 s mass difference.
The D − s invariant mass is also described by a sum of two Crystal Ball functions with a common mean. The signal PDFs are obtained from simulation separately for each D −    The fully and partially reconstructed backgrounds that contain a correctly reconstructed ) are assumed to have the same D − s invariant mass distribution as the signal. The shapes of the other backgrounds are KEYS templates taken from simulation.
The PDFs describing the L(K/π) distributions of pions, kaons and protons are obtained from dedicated data-driven calibration samples [42]. The L(K/π) shape of the companion track for the signal is obtained separately for each D − s decay mode to account for small kinematic differences between them. For the combinatorial background, the L(K/π) PDF is determined from a mixture of pion, proton, and kaon contributions, and its normalisation is left free in the multivariate fit. For fully or partially reconstructed backgrounds the L(K/π) PDF is obtained by weighting the PID calibration samples to match the event distributions of simulated events, separately for each background type.
The multivariate fit is performed simultaneously to the different D − s decay modes. For    each D − s decay mode the PDF is built from the sum of signal and background contributions. Each contribution consists of the product of three PDFs corresponding to the B 0 s and D s invariant masses and L(K/π), since their correlations are measured to be small in simulation. A systematic uncertainty is assigned to account for the impact of residual correlations.
Almost all background yields are left free to vary in the fit, except those that have an expected contribution below 2% of the signal yield, namely: Such background yields are fixed from known branching fractions and relative efficiencies measured using simulation.
The multivariate fit results in total signal yields of 96 942 ± 345 and 5955 ± 90 B 0 s → D − s π + and B 0 s → D ∓ s K ± signal candidates, respectively. Signal yields are increased by a factor of 3.4 with respect to the previous measurement [19], while the combinatorial background contribution is significantly reduced. The multivariate fit is found to be unbi-ased using large samples of data-like pseudoexperiments. The results of the multivariate fit are shown in Figs. 2 and 3 for the B 0 s → D − s π + and the B 0 s → D ∓ s K ± candidates, respectively, summed over all D − s decay modes.

Flavour tagging
The identification of the B 0 s initial flavour is performed by means of different flavourtagging algorithms. The same-side kaon (SS) tagger [43] searches for an additional charged kaon accompanying the fragmentation of the signal B 0 s or B 0 s . The opposite-side (OS) taggers [44] exploit the pair-wise production of b quarks that leads to a second b-hadron alongside the signal B 0 s . The flavour of the nonsignal b hadron is determined using the charge of the lepton (µ, e) produced in semileptonic B decays, or that of the kaon from the b → c → s decay chain, or the charge of the inclusive secondary vertex reconstructed from b-decay products. The different OS taggers are combined and used in this analysis.
Each of these algorithms has an intrinsic mistag rate ω = (wrong tags)/(all tags), for example due to selecting tracks from the underlying event, particle misidentifications, or flavour oscillations of neutral B mesons on the opposite side. The statistical precision of the CP -violating parameters that can be measured in B 0 s → D ∓ s K ± decays scales as the inverse square root of the effective tagging efficiency ε eff = ε tag (1 − 2ω) 2 , where ε tag is the fraction of signal having a tagging decision.
The tagging algorithms are optimised to obtain the highest possible value of ε eff on data. For each signal B 0 s candidate the tagging algorithms predict a mistag probability η through the combination of various inputs, such as kinematic variables of tagging particles and of the B 0 s candidate, into neural networks. The neural networks are trained on simulated samples of B 0 s → D − s π + decays for the SS tagger and on data samples of B + → J/ψK + decays for the OS taggers. For each tagger, the predicted mistag probability, η, is calibrated to match the mistag rate, ω, measured in data by using flavour-specific decays. A linear model is used as a calibration function, where the values of the parameters p 0 and p 1 are measured using the B 0 s → D − s π + decay mode and η is fixed to the mean of the estimated mistag probability η. For a perfectly calibrated tagger one expects p 1 = 1 and p 0 = η . The tagging calibration parameters depend on the B 0 s initial flavour, mainly due to the different interaction cross-sections of K + and K − mesons with matter. Therefore, the measured B 0 s -B 0 s tagging asymmetry is taken into account by introducing additional ∆p 0 , ∆p 1 and ∆ε tag parameters, which are defined as the difference of the corresponding B 0 s and B 0 s values. The calibrated mistag is treated as a per-candidate variable, thus adding an observable to the fit. The compatibility between the calibrations in B 0 s → D − s π + and B 0 s → D ∓ s K ± decays is verified using simulation.
The flavour-specific B 0 s → D − s π + decay mode is used for tagging calibration in order to minimize the systematic uncertainties due to the portability of the calibration from a different control channel to the signal one. The measured values of the OS and SS tagging calibration parameters and tagging asymmetries in the B 0 s → D − s π + sample are summarised in Table 1. They are obtained from a fit to the decay-time distribution of the B 0 s → D − s π + sample in which the background is statistically subtracted by weighting the  Systematic uncertainties on the calibration parameters have an impact on the CP parameters and they are added in quadrature with the statistical uncertainties and used to define the Gaussian constraints on the calibration parameters in the B 0 s → D ∓ s K ± fit. The largest systematic effect on the tagging calibration parameters is due to the decay-time resolution model, which also affects the B 0 s → D ∓ s K ± fit for CP observables. In order to avoid double counting, this source of systematic uncertainty is treated separately from the other systematic sources (see Sec. 9). Other relevant sources of systematic uncertainties are related to the calibration method and to the background description in the multivariate fit used to compute the weights for the sFit procedure. Uncertainties related to the decay-time acceptance and to the fixed values of ∆m s and ∆Γ s in the sFit procedure are found to be negligible. The total systematic uncertainties, reported in Table 1, are significantly smaller than the statistical.
The OS and SS tagging decisions and the mistag predictions are combined in the fit to the B 0 s → D ∓ s K ± decay-time distribution by using the same approach as described in Ref. [45]. The tagging performances for the OS and SS combination measured in the B 0 s → D − s π + channel are reported in Table 2. Three categories of tagged events are considered: OS only, SS only and both OS and SS. The estimated value of the effective tagging efficiency ε eff for the B 0 s → D ∓ s K ± decay mode is (5.7 ± 0.3)%, consistent with the value obtained for B 0 s → D − s π + decays, as expected.

Decay-time resolution
Due to the fast B 0 s -B 0 s oscillations, the CP -violation parameters related to the amplitudes of the sine and cosine terms are highly correlated to the decay-time resolution model. The signal decay-time PDF is convolved with a Gaussian resolution function that has a different width for each candidate, making use of the per-candidate decay-time uncertainty estimated from the kinematic fit of the B 0 s vertex. From the comparison to the measured decay-time resolution, a correction to the per-candidate decay-time uncertainty σ t is determined. This calibration is performed from a sample of "fake B 0 s " candidates with a known lifetime of zero obtained from the combination of prompt D − s mesons with a random track that originated from the PV. The spread of the observed decay times follows the shape of a double Gaussian distribution, where only the negative decay times are used to determine the resolution, to avoid biases in the determination of the decay-time resolution due to long-lived backgrounds. The resulting two widths are combined to calculate the corresponding dilution: where σ 1,2 are the widths, and f 1 and (1 − f 1 ) are the fractions of the two Gaussian components. The dilution, which represents the amplitude damping of the decay-time distribution, is used to obtain the effective decay-time resolution σ = (−2π/∆m 2 s ) ln(D). The effective decay-time resolution depends on the per-candidate decay-time uncertainty as σ(σ t ) = 1.28 σ t + 10.3 fs, and is shown in Fig. 4. The uncertainty on the decay-time resolution is dominated by the uncertainty on the modelling of the observed decay times of the "fake B 0 s " candidates. Modelling the spread by a single Gaussian distribution or by taking only the central Gaussian from the double Gaussian fit, results in the correction factors σ(σ t ) = 1.77 σ t and σ(σ t ) = 1.24 σ t , respectively, which are used to estimate the systematic uncertainty on the measured CP parameters.
The assumption that the measured decay-time resolution on "fake B 0 s " candidates can be used for true B 0 s candidates is justified, as the measured decay-time resolution does not significantly depend on the transverse momentum of the companion particle, which is the main kinematic difference between the samples. In addition, simulation shows that the "fake B 0 s " and signal B 0 s samples require compatible correction factors, varying in the range [1.19, 1.27].

Decay-time acceptance
The decay-time acceptance of B 0 s → D ∓ s K ± candidates is strongly correlated with the CP parameters, in particular with A ∆Γ f and A ∆Γ f . However, in the case of the flavourspecific B 0 s → D − s π + decays, the acceptance can be measured by fixing Γ s and floating the acceptance parameters. The decay-time acceptance in the B 0 s → D ∓ s K ± fit is fixed to that found in the fit to B 0 s → D − s π + data, corrected by the acceptance ratio in the two channels obtained from simulation, which is weighted as described in Sec. 4. In all cases, the acceptance is described using segments of cubic b-splines, which are implemented in an analytic way in the decay-time fit [46]. The spline boundaries, knots, are chosen in order to model reliably the features of the acceptance shape, and are placed at 0.5, 1.0, 1.5, 2.0, 3.0 and 12.0 ps. In the sFit procedure applied to the sample of B 0 s → D − s π + candidates, the CP -violation parameter C f is fixed to unity with are all fixed to zero. The spline parameters and ∆m s are free to vary. The result of the sFit procedure applied to the B 0 s → D − s π + candidates is shown in Fig. 5. Extensive studies with simulation have been performed and confirm the validity of the method. An alternative analytical decay-time acceptance parametrisation has been considered, and is in good agreement with the nominal spline description. Finally, doubling the number of knots results in negligible changes in the fit result.
In the sFit procedure applied to the B 0 s → D ∓ s K ± candidates, the following parameters ∆m s = (17.757 ± 0.021) ps −1 , are fixed to their central values. The values of B 0 s oscillation frequency and production asymmetry, A prod , are based on LHCb measurements [47,48]. The B 0 s decay width, Γ s , the decay-width difference, ∆Γ s , and their correlation, ρ(Γ s , ∆Γ s ), correspond to the HFLAV [15] world average. An estimate of the detection asymmetry A det based on Ref.
[49] is considered. The production asymmetry is defined as where σ denotes the production cross-section inside the LHCb acceptance. The detection asymmetry is defined as the difference in reconstruction efficiency between the D − s K + and the D + s K − final states. The detection and the production asymmetries contribute to the PDF with factors of (1 ± A prod ) and (1 ± A det ), depending on the tagged initial state and the reconstructed final state, respectively. The tagging calibration parameters and asymmetries are allowed to float within Gaussian constraints based on their statistical and systematic uncertainties given in Sec. 5. The decay-time PDF is convolved with a single Gaussian representing the per-candidate decay-time resolution, and multiplied by the decay-time acceptance described in Sec. 6 and Sec. 7, respectively.
The measured CP -violating parameters are given in Table 3, and the correlations of their statistical uncertainties are given in Table 4. The fit to the decay-time distribution is shown in Fig. 6. together with the two decay-time-dependent asymmetries, A mix (D + s K − ) and A mix (D − s K + ), that are defined as the difference of the decay rates (see Eqs. 1 and 2) of the tagged candidates. The asymmetries are obtained by folding the decay time in one mixing period 2π/∆m s . The central values of the CP parameters measured by the fit are used to determine the plotted asymmetries.

Systematic uncertainties
Systematic uncertainties arise from the fixed parameters ∆m s , Γ s , ∆Γ s , the detection A det and tagging efficiency ∆ε tag asymmetries, and from the limited knowledge of the decay-time resolution and acceptance. In addition, the impact of neglecting correlations among the observables for background candidates is estimated. Table 5 summarises the different contributions to the systematic uncertainties, which are detailed below. The systematic uncertainties are estimated using large sets of pseudoexperiments, in which the relevant parameters are varied. The pseudoexperiments are generated with central values of the CP parameters reported in Sec. 8. They are subsequently processed by the same fit procedure applied to data. The fitted values are compared between the nominal fit, where all fixed parameters are kept at their nominal values, and the systematic fit, where each parameter is varied according to its uncertainty. A distribution is formed by normalising the resulting differences to the uncertainties measured in the nominal fit, and the mean and width of this distribution are added in quadrature and assigned as the systematic uncertainty.
The systematic uncertainty related to the decay-time resolution model, together with its impact on the flavour tagging, is evaluated by fitting the B 0 s → D ∓ s K ± pseudoexperiments using the two alternative decay-time resolution models and their corresponding tagging calibration parameters. The latter are obtained with B 0 s → D − s π + pseudoexperiments that were generated with the nominal decay-time resolution, but fitted with the two alternative decay-time resolution models. The impact of neglecting the correlations Table 5: Systematic uncertainties on the CP parameters, relative to the statistical uncertainties.

Source
C among the observables in the background is accounted for by means of a dedicated set of pseudoexperiments in which the correlations are included at generation and neglected in the fit. The correlations between Γ s , ∆Γ s , and the decay-time acceptance parameters from the fit to B 0 s → D − s π + data are accounted for by fitting pseudoexperiments, where the values of the spline coefficients, Γ s and ∆Γ s are randomly generated according to multidimensional correlated Gaussian distributions centred at the nominal values. The combined correlated systematic uncertainty is listed as "acceptance data fit, Γ s , ∆Γ s ". The correlations between the spline coefficients among B 0 s → D − s π + and B 0 s → D ∓ s K ± simulation samples are accounted for by fitting pseudoexperiments with the parameters randomly generated as in the previous case, and the corresponding systematic uncertainty is listed as "acceptance, simulation ratio".
The nominal result is cross-checked by splitting the sample into subsets according to the two magnet polarities, the year of data taking, the B 0 s momentum, and the BDTG response. No dependencies are observed. In particular, the compatibility of the 1 fb −1 and the 2 fb −1 subsamples is at the level of 1 σ, where σ is the standard deviation. A closure test using the high-statistics fully simulated signal candidates provides an estimate of the intrinsic uncertainty related to the fit procedure. No bias is found and only the fit uncertainty is considered as a systematic uncertainty. The systematic effects due to the background subtraction in the sFit procedure are checked. Therefore, the nominal fitting procedure is applied to a mixture of the signal and the B 0 s → D − s π + simulation samples as well as combinatorial background data. The result is consistent with the values found by the fit to the signal only, as a consequence, no additional uncertainties are considered.
The resulting systematic uncertainties are shown in Table 5 relative to the corresponding statistical uncertainties. The total systematic correlation matrix, reported in Table 6, is obtained by adding the covariance matrices corresponding to each source.
A number of other possible systematic effects are studied, but found to be negligible. These include production asymmetries, missing or imperfectly modelled backgrounds, and fixed signal-shape parameters in the multivariate fit. Potential systematic effects due to fixed background yields are evaluated by generating pseudoexperiments with the nominal value for these yields, and fitting back with the yields fixed to twice or half their nominal value. No significant bias is observed and no systematic uncertainty assigned. The decay-time fit is repeated adding one or two additional spline functions to the decay-time acceptance description and no significant change in the fit result is observed. The multivariate and decay-time fits are repeated randomly removing multiple candidates, with no significant change observed in the fit result. No systematic uncertainty is assigned to the imperfect knowledge of the momentum and the longitudinal dimension of the detector since both effects are taken into account by the systematic uncertainty on ∆m s , as the world average is dominated by the LHCb measurement [47].

Interpretation
The measurement of the CP parameters is used to determine the values of γ − 2β s and, subsequently, of the angle γ. The following likelihood is maximised, replicating the procedure described in Ref. [6], where α = (γ, β s , r DsK , δ) is the vector of the physics parameters, A( α) is the vector of parameters expressed through Eq. 5, A obs is the vector of the measured CPviolating parameters and V is the experimental (statistical and systematic) uncertainty covariance matrix. Confidence intervals are computed by evaluating the test statistic ∆χ 2 ≡ χ 2 ( α min ) − χ 2 ( α min ), where χ 2 ( α) = −2 ln L( α), following Ref. [50]. Here, α min denotes the global maximum of Eq. 8, and α min is the conditional maximum when the parameter of interest is fixed to the tested value. The value of β s is constrained to the value obtained from [15], φ s = −0.030 ± 0.033 rad, assuming φ s = −2β s , i.e. neglecting contributions from penguin-loop diagrams or from processes beyond the SM. The results are where the values for the angles are expressed modulo 180 • . Figure 7 shows the 1 − CL curve for γ, and the two-dimensional contours of the profile likelihood L( α min ). The resulting value of γ is visualised in Fig. 7 by inspecting the complex plane for the measured amplitude coefficients. The points determined by (−A ∆Γ f , S f ) and (−A ∆Γ f , S f ) are proportional to r DsK e i(±δ−(γ−2βs)) , whilst an additional constraint on r DsK arises from C f . The value of γ measured in this analysis is compatible at the level of 2.3 σ, where σ is the standard deviation, with the value of γ found from the combination of all LHCb measurements [6] when all information from B 0 s → D ∓ s K ± decays is removed. The observed change in the fit log-likelihood between the combined best fit point and the origin in the complex plane indicates 3.8 σ evidence for CP violation in B 0 s → D ∓ s K ± .

Conclusion
The CP -violating parameters that describe the B 0 s → D ∓ s K ± decay rates have been measured using a data set corresponding to an integrated luminosity of 3.0 fb −1 of pp collisions recorded with the LHCb detector. Their values are found to be C f = 0.73 ± 0.14 ± 0.05 , where the first uncertainties are statistical and the second are systematic. The results are used to determine the CKM angle γ, the strong-phase difference δ and the amplitude ratio r DsK between the B 0 s → D − s K + and B 0 s → D − s K + amplitudes leading to γ = (128 +17    [9] R. Fleischer, New strategies to obtain insights into CP violation through B (s) → D ± (s) K ∓ , D * ± (s) K ∓ , ... and B (d) → D ± π ∓ , D * ± π ∓ , ... decays, Nucl. Phys. B671 (2003) 459, arXiv:hep-ph/0304027.   [49] LHCb collaboration, R. Aaij et al., Measurement of CP asymmetry in D 0 → K − K + and D 0 → π − π + decays, JHEP 07 (2014) 041, arXiv:1405.2797.