Study of $B_{(s)}^0 \to K_{\rm S}^0 h^{+} h^{\prime -}$ decays with first observation of $B_{s}^0 \to K_{\rm S}^0 K^{\pm} \pi^{\mp}$ and $B_{s}^0 \to K_{\rm S}^0 \pi^{+} \pi^{-}$

A search for charmless three-body decays of $B^0$ and $B_{s}^0$ mesons with a $K_{\rm S}^0$ meson in the final state is performed using the $pp$ collision data, corresponding to an integrated luminosity of $1.0\,\mbox{fb}^{-1}$, collected at a centre-of-mass energy of $7\mathrm{\,Te\kern -0.1em V}$ recorded by the LHCb experiment. Branching fractions of the $B_{(s)}^0 \to K_{\rm S}^0 h^{+} h^{\prime -}$ decay modes ($h^{(\prime)} = \pi, K$), relative to the well measured $B^0 \to K_{\rm S}^0 \pi^{+} \pi^{-}$ decay, are obtained. First observation of the decay modes $B_s^0 \to K_{\rm S}^0 K^{\pm} \pi^{\mp}$ and $B_s^0 \to K_{\rm S}^0 \pi^{+} \pi^{-}$ and confirmation of the decay $B^0 \to K_{\rm S}^0 K^{\pm} \pi^{\mp}$ are reported. The following relative branching fraction measurements or limits are obtained \begin{eqnarray*} \nonumber \frac{{\cal B}(B^0 \to K_{\rm S}^0 K^{\pm} \pi^{\mp})}{{\cal B}(B^0 \to K_{\rm S}^0 \pi^{+} \pi^{-})}&=&0.128 \pm 0.017 \, ({\rm stat.}) \pm 0.009 \, ({\rm syst.}) \,, \\ \nonumber \frac{{\cal B}(B^0 \to K_{\rm S}^0 K^{+} K^{-} )}{{\cal B}(B^0 \to K_{\rm S}^0 \pi^{+} \pi^{-})}&=&0.385 \pm 0.031 \, ({\rm stat.}) \pm 0.023 \, ({\rm syst.}) \,, \\ \nonumber \frac{{\cal B}(B_s^0 \to K_{\rm S}^0 \pi^{+} \pi^{-} )}{{\cal B}(B^0 \to K_{\rm S}^0 \pi^{+} \pi^{-})}&=&0.29\phantom{0} \pm 0.06\phantom{0} \, ({\rm stat.}) \pm 0.03\phantom{0} \, ({\rm syst.}) \pm 0.02 \, (f_s/f_d) \,, \\ \nonumber \frac{{\cal B}(B_s^0 \to K_{\rm S}^0 K^{\pm} \pi^{\mp})}{{\cal B}(B^0 \to K_{\rm S}^0 \pi^{+} \pi^{-})}&=&1.48\phantom{0} \pm 0.12\phantom{0} \, ({\rm stat.}) \pm 0.08\phantom{0} \, ({\rm syst.}) \pm 0.12 \, (f_s/f_d) \,, \\ \nonumber \frac{{\cal B}(B_s^0 \to K_{\rm S}^0 K^{+} K^{-} )}{{\cal B}(B^0 \to K_{\rm S}^0 \pi^{+} \pi^{-})}&\in&[0.004;0.068] \; {\rm at \;\; 90\% \; CL} \,. \end{eqnarray*}

The LHCb collaboration †

Abstract
A search for charmless three-body decays of B 0 and B 0 s mesons with a K 0 S meson in the final state is performed using the pp collision data, corresponding to an integrated luminosity of 1.0 fb −1 , collected at a centre-of-mass energy of 7 TeV recorded by the LHCb experiment. Branching fractions of the B 0 (s) → K 0 S h + h − decay modes (h ( ) = π, K), relative to the well measured B 0 → K 0 S π + π − decay, are obtained. First observation of the decay modes B 0 s → K 0 S K ± π ∓ and B 0 s → K 0 S π + π − and confirmation of the decay B 0 → K 0 S K ± π ∓ are reported. The following relative branching fraction measurements or limits are obtained = 0.128 ± 0.017 (stat.) ± 0.009 (syst.) , = 0.385 ± 0.031 (stat.) ± 0.023 (syst.) , = 0.29 ± 0.06 (stat.) ± 0.03 (syst.) ± 0.02 (f s /f d ) ,

Introduction
The study of the charmless three-body decays of neutral B mesons to final states including a K 0 S meson, namely B 0 (s) → K 0 S π + π − , B 0 (s) → K 0 S K ± π ∓ and B 0 (s) → K 0 S K + K − , has a number of theoretical applications. 1 The decays B 0 → K 0 S π + π − and B 0 → K 0 S K + K − are dominated by b → qqs (q = u, d, s) loop transitions. Mixing-induced CP asymmetries in such decays are predicted to be approximately equal to those in b → ccs transitions, e.g. B 0 → J/ψ K 0 S , by the Cabibbo-Kobayashi-Maskawa mechanism [1,2]. However, the loop diagrams that dominate the charmless decays can have contributions from new particles in several extensions of the Standard Model, which could introduce additional weak phases [3][4][5][6]. A time-dependent analysis of the three-body Dalitz plot allows measurements of the mixing-induced CP -violating phase [7][8][9][10]. The current experimental measurements of b → qqs decays [11] show fair agreement with the results from b → ccs decays (measuring the weak phase β) for each of the scrutinised CP eigenstates. There is, however, a global trend towards lower values than the weak phase measured from b → ccs decays. The interpretation of this deviation is made complicated by QCD corrections, which depend on the final state [12] and are difficult to handle. An analogous extraction of the mixinginduced CP -violating phase in the B 0 s system will, with a sufficiently large dataset, also be possible with the B 0 s → K 0 S K ± π ∓ decay, which can be compared with that from, e.g.
Much recent theoretical and experimental activity has focused on the determination of the CKM angle γ from B → Kππ decays, using and refining the methods proposed in Refs. [13,14]. The recent experimental results from BaBar [15] demonstrate the feasibility of the method, albeit with large statistical uncertainties. The decay B 0 s → K 0 S π + π − is of particular interest for this effort. Indeed, the ratio of the amplitudes of the isospin-related mode B 0 s → K − π + π 0 and its charge conjugate exhibits a direct dependence on the mixinginduced CP -violating phase, which would be interpreted in the Standard Model as (β s + γ). Unlike the equivalent B 0 decays, the B 0 s decays are dominated by tree amplitudes and the contributions from electroweak penguin diagrams are expected to be negligible, yielding a theoretically clean extraction of γ [16] provided that the strong phase can be determined from other measurements. The shared intermediate states between B 0 s → K − π + π 0 and B 0 s → K 0 S π + π − (specifically K * − π + ) offer that possibility, requiring an analysis of the B 0 s → K 0 S π + π − Dalitz plot. At LHCb, the first step towards this physics programme is to establish the signals of all the decay modes. In particular, the decay modes B 0 [17] is so far unconfirmed. In this paper the results of an analysis of all six B 0 (s) → K 0 S h + h − decay modes are presented. The branching fractions of the decay modes relative to that of B 0 → K 0 S π + π − are measured when the significance of the signals allow it, otherwise confidence intervals are quoted. Time-integrated branching fractions are computed, implying a non-trivial comparison of the B 0 and B 0 s decays at amplitude level [18].

Detector and dataset
The measurements described in this paper are performed with data, corresponding to an integrated luminosity of 1.0 fb −1 , from 7 TeV centre-of-mass pp collisions, collected with the LHCb detector during 2011. Samples of simulated events are used to estimate the efficiency of the selection requirements, to investigate possible sources of background contributions, and to model the event distributions in the likelihood fit. In the simulation, pp collisions are generated using Pythia 6.4 [19] with a specific LHCb configuration [20]. Decays of hadronic particles are described by EvtGen [21], in which final state radiation is generated using Photos [22]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [23] as described in Ref. [24]. The LHCb detector [25] 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 silicon-strip vertex detector (VELO) surrounding the pp interaction region, 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 placed downstream. The combined tracking system provides a momentum measurement with relative uncertainty that varies from 0.4% at 5 GeV/c to 0.6% at 100 GeV/c, and impact parameter resolution of 20 µm for tracks with high transverse momentum. Charged hadrons are identified using two ring-imaging Cherenkov (RICH) detectors [26]. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers.

Trigger and event selection
The trigger [27] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. To remove events with large occupancies, a requirement is made at the hardware stage on the number of hits in the scintillating-pad detector. The hadron trigger at the hardware stage also requires that there is at least one candidate with transverse energy E T > 3.5 GeV. In the offline selection, candidates are separated into two categories based on the hardware trigger decision. The first category are triggered by particles from candidate signal decays that have an associated cluster in the calorimeters above the threshold, while the second category are triggered independently of the particles associated with the signal decay. Events that do not fall into either of these categories are not used in the subsequent analysis.
The software trigger requires a two-, three-or four-track secondary vertex with a high sum of the transverse momentum, p T , of the tracks and significant displacement from the primary pp interaction vertices (PVs). At least one track should have p T > 1.7 GeV/c and χ 2 IP with respect to any primary interaction greater than 16, where χ 2 IP is defined as the difference in χ 2 of a given PV reconstructed with and without the considered track. A multivariate algorithm [28] is used for the identification of secondary vertices consistent with the decay of a b hadron.
The events passing the trigger requirements are then filtered in two stages. Initial requirements are applied to further reduce the size of the data sample, before a multivariate selection is implemented. In order to minimise the variation of the selection efficiency over the Dalitz plot it is necessary to place only loose requirements on the momenta of the daughter particles. As a consequence, selection requirements on topological variables such as the flight distance of the B candidate or the direction of its momentum vector are used as the main discriminants.
The K 0 S candidates are reconstructed in the π + π − final state. Approximately two thirds of the reconstructed K 0 S mesons decay downstream of the VELO. Since those K 0 S candidates decaying within the VELO, and those that have information only from the tracking stations, differ in their reconstruction and selection, they are separated into two categories labelled "Long" and "Downstream", respectively. The pions that form the K 0 S candidates are required to have momentum p > 2 GeV/c and χ 2 IP with respect to any PV greater than 9 (4) for Long (Downstream) K 0 S candidates. The K 0 S candidates are then required to form a vertex with χ 2 vtx < 12 and to have invariant mass within 20 MeV/c 2 (30 MeV/c 2 ) of the nominal K 0 S mass [29] for Long (Downstream) candidates. The square of the separation of the K 0 S vertex from the PV divided by the associated uncertainty (χ 2 VS ) must be greater than 80 (50) for Long (Downstream) candidates. Downstream K 0 S candidates are required, in addition, to have momentum p > 6 GeV/c. The B candidates are formed by combining the K 0 S candidates with two oppositely charged tracks. Selection requirements, common to both the Long and Downstream categories, are based on the topology and kinematics of the B candidate. The charged B-meson daughters are required to have p < 100 GeV/c, a momentum beyond which there is little pion/kaon discrimination. The scalar sum of the three daughters' transverse momenta must be greater than 3 GeV/c, and at least two of the daughters must have p T > 0.8 GeV/c. The impact parameter (IP) of the B-meson daughter with the largest p T is required to be greater than 0.05 mm relative to the PV associated to the B candidate. The χ 2 of the distance of closest approach of any two daughters must be less than 5. The B candidates are then required to form a vertex separated from any PV by at least 1 mm and that has χ 2 vtx < 12 and χ 2 VS > 50. The difference in χ 2 vtx when adding any track must be greater than 4. The candidates must have p T > 1.5 GeV/c and invariant mass within the range 4779 The cosine of the angle between the reconstructed momentum of the B meson and its direction of flight (pointing angle) is required to be greater than 0.9999. The candidates are further required to have a minimum χ 2 IP with respect to all PVs less than 4. Finally, the separation of the K 0 S and B vertices in the positive z direction 2 must be greater than 30 mm.
Multivariate discriminants based on a boosted decision tree (BDT) [30] with the AdaBoost algorithm [31] have been designed in order to complete the selection of the signal events and to further reject combinatorial backgrounds. Simulated B 0 (s) → K 0 S π + π − events and upper mass sidebands, 5420 < m K 0 S π + π − < 5866 MeV/c 2 , in the data are used as the signal and background training samples, respectively. The samples of events in each of the Long and Downstream K 0 S categories are further subdivided into two equally-sized subsamples. Each subsample is then used to train an independent discriminant. In the subsequent analysis the BDT trained on one subsample of a given K 0 S category is used to select events from the other subsample, in order to avoid bias. The input variables for the BDTs are the p T , η, χ 2 IP , χ 2 VS , pointing angle and χ 2 vtx of the B candidate; the sum χ 2 IP of the h + and h − ; the χ 2 IP , χ 2 VS and χ 2 vtx of the K 0 S candidate. The selection requirement placed on the output of the BDTs is independently optimised for events containing K 0 S candidates reconstructed in either Downstream or Long categories. Two different figures of merit are used to optimise the selection requirements, depending on whether the decay mode in question is favoured or suppressed. If favoured, the following is used where S (B) represents the number of expected signal (combinatorial background) events for a given selection. The value of S is estimated based on the known branching fractions and efficiencies, while B is calculated by fitting the sideband above the signal region and extrapolating into the signal region. If the mode is suppressed, an alternative figure of merit [32] is used where the signal efficiency (ε sig ) is estimated from the signal simulation. The value a = 5 is used in this analysis, which corresponds to optimising for 5σ significance to find the decay. This second figure of merit results in a more stringent requirement than the first. Hence, the requirements optimised with each figure of merit will from here on be referred to as the loose and tight BDT requirements, respectively. The fraction of selected events containing more than one candidate is at the percent level. The candidate to be retained in each event is chosen arbitrarily.
A number of background contributions consisting of fully reconstructed B meson decays into two-body Dh or ccK 0 S combinations, result in a K 0 S h + h − final state and hence are, in terms of their B candidate invariant mass distribution, indistinguishable from signal candidates. The decays of Λ 0 b baryons to Λ + c h with Λ + c → pK 0 S also peak under the signal when the proton is misidentified. Therefore, the following D, Λ + c and charmonia decays are explicitly reconstructed under the relevant particle hypotheses and vetoed in all the spectra: are applied to remove the handful of fully reconstructed and well identified peaking The veto for each reconstructed charm (charmonium) state R, |m − m R | < 30 (48) MeV/c 2 , is defined around the world average mass value m R [29] and the range is chosen according to the typical mass resolution obtained at LHCb.
Particle identification (PID) requirements are applied in addition to the selection described so far. The charged pion tracks from the K 0 S decay and the charged tracks from the B decay are all required to be inconsistent with the muon track hypothesis. The logarithm of the likelihood ratio between the kaon and pion hypotheses (DLL Kπ ), mostly based on information from the RICH detectors [26], is used to discriminate between pion and kaon candidates from the B decay. Pion (kaon) candidates are required to satisfy DLL Kπ < 0 (DLL Kπ > 5). These are also required to be inconsistent with the proton hypothesis, in order to remove the possible contributions from charmless b-baryon decays. Pion (kaon) candidates are required to satisfy DLL pπ < 10 (DLL pK < 10).

Fit model
A simultaneous unbinned extended maximum likelihood fit to the B-candidate invariant mass distributions of all decay channels is performed for each of the two BDT optimisations. In each simultaneous fit four types of components contribute, namely signal decays, crossfeed backgrounds, partially-reconstructed backgrounds, and combinatorial background.
Contributions from B 0 (s) → K 0 S h + h − decays with correct identification of the final state particles are modelled with sums of two Crystal Ball (CB) functions [33] that share common values for the peak position and width but have independent power law tails on opposite sides of the peak. The B 0 and B 0 s masses (peak positions of the double-CB functions) are free in the fit. Four parameters related to the widths of the double-CB function are also free parameters of the fit: the common width of the B 0 → K 0 S π + π − and B 0 s → K 0 S π + π − signals; the relative widths of K 0 S K ± π ∓ and K 0 S K + K − to K 0 S π + π − , which are the same for B 0 and B 0 s decay modes; the ratio of Long over Downstream widths, which is the same for all decay modes. These assumptions are made necessary by the otherwise poor determination of the width of the suppressed mode of each spectrum. The other parameters of the CB components are obtained by a simultaneous fit to simulated samples, constraining the fraction of events in the two CB components and the ratio of their tail parameters to be the same for all double-CB contributions.
Each selected candidate belongs uniquely to one reconstructed final state, by definition of the particle identification criteria. However, misidentified decays yield some cross-feed in the samples and are modelled empirically by single CB functions using simulated events. Only contributions from the decays B 0 → K 0 S π + π − and B 0 → K 0 S K + K − reconstructed and selected as K 0 S K ± π ∓ , or the decays B 0 s → K 0 S K ± π ∓ and B 0 → K 0 S K ± π ∓ reconstructed and selected as either K 0 S K + K − or K 0 S π + π − are considered. Other potential contributions are neglected. The relative yield of each misidentified decay is constrained with respect to the yield of the corresponding correctly identified decay. The constraints are implemented using Gaussian priors included in the likelihood. The mean values are obtained from the ratio of selection efficiencies and the resolutions include uncertainties originating from the finite size of the simulated events samples and the systematic uncertainties related to the determination of the PID efficiencies.
Partially reconstructed charmed transitions such as B − → D 0 π − (K − ) followed by D 0 → K 0 S π + π − , with a pion not reconstructed, are expected to dominate the background contribution in the lower invariant mass region. Charmless backgrounds such as from and B + → K 0 S π + π − π + decays are also expected to contribute with lower rates. These decays are modelled by means of generalised ARGUS functions [34] convolved with a Gaussian resolution function. Their parameters are determined from simulated samples. In order to reduce the number of components in the fit, only generic contributions for hadronic charmed and charmless decays are considered in each final state, however B 0 and B 0 s contributions are explicitly included. Radiative decays and those from B 0 → η (→ ρ 0 γ)K 0 S are considered separately and included only in the K 0 S π + π − final state. The normalisation of all such contributions is constrained with Gaussian priors using the ratio of efficiencies from the simulation and the ratio of branching fractions from world averages [29]. Relative uncertainties on these ratios of 100%, 20% and 10% are considered for charmless, charmed, and radiative and B 0 → η (→ ρ 0 γ)K 0 S decays, respectively. The combinatorial background is modelled by an exponential function, where the slope parameter is fitted for each of the two K 0 S reconstruction categories. The combinatorial backgrounds to the three final states B 0 (s)  Table 1 summarises the fitted yields of each decay mode for the optimisation used to determine the branching fractions. In the tight BDT optimisation the combinatorial background is negligible in the high invariant-mass region for the K 0 S π + π − and K 0 S K + K − final states, leading to a small systematic uncertainty related to the assumptions used to fit this component. An unambiguous first observation of B 0 s → K 0 S K ± π ∓ decays and a clear confirmation of the BaBar observation [17] of B 0 → K 0 S K ± π ∓ decays are obtained. Significant yields for the B 0 s → K 0 S π + π − decays are observed above negligible background with the tight optimisation of the selection. The likelihood profiles are shown in Fig. 3 for Downstream and Long K 0 S samples separately. The B 0 s → K 0 S π + π − decays are observed with a combined statistical significance of 6.2 σ, which becomes 5.9 σ including fit model systematic uncertainties. The statistical significance of the B 0 s → K 0 S K + K − signal is at the level of 2.1 σ combining Downstream and Long K 0 S reconstruction categories.

Determination of the efficiencies
The measurements of the branching fractions of the B 0 (s) → K 0 S h + h − decays relative to the well established B 0 → K 0 S π + π − decay mode proceed according to where ε sel is the selection efficiency (which includes acceptance, reconstruction, selection, trigger and particle identification components), N is the fitted signal yield, and f d and f s       are the hadronisation fractions of a b quark into a B 0 and B 0 s meson, respectively. The ratio f s /f d has been accurately determined by the LHCb experiment from hadronic and semileptonic measurements f s /f d = 0.256 ± 0.020 [35].
Three-body decays are composed of several quasi-two-body decays and non-resonant contributions, all of them possibly interfering. Hence, their dynamical structure, described by the Dalitz plot [36], must be accounted for to correct for non-flat efficiencies over the phase space. Since the dynamics of most of the modes under study are not known prior to this analysis, efficiencies are determined for each decay mode from simulated signal samples in bins of the "square Dalitz plot" [37], where the usual Dalitz-plot coordinates have been transformed into a rectangular space. The edges of the usual Dalitz plot are spread out in the square Dalitz plot, which permits a more precise modelling of the efficiency variations in the regions where they are most strongly varying and where most of the signal events are expected. Two complementary simulated samples have been produced, corresponding to events generated uniformly in phase space or uniformly in the square Dalitz plot. The square Dalitz-plot distribution of each signal mode is determined from the data using the sPlot technique [38]. The binning is chosen such that each bin is populated by approximately the same number of signal events. The average efficiency for each decay mode is calculated as the weighted harmonic mean over the bins. The average weighted selection efficiencies are summarised in Table 1 and depend on the final state, the K 0 S reconstruction category, and the choice of the BDT optimisation. Their relative uncertainties due to the finite size of the simulated event samples vary from 3% to 17%, reflecting the different dynamical structures of the decay modes.
The particle identification and misidentification efficiencies are determined from simulated signal events on an event-by-event basis by adjusting the DLL distributions measured from calibration events to match the kinematical properties of the tracks in the decay of interest. The reweighting is performed in bins of p and p T , accounting for kinematic correlations between the tracks. Calibration tracks are taken from D * + → D 0 π + s decays where the D 0 decays to the Cabibbo-favoured K − π + final state. The charge of the soft pion π + s hence provides the kaon or pion identity of the tracks. The dependence of the PID efficiency over the Dalitz plot is included in the procedure described above. This calibration is performed using samples from the same data taking period, accounting for the variation in the performance of the RICH detectors over time.

Systematic uncertainties
Most of the systematic uncertainties are eliminated or greatly reduced by normalising the branching fraction measurements with respect to the B 0 → K 0 S π + π − mode. The remaining sources of systematic effects and the methods used to estimate the corresponding uncertainties are described in this section. In addition to the systematic effects related to the measurements performed in this analysis, there is that associated with the measured value of f s /f d . A summary of the contributions, expressed as relative uncertainties, is given in Table 2.

Fit model
The fit model relies on a number of assumptions, both in the values of parameters being taken from simulation and in the choice of the functional forms describing the various contributions.
The uncertainties linked to the parameters fixed to values determined from simulated events are obtained by repeating the fit while the fixed parameters are varied according to their uncertainties using pseudo-experiments. For example, the five fixed parameters of the CB functions describing the signals, as well as the ratio of resolutions with respect to B 0 → K 0 S π + π − decays, are varied according to their correlation matrix determined from simulated events. The nominal fit is then performed on this sample of pseudoexperiments and the distribution of the difference between the yield determined in each of these fits and that of the nominal fit is fitted with a Gaussian function. The systematic uncertainty associated with the choice of the value of each signal parameter from simulated events is then assigned as the linear sum of the absolute value of the mean of the Gaussian and its resolution. An identical procedure is employed to obtain the systematic uncertainties related to the fixed parameters of the ARGUS functions describing the partially reconstructed backgrounds and the CB functions used for the cross-feeds.
The uncertainties related to the choice of the models used in the nominal fit are evaluated for the signal and combinatorial background models only. Both the partially reconstructed background and the cross-feed shapes suffer from a large statistical uncertainty from the simulated event samples and therefore the uncertainty related to the fixed parameters also covers any sensible variation of the shape. The B 0 s decay modes that are studied lie near large B 0 contributions for the K 0 S π + π − and K 0 S K + K − spectra. The impact of the modelling of the right hand side of the B 0 mass distribution is addressed by removing the second CB function, used as an alternative model.
For the combinatorial background, a unique slope parameter governs the shape of each K 0 S reconstruction category (Long or Downstream). Two alternative models are considered: allowing independent slopes for each of the six spectra (testing the assumption of a universal slope) and using a linear model in place of the exponential (testing the functional form of the combinatorial shape). Pseudo-experiments are again used to estimate the effect of these alternative models; in the former case, the value and uncertainties to be considered for the six slopes are determined from a fit to the data. The dataset is generated according to the substitute model and the fit is performed to the generated sample using the nominal model. The value of the uncertainty is again estimated as the linear sum of the absolute value of the resulting bias and its resolution. The total fit model systematic uncertainty is given by the sum in quadrature of all the contributions and is mostly dominated by the combinatorial background model uncertainty.

Selection and trigger efficiencies
The accuracy of the efficiency determination is limited in most cases by the finite size of the samples of simulated signal events, duly propagated as a systematic uncertainty. In addition, the effect related to the choice of binning for the square Dalitz plot is estimated from the spread of the average efficiencies determined from several alternative binning schemes. Good agreement between data and the simulation is obtained, hence no further systematic uncertainty is assigned.
Systematic uncertainties related to the hardware stage trigger have been studied. A data control sample of D * + → D 0 (→ K − π + )π + s decays is used to quantify differences between pions and kaons, separated by positive and negative hadron charges, as a function  [27]. Though they show an overall good agreement for the different types of tracks, the efficiency for pions is slightly smaller than for kaons at high p T . Simulated events are reweighted by these data-driven calibration curves in order to extract the hadron trigger efficiency for each mode, propagating properly the calibration-related uncertainties. Finally, the ageing of the calorimeters during the data taking period when the data sample analysed was recorded induced changes in the absolute scale of the trigger efficiencies. While this was mostly mitigated by periodic recalibration, relative variations occurred of order 10%. Since the kinematics vary marginally from one mode to the other, a systematic effect on the ratio of efficiencies arises. It is fully absorbed by increasing the trigger efficiency systematic uncertainty by 10%.

Particle identification efficiencies
The procedure to evaluate the efficiencies of the PID selections uses calibration tracks that differ from the signal tracks in terms of their kinematic distributions. While the binning procedure attempts to mitigate these differences there could be some remaining systematic effect. To quantify any bias due to the procedure, simulated samples of the control modes are used in place of the data samples. The average efficiency determined from these samples can then be compared with the efficiency determined from simply applying the selections to the simulated signal samples. The differences are found to be less than 1%, hence no correction is applied. The calibration procedure is assigned a systematic uncertainty. The observed differences in efficiencies are multiplied by the efficiency ratio and statistical uncertainties from the finite sample sizes are added in quadrature.

Results and conclusion
The 2011 LHCb dataset, corresponding to an integrated luminosity of 1.0 fb −1 recorded at a centre-of-mass energy of 7 TeV, has been analysed to search for the decays B 0 (s) are observed for the first time. The former is unambiguous, while for the latter the significance of the observation is 5.9 standard deviations, including statistical and systematic uncertainties. The decay mode B 0 → K 0 S K ± π ∓ , previously observed by the BaBar experiment [17], is confirmed. The efficiency-corrected Dalitz-plot distributions of the three decay modes B 0 Fig. 4. Some structure is evident at low K 0 S π ± and K ± π ∓ invariant masses in the B 0 s → K 0 S K ± π ∓ decay mode, while in the B 0 → K 0 S K ± π ∓ decay the largest structure is seen in the low K 0 S K ± invariant mass region. No significant evidence for B 0 s → K 0 S K + K − decays is obtained. A 90% confidence level (CL) interval based on the CL inferences described in Ref. [39] is hence placed on the branching fraction for this decay mode.