Improved limit on the branching fraction of the rare decay K0S→μ+μ−

Eur. Phys. J. C (2017) 77:678 DOI 10.1140/epjc/s10052-017-5230-x Regular Article - Experimental Physics Improved limit on the branching fraction of the rare decay K0 S→µ+µ− LHCb Collaboration CERN, 1211 Geneva 23, Switzerland Received: 12 June 2017 / Accepted: 18 September 2017 / Published online: 13 October 2017 © CERN for the benefit of the LHCb collaboration 2017. This article is an open access publication Abstract A search for the decay K0 S→μ+μ−is performed, based on a data sample of proton-proton collisions corresponding to an integrated luminosity of 3 fb−1, collected by the LHCb experiment at centre-of-mass energies of 7 and 8 TeV. The observed yield is consistent with the background-only hypothesis, yielding a limit on the branching fraction of B(K0 S→μ+μ−)<0.8(1.0)×10−9at 90% (95%)confidence level. This result improves the previous upper limit on the branching fraction by an order of magnitude. 1 Introduction In the Standard Model (SM), the unobserved K0 S→μ+μ− decay proceeds only through a Flavour-Changing Neutral Current (FCNC) transition, which cannot occur at tree level. It is further suppressed by the small amount of CP violation in kaon decays, since the S-wave component of the decay is forbidden when CP is conserved. In the SM, the decay amplitude is expected to be dominated by long distance contributions, which can be constrained using the observed decays K0 S→γγ and K0 L→π0γγ, leading to the prediction for the branching fraction B(K0 S→μ+μ−)= (5.0±1.5)×10−12 [1,2]. The predicted branching fraction for the K0 Ldecay is (6.85 ±0.32)×10−9[3], in excellent agreement with the experimental world average B(K0 L→μ+μ−)=(6.84 ±0.11)×10−9[4]. The prediction for K0 S→μ+μ−is currently being updated with a dispersive treatment, which leads to sizeable corrections in other K0 Sleptonic decays [5]. Due to its suppression in the SM, the K0 S→μ+μ− decay is sensitive to possible contributions from dynamics beyond the SM, notably from light scalars with CP-violating Yukawa couplings [1]. Contributions up to one order of magnitude above the SM branching fraction expectation naturally arise in many models and are compatible with the present bounds from other FCNC processes. An upper limit e-mail: [email protected] on B(K0 S→μ+μ−)close to 10−11 could be translated into model-independent bounds on the CP-violating phase of the s→d+−amplitude [2].Thiswouldbeveryusefultodiscriminate between scenarios beyond the SM if other modes, such as K+→π+νν, indicate a non-SM enhancement. The current experimental limit, B(K0 S→μ+μ−)< 9×10−9at 90% confidence level (CL), was obtained using pp collision data corresponding to 1.0fb −1of integrated luminosity at a centre-of-mass energy √s=7 TeV, collected with the LHCb detector in 2011 [6]. This result improved the previous upper limit [7] but is still three orders of magnitude above the predicted SM level. In this paper, an update of the search for the K0 S→μ+μ− decay is reported. Its branching fraction is measured using the known K0 S→π+π−decay as normalisation. The analysis is performed on a data sample corresponding to 2 fb−1of integrated luminosity at √s=8 TeV, collected in 2012, and the result is combined with that from the previous LHCb analysis [6]. Besides the gain in statistical precision due to the larger data sample, the sensitivity is noticeably increased with respect to the previous result due to a higher trigger efficiency, as well as other improvements to the analysis that are discussed in the following sections. An overview on how K0 S→μ+μ−decays are detected and triggered in LHCb is given in Sect. 2, while the strategy for this measurement is outlined in Sect. 3. Details of background suppression and the resulting sensitivity are given in Sects. 4and 5, respectively. The final result, taking into account the systematic uncertainties discussed in Sect. 6,is given in Sect. 7. 2K0 Sdecays in LHCb The LHCb detector [8,9] is a single-arm forward spectrometer covering the pseudorapidity range 2 <η<5, designed for the study of particles containing bor cquarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex locator (VELO) surrounding 123 678 Page 2 of 12 Eur. Phys. J. C (2017) 77 :678 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 of the magnet. 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/pT)µm, where pTis the component of the momentum transverse to the beam, in GeV/c. Different types of charged hadrons are distinguished using information from two ring-imaging Cherenkov detectors (RICH). Photons, electrons and hadrons are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by five stations which alternate layers of iron and multiwire proportional chambers. The online event selection is performed by the trigger [10], which consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a twostep software stage, which applies a full event reconstruction. Candidates are subsequently classified as TOS, if the event is triggered on the signal candidate, or TIS, if triggered by other activities in the detector, independently of signal. Only candidates that are classified as TOS at each trigger stage are used to search for K0 S→μ+μ−decays. The trigger selection constitutes the main limitation to the efficiency for detecting K0 Sdecays. A muon is only selected at the hardware stage when it is detected in all muon stations and a rough momentum estimation is provided. Trigger requirements at this stage imply a momentum larger than about 5 GeV/c, and a pTabove 1.76 GeV/c. These thresholds have an efficiency of order 1% for K0 S→μ+μ−decays. In the first step of the software trigger, all charged particles with pT>500 MeV/care reconstructed. At this stage most signal decays are triggered either by requiring a reconstructed track loosely identified as a muon [10,11], with IP >0.1mm and pT>1.0GeV/c, or by finding two oppositely charged muon candidates forming a detached secondary vertex (SV). Since these two categories, hereafter referred to as TOSμand TOSμμ, induce different kinematic biases on the signal and background candidates, the analysis steps described below are performed independently on each category. The two categories are made mutually exclusive by applying the TOSμμ selection only to candidates not already selected by TOSμ. In the second software trigger stage, an offline-quality event reconstruction is performed. Signal candidates are selected requiring a dimuon with pT>600 MeV/cdetached from the primary vertex, with both tracks having pT> 300 MeV/c. In the 2011 data taking, the dimuon mass was required to be larger than 1 GeV/c2in the second software trigger stage. This excluded the K0 Sregion, making the use of TIS candidates necessary. Due to the trigger reoptimisation, no mass requirements were applied during 2012 and a lower pTthreshold for reconstructed tracks was used. According to simulation, these changes improve the trigger efficiency over the previous analysis [6] by about a factor 2.5. Due to its large and well-known branching fraction and its similar topology, the K0 S→π+π−decay is taken as the normalisation mode. A large sample of candidates is obtained from an unbiased trigger, which does not apply any selection requirement. Despite the low trigger efficiency, the study detailed in this paper profits from the unprecedented number of K0 Sproduced at the LHC, O(1013)per fb−1of integrated luminosity within the LHCb acceptance, and from the fact that about 40% of these K0 Sdecays occur inside the VELO region. For such decays, the K0 Sinvariant mass is reconstructed with a resolution of about 4 MeV/c2. The analysis makes use of large samples of simulated collisions containing a signal decay, or background decays which can be reconstructed as the signal, and contaminate the μμ invariant mass distribution, such as K0 S→π+π− or K0 S→π+μ−¯νμ.1In the simulation, pp collisions are generated using Pythia [12,13] with a specific LHCb configuration [14]. Decays of hadronic particles are described by EvtGen [15], in which final-state radiation is generated using Photos [16]. The interaction of the generated particles with the detector, and its response, are implemented using the Geant4 toolkit [17,18] as described in Ref. [19]. 3 Selection and search strategy Common offline preselection criteria are applied to K0 S→ μ+μ−and K0 S→π+π−candidates to cancel many systematic effects in the ratio. Candidates are required to decay in the VELO region, where the best K0 Smass resolution is achieved. The two reconstructed tracks must have momentum smaller than 100 GeV/cand quality requirements are set on the track and secondary vertex fits. The SV must be well detached from the PV by requiring the K0 Sdecay time to be larger than 8.95 ps, 10% of the K0 Smean lifetime. The K0 SIP must be less than 0.4 mm, while the two charged tracks are required to be incompatible with originating from any PV, with IP χ2, defined as the difference of the χ2of the PV fit obtained with and without the considered track, to be larger than 100. Decays of Λbaryons to pπ−are suppressed by removing candidates close to the expected ellipses in the Armenteros– Podolanski (AP) plane [20]. In this plane the pTof the final-state particles under the pion mass hypothesis is plotted versus the longitudinal momentum asymmetry, defined 1The inclusion of charge-conjugate processes is implied throughout. 123 Eur. Phys. J. C (2017) 77 :678 Page 3 of 12 678 as α=(p+ L−p− L)/(p+ L+p− L), where p± Lis the longitudinal momentum of the charged tracks. Both pTand pLare considered with respect to the direction of the mother particle. The K0 Sdecays are symmetrically distributed on the AP plane while Λdecays produce two ellipses at low pTand |α|∼0.7. A kaon veto, based on the response of the RICH detector, is used to suppress K∗0→K+π−decays and other possible final states including a charged kaon. The preselection reduces the combinatorial background, arising from candidates formed from secondary hadronic collisions in the detector material or from spurious reconstructed SV. The purity of the K0 S→π+π−sample used for normalisation, whose mass distribution is shown in Fig. 1,is estimated from a fit to the mass spectrum to be 99.8%. The fraction of events with more than one candidate is less than 0.1% for signal and 4% for the normalisation channel, and all candidates are retained. Additional discrimination against backgrounds for the signal mode is achieved through the use of two multivariate discriminants. The first is designed to further suppress combinatorial background, and the second to reduce the number of K0 S→π+π−decays in which both pions are misidentified as muons. After requirements on the output of these discriminants have been applied, the number of signal candidates is obtained by fitting the K0 S→μ+μ−mass spectrum. The number of candidates is converted into a branching fraction using the yield of the K0 S→π+π−normalisation mode, and the estimated relative efficiency. Events in the K0 Smass region are scrutinised only after fixing the analysis strategy. 4 Backgrounds The K0 S→μ+μ−sample contains two main sources of background. Combinatorial background candidates are expected to exhibit a smooth mass distribution, and can therefore be estimated from the sidebands. The other relevant source of background is due to K0 S→π+π−decays where both pions pass the loose muon identification requirements after the trigger stage. This can be due either to π+→μ+νμ decays or to random association of muon detector hits with the pion trajectory. In such cases the K0 Smass, reconstructed with a wrong mass hypothesis for the final-state particles, is underestimated by 39 MeV/c2on average, as shown in Fig. 1. Despite the excellent mass resolution, the right-hand tail of the reconstructed mass distribution under the dimuon hypothesis extends intothe K0 Ssignal mass range and, given the large branching fraction of the K0 S→π+π−mode, constitutes a nonnegligible background. Two multivariate discriminants, based on a boosted decision tree (BDT) algorithm [21,22], are applied on the preselected candidates to improve the signal discrimination with respect to these backgrounds. ] 2 cInvariant mass [MeV/ 420 440 460 480 500 520 ) 2 c Candidates / (1 MeV/ 1 10 2 10 3 10 4 10 5 10 6 10 LHCb hypothesis - π + π hypothesis - μ + μ Fig. 1 Reconstructed mass for K0 S→π+π−decays in triggerunbiased events, computed assuming the muon (dashed red line) or pion (solid blue line) mass for the final-state tracks. Candidates satisfy the selection criteria described in the text The first discriminant, named hereafter BDTcb,aimsto reduce the combinatorial background, exploiting the different decay topologies, kinematic spectra and reconstruction qualities of signal and combinatorial candidates. It is optimised separately for each trigger category. The algorithm used for both categories is XGBoost [23], with a learning rate of 0.02 and a maximum depth of 4. The optimal number of estimators is 2000 and 800 for the TOSμand TOSμμ trigger categories, respectively. A set of ten input variables is used in BDTcb:theK0 SpTand IP, the minimum IP of the two charged tracks, the angle between the positively charged final-state particle in the K0 Srest frame and the axis defined by the K0 Sboost direction, the χ2of the SV fit, the distance of closest approach between the two tracks, an SV isolation variable, defined as the difference in vertex-fit χ2when the next nearest track is included in the vertex fit, and the SV absolute position coordinates. The SV position is particularly important, since a large fraction of the background is found to originate from interactions in the detector material. This set of variables does not distinguish between K0 S→μ+μ− and K0 S→π+π−decays as it does not contain quantities related to muon identification and ignores the K0 Scandidate invariant mass distribution. The signal training sample for BDTcb is composed of about 11800 (TOSμ) and 2400 (TOSμμ)K0 S→μ+μ−simulated candidates passing the trigger and preselection criteria. A signal training sample consisting of K0 S→π+π− decays in data is also used as a cross-check, as explained in Sect. 6. The background training sample is made from K0 S→μ+μ−data candidates surviving the trigger and preselection requirements with reconstructed mass in the range [520,600]MeV/c2, and contains about 15000 and 4000 candidates for the TOSμand TOSμμ trigger categories, respectively. Since candidates in the same mass region are also used 123 678 Page 4 of 12 Eur. Phys. J. C (2017) 77 :678 to estimate the residual background, the training is performed using a k-fold cross-validation technique [24] to avoid any possible effect of overtraining. A loose requirement on the BDTcb output is applied to suppress the combinatorial background. The cut is chosen to remove 99% of the background training candidates. The corresponding signal efficiency is about 56 and 66% for the TOSμand TOSμμ trigger categories, respectively. To exploit further the information provided by the discriminant, the candidates surviving this requirement are allocated to ten bins according to their BDTcb value, with bounds defined in order to have approximately equal population of signal training candidates in each bin. The background from misidentified K0 S→π+π−decays is further reduced with the second multivariate discriminant, called BDTμ. Its input includes the position, time and number of detector hits around the extrapolated track position to each muon detector station, a global match χ2between the muon hit positions and the track extrapolation, and other variables related to the tracking and the response of the RICH and calorimeter detectors. To train the BDTμdiscriminant, highly pure samples of 1.2 million pions and 0.68 million muons are obtained from TIS-triggered K0 S→π+π−and B+→J/ψ K+decays, respectively. In the latter case, a probe muon from the J/ψ is required to be TIS at all trigger stages, while stringent muon identification requirements are set on the other muon, reaching an estimated purity for muons above 99.9%. The multivariate AdaBoost algorithm implemented in the TMVA package [25] is used, with 850 trees and a maximum depth of 3. Before using it in the BDTμtraining, the muon sample is weighted to have the same two-dimensional distribution in pand pTas the pion sample, as well as the same distribution of number of tracks in the event. This is to prevent the BDTμ from discriminating pions and muons using these variables, which are included in the input because of their strong correlation with the identification variables. Weighting also allows optimisation of the discrimination power for the kinematic spectrum relevant to this search. The level of misidentification of the discriminant for a pion from K0 S→π+π−decay is found to be 0.4% for 90% muon efficiency. This reduces the level of double misidentification background, for a given efficiency, by about a factor of four with respect to the discriminant used in the previous publication [6], which was not tuned specifically for K0 S→μ+μ− searches. The BDTμdiscriminant is trained using half of the B+→ J/ψ K+sample, while the other half is used to evaluate the muon identification efficiency as a function of (p,pT). These values are used to compute the efficiency of a BDTμrequirement on the candidate K0 S→μ+μ−decays after selection and trigger requirements, in each bin of the BDTcb discriminant. The muon spectra assumed in this calculation are obtained from simulated decays, weighted to better reproduce the K0 SpTspectrum observed in K0 S→π+π−candidates. The BDTμrequirement on the signal candidates is optimised by maximising the figure of merit [26]μID/(Nbg + a/2), with a=3, where μID is the signal efficiency and Nbg the expected background yield. The latter is estimated from a fit to the mass distribution, after removing candidates in the range [492,504]MeV/c2around the K0 Smass, and extrapolating the result into this region. In the fit, the contribution of K0 S→π+π−decays is modelled with a Crystal Ball function [27] and the combinatorial background with an exponential function, where all the parameters are left free to vary. This optimisation is performed independently for the two trigger categories, with no significant difference found as a function of the BDTcb bin. The optimal threshold corresponds to a signal efficiency of μID ∼98% in both cases. Other possible sources of background have been explored and found to give negligible contribution to this search. The irreducible background due to K0 L→μ+μ−decays and from K0 S–K0 Linterference is evaluated from the known K0 L→μ+μ−branching fraction and lifetime, and by studying the decay-time dependence of the selection efficiency for K0 S→π+π−decays in data. The yield from this background becomes comparable to the signal for a branching fraction lower than 2 ×10−11, which is well below the sensitivity of this search. Semileptonic K0→π+μ−νμdecays with pion misidentification provide another possible source of background. Simulated events, where the pion is forced to decay to μν within the detector, are used to determine the efficiency of the offline selection requirements. No event survives the trigger selection. Under the very conservative hypothesis that the trigger efficiency is the same as in K0 S→μ+μ−decays, the expected yields from both K0 Land K0 Ssemileptonic decays are negligible. Decays including a dimuon from resonances, like ω→ π0μ+μ−and η→μ+μ−γ, do not produce peaking structures in the mass distribution, and are accounted for in the combinatorial background. 5 Search sensitivity The observed number of K0 S→μ+μ−candidates is converted into a branching fraction using the normalisation mode and its precisely known branching fraction B(K0 S→ π+π−)=0.6920 ±0.0005 [4]. The computation is made in every BDTcb bin iand trigger category jas follows B(K0 S→μ+μ−)=B(K0 S→π+π−)·ππ μμ ij · Nμμ ij Nππ ≡αijNμμ ij ,(1) 123 Eur. Phys. J. C (2017) 77 :678 Page 5 of 12 678 Table 1 Values of the single candidate sensitivity αij and the number of candidates NK ij compatible with the K0 Smass (reconstructed mass in the range [492,504]MeV/c2), for each BDTcb bin iand trigger category j. Only statistical uncertainties are given. The first uncertainty is uncorrelated, while the second is fully correlated among the BDTcb bins of the same trigger category Bin iαiTOSμ(×10−10)α iTOSμμ (×10−9)NK iTOSμNK iTOSμμ 17.48 ±0.84 ±0.16 5.30 ±0.72 ±0.12 49 13 27.72 ±0.87 ±0.17 4.71 ±0.63 ±0.10 28 9 37.85 ±0.89 ±0.18 4.88 ±0.65 ±0.11 9 14 47.93 ±0.89 ±0.19 4.66 ±0.62 ±0.10 18 10 57.53 ±0.85 ±0.18 4.65 ±0.61 ±0.10 6 3 67.78 ±0.88 ±0.19 4.95 ±0.66 ±0.11 2 2 77.56 ±0.85 ±0.19 4.60 ±0.61 ±0.10 3 1 87.90 ±0.89 ±0.19 5.00 ±0.67 ±0.11 2 1 97.81 ±0.88 ±0.18 4.72 ±0.63 ±0.11 1 1 10 7.75 ±0.87 ±0.17 4.66 ±0.62 ±0.11 0 0 where Nμμ ij and Nππ denote the background-subtracted yields for the signal and normalisation modes, respectively. The total selection efficiencies can be factorised as ππ μμ ij =ππ sel μμ sel × ππ trig μμ trig;j×1 μμ BDT;ij ×1 μID;ij .(2) The first factor refers to the offline selection requirements, which are applied identically to both modes and cancel to first order in the ratio; the residual difference is mainly due to the different interaction cross-sections for pions and muons with the detector material, and is estimated from simulation. The second factor is the ratio of trigger efficiencies; the efficiency for the signal is determined from simulation, with its systematic uncertainty estimated from data-driven checks, while that for the normalisation mode is the prescale factor of the random trigger used to select K0 S→π+π−, (9.38±1.01)×10−8. The third factor reflects the fraction of candidates in each BDTcb bin, and is also determined from simulation. Finally, the efficiency of the BDTμrequirement is obtained from the B+→J/ψ K+calibration sample described in Sect. 4, for each BDTcb bin and trigger category. To account for the difference between the kaon pTspectra observed in the K0 S→π+π−decays in data and simulation, all efficiencies obtained from simulation are computed in six roughly equally populated pTbins. A weighted average of the efficiencies is then performed, where the weights are determined from the yields in each bin observed in data for K0 S→π+π−candidates. The resulting values for the single candidate sensitivity αij are reported in Table 1. The quoted uncertainties are statistical only. They are separated between the uncertainty on μμ BDT;ij, due to the limited statistics of simulated data and uncorrelated among BDTcb bins, and all the other statistical uncertainties, which are conservatively considered as fully correlated among bins within the same trigger category. Table 1also presents the number of candidates around Table 2 Relevant systematic uncertainties on the branching fraction. They are separated, using horizontal lines, into relative uncertainties on (i) αij, (ii) on the signal yield from the signal model used in the mass fit, and (iii) on the branching fraction, obtained combining the two categories, from the background model Source TOSμTOSμμ Tracking (%) 0.4 0.4 Selection (%) 1.9 1.8 Trigger (%) 8.1 11.5 K0 SpTspectrum (%) 4.3 4.3 Muon identification (%) 0.2 0.3 Signal mass shape (%) 0.8 0.8 Background shape (%) 0.9 the K0 Smass. The separation between signal and background is presented in Sect. 7. 6 Systematic uncertainties Several systematic effects, summarised in Table 2, contribute to the uncertainty on the normalisation factors. Tracking efficiencies are not perfectly reproduced in simulated events. Corrections based on a J/ψ →μ+μ−data control sample are determined as a function of the muon pand η. The average effect of these corrections on the ratio ππ sel /μμ sel and its standard deviation, added in quadrature, leads to a systematic uncertainty of 0.4%. The distributions of all variables relevant to the selection are compared in data and simulation for K0 S→π+π− decays. The largest differences are found in the kaon pTand its decay vertex radial position. The effect on ππ sel /μμ sel of applying a two-dimensional weight to account for these discrepancies is taken as a systematic uncertainty, and amounts to a relative 1.9 and 1.8% for the TOSμand TOSμμ trigger categories, respectively. 123 678 Page 6 of 12 Eur. Phys. J. C (2017) 77 :678 The difference between data and simulation in the kaon pT spectrum could also affect the other factors in the computation of αij. An additional uncertainty is assigned by repeating the whole calculation with a finer binning in pT. Due to the limited size of the data samples, this is possible only in the TOSμcategory. The average relative change in αij,4.3%, is assigned as an uncertainty for both categories. A specific cross-check is performed to validate the efficiencies predicted by the simulation for the BDTcb requirements. An alternative discriminant is made using a signal training sample consisting of trigger-unbiased K0 S→π+π− decays, selected with additional kinematic criteria which mimic the effect of the muon trigger selections. The distributions of this alternative discriminant in data and simulation are found to agree within the statistical uncertainty, and no systematic uncertainty is assigned. The uncertainty due to the simulation of TOS selections in the first two trigger stages is assessed by comparing the trigger efficiency in simulation and data, using a control sample of B+→J/ψ K+decays. The resulting relative differences, 8.1% for TOSμand 11.5% for TOSμμ, are assigned as systematic uncertainties. No uncertainty is considered for the selection in the last trigger stage, which is based on the same offline kinematic variables used in the selection, for which a systematic uncertainty is already assigned. The uncertainty on μID;ij is estimated from half the difference between the values obtained with and without the weighting of the B+→J/ψ K+sample used in the determination of the muon identification efficiency. This results in an uncertainty of 0.2 and 0.3% for the TOSμand TOSμμ categories, respectively, which is comparable to the statistical uncertainties on these efficiencies due to the limited size of the B+→J/ψ K+samples. Systematic uncertainties on the signal yields Nμμ ij are related to the assumed models for the reconstructed K0 S mass distribution, determined from simulation. Possible discrepancies from the shape in data are estimated by comparing the shape of the invariant mass distribution in data and simulation for K0 S→π+π−decays, leading to a relative 0.8% systematic uncertainty on the signal yield. The final fit for the determination of the branching fraction is performed with two different background models, as discussed in Sect. 7. This leads to a relative variation on the branching fraction of 0.9%, which is assigned as a systematic uncertainty. 7 Results The μ+μ−mass distribution of the signal candidates is fitted in the range [470,600]MeV/c2to determine the signal and background yield in each trigger category and BDTcb bin. The mass distribution of simulated signal candidates is best described by a Hypatia function [28]. Its parameters are determined from simulation and fixed in the fit to data. In the background model, a power law function describes the tail of the double-misidentification background from K0 S→π+π− decays, affecting the mass region below the K0 Smass, while the combinatorial background mass distribution is described by an exponential function. The background model is validated on simulation, and its parameters are left free in the fit to data to account for possible discrepancies. An alternative combinatorial background shape, based on a linear function, is used instead of the exponential function to determine a systematic uncertainty due to the choice of the background shape. The signal yields in each BDT bin for the two trigger categories are all compatible with the absence of K0 S→μ+μ−candidates. The μ+μ−invariant mass distributions for the two highest BDTcb bins, which exhibit the best signal-to-background ratio and therefore the best sensitivity for a discovery, are shown in Fig. 2. A simultaneous maximum likelihood fit to the dimuon mass in all BDTcb bins is performed, using the values of αij given in Table 1and the normalization channel yield Nππ,to determine the branching fraction. The K0 S→π+π−candidates are counted within the mass region [460,530]MeV/c2, leading to Nππ =70 318 ±265. The quoted systematic uncertainties are included in the likelihood computation as nuisance parameters with Gaussian uncertainties. A posterior probability is obtained by multiplying the likelihood by a prior density, which is computed as the product of the likelihood from the 2011 analysis and a flat prior over the positive range of the branching fraction. Limits are obtained by integrating 90% (95%)of the area of the posterior probability distribution provided by the fit, as shown in Fig. 3. Due to the much larger sensitivity achieved with the 2012 data, the inclusion of the 2011 data result does not have a significant effect on the final limit, and a uniform prior would have provided very similar results. The expected upper limit, and the compatibility with background-only hypothesis have been computed by means of pseudoexperiments, where samples of background events are randomly generated according to the mass distribution obtained by the best fit to data. The median expected upper limit and its ±1σrange is B(K0 S→ μ+μ−)<0.95+0.42 −0.27 (1.17+0.45 −0.31)×10−9at 90% (95%)CL. The observed limit is B(K0 S→μ+μ−)<0.8(1.0)×10−9at 90% (95%)CL. The compatibility of the experimental measurement with the background-only model, expressed in terms of p value is 0.52. In conclusion, a search for the K0 S→μ+μ−decay based on a data sample corresponding to an integrated luminosity 123 Eur. Phys. J. C (2017) 77 :678 Page 7 of 12 678 ] 2 c [MeV/ - μ + μ m 500 550 600 ) 2 cCandidates / ( 1.3 MeV/ 2− 10 1− 10 1 10 2 10 500 550 600 Pull 5− 0 5 LHCb bin 9 μ TOS ] 2 c [MeV/ - μ + μ m 500 550 600 ) 2 cCandidates / ( 1.3 MeV/ 2− 10 1− 10 1 10 2 10 500 550 600 Pull 5− 0 5 LHCb bin 10 μ TOS ] 2 c [MeV/ - μ + μ m 500 550 600 ) 2 cCandidates / ( 1.3 MeV/ 2− 10 1− 10 1 10 2 10 500 550 600 Pull 5− 0 5 LHCb bin 9 μμ TOS ] 2 c [MeV/ - μ + μ m 500 550 600 ) 2 cCandidates / ( 1.3 MeV/ 2− 10 1− 10 1 10 2 10 500 550 600 Pull 5− 0 5 LHCb bin 10 μμ TOS Fig. 2 Fits to the reconstructed kaon mass distributions, for the two most sensitive BDTcb bins in the two trigger categories, TOSμand TOSμμ. The fitted model is shown as the solid blue line, while the combinatorial background and K0 S→π+π−double misidentification are overlaid with dotted red and dashed green lines, respectively. For each fit, the pulls are shown on the lower smaller plots 9 10×) - μ + μ 0 S B(K 0123 CL 0.8 0.85 0.9 0.95 1 LHCb Fig. 3 Confidence level of exclusion for each value of the K0 S→ μ+μ−branching fraction. The regions corresponding to 90% and 95% CL are emphasised in green (dark shading) and yellow (light shading), respectively of 3 fb−1of proton-proton collisions, collected by the LHCb experiment at centre-of-mass energies √s=7 and 8 TeV, improves the upper limit for this decay by a factor 11 with respect to the previous search published by LHCb [6], which is superseded by this result. Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFINHH (Romania), CBPF (Brazil), PL-GRID (Poland) and OSC (USA). 123 678 Page 8 of 12 Eur. Phys. J. C (2017) 77 :678 We are indebted to the communities behind the multiple open source software packages on which we depend. 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