Measurement of the time-integrated CP asymmetry in D 0 → K 0 S K 0 S decays
LHCb collaboration, R. Aaij, B. Adeva, M. Adinolfi, C.A. Aidala, Z. Ajaltouni, S. Akar, P. Albicocco, J. Albrecht, F. Alessio, M. Alexander, A. Alfonso Albero, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. An, L. Anderlini, G. Andreassi, M. Andreotti, J.E. Andrews, R.B. Appleby, F. Archilli, P. d'Argent, J. Arnau Romeu, A. Artamonov, M. Artuso, K. Arzymatov, E. Aslanides, M. Atzeni, S. Bachmann, J.J. Back, S. Baker, V. Balagura, W. Baldini, A. Baranov, R.J. Barlow, S. Barsuk, W. Barter, F. Baryshnikov, V. Batozskaya, B. Batsukh, V. Battista, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, A. Beiter, L.J. Bel, N. Beliy, V. Bellee, N. Belloli, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, S. Beranek, A. Berezhnoy, R. Bernet, D. Berninghoff, E. Bertholet, A. Bertolin, C. Betancourt, F. Betti, M.O. Bettler, M. van Beuzekom, Ia. Bezshyiko, L. Bian, S. Bifani, P. Billoir, A. Birnkraut, A. Bizzeti, M. Bjørn, T. Blake, F. Blanc, S. Blusk, D. Bobulska, V. Bocci, O. Boente Garcia, T. Boettcher, A. Bondar, N. Bondar, S. Borghi, M. Borisyak, M. Borsato, F. Bossu, M. Boubdir, T.J.V. Bowcock, C. Bozzi, S. Braun, M. Brodski, J. Brodzicka, D. Brundu, E. Buchanan, A. Buonaura, C. Burr, et al. (714 additional authors not shown)
EEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-EP-2018-133LHCb-PAPER-2018-01220 November 2018
Measurement of the time-integrated CP asymmetry in D → K S K S decays LHCb collaboration † Abstract
A measurement of the time-integrated CP asymmetry in D → K S K S decays isreported. The data correspond to an integrated luminosity of about 2 fb − collectedin 2015–2016 by the LHCb collaboration in pp collisions at a centre-of-mass energyof 13 TeV. The D candidate is required to originate from a D ∗ + → D π + decay,allowing the determination of the flavour of the D meson using the pion charge.The D → K + K − decay, which has a well measured CP asymmetry, is used as acalibration channel. The CP asymmetry for D → K S K S is measured to be A CP ( D → K S K S ) = (4 . ± . ± . , where the first uncertainty is statistical and the second is systematic. This result iscombined with the previous LHCb measurement at lower centre-of-mass energies toobtain A CP ( D → K S K S ) = (2 . ± . ± . . Published in JHEP 11 (2018) 048 c (cid:13) † Authors are listed at the end of this paper. a r X i v : . [ h e p - e x ] N ov i Introduction
In the Standard Model, violation of charge-parity ( CP ) symmetry originates from thepresence of a single phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1]. Experi-mental results support the CKM mechanism for CP violation, but additional sources of CP violation are needed to explain cosmological observations of the relative abundance ofmatter and antimatter in the universe [2]. In the charm sector, CP violation has not yetbeen observed, but measurements of CP asymmetries in Cabibbo-suppressed D → h + h − decays ( h = π, K ) have reached 0.2% and 0.03% precision for time-integrated [3] andindirect CP asymmetries [4], respectively.The D → K S K S decay is a promising discovery channel for CP violation in charmdecays [5]. Only loop-suppressed amplitudes and exchange diagrams that vanish in theSU(3) flavour limit contribute to this decay. These amplitudes can have different strongand weak phases and are of similar size. The time-integrated CP asymmetry, A CP , in D → K S K S decays may therefore be enhanced to an observable level [6], and could beas large as 1.1% [5]. Examples of such diagrams are shown in Fig. 1. The most precisemeasurement of this asymmetry to date, A CP ( K S K S ) = ( − . ± . ± . A CP in the decay D → K S K S using LHCb data collected in 2015 and 2016.¯ uc s ¯ dd ¯ s ¯ uc s ¯ dd ¯ s Figure 1: Exchange (left) and penguin annihilation (right) diagrams contributing to the D → K S K S amplitude. Based on Ref. [5]. The measurement of the CP asymmetry, defined as A CP ( K S K S ) ≡ Γ( D → K S K S ) − Γ( D → K S K S )Γ( D → K S K S ) + Γ( D → K S K S ) , (1)requires knowledge of the flavour of the D meson at production. A sample of flavour-tagged D → K S K S decays is obtained by selecting D ∗ + mesons that are produced in the primaryinteraction (hereafter referred to as prompt), with the subsequent decay D ∗ + → D π + . The charge of the pion in this decay identifies the flavour of the accompanying D meson.The effect of D − D mixing [10] is negligible compared to the precision of this analysisand is not considered further.The experimentally measured quantity is the raw asymmetry, defined as A raw ≡ N D − N D N D + N D , (2) The inclusion of charge-conjugate processes is implied throughout this document, unless explicitlyspecified. N D is the measured yield of D ∗ + → D π + , D → K S K S decays and N D is themeasured yield of D ∗− → D π − , D → K S K S decays. This observable is related to the CP asymmetry by the expression, valid for small asymmetries, A raw ≈ A CP + A prod + A det , (3)where A prod is the D ∗± production asymmetry, defined as A prod ≡ σ ( D ∗ + ) − σ ( D ∗− ) σ ( D ∗ + )+ σ ( D ∗− ) , and A det is the π ± tag detection asymmetry, defined as A det ≡ (cid:15) ( π +tag ) − (cid:15) ( π − tag ) (cid:15) ( π +tag )+ (cid:15) ( π − tag ) . The symbol π ± tag refers tothe pion in the D ∗± decay. To a very good approximation, knowledge of A det and A prod is unnecessary when using a calibration channel with the same production and taggingmechanism. The decay channel D → K + K − is used for this purpose. The productionand detection asymmetries cancel when taking the difference of the raw asymmetries:∆ A CP ≡ A raw ( K S K S ) − A raw ( K + K − ) (4)= A CP ( K S K S ) − A CP ( K + K − ) . (5)The quantity A CP ( K + K − ) has been measured with a precision of 0.2% [3], thus allowingthe determination of A CP ( K S K S ). The LHCb detector [11, 12] is a single-arm forward spectrometer covering thepseudorapidity 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 surrounding the pp interaction region, a large-area silicon-stripdetector (TT) 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 ofthe magnet. The tracking system provides a measurement of momentum, p , of chargedparticles 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 impactparameter (IP), is measured with a resolution of (15 + 29 /p T ) µ m, where p T is the com-ponent of the momentum transverse to the beam, in GeV /c . Different types of chargedhadrons are distinguished using information from two ring-imaging Cherenkov (RICH)detectors. Photons, electrons and hadrons are identified by a calorimeter system consistingof scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadroniccalorimeter. Muons are identified by a system composed of alternating layers of iron andmultiwire proportional chambers. The magnetic field deflects oppositely-charged particlesin opposite directions and this can lead to detection asymmetries. Periodically reversingthe magnetic field polarity throughout the data taking almost cancels the effect. Theconfiguration with the magnetic field pointing upwards (downwards), MagUp (MagDown),bends positively (negatively) charged particles in the horizontal plane towards the centreof the LHC ring.The online event selection is performed by a trigger, which consists of a hardware stage,based on information from the calorimeter and muon systems, followed by a softwarestage, which applies a full event reconstruction. At the hardware trigger stage, eventsare required to have a muon with high p T or a hadron, photon or electron with hightransverse-energy deposit in the calorimeters.2imulated events are used at various phases of the analysis. In the simulation, pp collisions are generated using Pythia [13] with a specific LHCb configuration [14]. Decaysof hadronic particles are described by
EvtGen [15], in which final-state radiation isgenerated using
Photos [16]. The interaction of the generated particles with the detector,and its response, are implemented using the
Geant4 toolkit [17] as described in Ref. [18].
The 2015 and 2016 data samples collected in pp collisions at 13 TeV, which correspondto about 2 fb − of integrated luminosity, are used in this analysis. Candidates arereconstructed in the decay D ∗ + → D π + , followed by D → K S K S and then K S → π + π − .The hardware trigger decision is required to be based either on the transverse energydeposited in the hadronic calorimeter by a charged particle from the decay of the D meson, or on signatures not associated with the D ∗ + decay, such as a high- p T muon, ora high transverse-energy deposit in the electromagnetic or hadronic calorimeters. Thefirst stage of the software trigger selects a sample with enhanced heavy-flavour content byrequiring the presence of a large IP, high- p T charged particle. In the second stage of thesoftware trigger, each selected event is required to contain at least one fully-reconstructedcandidate for the D ∗ + → D π + , D → K S K S decay.The decays K S → π + π − are reconstructed in two different categories: the first involving K S mesons that decay early enough for the decay products to be reconstructed in the vertexdetector; and the second containing K S candidates that decay outside the acceptance ofthe vertex detector, but within the TT acceptance. These categories are referred to as long and downstream , respectively. The long category has better mass, momentum anddecay-vertex resolution than the downstream category. In this analysis at least one K S ineach D decay is required to be of the long type. There are therefore two subsamples used:one where both K S candidates are long and the other where one is long and the otheris downstream. These are referred to as the LL and LD subsamples, and are analysedseparately, since they exhibit different resolutions. One or more of the charged decayproducts from a long K S meson is required to activate the first stage of the softwaretrigger. The pion candidates used in the K S reconstruction are required to be high-qualitytracks, using the χ /ndf of the track fit and the output P fake of a multivariate classifier,trained to identify fake tracks, that combines information from the particle identificationand tracking systems. To ensure that pion candidates do not originate from the PV, theyare required to satisfy χ >
36. The quantity χ for a given particle is defined as thedifference in the vertex fit χ of the PV associated to the particle, reconstructed withand without the particle being considered. For downstream K S candidates, the pions arerequired to satisfy p > /c and p T >
175 MeV /c .Two oppositely charged pions are used to form K S candidates. The vertex fit isrequired to satisfy χ <
30 and the χ is required to be greater than 9 (4) for long(downstream) K S candidates. Furthermore, long (downstream) K S candidates are requiredto satisfy p T >
500 (750) MeV /c .Two reconstructed K S candidates are paired to form D candidates, requiring χ < p T of the K S candidates is required to exceed 1500(2000) MeV /c for LL (LD) candidates. The angle between the D momentum and thevector connecting the PV to the D decay vertex is required to be less than 34.6 mrad.3he measured decay time of the D meson is required to be greater than 0.2 ps. Finally,the D mass is required to be within 20 MeV /c of the known value [10].A pion candidate ( π +tag ) is added to a reconstructed D meson to form a D ∗ + candidate,with a D ∗ + vertex fit which is required to have χ <
25. The π +tag candidate is requiredto have p T >
100 MeV /c , and to pass through regions of the detector that are known tohave a small detector asymmetry [8]. A small fraction of π ± tag candidates are reconstructedwith the wrong charge assignment, and are removed by a selection on track quality.An important source of background is due to the presence of D → K S π + π − decays,where the π + π − pair satisfies the K S selection. In principle, the contribution of thischannel can be substantial, due to its large branching fraction, but it is effectively reducedby placing a requirement on the K S flight distance (FD) and on the mass of the K S candidates. The quantity χ is the square of the measured K S flight distance dividedby the square of its uncertainty. Figure 2 shows a two-dimensional plot of the value ofthe quantity log χ for K S pairs in the LL sample. In the figure, four separate regionsare visible. The upper right part of the plot, where both K S candidates have significantflight distances, is the D → K S K S signal, while the upper left and lower right regionscorrespond to D → K S π + π − decays. The lower left is populated by D → π + π − π + π − decays and combinatorial background. A requirement on χ is only necessary for long K S candidates, since downstream K S candidates decay far from the PV by construction.For the LL subsample the requirement on the two K S candidates ( K S and K S ) is[log χ ( K S ) − + [log χ ( K S ) − < , (6)while for the LD sample log χ ( K S L ) > . K S candidate.The K S mass requirements are (cid:113) [ m ( K S ) − m K ] + [ m ( K S ) − m K ] < . /c , (7) - ) S10 K ( FD2 c log - ) S K ( F D c l og LHCb
Figure 2: Two-dimensional distribution of the logarithm of the K S flight distance signifi-cance (log χ ) for the two K S candidates in the LL subsample of D → K S K S decays. The D → K S K S signal can be observed in the upper right region of the plot. The contour correspondsto Eq. 6. m K = 497 . /c [10], and (cid:115)(cid:20) m ( K S L ) − m K . /c (cid:21) + (cid:20) m ( K S D ) − m K
15 MeV /c (cid:21) < , (8)for LD candidates. This selection takes into account the difference in resolution between m ( K S L ) and m ( K S D ). The log χ ( K S ) and m ( K S ) regions corresponding to signaland peaking-background candidates are identified using simulations. They are furtheroptimised on charge-integrated data by minimising the expected statistical uncertainty on A raw .Events in which the D ∗ + meson is not produced in the primary interaction, but insteadis the product of a b -hadron decay, are characterised by a different production asymmetryand are treated as background. These so-called secondary D ∗ + candidates tend to havelarger values of χ ( D ) than prompt D ∗ + candidates and are suppressed by requiringlog χ ( D ) < . χ ( π +tag ) < . k nearest neighbours (taken fromthe training sample of signal and background events), where the distance is calculated inthe n -dimensional space of the input variables and k is a positive integer. The trainingsample uses simulated events for the signal and data events from the D mass sidebandsfor the background. A wide range of input variables based on track and vertex quality, thetransverse momenta of K S and D candidates, helicity angles of the K S and D decays andparticle identification information on the pions in the D decays was initially considered.Variables depending on the π ± tag track are not included in the classifier to avoid introducingpossible bias on the asymmetry measurement. The actual variables used, the value of k , and the selection on the classifier output are optimised separately for the LL and LDsubsamples, using the expected statistical uncertainty on the raw asymmetry as a figureof merit.For the D → K + K − control channel, an attempt is made to keep the selection similarto the D → K S K S channel, although some selections made at the software trigger levelare different for the two channels. Charged tracks positively identified as kaons in theRICH detectors are selected to reconstruct D candidates. The kaons are required tosatisfy χ >
4. For the D candidates, at least one of the kaons is required to have p T > /c . The sum of the kaon momenta is required to exceed 5 GeV /c and the D p T is required to be at least 1 GeV /c . Furthermore, the angle between the D momentumvector and the vector connecting the primary and decay vertices is required to be lessthan 17.3 mrad. The following selections are the same as for the D → K S K S channel: π ± tag fiducial cuts, fake-track probability and χ selection; and requirements on D χ and invariant mass. The raw asymmetry for D → K S K S is determined by separating the selected candidatesinto subsets tagged by positively and negatively charged pions. A simultaneous unbinned5aximum likelihood fit to their ∆ m distributions is performed, where ∆ m is the differenceof the reconstructed invariant mass of the D ∗ + and the D candidates. The calculation of∆ m is made after the full decay chain has been reconstructed using a mass constraint onthe K S candidates and constraining the D ∗ + candidate to originate from the PV.The signal shape is modelled using the Johnson S U distribution [20], which consists ofa core Gaussian-like shape but allows for an asymmetric tail S ( x ; µ, σ, δ, γ ) ∝ (cid:34) (cid:18) x − µσ (cid:19) (cid:35) − × exp (cid:40) − (cid:20) γ + δ sinh − (cid:18) x − µσ (cid:19)(cid:21) (cid:41) . (9)The background shape is described with an exponential function multiplied by a thresholdfactor and is zero below a fixed endpoint, which is set to the pion mass m π B ( x ; m π , χ ) ∝ √ x − m π × exp (cid:18) χ xm π (cid:19) . (10)The likelihood function is parametrised in terms of A CP and the expected total number ofevents N exp = n sig + n bkg L = e − N exp N obs ! (cid:89) i (cid:20) n sig q i A rawsig S (∆ m ) + n bkg q i A rawbkg B q i (∆ m ) (cid:21) , (11)where n sig and n bkg are the signal and background yields, respectively, and the parameter q i = ± D ∗± candidate and N obs is the total number of candidates.The signal raw asymmetry A rawsig is a free parameter in the fit. The free parameter A rawbkg allows for a possible asymmetry in the combinatorial background. The four parametersin Eq. 9 defining the signal probability distribution function (PDF) are common to the D ∗ + and D ∗− samples, while the parameter describing the background shape is allowedto differ between the two subsamples. For the LL sample, there are ten free parameters.To achieve convergence of the fit in the smaller LD sample, it is necessary to fix the twoparameters that describe the asymmetric tail in the signal PDF to the values obtainedfrom the charge-integrated LL subsample. Based on studies of simulated events, the tailparameters of the LL and LD subsamples are expected to be compatible. Separate fitsare performed for the two magnet polarities.Table 1 shows the results of the simultaneous fits to the D → K S K S candidates. Theresults on each subset of the data are compatible with each other. The fit is shown inFig. 3 for the samples collected with the MagUp magnetic field configuration.For the D → K + K − channel, binned χ fits are performed to the ∆ m distributionsof the positively and negatively tagged D decays. The sample consists of 8 . × selected candidates for the MagDown magnet polarity and 5 . × candidates for theMagUp magnet polarity. The signal is modelled with a Johnson S U distribution plusa Gaussian distribution, while the background shape is described by a fourth-degreepolynomial multiplied by a √ ∆ m − m π threshold factor. There are 12 free parameters,and 150 bins, in each ∆ m fit. The χ probabilities associated to the fits are 28% (20%) forthe negatively (positively) tagged D decays, and 23% (3%) for the negatively (positively)tagged D decays, in the MagUp and MagDown magnet polarities, respectively. Figure 4shows the results for the MagUp magnet polarity fit. The results obtained for the two6 able 1: Fit results on the D → K S K S LL and LD samples for each magnet polarity, where N obs represents the number of candidates fitted. The purity is determined in the range144 . < ∆ m < . /c . For each sample, a χ test statistic for the fitted model andbinned data for positively and negatively charged candidates is constructed. The quantity P fit is the probability of observing a χ value greater than that observed in the fit to real data,determined using simulated pseudoexperiments sampled from the fitted model. A rawsig n sig A rawbkg Purity P fit (%) N obs LL MagUp 0 . ± .
057 346 ± − . ± .
069 0.92 48 589LL MagDown 0 . ± .
052 413 ± − . ± .
068 0.92 43 675LD MagUp − . ± .
102 156 ± − . ± .
044 0.67 93 758LD MagDown − . ± .
107 152 ± − . ± .
038 0.60 14 950 ] c /V [Me m D
140 142 144 146 148 150 152 154 ) c / V C a nd i d a t e s / ( . M e LHCb S0 K S0 K fi D LL DataTotalBkg (a) ] c /V [Me m D
140 142 144 146 148 150 152 154 ) c / V C a nd i d a t e s / ( . M e LHCb S0 K S0 K fi D LL DataTotalBkg (b) ] c /V [Me m D
140 142 144 146 148 150 152 154 ) c / V C a nd i d a t e s / ( . M e LHCb S0 K S0 K fi D LD DataTotalBkg (c) ] c /V [Me m D
140 142 144 146 148 150 152 154 ) c / V C a nd i d a t e s / ( . M e LHCb S0 K S0 K fi D LD DataTotalBkg (d)
Figure 3: Results of fits to ∆ m distributions of D → K S K S candidates for MagUp magnetpolarity. The fit to (a) D ∗ + → D π + and (b) D ∗− → D π − candidates for the LL sample andthe fit to (c) D ∗ + → D π + and (d) D ∗− → D π − candidates for the LD sample are shown. Theblack crosses represent the data points, the solid blue curve is the total fit function, and thedashed blue curve is the background component of the fit. magnet polarities are A raw ( K + K − ) MagUp = − . ± . , (12) A raw ( K + K − ) MagDown = 0 . ± . , where the uncertainties are statistical. The difference in the MagUp and MagDown valuesof A raw ( K + K − ) is an indication of a significant π ± tag detection asymmetry, which dependson the magnetic field orientation. 7 c /V [Me m D
140 142 144 146 148 150 152 154 ) c / V C a nd i d a t e s / ( . M e LHCb - K + K fi D DataTotalBkg (a) ] c /V [Me m D
140 142 144 146 148 150 152 154 ) c / V C a nd i d a t e s / ( . M e LHCb - K + K fi D DataTotalBkg (b)
Figure 4: Results of fits to ∆ m distributions of D → K + K − candidates for the MagUpmagnet polarity. The fits to (a) D ∗ + → D π + candidates and (b) D ∗− → D π − candidates areshown. The black points represent the data, the dashed blue and solid blue curves represent thebackground component and the total fit function, respectively. The main source of systematic uncertainty arises from the determination of A raw on the D → K S K S sample. Possible bias in the fitting procedure is evaluated using simulatedpseudoexperiments. In particular, the uncertainty related to the choice of the signal modelis evaluated by using the nominal model to fit samples generated with two alternativemodels for the signal PDF: either a sum of two Gaussians with a common mean (for theLL sample) or a single Gaussian (for the LD sample). The background PDF is varied bymodifying its behaviour at threshold. Systematic uncertainties of 5 × − and 0.01 forthe LL and LD samples, respectively, are assigned based on this study. As a cross-check,the background shapes are constrained to be the same for the D ∗ + and D ∗− samples,and the resulting asymmetry is compatible with the nominal. For the D → K + K − fit, an alternative procedure is used to evaluate the systematic uncertainty associatedwith the signal PDF. In this case, the signal region ( ± . /c around the signalmean) is excluded and only the background shape is fit. The yield is then determinedby estimating the background in the signal region by interpolating the fitted backgroundfunction. Additionally, alternative background shapes are tried, varying the degree of thepolynomial. Based on these studies a systematic uncertainty of 2 × − is assigned to A raw ( K + K − ).The contribution of the residual background of D → K S π + π − decays to the fittedLL and LD signal yields is estimated to be (3 . ± . . ± . K S π + π − background asymmetry, determined frombackground-dominated regions of the χ distributions, to estimate contributions to thesystematic uncertainty of 4 × − and 5 × − , for the LL and LD samples. Anothercontribution comes from the residual fraction of secondary decays, which leads to asystematic uncertainty for this source of 2 × − and 3 × − for the LL and LDsamples. In this case an upper limit of 0.02 for the maximum difference in the productionasymmetries of D ∗± mesons and b -hadrons is assumed [21–23].Potential trigger biases are studied using tagged D → K + K − decays, by comparingthe raw asymmetries obtained in the subsample in which the trigger decision is based onthe charged particles from the decay of the D meson, and in the subsample in which8 able 2: Systematic uncertainties on the quantities A raw and ∆ A CP . The total systematicuncertainties in the last row are obtained by summing the corresponding contributions in eachcolumn in quadrature. Uncertainties are expressed in units of 10 − . Source A raw (LL) A raw (LD) ∆ A CP (LL) ∆ A CP (LD)Fit procedure 5 10 5 10 K S π + π − background 4 5 4 5Secondaries 2 3 2 3Wrong π ± tag charge 2 2 – –Trigger selection 5 5 5 5 K + K − fit procedure – – 2 2Residual detection – – 2 2asymmetryTotal 9 13 9 13the trigger decision is not associated with the D ∗ + decay. The sum in quadrature of thedifference (albeit not statistically significant) and of its statistical uncertainty is assignedas a systematic uncertainty, which accounts for residual trigger-induced biases in thedifference of measured asymmetries for signal and control channels. This uncertaintyamounts to 5 × − for both the LL and LD samples. The small probability of assigningthe wrong charge to the π ± tag candidate results in a systematic uncertainty of 2 × − for both the LL and LD samples. This is obtained by varying the selection on the P fake value of π ± tag candidates. This uncertainty cancels for ∆ A CP . For each neutral kaon in thefinal state, asymmetries arising from regeneration and from mixing and CP violation inthe K − K system are suppressed at the O (10 − ) level [24]. Since they are expectedto affect D → K S K S and D → K S K S decays by the same amount, they cancel in A raw and therefore do not contribute to the systematic uncertainty.The cancellation of the production and detection asymmetries in the computationof ∆ A CP may not be perfect due to differences in the kinematics of the D → K S K S candidates and the D → K + K − candidates. The offline selection of the two channelsaims to keep the kinematics as similar as possible, but the different trigger selectionson the final states can introduce differences. The associated systematic uncertainty isevaluated by considering four kinematic variables: the transverse momentum and thepseudorapidity of the D ∗ + candidate and the π +tag candidate, respectively. For each variablea one-dimensional weighting is performed on the D → K + K − events such that they havethe same distribution as the D → K S K S sample. Then A raw ( K + K − ) is determined fromthe weighted sample. This is repeated for each of the four kinematic variables. The largestchange in A raw ( K + K − ) is taken as the systematic uncertainty and this is found to be2 × − for both the LL and LD samples. The systematic uncertainties are summarisedin Table 2. The procedure described in Sect. 1 is used to combine the results for the raw asymmetriesto obtain A CP ( K S K S ) for each of the LL and LD subsamples. For each of the subsamples,9 .2 - - ) S0 K S0 K ( CP A D AverageLD DownLD UpLL DownLL Up
Figure 5: Values of ∆ A CP obtained for both magnet polarities on the LL and LD samples, alongwith the average of these measurements. Only statistical uncertainties are shown. the difference ∆ A CP is calculated separately for the different magnet polarities using thefitted values of A raw (Table 1 and Eq. 12). The values of ∆ A CP corresponding to thetwo magnet polarities, which are found to be in good agreement (Fig. 5), are averagedby weighting with their statistical uncertainties. The systematic uncertainties are takenfrom Table 2. Using the LHCb measurement of A CP ( K + K − ) = (0 . ± . ± . A CP (LL) = 0 . ± . ± . , A CP (LD) = − . ± . ± . , where the first uncertainty is statistical and the second is systematic. These results arecombined by performing an average weighted by the total uncertainties and assumingthat the systematic uncertainties are fully correlated. The final result is A CP ( K S K S ) = 0 . ± . ± . . This measurement is systematically independent of the LHCb Run 1 measurement, A CP ( K S K S ) = − . ± . ± .
022 [8], and is compatible with it. An average, weightedby the total uncertainties, of the two measurements is performed to obtain A CP ( K S K S ) = 0 . ± . ± . . These results are compatible with the expectations of the Standard Model [5] and withprevious measurements [7, 9].
Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments for theexcellent performance of the LHC. We thank the technical and administrative staff at theLHCb 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 (Netherlands); MNiSWand 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),10IT and DESY (Germany), INFN (Italy), SURF (Netherlands), PIC (Spain), GridPP(United Kingdom), RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH(Romania), CBPF (Brazil), PL-GRID (Poland) and OSC (USA). We are indebted tothe communities behind the multiple open-source software packages on which we depend.Individual groups or members have received support from AvH Foundation (Germany),EPLANET, Marie Sk(cid:32)lodowska-Curie Actions and ERC (European Union), ANR, LabexP2IO and OCEVU, and R´egion Auvergne-Rhˆone-Alpes (France), Key Research Programof Frontier Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China),RFBR, RSF and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), HerchelSmith Fund, the Royal Society, the English-Speaking Union and the Leverhulme Trust(United Kingdom).
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Physikalisches Institut, RWTH Aachen University, Aachen, Germany Fakult¨at Physik, Technische Universit¨at Dortmund, Dortmund, Germany Max-Planck-Institut f¨ur Kernphysik (MPIK), Heidelberg, Germany Physikalisches Institut, Ruprecht-Karls-Universit¨at Heidelberg, Heidelberg, Germany School of Physics, University College Dublin, Dublin, Ireland INFN Sezione di Bari, Bari, Italy INFN Sezione di Bologna, Bologna, Italy INFN Sezione di Ferrara, Ferrara, Italy INFN Sezione di Firenze, Firenze, Italy INFN Laboratori Nazionali di Frascati, Frascati, Italy INFN Sezione di Genova, Genova, Italy INFN Sezione di Milano-Bicocca, Milano, Italy INFN Sezione di Milano, Milano, Italy INFN Sezione di Cagliari, Monserrato, Italy INFN Sezione di Padova, Padova, Italy INFN Sezione di Pisa, Pisa, Italy INFN Sezione di Roma Tor Vergata, Roma, Italy INFN Sezione di Roma La Sapienza, Roma, Italy Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam,Netherlands Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Krak´ow, Poland AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science,Krak´ow, Poland National Center for Nuclear Research (NCBJ), Warsaw, Poland Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS), Moscow, Russia Yandex School of Data Analysis, Moscow, Russia Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia Institute for High Energy Physics (IHEP), Protvino, Russia ICCUB, Universitat de Barcelona, Barcelona, Spain Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), Universidade de Santiago de Compostela,Santiago de Compostela, Spain European Organization for Nuclear Research (CERN), Geneva, Switzerland Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland Physik-Institut, Universit¨at Z¨urich, Z¨urich, Switzerland NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine University of Birmingham, Birmingham, United Kingdom H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom Department of Physics, University of Warwick, Coventry, United Kingdom STFC Rutherford Appleton Laboratory, Didcot, United Kingdom School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom Imperial College London, London, United Kingdom School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom Department of Physics, University of Oxford, Oxford, United Kingdom Massachusetts Institute of Technology, Cambridge, MA, United States University of Cincinnati, Cincinnati, OH, United States University of Maryland, College Park, MD, United States Syracuse University, Syracuse, NY, United States Pontif´ıcia Universidade Cat´olica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to
University of Chinese Academy of Sciences, Beijing, China, associated to
School of Physics and Technology, Wuhan University, Wuhan, China, associated to
Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China, associated to
Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to
Institut f¨ur Physik, Universit¨at Rostock, Rostock, Germany, associated to
Van Swinderen Institute, University of Groningen, Groningen, Netherlands, associated to National Research Centre Kurchatov Institute, Moscow, Russia, associated to
National University of Science and Technology ”MISIS”, Moscow, Russia, associated to
National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain,associated to
University of Michigan, Ann Arbor, United States, associated to