Contents lists available atScienceDirect
Physics
Letters
B
www.elsevier.com/locate/physletb
Measurement
of
CP
observables
in
B
±
→
D K
±
and
B
±
→
D
π
±
with two- and
four-body
D
decays
.
LHCb
Collaboration
a
r
t
i
c
l
e
i
n
f
o
a
b
s
t
r
a
c
t
Articlehistory: Received6April2016
Receivedinrevisedform8June2016 Accepted9June2016
Availableonline16June2016 Editor:W.-D.Schlatter
Measurements of CP observables in B±→D K± and B±→D
π
± decays are presented where the DmesonisreconstructedinthefinalstatesK±
π
∓,π
±K∓,K+K−,π
+π
−,K±π
∓π
+π
−,π
±K∓π
+π
−andπ
+π
−π
+π
−.ThisanalysisusesasampleofchargedBmesonsfromppcollisionscollectedbytheLHCb experimentin2011and2012,correspondingtoanintegratedluminosityof3.0 fb−1.VariousCP-violating effectsarereportedandtogetherthesemeasurementsprovideimportantinputforthedeterminationof theunitaritytriangleangleγ
.Theanalysisofthefour-pionD decaymodeisthefirstofitskind.©2016TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introduction
Asetofoverconstrainingmeasurementsoftheunitaritytriangle fromtheCKM matrixiscentral to thevalidationofthe Standard Model(SM)descriptionof
CP
violation[1].Ofthese,theleast-well measuredistheangleγ
≡
arg(
−
VudVub∗/
VcdVcb∗)
withaprecision, froma combination ofmeasurements, of about 7◦; this may be comparedwiththe 3◦ and<
1◦ precision on theother anglesα
and
β
[2,3].Amongstthethreeangles,γ
isuniqueinthatitdoes notdependonacouplingtothetopquarkandthusmaybe stud-iedattreelevel,largely avoidingpossibleinfluence fromnon-SMCPviolation.
Themostpowerfulmethodfordetermining
γ
intree-level de-cays isthrough measurement of relative partialwidths in B−→
D K− decays, where D represents a D0 or D0 meson.1 The am-plitude for the B−→
D0K− contribution is proportional to Vcb while the amplitude for B−
→
D0K− is proportional to Vub.By reconstructing hadronic D decays accessible to both D0 and D0mesons,phaseinformationmaybeextractedfromtheinterference ofthetwoamplitudes.Thesizeoftheresultingdirect
CP
violation isgovernedbythemagnitudeoftheratior
B oftheb
→
u¯
cs am-plitudetotheb
→
cus¯
amplitude.The relativelylargevalue ofr
B (about0.1)inB
−→
D K−decaysmeansthattherelativephaseof thetwointerferingamplitudescanbeobtained.Thisrelativephase hasaCP
-violating(weak)contributionandCP
-conserving(strong) contributionδ
B; a measurement of the total phase for both B+ and B− disentanglesγ
andδ
B.Similar interferenceeffects occur1 Theinclusionofcharge-conjugateprocessesisimpliedexceptinanydiscussion ofasymmetry.
in
B
±→
Dπ
±decays,albeitwithreducedsensitivitytothephases because,duetoadditionalCabibbosuppressionfactors,theratioof amplitudesisabout20timessmaller.The study of B−
→
D K− decays formeasurements ofγ
was first suggested for CP eigenstates of the D decay, for example the CP-even D→
K+K− and D→
π
+π
− decays, labelled here GLWmodes[4,5].TheargumenthasbeenextendedtosuppressedD
→
π
−K+decayswheretheinterplaybetweenthefavouredand suppressed decay paths in both the B− and the neutral D de-cays results in a large charge asymmetry. This is the so-called ADSmode[6],whichintroduces adependencyontheratioofthe suppressed andfavoured D decayamplitudesr
D andtheir phase differenceδ
D.The B−→ [
h+h−]
Dh− ADS/GLW decays(
h=
K,
π
)
havebeenstudiedattheB
factories[7,8]andatLHCb[9].This let-tercontainstheupdatedandimprovedresultusingboththe2011 and2012datasamples.The2012databenefitsfromahigher B±mesonproductioncross-sectionandamoreefficienttrigger,sothis updateisapproximatelyafactorfourincreaseinstatistics.
TheADS/GLWformalismcanbeextendedtofour-particle
D
de-cays. However, there are multiple intermediate resonances with differingamplituderatiosandstrongphaseswiththeconsequence that the interference in the B− decay, and hence the sensitivity toγ
,isdiluted[10].ForD
→
K−π
+π
+π
−andD
→
π
−K+π
+π
−decays this dilution is parameterised in terms of a coherence factor
κ
K3π, an effective strong phase difference averaged over all contributing resonancesδ
KD3π, and an overall suppressed-to-favoured amplitude ratio rKD3π. Best sensitivity toγ
is achieved using independent measurements of theκ
K3π andδ
K3πD param-eters, which have been determined using a sample of quantum-correlated D0D0 pairs [11,12], and by the study of D-mixing in
http://dx.doi.org/10.1016/j.physletb.2016.06.022
thisfinal state [13]. Asimilar dilutionparameter, labelledthe CP
fraction
F
4π+ canbedefinedfor
D
→
π
+π
−π
+π
−decays[14].For thisfinal state itis foundthat F+4π=
0.
737±
0.
028 [15],so that thedecaybehaveslikeaCP
-evenGLWmode,albeitwiththe inter-ferenceeffectsreducedbyafactor(
2F+4π−
1)
≈
0.
5.Thisletterincludesan analysisof B−
→ [
h+h−π
+π
−]
Dh− de-cays and supersedes the previous analysis of B−→
[
π
−K+π
+π
−]
Dh−[16] andcomplementsthestudyoftheB
−→
[
h+h−π
0]
Dh− modes [17]. The analysis of the four-pion D de-caymode isthefirstofits kind.In total,21measurementsof
CP
observablesare reported.Twooftheseare ratiosof thefavoured
B−
→
D0K−andB
−→
D0π
−partialwidths,RKf/π
=
(
B −→ [
f]
DK−
)
+
(
B+→ [ ¯
f]
DK+)
(
B−→ [
f]
Dπ
−)
+
(
B+→ [ ¯
f]
Dπ
+)
,
(1)where f is K−
π
+(
π
−π
+)
and¯
f is its charge-conjugate state. Three are double ratios that are sensitive to the partial widths of the (quasi-)GLW modes, f=
π
+π
−(
π
+π
−)
and K+K−, nor-malisedtothoseofthefavouredmodesofthesamemultiplicity,RK K
=
RK K K/π RKKπ/π
,
Rπ π
=
R π πK/π RKKπ/π
,
Rπ π π π
=
R π π π πK/π
RKKπ π π/π
.
(2)Fiveobservablesarechargeasymmetries,
Ahf
=
(
B −→ [
f]
Dh−
)
−
(
B+→ [ ¯
f]
Dh+)
(
B−→ [
f]
Dh−)
+
(
B+→ [ ¯
f]
Dh+)
,
(3)for h
=
K and f=
K−π
+(
π
−π
+),
π
+π
−(
π
+π
−)
and K+K−. There are a further three asymmetries for h=
π
and f=
π
+π
−(
π
+π
−)
andK
+K−.Fourobservablesarepartialwidthsof thesuppressedADSmodesrelativetotheircorrespondingfavoured decays,RADS¯f (h)
=
(
B −→ [ ¯
f]
Dh−
)
+
(
B+→ [
f]
Dh+)
(
B−→ [
f]
Dh−)
+
(
B+→ [ ¯
f]
Dh+)
,
(4)withwhichcomefourADS-modechargeasymmetries,
AADS¯f (h)
=
(
B−
→ [ ¯
f]
Dh−
)
−
(
B+→ [
f]
Dh+)
(
B−→ [ ¯
f]
Dh−)
+
(
B+→ [
f]
Dh+)
.
(5)An alternative formulationof the ADS observables measures the suppressedADSmodesrelativetotheirfavoured counterparts, in-dependentlyfor B+ andB− mesons,
R+(¯f h)
=
(
B +→ [
f]
Dh+
)
(
B+→ [ ¯
f]
Dh+)
,
R−(¯f h)=
(
B−
→ [ ¯
f]
Dh−
)
(
B−→ [
f]
Dh−)
.
(6)All the charge asymmetry measurements are affected by a pos-sible asymmetry in the B± production cross-section multiplied by any overall asymmetry from the LHCb detector, together de-noted as
σ
. This effective production asymmetry, defined asAB±
=
σ(B−)−σ(B+)
σ(B−)+σ(B+),ismeasuredinthisanalysisfromthecharge asymmetry ofthemostabundant B−
→ [
K−π
+]
Dπ
− and B−→
[
K−π
−π
+π
−]
Dπ
− modes.Thismeasurementisappliedasa cor-rection to all other CP asymmetry results. In these modes, the possibleCP
asymmetry,asderived fromexisting knowledgeofγ
and
r
B inthisdecay[18],issmallerthantheuncertaintyon exist-ing measurementsof the B± productionasymmetry [19].The CPasymmetryisthusassumedtobezerowithasmallsystematic un-certainty.Remainingdetection asymmetries,notably between K−
andK+,arecorrectedforusingcalibrationsamples.
2. Detector and simulation
The LHCb detector [20] is a single-arm forward spectrometer covering the pseudorapidity range 2
<
η
<
5, designed for the studyofparticles containing b orc
quarks.The detectorincludes a high-precision tracking systemconsisting ofa silicon-strip ver-tex detector surrounding the pp interaction region, a large-area silicon-strip detectorlocatedupstream ofadipole magnetwitha bending power ofabout 4 Tm, andthree stations of silicon-strip detectorsandstrawdrifttubesplaceddownstreamofthemagnet. Thetrackingsystemprovidesameasurementofmomentum, p,of chargedparticleswitharelativeuncertaintythatvariesfrom0.5% at low momentum to 1.0% at200 GeV/c. The minimum distance of atrackto aprimary vertex(PV), theimpact parameter (IP),is measured witha resolutionof(
15+
29/
pT)
μ
m,where pT isthe component ofthe momentum transverse to thebeam, inGeV/
c. Different types ofcharged hadronsare distinguished using infor-mationfromtworing-imagingCherenkovdetectors.Photons, elec-tronsandhadronsareidentifiedbyacalorimetersystemconsisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadroniccalorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The trigger consists of a hardware stage, based on information from the calorimeter and muon systems, followed by asoftware stage,in whichall chargedparticles withpT
>
500(
300)
MeV arereconstructedfor2011 (2012)data. At the hardware trigger stage, events are required to have a muon with high pT or a hadron, photon or electron with high transverse energy in the calorimeters. For hadrons, the trans-verse energy threshold is 3.5 GeV. The software trigger requires a two-,three- orfour-track secondaryvertexwithsignificant dis-placement fromthe primary pp interactionvertices. Atleast one chargedparticlemusthavetransversemomentum pT>
1.
7 GeV/
c andbeinconsistentwithoriginatingfroma PV.Amultivariate al-gorithm [21] is used for the identification of secondary vertices consistentwiththedecayofab
hadron.In the simulation, pp collisions are generated using Pythia8
[22] with a specific LHCb configuration [23]. Decays of hadronic particles aredescribedby EvtGen[24],inwhichfinal-state radia-tionisgeneratedusingPhotos[25].The interactionofthe gener-atedparticleswiththedetector,anditsresponse,areimplemented usingtheGeant4 toolkit[26]asdescribedinRef.[27].
3. Event selection
Afterreconstructionofthe D mesoncandidatefromeithertwo or four charged particles, the same basic event selection is ap-plied toall B−
→
Dh− channelsofinterest.The reconstructed Dmeson candidate mass is required to be within
±
25 MeV/
c2 of its known value [28]. This mass range corresponds to approxi-mately three times the mass resolution of the signal peaks. The kaonorpionoriginatingfromtheB
±decay,subsequentlyreferred to as the bachelor particle, is required to have pT in the range 0.
5–10.
0 GeV/
c and p in the range5–100 GeV/
c. These require-ments ensure that the track iswithin the kinematic coverage of the RICHdetectors, whichare usedto provideparticle identifica-tion(PID)information.DetailsofthePIDcalibrationprocedureare givenin Sect.4. Inaddition,a kinematicfitisperformedtoeach decaychain,withvertexconstraintsappliedtoboththeB
±andDFor both the two- and four-body D-mode selections, a pair of boosted decision tree (BDT) discriminators, implementing the gradient boost algorithm [30], are employed to achieve further background suppression. The BDTs are trained using simulated
B−
→ [
K−π
+(
π
+π
−)
]
DK− decays together with a background sample of Kπ
± combinations with invariant mass in the range 5900–7200 MeV/
c2.ForthefirstBDT,thosebackgroundswithaD
candidatemassmorethan
±
30 MeV/
c2 awayfromtheknown D0 massareusedinthetraining.InthesecondBDT,backgroundswith a D candidatemass within±
25 MeV/
c2 ofthe known D0 mass areused.Aloosecut ontheclassifierresponseofthefirstBDTis appliedpriortotrainingthesecondone.Thisfocussesthesecond BDTtraining onbackgroundsenriched withfullyreconstructed Dmesons.
TheinputtotheBDTisasetofquantitiesthatcharacterisethe signaldecay.Thesequantitiescan bedivided intotwocategories: (1)propertiesofanyparticleand(2)propertiesofcomposite par-ticlesonly(the
D
andB
±candidates).Specifically:1. p,
p
T andthesquareoftheIPsignificance;2. decaytime,flightdistance,decayvertexquality,radialdistance betweenthedecayvertexandthePV, andtheanglebetween the particle’s momentum vector andthe lineconnecting the productionanddecayvertex.
Signal purity is improved by usinga variable that estimates the imbalanceof
p
T aroundtheB
±candidate,definedasI pT
=
pT
(
B±)
−
pT pT
(
B±)
+
pT
,
(7)where the sum is taken over tracks lying within a cone around the B± candidate, excluding thetracks relatedto the signal. The coneisdefinedby acirclewitharadiusof1.5unitsintheplane ofpseudorapidityandazimuthalangle(expressed inradians).The BDT thus gives preference to B± candidates that are either iso-latedfromtherestoftheevent,orconsistentwitharecoilagainst another
b
hadron.NoPIDinformationisusedintheBDTtrainingsotheefficiency for B−
→
D K− and B−→
Dπ
− decaysis similar, with insignifi-cantvariationsarisingfromthesmalldifferencesinthekinematics. Thecutson thetwo BDT selectionsare optimisedby minimising theexpecteduncertaintyon AADSπK((π πK)),asmeasured inthe invari-antmassfitdescribed below. Thepurityofthe sample isfurther improvedwithRICHinformationbyrequiringallkaonsandpions inthe D decaytobecorrectlyidentifiedwithaPIDselectionthat hasanefficiencyofabout85%perparticle.Peakingbackgroundsfromcharmlessdecaysaresuppressedby requiringthat the flight distance significance ofthe D candidate fromthe B± decayvertexislarger thantwo standarddeviations. Theresidualcharmlesscontributionisinterpolatedfromfitstothe
B± massspectrum(without thekinematicfitofthedecaychain) in both the lower and upper D-mass sidebands. The charmless yields are determined independently for B+ and B− candidates andare later usedin the massfit asfixed terms,with their un-certainties included in the systematic uncertainties of the final results. The largest residual charmless contributions are in the
B−
→ [
π
+π
−(
π
+π
−)
]
DK−modeswhichshowcharge-integrated yieldsof88±
11 and115±
11 forthetwo- andfour-pionmodes. Thisis7%and8%ofthemeasuredsignalyields.EvenwithPIDrequirements,thesuppressedADSsamples con-tain significantcross-feedfromfavoured signal decays wherethe
K− and a
π
+ from the D decay are misidentified as aπ
−and K+. This contamination is reduced by removing any candi-date whose reconstructed D mass, under the exchange of mass
hypotheses between the kaon and an opposite-sign pion, lies within
±
15 MeV/
c2 of theknown D0 mass.Thisvetoisalso ap-pliedtothefavouredmode,withthesameefficiency.Theresidual cross-feed rates are estimated in data from the favoured sam-ple, assuming the veto and PID efficiencies factorise; they are(
4.
3±
0.
2)
×
10−5 for B−→ [
K−π
+]
Dh− and(
2.
7±
0.
1)
×
10−4 forB
−→ [
K−π
+π
+π
−]
Dh−.Aftertheaboveselections,multiplecandidatesexistin0.1%and 1%ofeventsinthe
B
−→ [
h+h−]
Dh−andB
−→ [
h+h−π
+π
−]
Dh− samples,respectively.Onlyonecandidatepereventisretainedfor themain fit.When morethanone candidateis selected,theone withthebestB
±vertexqualityisretained.4. Signal yields and systematic uncertainties
Thevaluesofthe
CP
observablesare determinedusingbinned maximum-likelihoodfitstotheinvariantmassdistributionsof se-lected B± candidates. Independent fits are used for the B−→
[
h+h−]
Dh− and B−→ [
h+h−π
+π
−]
Dh− samples. Information from the RICH detectors is used to separate B−→
D K−from B−
→
Dπ
− decays with a PID requirement on the bach-elor particle. Distinguishing between B+ and B− candidates, bachelor particle hypotheses, and four (three) D daughter final states, yields 16 (12) disjoint samples in the B−→ [
h+h−]
Dh− (B−→ [
h+h−π
+π
−]
Dh−)fit,whicharefittedsimultaneously.The total probability density function (PDF) is built from two signal PDFs,for B−→
D K− andB−→
Dπ
− decays,andthreetypesof backgroundPDF.AllPDFsareidenticalforB+andB
−decays.1. B−
→
Dπ
−Inthe
D
π
−samplesanasymmetricdouble-Gaussian-like func-tionisusedforthe B±→
Dπ
±signal,f
(
m)
=
fcoreexp−
(
m−
μ
)
22
σ
2c
+
(
m−
μ
)
2α
L,R+
(
1−
fcore)
exp−
(
m−
μ
)
22
σ
2 w,
(8)which has a peak position
μ
and core widthσ
c, whereα
L(
m<
μ
)
andα
R(
m>
μ
)
parameterisethe tails.Theμ
andα
parametersaresharedacrossallsamplesbutthecorewidth parametervariesindependentlyforeach D finalstate, except in the suppressedπ
K(
π π
)
PDFs which are required to be identical to their favoured Kπ
(
π π
)
counterpart. The addi-tional Gaussian function with a small fractional contribution ofabout1%isfoundnecessarytomodelsatisfactorilythetails ofthepeak.The B−
→
Dπ
− decaysmisidentifiedas B−→
D K− are dis-placedtohighermassintheD K
−subsamples.These misiden-tified candidates are modelled by the sum of two Gaussian functions with common mean but modified to include tail componentsasinEq.(8).The mean,widths andonetail pa-rameterarelefttovaryfreely.2. B−
→
D K−Inthe D K− samples,Eq.(8)isagainusedforthesignalPDF. The peakposition
μ
andthe two tailparametersα
L andα
R are fixed to those ofthe B−→
Dπ
− signal function, asare thewidecomponentparameters fcoreandσ
w.Thecorewidth parameter in each D mode is related to the correspondingB−
→
Dπ
− widthby afreelyvarying ratiocommontoall Dfinalstates.Misidentified
B
±→
D K±candidatesappearinthe3. Combinatorial background
Duetothelowbackgroundlevel,alinearfunctionissufficient todescribetheentireinvariantmassspectrum.Twocommon slopeparametersareused,oneforD
π
−andanotherforD K
−subsamplesbutyieldsvaryindependently. 4. Peaking backgrounds
Charmless B± decayandthefavouredmode cross-feed back-groundsboth peak atthe B± mass andare indistinguishable from the signal. Their residual yields are estimated in data, entering the fit as fixed proportions of the favoured B−
→
Dπ
− yield. A Gaussian function is used for the PDF, witha(
25±
2)
MeV/
c2 widthparameter thatis takenfrom simula-tion;thisisabout50%widerthanthesignalPDF.5. Partially reconstructed b-hadron decays
Partially reconstructed backgrounds generally have lower in-variantmassthanthesignalpeak.Thedominantcontributions arefrom B−
→
Dh−π
0, B−→
D∗0π
− and B0→
D∗+π
− de-cays where either a photon or a pion is missed in the re-construction.The distributionofeach ofthesesources inthe invariant mass spectrum depends on the spin and mass of the missing particle. If the missing particle has spin-parity 0−(
1−)
,thedistributionisparameterised byaparabolawith positive (negative) curvature convolved witha Gaussian res-olutionfunction. The kinematics of the decay that produced themissingparticle definethe endpointsofthe rangeofthe parabola. Decays in which both a particle is missed and a bachelor pion is misidentified as a kaon are parameterised witha semi-empiricalPDF,formed fromthesumofGaussian anderror functions. The parameters of each partially recon-structed PDF are fixed to the values found in fits to simu-lated events, and are varied as a source of systematic un-certainty.The yields ofeach contribution varyindependently in each subsample, where all partially reconstructed decay modesshareacommoneffectivechargeasymmetryacrossallD modes. Thoughits effectismitigatedby the limitedrange ofthemassfit,large
CP
violationinthelow-massbackground ispossibleintheGLWandADSsamples,soasystematic un-certaintyisassigned.Inthe
B
−→ [
K+K−]
Dh−samples,Λ
0b→ [
p+K−π
+]
Λ+ch−
de-cays contribute to backgroundwhen the pion is missed and theprotonismisidentifiedasthesecondkaon.ThewidePDF of this componentis fixed from simulation but the yield in the B−
→ [
K+K−]
Dπ
− subsample varies freely. TheΛ
0b→
[
p+K−π
+]
Λ+cK−yieldisconstrainedusingarecent
measure-mentof
B(Λ
b0→
Λ
+cK−)/B(Λ
0b→
Λ
+cπ
−)
[31].Furthermore,B0s
→
D0K−π
+decays,wherethepionismissed,forma back-ground for the suppressed B−→
D K− modes. The yield of thiscomponentvariesinthe fitbutthe PDFis takenfroma simulationmodelofthethree-bodyB
0s decay[32],smearedto matchtheresolutionmeasuredindata.
Inthe D K−subsamples,the B±
→
Dπ
−cross-feedcanbe de-termined by the fit to data. The B−→
D K− cross-feedinto theD
π
− subsamples is not well separated frombackground, so the expectedyieldisdeterminedbyaPIDcalibrationprocedureusing approximately 20million D∗+→ [
K−π
+]
Dπ
+ decays.The clean reconstruction ofthis charm decayis performedusingkinematic variablesonlyandthus providesahighpuritysample of K∓ and [image:4.612.339.514.108.265.2]π
± tracks, unbiased in the PID variables. The PID efficiency de-pends on track momentum and pseudorapidity, as well as the numberoftracks intheevent. The effectivePIDefficiencyofthe signalisdeterminedbyweightingthecalibrationsamplesuchthat the distributions of these variables match those of the selected candidates in the B−→
Dπ
− mass distribution. It is found thatTable 1
Signalyieldsasmeasuredinthe B−→ [h+h−]Dh− and
B−→ [h+h−π+π−]Dh−invariantmassfits,togetherwith
theirstatisticaluncertainties.
Decay mode Yield
B±→K±π∓Dπ± 378,050±650
B±→K±π∓DK± 29,470±230
B±→K+K−Dπ± 50,140±270 B±→K+K−DK± 3816±92 B±→π+π−Dπ± 14,680±130 B±→π+π−DK± 1162±48
B±→π±K∓Dπ± 1360±44
B±→π±K∓DK± 553±34 B±→K±π∓π+π−Dπ± 142,910±390
B±→K±π∓π+π−DK± 11,330±140 B±→π+π−π+π−Dπ± 19,360±150
B±→π+π−π+π−DK± 1497±60 B±→π±K∓π+π−Dπ± 539±26
B±→π±K∓π+π−DK± 159±17
68.0% (
PID(K)) of B−
→
D K− decays passthe bachelor kaon PID requirement;theremaining32.0%cross-feedintothe B−→
Dπ
−sample.Withthisselection,approximately98%ofthe
B
−→
Dπ
−decays are correctlyidentified. Due to the size ofthe calibration sample,thestatisticaluncertaintyisnegligible;thesystematic un-certainty of the method is determined by the size of the signal track samples used, and thus increases for the lower statistics modes. Thesystematicuncertaintyon
PID(K) ranges from0.3%in B−
→ [
K−π
+]
DK−to1.5%in B−→ [
π
+π
−π
+π
−]
DK−.Inordertomeasure
CP
asymmetries,thedetectionasymmetries for K± andπ
± must be taken into account. A detection asym-metry of(
−
0.
96±
0.
10)
% is assignedfor each kaon in the final state, arising from thefact that the nuclear interaction length ofK− mesonsis shorterthanthat of K+ mesons.Thisiscomputed by comparing the charge asymmetries in D−
→
K+π
−π
− andD−
→
KS0π
− calibrationsamplesandweightingtothekinematics ofthesignalkaons.Theequivalentasymmetryforpionsissmaller(
−
0.
17±
0.
10)
% andistakenfromRef.[33].TheCP
asymmetries in the favoured B−→ [
K−π
+(
π
+π
−)
]
Dπ
− decays are fixed to zero,withasystematicuncertaintyof0.16%calculatedfrom exist-ingknowledgeofγ
andr
B inthisdecay[18],withnoassumption madeaboutthestrongphase.Thisenablestheeffectiveproduction asymmetry, AB±, to be measured and simultaneously subtracted from the charge asymmetry measurements in other modes. The signal yield foreachmode isa sumofthenumberofsignal and cross-feedcandidates;theirvaluesaregiveninTable 1.The corre-sponding invariant mass spectra,separatedby charge, are shown inFigs. 1–7.To obtain the observables RKf/π, the ratio of yields must be corrected by the relative efficiency withwhich B−
→
D K− andB−
→
Dπ
− decaysare reconstructed andselected. From simula-tion, this ratio is found to be 1.
017±
0.
017 and 1.
018±
0.
026 forthe two- and four-body D decayselections. Theuncertainties are calculatedfrom the finite size of the simulatedsamples and accountforimperfectmodellingoftherelativepionandkaon ab-sorptioninthedetectormaterial.Fig. 1.InvariantmassdistributionsofselectedB±→ [K±π∓]Dh± candidates,separatedbycharge,with B−(B+)candidatesontheleft(right).Thetopplotscontainthe
[image:5.612.104.502.318.508.2]B±→D K±candidatesample,asdefinedbyaPIDrequirementonthebachelorparticle.Theremainingcandidatesareplacedinthebottomrow,reconstructedwithapion hypothesisforthebachelor.Thered(thick,open)andgreen(hatched-area)curvesrepresenttheB±→D K±and B±→Dπ±signals.Theshadedpartindicatespartially reconstructeddecays,thedottedline,wherevisible,showsthecombinatorialcomponent,andthetotalPDFisdrawnasathinblueline.(Forinterpretationofthereferences tocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
Fig. 2.InvariantmassdistributionsofselectedB±→ [π±K∓]Dh±decays,separatedbycharge.Thedashedpinklineleftofthesignalpeakshowspartiallyreconstructed
B0
s→ [K+π−]DK−π+ decays,wherethebachelorpionismissed.Thefavouredmodecross-feedisalsoincludedinthefit,butistoosmalltobeseen.Seethecaptionof
Fig. 1forotherdefinitions.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
Table 2
SystematicuncertaintiesfortheB−→ [h+h−]Dh−CPobservablesquotedasapercentageofthestatisticaluncertaintyontheobservable.PIDreferstothePIDcalibration
procedure.Bkgreferstothechoiceofbackgroundshapesandyieldsinthefit.Simreferstotheuseoffinitesamplesofsimulatedeventstodetermineefficiencyratios.Asym referstothefixedpionandkaondetectionasymmetries,andtheassumptionofnoCPviolationinB−→D0π−decays.
[%] AKπ
K RKKπ/π AK KK AπK K RK K Aπ πK Aπ ππ Rπ π RπADSK(π ) RπADSK(K) Aπ K
ADS(π ) AπADSK(K)
PID 42 95 11 1 38 9 9 39 29 25 15 5
Bkg 65 190 34 3 84 30 28 48 69 74 24 15
Sim 21 250 14 0 24 8 7 13 29 30 8 5
Asym 23 27 11 34 6 7 20 5 12 13 7 8
Total 83 330 40 34 96 33 36 64 81 85 30 19
Table 3
SystematicuncertaintiesfortheB−→ [h+h−π+π−]Dh− CPobservablesquotedasapercentageofthestatisticaluncertaintyontheobservable.SeetheTable 2captionfor
definitions.
[%] RKπ π π
K/π Rπ π π π RπADSKπ π(K) RπADSKπ π(π ) AKKπ π π AπADSKπ π(K) AADSπKπ π(π ) Aπ π π πK Aπ π π ππ
PID 37 43 1 2 1 1 0 0 1
Bkg 63 28 40 33 2 36 8 54 21
Sim 160 0 0 0 0 1 0 0 0
Asym 20 5 7 6 16 5 5 8 22
[image:5.612.39.572.589.649.2] [image:5.612.44.563.686.748.2]Fig. 3.InvariantmassdistributionsofselectedB±→ [π+π−]Dh±candidates,separatedbycharge.Thedashedblacklinerepresentstheresidualcontributionfromcharmless
decays.ThiscomponentispresentintheDfinalstatesconsidered,butismostvisibleinthiscase.SeethecaptionofFig. 1forotherdefinitions.(Forinterpretationofthe coloursinthisfigure,thereaderisreferredtothewebversionofthisarticle.)
Fig. 4. Invariant mass distributions ofselected B±→ [K+K−]Dh± candidates, separated by charge. The dashed cyan line represents partially reconstructed Λ0b→
[p+K−π+]Λ+ch
− decays,wherethepionismissedandthe protonismisidentifiedasakaon.SeethecaptionofFig. 1 forotherdefinitions. (Forinterpretationofthe
referencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
thetwo-bodyandfour-body
D
modefits.Correlationsbetweenthe categoriesarenegligibleandthetotalsystematicuncertaintiesare givenbythesumsinquadrature.5. Results
The results of the fits to data, withstatistical and systematic uncertainties,are:
AKKπ
= −
0.
0194±
0.
0072±
0.
0060RKKπ/π
=
0.
0779±
0.
0006±
0.
0019AK KK
=
0.
087±
0.
020±
0.
008AπK K
= −
0.
0145±
0.
0050±
0.
0017RK K
=
0.
968±
0.
022±
0.
021Aπ πK
=
0.
128±
0.
037±
0.
012Aπ ππ
=
0.
0043±
0.
0086±
0.
0031Rπ π
=
1.
002±
0.
040±
0.
026RπADSK(π)
=
0.
00360±
0.
00012±
0.
00009RπADSK(K)
=
0.
0188±
0.
0011±
0.
0010AπADSK(π)
=
0.
100±
0.
031±
0.
009AπADSK(K)
= −
0.
403±
0.
056±
0.
011,
RKKπ π π/π
=
0.
0793±
0.
0010±
0.
0018Rπ π π π
=
0.
975±
0.
037±
0.
019 [image:6.612.97.493.318.508.2]Fig. 5.InvariantmassdistributionsofselectedB±→ [K±π∓π+π−]Dh±candidates,separatedbycharge.SeethecaptionofFig. 1forthedefinitions.(Forinterpretationof
[image:7.612.105.503.307.499.2]thecoloursinthisfigure,thereaderisreferredtothewebversionofthisarticle.)
Fig. 6.Invariant massdistributionsofselected B±→ [π±K∓π+π−]Dh± candidates, separatedbycharge.Thedashed pinklineleftofthe signalpeakshowspartially
reconstructedB0
s→ [K+π−π+π−]DK−π+decays,wherethebachelorpionismissed.SeethecaptionofFig. 1forotherdefinitions.(Forinterpretationofthereferencesto
colourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
RπADSKπ π(π)
=
0.
00377±
0.
00018±
0.
00006AKKπ π π
=
0.
000±
0.
012±
0.
002AπADSKπ π(K)
= −
0.
313±
0.
102±
0.
038AπADSKπ π(π)
=
0.
023±
0.
048±
0.
005Aπ π π πK
=
0.
100±
0.
034±
0.
018Aπ π π ππ
= −
0.
0041±
0.
0079±
0.
0024.
These results supersede those in Refs. [9] and [17] except for theresultsrelating tothe four-pion D decay,which arereported forthe first time. The correlation matricesare given in the Ap-pendix. The statistical correlations for the observables RADS and
AADS are small. The alternative ADS observables are calculated:
Rπ+(KK)
=
(
2.
58±
0.
23)
%; Rπ−(KK)=
(
1.
15±
0.
14)
%; R+(πKπ)=
(
3.
22±
0.
18)
×
10−3; RπK−(π)
=
(
3.
98±
0.
19)
×
10−3; Rπ+K(Kπ π)=
(
1.
82±
0
.
25)
%; Rπ−(KKπ π)=
(
0.
98±
0.
20)
%; Rπ+(Kππ π)=
(
3.
68±
0.
26)
×
10−3; and Rπ−(Kππ π)=
(
3.
87±
0.
26)
×
10−3, wherethe quoted uncertain-tiescombinestatisticalandsystematiceffects.Theasymmetriesinthetwo
CP
-evenD
decays,D
→
K+K−andD
→
π
+π
−, are averaged by notingthat their systematic uncer-taintiesarenearlyfullycorrelated,ACP(K)
=
0.
097±
0.
018±
0.
009,
ACP(π)= −
0.
0098±
0.
0043±
0.
0021.
Similarly the average ratio of partial widths from these D
modesis
R K K,π π
=
0.
978±
0.
019±
0.
018(
±
0.
010)
;
Fig. 7.InvariantmassdistributionsofselectedB±→ [π+π−π+π−]Dh±candidates,separatedbycharge.Thedashedblacklinerepresentstheresidualcontributionfrom
charmlessdecays.SeethecaptionofFig. 1forotherdefinitions.(Forinterpretationofthecoloursinthisfigure,thereaderisreferredtothewebversionofthisarticle.)
becomes equaltothe classicGLW observable RCP(K) [5].This ad-ditionaluncertaintyis applicabletothe K K and
π π
modes indi-viduallybutthe equivalent uncertaintyforthefour-pion modeis lower,±
0.
005,duetothereducedcoherenceinthatD
decay.The significance of these measurements may be quantified from the likelihood ratio with respect to a CP-symmetric null hypothesis,
−
2 log(L
0/L)
. The significance of CP violation in the ADSmode B−→ [
π
−K+]
DK− is 8.
0σ
(standarddeviations) and represents the first observation of CP violation in a singleB−
→
Dh−decaymode.Thecombinationofthetwo GLWmodesB−
→ [
K+K−]
DK− and B−→ [
π
+π
−]
DK− also demonstrates aCP-violationeffectwith5
.
0σ
significance.Takentogether,theADS andGLW modes ofthe B−→
Dπ
− decays show evidence of CPviolation with 3
.
9σ
significance after accounting for systematic uncertainties. The B−→ [
π
+π
−π
+π
−]
DK− data shows a 2.
7σ
CP-violationeffect.
6. Acceptance effects
Thenon-uniformacceptanceacrossthephasespaceofthe four-body modes affects the applicability of the external coherence factor andstrong phase difference measurements [11] in the in-terpretation of these results. With an acceptance model for the four-body D decays from simulation, the effective values of the
D
→
K+π
−π
+π
− coherence parameters are calculated using a rangeof plausible amplitude models. The acceptanceis found to be almost uniformand theeffective valuesof the coherence pa-rametersare closetothoseforperfectacceptance.The additional systematicuncertaintiesonκ
K3π andδ
K3πD ,wheninterpretingthe four-bodyresultsreportedhere, are
±
0.
01 and±
2.
3◦.Ina simi-larstudy,theadditionalsystematicuncertaintyassociatedwiththe modulationoftheCP
fractionF+4π bytheLHCbacceptanceis esti-matedtobe±
0.
02.It has been shown that D-mixing effects must be takeninto account when using these CP observables in the determination of
γ
[34].ThecorrectionismostimportantintheADSobservables ofB
−→
Dπ
− decaysandiscorrectedforusingknowledgeofthe decay-timeacceptance.Fromsimulationsamples,adecay-time ac-ceptancefunctionisdefinedforboththetwo-bodyandfour-bodyD-modeselections.The D-mixingcoefficient
α
,definedin[34],is found to be−
0.
59 and−
0.
57 for the two- and four-body cases,with negligible uncertainties compared to those of the x and y D-mixingparameters.
7. Discussion and conclusions
World-bestmeasurementsof
CP
observablesinB
−→
Dh− de-cays are obtained with the D meson reconstructed in K−π
+,K+K−,
π
+π
−,π
−K+, K−π
+π
+π
− andπ
−K+π
+π
− final states;thissupersedesearlierwork[9,16].Measurements exploit-ingthefour-pion D decayarereportedforthefirsttime withtheB−
→ [
π
+π
−π
+π
−]
DK− decayshowingan indication ofCP
vi-olationat the2.
7σ
level.The charge asymmetry inthismode is positive,similartotheclassicGLWmodes,B
−→ [
K+K−]
DK−andB−
→ [
π
+π
−]
DK−, in linewithexpectation fora multi-body D modewithaCP
fractiongreaterthan0.5[15].AcomparisonwithSM expectationismadeby calculatingthe
CP observablesfromcurrent best-fitvaluesof
γ
=
(
73.
2+−67..30)
◦ as well asδ
B=
(
125.
4−+77..08)
◦ andrB=
(
9.
70−+00..6263)
% for B±→
D K± decays [2]. For B±→
Dπ
± decays, where no independent in-formation on rB andδ
B is available, uniform PDFs are used, 180◦< δ
B<
360◦ and 0.
004<
rB<
0.
008. The D-decay param-eters are taken from the literature: r2D=
(
0.
349±
0.
004)
% andδ
D=
(
191.
8+−149..57)
◦[35];F
4π
+
=
0.
737±
0.
028[15];r
KD3π=
(
5.
52±
0.
007)
%,δ
KD3π=
(
170+−3739)
◦andκ
K3πD
=
0.
32+ 0.12−0.08 [11].Thecurrent worldaveragesofthe
D
-mixingparametersarex
=
(
0.
37±
0.
16)
% andy
=
(
0.
66−+00..0710)
%[35],andtheα
coefficientsreportedinSec.6are usedfor thesmall D-mixingcorrection. Fortheseinputs, the central 68% confidence-level expectation interval is displayed in
Fig. 8,togetherwiththeresultspresentedherein.Itisseenthatthe
B−
→
D K− measurements are compatible with expectation and that the improvement inthe knowledge ofthe AADS observables isparticularlysignificant.Themeasurementspresentedinthis pa-per improve many of the CP observables used in global fits for the unitaritytriangleangleγ
aswell asthe hadronicparametersrB and
δ
B forthesedecays.Animprovementintheglobalbest-fit precisiononγ
ofaround15%isanticipatedfromthiswork.Acknowledgements
Fig. 8.ComparisonofselectedresultswithanSMexpectation(shaded)basedonexistingknowledgeoftheunderlyingparametersandD-decaymeasurements,asdescribed inthetext.ThefirstfiverowsshowB−→D K−observables,two-bodyDdecayontheleftandfour-bodyontheright.ThelastthreerowsshowB−→Dπ−observables.
We thank the technical andadministrative staff at the LHCb in-stitutes. We acknowledge support from CERN and from the na-tional agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN(Italy); FOMandNWO (TheNetherlands);MNiSWandNCN (Poland);MEN/IFA (Romania);MinESandFANO(Russia);MINECO (Spain);SNSFandSER(Switzerland);NASU(Ukraine);STFC(United Kingdom);NSF (USA). We acknowledge the computing resources that are provided by CERN, IN2P3 (France), KIT and DESY (Ger-many), INFN (Italy), SURF (The Netherlands), PIC (Spain), GridPP (UnitedKingdom), RRCKIandYandexLLC(Russia), CSCS (Switzer-land),IFIN-HH(Romania),CBPF(Brazil),PL-GRID(Poland)andOSC (USA). We are indebted to the communities behind the multi-pleopensource softwarepackageson whichwedepend.
Individ-ualgroupsor membershavereceived supportfromAvH Founda-tion(Germany),EPLANET,MarieSkłodowska-CurieActionsandERC (European Union), Conseil Général de Haute-Savoie, Labex ENIG-MASSandOCEVU,RégionAuvergne(France),RFBRandYandexLLC (Russia),GVA,XuntaGalandGENCAT(Spain),HerchelSmithFund, The Royal Society, Royal Commission for the Exhibition of 1851 andtheLeverhulmeTrust(UnitedKingdom).
Appendix A. Correlation matrices
Table A.4
Statisticalcorrelationmatrixfromforthe2-bodyfit.
AKKπ R Kπ
K/π AK KK AK Kπ RK K Aπ πK Aπ ππ Rπ π RπADSK(π ) Rπ K
ADS(K) Aπ K
ADS(π ) Aπ K
ADS(K)
AKπ
K 1 0.00 0.02 0.09 0.00 0.01 0.05 0.00 0.00 0.00 0.01 0.01
RKKπ/π 1 0.00 0.00 −0.32 0.00 0.00 −0.18 0.01 −0.11 0.00 0.00
AK K
K 1 −0.01 −0.01 0.00 0.02 0.00 0.00 0.00 0.00 0.00
AK K
π 1 0.00 0.02 0.06 0.00 0.00 0.00 0.02 0.01
RK K 1 0.00 0.00 0.06 −0.01 0.04 0.00 0.00
Aπ πK 1 −0.04 −0.04 0.00 0.00 0.00 0.00
Aπ π
π 1 0.00 0.00 0.00 0.01 0.01
Rπ π 1 0.00 0.02 0.00 0.00
RπADSK(π ) 1 −0.02 −0.04 0.00
RπK
ADS(K) 1 0.02 0.10
AπK
ADS(π ) 1 −0.05
AπK
ADS(K) 1
Table A.5
Statisticalcorrelationmatrixfromforthe4-bodyfit.
RKKπ π π/π A
Kπ π π
K Rπ π π πC P Aπ π π ππ Aπ π π πK Rπ
Kπ π
A D S(K) Rπ Kπ π
A D S(π ) Aπ Kπ π
A D S(K) Aπ Kπ π A D S(π )
RKπ π π
K/π 1 0.00 −0.31 0.00 0.00 −0.10 0.01 0.00 0.00
AKKπ π π 1 0.00 0.10 0.02 0.00 0.00 0.01 0.02
Rπ π π π
C P 1 −0.00 −0.02 0.04 0.00 0.00 0.00
Aπ π π π
π 1 −0.02 0.00 0.00 0.01 0.02
Aπ π π πK 1 0.00 0.00 0.00 0.00
RπA D SKπ π(K) 1 −0.05 0.08 0.01
RπKπ π
A D S(π ) 1 0.00 −0.02
AπKπ π
A D S(K) 1 −0.06
[image:10.612.37.553.88.738.2]AπA D SKπ π(π ) 1
Table A.6
Correlationmatrixforthesystematicuncertaintiesinthe2-bodyanalysis.
AKKπ R Kπ
K/π AK KK AπK K RK K Aπ πK Aπ ππ Rπ π RADS(π ) RADS(K) AADS(π ) AADS(K)
AKπ
K 1 −0.03 0.20 0.21 0.00 0.01 0.36 −0.00 0.07 −0.09 −0.52 0.09
RKπ
K/π 1 0.09 −0.10 −0.10 −0.06 −0.03 0.24 −0.01 −0.22 −0.03 −0.13
AK K
K 1 −0.46 −0.29 0.03 0.24 0.01 −0.06 −0.10 0.05 −0.04
AK K
π 1 0.18 0.32 0.50 −0.11 −0.15 0.31 0.42 0.33
RK K 1 0.12 −0.05 0.19 0.12 0.32 0.07 0.23
Aπ πK 1 0.05 −0.30 −0.33 0.39 0.38 0.30
Aπ π
π 1 −0.03 0.07 −0.06 0.25 0.11
Rπ π 1 0.18 −0.16 −0.11 −0.06
RADS(π ) 1 −0.57 −0.56 −0.44
RADS(K) 1 0.56 0.76
AADS(π ) 1 0.40
AADS(K) 1
Table A.7
Correlationmatrixforthesystematicuncertaintiesinthe4-bodyanalysis.
RKKπ π π/π Rπ π π πC P Rπ
Kπ π
A D S(K) Rπ Kπ π
A D S(π ) ABu A
Kπ π π
K Aπ
Kπ π
A D S(K) Aπ Kπ π
A D S(π ) Aπ π π πD K Aπ π π πDπ
RKπ π π
K/π 1 0.11 −0.04 0.13 −0.01 0.00 −0.13 0.17 −0.14 0.08
Rπ π π πC P 1 0.04 −0.06 0.01 0.02 −0.04 −0.04 0.07 −0.07
RπKπ π
A D S(K) 1 0.14 −0.00 0.02 0.87 0.10 0.03 0.01
RπKπ π
A D S(π ) 1 −0.04 −0.02 0.05 0.46 −0.35 0.24
ABu 1 −0.36 −0.05 −0.42 −0.03 −0.64
AKπ π π
K 1 0.02 0.05 0.09 0.32
AπKπ π
A D S(K) 1 −0.09 −0.04 0.02
AπKπ π
A D S(π ) 1 −0.34 0.43
Aπ π π πD K 1 0.31
Aπ π π π
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LHCb Collaboration