Measurement of CP observables in B ± → D (⁎) K ± and B ± → D (⁎) π ± decays

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Measurement

of

CP observables

in

B

±

→

D

(

∗)

K

±

and

B

±

→

D

(

∗)

π

±

decays

.

The

LHCb

Collaboration

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received23August2017

Receivedinrevisedform1November2017 Accepted27November2017

Availableonline5December2017 Editor:M.Doser

MeasurementsofCP observablesinB±→D(∗)_K±_andB±_→D(∗)

_π

±_{decaysarepresented,whereD}(∗)

indicates aneutral D or D∗ mesonthat isan admixture ofD(∗)0 _{and D}¯(∗)0 _{states. Decays ofthe D}∗

mesontotheD

π

0_andD

_γ

_{finalstatesarepartiallyreconstructedwithoutinclusionoftheneutralpion} orphoton,resultingindistinctiveshapesintheB candidateinvariantmassdistribution.DecaysoftheD

mesonarefullyreconstructedinthe K±

π

∓,K+K− and

π

+

π

−finalstates.Theanalysisusesasample ofcharged B mesonsproducedin pp collisionscollectedbytheLHCbexperiment,correspondingtoan integratedluminosityof2.0,1.0and2.0 fb−1takenatcentre-of-massenergiesof√s=7,8and13 TeV, respectively. ThestudyofB±→D∗K± and B±→D∗

π

± decaysusingapartialreconstructionmethod isthe firstofitskind,whilethe measurementofB±_→D K± and B±_→D

π

± decaysisanupdateof previousLHCbmeasurements.TheB±→D K±resultsarethemostprecisetodate.

©2017TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Overconstraining the Unitarity Triangle(UT) derived fromthe Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix is cen-tral to testing the Standard Model (SM) description of

CP violation [1]. The least well known angle of the UT is

γ

≡

arg

(

−

VudV∗_ub

/

VcdV_cb∗

)

, which has been determined with a

precision of about 7◦ from a combination of measurements [2, 3](c f

.

3◦ and

<

1◦ ontheangles

α

and

β

[4,5]).Among theUT angles,

γ

isunique in that itdoesnot depend onany top-quark coupling,andcanthusbemeasuredindecaysthat aredominated by tree-level contributions. In such decays, the interpretation of physicalobservables (rates and CP asymmetries) in terms ofthe underlyingUTparametersissubjecttosmalltheoretical uncertain-ties[6].Anydisagreementbetweenthesemeasurementsof

γ

and thevalueinferredfromglobalCKMfitsperformedwithoutany

γ

informationwouldinvalidatetheSMdescriptionof

CP violation.

Themostpowerfulmethodfordetermining

γ

indecays domi-nated by tree-level contributions is through the measurement of relative partial widths in B−

→

D K− decays, where D

repre-sentsan admixture ofthe D0 and D0 states.1 The amplitude for the B−

→

D0_K− _{decay, which atthe quark level proceedsvia a}

b

→

cus transition,

¯

isproportional to Vcb.Thecorresponding

am-plitudefortheB−

→

D0K−decay,whichproceedsviaa

b

→

ucs

¯

1 _{Theinclusionofcharge-conjugateprocessesisimpliedexceptinanydiscussion}

ofasymmetries.

transition, is proportional to Vub. Bystudyinghadronic D decays

accessible toboth D0 and D0 mesons,phase informationcan be extractedfromtheinterferencebetweenthesetwoamplitudes.The degree of the resulting CP violation is governed by the size of

rD K

B ,theratio ofthe magnitudesofthe B−

→

D 0

K− and B−

→

D0K− amplitudes. Therelatively large value ofr_BD K

≈

0

.

10[3]in

B−

→

D K− decaysallowsthedeterminationoftherelativephase of the two interfering amplitudes. This relative phase has both

CP -violating

(

γ

)

and CP -conserving (

δ

D K_B ) contributions; a mea-surementofthedecayratesforboth B+ and B−givessensitivity to

γ

.SimilarinterferenceeffectsalsooccurinB−

→

D

π

−decays, albeit withlower sensitivity tothe phases. The reduced sensitiv-ity is theresult ofadditional Cabibbo suppression factors, which decreasetheratioofamplitudesrelativeto B−

→

D K− decaysby aroundafactorof20.

The B−

→

D∗K− decay, in which the vector D∗ meson2 de-caystoeitherthe

D

π

0_or

_D

_γ

_{finalstate,alsoexhibits}

_{CP -violating}

effects when hadronic D decays accessible to both D0 and D0

mesonsarestudied.Inthisdecay,theexactstrongphasedifference of

π

betweenD∗

→

D

π

0 _{and D}∗

_→

_D

_γ

_{decayscanbeexploited}

to measure CP observables for states with opposite CP

eigen-values [7]. The degree of CP violation observed in B−

→

D∗K−

decays is set by the magnitudeof the ratior_BD∗K

≈

0

.

12 [3],and

2 _D∗_{representsanadmixtureoftheD}∗₍₂₀₀₇₎0_andD¯∗₍₂₀₀₇₎0_states.

https://doi.org/10.1016/j.physletb.2017.11.070

0370-2693/©2017TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

measurementofthephaseforboth

B

+and

B

−allows

γ

and

δ

_BD∗K tobedisentangled.

Thestudyof B−

→

D(∗)_K−_{decaysformeasurementsof}

_γ

_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 asGLW modes[8,9]. In thiswork, the GLW decays D

→

K+K−

andD

→

π

+

π

− are consideredalong withtheCabibbo-favoured

D

→

K−

π

+ decay,wherethe latterdecayis usedfor normalisa-tion purposes and to define shape parameters in the fit to data (seeSec.4).

The

B

−

→ [

h+₁h−₂

]

Dh−decays,inwhich

h

+1,

h

−2 and

h

−caneach representeithera chargedkaon orpionandthe D-meson decay productsaredenotedinsidesquarebrackets,havebeenstudiedat the B factories [10,11] andat LHCb [12]. ThisLetter reports up-datedandimprovedresults usingasample of charged B mesons

frompp collisions collectedby theLHCbexperiment, correspond-ing to an integratedluminosity of 2.0, 1.0and 2.0 fb−1 takenat centre-of-massenergiesof

√

s

=

7,8and13 TeV,respectively.The datatakenat

√

s

=

13 TeVbenefitsfromahigher B± meson pro-ductioncross-sectionandamoreefficienttrigger,sothisupdateof the B−

→ [

h+₁h−₂

]

Dh− modesgainsapproximatelyafactoroftwo

insignal yield relative to Ref. [12].The B−

→ ([

h+₁h−₂

]

Dπ0

₎

D∗h−

and B−

→ ([

h+₁h−₂

]

D

γ

)

D∗h− decays, where the D∗-mesondecay

products are denoted in parentheses, have also been studied by the B factories[13,14],whilethisworkpresentsthefirstanalysis ofthesedecaysatLHCb.

Thesmall

D

∗

−

D mass differenceandtheconservationof angu-larmomentuminD∗

→

D

π

0_and

_D

∗

_→

_D

_γ

_{decaysresultsin}

dis-tinctivesignaturesforthe B−

→

D∗K−signal inthe D K− invari-antmass,allowingyieldstobeobtainedwithapartial reconstruc-tiontechnique.Sincethereconstructionefficiencyforlow momen-tumneutralpionsandphotonsisrelativelylow inLHCb[15],the partial reconstruction method provides significantly larger yields comparedto full reconstruction,but thestatisticalsensitivity per signal decay is reduced due to the need to distinguish several signal and background components in the same region of D K−

invariantmass.

A total of 19 measurements of CP observables are reported, eight of which correspond to the fully reconstructed

B−

→ [

h+₁h−₂

]

Dh− decays while the remaining 11 relate to the

partiallyreconstructed B−

→ ([

h+₁h−₂

]

D

π

0

/

γ

)

D∗h− decays. Inthe

lattercase,theneutralpionorphotonproducedinthedecayofthe

D∗vectormesonisnotreconstructedinthefinalstate.Asummary ofallmeasured

CP observables

isprovidedinTable 1.Inaddition, the branching fractions

B(

B−

→

D∗0

_π

−

₎

_and

_B(

_D∗0

_→

_D0

_π

0

₎

_, along withthe ratioof branching fractions B(B−→D∗0K−)

B(B−→D0_K−₎,are

re-ported.

Allofthechargeasymmetry measurementsare affectedbyan asymmetry in the B± production cross-section and any charge asymmetry arising from the LHCb detector efficiency, together denoted as

σ

. This effective production asymmetry, defined as

Aeff_B±

=

σ

_(B−_{)−σ (B}+₎

σ (B−)+σ (B+), is measured from the charge asymmetry of the mostabundant B−

→ [

K−

π

+

]

Dπ− mode. In this mode, the

CP asymmetry isfixedtohavethevalue A_πKπ

= (+

0

.

09

±

0

.

05

)

%, whichisdeterminedusingknowledgeof

γ

and

r

D K

B fromRef.[2],

where A_πKπ was notusedasaninputobservable.Thisuncertainty is smaller than that of previous measurements of the B± pro-duction asymmetry measured at

√

s

=

7 and 8 TeV [16,17], and reducesthesystematicuncertainties ofthe asymmetrieslisted in Table 1. The value of Aeff_B± is applied asa correction toall other

charge asymmetries. The remaining detection asymmetries, most notablyduetodifferentnumbersof

K

+and

K

−mesonsappearing ineachfinal state,arecorrectedforusingindependentcalibration

Table 1

Summarytableofthe19measuredCP observables,definedintermsofB meson decaywidths.Whereindicated,CP representsanaverageofthe D→K+K−and D→π+π−modes.The R observablesrepresentpartialwidthratiosand double ratios,whereRKKπ ,π/π 0/γ isanaverageoverthe D∗→Dπ0 and D∗→Dγ modes. TheA observablesrepresentCP asymmetries.

Observable Definition RKπ K/π (B−→[K−π+]DK−)+(B+→[K+π−]DK+) (B−→[K−π+_]Dπ−)+(B+→[K+π−_]Dπ+) RK K (B−→[K−K+]DK−)+(B+→[K+K−]DK+) (B−→[K−K+]Dπ−)+(B+→[K+K−]Dπ+)× 1 RKπ K/π Rπ π (B−→[π−_π+_] DK−)+(B+→[π+π−]DK+) (B−→[π−_π+_]Dπ−)+(B+→[π+_π−_]Dπ+)× 1 RKπ K/π AKπ K (B−→[K−π+]DK−)−(B+→[K+π−]DK+) (B−→[K−π+]DK−)+(B+→[K+π−]DK+) AK K K (B−→[K−K+]DK−)−(B+→[K+K−]DK+) (B−→[K−K+]DK−)+(B+→[K+K−]DK+) Aπ π K (B−→[π−_π+_] DK−)−(B+→[π+_π−_] DK+) (B−_→[π−π+_]DK−)+(B+_→[π+π−_]DK+) AK K π (B −_→[K−K+]Dπ−)−(B+→[K+K−]Dπ+) (B−→[K−K+]Dπ−)+(B+→[K+K−]Dπ+) Aπ π π (B −_→[π−_π+_] Dπ−)−(B+→[π+π−]Dπ+) (B−→[π−_π+_]Dπ−)+(B+→[π+_π−_]Dπ+) RK_Kπ ,π_/π 0/γ (B−→([K−π+]Dπ0_{/γ )} D∗K−)+(B+→([K+π−]Dπ0_{/γ )} D∗K+) (B−→([K−π+_]Dπ0/γ )D∗π−)+(B+→([K+π−]Dπ0/γ )D∗π+) RCP,π0 (B−→([CP]Dπ0₎ D∗K−)+(B+→([CP]Dπ0₎ D∗K+) (B−→([CP]Dπ0)D∗π−)+(B+→([CP]Dπ0)D∗π+)× 1 RK_Kπ ,π/π0/γ RCP,γ (B−→([CP]Dγ )D∗K−)+(B+→([CP]Dγ )D∗K+) (B−_→([CP]Dγ )D∗π−)+(B+_→([CP]Dγ )D∗π+)× 1 RK_Kπ ,π_/π0/γ AKKπ ,π0 (B−→([K−π+]Dπ0)D∗K−)−(B+→([K+π−]Dπ0)D∗K+) (B−→([K−π+]Dπ0)D∗K−)+(B+→([K+π−]Dπ0)D∗K+) AπKπ ,π0 (B −_→([K−π+]Dπ0)D∗π−)−(B+→([K+π−]Dπ0)D∗π+) (B−→([K−π+]Dπ0)D∗π−)+(B+→([K+π−]Dπ0)D∗π+) AKKπ ,γ (B−→([K−π+]Dγ )D∗K−)−(B+→([K+π−]Dγ )D∗K+) (B−→([K−π+_]Dγ )D∗K−)+(B+→([K+π−]Dγ )D∗K+) AπKπ ,γ (B −_→([K−π+_]Dγ )D∗π−)−(B+→([K+π−]Dγ )D∗π+) (B−→([K−π+]Dγ )D∗π−)+(B+→([K+π−]Dγ )D∗π+) ACPK,π0 (B−→([CP]Dπ0)D∗K−)−(B+→([CP]Dπ0)D∗K+) (B−→([CP]Dπ0)D∗K−)+(B+→([CP]Dπ0)D∗K+) ACPπ,π0 (B −_→([CP]Dπ0)D∗π−)−(B+→([CP]Dπ0)D∗π+) (B−→([CP]Dπ0)D∗π−)+(B+→([CP]Dπ0)D∗π+) ACPK,γ (B−→([CP]Dγ )D∗K−)−(B+→([CP]Dγ )D∗K+) (B−→([CP]Dγ )D∗K−)+(B+→([CP]Dγ )D∗K+) ACPπ,γ (B −_→([CP] Dγ )D∗π−)−(B+→([CP]Dγ )D∗π+) (B−→([CP]Dγ )D∗π−)+(B+→([CP]Dγ )D∗π+)

samples.Thesecorrectionstransformthe measured charge asym-metriesinto

CP asymmetries.

2. Detector and simulation

The LHCb detector[15,18] isa single-armforward spectrome-tercoveringthepseudorapidity range2

<

η

<

5,designedforthe studyofparticles containing

b or

c quarks. The detectorincludes a high-precision trackingsystem consistingof a silicon-strip ver-tex detector surrounding the pp interaction region, a large-area silicon-stripdetectorlocated upstreamofa dipole magnetwitha bending powerof about4 Tm, andthree stations ofsilicon-strip detectorsandstrawdrifttubesplaceddownstreamofthemagnet. Thetrackingsystemprovidesameasurementofmomentum, p, of chargedparticleswitharelativeuncertaintythatvariesfrom0.5% atlow momentum to 1.0%at 200 GeV

/

c. Theminimum distance ofa tracktoa primary vertex(PV), theimpact parameter(IP), is measured witha resolutionof

(

15

+

29

/

pT)μm,where pT is the componentof the momentum transverseto the beam, in GeV

/

c.

Different types ofcharged hadronsare distinguished using infor-mation from two ring-imaging Cherenkov detectors (RICH) [19, 20]. Photons,electrons andhadronsare identified by a calorime-tersystemconsistingofscintillating-padandpreshowerdetectors, anelectromagneticcalorimeterandahadroniccalorimeter.Muons are identified bya systemcomposed ofalternating layers ofiron andmultiwireproportionalchambers.

Thetriggerconsistsofahardwarestage, basedoninformation from the calorimeterandmuon systems,followed by a software stage,inwhichallchargedparticleswithpT

>

500

(

300

)

MeVare reconstructedfor2011 (2012)data,and

p

T

>

70 MeVfor2015and 2016 data.At the hardware trigger stage, eventsare requiredto

containamuonwithhigh

p

Torahadron,photonorelectronwith hightransverseenergyinthecalorimeters.Forhadrons,the trans-verseenergythresholdvariedbetween3and4GeVbetween2011 and2016.Thesoftwaretriggerrequiresatwo-,three- orfour-track secondary vertex with significant displacement from all primary

pp interaction vertices.Amultivariatealgorithm[21,22]isusedfor theidentificationof secondaryvertices consistentwiththe decay ofa

b hadron.

Inthesimulation,

pp collisions

aregeneratedusing Pythia8[23] withaspecificLHCb configuration [24].Decaysofhadronic parti-clesaredescribedbyEvtGen[25],inwhichfinal-stateradiationis generatedusingPhotos_{[26].Theinteractionofthegenerated} par-ticleswiththe detector,andits response,are implemented using theGeant4 toolkit[27]asdescribedinRef.[28].

3. Event selection

After reconstruction of the D-meson candidate from two op-positely chargedparticles, the sameevent selection isapplied to all B−

→

D(∗)_h−_{channels. Sincetheneutralpionorphoton from} the vector D∗ decay is not reconstructed, partially reconstructed

B−

→

D∗h− decays and fully reconstructed B−

→

Dh− decays contain thesame reconstructed particles,andthus appear inthe samesample.Thesedecaysare distinguishedaccordingtothe re-constructedinvariantmass

m

(

Dh

)

,asdescribedinSec.4.

The reconstructed D-meson candidate mass is required to be within

±

25 MeV

/

c2 _{of the known D}0 _{mass [29], which} corre-spondstoapproximatelythreetimesthemassresolution.Thekaon or pion originatingfrom the B− decay, subsequently referred to as the companion particle, is required to have pT in the range 0.5–10 GeV

/

c and p in the range 5–100 GeV

/

c. These require-ments ensure that the trackis within the kinematic coverage of the RICH detectors, which are used to provide particle identifi-cation (PID)information.Details ofthe PID calibrationprocedure are given in Sec. 4. A kinematic fit is performed to each de-cay chain,withvertexconstraintsapplied to boththe B− and D

decay products, and the D candidate constrained to its known mass [30]. Events are required to have been triggered by either the decay products of the signal candidate, or by particles pro-ducedelsewhere in the pp collision. Each B− candidate is asso-ciated to the primary vertex (PV) to which it has the smallest

χ

2

IP, which isquantified asthe difference inthe vertex fit

χ

2 of a givenPVreconstructed withandwithout theconsidered parti-cle. The B− meson candidates with invariant masses in the in-terval 4900–5900 MeV

/

c2 are retained. This rangeis wider than that considered in Ref. [12],in order to include thepartially re-constructed B−

→ ([

h+₁h−₂

]

Dπ0

₎

D∗h− and B−

→ ([

h+1h−2

]

D

γ

)

D∗h− decays,which fall at m

(

Dh

)

values belowthe known B− meson mass.

A pair of boosted decision tree (BDT) classifiers, implement-ing the gradient boost algorithm [31], is employed to achieve further background suppression. The BDTs are trained using simulated B−

→ [

K−

π

+

]

DK− decays and a background

sam-ple of K−

π

+K− combinations in data with invariant mass in the range 5900–7200 MeV

/

c2_{; the training was also repeated} using partially reconstructed B−

→ ([

K−

π

+

]

Dπ0

₎

D∗K− and B−

→ ([

K−

π

+

]

D

π

0

)

D∗K− decays, and the difference in

perfor-mance found to be negligible. No evidence of overtraining was found in the training of either BDT. For the first BDT, back-groundcandidateswithareconstructed D-meson massmorethan 30 MeV

/

c2 from the known D0 mass are used in the training. In the second BDT, background candidates with a reconstructed

D-meson mass within

±

25 MeV

/

c2 of the known D0 mass are used. A loose requirement on the classifier response of the first BDT isapplied prior to training thesecond one. Thisfocusesthe

second BDT training ona backgroundsample enriched withfully reconstructed D mesons. Both BDT classifier responses arefound tobeuncorrelatedwiththe

B-candidate

invariantmass.

TheinputtobothBDTsisasetoffeaturesthatcharacterisethe signal decay. These features can be divided into two categories: (1) properties ofanyparticleand(2)propertiesofcomposite par-ticlesonly(the

D and

B−candidates).Specifically:

1. p,pT and

χ

IP2;

2. decaytime,flightdistance,decayvertexquality,radialdistance betweenthedecayvertexandthePV,andtheanglebetween the particle’s momentum vector and the lineconnecting the productionanddecayvertices.

In addition, a featurethat estimates the imbalanceof pT around the B− candidatemomentumvector isalsousedinbothBDTs. It isdefinedas

IpT

=

pT

(

B−

)

− 

pT

pT

(

B−

)

+ 

pT

,

(1)

where thesumis takenover tracksinconsistent withoriginating from the PV which lie within a cone around the B− candidate, excluding tracks used to make the signal candidate.The cone is definedbyacirclewitharadiusof1.5unitsintheplane of pseu-dorapidity and azimuthal angle (expressed in radians). Including the IpT featureinthe BDTtraining givespreferenceto B−

candi-datesthatareeitherisolatedfromtherestoftheevent,or consis-tentwitharecoilagainstanother

b hadron.

SincenoPIDinformationisusedintheBDTclassifier,the effi-ciencyforB−

→

D(∗)_K−_and

_B

−

_→

_D(∗)

_π

−_{decaysissimilar,with}

insignificantvariations arisingfromsmalldifferencesinthedecay kinematics.ThecriteriaappliedtothetwoBDTresponsesare opti-misedbyminimisingtheexpectedstatisticaluncertaintyon

R

CP,π0

and

R

CP,_{γ , asmeasuredwiththemethoddescribedbelow.The} pu-rityofthesampleis furtherimprovedbyrequiringthatall kaons and pions in the D decay are positively identified by the RICH. This PIDselection usedtoseparate the D

π

and D K samples has anefficiencyofabout85%perfinal-stateparticle.

Peaking background contributions fromcharmless decays that resultin thesame final stateasthe signal are suppressedby re-quiring that the flight distance ofthe D candidate fromthe B−

decay vertex is larger than two times its uncertainty. After the aboveselections,multiplecandidatesexistin0.1%oftheeventsin the sample. When morethan one candidateis selected,only the candidatewiththebest B− vertexquality isretained.The overall effectofthemultiple-candidateselectionisnegligible.

4. Fit to data

Thevaluesofthe

CP observables

aredeterminedusingabinned extended maximum likelihood fit to the data. Distinguishing be-tween

B

+and

B

−candidates,companionparticlehypotheses,and thethree D decay productfinalstates,yields12independent sam-ples whicharefittedsimultaneously.Thetotal probabilitydensity function(PDF)isbuiltfromsixsignalfunctions,oneforeachofthe

B−

→

D

π

−, B−

→

D K−, B−

→ (

D

π

0

₎

D∗

π

−, B−

→ (

D

π

0

)

D∗K−, B−

→ (

D

γ

)

D∗

π

−,andB−

→ (

D

γ

)

D∗K−decays.Inaddition,there

are functionswhich describe thecombinatorial background com-ponents, background contributions from B decays to charmless final states and background contributions from partially recon-structeddecays.AllfunctionsareidenticalforB+ andB−decays.

4.1. B−

→

D

π

−

The B−

→

D

π

− signalcomponentismodelledusingan asym-metricdouble-Gaussian-likefunction

Fig. 1. InvariantmassdistributionsofselectedB±→ [K±π∓]Dh±candidates,separatedbycharge,withB−(B+)candidatesontheleft (right).Thetoppanelscontainthe B±→D(∗)0_K± _{candidatesamples,asdefinedbyaPIDrequirementonthecompanionparticle.Theremainingcandidatesareplacedinthebottompanels,reconstructed}

withapionhypothesisforthecompanion.Theresultofthefitisshownbythethinsolidblackline,andeachcomponentislistedinthelegend.Thecomponentreferredto as‘Part.reco.mis-ID’isthetotalcontributionfromallpartiallyreconstructedandmisidentifieddecays.

f

(

m

)

=

fcoreexp



−(

m

−

μ

)

2 2

σ

2 c

+ (

m

−

μ

)

2

α

L,R



+ (

1

−

fcore

)

exp



₋₍

m

−

μ

)

2 2

σ

2 w



(2)

whichhasapeakposition

μ

andcorewidth

σ

c,where

α

L

(

m

<

μ

)

and

α

R

(

m

>

μ

)

parameterise the tails. The

μ

and

α

parameters

aresharedacrossallsamplesbutthecorewidthparametervaries independentlyforeach

D final

state.TheadditionalGaussian func-tion, with a small fractional contribution, is necessary to model satisfactorilythetailsofthepeak.

The B−

→

D

π

− decays misidentified as B−

→

D K− are dis-placedtohighermassinthe

D K

−subsamples.Thesemisidentified candidates are modelled by the sum of two Gaussian functions witha commonmean,modified toincludetailcomponentsasin Eq.(2).The mean,widthsand

α

R areleft tovaryfreely,while

α

L

isfixedtothevaluefoundinsimulation.

4.2. B−

→

D K−

Inthe

D

(∗)0_K−_{samples,Eq.(2)isusedforthe}

_B

−

_→

_{D K}−

sig-nalfunction.The peakposition

μ

andthetwo tailparameters

α

L

and

α

R aresharedwiththe

B

−

→

D

π

−signalfunction,asarethe

widecomponentparameters fcoreand

σ

w.Thecorewidth

param-eterineach D mode isrelatedto thecorresponding B−

→

D

π

−

widthbyafreelyvaryingratiocommontoall

D final

states.

Misidentified B−

→

D K−candidatesappearinginthe D(∗)0

π

−

subsamplesare describedbya fixedshapeobtainedfrom simula-tion, which islater varied to determinea systematic uncertainty associatedwiththischoice.

4.3. B−

→ (

D

π

0

₎

D∗

π

−

Inpartiallyreconstructeddecaysinvolvingavectormeson,the

Dh− invariantmassdistributiondependsuponthespinandmass ofthe missingparticle. Inthe caseof B−

→ (

D

π

0

₎

D∗

π

− decays,

themissingneutralpionhasspin-parity0−.Thedistributionis pa-rameterisedby anupward-openparabola, whoserangeisdefined by the kinematic endpoints of the decay. It is convolved witha Gaussianresolutionfunction,resultingin

f

(

m

)

=

b



a



μ

−

a

+

b 2



2



₁

_{− ξ}

b

−

a

μ

+

b

ξ

−

a b

−

a



e− (μ−m)2 2σ2 _d

_μ

_.

₍₃₎

Theresultingdistributionhasacharacteristicdouble-peakedshape, visibleinFigs. 1–3asthelightgreyfilledregionsappearingtothe left of thefully reconstructed B−

→

D0h− peaks.The lower and upper endpoints ofthe parabola are a and b, respectively, while the relative height of the lower and upper peaks is determined by the

ξ

term. When

ξ

=

1,both peaks areof equalheight, and

Fig. 2. Invariant mass distributions of selected B±→ [K+K−]Dh±candidates, separated by charge. SeeFig. 1for details of each component.

Fig. 3. Invariant mass distributions of selected B±→ [π+π−]Dh±candidates, separated by charge. SeeFig. 1for details of each component.

deviationof

ξ

fromunityaccountsformass-dependent reconstruc-tionandselection efficiencyeffects. Thevalues ofa, b and

ξ

are takenfrom fitsto simulated events,while theconvolution Gaus-sian width

σ

isallowed to varyfreely inthe massfit ineach D

modesubsample.

Partially reconstructed B−

→ (

D

π

0

₎

D∗

π

− decays, where the

companionpionismisidentifiedasakaon,areparameterisedwith a semiempirical function, formed fromthe sum ofGaussian and error functions. The parameters of thisfunction are fixed to the valuesfound infits to simulatedevents,andare varied to deter-minetheassociatedsystematicuncertainty.

4.4. B−

→ (

D

π

0

₎

D∗K−

Equation (3) is also used to describe partially reconstructed

B−

→ (

D

π

0

₎

D∗K−decays,wherethewidth

σ

ineachofthe

D K

−

samples is related to the D

π

− width by a freely varying ratio

rσ , whichissharedacrossall functionsdescribingpartially recon-structed decays. All other shape parameters are shared withthe

B−

→ (

D

π

0

₎

D∗

π

−function.

Partially reconstructed B−

→ (

D

π

0

₎

D∗K− decays, where the

companionkaonismisidentifiedasapion,areparameterisedwith a semiempirical function, formed fromthe sum ofGaussian and error functions. The parameters of thisfunction are fixed to the

values foundin fitsto simulatedevents,andare varied to deter-minetheassociatedsystematicuncertainty.

4.5. B−

→ (

D

γ

)

D∗

π

−

Partiallyreconstructed

B

−

→ (

D

γ

)

D∗

π

−decaysinvolvea

miss-ing particle ofzeromass andspin-parity 1−.The D

π

− invariant mass distribution is described by a parabola exhibiting a maxi-mum, convolved with a Gaussian resolution function. The func-tionalformofthiscomponentis

f

(

m

)

=

b



a

−(

μ

−

a

)(

μ

−

b

)



1

− ξ

b

−

a

μ

+

b

ξ

−

a b

−

a



e− (μ−m)2 2σ2 _d

_μ

_.

(4)

This distribution exhibits a broad single peak, as opposed to the double-peaked B−

→ (

D

π

0

₎

D∗

π

− distribution described by

Eq.(3).InFigs. 1–3,thiscomponentisvisibleasthewidehatched regions bounded by solid black curves,which appear below the fullyreconstructed

B

−

→

D0h− peaks.

The valuesof a, b,

ξ

and

σ

are fixed using fits to simulated events. Theclear difference betweentheinvariant mass distribu-tions of B−

→ (

D

γ

)

D∗

π

− and B−

→ (

D

π

0

)

D∗

π

− decaysenables

theirstatisticalseparation,andhencethedeterminationof

CP

ob-servablesforeachmodeindependently.

Partially reconstructed B−

→ (

D

γ

)

D∗

π

− decays where the

companionpionismisidentifiedasakaonaretreatedinan equiv-alent manner to misidentified B−

→ (

D

π

0

₎

D∗

π

− decays, as

de-scribedabove.

4.6. B−

→ (

D

γ

)

D∗K−

Equation (4) is also used to describe partially reconstructed

B−

→ (

D

γ

)

D∗K− decays,wherethewidth

σ

ineachofthe D K−

samples is related to the D

π

− width by the ratio rσ . All other shapeparametersareshared withthe B−

→ (

D

γ

)

D∗

π

− function.

Partiallyreconstructed B−

→ (

D

π

0

₎

D∗K− decayswherethe

com-panionkaonismisidentifiedasapionaretreatedinanequivalent mannertomisidentified

B

−

→ (

D

π

0

₎

D∗K−decays. 4.7. Combinatorial background

Anexponential function isused to describe thecombinatorial background.The exponential function is widely used to describe combinatorial backgrounds to B− decays in LHCb, andhas been validated for numerous different decay modes. Independent and freelyvaryingexponentialparametersandyieldsareusedtomodel thiscomponentineachsubsample,withtheconstraintthatthe

B

+

and

B

−yieldsarerequiredtobeequal.Thesystematicuncertainty associatedwiththisconstraintisnegligible.

4.8. Charmless background

Charmless B−

→

h+₁h−₂h− decays, where

h

+₁, h−₂ and

h

− each representa chargedkaonorpion,peakatthe B− massand can-notbedistinguishedeffectivelyfromthefullyreconstructed

B

−

→

Dh− signalsintheinvariant massfit.AGaussianfunctionisused tomodel thiscomponent, witha 25

±

2 MeV

/

c2 width parame-terthatistakenfromsimulation;thisisabout50%widerthanthe

B−

→

Dh− signal function, due to the application of a D mass

constraint in the calculation of the B-candidate invariant mass. Thisconstraint improves the invariant mass resolution for signal decays,butworsensitforcharmlessbackgroundcontributions.

Partially reconstructed charmless decays of the type

B

→

h+₁h−₂h−X , where X is achargedpion,neutralpionorphoton thathasnotbeenreconstructed,contribute atlowinvariantmass. Their contributions are fixed to the fully reconstructed charm-less components scaled by relative branching fractions [29] and efficiencies determined fromsimulated samples.A parabolawith negative curvatureconvolvedwitha Gaussian resolutionfunction is used to model this component, with shape parameter values takenfromsimulation[32].

Thecharmlesscontributionisinterpolatedfromfitsto the B−

mass spectrum in both the lower and upper D-mass sidebands, withoutthekinematicfitofthedecaychain.Thecharmlessyields aredeterminedindependently forB+ and B− candidatesandare then fixed in the analysis. Their uncertainties contribute to the systematicuncertainties ofthefinal results.The largestcharmless contributionisinthe

B

−

→ [

π

+

π

−

]

DK−mode,whichhasayield

correspondingto7%ofthemeasuredsignalyield.

4.9. Partially reconstructed background

Severaladditionalpartiallyreconstructed

b-hadron

decays con-tributeatlow invariant mass values.The dominantcontributions arefrom B−

→

Dh−

π

0 _{and B}0

_{→ (}

_D

_π

+

₎

D∗+

π

− decays,wherea

neutral pion or positively charged pion is missed in the recon-struction.3 _{The invariant mass distribution of these sources}

de-3 _{Whenconsidering partiallyreconstructed background contributions, the}

as-sumptionismadethattheproductionfractions fuand fdareequal.

pends upon the spin and mass of the missing particle, as with the B−

→

D∗h− signals. In both cases, the missing particle has spin-parity 0−, such that the Dh− distribution is parameterised usingEq.(3),withshapeparametervaluestakenfromsimulation. The Dalitz structure of B−

→

Dh−

π

0 _{decays is modelled using} Laura++[33].

Decays inwhichaparticleismissed anda companionpionis misidentified as a kaon are parameterised with a semiempirical function, formed from the sum of Gaussian and error functions. The parameters ofeach partially reconstructed functionare fixed to the values found in fits to simulated events, and are varied to determine theassociated systematicuncertainty. The yields of the

B

−

→

D

π

−

π

0_and

_B

−

_→

_{D K}−

_π

0_{contributionsvary} indepen-dently in each subsample,with a CP asymmetry that isfixed to zerointhe caseof thefavouredmode butallowed tovaryfreely in the GLW samples. The yields of the B0

→ (

D

π

+

)

D∗+

π

− and B0

→ (

D

π

+

)

D∗+K− contributions, where the

π

+ is not

recon-structed,arefixedrelativetothecorresponding

B

−

→

D

π

−yields usingbranching fractions[29,34,35] andefficiencies derived from simulation.TheirCP asymmetries arefixedtozeroinall subsam-plesasno

CP violation

isexpected.

Further contributions from partially reconstructed

B−

→ (

D

π

0

_/

_γ

₎

D∗h−

π

0 and B 0

→ (

D

π

+

)

D∗+h−

π

0 decays occur

at the lowest values of invariant mass, where two particles are not reconstructed.Thesedecaysaredescribed bythesumof sev-eral parabolas convolved with resolution functions according to Eqs.(3)and(4),withshapeparameters fixedtothevaluesfound in fits to simulated samples. The yields and CP asymmetries of thesecontributionsvaryfreelyineachsubsample.

Colour-suppressed B0

_→

_Dh−

_π

+ _{and B}0

_→

_D∗_h−

_π

+ _decays alsocontributetothebackground.Theratesofthesesmall contri-butions are fixed relative to their corresponding colour-favoured mode yields using the known relative branching fractions [29, 36–39].Inthe

B

−

→ [

K+K−

]

Dh−samples,

_b0

→ [

p+K−

π

+

]

+

ch

−

decays contribute to the background when the pion is missed and the proton is misidentified as the second kaon. The wide function describing thiscomponent is fixed fromsimulation, but the yield in the B−

→ [

K+K−

]

D

π

− subsample varies freely.

The

0_b

→ [

p+K−

π

+

]

+cK

− _{yield is constrained using a}

mea-surement of

B(

0

b

→

+cK−

)/B(

0b

→

+c

π

−

)

[40].In both the B−

→ [

K+K−

]

DK−and

B

−

→[

π

+

π

−

]

DK−samples,

B

0s

→

D K−

π

+

decays in which the companion pion is missed contribute to the background. The function describing this componentis fixed fromfits to simulated samplesgenerated according tothe Dalitz modelin Ref.[33,41],andthe yieldis constrainedrelativeto the corresponding B−

→

D

π

− mode yield scaled by branching frac-tions [29,34,42],efficiencies determinedfromsimulation, andthe relativeproductionratesof

B

0

s andB0 mesonsat

√

s

=

7 TeV[43]. The increase in relative production rate at 13 TeV is small [44], andsothe7 TeVvalueisusedtodescribealldataintheanalysis.

4.10. PID efficiencies

Inthe

D

(∗)_K− _{subsamples,the B}−

_→

_D(∗)

_π

−_cross-feedis

de-termined by the fit to data. The B−

→

D(∗)_K− _{cross-feed into}

the D(∗)

_π

− _{subsamples is not well separated from background,} so the expected yield is determined by a PID calibration proce-dure usingapproximately20million D∗+

→ [

K−

π

+

]

D

π

+ decays.

The reconstruction of this decay is performed using kinematic variables only, and thus provides a pure sample of K∓ and

π

±

particlesunbiasedinthePIDvariables.ThePIDefficiencyis param-eterised asa functionof particlemomentum andpseudorapidity, as well asthe charged-particle multiplicity in the event. The ef-fectivePIDefficiencyofthesignalisdeterminedbyweightingthe

Table 2

Signalyieldsasmeasuredinthefittothedata.

Mode Yield B±→ [Kπ]Dπ± 862 785±945 B±→ [K K]Dπ± 105 923±368 B±→ [π π]Dπ± 33 381±173 B±→ [Kπ]DK± 66 987±326 B±→ [K K]DK± 8125±129 B±→ [π π]DK± 2571±70 B±→ ([Kπ]Dπ0)D∗π± 519 211±3747 B±→ ([K K]Dπ0)D∗π± 63 742±460 B±→ ([π π]Dπ0)D∗π± 20 088±145 B±→ ([Kπ]Dπ0)D∗K± 40 988±569 B±→ ([K K]Dπ0)D∗K± 5725±165 B±→ ([π π]Dπ0)D∗K± 1804±52 B±→ ([Kπ]Dγ)D∗π± 291 372±2103 B±→ ([K K]Dγ)D∗π± 35 771±258 B±→ ([π π]Dγ)D∗π± 11 273±81 B±→ ([Kπ]Dγ)D∗K± 22 752±316 B±→ ([K K]Dγ)D∗K± 2520±245 B±→ ([π π]Dγ)D∗K± 794±77

calibration sample such that the distributions of these variables match those of selected B−

→

D0

_π

− _{signal decays. It is found} that 71.2% of B−

→

D K− decays pass the companion kaon PID requirement,withnegligiblestatisticaluncertaintyduetothesize ofthecalibrationsample;theremaining28.8%cross-feedintothe

B−

→

D(∗)

_π

− _{sample. With the same PID requirement,} approx-imately 99.5% of the B−

→

D

π

− decays are correctlyidentified. These efficiencies are also taken to represent B−

→ (

D

π

0

₎

D∗h−

andB−

→ (

D

γ

)

D∗h−signaldecaysinthefit,sincethecompanion

kinematicsaresimilar acrossalldecaymodesconsidered.The re-latedsystematicuncertaintyisdeterminedbythesizeofthesignal samplesused, andthusincreases forthelower yield modes. The systematicuncertaintyrangesfrom0.1%in B−

→ [

K−

π

+

]

DK−to

0.4%in B−

→ [

π

+

π

−

]

DK−.

4.11. Production and detection asymmetries

In order to measure CP asymmetries, the detection asymme-triesfor K± and

π

± mesonsmust be taken intoaccount. A de-tectionasymmetryof

(

−

0

.

87

±

0

.

17

)

%isassignedforeachkaonin thefinalstate,primarilyduetothefactthatthenuclearinteraction lengthof

K

−mesonsisshorterthanthatof

K

+mesons.Itis com-puted by comparingthe charge asymmetries in D−

→

K+

π

−

π

−

and

D

−

→

K0

S

π

−calibrationsamples,weightedtomatchthe kine-maticsofthesignalkaons.Theequivalent asymmetryforpionsis smaller

(

−

0

.

17

±

0

.

10

)

%[16].The

CP asymmetry

inthefavoured

B−

→ [

K−

π

+

]

D

π

− decay isfixed to

(

+

0

.

09

±

0

.

05

)

%, calculated

from currentknowledge of

γ

and rB in this decay [2], withno

assumption made aboutthe strong phase,

δ

D_Bπ. This enablesthe effectiveproductionasymmetry, Aeff_B±,tobe measured and

simul-Table 4

SystematicuncertaintiesfortheCP observablesmeasuredinafullyreconstructed manner,quotedasapercentageofthestatisticaluncertaintyontheobservable.The Sim uncertaintyonRKKπ/π isduetothelimitedsizeofthesimulatedsamplesused todeterminetherelativeefficiencyforreconstructingandselectingB−→Dπ−and B−→D K−decays. [%] AKπ K AπK K AK KK Aπ ππ Aπ πK RK K Rπ π RKKπ/π PID 6.0 4.3 2.0 2.7 10.3 13.8 18.8 0.0 Bkg rate 7.5 1.8 10.2 4.1 18.9 68.7 46.0 0.0 Bkg func 7.6 0.4 4.2 0.4 7.2 9.5 16.7 0.0 Sig func 11.1 0.9 0.8 0.9 14.3 7.9 20.9 0.0 Sim 7.1 0.5 0.2 0.4 5.6 3.5 7.6 174.2 Asym 37.4 52.7 3.7 31.2 2.3 0.1 0.1 0.0 Total 41.5 52.9 11.9 31.6 27.5 71.2 56.9 174.2

taneouslysubtractedfromthechargeasymmetrymeasurementsin othermodes.

4.12. Yields and selection efficiencies

The totalyield foreach modeis asumofthe numberof cor-rectly identified andcross-feedcandidates;their valuesare given inTable 2.Thecorrespondinginvariantmassspectra,separatedby charge,areshowninFigs. 1–3.

Toobtaintheobservable RK_Kπ_{/π (RK}Kπ_/π,π0/γ),whichisdefinedin Table 1, theratioofyieldsmustbe correctedby therelative effi-ciencywithwhich

B

−

→

D K−and

B

−

→

D

π

−(B−

→

D∗K−and

B−

→

D∗

π

−) decays are reconstructed and selected. Both ratios arefoundtobeconsistentwithunitywithintheirassigned uncer-tainties,whichtakeintoaccountthesizeofthesimulatedsamples andtheimperfectmodellingoftherelativepionandkaon absorp-tioninthedetectormaterial.

Todeterminethebranchingfraction

B(

D∗0

_→

_D0

_π

0

₎

_,theyields of the B−

→ (

D

π

0

₎

D∗

π

− and B−

→ (

D

γ

)

D∗

π

− modes are

cor-rectedfortherelative efficienciesofthe neutralpionandphoton modesasdeterminedfromsimulation.Asbothofthesemodesare partially reconstructed with identicalselection requirements, the relative efficiencyis foundto be unitywithin its assigned uncer-tainty,andisvariedtodeterminetheassociatedsystematic uncer-tainty. In the measurement of

B(

D∗

→

D

π

0

₎

_{, the assumption is} madethat

B(

D∗

→

D

π

0

₎

₊

_B(

_D∗

_→

_D

_γ

₎

₌

_1[29].

The branching fraction

B(

B−

→

D∗0

π

−

)

is determined from the total B−

→

D∗

π

− yield, the total B−

→

D

π

− yield, the rel-ativeefficienciesdeterminedfromsimulation,andthe

B

−

→

D

π

−

branchingfraction[29,34].Boththeefficienciesandexternalinput branching fractionare variedtodeterminetheassociated system-aticuncertainty.

5. Systematic uncertainties

The21observablesofinterestarefreeparametersofthefit,and each ofthem is subject to a set of systematicuncertainties that resultfromtheuseoffixedtermsinthefit.Thesystematic

uncer-Table 3

SystematicuncertaintiesfortheCP observablesmeasuredinapartiallyreconstructedmanner,quotedasapercentageofthestatisticaluncertaintyontheobservable.

[%] AKKπ ,γ A Kπ ,γ π AKπ ,π 0 K A Kπ ,π0 π A C P,γ K A C P,γ π AC P,π 0 K A C P,π0 π RC P,γ RC P,π 0 RKKπ ,π/π 0/γ PID 4.0 11.4 4.4 3.8 9.1 5.0 4.7 4.4 22.0 16.9 74.8 Bkg rate 3.5 1.6 3.2 3.6 40.8 3.5 16.5 5.7 114.0 41.9 180.3 Bkg func 8.9 1.0 3.7 0.7 24.4 1.6 27.1 1.3 42.6 25.0 417.3 Sig func 4.8 3.9 2.9 3.9 10.9 3.6 3.7 4.3 24.6 13.8 148.4 Sim 3.1 1.1 2.1 1.9 6.5 0.9 4.3 2.9 23.5 15.3 153.8 Asym 29.9 6.8 34.1 19.4 1.0 9.4 2.2 26.1 1.4 0.6 1.9 Total 32.1 14.0 35.0 20.6 50.0 11.9 32.7 27.6 128.3 55.6 507.9

Table 5

Systematicuncertaintiesforthebranchingfractionmeasurements,quotedasa per-centageofthestatisticaluncertaintyontheobservable.

[%] B(D∗0_→D0_π0_{) B(B}−_→D∗0_π−₎ PID 85.3 117.7 Bkg rate 364.4 672.1 Bkg func 52.2 29.0 Sig func 417.2 379.7 Sim 295.4 509.3 Asym 0.2 0.3 Total 635.7 932.7

taintiesassociatedwithusingthesefixedparameters areassessed byrepeatingthefitmanytimes,varyingthevalueofeachexternal parameterwithinits uncertaintyaccordingtoaGaussian distribu-tion.The resulting spread (RMS)in the value of each observable istaken asthe systematicuncertainty onthat observable dueto theexternalsource.Thesystematicuncertainties,groupedintosix categories, are listed in Tables 3 and 4 for the CP observables

measuredinapartiallyreconstructedandfullyreconstructed man-ner, respectively. The systematic uncertainties for the branching fractionmeasurements arelistedinTable 5.Correlations between thecategoriesarenegligible,butcorrelationswithincategoriesare accountedfor. The total systematic uncertainties are summed in quadrature.

Thefirstsystematiccategory,referred toas

PID in

Tables 3

−

5, accountsfortheuncertaintyduetotheuseoffixedPIDefficiency valuesinthefit.Thesecond category

Bkg rate corresponds

tothe use of fixed background yields in the fit. For example, the rate ofB0

_→

_D∗−

_π

+ _{decaysisfixedinthefitusingknownbranching}

fractionsasexternalinputs.Thiscategoryalsoaccountsfor charm-lessbackground contributions, each of whichhave fixed rates in thefit.The

Bkg func and Sig func categories

refertotheuseoffixed shapeparametersinbackgroundandsignalfunctions,respectively; eachoftheseparameters isdetermined usingsimulatedsamples. Thecategory

Sim accounts

fortheuse offixed selection efficien-ciesderivedfromsimulation,forinstancetherelativeefficiencyof selecting

B

−

→ (

D

π

0

₎

D∗

π

−and

B

−

→

D

π

−decays.Thefinal

cat-egory,

Asym,

referstotheuseoffixedasymmetriesinthefit.This categoryaccountsfortheuseoffixed

CP asymmetries

and detec-tionasymmetriesinthefit,asdescribedearlier.

6. Results

Theresultsare

AK_Kπ,γ

= +

0.001

±

0.021 (stat)

±

0.007 (syst) AπKπ,γ

= +

0.000

±

0.006 (stat)

±

0.001 (syst) AK_Kπ,π0

= +

0.006

±

0.012 (stat)

±

0.004 (syst) A_πKπ,π0

= +

0.002

±

0.003 (stat)

±

0.001 (syst) ACP_K,γ

= +

0.276

±

0.094 (stat)

±

0.047 (syst) ACP_π,γ

= −

0.003

±

0.017 (stat)

±

0.002 (syst) ACP_K,π0

= −

0.151

±

0.033 (stat)

±

0.011 (syst) ACP_π,π0

= +

0.025

±

0.010 (stat)

±

0.003 (syst) RCP,γ

=

0.902

±

0.087 (stat)

±

0.112 (syst) RCP,π0

=

1.138

±

0.029 (stat)

±

0.016 (syst) R_KKπ_/π,π0/γ

= (

7.930

±

0.110 (stat)

±

0.560 (syst))

×

10−2

B

(

D∗0

→

D0

π

0

)

=

0.636

±

0.002 (stat)

±

0.015 (syst)

B

(

B−

→

D∗0

π

−

)

= (

4.664

±

0.029 (stat)

±

0.268 (syst))

×

10−3 AK_Kπ

= −

0.019

±

0.005 (stat)

±

0.002 (syst) AK K_π

= −

0.008

±

0.003 (stat)

±

0.002 (syst) AK K_K

= +

0.126

±

0.014 (stat)

±

0.002 (syst) Aπ π_π

= −

0.008

±

0.006 (stat)

±

0.002 (syst) Aπ π_K

= +

0.115

±

0.025 (stat)

±

0.007 (syst) RK K

=

0.988

±

0.015 (stat)

±

0.011 (syst) Rπ π

=

0.992

±

0.027 (stat)

±

0.015 (syst) RK_Kπ_/π

= (

7.768

±

0.038 (stat)

±

0.066 (syst))

×

10−2

.

The resultsobtainedusing fullyreconstructed B−

→

Dh− decays supersede those in Ref. [12], while the B−

→

D∗h− results are reported for the first time. The statistical and systematic corre-lation matrices are given in the appendix. There is a high de-greeofanticorrelation betweenpartially reconstructedsignal and backgroundcomponentsinthefit,whichall compete foryield in the same invariant mass region.The anticorrelation betweenthe

B−

→ (

D

π

0

₎

D∗h− and B−

→ (

D

γ

)

D∗h−CP observables is visible

inTable 6oftheappendix.Thepresenceofsuchanticorrelationsis anaturalconsequenceofthemethodofpartialreconstruction,and limitsthe precision with whichthe CP observables can be mea-suredusingthisapproach.

ThevalueofAK K_K hasincreasedwithrespecttotheprevious re-sult[12],duetoalargervaluebeingmeasuredinthe

√

s

=

13 TeV data. The values measured in the independent

√

s

=

7, 8 and 13 TeVdatasetsareconsistentwithin 2.6standarddeviations.All other updated measurements are consistent within one standard deviationwiththoseinRef.[12].

Observablesinvolving D

→

K+K−and D

→

π

+

π

− decayscan differ dueto CP violation inthe D decays oracceptance effects. The latest LHCb results [45] show that charm CP -violation ef-fectsare negligible forthe determination of

γ

, andthat there is alsonosignificantdifferenceintheacceptanceforthetwomodes. Therefore,whileseparateresultsarepresentedforthe B−

→

Dh−

modestoallowcomparisonwithpreviousmeasurements,the com-binedresultismostrelevantforthedeterminationof

γ

.The RK K

and

Rππ observables

havestatisticalandsystematiccorrelationsof

+

0.07 and

+

0.18, respectively. Taking thesecorrelations into ac-count,acombinedweightedaverage

R

CP _isobtained

RCP

=

0.989

±

0.013

(stat)

±

0.010

(syst) .

The sameprocedureis carriedout forthe AK K

K and AππK

observ-ables, whichhavestatisticalandsystematiccorrelationsof

+

0.01 and

+

0.05,respectively.Thecombinedaverageis

ACP_K

= +

0.124

±

0.012

(stat)

±

0.002

(syst) .

The observables RCP,π0

and ACP,π0

(RCP,_{γ and A}CP,_{γ ),} mea-suredusingpartiallyreconstructed

B

−

→

D∗h−decays,canbe di-rectlycomparedwiththeworldaveragevaluesforRCP+

≡

RCP,π0

and ACP+

≡

ACP,π0

(RCP−

≡

RCP,_{γ and A}

CP−

≡

ACP,_{γ ) reported} by the Heavy Flavor Averaging Group [3]; agreement is found at the levelof1.5 and0.4(1.1and 1.4)standard deviations, respec-tively.ThevaluesofRCP,π0

and ACP,π0

considerablyimproveupon the world average precision of RCP+ and ACP+, while the mea-surementsofRCP,_{γ and A}CP,_{γ have aprecision comparabletothe} previousworldaverage.

The value of RK_Kπ_/π,π0/γ is in agreement with, and substan-tially more precise than, the current world average [29,34,46].

Fig. 4. 1σ,2σand3σprofilelikelihoodcontoursforrD∗K B ,δD

∗K

B and

γ

,correspondingto68.3%,95.5%and99.7%confidencelevel(CL),respectively.Thecontoursaremeasured

usingB−→ (Dπ0₎

D∗K−andB−→ (Dγ)D∗K−decays.Thecolourscalerepresents1−CL.(Forinterpretationofthecoloursinthisfigure,thereaderisreferredtotheweb versionofthisarticle.)

The branching fraction measurements of

B(

D∗0

_→

_D0

_π

0

₎

_and

B(

B−

→

D∗0

π

−

)

arefound toagreewiththecurrentworld aver-agevalueswithin0.6and1.3standarddeviations,respectively[29, 34,47].A valuefortheratioofbranchingfractions B(_B(B_B−₋→_→D_D∗00_KK−−)) is alsoobtainedusingthemeasuredresults

B

(

B−

→

D∗0K−

)

B

(

B−

→

D0_K−

₎

=

RK_Kπ_/π,π0/γ RK_Kπ_/π

×

B

(

B−

→

D∗0

π

−

)

B

(

B−

→

D0

_π

−

₎

=

0.992

±

0.077, (5)

wherethe uncertaintyquoted is dominatedby systematic uncer-tainties, and the statistical and systematic correlations between the input observables are fully taken into account. This value is in agreement with, and improves upon, the current world average. The ratios RK_Kπ_{/π and} R_KKπ_/π,π0/γ can be interpreted as

B−

→

D(∗)0_K−

_/

_B−

_→

_D(∗)0

_π

− _{branching fraction ratios, in the}

limit that the suppressed contributions are neglected, which is thesameassumptionthatismadewhenreportingtheresultsfor

B(

B−

→

D∗0

π

−

)

and B(_B(B_B−₋→_→D_D∗00_KK−−)).The branching fraction mea-surementsdemonstratethat the methodofpartial reconstruction is able to measure the B−

→ (

D

π

0

₎

D∗h− and B−

→ (

D

γ

)

D∗h−

signals,despitethecorrelationspresentinthemassfit.

7. Conclusion

World-bestmeasurementsof

CP observables

in

B

−

→

Dh− de-caysare obtainedwiththe D meson reconstructedinthe K−

π

+,

K+K− and

π

+

π

− final states; thesesupersede earlier work on

the GLWmodespresentedinRef.[12].Studiesofpartially recon-structed B−

→

D∗h− decays are also reportedfor the first time, where the measurements of CP observables in B−

→ (

D

γ

)

D∗K−

decaysarecomparableinprecisiontothecurrentworldaverages; theequivalentobservablesmeasuredin

B

−

→ (

D

π

0

₎

D∗K−decays

substantially improve upon the world averages. Evidence of CP

violation in B−

→ (

D

π

0

₎

D∗K− decays is found with a statistical

significanceof4.3standarddeviations,whilethesignificanceofa nonzerovalueof ACP_K,γ is2.4standarddeviations.The

K

−

π

+final state, which offers highersensitivity to

γ

dueto larger interfer-enceeffects[48],hasnotbeenconsideredinthiswork,duetothe presence ofa large backgroundcontribution fromthepoorly un-derstood

B

s0

→

D∗0K+

π

−decay.

Usingtheobservables

A

K_Kπ,γ,

A

Kπ,π0

K , A CP,γ K ,A CP,π0 K ,

R

CP,γ and RCP,π0

asinput,aderivationofthefundamentalparameters

r

D∗K B ,

δ

D_B∗K and

γ

has been performed using the approach detailed in Ref. [2]. The profile likelihood contours at 1

σ

, 2

σ

and 3

σ

are shown inFig. 4. The preferredvaluesof

r

D∗K

B are lower than the

current world average values, owing to the fact that the values of RCP,_{γ and R}CP,π0

measured inthisworkare belowandabove unity, respectively, in contrast to the world averages which are bothlargerthanunity[3].Thepreferredvaluesof

γ

and

δ

_BD∗K are consistent within 1 standard deviation with the LHCb combina-tion[2]andtheworldaverage.

Acknowledgements

We express our gratitude to our colleagues in the CERN ac-celerator departments for the excellent performance of the LHC. We thank the technical andadministrative staff at the LHCb