Observation of the B-0 -> rho(0)rho(0) decay from an amplitude analysis of B-0 -> (pi(+)pi(-))(pi(+)pi(-)) decays

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Observation

of

the

B

0

→

ρ

0

ρ

0

decay

from

an

amplitude

analysis

of

B

0

→ (

π

+

π

−

)(

π

+

π

−

)

decays

.

LHCb

Collaboration

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received26March2015

Receivedinrevisedform18May2015 Accepted10June2015

Availableonline15June2015 Editor:M.Doser

Proton–proton collision data recorded in 2011 and 2012 by the LHCb experiment, corresponding to an integrated luminosity of 3.0 fb−1, are analysed to search for the charmless B0_→

_ρ

0

_ρ

0 _decay. More than 600 B0_{→ (}

_π

+

_π

−₎₍

_π

+

_π

−_{) signal decays are selected and used to performan amplitude} analysis, under theassumption ofnoCPviolationin thedecay, from whichthe B0→

ρ

0

_ρ

0 _decayis observedforthefirsttimewith7.1standarddeviationssignificance.ThefractionofB0_→

_ρ

0

_ρ

0_decays yielding alongitudinally polarised final stateis measured to be fL=0.745+₋00..048058(stat)±0.034(syst). The B0_→

_ρ

0

_ρ

0_{branchingfraction,using the B}0_{→ φK}∗₍₈₉₂₎0 _{decayas reference,isalsoreported as}

B(B0_→

_ρ

0

_ρ

0_{)= (0.94±0.17(stat)±0.09(syst)±0.06(BF))×10}−6_.

©2015CERNforthebenefitoftheLHCbCollaboration.PublishedbyElsevierB.V.Thisisanopenaccess articleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

The study of B meson decays to

ρρ

final states provides themostpowerfulconstraintto datefortheCabibbo–Kobayashi– Maskawa(CKM)angle

α

≡

arg



(

VtdVtb∗

)/(

VudVub∗

)



[1–3].Mostof thephysics informationis providedby thedecay B0

_→

_ρ

+

_ρ

− _as

measuredatthee+e− collidersatthe

ϒ(

4S

)

resonance[4,5],1 for whichthedominantdecayamplitude,involvingtheemissionofa

W boson only (tree), exhibitsa phase difference that can be in-terpreted asthe sum of the CKM angles

β

+

γ

=

π

−

α

in the StandardModel.Thesubleadingamplitudeassociatedwiththe ex-changeofaW bosonandaquark (penguin)mustbedetermined inordertointerprettheelectroweakphase differenceintermsof theangle

α

.Thisisrealisedbymeansofanisospinanalysis involv-ing the companionmodes B+

→

ρ

+

ρ

0 _{[6,7] and B}0

_→

_ρ

0

_ρ

0 _[8, 9].2Inparticular,thesmallnessoftheamplitudeofthelatterleads toabetterconstrainton

α

.

The BaBar and Belle experiments reported evidence for the

B0

→

ρ

0

_ρ

0 _{decay [8,9] with an average branching fraction of}

B(

B0

→

ρ

0

_ρ

0

₎

_{= (}

₀

_.

₉₇

_±

₀

_.

₂₄

₎

_×

₁₀−6 _{[8,9]. Despite small}

ob-served signal yields, each experiment measured the fraction fL

ofdecaysyieldingalongitudinallypolarisedfinalstatethroughan angularanalysis.TheBelle Collaboration didnotfindevidencefor polarisation, fL

=

0

.

21+₋00..2226[9],whiletheBaBar experiment

mea-sureda mostly longitudinallypolarised decay, fL

=

0

.

75+₋00..1215 [8].

These results differ at the level of 2

.

0 standard deviations. The

1 _{Chargeconjugationisimplicitthroughoutthetextunlessotherwisestated.} 2 _ρ0_standsforρ0_{(770)throughoutthetext.}

largeLHCb datasetmayshedlightonthisdiscrepancy.Inaddition, LHCb mayconfirmthehintofB0

_→

_ρ

0_f

0

(

980

)

decaysreportedby

Belle[9].MeasurementsoftheB0

→

ρ

0

_ρ

0 _{branchingfractionand}

longitudinalpolarisationfractionatLHCb canbeusedasinputsin thedeterminationof

α

[2,3].

This work focuses on the search and study of the B0

→

(

π

+

π

−

)(

π

+

π

−

)

decay in which the two

(

π

+

π

−

)

pairs are se-lected in the low invariant mass range (

<

1100 MeV

/

c2_{). The}

B0

_→

_ρ

0

_ρ

0 _{isexpectedtodominatethe}

₍

_π

+

_π

−

₎

_{massspectrum.}

The

(

π

+

π

−

)

combinationscanactuallyemergefromS-wave non-resonant andresonant contributions or other P- orD-wave reso-nancesinterferingwiththesignal.Hence,thedeterminationofthe

B0

→

ρ

0

_ρ

0 _{yieldsrequiresatwo-bodymassandangularanalysis,}

fromwhichthe fractionofthelongitudinallypolarisedfinal state canbemeasured.

The branching fraction is measured relative to the B0

_→

φ

K∗

(

892

)

0 mode. The B0

→ φ

K∗

(

892

)

0 decay, which results in four light mesons inthe final state, issimilar to the signal, thus allowing foracancellationoftheuncertainties intheratioof se-lectionefficiencies.

2. Datasetsandselectionrequirements

The analysed data correspond to an integrated luminosity of 1

.

0 fb−1 and2

.

0 fb−1 from pp collisionsrecordedata centre-of-mass energy of 7 TeV,collected in 2011, and 8 TeV, collected in 2012,bytheLHCb experimentatCERN.

The LHCb detector[10,11] isa single-armforward spectrome-tercoveringthepseudorapidity range 2

<

η

<

5,designedforthe study of particles containing b or c quarks. It includes a

high-http://dx.doi.org/10.1016/j.physletb.2015.06.027

0370-2693/©2015CERNforthebenefitoftheLHCbCollaboration.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

precisiontrackingsystemconsistingofasilicon-stripvertex detec-torsurroundingthepp interactionregion[12],alarge-area silicon-stripdetectorlocatedupstreamofadipolemagnetwithabending power of about 4 Tm, and three stations of silicon-strip detec-torsandstraw drifttubes[13]placeddownstreamofthemagnet. Thetrackingsystemprovidesameasurementofmomentum,p,of chargedparticleswitharelativeuncertaintythatvariesfrom0.5% atlow momentum to 1.0% at200 GeV

/

c. The minimumdistance ofatracktoaprimary vertex,theimpactparameter,ismeasured with a resolution of

(

15

+

29

/

pT

)

μm, where pT is the

compo-nentofthemomentumtransversetothebeam,inGeV

/

c.Different typesofchargedhadronsaredistinguishedusinginformationfrom tworing-imaging Cherenkov(RICH)detectors [14]. Photons, elec-tronsandhadronsareidentifiedbyacalorimetersystemconsisting of scintillating-pad and preshower detectors, an electromagnetic calorimeterandahadroniccalorimeter.Muonsare identifiedbya systemcomposed ofalternatinglayers ofironandmultiwire pro-portionalchambers[15].The onlineeventselection is performed byatrigger[16],whichconsistsofahardwarestage,basedon in-formationfromthe calorimeterandmuonsystems,followedby a softwarestage,whichappliesafulleventreconstruction.

Inthisanalysistwocategoriesofeventsthatpassthehardware trigger stage are considered: those where the trigger decisionis satisfied by the signal b-hadron decayproducts (TOS) and those whereonlytheother activityinthe eventdetermines thetrigger decision(TIS).Thesoftwaretriggerrequiresatwo-,three- or four-tracksecondary vertexwithlargetransversemomentaofcharged particles anda significant displacementfrom theprimary pp

in-teractionvertices (PVs).Atleastonechargedparticleshouldhave

pT

>

1

.

7 GeV

/

c andisrequiredtobeinconsistentwithoriginating

fromanyprimaryinteraction.Amultivariatealgorithm[17]isused fortheidentificationofsecondaryverticesconsistentwiththe de-cayofab hadron.

Further selection criteria are applied offline to reduce the number of background events with respect to the signal. The

(

π

+

π

−

)

candidatesmusthavetransversemomentum largerthan 600 MeV

/

c, with at least one charged decay product with pT

>

1000 MeV

/

c.Thetwo

(

π

+

π

−

)

pairsarethencombinedtoforma

B0 _{candidatewith a goodvertex quality andtransverse}

momen-tumlarger than 2500 MeV

/

c.The invariant mass of each pairof opposite-chargepionsformingthe B0 candidateisrequiredto be intherange300–1100 MeV

/

c2_{.Theidentificationofthefinal-state}

particles(PID)isperformedwithdedicatedneural-networks-based discriminatingvariables that combineinformation fromthe RICH detectorsandotherpropertiesoftheevent[14].Thecombinatorial backgroundisfurthersuppressedwithmultivariatediscriminators based on a boosted decision tree algorithm (BDT) [18,19]. The BDT is trainedwith simulated B0

→

ρ

0

_ρ

0 _(where

_ρ

0

_→

_π

+

_π

−₎ eventsas signal sample and candidates reconstructed with four-bodymass inexcess of5420 MeV

/

c2 asbackgroundsample. The discriminatingvariablesarebasedonthekinematicsofthe B

de-cay candidate(B pT and theminimum pT of thetwo

ρ

0

candi-dates)andongeometrical vertexmeasurements(qualityofthe B

candidatevertex,impactparametersignificancesofthedaughters,

B flight distancesignificance and B pointing to theprimary ver-tex). The optimal thresholds for the BDT and PID discriminating variables are determined simultaneously by means ofa frequen-tistestimator for which no hypothesis on the signal yield is as-sumed [20]. The B0 _{meson candidates are accepted in the mass}

range5050–5500 MeV

/

c2.

Thenormalisationmode B0

→ φ

K∗

(

892

)

0 isselectedwith sim-ilar criteria, requiring in addition that the invariant mass of the

(

K+

π

−

)

candidate is found in a range of

±

150 MeV

/

c2 around theknownvalueoftheK∗

(

892

)

0mesonmass[21]andthe invari-antmassofthe

(

K+K−

)

pairisinarangeof

±

15 MeV

/

c2_centred

attheknownvalueofthe

φ

mesonmass[21].Asampleenriched inB0

→ (

K+

π

−

)(

π

+

π

−

)

eventsisselectedusingthesameranges in

(

π

+

π

−

)

and

(

K+

π

−

)

massestoestimate thebackgroundwith onemisidentifiedkaon.

The presenceof

(

π

+

π

−

)

pairsoriginatingfrom J

/ψ

,

χ

c0 and

χc2

charmoniadecaysisvetoedbyrequiringtheinvariantmasses

M of all possible

(

π

+

π

−

)

pairs to be

|

M

−

M0

|

>

30 MeV

/

c2,

whereM0 standsforthecorrespondingknownvaluesofthe J

/ψ

,

χc0

and

χ

c2mesonmasses[21].Similarly,thedecaysD0

→

K−

π

+

and D0

_→

_π

+

_π

− _{are vetoed by requiring the corresponding} in-variantmassestodifferby25 MeV

/

c2_{ormorefromtheknownD}0

mesonmass[21].Toreducecontaminationfromothercharm back-groundsandfromthe B0

_→

_a+

1

(

→

ρ

0

π

+

)

π

−decay,theinvariant

massofanythree-bodycombinationintheeventisrequiredtobe largerthan2100 MeV

/

c2.

Simulated B0

→

ρ

0

_ρ

0 _{and B}0

_{→ φ}

_K∗

₍

₈₉₂

₎

0 _{decays are also}

used for determining the relative reconstruction efficiencies. The

pp collisionsaregeneratedusingPythia[22]withaspecificLHCb configuration [23]. Decays of hadronicparticles are described by EvtGen [24]. The interaction ofthe generated particles with the detector and its response are implemented using the Geant4 toolkit[25]asdescribedinRef.[26].

3. Four-bodymassfit

Thefour-bodymassspectrum M

(

π

+

π

−

)(

π

+

π

−

)

isfitwithan unbinnedextendedlikelihood.Thefitisperformedsimultaneously for the two data taking periods together with the normalisation channelM

(

K+K−

)(

K+

π

−

)

andPIDmisidentificationcontrol chan-nelM

(

K+

π

−

)(

π

+

π

−

)

massspectra.Thefour-bodyinvariantmass modelsaccountforB0andpossibleB0_s signals,combinatorial back-grounds, signal cross-feeds and background contributions arising frompartiallyreconstructedb-hadrondecaysinwhichoneormore particlesarenotreconstructed.

The B0 and B0_s meson shapes are modelled witha modified CrystalBalldistribution[27].Asecondpower-lawtailisaddedon the high-mass side of the signal shape to account for imperfec-tionsofthetrackingsystem.Themodelparametersaredetermined froma simultaneous fit ofsimulated signal events that fulfil the trigger,reconstructionandselectionchain,foreachdatataking pe-riod.Thevaluesofthetailparametersareidenticalforthe B0 and

B0

s mesons.Theirmassdifferenceisconstrainedtothevaluefrom

Ref.[21].ThemeanandwidthofthemodifiedCrystalBallfunction arefreeparametersofthefittothedata.

The combinatorial background in each four-body spectrum is described by an exponential function wherethe slopeis allowed tovaryinthefit.

The misidentification of one or more final-state hadrons may resultinafullyreconstructedbackgroundcontributiontothe cor-responding signal spectrum, denoted signal cross-feed. The mag-nitudeofthebranchingfractionsofthesignal andcontrol modes as well as the two-body mass selection criteria make these sig-nal cross-feeds negligible, with one exception: the misidentifica-tion of the kaon of the decay B0

→ (

K+

π

−

)(

π

+

π

−

)

as a pion yields a significant contribution in the M

(

π

+

π

−

)(

π

+

π

−

)

mass spectrum. The mass shape of B0

_{→ (}

_K+

_π

−

₎₍

_π

+

_π

−

₎

_decays

re-constructed as B0

→ (

π

+

π

−

)(

π

+

π

−

)

is modelled by a Crystal Ball function, whose parameters are determined from simulated events. The yield of this signal cross-feed is allowed to vary in the fit. The measurement ofthe actual number ofreconstructed

B0

→ (

K+

π

−

)(

π

+

π

−

)

events multiplied by the data-driven es-timate of the misidentification efficiency is consistent with the measuredyield.

The partiallyreconstructed backgroundis modelledby an AR-GUSfunction [28]convolvedwithaGaussian functionaccounting

Fig. 1. Reconstructedinvariantmassspectrumof(left)

(

π+π−)(π+π−)and(right)

(

K+K−)(K+π−).Thedataarerepresentedbytheblackdots.Thefitisrepresentedby thesolidblueline,theB0_{signalbythesolidredlineandtheB}0

sbythesolidgreenline.Thecombinatorialbackgroundisrepresentedbythepinkdottedline,thepartially

reconstructedbackgroundbythecyandottedlineandthecross-feedbythedarkbluedashedline.(Forinterpretationofthereferencestocolourinthisfigurelegend,the readerisreferredtothewebversionofthisarticle.)

Table 1

Yieldsfrom thesimultaneousfit forthe 2011and2012datasets.Thefirstand seconduncertaintiesarethestatisticalandsystematiccontributions,respectively.

Decay mode Signal yields 2011 Signal yields 2012 B0_{→ (π}+_π−_)(π+_π−_{) 185±15±4 449±24±7} B0_{→ (K}+_π−_)(π+_π−_{) 1610±42±5 3478±62±10} B0_{→ (K}+_K−_)(K+_π−_{) 1513±40±8 3602±62±10} B0 s→ (π+π−)(π+π−) 30±7±1 71±11±1 B0 s→ (K−π+)(π+π−) 40±10±3 96±14±6 B0 s→ (K+K−)(K−π+) 42±10±3 66±13±4

for resolution effects. Various mass shape parameterisations are examined.The bestfit isobtainedwhenthe endpointofthe AR-GUSfunctionisfixedtothevalue expectedwhenonepionisnot attributedtothedecay.TheothershapeparametersoftheARGUS function are free parameters ofthe fit, commonto the two data taking periods. The floating width parameter of the signal mass shapeisconstrainedtobeequaltothewidthoftheGaussian func-tionusedintheconvolution.

Fig. 1 displays the M

(

π

+

π

−

)(

π

+

π

−

)

and M

(

K+K−

)(

K+

π

−

)

spectra with the fit results overlaid. The signal event yields are shownin Table 1. Aside fromthe prominentsignal of the B0

_→

(

π

+

π

−

)(

π

+

π

−

)

decays,the decay mode B0_s

→ (

π

+

π

−

)(

π

+

π

−

)

isobservedwithastatisticalsignificanceofmorethan10standard deviations. The statisticalsignificance is evaluated by taking the ratioof the likelihood ofthe nominal fitand ofthe fit withthe signalyieldfixedtozero.

Asystematicuncertainty duetothe fitmodelis associatedto the measured yields. The dominant uncertainties arise from the knowledge of the signal and signal cross-feed shape parameters determinedfromsimulatedevents.Severalpseudoexperimentsare generatedwhilevaryingtheshapeparameterswithintheir uncer-tainties, and the systematic uncertainties on the yields are esti-matedfromthedifferencesinresultswithrespecttothenominal fit.

4. Amplitudeanalysis

An amplitude analysis isused to determine the vector–vector (VV)contributionB0

→

ρ

0

_ρ

0 _{byusingtwo-bodymassspectraand}

angular variables. The four-body mass spectrum is first analysed with the sPlot technique [29] to subtract statistically the back-groundundertheB0

→ (

π

+

π

−

)(

π

+

π

−

)

signal.

For the two-body mass spectra, contributions from resonant andnon-resonant scalar (S), resonant vector (V ) and tensor (T ) components are considered in the amplitude fit model through complexmasspropagators,M

(

mi

)

,wherethelabeli

=

1

,

2 arethe

first and second pionpairs, which are assignedrandomly in ev-ery decay since they are indistinguishable.The P-wave lineshape model comprises the

ρ

0 _{meson, described using the Gounaris–}

Sakuraiparameterisation Mρ

(

mi

)

[30],andthe

ω

meson,

parame-terisedwitharelativisticspin-1Breit–WignerMω

(

mi

)

.TheD-wave

lineshape Mf2

(

mi

)

accounts for the f2

(

1270

)

, modelled with a

relativistic spin-2 Breit–Wigner. The S-wave model includes the

f0

(

980

)

propagator Mf(980)

(

mi

)

, described usinga Flatté

param-eterisation[31,32],andalow-masscomponent.Thelatterincludes the broad low-mass resonance f0

(

500

)

and a non-resonant

con-tributions, which are jointly modelled in the framework of the

K -matrix formalism [33] and referred as M(π π)0

(

mi

)

. Following

the K -matrix formalism, the amplitude for the low-mass

π

+

π

−

S-wavecanbewrittenas

A

(

m

)

∝

K

ˆ

1

−

i

ρ

K

ˆ

,

(1) with

ˆ

K

≡ ˆ

Kres

+ ˆ

Knon-res

=

m0

(

m

)

(

m2 0

−

m2

)

ρ

(

m

)

+

κ

,

(2)

ρ

(

m

)

=

2



q

(

m

)

m



,

(3)

where

κ

is measured to be

−

0

.

07

±

0

.

24 from a fit to the in-clusive

π

+

π

− mass distribution andm0 and

are the nominal

massandmass-dependentwidthofthe f0

(

500

)

,asdeterminedin

Ref.[34].Thefunctions

ρ

(

m

)

andq

(

m

)

,definedinRef.[33],arethe phasespacefactorandtherelativemomentumofapioninthe

ρ

0

centre-of-masssystem.Byconvention,thephaseoftheM(π π)0

(

mi

)

masspropagatorissettozeroatthe

ρ

0 _nominalmass.

The signal sample is described by considering the dominant amplitudesofthesignaldecay.The B

→

V V componentcontains the B

→

ρ

0

_ρ

0 _{and B}0

_→

_ρ

0

_ω

_{amplitudes.The B}

_→

_{V S}

compo-nent accounts for B0

→

ρ

0

₍

_π

+

_π

−

₎

_{0 and B}0

_→

_ρ

0_f

0

(

980

)

am-plitudes and the B

→

V T contribution is limited to the purely longitudinalamplitudeoftheB0

_→

_ρ

0_f

2

(

1270

)

transition.Because

of the broad natural width of the a±₁ particle, a small contami-nation fromthedecays B0

_→

_a±

1

π

∓ remains inthe sample. This

contributionwitha±₁

→

ρ

0

_π

±_{inS-waveisconsideredalongwith} its interference with the other amplitudes. This is done by in-troducing the CP-eveneigenstate from the linear combinationof individualamplitudesofthedecaysB0

_→

_a+

1

π

−andB0

→

a−1

π

+,

as defined in Ref. [35]. The contribution of the decays B0

→

ωω

, B0

→

f0

(

980

)

f0

(

980

)

, B0

→

ω

S, B0

→

ω

T ,B0

→

f2

(

1270

)

S,

B0

_→

_f

Fig. 2. Helicity angles for the(π+π−)(π+π−)system.

tobe negligible, wherethe



and

⊥

subindicesindicate the par-allelandperpendicularamplitudesofthedecay.Thechoiceofthe baselinemodelwasmadepriortothemeasurementofthephysical parametersofinterestaftercomparingasetofalternative param-eterisationsaccordingtoadissimilaritystatisticaltest[36].

ThedifferentialdecayrateforB0

→ (

π

+

π

−

)(

π

+

π

−

)

decaysat theB0_{productiontimet}

₌

_{0 isgivenby}

d5

d cos

θ

1d cos

θ

2d

ϕ

dm2₁dm2₂

∝ 

4

(

m1

,

m2

)







11



i=1 Aifi

(

m1

,

m2

, θ

1

, θ

2

,

ϕ

)







2

,

(4)

wherethevariables

θ

1,

θ

2and

ϕ

arethehelicityangles,described

inFig. 2,and



4isthefour-bodyphasespacefactor.Thenotations

ofthecomplexamplitudes,Ai,andtheexpressionsoftheirrelated

angulardistributions, fi,are displayedinTable 2.The mass

prop-agatorsincludedinthe fi functionsarenormalisedtounityinthe

fitrange.

FortheCP conjugatedmode, B0

→ (

π

+

π

−

)(

π

+

π

−

)

,thedecay rateisobtainedunderthe transformation Ai

→

ηi

Ai,where

η

i is

theCP eigenvalueoftheCP eigenstatei,showninTable 2. Theuntaggedtime-integrated decayrateof B0 and B0 tofour pions,assumingnoCP violation,canbewrittenas

d5

(

+ )

d cos

θ

1d cos

θ

2d

ϕ

dm21dm22

∝

11



j=1



i≤j

R

e

[

AiA∗jfif∗j

](

2

− δ

i j)(1

+

η

i

η

j)4

(

m1

,

m2

) ,

(5)

where

δ

i j

=

1 wheni

=

j and

δ

i j

=

0 otherwise.

The efficiency of the selection of the final state B0

_→

(

π

+

π

−

)(

π

+

π

−

)

varies as a function of the helicity angles and thetwo-bodyinvariant masses.Totake intoaccountvariations in the efficiencies, fourevent categories k are defined according to their hardware triggerdecisions(TIS orTOS)anddatataking pe-riod(2011and2012).

Theacceptanceisaccountedforthroughthecomplexintegrals

ω

k_{i j}

=





(θ

1

, θ

2

,

ϕ

,

m1

,

m2

)

fif∗j

(

2

− δ

i j

)

× 

4

(

m1

,

m2

)

d cos

θ

1d cos

θ

2d

ϕ

dm21dm22

,

(6)

where fi arethefunctionsgiveninTable 2and



theoverall

effi-ciency. Theintegralsarecomputedwithsimulatedeventsofeach ofthefourconsideredcategories,selectedwiththesamecriteriaas thoseappliedtodata,followingthemethoddescribedinRef.[38]. The coefficients

ω

k

i j are used to determine the efficiency and to

buildaprobabilitydensityfunctionforeachcategory,whichis de-finedas Sk

(

m1

,

m2

, θ

1

, θ

2

,

ϕ

)

=

11 j=1

i≤j

R

e

[

AiA∗jfif∗j

](

2

− δ

i j)(1

+

η

i

η

j)4

(

m1

,

m2

)

11 j=1

i≤j

R

e

[

AiA∗j

ω

ki j

](

1

+

η

i

η

j)

.

(7)

The four event categories are used in the simultaneous un-binned maximum likelihood fit which depends on the 19 free parametersindicatedinTable 3.

Systematic effects are estimated by fitting with the angular model and ensemble of1000 pseudoexperimentsgenerated with the same number of events asobserved in data. The biases are for theparameters ofinterest consistent with zero.A systematic uncertaintyisassignedbytaking50%ofthefitbiasorthe uncer-taintyonthermswhenthelatterisbiggerinordertoaccountfor possiblestatisticalfluctuations.

Several model related uncertainties are envisaged. The B0

→

a±₁

π

∓ angularmodelrequires knowledge ofthe lineshapeofthe

a±₁ meson.Thea±₁ naturalwidthischosentobe400 MeV

/

c2_.The

difference to the fit results obtained by varying the width from 250to 600 MeV

/

c2 _{is takenasthecorresponding systematic}

un-certainty.Inaddition,asystematicuncertaintyisobtainedby intro-ducingtheCP-oddcomponentinthefitmodelofthedecay ampli-tude B0

→

a±₁

π

∓byfixingtherelativeamplitudesof B0

→

a+₁

π

−

andB0

_→

_a−

1

π

+ componentsto thevaluesmeasuredinRef.[39]. Table 2

Amplitudes,Ai,CP eigenvalues,ηi,andmass-angledistributions, fi,oftheB0→ (π+π−)(π+π−)model.Theindicesi jkl indicate

theeightpossiblecombinationsofpairsofopposite-chargepions.Theanglesαkl,

β

i jand



klaredefinedinRef.[37].

Ai ηi fi

A0

ρρ 1 Mρ(m1)Mρ(m2)cosθ1cosθ2

Aρρ 1 Mρ(m1)Mρ(m2)√1₂sinθ1sinθ2cosϕ

A⊥ρρ −1 Mρ(m1)Mρ(m2)√i₂sinθ1sinθ2sinϕ

A0

ρω 1 √12[Mρ(m1)Mω(m2)+Mω(m1)Mρ(m2)]cosθ1cosθ2

Aρω 1 √1₂[Mρ(m1)Mω(m2)+Mω(m1)Mρ(m2)]√1₂sinθ1sinθ2cosϕ

A⊥ρω −1 √12[Mρ(m1)Mω(m2)+Mω(m1)Mρ(m2)]

i

√

2sinθ1sinθ2sinϕ

Aρ(π π )0 −1 1 √ 6[Mρ(m1)M(π π )0(m2)cosθ1+M(π π )0(m1)Mρ(m2)cosθ2] Aρf(980) −1 √1₆[Mρ(m1)Mf(980)(m2)cosθ1+Mf(980)(m1)Mρ(m2)cosθ2] A(π π )0(π π )0 1 M(π π )0(m1)M(π π )0(m2) 1 3 A0 ρf2 −1 5 24  Mρ(m1)Mf2(m2)cosθ1(3 cos 2_θ 2−1)+Mf2(m1)Mρ(m2)cosθ2(3 cos 2_θ 1−1) AS+ a1π 1 1 √ 8

Table 3

Resultsoftheunbinnedmaximumlikelihoodfittotheangularandtwo-bodyinvariantmassdistributions.Thefirstuncertaintyis statistical,thesecondsystematic.

Parameter Definition Fit result

fL |A0ρρ|2/(|A0ρρ|2+ |Aρρ|2+ |Aρρ⊥|2) 0.745−+00..048058±0.034 f_ |Aρρ|2/(|Aρρ|2+ |A⊥ρρ|2) 0.50±0.09±0.05 δ− δ0 arg(AρρAρρ0∗) 1.84±0.20±0.14 Fρ(π π )0 |Aρ(π π )0| 2_/(|A0 ρρ|2+ |Aρρ|2+ |A⊥ρρ|2) 0.30+−00..1109±0.08 Fρf(980) |Aρf(980)|2/(|A0ρρ|2+ |Aρρ |2+ |A⊥ρρ|2) 0.29+−00..1209±0.08 F(π π )0(π π )0 |A(π π )0(π π )0| 2 /(|A0 ρρ|2+ |Aρρ|2+ |A⊥ρρ|2) 0.21+−00..0604±0.08 δ⊥− δρ(π π )0 arg(Aρρ⊥ A∗ρ(π π )0) −1.13 +0.33 −0.22±0.24 δ⊥− δρf(980) arg(A⊥ρρA∗ρf(980)) 1.92±0.24±0.16 δ(π π )0(π π )0− δ0 arg(A(π π )0(π π )0A 0∗ ρρ) 3.14+ 0.36 −0.38±0.39 Fρω (|A0ρω|2+ |Aρω|2+ |A⊥ρω|2)/(|A0ρρ|2+ |Aρρ|2+ |A⊥ρρ|2) 0.025+−00..048022±0.020 fLρω |A0ρω|2/(|A0ρω|2+ |Aρω|2+ |Aρω⊥ |2) 0.70−+00..2360±0.13 f_ρω |Aρω|2/(|Aρω|2+ |A⊥ρω|2) 0.97+−00..6956±0.15 δω0− δ0 arg(A0ρωA0ρρ∗) −2.56+−00..7692±0.22 δω_− δ0 arg(AρωA0ρρ∗) −0.71+−00..7167±0.32 δω_⊥− δρ(π π )0 arg(A⊥ρωA∗ρ(π π )0) −1.72±2.62±0.80 F0 ρf2 |A 0 ρf2| 2 /(|A0 ρρ|2+ |Aρρ|2+ |A⊥ρρ|2) 0.01+−00..0402±0.03 δ0 ρf2− δρ(π π )0 arg(A 0 ρf2A ∗ ρ(π π )0) −0.56±1.48±0.80 FS+ a1π |A S+ a1π| 2_/(|A0 ρρ|2+ |Aρρ|2+ |A⊥ρρ|2) 1.4+−10..07+ 1.2 −0.8 δaS1+π− δρ(π π )0 arg(A S+ a1πA ∗ ρ(π π )0) −0.09 +0.30 −0.36±0.38

Anothersource of uncertainty originatesin the modelling ofthe lowmass

(

π

+

π

−

)

S-wavelineshape.The f0

(

500

)

massand

natu-ral widthuncertaintiesfromRef. [34] andtheuncertaintyonthe parameterthatquantifiesthenon-resonantcontributionare propa-gatedtotheangularanalysisparametersbygeneratingandfitting 1000 pseudoexperiments in which these input values are varied accordingtoa Gaussian distributionhaving their uncertainties as widths.Therootmeansquare ofthedistributionoftheresultsis assignedasasystematicuncertainty.Thesamestrategyisfollowed to estimate the systematicuncertainties originatingfrom the

ρ

0_,

f0

(

500

)

and

ω

lineshapeparameters.

Theuncertaintyrelatedtothe backgroundsubtractionmethod is estimated by varying within their uncertainties the fixed pa-rametersofthemassfitmodelandstudyingtheresultingangular distributionsandtwo-bodymassspectra.Thedifferencetothefit resultsistakenasasystematicuncertainty.Analternative subtrac-tionofthebackgroundestimatedfromthehigh-mass sidebandis performed,yieldingcompatibleresults.

The knowledge of the acceptance model described in Eq. (6) comes froma finite sample ofsimulated events.An ensemble of pseudoexperimentsisgeneratedbyvaryingtheacceptanceweights accordingtotheircovariancematrix.Therootmeansquare ofthe distributionoftheresultsisassignedasasystematicuncertainty.

Theresolutiononthehelicityanglesisevaluatedwith pseudo-experiments resulting in a negligible systematic uncertainty. The systematicuncertainty relatedto the

(

π

+

π

−

)

mass resolutionis estimatedwith pseudoexperiments by introducing a smearing of the

(

π

+

π

−

)

mass.Differencesin theparameters betweenthefit withandwithoutsmearingaretakenasasystematicuncertainty.

Table 4details thecontributions tothe systematicuncertainty inthemeasurementofthefractionofB0

→

ρ

0

_ρ

0_{signaldecaysin}

the B0

→ (

π

+

π

−

)(

π

+

π

−

)

andits longitudinal polarisation frac-tion.

The final resultsofthe combinedtwo-bodymassand angular analysisare shownin Fig. 3 and Table 3.The fit alsoallows for

Table 4

Relativesystematicuncertaintiesonthelongitudinalpolarisationparameter, fL,and

thefractionofB0_→ρ0_ρ0_decaysintheB0_{→ (π}+_π−_)(π+_π−_{)sample.Themodel} uncertaintyincludesthethreeuncertaintiesbelow.

Systematic effect Uncertainty onfL(%) Uncertaintyon P(B0_→ρ0_ρ0 )(%) Fit bias 0.1 0.8 Model 3.6 6.2 B0_→a 1(1260)+π− 1.2 1.1 S-wave lineshape 3.4 6.1 Lineshapes <0.1 0.1 Background subtraction 0.1 0.5 Acceptance integrals 2.7 4.5 Angular/Mass resolution 0.8 1.5

the extractionof thefraction of B0

_→

_ρ

0

_ρ

0 _{decaysin the B}0

_→

(

π

+

π

−

)(

π

+

π

−

)

sample,definedas P

(

B0

→

ρ

0

ρ

0

)

=

3 j=1

i≤j

R

e

[

AiA∗_j

ω

i j

]

11 j=1

i≤j

R

e

[

AiA∗_j

ω

i j

]

,

(8) whichis P

(

B0

→

ρ

0

ρ

0

)

=

0

.

619

±

0

.

072

(

stat

)

±

0

.

049

(

syst

).

The B0

_→

_ρ

0

_ρ

0_{signalsignificanceismeasuredtobe7}

_.

_{1 standard}

deviations.Thesignificanceisobtainedbydividingthevalueofthe purity by the quadrature ofthe statistical and systematic uncer-tainties. No evidenceforthe B0

_→

_ρ

0_f

0

(

980

)

decaymode is

ob-tained.Thefractionoflongitudinalpolarisationofthe B0

_→

_ρ

0

_ρ

0

decayismeasuredtobe

Fig. 3. Background-subtractedM(π+π−)1,2,cosθ1,2andϕdistributions.Theblackdotscorrespondtothefour-bodybackground-subtracteddataandtheblacklineisthe

projectionofthefitmodel.ThespecificdecaysB0_→ρ0_ρ0_(brown),B0_→ωρ0_{(dashedbrown),B}0_{→V S (dashedblue),B}0_{→S S (longdashedgreen),B}0_{→V T (orange)}

andB0_→a±

1π∓ (lightblue)arealsodisplayed.TheB0→ρ0ρ0contributionissplitintolongitudinal(dashedred)andtransverse(dottedred)components.Interference

contributionsareonlyplottedforthetotal(black)model.Theefficiencyforlongitudinallypolarised B0_→ρ0_ρ0_{eventsis∼5timessmallerthanforthetransversecomponent.}

(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.) 5. Branchingfractiondetermination

Thebranchingfractionofthe decaymode B0

→

ρ

0

_ρ

0 _relative

tothedecayB0

→ φ

K∗

(

892

)

0 canbeexpressedas

B

(

B0

→

ρ

0

_ρ

0

₎

B

(

B0

_{→ φ}

_K∗

₍

₈₉₂

₎

0

₎

=

λ

fL

·

P

(

B 0

_→

_ρ

0

_ρ

0

₎

P

(

B0

_{→ φ}

_K∗

₍

₈₉₂

₎

0

₎

×

N

(

B0

→ (

π

+

π

−

)(

π

+

π

−

))

N

(

B0

_{→ (}

_K+_K−

₎₍

_K+

_π

−

₎₎

×

B

(φ

→

K+K−

)

B

(

K∗

→

K+

π

−

)

B

(

ρ

0

_→

_π

+

_π

−

₎

2

,

(9)

wherethefactor

λ

fL correctsfordifferencesindetection

efficien-ciesbetweenexperimental andsimulateddatadueto the polari-sationhypothesis of the B0

_→

_ρ

0

_ρ

0 _{sample, P}

₍

_B0

_→

_ρ

0

_ρ

0

₎

_and

P

(

B0

_{→ φ}

_K∗

₍

₈₉₂

₎

0

₎

_{are the fractions of B}0

_→

_ρ

0

_ρ

0 _{and B}0

_→

φ

K∗

(

892

)

0 signals in the samplesof B0

→ (

π

+

π

−

)(

π

+

π

−

)

and

B0

→ (

K+K−

)(

K+

π

−

)

decays,respectively.ThequantitiesN

(

B0

→

(

π

+

π

−

)(

π

+

π

−

))

andN

(

B0

→ (

K+K−

)(

K+

π

−

))

aretheyieldsof

B0

_{→ (}

_π

+

_π

−

₎₍

_π

+

_π

−

₎

_{and B}0

_{→ (}

_K+_K−

₎₍

_K+

_π

−

₎

_{decays as} de-terminedfromafittothefour-body massdistributions, weighted foreach data-taking period by the efficiencies of the signal and normalisation channels obtained from their respective simulated data.Finally,

B(φ →

K+K−

)

,

B(

K∗

(

892

)

0

→

K+

π

−

)

and

B(

ρ

0

_→

π

+

π

−

)

denoteknownbranchingfractions[21].

The product

λ

fL

·

P

(

B

0

_→

_ρ

0

_ρ

0

₎

_{is determined from the}

am-plitudeanalysistobe1

.

13

±

0

.

19

(

stat

)

±

0

.

10

(

syst

)

.Thisquantity ismainlyrelatedtothemodellingoftheS-wavecomponent, and dominatesthesystematicuncertaintyoftheparametersofinterest. The fraction of B0

_{→ φ}

_K∗

₍

₈₉₂

₎

0 _{present in the B}0

_→

(

K+K−

)(

K+

π

−

)

sample is takenfromRef. [40].A 1% systematic uncertainty is added, accounting for differences in the selection acceptanceforP- andS-wavecontributions.

The amounts of B0

→ (

π

+

π

−

)(

π

+

π

−

)

and B0

→

(

K+K−

)(

K+

π

−

)

candidates are determined from the four-body massspectraanalysis andtheir associatedstatisticaland system-aticaluncertainties are propagatedquadratically tothe branching fractionuncertaintyestimate.

Thelimitedsizeofthesimulatedeventssamplesthat meetall selectioncriteriaresult ina systematicuncertaintyof 1.7%(2.6%) on the measurement of the relative branching fraction for the 2011(2012) data-taking period. The impact of the discrepancies betweenexperimental andsimulated datarelated tothe B0 me-son kinematical properties is 0.6% (1.2%). The efficiencies of the particle-identification requirements are determined from control

samplesofdatawithasystematicuncertaintyof0.5%,mostly orig-inatingfromthelimitedsize ofthecalibrationsamples.An addi-tional1%systematicuncertaintyonthetrackingefficiencyisadded accountingfordifferentinteractionlengths between

π

andK .

Therelativebranchingfractionismeasuredtobe

B

(

B0

→

ρ

0

_ρ

0

₎

B

(

B0

_{→ φ}

_K∗

₍

₈₉₂

₎

0

₎

=

0

.

094

±

0

.

017

(

stat

)

±

0

.

009

(

syst

).

(10)

The agreement between the results obtained in the two data-taking periods is tested with the best linear estimator tech-nique[41]yieldingcompatibleresults.

The average branching fraction of B0

_{→ φ}

_K∗

₍

₈₉₂

₎

0 _as

deter-mined in Ref. [21] does not take into account the correlations betweensystematicuncertaintiesduetotheS-wavemodelling. In-stead, we average the results fromRefs. [42–44] including these correlationstoobtain

B(

B0

→ φ

K∗

(

892

)

0

)

= (

1

.

00

±

0

.

04

±

0

.

05

)

×

10−5_{.Usingthisvaluein Eq.(10),thebranchingfractionofB}0

_→

ρ

0

_ρ

0_is

B

(

B0

→

ρ

0

ρ

0

)

= (

0

.

94

±

0

.

17

(

stat

)

±

0

.

09

(

syst

)

±

0

.

06

(

BF

))

×

10−6

,

where the last uncertainty is due to the normalisation channel branching fraction. Using the B0

_→

_ρ

0

_ρ

0 _{branching fraction, the}

ρ

0_f

0

(

980

)

amplitude,aphasespacecorrectionandassuming100%

correlated uncertainties, an upper limit for the B0

→

ρ

0_f 0

(

980

)

decay,at90%confidencelevel,isobtained

B

(

B0

→

ρ

0f0

(

980

))

×

B

(

f0

(

980

)

→

π

+

π

−

) <

0

.

81

×

10−6

.

(11) 6. Conclusions

The full data set collected by the LHCb experiment in 2011 and2012, corresponding to an integratedluminosity of 3

.

0 fb−1, isanalysed to searchforthe B0

_→

_ρ

0

_ρ

0 _{decay.A yieldof634}

_±

28

±

8 B0

→ (

π

+

π

−

)(

π

+

π

−

)

signal decayswith

π

+

π

− pairs in the300–1100 MeV

/

c2 _{massrangeisobtained.Anamplitude}

anal-ysisisconductedto determinethecontributionfrom B0

_→

_ρ

0

_ρ

0

decays.Thisdecaymodeisobservedforthefirsttime witha sig-nificanceof7.1standarddeviations.Inthesame

π

+

π

−pairsmass range,B0

s

→ (

π

+

π

−

)(

π

+

π

−

)

decaysarealsoobservedwitha

sta-tisticalsignificanceofmorethan10standarddeviations.

Thelongitudinal polarisationfractionofthe B0

_→

_ρ

0

_ρ

0 _decay

is measuredto be fL

=

0

.

745₋+₀0._.048₀₅₈

(

stat

)

±

0

.

034

(

syst

)

.The

B

(

B0

→

ρ

0

ρ

0

)

= (

0

.

94

±

0

.

17

(

stat

)

±

0

.

09

(

syst

)

±

0

.

06

(

BF

))

×

10−6

,

where the last uncertainty is due to the normalisation channel. These resultsare the mostprecise to date and will improvethe precisionofthedeterminationoftheCKMangle

α

.

The measured longitudinal polarisation fraction is consistent with the measured value from BaBar [8] while it differs by 2

.

3 standard deviations from the value obtained by Belle [9]. The branching fractionmeasurement isin agreement withthe values measuredbyboth BaBar[8]and Belle[9]Collaborations.

Theevidence ofthe B0

→

ρ

0_f

0

(

980

)

decaymodereportedby

the Belle Collaboration[9]isnotconfirmed,andanupperlimitat 90%confidencelevelisestablished

B

(

B0

→

ρ

0f0

(

980

))

×

B

(

f0

(

980

)

→

π

+

π

−

) <

0

.

81

×

10−6

.

Acknowledgements

We express our gratitude to our colleagues in the CERN ac-celerator departments forthe excellent performance of the LHC. We thank the technical and administrative staff atthe 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, HGF and MPG (Ger-many); INFN (Italy); FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES and FANO (Rus-sia);MinECo(Spain);SNSFandSER(Switzerland);NASU(Ukraine); STFC (United Kingdom); NSF (USA). The Tier1 computing cen-tres are supported by IN2P3 (France),KIT and BMBF(Germany), INFN(Italy),NWOandSURF(TheNetherlands),PIC(Spain),GridPP (United Kingdom). We are indebted to the communities behind the multiple open source software packages on which we de-pend.We are also thankful for thecomputing resources andthe accesstosoftwareR&Dtoolsprovidedby YandexLLC(Russia). In-dividualgroupsormembershavereceivedsupportfromEPLANET, Marie Skłodowska-Curie ActionsandERC (EuropeanUnion), Con-seilgénéraldeHaute-Savoie,LabexENIGMASSandOCEVU,Région Auvergne (France), RFBR (Russia), XuntaGal and GENCAT (Spain), Royal Society and Royal Commission for the Exhibition of 1851 (UnitedKingdom).

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