Measurement of the absolute branching ratio of the $K^{+} \rightarrow \pi^{+}\pi^{-}\pi^{+} (\gamma)$ decay with the KLOE detector

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Measurement of the absolute branching ratio of the K ⁺ → π ⁺ π ⁻ π ⁺ ( γ ) decay with the KLOE detector

KLOE/KLOE-2 Collaboration

D. Babusci

^h

, I. Balwierz-Pytko

^g

, G. Bencivenni

^h

, C. Bloise

^h

, F. Bossi

^h

, P. Branchini

^r

, A. Budano

^q,r

, L. Caldeira Balkeståhl

^u

, F. Ceradini

^q,r

, P. Ciambrone

^h

, F. Curciarello

^i,d

, E. Czerwi ´nski

^g

, E. Danè

^h

, V. De Leo

^i,d

, E. De Lucia

^h

, G. De Robertis

^b

, A. De Santis

^h

, P. De Simone

^h,∗

, A. Di Cicco

^q,r

, A. Di Domenico

^m,n

, R. Di Salvo

^p

, D. Domenici

^h

,

O. Erriquez

^a,b

, G. Fanizzi

^a,b

, A. Fantini

^o,p

, G. Felici

^h

, S. Fiore

^s,n

, P. Franzini

^m,n

, A. Gajos

^g

, P. Gauzzi

^m,n

, G. Giardina

^i,d

, S. Giovannella

^h

, E. Graziani

^r

, F. Happacher

^h

,

L. Heijkenskjöld

^u

, B. Höistad

^u

, T. Johansson

^u

, D. Kami ´nska

^g

, W. Krzemien

^g

, A. Kupsc

^u

, J. Lee-Franzini

^h,t

, F. Loddo

^b

, S. Loffredo

^q,r

, G. Mandaglio

^i,d,c

, M. Martemianov

^j

,

M. Martini

^h,l

, M. Mascolo

^o,p

, R. Messi

^o,p

, S. Miscetti

^h

, G. Morello

^h

, D. Moricciani

^p

, P. Moskal

^g

, A. Palladino

^h

, A. Passeri

^r

, V. Patera

^k,h

, I. Prado Longhi

^q,r

, A. Ranieri

^b

, P. Santangelo

^h

, I. Sarra

^h

, M. Schioppa

^e,f

, B. Sciascia

^h

, M. Silarski

^g

, L. Tortora

^r

, G. Venanzoni

^h

, W. Wi´slicki

^v

, M. Wolke

^u

aDipartimentodiFisicadell’UniversitàdiBari,Bari,Italy bINFNSezionediBari,Bari,Italy

cCentroSicilianodiFisicaNucleareeStrutturadellaMateria,Catania,Italy dINFNSezionediCatania,Catania,Italy

eDipartimentodiFisicadell’UniversitàdellaCalabria,Cosenza,Italy fINFNGruppocollegatodiCosenza,Cosenza,Italy

gInstituteofPhysics,JagiellonianUniversity,Cracow,Poland hLaboratoriNazionalidiFrascatidell’INFN,Frascati,Italy

iDipartimentodiFisicaeScienzedellaTerradell’UniversitàdiMessina,Messina,Italy jInstituteforTheoreticalandExperimentalPhysics(ITEP),Moscow,Russia

kDipartimentodiScienzediBaseedApplicateperl’Ingegneriadell’Università“Sapienza”,Roma,Italy lDipartimentodiScienzeeTecnologieapplicate,Università“GuglielmoMarconi”,Roma,Italy mDipartimentodiFisicadell’Università“Sapienza”,Roma,Italy

nINFNSezionediRoma,Roma,Italy

oDipartimentodiFisicadell’Università“TorVergata”,Roma,Italy pINFNSezionediRomaTorVergata,Roma,Italy

qDipartimentodiMatematicaeFisicadell’Università“RomaTre”,Roma,Italy rINFNSezionediRomaTre,Roma,Italy

sENEAUTTMAT-IRR,CasacciaR.C.,Roma,Italy

tPhysicsDepartment,StateUniversityofNewYorkatStonyBrook,USA uDepartmentofPhysicsandAstronomy,UppsalaUniversity,Uppsala,Sweden vNationalCentreforNuclearResearch,Warsaw,Poland

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received10July2014

Receivedinrevisedform12September 2014

Accepted14September2014

TheabsolutebranchingratiooftheK⁺→π⁺π⁻π⁺(γ)decay,inclusiveoffinal-stateradiation,hasbeen measuredusing∼^{17 milliontaggedK+mesonscollectedwiththeKLOEdetectoratDA}NE,theFrascati φ-factory.Theresultis:

BR

K⁺→π⁺π⁻π⁺(γ)

=0.05565±0.00031stat±0.00025syst

*

Correspondingauthor.

E-mailaddress:patrizia.desimone@lnf.infn.it(P. De Simone).

http://dx.doi.org/10.1016/j.physletb.2014.09.033

0370-2693/©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP³.

Availableonline19September2014 Editor:L.Rolandi

Keywords:

e⁺e⁻experiments Kaondecays

a factor ^{5 more precise with respect tothe previous result. Thiswork completes the program of} precisionmeasurementsofthedominantkaonbranchingratiosatKLOE.

©^{2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP³.

1. Introduction

The measurement of the branching ratio (BR) of K⁺ →

π

⁺

π

⁻

π

⁺(

γ

) decay completes the KLOE program of precision measurements of the dominant kaon branching ratios, fully in- clusive of radiation effects. We have already published an eval- uation, from a fit to the KLOE measurements of the charged kaon lifetime [1], and BRs, [2–5], constraining the BR sum to unity:BR(K^±→

π

^±

π

⁺

π

⁻)= (⁵.68±⁰.22)% [5].Themostrecent BR(K^±→

π

^±

π

⁺

π

⁻)measurement,basedon2330eventsfroma sampleof∼^{105 kaondecays,datesbackto1972andgivesnoin-} formationontheradiationcut-off:BR(K^±→

π

^±

π

⁺

π

⁻)= (⁵.56± 0.20)% [6].The PDGvalue, BR(K^±→

π

^±

π

⁺

π

⁻)= (⁵.59±⁰.04)% [7], is obtained from a global fit that does not use any of the available BR(K^±→

π

^±

π

⁺

π

⁻) measurements but the rate mea- surementΓ (

π

⁺

π

⁺

π

⁻)= (⁴.511±⁰.024)×^{106 s−1 publishedin} 1970 [8]. Furthermore the BR(K^±→

π

^±

π

⁺

π

⁻) value enters in the evaluation of the difference a0−^a2 between the I=^{0 and} I=2 S-wave

π π

scatteringlengths[16,19];thiswillbediscussed inSection5.

In the following we report the measurement of the absolute branchingratioBR(K⁺→

π

⁺

π

⁻

π

⁺(

γ

))performedwiththeKLOE detector using data corresponding to an integrated luminosity

Ldt^{174 pb−1 collected atDA}NE,theFrascati φ-factory [9].

DANE isan e⁺e⁻ collideroperatedatthe energyof1020 MeV, the mass of the φ-meson. The beams collide at the interac- tion point (IP) with a crossing angle θx^{25 mrad,1 producing} φ-mesons with a small momentum of ∼¹².5 MeV in the hori- zontalplane.Theφ-mesonsdecayinanti-collinearandmonochro- maticneutral(34%)andcharged(49%)kaonpairs.Theuniquefea- tureofa φ-factory isthetagging: detectionofa K^± (thetagging kaon) tags the presence of a K^∓ (the tagged kaon) withknown momentumanddirection.Theavailabilityoftaggedkaonsenables theprecision measurementofabsoluteBRsprovidingthenormal- izationsample.The decayproductsofthe K⁺K⁻ pairdefine two spatiallywellseparatedregionscalledinthefollowingthetagand thesignalhemispheres.

2. TheKLOEdetector

TheKLOEdetectorconsistsofa largecylindricaldriftchamber (DC)[10],surroundedbyaleadscintillatingfiberelectromagnetic calorimeter (EMC) [11] both immersed in an axial 0.52 T mag- netic field produced by a superconducting coil. At the beams IP thesphericalbeampipe of10 cmradius ismadeofaberyllium–

aluminumalloyof0.5 mmthickness.

The DC tracking system has 25 cm internal radius, 4 m dia- meterand3.3 mlength,witha totalof∼^{52000 wires,ofwhich}

∼^{12000 are sensewires arrangedina stereogeometry.Inorder} to minimize the multiple scatteringand KL regeneration, andto maximizethedetectionefficiencyforlowenergyphotons,theDC workswithahelium-basedgasmixtureanditswallsaremadeof lightmaterials,mostlycarbonfibercomposites.Spatialresolutions

1 Weuseleft-handedcoordinatessystemwiththez-axisdefinedasthebisectrix ofthe e⁺e⁻beamsandthey-axisvertical.

are

σ

xy^{150 μm and}

σ

z^{2 mm andthetransversemomentum} resolutionis

σ

(p^T)/p^T≤⁰.4%.

The calorimeter covers 98% of the solid angle and is com- posed by a barrel and two endcaps. Particles showering in the lead-scintillator-fiber EMC structure are detected as local energy deposits by clustering signals from read-out elements. For each impinging particlethecalorimeterinformationconsistsofenergy, position of impact point and time of arrival with accuracies of

σ

E/E=⁵.7%/√

E(GeV),

σ

z=¹.2 cm/√

E (GeV),

σ

φ=¹.2 cm,and

σ

t=^57ps/√

E (GeV)⊕^{100ps.Energyclustersnotassociatedwith} reconstructed tracks in the DC (neutral clusters) identify neutral particles. Thedefinitionofenergy clustersassociatedwithrecon- structedtracksisrelatedtothetrack-to-clusterassociationproce- duredescribedinRef.[12].

Thetrigger[13]isbasedonenergydepositsinthecalorimeter andonhitmultiplicityinthedriftchamber.Onlyeventstriggered by the calorimeter have been used in the present analysis. The triggersystemincludes asecond-level vetoforcosmic-ray muons (cosmic-ray veto or CRV) based on energy deposits in the out- ermost layers of the calorimeter and followed by a third-level softwaretrigger.A softwarefilter(SF),basedonthetopologyand multiplicityofenergyclustersanddriftchamberhits,isappliedto filterout machine background.Both CRV and SF maybe sources ofeventsloss.TheireffectontheBRmeasurementhasbeenstud- iedoncontroldatasamplesacquiredrespectivelywithouttheCRV andtheSFfilters.

Thedatasampleusedforthisanalysishasbeenprocessedand filtered with the KLOE standard reconstruction software and the event classification procedure [12]. The KLOE Monte Carlo (MC) simulation package, GEANFI, has beenused to produce a sample equivalenttodata,accountingforthedetectorstatusandthema- chineoperationonarun-by-runbasis.

3. Analysisstrategy

Tagging with K^±→

μ

^±

ν

(

γ

) _(Kμ2 tags) and K^±→

π

^±

π

⁰(

γ

) (Kπ2 tags) provides two indipendent samples of pure kaons for the signal selection useful for systematic uncertainties evalua- tion and cross-checks [3]. These decays are easily identified as clear peaks in the distribution of p^∗_m

π, the momentum of the charged secondary track in the kaon rest frame evaluated using the pion mass.² The selection efficiency of the two taggingnor- malizationsamples aresimilar, about36%. Then theseeventsare classifiedasφ→K⁺K⁻andarchivedindedicateddatasummary tapes, as described in Ref. [12]. MC studies show that the con- tamination dueto φ-meson decays other than K⁺K⁻ isnegligi- ble.

Tominimize the impactofthe triggerefficiencyon thesignal side, we choose as normalization sample Kμ2 or Kπ2 tags pro- viding the trigger of the event(self-triggering two-body decays).

AfterthisrequesttheKμ2 sampleisreducedbyafactorof∼^33%, while the Kπ2 sample by a factorof ∼^{43%. The residual depen-} dence ofthe signal sample on the tag selection, which we refer

2 Thecontributiontothep^∗_mπ distributionfromK_μ2decaysisslightlybroadened duetothepionmasshypothesis[3].

toastagbias,hasbeenevaluatedfortheBRmeasurement.More- overweuseK⁻ asthetaggingkaon(Kμ2 orKπ2)and K⁺asthe tagged kaon (signal), since the nuclear cross section for positive kaons with momenta ^{100 MeV is lower by a factor of} ∼¹⁰³ withrespecttothatofnegativekaons[14].

Thetrackofthetaggingkaonisbackwardextrapolatedfromits firsthitintheDCtotheIP.Weusethemomentumofthetagging kaonattheIP,p^IP_K−,andthemomentumoftheφ-mesonmeasured runbyrunwithBhabhascatteringevents,p_φ,toevaluatethemo- mentumofthetaggedkaonattheIP,p^IP_K+=^pφ−^pIP_K−.Finallywe extrapolatep^IP_K+ insidetheDC(signalkaonpath).

Thekaonandthethreechargedpionsfromitsdecayhavelow momenta,lessthan∼^{200 MeV,andcurlupintheKLOEmagnetic} field; this increasesthe probability to have poorly reconstructed tracks brokenin more segments (the trackreconstruction proce- dure inKLOE is described in Ref. [12]). We significantly improve thequalityofthereconstruction requiringthe K⁺ decaytooccur beforeit reachestheDCsensitive volume,i.e.insidea cylindrical fiducialvolumecenteredattheIPandwithatransverseradius

ρ

xy closetotheDCinnerwall(detectoracceptance∼^{26%).Inthisway} only the pion tracks are reconstructed, and we extrapolate only twoofthemtosearchforavertexalongthesignal kaonpath.No furtherrequeston thecharge oftheparticles isappliedto maxi- mizetheselectionefficiency.

ToextractthenumberofK⁺→

π

⁺

π

⁻

π

⁺(

γ

)we fitthemiss- ingmassspectrumm²_miss=^E2_miss− (^pK⁺−^p1−^p2)²wherep₁ and p2 are the momenta of the selected tracks, with MC-predicted shapesforthe signal andthe background.The branching ratiois givenby:

BR



K⁺

→ π

⁺

π

⁻

π

⁺

( γ ) 

=

^NK→3π

N_tag

×

¹



selC_TBC_SFC_CRV (1) whereNK→³π isthenumberofsignalevents,Ntagisthenumber oftaggedeventsand



sel istheoverall signal selectionefficiency, includingthedetectoracceptanceandthereconstructionefficiency.

CSFandCCRV arethecorrectionsforthemachinebackgroundfilter andthecosmic-rayveto.C_TB accountsforthetagbiaseffect.

3.1. BRmeasurementwiththeKμ⁻2normalizationsample

The normalization sample is given by Ntag=^{12065087 K}_μ⁻2 taggingevents. The K⁺→

π

⁺

π

⁻

π

⁺(

γ

) signal selection uses DC informationonly.

Anyreconstructed trackidentified asa K⁺ (andthereforecor- respondingtoaK⁺outsidethefiducialvolume)isrejected.More specifically we reject tracks with the point of closest approach (PCA)totheIPsatisfyingtheconditions



x²_PCA+^y2PCA<10 cm,and

|^zPCA|<20 cm,withthemomentumwithin70<p_K<130 MeV, andwithagoodmatchingwiththe positionandthemomentum extrapolatedfromthetaggingkaon.

Toselect the decayvertex K⁺→

π

⁺

π

⁻

π

⁺(

γ

) we require at leasttworeconstructedtracksthathave:

(1) momentumin thekaonrest frame,p_m^∗

π <190 MeV,thiscut removesthebackgroundfromtwo-bodydecays;

(2) distanceofclosestapproach(DCA) betweeneachextrapolated trackandthesignalkaonpath,DCA<3 cm;

(3) distanceofclosestapproachbetweenthetwotracks,DCA12<

3 cm;

(4) the opening angle between the momenta of the two tracks,

|^cos(θ12)|<0.9,thiscutremovesthebackgroundduetoresi- dualkaonbrokentracks;

(5) the decay vertex is accepted in the fiducial volume,

ρ

xy≤ 26 cm.

Fig. 1. MC(dashed)anddata(points)missingmassspectrumoftheselectedevents.

Thearrowsshowthemissingmasswindowforsignalcounting.

Fig. 2. Topplot:fitofthemissingmassspectrumsuperimposedwithdatapoints.

Bottomplot:residualsbetweentheoutputofthefitanddatadistributionnormali- zedtotheirerrors.

Fig. 1 shows the comparison between MC and data missing massdistributionsfortheselected K⁺decays.Wecountthenum- berofsignaleventsinthemissingmasswindow10000<m²_miss<

30000MeV²,wherethesignaloverbackgroundratioisS/B^88.

The background composition is given by K⁺ in two-body

μ

⁺

ν

and

π

⁺

π

⁰_0.1%, semileptonic

π

⁰e⁺

ν

and

π

⁰

μ

⁺

ν

_0.5%, and

π

⁺

π

⁰

π

⁰_0.4% decays. These single track events pass the se- lection criteriabecause asecondary charged trackis wrongly re- constructed astwo separatetracks.Thetop panelofFig. 2shows theresultofthefitofthemissingmassdistribution comparedto data. The fit gives NK→³π=⁴⁸⁰³²±^{286 signal events (the er-} ror accountingfordataandMCstatistics),with

χ

²/ndf=⁴⁴.8/46 (P(

χ

²)=⁰.52). The bottom panel of Fig. 2showsthe fit norma- lizedresiduals.

The signal selection efficiency,



sel, is related to the track re- construction efficiency of two charged secondaries from K⁺ de- cays. We evaluate theselection efficiencyfrom MC,and thenwe correct it to take into account data-MC differences in the track

Table 1

CorrectionstoBR(K⁺→π⁺π⁺π⁻(γ))measurement.Theeventsselectedbythe twotagshavedifferenttopologiesintheKLOEdetectordeterminingdifferentcor- rectionsfactors.

Table of corrections K_μ⁻₂tags K_π⁻₂tags cosmic ray veto correction CCRV 1.00125±⁰.00002 1.00049±⁰.00001 software filter correction CSF 1.0144±⁰.0013 1.0003±⁰.0005 tag bias correction CTB 0.839±0.001 0.802±0.002

Table 2

Summarytableofthefractionalstatisticaluncertainties.

Source of statistical uncertainties K_μ⁻₂tags (%) K_π⁻₂tags (%)

signal counting 0.45 0.70

selection efficiency 0.38 0.60

tag bias 0.11 0.18

software filter 0.13 0.05

cosmic ray veto 0.002 0.0005

Total fractional statistical uncertainty 0.62 0.95

reconstruction.Tothisaimweselect,bothondataandMC,acon- trolsampleofK⁺→

π

⁻X decays(forsignaleventsX corresponds to

π

⁺

π

⁺).Thefirstrequirementisthepresenceofaself-triggering K⁻μ2 inthetaghemisphere.Thenthetrackofthe

π

⁻ candidateis selectedwiththefollowingrequirements:

(1) thenumberofneutralclusterswithan energy Eγ ≥^{30 MeV} mustbe,N_clusters≤^1;

(2) themomentum of the selected track inthe kaon rest frame mustbe,p^∗_m

π ≤^{130 MeV;}

(3) the distance of closest approach between the extrapolated trackandthesignalkaonpathmustbe,DCAπ−<7 cm;

(4) thecosineoftheopeninganglebetweenthemomentaofthe signal kaon andthe momentaof theselected trackmust be, cos(θKπ)≤ −⁰.85.

Thecontrolsample K⁺→

π

⁻X , selectedwitha backgroundcon- tamination of ¹⁰.7%, is used to measure the efficiency cor- rections as function of the total transverse momentum p^T_X, and of the total longitudinal momentum p^L_X of the

π

⁺

π

⁺ pair (the average efficiencycorrectionis∼⁰.92).Theselectionefficiency,is finallyobtainedfoldingtheMCselection efficiencywiththemea- suredcorrections:



sel=⁰.0842±⁰.0003.

The corrections CCRV and CSF have been measured with data takenwithoutthe cosmic-ray vetoandthe softwarefilter,respe- ctively. The correction for the tag bias, CTB, has been evaluated usingMC.AllcorrectionvaluesarereportedinTable 1.

The summary of the fractional statistical uncertainties is re- portedinTable 2.Thetotalstatisticalfractionaluncertaintyonthe branchingratiomeasurementis0.62%.

3.2.BRmeasurementwiththeKπ⁻2normalizationsample

ThenormalizationsampleisgivenbyNtag=^{5171239 K}_π⁻2tag- gingevents.

Thesignalselectiondescribed inSubsection3.1isalsoapplied tothesample taggedby Kπ⁻2 decays.The fittothemissingmass spectrumoftheselectedeventsgives NK→³π=²⁰⁰⁶³±^{186 sig-} naleventswith

χ

²/ndf=⁴².9/45 (P(

χ

²)=⁰.56). Thesignalover background ratio in the missing mass window 10000<m²_miss<

30000 MeV²isevaluatedwithMC:S/B^84.

To evaluate the selection efficiency, we used the corrections measured with the control sample K⁺→

π

⁻X tagged by K⁻μ2 events. The selection efficiency for signal events tagged by Kπ⁻2 events,is:



sel=⁰.0866±⁰.0005.

Table 3

Summarytableofthefractionalsystematicuncertainties.

Source of systematic uncertainties K_μ⁻₂tags (%) K⁻_π2tags (%) DCA, DCA12, cos(θ12)cuts 0.52 0.41

p^∗_mπ cut 0.08 0.11

m²_misscut 0.05 0.14

fiducial volume 0.11 0.10

selection efficiency estimate 0.16 0.16

tag bias 0.16 0.32

K^±lifetime 0.12 0.12

Total fractional systematic uncertainty 0.60 0.59

The summary of the fractional statistical uncertainties is re- ported in Table 2. The total statistical fractional uncertainty on the branching ratio measurement using the Kπ⁻2 tagging sample is0.95%.

4. Systematicuncertainties

Thefollowingsourcesofsystematicuncertaintiesonthebranch- ing ratios measured using both tags, Kμ⁻2 and Kπ⁻2, have been considered:

(1) thecutsusedtoselectthesignalsample;

(2) thefiducialvolume;

(3) thecutsusedtoselectthecontrolsample K⁺→

π

⁻X ; (4) thecutsusedtoselectthetaggingsamples Kμ⁻2andK⁻π2; (5) thechargedkaonlifetime.

ThecorrespondingsystematicuncertaintiesarelistedinTable 3.

The contributions to the systematic error due to points (1), (2), and(3) have been evaluated varying the selection cuts. The DCA,DCA₁₂variablesandthefiducialvolume

ρ

xy havebeenvar- ied within fewsigmas, thecuts oncos(θ12), p^∗_m

π andm²_miss have been varied todecrease the S/B ratio to ^{64. The cutsused to} selectthecontrolsample K⁺→

π

⁻X havebeenvariedtoincrease thebackgroundcontaminationupto^20%.

Concerning the selection of the normalization samples (point (4))wehaveevaluatedtheeffectofaCTBvariationontheBRmea- surements.Thishasbeendonemodifyingtheselectionofthedata andMCnormalizationsamplesaddingacutontheopeningangle between the K⁻ track and the secondary track retaining events with cos(θK t)≥^{0, where t is the}

μ

⁻ (

π

⁻) track in case of the Kμ⁻2 (Kπ⁻2) sample. Using MC we found that the fractional varia- tions of the tag bias corrections are δCTB/CTB(K⁻μ2)=⁰.26% and δC_TB/C_TB(Kπ⁻2)=⁰.63%. Consequently the branching ratios mea- suredvalueschangeofδBR/BR(Kμ⁻2)=⁰.32% andδBR/BR(Kπ⁻2)= 0.64%;halfofthesevariationshavebeenassignedasconservative values for the fractional systematic uncertainties due to the tag bias(seeTable 3).

The BR(K⁺→

π

⁺

π

⁺

π

⁻(

γ

)) depends on the charged kaon lifetime

τ

K^± through the detector acceptance, that is evaluated with MC simulation (point (5)). The systematic effect has been obtained varying

τ

K^± within the uncertainty of the KLOE result

τ

_K±=¹².347±⁰.030 ns[1].Thishasbeendonere-weighting the MC events witha hit-or-miss procedure,both for the signal and the control sample selection procedures. The corresponding sis- tematicerrorsarelistedinTable 3.

Theanalysisisfullyinclusiveofradiativedecays.Onlytheeffi- ciencyevaluationcouldbeaffectedbyasystematicuncertantydue tothecut Nclusters≤1 (seeSubsection3.1).WehaveusedPHOTOS [15] toevaluatesuch aneffectandwe obtaineda negligiblecon- tribution,beingO(10⁻⁶)thefractionofdecaysremovedbythecut N_clusters≤^1.

Table 4

Resultsofthefit:K^±BRsandcorrelationcoefficients.

Parameter Value Correlation coefficients BR(K_μ^±₂) 0.6372(11)

BR(K_π^±₂) 0.2070(9) 0.55 BR(π^±π⁻π⁺) 0.0558(4) −⁰.23 −⁰.05 BR(K_e3^±) 0.0498(5) 0.42 −⁰.15 0.06 BR(K_μ^±₃) 0.0324(4) −0.39 0.14 −0.05 −0.58 BR(π^±π⁰π⁰) 0.01764(25) −0.13 0.05 −0.02 0.04 −0.04 τK^± (ns) 12.344(29) 0.20 0.19 −0.14 0.05 −0.04 0.02

The fractionof K⁺ undergoing nuclearinteractions isnegligi- ble,∼ ^{10−5,asevaluatedusingtheMC}simulation,basedondata availableinliterature[14].Thereforetherelatedsystematicuncer- taintyisnegligible.

Furthermorewehavecheckedontwoindependentsub-samples ofabout88 pb⁻¹ and86 pb⁻¹ thattheefficiencycorrectionsand theBRevaluationsarenotcorrelated.

Finallythestabilityofthemeasurementswithrespecttodiffer- entdatatakingperiodsandconditionshasbeenchecked.

5. Results

With a sample of K⁻ →

μ

⁻

ν

¯(

γ

) tagging events N_tag=^{12065087 we found N}K→3π=⁴⁸⁰³²±^{286 signal events.}

UsingEq.(1),weobtainthebranchingratio:

BR



K⁺

→ π

⁺

π

⁻

π

⁺

( γ ) 

Tag Kμ2

=

⁰

.

05552

±

⁰

.

00034_stat

±

⁰

.

00034_syst

.

(2) Withasampleof K⁻→

π

⁻

π

⁰(

γ

)taggingeventsNtag=^5171239 wefoundN_K→3π=²⁰⁰⁶³±^{186 signalevents,}correspondingto:

BR



K⁺

→ π

⁺

π

⁻

π

⁺

( γ ) 

Tag Kπ2

=

0

.

05587

±

0

.

00053stat

±

0

.

00033syst

.

(3) Averagingthesetworesults,accountingforcorrelations,weobtain:

BR



K⁺

→ π

⁺

π

⁻

π

⁺

( γ ) 

=

⁰

.

05565

±

⁰

.

00031_stat

±

⁰

.

00025_syst

.

(4) This absolute branching ratio measurement is fully inclusive of final-state radiation and has a 0.72% accuracy, a factor ^{5 bet-} terwithrespecttothepreviousmeasurement[6].

We fit the six largest K^± BRs and the lifetime

τ

_K^± using the KLOE measurements of

τ

K^± [1], BR(K⁺μ2) [2], BR(Kπ⁺2) [5], BR(K⁺ →

π

⁺

π

⁻

π

⁺(

γ

)) (Eq. (4)), BR(K_l3^±) [3], and BR(K^± →

π

^±

π

⁰

π

⁰) [4], with their dependence on

τ

_K^±, and imposing the constraint 

BR(K^±→ ^f)=^{1. The fit results, with}

χ

²/ndf= 0.24/1 (CL=⁰.63),showacoherentsetofmeasurements(seeTa- ble 4).

TheNA48experimentobservedinthe

π

⁰

π

⁰invariantmassdis- tributionacusp-likeanomalyatM₀₀=2mπ⁺ [16],whichhasbeen interpretedasmainlyduetothefinalstatecharge–exchangereac- tion

π

⁺

π

⁻_→

π

⁰

π

⁰ in K^±→

π

^±

π

⁺

π

⁻ decay[17,18].Thefitto theM²₀₀ distribution[19]withtwodifferentmodels[20]and[21, 22] determines a0−^a2, the difference between the S-wave

π π

scatteringlengthsintheisospinI=^{0 andI}=^{2 states.Inthiscal-} culationthe main source of uncertaintyis the ratio ofthe weak amplitudesof K^±→

π

^±

π

⁻

π

⁺andK^±→

π

^±

π

⁰

π

⁰ decay,thatis obtainedfromtheratio R ofthebranchingratiovalues.Usingthe BR(

π

^±

π

⁻

π

⁺), BR(

π

^±

π

⁰

π

⁰) and their correlation shown in Ta- ble 4weevaluateR=³.161±⁰.049,inagreementwiththevalue

R=³.175±⁰.050 obtained byNA48[19]withBRsfromthePDG fit[7].

6. Conclusions

We havemeasured theabsolute branchingratio ofthe K⁺→

π

⁺

π

⁻

π

⁺(

γ

)decay,inclusiveoffinal-stateradiation,usingtwoin- dipendentnormalizationsamplesfromKμ⁻2andK⁻π2 tags:

BR



K⁺

→ π

⁺

π

⁻

π

⁺

( γ ) 

=

⁰

.

05565

±

⁰

.

00031stat

±

⁰

.

00025syst withanoverallaccuracyof0.72%.Thismeasurementcompletesthe KLOE program ofprecision measurements of the dominant kaon branchingratios.

Acknowledgements

We warmly thank our former KLOE colleagues for the access to the data collected during the KLOE data taking campaign.

We thank the DANE team for their efforts in maintaining low background running conditionsand their collaboration during all data taking. We want to thank our technicalstaff: G.F. Fortugno and F. Sborzacchi for their dedication in ensuring efficient op- eration of the KLOE computing facilities; M. Anelli for his con- tinuous attentiontothe gassystemanddetector safety;A.Balla, M. Gatta, G. CorradiandG. Papalinoforelectronicsmaintenance;

M. Santoni, G. Paoluzzi andR. Rosellinifor generaldetectorsup- port; C.Piscitelli forhis help during major maintenance periods.

This work was supported in part by the EU Integrated Infras- tructure Initiative Hadron Physics Project under contract number RII3-CT-2004-506078;bytheEuropeanCommissionunderthe7th Framework Programme through the ‘Research Infrastructures’ ac- tion of the ‘Capacities’ Programme, Call: FP7-INFRASTRUCTURES- 2008-1, GrantAgreementNo. 227431;by thePolishNationalSci- ence Centre through the Grants No. DEC-2011/03/N/ST2/02641, 2011/01/D/ST2/00748,2011/03/N/ST2/02652,2013/08/M/ST2/00323, and by the Foundation ForPolish Science through the MPD pro- grammeandtheprojectHOMINGPLUSBIS/2011-4/3.

References

[1]KLOECollaboration,F.Ambrosino,etal.,J.HighEnergyPhys.01(2008)73.

[2]KLOECollaboration,F.Ambrosino,etal.,Phys.Lett.B632(2006)76.

[3]KLOECollaboration,F.Ambrosino,etal.,J.HighEnergyPhys.02(2008)98.

[4]KLOECollaboration,A.Aloisio,etal.,Phys.Lett.B597(2004)139.

[5]KLOECollaboration,F.Ambrosino,etal.,Phys.Lett.B666(2008)15.

[6]I.H.Chiang,etal.,Phys.Rev.D6(1972)1254.

[7]PDG,Phys.Rev.D86(2012)010001.

[8]W.T.Ford,etal.,Phys.Rev.Lett.25(1970)1370.

[9] A.Drago,etal.,LNF-03/012,2003.

[10]KLOECollaboration,M.Adinolfi,etal.,Nucl.Instrum.Methods488(2002)51.

[11]KLOECollaboration,M.Adinolfi,etal.,Nucl.Instrum.Methods482(2002)364.

[12]KLOECollaboration,F.Ambrosino, etal.,Nucl.Instrum.Methods534(2004) 403.

[13]KLOECollaboration,M.Adinolfi,etal.,Nucl.Instrum.Methods492(2002)134.

[14]C.B.Dover,G.E.Walker,Theinteraction ofkaonswith nucleonsand nuclei, Phys.Rep.89(1982)1–177.

[15]E.Barberio,B.vanEijk,Z.Was,Comput.Phys.Commun.66(1991)115.

[16]NA48Collaboration,J.R.Batley,etal.,Phys.Lett.B633(2006)173.

[17]P.Budini,L.Fonda,Phys.Rev.Lett.6(1961)419.

[18]N.Cabibbo,Phys.Rev.Lett.93(2004)121801.

[19]NA48Collaboration,J.R.Batley,etal.,Eur.Phys.J.B64(2009)589.

[20]N.Cabibbo,G.Isidori,J.HighEnergyPhys.0503(2005)21.

[21]G.Colangelo,J.Gasser,B.Kubis,A.Rusetsky,Phys.Lett.B638(2006)187.

[22]M.Bissegger,A.Fuhrer,J.Gasser,B.Kubis,A.Rusetsky,Nucl.Phys.B806(2009) 178.