Measurement of the π0 electromagnetic transition form factor slope

(1)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Measurement

of

the

π

0 electromagnetic

transition

form

factor

slope

The

NA62

Collaboration

C. Lazzeroni

1

,

N. Lurkin

∗

,2

,

A. Romano

UniversityofBirmingham,Edgbaston,Birmingham,B152TT,UnitedKingdom

T. Blazek,

M. Koval

∗∗

,3

FacultyofMathematics,PhysicsandInformatics,ComeniusUniversityinBratislava,84248Bratislava,Slovakia

A. Ceccucci,

H. Danielsson,

V. Falaleev,

L. Gatignon,

S. Goy

Lopez

4

,

B. Hallgren

5

,

A. Maier,

A. Peters,

M. Piccini

6

,

P. Riedler

CERN,CH-1211Genève23,Switzerland

P.L. Frabetti,

E. Gersabeck

7

,

V. Kekelidze,

D. Madigozhin,

M. Misheva

8

,

N. Molokanova,

S. Movchan,

Yu. Potrebenikov,

S. Shkarovskiy,

A. Zinchenko

†

JointInstituteforNuclearResearch,141980Dubna(MO),Russia

P. Rubin

9

GeorgeMasonUniversity,Fairfax,VA22030,USA

W. Baldini,

A. Cotta

Ramusino,

P. Dalpiaz,

M. Fiorini,

A. Gianoli,

A. Norton,

F. Petrucci,

M. Savrié,

H. Wahl

DipartimentodiFisicaeScienzedellaTerradell’UniversitàeSezionedell’INFNdiFerrara,I-44122Ferrara,Italy

A. Bizzeti

10

,

F. Bucci

11

,

E. Iacopini

11

,

M. Lenti,

M. Veltri

12

Sezionedell’INFNdiFirenze,I-50019SestoFiorentino,Italy

A. Antonelli,

M. Moulson,

M. Raggi

13

,

T. Spadaro

LaboratoriNazionalidiFrascati,I-00044Frascati,Italy

K. Eppard,

M. Hita-Hochgesand,

K. Kleinknecht,

B. Renk,

R. Wanke,

A. Winhart

5

InstitutfürPhysik,UniversitätMainz,D-55099Mainz,Germany15

R. Winston

UniversityofCalifornia,Merced,CA95344,USA

V. Bolotov

†

,

V. Duk

6

,

E. Gushchin

InstituteforNuclearResearch,117312Moscow,Russia

http://dx.doi.org/10.1016/j.physletb.2017.02.042

(2)

F. Ambrosino,

D. Di

Filippo,

P. Massarotti,

M. Napolitano,

V. Palladino

15

,

G. Saracino

DipartimentodiFisicadell’UniversitàeSezionedell’INFNdiNapoli,I-80126Napoli,Italy

G. Anzivino,

E. Imbergamo,

R. Piandani

16

,

A. Sergi

5

DipartimentodiFisicadell’UniversitàeSezionedell’INFNdiPerugia,I-06100Perugia,Italy

P. Cenci,

M. Pepe

Sezionedell’INFNdiPerugia,I-06100Perugia,Italy

F. Costantini,

N. Doble,

S. Giudici,

G. Pierazzini

†

,

M. Sozzi,

S. Venditti

DipartimentodiFisicadell’UniversitàeSezionedell’INFNdiPisa,I-56100Pisa,Italy

S. Balev

†

,

G. Collazuol

17

,

L. DiLella,

S. Gallorini

17

,

E. Goudzovski

1,2,5

,

G. Lamanna

18

,

I. Mannelli,

G. Ruggiero

19

ScuolaNormaleSuperioreeSezionedell’INFNdiPisa,I-56100Pisa,Italy

C. Cerri,

R. Fantechi

Sezionedell’INFNdiPisa,I-56100Pisa,Italy

S. Kholodenko,

V. Kurshetsov,

V. Obraztsov,

V. Semenov,

O. Yushchenko

InstituteforHighEnergyPhysics,142281Protvino(MO),Russia20

G. D’Agostini

DipartimentodiFisica,SapienzaUniversitàdiRomaandSezionedell’INFNdiRomaI,I-00185Roma,Italy

E. Leonardi,

M. Serra,

P. Valente

Sezionedell’INFNdiRomaI,I-00185Roma,Italy

A. Fucci,

A. Salamon

Sezionedell’INFNdiRomaTorVergata,I-00133Roma,Italy

B. Bloch-Devaux

21

,

B. Peyaud

DSM/IRFU–CEASaclay,F-91191Gif-sur-Yvette,France

J. Engelfried

InstitutodeFísica,UniversidadAutónomadeSanLuisPotosí,78240SanLuisPotosí,Mexico22

D. Coward

SLACNationalAcceleratorLaboratory,StanfordUniversity,MenloPark,CA94025,USA

V. Kozhuharov

23

,

L. Litov

FacultyofPhysics,UniversityofSoﬁa,1164Soﬁa,Bulgaria24

R. Arcidiacono

25

,

S. Bifani

5

DipartimentodiFisicadell’UniversitàeSezionedell’INFNdiTorino,I-10125Torino,Italy

C. Biino,

G. Dellacasa,

F. Marchetto

Sezionedell’INFNdiTorino,I-10125Torino,Italy

T. Numao,

F. Retière

(3)

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received28December2016

Receivedinrevisedform14February2017 Accepted15February2017

Availableonline22February2017 Editor:W.-D.Schlatter

The NA62experimentcollectedalarge sampleofchargedkaondecays in2007withahighlyeﬃcient triggerfordecaysintoelectrons.A measurementofthe

π

0_{electromagnetic}_transition_form_factor_slope

parameter from 1.11×106 fully reconstructed K±→

π

±

π

0

D,

π

0

D→e+e−

γ

events is reported. The

measuredvaluea= (3.68±0.57)×10−2_is_in_good_agreement_with_theoretical_expectations_and_previous

measurements,andrepresentsthemostpreciseexperimentaldeterminationoftheslopeinthetime-like momentumtransferregion.

0. Introduction

The Dalitz decay

π

0

D

→

e+e−

γ

with a branching fraction of

B = (

1

.

174

±

0

.

035

)

% [1]proceedsthrough a

π

0

_→

_{γ γ}

∗ _process

with an off-shell photon converting into an e+e− pair. The

π

0 electromagnetic transition form factor (TFF) describes the devia-tionof thistransitionfromapoint-like interaction.It isan input tothecomputationofthe

π

0

_→

_e+_e−_decay_rate_[2]_,_as_well_as_the

hadroniclight-by-lightscatteringcontributiontothemuon anoma-lousmagneticmoment

(

g

−

2

)

μ which atpresentcontributesthe second largest uncertainty on its Standard Model value [3]. The commonlyusedkinematicvariablesaredeﬁnedintermsofthee± and

π

0 _four-momenta_(p

e±,pπ0)as

*

Correspondingauthor.

**

Correspondingauthor.

E-mailaddresses:nicolas.lurkin@cern.ch(N. Lurkin),michal.koval@cern.ch (M. Koval).

1 _Supported_by_a_Royal_Society_University_Research_Fellowship. 2 _Supported_by_ERC_Starting_Grant_336581.

3 _Present_address:_CERN,_CH-1211_Genève_23,_Switzerland. 4 _Present_address:_CIEMAT,_E-28040_Madrid,_Spain.

5 _Present_address:_School_of_Physics_and_Astronomy,_University_of_Birmingham, Birmingham,B152TT,UK.

6 _Present_address:_Sezione_dell’INFN_di_Perugia,_I-06100_Perugia,_Italy.

7 _Present _address: _{Ruprecht-Karls-Universität} _Heidelberg, _D-69120 _Heidelberg, Germany.

8

Presentaddress:InstituteofNuclearResearchandNuclearEnergyofBulgarian AcademyofScience(INRNE–BAS),Soﬁa,Bulgaria.

9 _Funded_by_the_National_Science_Foundation_under_award_No._0338597. 10 _Also_at_Dipartimento_di_Fisica,_Università_di_Modena_e_Reggio_Emilia,_I-41125 Modena,Italy.

11 _Also_at_Dipartimento_di_Fisica,_Università_di_Firenze,_I-50019_Sesto_Fiorentino, Italy.

12 _Also_at_Istituto_di_Fisica,_Università_di_Urbino,_I-61029_Urbino,_Italy. 13 _Present_address:_Università_di_Roma_“La_Sapienza”,_Roma,_Italy.

14 _Funded_by_the_German_Federal_Minister_for_Education_and_Research_(BMBF) un-dercontract05HA6UMA.

15 _Present_address:_Physics_Department, _Imperial_College_London, _London,_SW7 2BW,UK.

16 _Present_address:_Sezione_dell’INFN_di_Pisa,_I-56100_Pisa,_Italy.

17 _Present_address:_Dipartimento_di_Fisica_{dell’Università}_e _Sezione_dell’INFN_di Padova,I-35131Padova,Italy.

18 _Present_address:_Dipartimento_di_Fisica_{dell’Università}_e _Sezione_dell’INFN_di Pisa,I-56100Pisa,Italy.

19 _Present_address:_Department_of_Physics,_University_of_Liverpool,_Liverpool,_L69 7ZE,UK.

20 _Partly_funded_by_the_Russian_Foundation_for_Basic_Research_grant_12-02-91513. 21 _Present_address:_Dipartimento_di_Fisica_{dell’Università}_di_Torino,_I-10125_Torino, Italy.

22 _Funded_by_Consejo_Nacional_de_Ciencia_y_Tecnología_(CONACyT)_and_Fondo_de ApoyoalaInvestigación(UASLP).

23 _Also_at_Laboratori_Nazionali_di_Frascati_dell’INFN,_Italy.

24 _Funded_by_the_Bulgarian_National_Science_Fund_under_contract_DID02-22. 25 _Also_at_Università_degli_Studi_del_Piemonte_Orientale,_I-13100_Vercelli,_Italy.

† _Deceased. x

=

Mee m_π0

2

=

(

pe+

+

pe−

)

2 m2_π0

,

y

=

2 pπ0

· (

pe+

−

pe−

)

m2_π0

(

1

−

x

)

,

withtheallowedkinematicregiondeﬁnedas

r2

=

2me m_π0

2

≤

x

≤

1

,

|

y

| ≤

1

−

r 2 x

,

whereme andmπ0 arethecorrespondingPDG[1]masses,andMee istheinvariantmassofthee+e−pair.Thedifferentialdecaywidth reads[4] d2

(π

0 D

)

dxd y

=

α

4

π

(π

0 2γ

)

(

1

−

x

)

3 x

1

+

y2

+

r 2 x

× (

1

+ δ(

x

,

y

))

|

F

(

x

)

|

2

,

where

(

π

0

2γ

)

is the

π

0

→

γ γ

decaywidth, the function

δ(

x

,

y

)

describestheradiativecorrectionsand

F(

x

)

istheelectromagnetic transitionformfactorofthe

π

0 _to_a_real_and_virtual_photon. _The function

F(

x

)

isexpected tovary slowly inthe kinematicregion ofthe

π

0

D decayandisusuallyapproximatedbyalinearexpansion

F(

x

)

=

1

+

ax, wherea is theslope parameter. The vectormeson dominance (VMD) model [5,6] predicts a

π

0 _TFF _slope _value _of a

≈

0

.

03,inagreementwithfurthertheoreticalestimates[7–10].

TheTFFslopehasbeendeterminedinthetime-likemomentum transferregionbymeasuringthe

π

0

D decayrate[11–15],all includ-ingradiativecorrections.TheTFFhasbeenmeasuredinthe space-like momentum transfer regionin the reactione+e−

→

e+e−

π

0_, wherethe

π

0 _is_produced_by _the_fusion _of_two_photons_radiated by theincomingbeamsanddecaystotwodetected photons[16]. The currentworldaveragea

=

0

.

032

±

0

.

004[1]isobtainedfrom time-like measurements [12–14] andthe extrapolation of space-likedata[16]usingaVMDmodel.

The NA62 experiment at the CERN SPS collected in 2007 a

largesampleofchargedkaonsdecayinginﬂightinvacuumwitha minimum-biastriggerconﬁguration[17].TheK±decaysrepresent a source of tagged neutral pions; the K±

→

π

±

π

0 _(K

2π ) decay channelaccountsfor63%of

π

0 _production._The_mean_free_path_of theneutralpionintheNA62experimentalconditionsisnegligible (fewμm).Thisletterreportsamodel-independentmeasurementof the

π

0_TFF_slope_parameter_from_an_analysis_of₁

_.

₁₁

_×

₁₀6_K

2π de-cays followedbythe prompt

π

0

D decay(denoted K2πD) usingthe fullNA622007dataset.

1. Beamanddetector

The NA62 experimental setup used in2007 was composed of

theNA48detector[18]andamodiﬁedbeamline[19]oftheearlier NA48/2experiment.

ThebeamlinewasdesignedtoprovidesimultaneouslyK+and K− beams.Theprimary 400 GeV/c protonbeamdeliveredby the

(4)

SPSimpingedonaberylliumtargetof40 cmlengthand0.2 cm

di-ameter.The secondarybeammomentawere selected bymagnets

in a four dipole achromat and a momentum-deﬁning slit incor-porated into a beam dump. This 3.2 m thick copper/iron block providedthe possibility toblock eitherofthe K+ or K− beams. Theselectedparticleshadacentral momentumof74 GeV/c with a spreadof

±

1.4 GeV/c (rms). The beamswere focused and col-limated before entering a 114 m long cylindrical vacuum tank

containing the ﬁducial decay volume. The beams were mostly

composed of

π

±, witha K± fractionof approximately6%. Since the muon halo sweeping system was optimised for the positive beam in 2007, most of the data were recorded with the single K+ beamtoreducethehalobackground.The K+ andK− beams weredeﬂected horizontallyby asteering magnetatthe entrance of the ﬁducial decay volume at angles of

±(

0

.

23 to 0

.

30

)

mrad withrespect tothe detectoraxis,to compensateforthe opposite

∓

3

.

58 mrad deﬂection by thedownstream spectrometer magnet. The polaritiesof those magnetic ﬁeldswere regularly simultane-ously reversed to reduce theeffects caused by an asymmetry in thedetectoracceptance.

The momenta ofcharged particles were measured by a

spec-trometer composed of four drift chambers (DCH) and a dipole

magnet placed between the second and third chamber

provid-ing a horizontal transverse momentum kick of 265 MeV/c to

singly-chargedparticles.Themeasuredmomentumresolutionwas

σ

p

/

p

=

0

.

48%

⊕

0

.

009%

·

p,wherethemomentump isexpressedin GeV/c.Thespectrometer was housedinatank ﬁlledwithhelium atnearly atmosphericpressure,separatedfromthedecayvolume byathin(3

×

10−3_X

0)Kevlar™window.

Thephotonswere detected andmeasured by aliquid krypton (LKr) electromagneticcalorimeter, whichis a quasi-homogeneous ionisationchamberwithan activevolume of6.7 m3 ofoctagonal cross-sectionandathicknessof127 cm,corresponding to27 X0. The LKr volume is divided into 13,248 cells of about 2

×

2 cm2 crosssectionwithoutlongitudinalsegmentation.Themeasured en-ergy resolution was

σ

E

/

E

=

3

.

2%

/

√

E

⊕

9%

/

E

⊕

0

.

42%, and the spatial resolution for the transverse coordinates x and y was 0

.

42 cm

/

√

E

⊕

0

.

06 cm,wheretheenergyisgiveninGeVinboth cases.

Ascintillatorhodoscope(HOD)was locatedbetweenthe spec-trometerandtheLKrcalorimeter.Itconsistsofasetofscintillators arrangedintoaplaneof64verticalcountersfollowedbyaplaneof 64horizontalcounters.Eachplanewasdividedintofourquadrants of16countersprovidingafasttriggersignalforchargedparticles.

2. Datasampleandtriggerlogic

The analysis is based on the full data set collected during 4 monthsin2007,correspondingtoabout2

×

1010_K±_decays_in_the vacuumtank.A totalof65%(8%)oftheK+(K−)ﬂuxwascollected insingle-beammodewhiletheremaining27%werecollectedwith simultaneous K± beams witha K+

/

K− ﬂux ratio of2.0. During partofthedatataking(55%ofthe K±ﬂux), a 9.2 X0 thick trans-versehorizontalelectronabsorber lead(Pb) barwasinstalled be-tweenthetwoHODplanes,approximately1.2 minfrontoftheLKr calorimeter,tostudymuon-inducedelectromagneticshowers[17]. A totalof11rowsofLKrcalorimetercellswere shadowedbythe bar,correspondingtoabout10%ofthetotalnumberofcells.

The100 kHzkaondecayrateinthevacuumvolumeduringthe spillenabledtheuseofaminimum-biastriggerconﬁgurationwith ahighly eﬃcienttrigger chainoptimised toselectevents withat leastoneelectron(e±)track.

The low level hardware trigger required a coincidence ofhits inat least one hodoscope quadrant in both planes (the Q1 con-dition),upper andlower cutson the hit multiplicity in the drift

chambers(the1-trackcondition),andaminimumtotalenergy de-posit of 10 GeV in the LKr calorimeter(the ELKr condition).The high level software trigger (HLT) condition required atleast one trackwith5 GeV/c

<

p

<

90 GeV/c and E

/

p

>

0

.

6,where E isthe energyreconstructed inthe calorimeterand p isthemomentum reconstructedinthespectrometer.Downscaledminimumbias trig-gerstreamswerecollectedtoevaluatethetriggereﬃciencies.

3. Simulatedsamplesand

π

0

Ddecaysimulation

Monte Carlo (MC) simulations of the K2πD decay chain and two other K± decay chains producing

π

0 _Dalitz _decays, _K±

_→

π

0

De±

ν

andK±

→

π

D0

μ

±

ν

(denoted Ke3D andKμ3D,respectively), were performedwitha

π

0 _TFF_slope _a

MC

=

3

.

2

×

10−2.Separate simulated samples,proportionally to the numberof kaon decays recorded, were produced foreach datataking condition. The to-tal simulatedsample amounts to386 M K2πD,105 M Kμ3D and 103 M Ke3D eventswithinthe97 mlongﬁducialdecayregion.All thesemodescontribute tothe

π

0

D sample, althoughtheselection isoptimizedforK2πD.

Theradiativecorrectionstothetotal[4]anddifferential[20–22]

π

_D0 decaywidthshave beenstudied extensively.Theyhave tobe considered for the TFF measurement since their effect on the x spectrumiscomparabletotheeffectoftheTFF.Thecalculationof theradiativecorrections[22]implementedintheMCsimulationof the

π

0

D decayforthepresentanalysisincludes realphoton emis-sion from the

π

0

D decay vertex. It also includes the one-photon irreduciblecontribution,neglectedinearlierstudies,whichhasan effectof

|

a

|

≈

0

.

5

×

10−2 ontheslopeofthex spectrum.Higher order correction terms not included inthe simulation contribute totheslopeby

|

a

|

<

0

.

01

×

10−2,whichisconsideredasa sys-tematicuncertainty(Table 1).

4. Dataanalysis

4.1. Eventreconstructionandselection

Hits and drift times in the DCH and a detailed map of the magnetic field are used to reconstruct track directions and mo-menta. Three-track vertices are reconstructed by a Kalman filter algorithmextrapolatingtracksegmentsfromtheupstreampartof thespectrometerintothedecayvolume,takingintoaccount mul-tiplescatteringinthe heliumandtheKevlarwindow, theEarth’s magnetic field and residual vacuum tank magnetization. The re-constructedK±

→

π

±

π

+

π

−invariantmassandthemissingmass in the K±

→

μ

±

ν

decay are monitored and used for ﬁne cali-brationofthespectrometermomentumscaleandDCHalignment. Clustersofenergydeposition intheLKrcalorimeterare foundby locating maxima in space and time in the digitized pulses from individual cells. Reconstructed energies are corrected for energy outsidetheclusterboundaries,energylostinisolatedinactivecells (0.8%ofthetotalnumber),sharingofenergybetweenclusters,and non-linearityforclusterswithenergybelow11 GeV.Electrons pro-duced in K±

→

π

0_e±

_ν

_decays _are _used _to _calibrate _the _energy response.

ThemainK2πD selectioncriteriaarethefollowing.

•

The event should contain exactly one reconstructed 3-track vertex, whichshouldbe locatedwithin theﬁducial decay re-gionandbegeometricallycompatiblewithabeamkaondecay. Thevertexchargeqvtx,deﬁnedasthesumofthetrackcharges, shouldmatchthebeamchargeinthesingle-beammode. Oth-erwiseitshouldsatisfyarelaxedcondition

|

qvtx

|

=

1.Thetrack withthechargeoppositetoqvtxisnecessarilyane±candidate, whilethesame-signtrackscanbeeither

π

±ore±candidates.

(5)

Fig. 1. Reconstructed(a)π±π0_and_(b)_e+_e−_γ _mass_{distributions}_for_data_and_simulated_components._The_radiative_shoulders_in_the_{reconstructed}_masses_are_well

repro-ducedintheMCthankstothesimulationoftheradiativephoton.Theadoptedmassselectioncriteriarepresentedbythearrowsareasymmetricwithrespecttothenominal K±andπ0_masses.

•

The tracks are required to be in time (within 25 ns of the trigger time and 15 ns of each other), and within the geo-metricalacceptance ofthe drift chambers. The allowed track momentumrangeis(2–74) GeV/c,excludingthelow

momen-tum range where a 2% deﬁcit of data with respect to MC

simulation is seen. Events with a photon converting into an e+e−pairinthematerialinorinfrontofDCH1(Kevlar win-dow,helium)aresuppressedbyrequiringaminimumdistance of2 cmbetweentheimpactpoints ofevery trackpairinthe ﬁrstdriftchamber,asveriﬁedbysimulationofK2π decays.

•

Reconstructedclustersofenergydeposition intheLKr calori-meterareusedtoidentifyphotoncandidates.A photon candi-date cluster should be geometrically isolated fromthe track impact points in the LKr calorimeter (dt

>

20 cm from the same-sign tracks and dt

>

10 cm from the remaining track), within10 nsofeach trackandwithmorethan2 GeVof en-ergy.Thephoton4-momentumisreconstructedassumingthat the photon originates fromthe same vertexas the tracks. If more than one photon candidate is found, the event is re-jected.

•

Thetotalreconstructedmomentumshouldbecompatiblewith thebeammomentum,in therange(70–78) GeV/c, andthere should be no missingtransverse momentum withrespect to thebeamaxiswithintheresolution: p2

t

<

10−5

(

GeV/c

)

2. Par-ticleidentiﬁcation usingan E

/

p ratio isnot requiredthanks tothelow backgroundinthesample,reducing the systemat-icsassociatedtothemisidentiﬁcationandincreasingtheK2πD acceptance by more than a factor of two. The

π

/

e ambigu-ity for the two same-sign tracks is resolved by testing the two possible mass assignments. For each hypothesis, the re-constructedkinematicvariablesshouldbe

|

x

|,

|

y

|

<

1,andthe reconstructed e+e−

γ

and

π

±

π

0 _masses _should _be _close _to thenominal ones: Meeγ inthe range(115–145) MeV/c2 and Mππ intherange(460–520) MeV/c2_._Only_events_with_a_single validhypothesisareselected.Theprobabilityofcorrect (incor-rect)massassignmentevaluatedwiththe K2πD MCsample is 99.62%(0.02%).Theremaining0.36%ofeventshaveeitherzero ortwovalidhypothesesandarerejected.

•

The triggerconditions described inSection 2are reproduced on simulated samples. To eliminate edge effects due to dif-ferentcalibrationandresolution betweenthetriggerandthe

offline analysis, tightervariants ofthe triggercriteriaare ap-pliedtobothdataandMCsamples.TheofflineELKrcondition requires a minimumof 14 GeV ofelectromagnetic energyin the LKr calorimeter summed over the reconstructed photon and e± clusters. The offline condition corresponding to the HLTrequiresatleastonetrackwhoseimpactpointontheLKr calorimeterfrontplaneiswithinitsacceptanceandnotbehind thePbbar,p

>

5

.

5 GeV/c andE

/

p

>

0

.

8,effectivelyrequesting thatatleastonee± trackisdetectedinthecalorimeter.

•

A 1% deﬁcit in the data/MC ratio is seen for events with x

<

0

.

01 due to the steeply falling acceptance. For this rea-son the signal region is deﬁned as x

>

0

.

01, equivalent to Mee

>

13

.

5 MeV/c2.

The selected K2πD sample amountsto 1

.

11

×

106 events.The overallacceptancesoftheselectionevaluatedwithMCsimulations are 1.90% for K2πD decays, 0.02%for Kμ3D decaysand 0.01%for Ke3D decays.The K2πD acceptancesforperiodswithandwithout thePbbarinstalledare1.64%and2.23%,respectively.

Thereconstructed

π

±

π

0_and_e+_e−

_γ

_invariant_mass_spectra_are

shown in Fig. 1; the mass resolutions obtainedfrom a Gaussian ﬁt are 3.7 MeV/c2 _and_{1.5 MeV/c}2 _(rms), _{respectively.} _The recon-structed spectrum of the x variable and the acceptances for the decay channels considered are shown in Fig. 2. The e+e− mass resolutiondeterminedfromthe K2πD MC samplecanbe approxi-matedby

σ

ee

=

0

.

9%

·

Mee,whichtranslatesintotheresolutionon thex variableas

σ

x

=

1

.

8%

·

x.

4.2. Fitprocedure

A

χ

2 _ﬁt_with_free _MC_{normalisation}_in_equally_populated_bins comparing the data andMC reconstructed x distributions is per-formedtoextracttheTFFslope.A numberofslopehypothesesah aretestedbyreweightingasinglesetofMCeventssimulatedwith aslopeaMC

=

3

.

2

×

10−2 usingtheweights

w

(

ah

)

=

(

1

+

ahxMC

)

2

(

1

+

aMCxMC

)

2

,

wherexMCisthetruex valueforeachevent.Theminimizationof the

χ

2 _test_statistics_yields_the_following_result:

(6)

Fig. 2. (a)Spectraofthereconstructedx variablefordataandMCcomponents.(b)Acceptancesofthe K2πD selectionforeachdecayasfunctionsofthex variable.The

acceptancesforK3Ddecays(=e;μ)arescaledupbyafactorof50.Thedropintheﬁrstbinisduetothesignalregiondeﬁnition(x>0.01).

Fig. 3. Ratioofthereconstructedx distributionsfordataandMC,wheretheMC samplecorrespondstoa=0.TheeffectofapositiveTFFslope(a>0)isclearlyseen inthisillustration.DataandMCeventsaredistributedinto25equallypopulated bins;the horizontalpositionsofthemarkerscorrespondtothebinbarycentres. Thesolidlinerepresents|F(x)|2_with_the_measured_central_slope_value:_a₌₃_.₆₈_× 10−2_._The_dashed_lines_indicate_the_±₁_σ_band._Only_the_statistical_{uncertainties}_are shown.

a

= (

3

.

68

±

0

.

48

±

0

.

18

)

×

10−2

,

where the uncertainties are statistical due to the limited data andMC sample sizes. The ﬁtgives

χ

2

_/

_ndf

₌

₅₄

_.

₈

_/

_49, _which_has a p-value of 26.4%. The ﬁt result is illustrated in Fig. 3. Using a quadratic function

|

F(

x

)

|

2

=

1

+

2bx

+

cx2_, _the _ﬁt _results _are b

= (

3

.

71

±

0

.

51

)

×

10−2_and_c

_{= (}

₀

_.

₀₀

_±

₀

_.

₁₉

₎

_.

4.3.Systematiceffects

4.3.1. Calibration,resolutionandbeamsimulation

Thespectrometermomentumscalemodifiesproportionallythe x variable.The corrections applied to the momentum calibration haveatypicalrelativesizeoftheorderof10−3_._The_sensitivity_of thefittoaresidualmiscalibrationhasbeenassessedconservatively byturningthecorrectionsoff,leadingtoashiftofthefitresultof

a

= −

0

.

16

×

10−2 _considered _as_the _systematic _uncertainty _on thespectrometercalibration.A similarprocedureisappliedforthe chambermisalignmentcorrectionwithnosigniﬁcanteffectonthe ﬁtresult.

The spectrometer mass resolution has been evaluated sepa-rately for individual data-taking periods usingsamples of K±

→

π

±

π

+

π

− decays. The maximum relative difference observed on theresolutionofthereconstructedsquared3-pionsmassbetween dataandMC is2%. ScalingtheMC resolutionofthex variableby 0.98resultsinashiftof

a

=

0

.

05

×

10−2_,_which_is_considered_as asystematicuncertainty.

The corrections applied to the energies measured in the LKr calorimeteraffecttheTFFsloperesultindirectlythroughthe pho-ton selectionacceptance.A correctionforthenon-linearityinthe energyresponseinthedatasamplewithanalternativefunctionis usedto evaluatethe sensitivitytothecorrection function, result-inginashiftof

a

=

0

.

03

×

10−2.A globalphotonenergyscaling factorof1.001,whichisthetypicalsizeoftheenergycorrections, appliedonlyintheMCsamplecausesashiftof

a

=

0

.

02

×

10−2. The overallsystematicuncertainty duetothe LKrenergy calibra-tion is assigned as the sum of these two effects in quadrature:

a

=

0

.

04

×

10−2.

The beam momentum is simulated according to the central

value measured separately fordifferent data takingperiods from fullyreconstructed K±

→

π

±

π

+

π

− decays.A remaining

discrep-ancy betweendata andMC in the tails ofthe beammomentum

spectrumaffectstheTFFslopemeasurementthroughthe K± mo-mentumdependenceoftheacceptance.Afterapplyingacorrection toimprovethespectrumdata/MCagreement,themeasuredslope shifts by

a

=

0

.

03

×

10−2_, _which_is _considered _as_a _systematic uncertainty.

4.3.2. Triggereﬃciency

The efficiencies ofindividual components ofthe signal trigger chain have been measured using control data samples collected viaalternativetriggerchains.Sincenoinefficienteventshavebeen found,upperlimitsontheinefficienciesat90%CLhavebeen eval-uatedforeachtriggerconditions:0.06%(Q1),0.10%(1-track),0.03% (ELKr)and0.03%(HLT).

Possible systematic effects caused by each trigger condition have been investigated separately by removing potentially ineﬃ-cient events either fromthe data or the MC sample. The Q1

(7)

ef-Table 1

Summaryoftheuncertainties.

Source a×102

Statistical – data 0.48 Statistical – MC 0.18 Total statistical 0.51 Spectrometer momentum scale 0.16 Spectrometer resolution 0.05 LKr calibration 0.04 Beam momentum spectrum simulation 0.03 Calorimeter trigger ineﬃciency 0.06 Accidental background 0.15 Particle misidentiﬁcation 0.06 Neglectedπ0

Dsources 0.01

Higher order radiative contributions <0.01 Total systematic 0.25

ficiencyis modeled by introducing fully inefficientgaps between theHODquadrantswith0.2 mmwidthtunedusingdata/MC com-parison in other decay channels. This leads to a Q1 inefficiency of0.02%forK2πD events.Energeticphotonsmayinitiate showers by interactingwiththe beampipe material, causingthe DCHhit multiplicities to exceed the limitsallowed by the 1-track trigger condition.Thesensitivitytothiseffectistestedbyremovingfrom theMC K2πD sample0.10%ofeventswitharadiativephotonwith an energy above 0.5 GeV traversing the beampipe. Forthe ELKr andHLT triggers, events closest to failinga trigger condition are removed fromthe datasample.Those areeventswiththelowest reconstructedenergyintheLKrcalorimeterfortheELKrcondition, andeventswiththelowestmaximumtrack E

/

p ratiofortheHLT condition.Inbothcasesthefractionofremovedeventsisequalto theupperlimitsonineﬃcienciesquotedabove. Theonlysizeable change inthe TFF slope resulthas been observed by testingthe ELKr trigger condition, resulting in a systematic uncertainty esti-mateof

a

=

0

.

06

×

10−2.

4.3.3. Backgrounds

The effect of accidental background is investigated by releas-ingindependentlythetimingcutsandconstraintsonthenumbers oftracks andvertices in the selection.The number ofadditional eventsincludedintothedatasampleforeachvariationofthe se-lectionislessthan 6

×

103_._The _total_systematic_uncertainty _due toaccidentalsisevaluatedtobe

a

=

0

.

15

×

10−2.

Themisidentificationofchargedparticlesisstudiedbya mod-ificationoftheselectioncriteria.Thepionmassisassignedtothe trackwiththechargeoppositetoqvtxinthekinematicevent iden-tificationtoselect K±

→

e±e±

π

∓

γ

candidates.Sincethisprocess violatesleptonnumberconservation,all eventspassing this“LNV selection” are considered to be events with misidentiﬁed tracks. A totalof188eventsfromthefulldatasetpasstheLNVselection. Usingthesame selectionontheMC samples,itisestimatedthat mostofthoseeventsaregenuine K2πD decayswithmisidentiﬁed

π

± ande∓tracks,while

(

42

±

18

)

dataeventsarenot accounted for.Thex distributionoftheseeventsisaddedtothereconstructed K2πD MCone.TheTFFslopeshiftsby

a

=

0

.

06

×

10−2,whichis consideredasasystematicuncertainty.

Removingthe Kμ3D andKe3D MC samplesfromtheﬁt proce-dure results in a shift ofthe slope of

a

=

0

.

01

×

10−2_. _This _is consideredasanestimateofthesystematicuncertaintyontheTFF slopeduetotheneglected

π

0_sources_as_the_other_neglected_kaon decaymodesproducingneutralpionsaccount forlessthan 4% of

π

0 _production.

The acceptance of the K2πD selection for the K2π decay

fol-lowed by

π

0

_→

_{γ γ}

_is _estimated _with _MC _simulations _to _be

Fig. 4. Comparisonoftheπ0_TFF_slope_measurements_in_the_time-like_momentum transferregion[11–15].

smaller than10−7,conﬁrming thatthe minimaldistance

require-ment between tracks in the ﬁrst DCH eﬃciently removes the

events withphoton conversion.The reduction ofdetector accep-tance by the Pb bar (Section 2) doesnot lead to anysystematic uncertainties since eventswithaparticle withinthe leadbar ac-ceptancearediscarded.

5. Result

The statistical and systematic uncertainties discussed in the previous sections are summarised in Table 1. The result of the measurementofthe

π

0 _TFF_slope_parameter_is

a

=

3

.

68

±

0

.

51stat

±

0

.

25syst

×

10−2

= (

3

.

68

±

0

.

57

)

×

10−2

,

which is in good agreement with the theoretical predictions

[5–10].A comparison withprevious

π

0

D measurements is shown

inFig. 4.

6. Conclusions

The slope ofthe electromagnetictransition formfactorof the

π

0 _is _measured _from _a _sample _of ₁

_.

₁₁

_×

₁₀6

_π

0 _Dalitz _decays. The resulta

= (

3

.

68

±

0

.

57

)

×

10−2 _represents _the _most _precise measurementoftheformfactorslopeinthetime-likemomentum region.The15%relativeuncertaintyrepresentsanimprovementby afactorof2withrespecttothepreviousbestmeasurement[15].

Acknowledgements

We expressour gratitude to the staff of the CERN laboratory and the technical staff of the participating laboratories and uni-versities for their efforts in the operation of the SPS accelerator, theexperimentanddataprocessing.WethankT.Husekforfruitful discussions andcollaborationonthe

π

0

D decaygenerator develop-ment.

(8)

References

[1]C.Patrignani,etal.,Chin.Phys.C40(2016)100001. [2]T.Husek,K.Kampf,J.Novotný,Eur.Phys.J.C74(2014)3010. [3]A.Nyffeler,Phys.Rev.D94(2016)053006.

[4]D.W.Joseph,NuovoCimento16(1960)997.

[5]M.Gell-Mann,F.Zachariasen,Phys.Rev.124(1961)953. [6]P.Lichard,Phys.Rev.D83(2011)037503.

[7]K.Kampf,M.Knecht,J.Novotný,Eur.Phys.J.C46(2006)191. [8]P.Masjuan,Phys.Rev.D86(2012)094021.

[9]M.Hoferichter,B.Kubis,S.Leupold,F.Niecknig,S.P.Schneider,Eur.Phys.J.C 74(2014)3180.

[10]T.Husek,S.Leupold,Eur.Phys.J.C75(2015)586.

[11]J.Fischer,etal.,Phys.Lett.B73(1978)359. [12]H.Fonvieille,etal.,Phys.Lett.B233(1989)65. [13]F.Farzanpay,etal.,Phys.Lett.B278(1992)413. [14]R.MeijerDrees,etal.,Phys.Rev.D45(1992)1439. [15]P.Adlarson,etal.,arXiv:1611.04739[hep-ex],2016. [16]H.J.Behrend,etal.,Z.Phys.C49(1991)401. [17]C.Lazzeroni,etal.,Phys.Lett.B719(2013)326. [18]V.Fanti,etal.,Nucl.Instrum.MethodsA574(2007)433. [19]J.R.Batley,etal.,Eur.Phys.J.C52(2007)875. [20]B.E.Lautrup,J.Smith,Phys.Rev.D3(1971)1122. [21]K.O.Mikaelian,J.Smith,Phys.Rev.D5(1972)1763. [22]T.Husek,K.Kampf,J.Novotný,Phys.Rev.D92(2015)054027.