Measurement of the running of the fine structure constant below 1 GeV with the KLOE detector

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Measurement of the running of the fine structure constant below 1 GeV with the KLOE detector

The KLOE-2 Collaboration

A. Anastasi

^e,d

, D. Babusci

^d

, G. Bencivenni

^d

, M. Berlowski

^u

, C. Bloise

^d

, F. Bossi

^d

,

P. Branchini

^r

, A. Budano

^q,r

, L. Caldeira Balkeståhl

^t

, B. Cao

^t

, F. Ceradini

^q,r

, P. Ciambrone

^d

, F. Curciarello

^e,b,l

, E. Czerwi ´nski

^c

, G. D’Agostini

^m,n

, E. Dané

^d

, V. De Leo

^r,∗

, E. De Lucia

^d

, A. De Santis

^d

, P. De Simone

^d

, A. Di Cicco

^q,r

, A. Di Domenico

^m,n

, R. Di Salvo

^p

,

D. Domenici

^d

, A. D’Uffizi

^d

, A. Fantini

^o,p

, G. Felici

^d

, S. Fiore

^s,n

, A. Gajos

^c

, P. Gauzzi

^m,n

, G. Giardina

^e,b

, S. Giovannella

^d

, E. Graziani

^r

, F. Happacher

^d

, L. Heijkenskjöld

^t

,

T. Johansson

^t

, D. Kami ´nska

^c

, W. Krzemien

^u

, A. Kupsc

^t

, S. Loffredo

^q,r

, P.A. Lukin

^v,w

, G. Mandaglio

^f,g

, M. Martini

^d,k

, M. Mascolo

^d

, R. Messi

^o,p

, S. Miscetti

^d

, G. Morello

^d

, D. Moricciani

^p

, P. Moskal

^c

, M. Papenbrock

^t

, A. Passeri

^r

, V. Patera

^j,n

, E. Perez del Rio

^d

, A. Ranieri

^a

, P. Santangelo

^d

, I. Sarra

^d

, M. Schioppa

^h,i

, A. Selce

^q

, M. Silarski

^d

, F. Sirghi

^d

, L. Tortora

^r

, G. Venanzoni

^d,∗

, W. Wi´slicki

^u

, M. Wolke

^t

aINFNSezionediBari,Bari,Italy bINFNSezionediCatania,Catania,Italy

cInstituteofPhysics,JagiellonianUniversity,Cracow,Poland dLaboratoriNazionalidiFrascatidell’INFN,Frascati,Italy

eDipartimentodiScienzeMatematicheeInformatiche,ScienzeFisicheeScienzedellaTerradell’UniversitàdiMessina,Messina,Italy fDipartimentodiScienzeChimiche,Biologiche,FarmaceuticheedAmbientalidell’UniversitàdiMessina,Messina,Italy

gINFNGruppocollegatodiMessina,Messina,Italy

hDipartimentodiFisicadell’UniversitàdellaCalabria,Rende,Italy iINFNGruppocollegatodiCosenza,Rende,Italy

jDipartimentodiScienzediBaseedApplicateperl’Ingegneriadell’Università“Sapienza”,Roma,Italy kDipartimentodiScienzeeTecnologieapplicate,Università“GuglielmoMarconi”,Roma,Italy lNovosibirskStateUniversity,630090Novosibirsk,Russia

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

tDepartmentofPhysicsandAstronomy,UppsalaUniversity,Uppsala,Sweden uNationalCentreforNuclearResearch,Warsaw,Poland

vBudkerInstituteofNuclearPhysics,Novosibirsk,630090,Russia wNovosibirskStateUniversity,Novosibirsk,630090,Russia

F. Jegerlehner

^x,y

xInstituteofPhysics, Humboldt-UniversityofBerlin,Berlin,Germany

yDeutschesElektronen-Synchrotron(DESY),Platanenallee6,D-15738Zeuthen,Germany

*

Correspondingauthors.

E-mailaddresses:veronica.deleo@roma3.infn.it(V. De Leo),graziano.venanzoni@lnf.infn.it(G. Venanzoni).

http://dx.doi.org/10.1016/j.physletb.2016.12.016

0370-2693/©^{2016TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

1. Introduction

PrecisiontestsoftheStandardModel(SM)requireanappropri- ateinclusion of higher-order effectsand the very preciseknowl- edge of input parameters [1].One of the basic input parameters is the effective QED coupling constant

α

, determined from the anomalousmagneticmoment oftheelectronwiththeimpressive accuracyof0.37partsperbillion[2].However,physicsatnon-zero momentumtransferrequiresaneffectiveelectromagneticcoupling

α

(s).¹ The shift of the fine-structure constant from the Thom- son limit to high energy involves low energy non-perturbative hadronic effects which affect the precision. These effects repre- sentthelargestuncertainty(andthemainlimitation)fortheelec- troweak precision testsas the determination ofsin²θW at the Z poleortheSMpredictionofthemuong−^2[3].

The QED coupling constant is predicted and observed [4,5]

to increase with rising momentum transfer (differently fromthe strongcouplingconstant

α

S whichdecreaseswithrisingmomen- tumtransfer),whichcanbeunderstoodasaresultofthescreening ofthebarecharge causedby thepolarizedcloudofvirtual parti- cles. The vacuum polarization (VP) effects can be absorbed in a redefinitionofthefine-structureconstant,makingits dependent:

α (

s

) = α (

0

)

1

−  α (

s

) .

(1)

The shift

α

(s) interms ofthe vacuumpolarization function ^_γ(s)isgivenby:

 α (

s

) = −

⁴

π α (

0

)

Re

[

^_γ

(

s

) −

^_γ

(

0

)]

⁽²⁾

anditisthesumofthelepton(e,

μ

,

τ

) contributions,thecontri- butionfromthefivequark flavours(u, d, s, c, b), andthe contribu- tion of the top quark (which can be neglected at low energies):



α

(s)= 

α

lep(s)+ 

α

_had⁽⁵⁾(s)+ 

α

top(s)[1].

Theleptoniccontributionscanbecalculatedwithveryhighpre- cision in QED using the perturbation theory [6,7]. However, due tothenon-perturbativebehaviourofthestronginteractionatlow energies, perturbative QCD only allows us to calculate the high energytailofthehadronic(quark) contributions.Intheloweren- ergy region the hadronic contribution can be evaluated through a dispersion integral over the measured e⁺e⁻→ ^{hadrons cross-} section:

 α

_had

(

s

) = −  α (

0

)

s 3

π



Re



∞

m²π

ds^ R_had

(

s^

)

s^

(

s^

−

s

−

i

 ) ,

(3)

1 Inthefollowingwewillindicatewiths themomentumtransfersquaredofthe reaction.

where Rhad(s) is defined as the cross section ratio Rhad(s)= σ(e⁺e⁻→γ∗→^hadrons)

σ(e⁺e⁻→γ∗→μ⁺μ⁻).

Inthisapproachthedominantuncertaintyintheevaluationof



α

isgivenbytheexperimentaldataaccuracy.

In the Eq. (2)Im

α

relatedto the imaginary part ofthe VP function ^_{γ is}completelyneglected,whichisagoodapproxima- tioninthecontinuumasthecontributionsfromtheimaginarypart aresuppressed.However,thisapproximationisnotsufficientinthe presence of resonances likethe

ρ

meson, where theaccuracy of thecrosssectionmeasurementsreachestheorderof(orevenless than)1%,andtheimaginarypartshouldbetakenintoaccount.

Inthispaperwepresentameasurementoftherunningofthe effective QED coupling constant

α

in the time-like region 0.6<

√s<0.975 GeV.Thestrengthofthecouplingconstantismeasured asafunctionofthemomentum transferoftheexchangedphoton

√s=Mμμ where Mμμ isthe

μ

⁺

μ

⁻ invariant mass. Thevalueof

α

(s)isextractedfromtheratioofthedifferentialcrosssectionfor the process e⁺e⁻→

μ

⁺

μ

⁻

γ

(

γ

) withthe photon emittedin the InitialState(ISR)tothecorrespondingcrosssectionobtainedfrom MonteCarlo(MC)simulationwiththecouplingsettotheconstant value

α

(s)=

α

(0):

| α (

s

)

α (

0

) |

²

=

^d

σ

data

(

e⁺e⁻

→ μ

⁺

μ

⁻

γ ( γ ))|

I S R

/

d

√

s d

σ

_MC⁰

(

e⁺e⁻

→ μ

⁺

μ

⁻

γ ( γ ))|

I S R

/

d

√

s (4)

ToobtaintheISRcrosssection,theobservedcrosssectionmustbe corrected forevents withone ormore photonsin thefinal state (FSR).ThishasbeendonebyusingthePHOKHARAMCeventgen- erator,whichincludesnext-to-leading-orderISRandFSRcontribu- tions [8]. Inthe following we onlyuse eventswhere the photon is emitted atsmallangles, whichresults ina large enhancement of the ISR with respect to the FSR contribution. From the mea- surement ofthe effectivecoupling constant andthe dipion cross section [9],we extractedforthe firsttimeina singleexperiment therealandimaginarypartof

α

.

The analysis has been performedby using the data collected with the KLOE detector at DANE [10], the e⁺e⁻ collider run- ning atthe φ meson mass,with a totalintegrated luminosity of 1.7 fb⁻¹.

2. TheKLOEdetector

The KLOEdetectorconsistsofa cylindricaldriftchamber (DC) [11] andanelectromagneticcalorimeter(EMC)[12].TheDChasa momentumresolutionof

σ

p_⊥/p_⊥∼⁰.4% fortrackswithpolaran- gleθ >45^◦.TrackpointsaremeasuredintheDCwitharesolution inr− φ^of∼⁰.15 mm and∼^{2 mm inz.TheEMChasanenergy} resolution of

σ

E/E∼⁵.7%/√

E (GeV) andan excellent time reso- lutionof

σ

t∼^{54 ps}/√

E (GeV)⊕^{100 ps.}Calorimeter clustersare reconstructedgroupingtogetherenergydepositscloseinspaceand

Fig. 1. Detectorsectionwith theacceptance regionforthe chargedtracks(wide cones)andforthephoton(narrowcones).

time. A superconducting coil provides an axial magnetic field of 0.52Talongthebisectorofthecollidingbeamdirections.Thebi- sectoristakenasthezaxisofourcoordinatesystem.Thexaxisis horizontalandtheyaxisisvertical,directedupwards.Acrosssec- tionofthedetectorinthey,zplaneisshowninFig. 1.Thetrigger uses both EMC andDC information. Events used in thisanalysis aretriggered by two energy depositslarger than 50MeV in two sectorsofthebarrelcalorimeter.

2.1.Eventselection

A photon and two tracks of opposite curvature are required toidentifya

μμγ

event.Events areselectedwitha(undetected) photonemittedatsmallangle(SA),i.e. withinaconeofθγ<15^◦ aroundthebeamline(narrowconesinFig. 1)andthetwocharged muons are emitted at large polar angle, 50^◦< θ_μ<130^◦. High statisticsfortheISRsignalandsignificantreductionofbackground events as φ→

π

⁺

π

⁻

π

⁰ in which the

π

⁰ mimics the missing momentumof thephoton(s) andfrom theFSRradiation process, e⁺e⁻→

μ

⁺

μ

⁻

γ

F S R, are guaranteed by this selection. However, this requirement results in a kinematical suppression of events with √

s<0.6 GeV, since a highly energetic photon emitted at smallangleforcesthemuonsalsotobeatsmallangles(andthus outsidetheacceptance).

Toavoidspirallingtracksinthedriftchamber,thereconstructed momentamusthavepT>160MeVor|^pz|>90MeV.Thisensures goodreconstructionandefficiency.

Themainbackgroundreactionsaregivenby:

• ^e+e−→

π

⁺

π

⁻

γ

(

γ

)

• ^e+e−→

π

⁺

π

⁻

π

⁰

• ^e+e−→^e+e−

γ

(

γ

).

AparticleIDestimator(PID)basedonapseudo-likelihoodfunc- tion (L_±) using time-of-flight and calorimeter information (size andshapeoftheenergydeposit)isusedtoobtainseparationbe- tweenelectronsandpionsormuons.Eventswithbothtrackssat- isfyingL_±<0 arerejectedase⁺e⁻

γ

.Toseparatethemuonsfrom thepionsweappliedmainlytwocuts:thefirstonthetrackmass (M_{T R K})variableandthesecondonthe

σ

M T R K,theestimatederror

Fig. 2.π π γ andμμγ MT R Kdistributions.Theverticallineshowstheμμγselec- tioncut(MT R K<115 MeV).TheeffectoftheσM_{T R K} cutonthetwodistributionsis clearlyvisible.

on M_{T R K}.Assumingthepresenceofonlyone unobservedphoton andthatthetracksbelongtoparticlesofthesamemass,MT R K is computedfromenergy andmomentumconservation.The

σ

M T R K variableisconstructedeventbyeventwiththeerrormatrixofthe fitted tracks at the point of closest approach (PCA) [13]. Cutting the highvalues of thisvariable the bad reconstructed tracks are rejected allowing a reduction of the

π π γ

events contamination (showninFig. 2).

ResidualbackgroundisevaluatedbyfittingtheobservedMT R K spectrumwithasuperpositionofMCsimulationdistributionsde- scribing signal and

π

⁺

π

⁻

γ

,

π

⁺

π

⁻

π

⁰ and e⁺e⁻

γ

events. The normalizationfactorsfromsignalandbackgroundsarefreeparam- etersofthefit,performedfor30intervalsins of0.02 GeV² width for0.35<s<0.95 GeV².Additionalbackgroundfromthee⁺e⁻→ e⁺e⁻

μ

⁺

μ

⁻ process has been evaluated using the NEXTCALIBUR MC generator [14]. The maximum contribution is 0.7% at √

s= 0.6 GeV. The uncertainty on this background has been taken as 50% of the total contribution and added to the systematic error.

The contribution from e⁺e⁻→^e+e−

π

⁺

π

⁻ has been evaluated withtheEKHARAgenerator[15]andfoundtobenegligible.

Thetotalfractional systematicuncertaintyonbackgroundsub- traction,obtainedbyaddinginquadraturetheuncertaintiesonthe fit normalization parameters and the e⁺e⁻

μ

⁺

μ

⁻ residual back- ground,rangesfrom0.2%to0.05%decreasingwiths.

About4.5·¹⁰⁶

μμγ

eventspasstheseselectioncriteria.

3. Measurementofthe

μμγ

crosssection

The experimental ISR

μ

⁺

μ

⁻

γ

cross section is obtained from theobservednumberofevents(Nobs)andthebackgroundestimate (N_bckg)as:

d

σ (

e⁺e⁻

→ μ

⁺

μ

⁻

γ ( γ ))

d

√

s

 

I S R

=

^Nobs

−

^Nbkg

 √

s

· (

1

− δ

F S R

)  ( √

s

) ·

^L

,

(5) where (1− δF S R) is the correction applied to remove the FSR contribution (which increases with the energy from 0.998 at 0.605 GeV to1.032 at0.975GeV),



isthe efficiency(seesection below)andL istheintegratedluminosity.

We firstly compare the

μ

⁺

μ

⁻

γ

cross-section with only ISR withthecorrespondingNLOQEDcalculationfromPHOKHARAgen- eratorincludingtheVPeffects.

IntheupperplotofFig. 3themeasured

μ

⁺

μ

⁻

γ

cross-section as a function of √

s for both experimental (red points) and MC (bluepoints)dataisshown.Theagreementbetweenthetwocross sectionsisexcellent.Thesamefigureshowsaninteresting feature around 0.78 GeV (corresponding to the mass of the

ω

meson),

Fig. 3. Upperplot:comparisonofthemeasureddifferentialcrosssection(redpoints) andPHOKHARAMCprediction(bluepoints)ofthe μ⁺μ⁻γ crosssection.Lower plot:theratioofthetwo.Thegreenbandshowsthesystematicerror.(Forinterpre- tationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothe webversionofthisarticle.)

where a small step appears in the cross section. This step be- haviourisduetothe

ρ

−

ω

interferenceinthephotonpropagator, asitwillbeshowninthefollowing.Inthelowerplotthedatato MCratioisshowntogetherwiththesystematicerror(greenband) oftheorderof1%.

4. Efficienciesandsystematicerrors

Theglobalefficiency,whichrangesfrom0.086at0.605 GeVto 0.27at0.975 GeV,hasbeenobtainedfroma

μ

⁺

μ

⁻

γ

eventsgen- eration with PHOKHARA interfaced with the detector simulation codeGEANFI[16].It includescontributions fromtrigger, tracking, PID,

σ

M T R K,MT R K andacceptance.

Trigger:thetriggerefficiencyhasbeenobtainedfromasample of

μ

⁺

μ

⁻

γ

events where a single muon satisfies the trigger re- quirement. Triggerresponse fortheother muon isparameterized asafunction ofits momentum anddirection.The efficiencyasa functionofs isobtainedusingtheMC eventdistributionanddif- fersfromonebylessthan10⁻⁴,withnegligiblesystematicerror.

Tracking: the single muon track efficiencyhas been obtained asafunctionoftheparticlemomentumandpolaranglebymeans ofa highpurity

μ

⁺

μ

⁻

γ

sample obtainedby usingone muonto tag the presence of the other. The combined efficiency is about

in s)whichwetakeassystematicerror.

•^{The systematic} uncertainty on

σ

MTRK cut has beenevaluated asthe maximumdifferencebetweenthe

μμγ

normalization parametersofthebackgroundfittingprocedure,obtainedwith standard cuts, and thoseobtained by shifting

σ

MTRK by ±^2%.

Thecontributionislessthan1%inthewholeenergyrange.

•^{Systematic effects due to polar angle} requirements for the muonsand forthephoton, are estimatedby varying the an- gularacceptanceby±^{1◦ (morethantwotimestheresolution} onthepolarangle)aroundthenominalvalue.Theuncertainty rangesfrom0.1to0.6%.

Softwaretrigger:Athird-level triggerisimplementedto keep the physics eventswhichare misidentified ascosmic rays. Its ef- ficiencyfor

μμγ

events,evaluatedfromanunbiaseddownscaled sample,isconsistentwithonewithin10⁻³ whichistakenassys- tematicerror.

Table 1givesthesystematicerrorsatthe

ρ

-peakmassvalue.

5. Luminosityandradiativecorrections

Large angleBhabha scatteringis used to determine the lumi- nosity,withareferencecrosssectionobtainedwithBabayaga@NLO MC eventgenerator[17],convolvedwithdetectorandbeamcon- ditions[18].Twosourcescontribute tothesystematicuncertainty intheevaluationoftheluminosity:

•^the theoreticalaccuracy ofBabayaga@NLO,quoted as0.1%by theauthors;

•^{the systematic error associated to the counting of Bhabha} eventswhichis0.3%[18]

When extractingtherunningof

α

(see followingSection),the dependenceoftheBhabhacrosssectionontheVPeffectmustbe taken into account. By switching off the hadronic corrections to theVP,wecheckedthatthepresenceofthehadroniccontribution to 

α

forboth s andt channelsinthecrosssection givesa 0.2%

contributionwhichweconsiderasasystematicerrorofourmea- surement(

α

haddep. inTable 1).

TheuncertaintyonPHOKHARAMCgenerator(Rad.functionH in Table 1) is0.5% constant in s, mostly dueto missingISR higher- order terms[8]. Theuncertaintyinthe procedureto subtractthe FSRcontributionis0.2%,mostlyduetomissingFSRdiagrams[19].

6. Measurementoftherunningof

α

We use Eq.(4)and Eq.(5) inthe angularregion θ_γ <15^◦ to extract the running of the effectiveQED coupling constant

α

(s). By setting in the MC the electromagnetic coupling to the con- stant value

α

(s)=

α

(0), thehadronic contributionto the photon propagator,withits characteristic

ρ

−

ω

interferencestructure,is clearly visible in the data to MC ratio, as shown in Fig. 4. The

Table 1

Listofsystematicerrors.

Source σμμγ |α(s)/α(0)|²

Trigger <0.1%

Tracking s dep. (0.5% atρ-peak) Particle ID <0.1%

Background subtraction s dep. (0.1% atρ-peak)

MT R K 0.4%

σM T R K s dep. (<0.1% atρ-peak)

Acceptance s dep. (0.3% atρ-peak) Software Trigger 0.1%

Luminosity 0.3%

αhaddep. (normalization) – 0.2%

FSR treatment 0.2%

Rad. function H – 0.5%

Total systematic error s dep. (0.8% atρ-peak) (1% atρ-peak)

Fig. 4. Thesquareofthemodulusoftherunningα(s)inunitsofα(0)compared withtheprediction(providedbythealphaQEDpackage[20])asafunctionofthe dimuoninvariantmass.TheredpointsaretheKLOEdatawithstatisticalerrors;

thevioletpointsarethetheoreticalpredictionforafixedcoupling(α(s)=α(0));

theyellowpointsarethepredictionwithonlyvirtualleptonpairscontributingto theshiftα(s)= α(s)lep,andfinallythepointswiththesolidlinearethefull QEDpredictionwithbothleptonandquarkpairscontributingtotheshiftα(s)=

α(s)lep+had.(Forinterpretationofthereferencestocolourinthisfigurelegend, thereaderisreferredtothewebversionofthisarticle.)

prediction fromRef. [20] is also shown. While the leptonic part is obtained by perturbation theory, the hadronic contribution to

α

(s) is obtained via an evaluation in terms of a weighted aver- age compilation of Rhad(s), based on the available experimental e⁺e⁻→hadrons annihilationdata(foran up todatecompilation see[21]andreferencestherein).

Forcomparison,thepredictionwithconstantcoupling(norun- ning) and with only lepton pairs contributing to the running of

α

(s)isgiven.

The value of |

α

(s)/

α

(0)|^{2 with the} statistical and systematic uncertaintyis reported inTable 2. As can be seen, the total un- certaintyisatthe1%level.

Inordertoevaluate thestatisticalsignificanceofthehadronic contributiontotherunningof

α

(s),a

χ

² basedstatisticaltestfor twodifferentrunninghypotheses:(a)norunning;(b)running due toleptonpairsonlyisperformed.

Byincludingstatisticalandsystematicserrors, weexclude the only-leptonic hypothesis at 6

σ

which is the strongest direct ev- idence ever achieved by a single experiment. Our result is also consistentwiththeestimateof

α

(s)ofRef.[22]witha

χ

² prob- abilityof0.3(

χ

²/ndf=⁴¹.2/37).

Fig. 5. ImαextractedfromtheKLOEdatacomparedwiththevaluesprovidedby alphaQEDroutine(withouttheKLOEdata)forImα=Imαlep(yellowpoints) andImα=^Imαlep+hadonlyforπ π channels(bluesolidline).(Forinterpreta- tionofthereferencestocolourinthisfigurelegend,thereaderisreferredtothe webversionofthisarticle.)

Similar results are obtained using different

α

(s) predictions inRef.[22,23].

7. Extractionofrealandimaginarypartof

α

(s)

Inthe contributionto therunningof

α

,theimaginary partis usuallyneglected.Thisisagoodapproximationasthecontribution fromthe imaginary partof 

α

enters atorder O(

α

²) compared to O(

α

) for the real part, and is suppressed [26]. However, the imaginary part should be taken into account in the presence of resonanceslikethe

ρ

meson, wherethecrosssectionismeasured withanaccuracybetterthan1%.

By usingthe definition of therunning of

α

(Eq. (1)) the real partoftheshift

α

(s)canbeexpressedintermsofitsimaginary partand|

α

(s)/

α

(0)|^2:

Re

 α =

¹

− 

| α (

0

)/ α (

s

)|

²

− (

^Im

 α )

²

.

(6)

The imaginary part of 

α

(s) can be related to the total cross section

σ

(e⁺e⁻→

γ

^∗ _→anything), where the precise relation reads[3,24,25]:Im

α

= −^α₃^R(s),withR(s)=

σ

tot/^4π|α_3s^(s)|2. R(s) takes into account leptonic and hadronic contribution R(s)= Rlep(s)+^Rhad(s),wheretheleptonicpartcorresponds tothepro- ductionofalepton pairatlowest ordertakingintoaccount mass effects:

R_lep

(

s

) =



1

−

^4m

2 l

s



1

+

^2m

2 l

s

, (

l

=

^e

, μ , τ ).

(7)

Intheenergyregionaroundthe

ρ

-mesonwecanapproximate thehadroniccrosssectionbythe2

π

dominantcontribution:

R_had

(

s

) =

¹ 4

1

−

^4m2π s



³₂

|

^F_π⁰

(

s

) |

²

,

(8)

where Fπ⁰ is the pion form factor deconvolved: |^F_π⁰(s)|² =

|Fπ(s)|²^α_α⁽₍⁰_s)⁾^2.

The resultsobtainedforthe2

π

contributiontothe imaginary part of

α

(s) by using the KLOE pionform factormeasurement [9],are shown in Fig. 5and compared withthe values givenby the R_had(s) compilation of Ref. [20] using only the 2

π

channel,

0.705 1.004±0.007±0.010 2.1±3.4±5.1 −16.89±0.23±0.07 0.715 1.005±0.007±0.010 2.7±3.3±5 −19.46±0.26±0.09 0.725 1.017±⁰.007±⁰.010 8.6±³.3±⁴.9 −²⁰.54±⁰.28±⁰.11 0.735 1.018±⁰.007±⁰.010 9.3±³.3±⁵.1 −²³.04±⁰.33±⁰.11 0.745 1.009±⁰.006±⁰.010 4.8±³.2±⁴.7 −²⁴.15±⁰.34±⁰.23 0.755 1.006±⁰.006±⁰.010 3.3±³.1±⁵.1 −²⁵.76±⁰.37±⁰.25 0.765 1.013±0.006±0.010 6.5±3.1±5.1 −25.89±0.37±0.25 0.775 1.018±0.006±0.010 9.2±3.0±4.8 −25.93±0.36±0.51 0.785 1.065±0.007±0.011 31.1±3.0±4.9 −21.36±0.27±0.69 0.795 1.066±⁰.007±⁰.011 31.8±³.0±⁵.0 −¹⁹.49±⁰.22±⁰.16 0.805 1.049±⁰.006±⁰.011 23.8±².8±⁵.0 −¹⁸.69±⁰.22±⁰.17 0.815 1.050±⁰.006±⁰.011 24.2±².8±⁵.1 −¹⁶.2±⁰.17±⁰.12 0.825 1.049±⁰.006±⁰.011 23.6±².7±⁵.0 −¹⁴.79±⁰.15±⁰.11 0.835 1.057±0.006±0.011 27.2±2.7±5.0 −13.93±0.13±0.1 0.845 1.044±0.006±0.011 21.3±2.6±5.0 −12.49±0.11±0.08 0.855 1.044±0.005±0.011 21.4±2.5±4.9 −11.65±0.09±0.06 0.865 1.041±⁰.005±⁰.011 20.2±².5±⁵.0 −¹¹.25±⁰.09±⁰.05 0.875 1.044±⁰.005±⁰.011 21.6±².4±⁵.1 −¹⁰.16±⁰.07±⁰.04 0.885 1.050±⁰.005±⁰.011 24.4±².3±⁴.9 −⁹.53±⁰.06±⁰.03 0.895 1.048±⁰.005±⁰.011 23.1±².2±⁵ −⁹.03±⁰.05±⁰.03 0.905 1.045±0.004±0.011 22.0±2.1±5.1 −8.81±0.05±0.02 0.915 1.035±0.004±0.011 17.2±1.9±5.3 −8.35±0.04±0.02 0.925 1.046±⁰.004±⁰.011 22.3±¹.8±⁵.1 −⁷.89±⁰.03±⁰.02 0.935 1.035±⁰.003±⁰.011 17.0±¹.7±⁵.1 −⁷.62±⁰.03±⁰.01 0.945 1.038±⁰.003±⁰.010 18.3±¹.5±⁴.8 −⁷.33±⁰.02±⁰.01 0.955 1.039±⁰.003±⁰.010 18.8±¹.4±⁴.8 −⁷.13±⁰.02±⁰.01 0.965 1.029±0.003±0.010 14.2±1.3±4.8 −6.94±0.02±0.01 0.975 1.030±0.002±0.010 14.6±1.1±4.7 −6.82±0.02±0.01

Fig. 6. Reαextractedfromtheexperimentaldatawithonlythestatisticalerror includedcomparedwiththealphaQEDprediction(withouttheKLOEdata)when Reα=Reαlep (yellowpoints)andReα=Reαlep+had (bluesolidline).(For interpretationofthereferencestocolourinthisfigurelegend,thereaderisreferred tothewebversionofthisarticle.)

withtheKLOEdataremoved (toavoidcorrelations).Table 2gives the 2

π

contribution to Im

α

(s) with statistical and systematic errors.

The extraction of the Re

α

has been performed using the Eq.(6)anditisshowninFig. 6.Theexperimental datawithonly the statistical error included have been compared with the al- phaQEDpredictionwhen Re

α

₌Re

α

lep (yellowpoints inthe colourfigure)andRe

α

=^Re

α

lep+^had(dotswithsolidline).The Re

α

(s)valueswithstatisticalandsystematicerrorsaregivenin Table 2.Thesystematicerrorsincludethemissinghadroniccontri- butions (3

π

,4

π

,...)which werenot includedintheevaluationof Im

α

(s).Ascanbeseen,anexcellentagreementforRe

α

(s)has beenobtainedwiththedata-basedcompilation.

8. FitofRe

α

andextractionofB R(

ω

→

μ

⁺

μ

⁻)B R(

ω

→^e+e−)

We fitRe

α

bya sumoftheleptonic andhadroniccontribu- tions, where the hadroniccontribution isparametrized asa sum ofthe

ρ

(770),

ω

(782)andφ (1020) resonancecomponentsanda non-resonant term. We usea Breit–Wigner descriptionforthe

ω

andφresonances[3,26,27]:

Re

 α

_V=ω,φ

₌

³



BR

(

V

→

^e+e−

) ·

^BR

(

V

→ μ

⁺

μ

⁻

) α

M_V

×

^s

(

s

−

^M2_V

)

V

(

s

−

^M2_V

)

²

+

^s



_V²

,

(9) whereMV andV arethemassandthetotalwidthofthemesons V =

ω

andφ.Forthe

ρ

weusea Gounaris–Sakuraiparametriza-

Table 3

ResultsfromthefitofReαcomparedwiththeworldaveragevalues(PDG[31]).Second(third)column:without(with) theρ−ωinterference.Onlystatisticalerrorsarereportedforthefitvalues.

Parameter Result from the fit Result from the fit withρ−ωinterf. PDG

Mρ,MeV 775±6 778±7 775.26±0.25

ρ,MeV 146±9 147±10 147±0.9

M_ω,MeV 782.7±¹.1 783.4±⁰.8 782.65±⁰.12

B R(ω→μ⁺μ⁻)B R(ω→e⁺e⁻) (4.3±1.8)·10⁻⁹ (4.4±1.8)·10⁻⁹ (6.5±2.3)·10⁻⁹

χ²/ndf 1.19 1.15 –

Fig. 7. Fit of the Reαdata. Only statistical errors are shown.

tionB Wρ^{G S}(s)[28,29]ofthepionformfactor,whereweneglectthe interferencewiththe

ω

andthehigherexcitedstatesofthe

ρ

:

F_π

(

s

) =

^{B W}_ρ^{G S}₍s)

=

^M

2ρ

(

1

+

^d



_ρ

/

M_ρ

)

M_ρ²

−

^s

+

^f

(

s

) −

^iMρ



_ρ

(

s

)

⁽¹⁰⁾

Theterms d and f(s) are described inRef. [29]. As itwill be shown in the following, this approximation turns out to be ap- propriategiventhelimitedstatisticsofthedata.Inparticular,the inclusionoftheenergydependenceonthetotalwidthsof

ω

and φresonances[30]givesnegligiblecontributions.Thenon-resonant termhasbeenparametrizedasafirst-orderpolynomial p0+^p1√

s.

The following parameters have been fixed to the PDG val- ues[31]:_ω= (⁸.49±⁰.08)MeV, Mφ= (¹⁰¹⁹.461±⁰.019)MeV,

φ= (⁴.266±⁰.031)MeV,and B R(φ→^e+e−)B R(φ→

μ

⁺

μ

⁻)= (8.5⁺₋⁰₀^._.⁵₆)·^10−8.

ResultsofthefitareshowninFig. 7andcomparedinTable 3 (second column) with the corresponding values from PDG [31].

Onlystatisticalerrorsarereported.

The parameters of thenon-resonant term are consistent with zero within the statistical uncertainties: p₀= (².4±⁴.5)·^10−3, p1= (−².8±⁵.3)·^10−3.The

χ

²/ndf ofthefitis36.85/31=¹.19.

Tostudytheeffectofthe

ρ

−

ω

interferenceinestimating

α

, an additional term δ ^s

M²_ωB W ω(s)B Wρ^{G S} has been included in the fit.Results are shown in the third column of Table 3 where we fix |δ|=¹.45·^{10−3 and arg} δ=¹⁰.2^◦ [32]. As it can be seen, results withthe interference term are well within the statistical uncertainties,andinthefollowingwewillusetheresultswithout theinterferenceterm.

Byincludingthesystematicerrors(takingalsointoaccountthe correlationsofthe systematic uncertainties onthe parameters of thefit,andtheuncertaintyofthePDGvaluesforfixedparameters) theproductofthebranchingfractionsreads:

B R

( ω _→ μ

⁺

μ

⁻

)

B R

( ω _→

e⁺e⁻

) = (

⁴

.

3

±

¹

.

8

±

²

.

2

) ·

¹⁰⁻⁹

,

(11)

where the first error is statisticaland the second systematic. By multiplyingbythephasespacefactorξ=

1+^2m_m^2μ2 ω

 1−^4m_m^2μ2

ω

1/2 and assuming lepton universality, B R(

ω

_→

μ

⁺

μ

⁻) can be ex- tracted:

B R

( ω → μ

⁺

μ

⁻

) = (

⁶

.

6

±

¹

.

4_stat

±

¹

.

7_syst

) ·

^{10−5 (12)}

comparedtoB R(

ω

→

μ

⁺

μ

⁻)= (⁹.0±³.1)·^{10−5fromPDG[31].}

9. Conclusions

We have measured the hadronic contribution to the running ofthe effectiveQEDcouplingconstant

α

(s) usingthe differential crosssectiond

σ

(e⁺e⁻→

μ

⁺

μ

⁻

γ

)/d√

s intheregion0.6<√ s<

0.975 GeV,withthephotonemittedintheinitialstate.Ourresults show aclearcontributionofthe

ρ

−

ω

resonancestothephoton propagator, which results in a more than 5

σ

significance of the hadroniccontributiontotherunningof

α

(s).Thisisthestrongest directevidenceachievedinbothtime- andspace-likeregionsbya single experiment.Forthefirsttime therealandimaginaryparts of 

α

(s) havealso beenextracted. From a fitof therealpart of



α

(s) and assuming the lepton universality the branching ratio B R(

ω

_→

μ

⁺

μ

⁻)= (⁶.6±¹.4stat±¹.7syst)·^{10−5hasbeenobtained.}

Acknowledgements

We thank F. Ignatov and C.M. Carloni Calame for useful dis- cussions. We warmly thank our former KLOE colleagues for the access to the data collected during the KLOE data taking cam- paign.We thanktheDANEteamfortheireffortsinmaintaining lowbackgroundrunningconditionsandtheir collaborationduring all data taking. We want to thank our technical staff: G.F. For- tugno andF. Sborzacchi for their dedication in ensuring efficient operation oftheKLOEcomputingfacilities; M.Anelli forhis con- tinuousattentionto the gassystemanddetectorsafety;A. Balla, M.Gatta, G. CorradiandG.Papalino forelectronicsmaintenance;

M. Santoni, G. Paoluzziand R.Rosellini for generaldetectorsup- port; C. Piscitelli for his 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, Grant Agreement No. 227431; by the Polish National Science Centre through the Grants No. 2011/03/N/ST2/02652, 2013/08/M/ST2/00323,2013/11/B/ST2/04245,2014/14/E/ST2/00262, 2014/12/S/ST2/00459.

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