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

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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/©²⁰¹⁶^TheÂuthor.^Published^byÊlsevier^B.V.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

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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 contribution 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 ^crosssection:

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 measurement 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 running 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− φôf∼⁰.15 mm and∼^{2 mm in}^z.^TheÊMC^hasânênergy 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

(3)

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

• ê⁺ê⁻→ê⁺ê⁻

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 background,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

=

^N^obs

−

^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),

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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 requirement. 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^◦ ^(more^than^two^times^the^resolution 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 luminosity,withareferencecrosssectionobtainedwithBabayaga@NLO MC eventgenerator[17],convolvedwithdetectorandbeamcon- ditions[18].Twosourcescontribute tothesystematicuncertainty intheevaluationoftheluminosity:

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

•^the ^systematic êrror âssociated ^to ^the ^counting ôf ^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 constant 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

(5)

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/⁴^π^|^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

−

^4m²^π 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,

(6)

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-

(7)

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 values[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 ând ârg 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+²^m_m²^μ2 ω

1−⁴^m_m²^μ2

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)·¹⁰⁻⁵^from^PDG^[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)·¹⁰⁻⁵^has^been^obtained.

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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