Combined limit on the production of a light gauge boson decaying into $\mu ^{+}\mu ^{-}$ and $\pi ^{+}\pi ^{-}$

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Combined limit on the production of a light gauge boson decaying into

μ ⁺ μ ⁻ and π ⁺ π ⁻

The KLOE-2 Collaboration

A. Anastasi

^f,c

, D. Babusci

^c

, M. Berlowski

^c,w

, C. Bloise

^c

, F. Bossi

^c

, P. Branchini

^t

, A. Budano

^s,t

, B. Cao

^v

, F. Ceradini

^s,t

, P. Ciambrone

^c

, F. Curciarello

^c,∗

, E. Czerwi ´nski

^b

, G. D’Agostini

^o,p

, E. Danè

^c

, V. De Leo

^r

, E. De Lucia

^c

, A. De Santis

^c

, P. De Simone

^c

, A. Di Cicco

^s,t

, A. Di Domenico

^o,p

, D. Domenici

^c

, A. D’Uffizi

^c

, A. Fantini

^q,r

, G. Fantini

^d

, P. Fermani

^c

, S. Fiore

^u,p

, A. Gajos

^b

, P. Gauzzi

^o,p

, S. Giovannella

^c

, E. Graziani

^t

,

V.L. Ivanov

^h,i

, T. Johansson

^v

, X. Kang

^c

, D. Kisielewska-Kami ´nska

^b

, E.A. Kozyrev

^h,i

, W. Krzemien

^w

, A. Kupsc

^v

, P.A. Lukin

^h,i

, G. Mandaglio

^g,a,∗

, M. Martini

^c,n

, R. Messi

^q,r

, S. Miscetti

^c

, D. Moricciani

^r

, P. Moskal

^b

, A. Passeri

^t

, V. Patera

^m,p

, E. Perez del Rio

^c

, N. Raha

^r

, P. Santangelo

^c

, M. Schioppa

^k,l

, A. Selce

^s,t

, M. Silarski

^b

, F. Sirghi

^c,e

, E.P. Solodov

^h,i

, L. Tortora

^t

, G. Venanzoni

^j

, W. Wi´slicki

^w

, M. Wolke

^v

aINFNSezionediCatania,Catania,Italy

bInstituteofPhysics,JagiellonianUniversity,Cracow,Poland cLaboratoriNazionalidiFrascatidell’INFN,Frascati,Italy dGranSassoScienceInstitute,L’Aquila,Italy

eHoriaHulubeiNationalInstituteofPhysicsandNuclearEngineering,Mˇagurele,Romania

fDipartimentodiScienzeMatematicheeInformatiche,ScienzeFisicheeScienzedellaTerradell’UniversitàdiMessina,Messina,Italy gDipartimentodiScienzeChimiche,Biologiche,FarmaceuticheedAmbientalidell’UniversitàdiMessina,Messina,Italy

hBudkerInstituteofNuclearPhysics,Novosibirsk,Russia iNovosibirskStateUniversity,Novosibirsk,Russia jINFNSezionediPisa,Pisa,Italy

kDipartimentodiFisicadell’UniversitàdellaCalabria,Rende,Italy lINFNGruppocollegatodiCosenza,Rende,Italy

mDipartimentodiScienzediBaseedApplicateperl’Ingegneriadell’Università“Sapienza”,Roma,Italy nDipartimentodiScienzeeTecnologieapplicate,Università“GuglielmoMarconi”,Roma,Italy oDipartimentodiFisicadell’Università“Sapienza”,Roma,Italy

pINFNSezionediRoma,Roma,Italy

qDipartimentodiFisicadell’Università“TorVergata”,Roma,Italy rINFNSezionediRomaTorVergata,Roma,Italy

sDipartimentodiMatematicaeFisicadell’Università“RomaTre”,Roma,Italy tINFNSezionediRomaTre,Roma,Italy

uENEA,DepartmentofFusionandTechnologyforNuclearSafetyandSecurity,Frascati(RM),Italy vDepartmentofPhysicsandAstronomy,UppsalaUniversity,Uppsala,Sweden

wNationalCentreforNuclearResearch,Warsaw,Poland

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

Articlehistory:

Received6July2018

Receivedinrevisedform3August2018 Accepted9August2018

Availableonline13August2018 Editor:L.Rolandi

We searched for theμ⁺μ⁻ decayofalightvector gaugeboson, alsoknown as dark photon,in the e⁺e⁻→μ⁺μ⁻γISRprocessbymeansoftheInitialStateRadiation(ISR)method.Weused 1.93 fb⁻¹of datacollectedbytheKLOEexperimentattheDANEφ-factory.Nostructureshavebeenobservedover the irreducibleμ⁺μ⁻ background.A 90%CLlimitontheratio ε²₌α^/α betweenthe darkcoupling constantand thefinestructureconstantof3×^10−6–2×^{10−7 hasbeenset inthedarkphoton mass} regionbetween519MeVand 973MeV.Thisnewlimithasbeencombined withthe publishedresult

*

Correspondingauthors.

E-mailaddresses:francesca.curciarello@lnf.infn.it(F. Curciarello),gmandaglio@unime.it(G. Mandaglio).

https://doi.org/10.1016/j.physletb.2018.08.012

0370-2693/©^{2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

Keywords:

e⁺e⁻Collisions Darkforces Gaugevectorboson Upperlimits

obtainedinvestigatingthehypothesis ofthedark photondecayingintohadronsine⁺e⁻→π⁺π⁻γISR events.Thecombined90%CLlimitincreasesthesensitivityespeciallyintheρ–ωinterferenceregionand excludesε²greaterthan(13−²)×^{10−7.Fordarkphotonmassesgreaterthan600MeVthecombined} limitislowerthan8×^{10−7resultingmorestringentthanpresent}constraintsfromotherexperiments.

©^{2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Manygravitationalanomalies observed since thefirst decades ofthe twentieth century, aswell aslarge-scale structure forma- tionintheearlyUniverse,canbe explainedbytheexistence ofa non-baryonicmatterknownasdarkmatter(DM) [1].Darkmatter motivatesextendingtheStandardModelofparticlephysics(SM)to includeadarksectorconsistingoffieldsandparticleswithnoSM gaugechargesandincludingextragaugesymmetries.Theminimal extensionofthe SM consistsofjustone additionalabelian gauge symmetry UD(1) with associateda light vector gauge boson,the darkphoton –knownalsoasU boson,

γ

^ or A^–asmediatorof thenewforce,calledforthisreasondarkforce.Inthesimplestsce- nario [2],thecouplingwithSMparticlesarisesfromavectorportal knownaskinetic mixingconsisting inloopsof heavy darkparti- clescharged under both the electromagnetic andthe dark force.

Theportalallowsthemixingofthedarkphotonbelongingtothe UD(1) groupwiththeSM photon oftheUem(1)symmetryintro- ducingtheLagrangianterm:

L_mix

= − ε

2F_{i j}^emF_dark^{i j}

.

(1)

Here

ε

isadimensionlessparameterwhichgovernsthestrengthof themixing(

ε

²=

α

^/

α

,

α

=

α

em,

α

^ istheeffectivedarkcoupling constant)whileF_{i j}^emandF_dark^{i j} arethefieldstrengthtensorsofthe SM Uem(1) and dark UD(1) gauge groups, respectively. Through theportaltheU bosoncancoupletotheelectromagneticcurrent witha strength proportional to the SM particles electric charge.

The process is responsible for both production and decayof the darkphoton inSM interactions thus resulting in an

ε

² suppres- sion.Ifthekineticmixingappearsattheone-loop level,

ε

canbe estimatedtobeintherange10⁻²–10⁻⁶ allowingvisibleeffectsat highluminositye⁺e⁻ colliders [3].

Duringthelastdecade,thedarkphotonhasbeenthefocusofa world-wideintensive researchbecause consideredaspossibleex- planationofmanyastrophysicalpuzzlingevidences [4].

Inthisworkweinvestigatethesimplesthypothesisofavisibly decaying dark photon looking for resonant production of U bo- sonfromthecontinuum, consideringasallowed onlydecaysinto SMparticles.The U signal shouldappearasapeak intheinvari- ant mass of the final state particles with a widthmainly domi- natedbytheinvariantmassresolutionsincetheexpectedU -decay widthcan beconsidered negligible [5].KLOEalreadyinvestigated e⁺e⁻→^{U h(dark} Higgsstrahlung) [6], U bosonindecaysofvec- torparticlesto pseudoscalars [7,8],andthevisibledecayhypoth- esis publishing three searches for radiative U production in the e⁺e⁻→^U

γ

process, withthe U boson decayinginto: a)

μ

⁺

μ

⁻

[9], using 240 pb⁻¹ of data; b) e⁺e⁻ [10], using a sample of 1.54 fb⁻¹; c)

π

⁺

π

⁻ [11] analyzing thewholeKLOE datasetcor- respondingto anintegrated luminosityof 1.93 fb⁻¹.Searches for muon andpion pairs, withthe ISR photon selected atsmall an- gle(θ < 15^◦,θ > 165^◦),coverapproximatelythesameU -boson massrangeof520–990MeV,whilefortheelectronpairsthepho- tonselection was atlarge angle(55^◦ < θ < 125^◦) allowing to

reachalowest U -bosonmassof5MeVandprobingthe (g−²)_μ favored region [12].

InthepresentworkweextendthestatisticsoftheU→

μ

⁺

μ

⁻

search to the wholedata sample andupdate the analysiswitha newestimateofthebackground,analogoustotheoneusedforthe U →

π

⁺

π

⁻ search. The new search confirms no U -bosonsignal inthe dimuoninvariant mass spectrum:a new90% CL exclusion limit in

ε

² is estimated. This limit is of comparable magnitude withrespecttothepreviousones,thusacombinedsearchofdark photondecaysintobothmuon andpionpairswouldincrease the sensitivity ofthe single channel searches, particularly, it is more effectivein theregion ofthe

ρ

–

ω

interferencewhere thesearch forU→

μ

⁺

μ

⁻losessensitivity.

2. TheKLOEdetector

TheKLOEdetectoroperatesatDANE[13],theFrascatiφ-fac- tory. DANE is an e⁺e⁻ collider working at a center of mass energy m_φ¹.019 GeV. Positron and electron beams collide at an angleof

π

−^{25 mrad,producing}φ mesonsnearlyatrest.The detectorconsistsofalargecylindricaldriftchamber(DC) [14],sur- rounded by a lead scintillating-fiber electromagnetic calorimeter (EMC)[15].A superconductingcoilaroundtheEMCprovidesa0.52 Tmagneticfieldalongthebisectorofthecollidingbeamswhichis takenasthez axisofourcoordinatesystem.

The EMC barrel and end-caps cover 98% of the solid angle.

Calorimetermodulesarereadoutatbothendsby4880photomul- tipliers. Energy andtime resolutions are

σ

E/E=⁰.057/√

E(GeV) and

σ

t=^{57 ps}/√

E(GeV)⊕^{100 ps,}respectively. The drift cham- berhasonlystereowiresandis4 mindiameter,3.3 mlong.Itis builtout ofcarbon-fibersandoperates witha low- Z gasmixture (heliumwith10%isobutane).Spatialresolutionsare

σ

xy∼^150μm and

σ

z∼^{2 mm.Themomentumresolutionforlargeangletracksis}

σ

(p_⊥)/p_⊥∼⁰.4%.ThetriggerusesbothEMCandDCinformation.

Events used inthis analysisare triggered by atleast two energy depositslargerthan50 MeVintwosectorsofthebarrelcalorime- ter [16].

3. e⁺e⁻→

μ

⁺

μ

⁻

γ

dataanalysis 3.1. Eventselection

We selected

μ

⁺

μ

⁻

γ

candidates by requiring events with two oppositely-charged tracks emitted at large polar angles, 50^◦ < θ < 130^◦,withtheundetectedISR photon missingmo- mentumpointing–accordingtothe

μ

⁺

μ

⁻

γ

kinematics–atsmall polarangles(θ < 15^◦,θ > 165^◦).Thetracksarerequiredtohave thepointofclosestapproachtothez axiswithinacylinderofra- dius 8 cm and length 15 cm centered at the interaction point.

Inordertoensuregoodreconstructionandefficiency,weselected trackswithtransverseandlongitudinalmomentump_⊥>160 MeV or|^pz|>90 MeV,respectively.Thisseparationoftrackandphoton selectionregionsintheanalysis,greatlyreducesthecontamination fromtheresonantprocesse⁺e⁻→ φ →

π

⁺

π

⁻

π

⁰,fromtheFinal State Radiation (FSR)processes e⁺e⁻→

π

⁺

π

⁻

γ

FSR ande⁺e⁻→

μ

⁺

μ

⁻

γ

FSR, since the

μ

⁺

μ

⁻

γ

cross section diverges at small

Fig. 1. Mtrkdistributionsforμ⁺μ⁺γ andπ⁺π⁻γ.Dataarerepresentedinblack, theMCsimulationsofπ⁺π⁻γ and μ⁺μ⁻γ channelsareingreenandred,re- spectively,whiletheirsumisinblue;theverticalbacklinerepresentstheselection cutappliedtoseparatethetwochannels.(Forinterpretationofthecolorsinthe figure(s),thereaderisreferredtothewebversionofthisarticle.)

ISR photon angle making FSR processes and φ decays relatively unimportant [17–20].Consequently, since ISR-photons are mostly collinear with the beam line, a high statistics for the ISR signal eventsremains. The main background contributions affectingthe ISR

μ

⁺

μ

⁻

γ

samplearetheresonante⁺e⁻→ φ →

π

⁺

π

⁻

π

⁰ pro- cess andthe ISR and FSR e⁺e⁻→^x+x−γ(γ),^x=^e,

π

processes.

Their contributions have beenevaluated by applying kinematical cutsintheMtrk,M²π π plane,¹ withMππ theinvariantmassofthe trackpairinthepionmasshypothesis.

Aparticleidentificationestimator(PID),L_±,basedonapseudo- likelihood function usingthe charged particles time-of-flight and energy depositionsin the five calorimeterlayers is used to sup- pressradiative Bhabhaevents [19,21,22]. Events withbothtracks having L_±<0 are identified as e⁺e⁻

γ

events andrejected (see Fig.2).

Finally,acutonthetrack-massvariable M_trk selectsmuonsby requiringM_trk < 115 MeV asshowninFig.1.Attheendofthe selectiondescribedaboveabout7.16×^{106eventssurvive.}

In order to evaluate the residual background contributions, the sameanalysis chain was applied to simulatedevents forthe

π

⁺

π

⁻

γ

and

π

⁺

π

⁻

π

⁰ channelswhilethe radiativeBhabha con- tributionhas beenevaluated directlyfrommeasured data.Distri- butionsofthefractionalresidualbackgroundFBGforeachchannel andtheir sumare shownin Fig. 3asa function ofthe invariant massofthetrackpairinthemuonmasshypothesis,Mμμ.

Thetotalresidualbackgroundrisesuptoabout9%inthe

ρ

–

ω

regionanddecreases downtoabout3%atlowandhighinvariant massvalues.

4. Parametrizationoftheirreducible

μ

⁺

μ

⁻

γ

background

Tominimizethesystematicuncertaintiesaffectingtheanalysis, weevaluatedtheirreducible

μ

⁺

μ

⁻

γ

backgrounddirectlyfromthe data.InFig. 4,we report thecomparisonbetween dataandesti-

1 Mtrk is computed from energy and momentum conservation, assumingthe presenceofoneundetectedphotonandthatthetracksbelongtoparticlesofthe samemass:

√s−

|^p+|²+^M2trk−

|^p−|²+^M2trk

2

−^p++ ^p−2

=⁰

where^p+(^p−)isthemeasuredmomentumofthepositive(negative)particle,and onlyoneofthefoursolutionsisphysical.

Fig. 2. MCL₊vs.L₋PIDdistributionsforbothtracks.Eventscontainedinthelow leftrectangle(havingbothtrackswithL_±<0)areregardedase⁺e⁻γ eventsand rejectedintheselection.

Fig. 3. Fractional residual backgrounds as function of Mμμ.

matedbackgrounddistributions(toppanel)andtheirratio(bottom panel),whichareingoodagreementwithinerrors.

We estimated theirreducible

μ

⁺

μ

⁻

γ

background by using a sidebandfittotheobservedspectrum,keeping,foreachiteration, thefitwiththebestreduced

χ

².Thefittosidebandsinthewhole mass range has been performed considering sub ranges ±¹²

σ

wide, where

σ

is thedimuon invariant massresolution ofabout 2MeV[11].ForeachU-masshypothesisaregioncorrespondingto

±³

σ

isexcludedfromthefit.Wefitthedatadistributionsbyus- ing Chebyshevpolynomials(asinRef. [9])up to6thorderinthe massranges519–757 MeVand811–973 MeV.Inthemassinterval between759and809 MeV,wheretheeffectofthe

ρ

–

ω

interfer- enceispresent[23],weusedanotherparametrization:

f

(

x

) =

^pol2

(

x

) · [

¹

+

^A

· (

^x

−

^M

) ·

^exp

(−

⁰

.

5

· ((

^x

−

^M

)/λ)

²

)].

(2) The parametrization (2) has beenused because found to best fitthe

μμ

invariantmasssimulatedspectrum(PHOKHARA gener- ator [24–27] with vacuumpolarization correction included anda full descriptionofthe detectorperformedwiththeGEANFIpack- age [28])asshowninFig. 5.As afirststep,thethreecoefficients ofthesecond orderpolynomial pol2(x)andthe parameters A, M andλarecomputedbyfittingthefunctioninEq.(2) overthefull

μ

⁺

μ

⁻

γ

simulatedspectrum:valuesof782.24MeVand6.09MeV wereobtainedfortheparametersM andλ,respectively.Then,the

Fig. 4. Toppanel:μ⁺μ⁻γ observedspectrum(fullsquares)and estimatedirre- duciblebackground(opensquares).Bottompanel:dataandestimatedbackground ratio.

Fig. 5. FitofreconstructedPHOKHARAMCwithvacuumpolarization correctionin- cluded.

fitsinthe consideredmassrange(759–809 MeV)ofthe

μ

⁺

μ

⁻

γ

observedspectrumhavebeenperformedbyusingagainthefunc- tion (2), keeping the parameters M and λ fixed at the values 782.24MeVand6.09MeV,andleavingfreeall theother parame- ters.

ExamplesofthefitsperformedbyusingChebyshevpolynomials ortheparametrizationineq. (2) areshowninFig.6.

Thereduced

χ

² ofthefittosidebandsforbothparameteriza- tionsremainsbelow2inthewholemassrange.Thefitprocedure isstableinthe wholedata rangeandnoanomaly is observedin thefittedbackground.

5. Systematicuncertainties

Inthefollowingwe report thesystematicuncertaintiesaffect- ing the analysis, mainly dueto the evaluationof the irreducible

Fig. 6. Examplesoffitsperformedintwosub-rangesoftheμ⁺μ⁻γ spectrumby usingChebyshevpolynomials(upperpanel)andparametrization(2) (lowerpanel).

Fig. 7. Bin-by-bin total fractional systematic error of the background estimate.

backgroundandtotheeventselectionappliedtothe

μ

⁺

μ

⁻

γ

can- didates.

5.1. Systematicuncertaintiesontheirreduciblebackground

Thefractionalsystematicerrorontheirreducible

μ

⁺

μ

⁻

γ

back- groundisshowninFig.7.Theevaluationofthesystematicuncer- taintieshasbeenderivedforeachmassbinbyestimatingtheerror ofthefit.Thetotalsystematicerrorislessthan1%inmostofthe massrange.

The systematic error dueto the side bands fit procedure has been also evaluated by varying the range of the fit interval of

±¹

σ

andcomputingthemaximumdifferencebetweennominalfit

Fig. 8. Global efficiency as function of Mμμ.

Table 1

Summaryofthesystematicuncertainties.

Systematic source Relative uncertainty (%)

Mtrkcut 0.4

Acceptance 0.6–0.1 as Mμμincreases

Trigger 0.1

Tracking 0.3–0.6 as Mμμincreases

Generator 0.5

Luminosity 0.3

PID negligible

Total 0.98–0.94 as Mμμincreases

andthefitderived bychangingthefitinterval.Itscontributionis

<<1%andthereforeresultsnegligibleinthewholemassrange.

5.2. Systematicuncertaintiesoftheglobalefficiency

Fig.8showstheglobalanalysisefficiencythathasbeenevalu- atedfromafull

μ

⁺

μ

⁻

γ

simulation.Thisefficiencyincludes con- tributions from kinematic selection, trigger, tracking, acceptance andPID-likelihoodefficiencies.

Table 1 lists all the systematic errors affecting the

μ

⁺

μ

⁻

γ

analysis. We evaluated the corresponding uncertainties by using the same procedures described in Ref. [9]. These systematic un- certaintiesdonotaffecttheirreduciblebackgroundestimationbut enterin thedetermination oftheselection efficiencyandthe lu- minositymeasurement.

6. LimitsonU -bosonproductionin

μμγ

events

The

μ

⁺

μ

⁻

γ

observed spectrum doesnot reveal thepresence of any visible structure (see Fig. 4) within the mass-dependent systematicuncertainties.Forthisreason,aprocedurehasbeenap- plied to evaluate the statisticalsignificance of theobserved data fluctuationsandeventually set a limit on the e⁺e⁻→^U

γ

,U→

μ

⁺

μ

⁻ process. The following subsection describesthe results of thelimitextractionprocedure.

6.1. Upperlimitextractionon

ε

²

Toextractthe upperlimit(UL) on

ε

² weused theConfidence LevelSignal(CLS)technique [29].Theprocedurerequiresasinputs theinvariantmassdataspectrum,thebackground(theirreducible

μ

⁺

μ

⁻

γ

background),theU -bosonsignalandthesystematicfrac- tional uncertainties on the background estimation foreach Mμμ bin.ThesignalhasbeengeneratedwithatoyMCinstepsof2MeV fortheU -bosonmass.Ateachstep,aGaussiandistributionisbuilt

withawidthcorrespondingtotheinvariantmassresolutionofthe dimuonsystemofabout2 MeV.Thesignalisthenintegratedover Mμμ around MU.Thenumberof signalevents,givenasinput to theprocedure,isinitiallyarbitraryandveryhigh(abouttentimes the square rootof the estimatedbackground value inthe corre- spondingmassbin)andtheniterativelyscaleduntiltheconfidence levelCL_S reaches0.1within±^{0.01.Theintegralofthesignalcor-} respondingtothedefinedlevelofconfidencerepresentsthelimit on the numberof U -bosoneventsexcluded at 90%CL. Since the limit isstronglydependent ontheirreduciblebackgroundevalua- tion,thelimitextractionaccountsforthesystematicuncertainties of the background estimate. The limit extraction procedure uses the total bin-by-bin fractional systematicuncertainty, reportedin Fig. 7, to perform a Gaussian smearing of the

μ

⁺

μ

⁻

γ

expected backgroundgivenasinput.

TheULonthekinetic mixingparameterhasbeenextractedby using,foreachU -bosonmassvalue,thefollowingformula [9–11]:

ε

²

= α

^

α ⁼

N_CLS



eff

·

^L

·

^H

·

^{I (3)}

where NCLS is thelimit on thenumber ofevents,



eff represents the global efficiency (shown in Fig. 8), L is the integrated lumi- nosity (1.93 fb⁻¹ with an uncertainty of 0.3% [18,19]), H is the radiator function calculated at QED next-to-leading-order correc- tionswithanuncertaintyof0.5% [25–27,30] andgivenby:

H

=

^d

σ

_μμγ

/

dM_μμ

σ (

e⁺e⁻

→ μ

⁺

μ

⁻

,

M_μμ

) .

(4) Here d

σ

μμγ/dMμμ is the differential cross section of e⁺e⁻→

μ

⁺

μ

⁻

γ

,

σ

(e⁺e⁻→

μ

⁺

μ

⁻,_Mμμ)isthetotalcrosssectionofthe e⁺e⁻→

μ

⁺

μ

⁻ process.InEq. (3), I isgivenbythefollowingin- tegralaround M_U:

I

=



σ

_U^μμd

√

s

,

(5)

where

σ

_U^μμ=

σ

(e⁺e⁻→^U→

μ

⁺

μ

⁻,s)isthetotalcrosssection of U -bosonproductiondecaying inthe

μ

⁺

μ

⁻ channelwhen the kineticmixingparameter

ε

isequalto1,s=^M2_U^.Theuncertainties on H ,



eff, L, and I, propagate tothe systematicerror on

ε

² via eq. (3).The resultinguncertaintyon

ε

² islowerthan 1%andhas beentakenintoaccountintheestimatedlimit.

The exclusion plot on

ε

² is shown asa dashed linein Fig. 9 comparedwiththeexistinglimitsinthemassrangebelow1 GeV.

Our90%CLULrangesfrom3×^{10−6 to2}×^{10−7 inthe519–973} MeVmassinterval.

7. CombinedlimitonU -bosonproductionin

μμγ

and

π π γ

events

Inthissectionwepresentthecombinationprocedureofthefull statistics

π

⁺

π

⁻

γ

and

μ

⁺

μ

⁻

γ

limits. As forthe previous analy- ses, weuse theCLS technique toestimate a 90%CL limit forthe e⁺e⁻→^U

γ

ISR,U →

μ

⁺

μ

⁻,

π

⁺

π

⁻ process. Toextract the limit, weusethealreadyestimatedbackgroundandobservedspectrafor both

π π γ

[11] and

μμγ

channels in a combined way. A total systematicerrorontheirreduciblebackgroundestimate,givenby the combinationofthecorresponding estimateduncertainties for bothU -bosondecaymodes,isalsogivenasinputtotheprocedure.

A combinedU -bosonsignal isgeneratedforboth decaychannels taking into account the differences in global efficiencyand rela- tive branching ratio [3]. Thesignal inputsare generatedwiththe same toy MC procedure performedfor the

μ

⁺

μ

⁻

γ

limit extrac- tion,then,eachsignalisintegratedandnormalized tothenumber

Fig. 9. 90%CLexclusionplot forε² as afunctionofthe U -boson massfor the e⁺e⁻→Uγ process.The U→μ⁺μ⁻ limit(dashedline), theU→π⁺π⁻ [11]

constraint(dotted line),and the U→μ⁺μ⁻,π⁺π⁻ combination (solidline) at fullKLOE statistics,arepresentedincomparisonwiththe competitivelimits by BaBar [31],NA48/2 [32],andLHCbexperiments[33].

of events estimated from Eq. (3), for a given hypothesis of the kinetic mixingparameter

ε

². Thelimit computation proceedsac- cordingtothefollowingsteps:itmakesahypothesisofthe

ε

² ki- neticmixingparameter,startingfromanarbitraryverylowvalue;

thecorresponding numberofeventsfor

π π γ

and

μμγ

channels aregeneratedaccordingtoEq. (3) inordertobuild thesignalin- puthistogram, then, the procedure runsas before by comparing data andexpected irreducible background. The search procedure endswhentheestimatedCLS becomescloseto 0.1within±^0.01, providingdirectlythecorrespondingexclusionon

ε

².

Thecombinedupperlimit,obtainedafteraveraging thestatis- ticalfluctuationsby asmoothingprocedure,excludesvaluesof

ε

²

greaterthan(13−²)×^{10−7 intheU -massrange519–987 MeV.}

ItisshowninFig.9,comparedtothemostcompetitivelimits.The other existing limits [7–10,34–37] are not reported to make the figure more readable. The combined limit is represented by the blue area and is more stringent with respect to the already set limitsinthemassregion600–987 MeV,whileitiscomparableto BaBarandLHCbresultsformasseslowerthan600 MeV.

8. Conclusions

We analyzed 1.93 fb⁻¹ of KLOE data to investigate the hy- pothesis ofa light vector gauge boson decaying into muonsand pions by means of the ISR method in the e⁺e⁻→^U

γ

ISR,U →

μ

⁺

μ

⁻,

π

⁺

π

⁻ process.No U -bosonevidencehasbeenfound and acombinedlimitat90%CLusingthetwoU -decaymodeshasbeen extractedonthekineticmixingparameter

ε

² intheenergyrange between519and987 MeV.Thenewcombinedlimitismorestrin- gent than the alreadyset constraints in the region between600 and987MeVbyexcludingvaluesof

ε

² higherthan(8−²)×^10−7. Acknowledgements

We warmly thank our former KLOE colleagues for the ac- cess to the data collected during the KLOE data taking cam- paign.WethanktheDANEteamfortheir effortsinmaintaining

low background running conditions and their collaboration dur- ing all data taking. We want to thank our technical staff: G.F.

Fortugno and F. Sborzacchi for their dedication in ensuring ef- ficient operation of the KLOE computing facilities; M. Anelli for hiscontinuous attentiontothegassystemanddetectorsafety;A.

Balla, M. Gatta, G. Corradiand G. Papalino for electronicsmain- tenance; C. Piscitelli for his help during major maintenance pe- riods. This work was supported in part by the Polish National Science Centre through the Grants Nos. 2013/11/B/ST2/04245, 2014/14/E/ST2/00262,2014/12/S/ST2/00459,2016/21/N/ST2/01727, 2016/23/N/ST2/01293,2017/26/M/ST2/00697.

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