Contributions of LEP1.5, LEP2 and linear-collider data to indirect constraints on non-Abeli

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

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aKEK-TH-531KEKPreprint97-107hep-ph/9708218July1997ContributionsofLEP1.5,LEP2andlinear-colliderdatatoindirectconstraintsonnon-Abeliangauge-bosoncouplingsR.SzalapskiTheoryGroup,KEK,Tsukuba,Ibaraki305,JapanAbstractItispossibletoplacedirectconstraintsonWWγandWWZcouplingsbystudyingtheirtree-levelcontributionstotheprocesse+e →W+W .How-ever,thesecouplingsalsocontributeattheloopleveltoe+e →f

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

I.INTRODUCTIONANDPARAMETERIZATIONS

Non-Abeliangauge-bosoncouplingsareanessentialandfascinatingaspectoftheStan-dardModel(SM).WiththeaimofverifyingtheSMordetectingnewphysicsitisimportanttomeasuresuchcouplings.Thebasicstrategyistointroduceandthenmeasurethemostgeneralcouplingsallowedunderaparticularsetofphysicalassumptions.If,forexample,onerequiresthenewphysicstobeinvariantunderU(1)em,thenthemostgeneralWWγandWWZcouplingsallowedareparameterizedbythee ectiveLagrangian[1]

LWWV=

igWWV gV1 W µ

µν+WVν W µν

µ+VνW +κ+ VWµWνVµν+λV

1The‘hatted’couplingsarethe

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

ffLlinear=LW

e SM+WWW

Λ2OB+

m2Z

2

m2W

2

m2Z

2

2g 2m2

W

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

separatelyconserveCandP,andonlyL9breaksthecustodialSU(2)Csymmetry.NotethatthereisoneCP-conservingoperator,L11,thatconservesneitherCnorP.ThisadditionaloperatorcontributestoaP-violatingWWZcoupling,butitisnoteasilyincorporatedinthecurrentanalysis;ithasbeendiscussedelsewhere[7].FromEqn.(4)onemayobtainasetofrelationshipssimilartothoseofEqn.(3).Inparticular[4,7,8],

Z22g1(q)=1+g Zα3,

22(5a) κγ(q)=1+g α2+α3+α9,

κZ(q)=1+22g Z (5b)2

λγ(q2)=λZ(q2)≈0. s α2+c α3+c α9,22 (5c)(5d)

ThecouplingsλγandλZareequalonlyinthesensethatthattheyarebothzero,andathigherordersitisexpectedthatthesetwoparameterswillbenonzeroandunequal.Notice

Zthecontributionofα9maybedropped.Thentherelationsamong{κγ,κZ,g1}arethesame1

Λ2Zthat,intheset{κγ,κZ,g1},allthreeareindependent.IfthenewphysicsisSU(2)Cinvariant,asthoseofEqns.(3b)-(3d).Thisisclari edbymakingtheidenti cationsg 2α2=

andg 2α3=1

Λ2fBfW.

II.CONTRIBUTIONSTOOBSERVABLES

ToplaceconstraintsuponfW,fBandfWWW,itisnecessarytocalculatethecontributionsoftheoperatorsOW,OBandOWWWtoamplitudeswithfourexternallightfermions.Toconstrainα2,α3andα9itisnecessarytorepeattheprocedureforL2,L3andL9.ItisparticularlyconvenienttoorganisetheoverallcalculationaccordingtoRef.[9].Thepropagatorcorrectionsandthepinch-portionsofthevertexcorrectionsareabsorbedintothechargeformfactors2g2Z(q),2g2W(q).(Thepinchtechnique[10–13]rendersthepropagatorandvertexcorrectionsseparatelygaugeinvariant.)Withtheexceptionofanynon-SMvertexcorrectionswhichmustbeaddedexplicitly,theSMvertexandboxcorrectionsareemployed.Thetwo-point-functionsinthelinearrealizationwerecalculatedinRef.[3];forthenonlinearrealization,seeRefs.[3,14,15].Expressionsforthe‘barred’chargesappearinRef.[15];beingratherlengthy,theyarenotreproducedhere.

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

FIG.1.Non-SMcorrectionstotheZbbandγbbvertex.Thenewphysicsentersthroughthe‘blob’vertex.

Non-SMcontributionstotheZbbvertexwerepresentedinRef.[15].Inthecurrentcontextitisnecessarytoconsidernon-SMcontributionstotheγbbvertexaswell.SeeFig.1.Inordertoavoidacon ictwithexistingnotationfoundinthereferences,butatthesametimewishingtoavoidalengthydiscussionofvertexcorrections,twonewparameters,

2NP2 ΓNPγbb(q)and ΓZbb(q)areintroduced.(Thesuperscript‘NP’indicatesnewphysics.)Theyarede nedaccordingtohowtheymodifytheSMFeynmanrules:

ie γ µ QbP++

2 2QbP++(1+ ΓNP ig Zγµ s 2QbP ,Zbb(q))I3 s 2 ΓNPγbb(q)I3+QbP ,

(6a)

(6b)

fortheγbbandZbbverticesrespectively.Theone-loopcontributionsoftheSMarenotshown.Theprojectionoperatorsarede nedbyP±=(1±γ5)/2,andQb= 1/3andI3= 1/2arethechargeandweak-isospinquantumnumbersofthebquark.Also, ΓNPγbb(0)=0by

2Wardidentities[9]while ΓNPZbb(q)maycontainaconstantterm.Bothoftheseformfactors

receivecontributionsfromOWandOBthroughanew-physicscontributiontotheWWγandWWZcouplingsasdepictedinFig.1.(TheyreceivecorrectionsfromL2,L3andL9inthenonlinearrealizationofSSB.)Recallthatthepinchtermisremovedfromthevertexcorrectionformfactorbutincludedinthebarredcharges.Anexplicitcalculationyields

2 ΓNPγbb(q)= g 2

m2W

g 2

m2Z2 ΓNPZbb(q)= q2

m2Z

q2

m2Z ,,(7a)(7b)

γγZZwhere κγ=κγ 1, κZ=κZ 1, g1=g1 1and g1=g1 1.Recallthatγg1(0)=1.AnexpressionequivalenttoEqn.(7b)waspresentedinRef.[15].Replacing

aWWZvertexinFig.1withaWWγvertexintroducesafactorofs /c andrequiresthe

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

replacements κZ→ κγand gZγ

1→ g1.Takingintotheaccountdi erentoverallfactors

inthede nitionsofEqn.(6),theexpressionfor ΓNPγbb(q2)isobtainedfromtheexpressionfor ΓNPZbb(q2).Explicitexpressionsforκγ,κZ,gγ1andgZ1fromEqns.(3)andEqns.(5)maybeusedtoobtaintheappropriateformfactorsforthelinearandnonlinearrealizationsofSSBrespectively.

III.THEDATA

FromtherecentanalysisofRef.[16],thedataforlow-energyandZ-polemeasurementsisnicelysummarizedasmeasurementsofthevariouschargeformfactors.Thatglobalanalysisyielded,formeasurementsontheZ-pole,

δb(m2Z) 0.1065

s2(m20.24,(8)Z)=0.23065+0.00003αs+1.54

0.0038±0.00024

whereδbSM(m2 Z)+ ΓNPZbb(m2Z);theprecisede nitionof ρcorr=

g2W(0)=0.4237±0.0013.(9)

And nally,fromthelow-energydata,

s2(0)=0.2362±0.0044 ρcorr=0.70.(10)

Theimplicationsofthisdatafornon-Abeliangauge-bosoncouplingswasconsideredinRef.[15].Notethattheabovedataisinsensitiveto ΓNPγbb(q2);because ΓNPγbb(0)=0,thereisanegligiblecontributionatlowenergies,andontheZ-polethee ectsofphotonexchangearenegligible.Alsonoticethattherearenoconstraintsong2W(q2)awayfromq2=0.

DataisnowavailablefrommeasurementsatLEP1.5andLEP2.Thishasimplicationsfortheindirectconstraintsonnewphysics.Inparticular,thereisnowsensitivityto ΓNPγbb(q2)and

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

combinationsoftheparameters.SomeofthecontributionsareenhancedastheCMenergyincreases,butunfortunatelythereisalsoalossinstatistics.

ThethreeLEPdetectorcollaborationshavepublishedmeasurementsfore+e →fDetector

ALEPH

ALEPH√130136L(pb 1)74.2±6.257.4±4.8

2.7

2.3

0.05

10.9

1.0

10.2σµ+µ (pb)11.8±2.35.8±1.77.7±1.86.1±1.7µAfb+µ Rb.080.91+0 0.20.180.70+0 0.42.160.83+0 0.22.080.92+0 0.274.59±0.86.140.59+0 0.183.60±0.76

66±6

60±56.0±2.07.6±2.3.190.31+0 0.22OPALOPAL

OPAL

OPAL

OPAL130.26136.23133140161.30.31±0.1650±3635.3±2.16.7±1.00.52±0.14

TABLEI.LEP1.5andLEP2datafromthethreeLEPdetectorcollaborations.Statisticalandsystematicerrors,wherereportedseparately,havebeencombinedinquadrature.OPALentriesat133GeVareactuallytheaverageofmeasurementsat130GeVand136GeV.

thatthetableissomewhatincomplete.ALEPHmeasurementsatLEP2energiesarenotyetavailable,andonlyL3hasreportedmeasurementsabove170GeV.OnlyOPALreportsonmeasurementsspeci ctobquarks.TheresultsforOPALat133GeVwereobtainedbycombiningdataat130GeVand136GeV;nodatawasactuallytakenat133GeV.When-everstatisticalandsystematicerrorswerereportedseparately,theyhavebeencombinedinquadraturebeforebeingenteredintothetable.Thevarioustableentriesaretreatedasseparateandindependentmeasurements;noattemptismadetodirectlycombinetheresults

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

ofthedi erentcollaborations.

Asigni cantportionoftheeventsabovetheZpeakare“radiativereturn”events.Thatis,aphotonisradiatedwhichreducesthee ectiveCMenergyto√

fγeventsatLEP1.5andLEP2whenthe

photoncarriesawayasigni cantportionoftheenergy.Forthisreasononlytheexclusivemodesarereportedinthetable.

Additionallyitisinterestingtoanticipatewhatdatamightbeavailableinthefuture.Fourdi erentdatasetswillbeconsidered:

1.Fit1:Alllow-energyandZ-poledataassummarisedbyEqns.(8)-(10).

2.Fit2:ThedatafromFit1plustheLEP1.5andearlyLEP2datasummarisedinTableI.

3.Fit3:ThedatafromFit2plusanestimateoffutureLEP2data.Aluminosityof125pb 1/year/detectorisassumedfor2detectorsand3years.Theluminosityisevenlydividedbetween188GeVand192GeV.ItisalsoassumedthatLEP2willobtainameasurementof mW=50MeV.

4.Fit4:ThedatafromFit3plus50fb 1ofdatacollectedata√

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

IV.NUMERICALSTUDIESANDDISCUSSION

InallnumericalstudiesthescaleofnewphysicsistakentobeΛ=1TeV,andtherenormalizationscaleischosentobeµ=mZ.Theone-sigmaboundsonfWWW,fWandfBarepresentedinTableII.ThecentralvaluesforfWWWdependuponmHonlythrough

mH=200GeVmH=800GeV

fWWW-19±1025±10

Fit25±1044±10

-22±1024±10

Fit42±851±8

Fit1-5.0±3.71.9±3.8

2.1±3.2-7.3±4.4

Fit3-4.3±3.63.2±3.7

6.1±2.4-8.0±4.3

fB-3±10-0.5±4.0

Fit28.2±7.4-4.1±2.5

-1±10-1.4±3.4

Fit42.9±4.2-6.9±1.3

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

0.15fB= fW

0.1

κγ

0.05

λ

FIG.2.Constraintsatthe95%con dencelevelinthe κγversusλplaneformH=300GeVsubjecttotheconstraintsofEqns.3withfB=fW.Thesolid,dashed,dottedanddot-dashedcurvescorrespondtoFit1(low-energyandZ-poledata),Fit2(LEP1.5andearlyLEP2dataincluded),Fit3(LEP2included)andFit4(linearcolliderincluded)respectively.Forthescaleofnewphysicsandtherenormalizationscale,Λ=1TeVandµ=mZhavebeenemployed.

projectioninthe κZversusλplaneformH=300GeVhasbeenobtainedbychoosingfB= fW(andtherefore κγ=0).TheboundsonλfromFigure3arenoticeablyweakerthanthoseofFig.2.Thisillustratesanimportantaspectoftheseindirectbounds.Withoutrestrictingthediscussiontoaparticularmodel,itisimpossibletopredictrelationsbetweenthevariousparameters,andhencethepossibilityofcancellationsbetweenthemmustbeconsidered.Furthermore,thecentralvalueswhichareobtainedinthe tsaresensitivetoαs,theerroronthehadroniccontributiontotherunningofαQED,thetop-quarkmassand,asshowninTableII,theunknownHiggs-bosonmass.Allfour tsarere ectedineachofthese gures.LookingatthedashedcurveweseethattheLEP1.5andavailableLEP2dataleadstoonlyatinyshiftinthecentralvaluesofthesolidcurves.Thedottedcurvere ectsanimportantreductionintheavailableparameterspacebytheendofLEP2,andthedot-dashedcurvesshowseriousimprovementatthelinearcollider.

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

0.15

fB= -fW

0.1

κZ0.05

-0.05

-0.08-0.06-0.04-0.020

λ

FIG.3.Constraintsatthe95%con dencelevelinthe κZversusλplaneformH=300GeVsubjecttotheconstraintsofEqns.3withfB= fW.Thesolid,dashed,dottedanddot-dashedcurvescorrespondtoFit1(low-energyandZ-poledata),Fit2(LEP1.5andearlyLEP2dataincluded),Fit3(LEP2included)andFit4(linearcolliderincluded)respectively.Forthescaleofnewphysicsandtherenormalizationscale,Λ=1TeVandµ=mZhavebeenemployed.

TheseindirectboundsareparticularlyinterestingwhencomparedtothedirectboundswhichmaybeobtainedbystudyingW-bosonpairproduction.Suchbounds,takenfromRef.[4],aresummarisedinTableIII.ThenumbersinthistablemakethesameassumptionsaboutalinearcolliderasdoesFit4.However,forLEP2,Ref.[4]consideredonlyLint=√500pb 1at

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

fW

7.1

LC0.230.25

Fit2

α2

-0.119±0.023

α9-0.03±0.160.250±0.053-0.121±0.022Fit40.211±0.037-0.242±0.040

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

α3

0.053

LC0.00180.00078

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

0.040.02

00.030.020.01

0 κγ-0.02

-0.06-0.04

-0.08

0.05

0.040.03

0.02

0.01

-0.01

-0.02 κγ-0.01-0.02-0.03 κZ κZ-0.03-0.04 κγ κZ κγ-0.05-0.06-0.07-0.08-0.0500.050.1 κZ

FIG.4.Constraintsatthe95%con dencelevelinthe κγ– κZplane.Fig.(a)comparesthelinearrealizationofSSB(withfWWW=0)formH=100GeV(solidcurve),mH=300GeV(dashedcurve)andmH=700GeV(dottedcurve)withthenonlinearrealizationofSSBwithα9=0(dot-dashedcurve).Then,Fig.(b)and(c)specializetomH=100GeVandmH=700GeV,respectively,andFig.(d)concernsthenonlinearrealizationofSSB.InFig.(b)-(d),thesolid,dashed,dottedanddot-dashedcurvescorrespondtoFit1,Fit2,Fit3andFit4respectively.Λ=1TeVandµ=mZhavebeenassumed.

signi cantlyasdataisaccumulatedattheongoingLEP2experiments.Theresultingdirectconstraintsandthecurrentindirectconstraints,allowingfortheoreticaluncertaintiesinthelatter,probenewphysicsatroughlythesamelevel.Thereforeitisdesirabletoalsoimprovetheindirectconstraintsasmuchaspossible.TheLEP1.5andLEP2datawhichispresentlyavailableleadstoonlyaverytinyimprovement,butitisanticipatedthatsigni cantgainswillhavebeenaccomplishedbeforetheendoftheLEP2experiments.

Trulysigni cantimprovementsintheloop-levelconstraintsareexpectedfromtheinclu-√sionofdatacollectedatafuturelinearcollideroperatingat

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

beobtainedfromthesedirectstudies.

ACKNOWLEDGEMENTS

DiscussionswithKaoruHagiwaraaregratefullyacknowledged.TheauthorisisthankfultoSeijiMatsumoto,SallyDawsonandSherAlamforpriorcollaborationsonrelatedtopics.ThisworkwassupportedinpartbytheNationalScienceFoundation(NSF)throughgrantno.INT9600243,andinpartbytheJapanSocietyforthePromotionofScience(JSPS).

It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

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It is possible to place direct constraints on WW-photon and WWZ couplings by studying their tree-level contributions to the process e^+e^- ---> W^+W^-. However, these couplings also contribute at the loop level to e^+e^- ---> f fbar processes where `f' is

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