hep-ph/0107247arXiv:hep-ph/0107247v2 6 Oct 2001Neutrinodeep-inelasticscattering:newexperimentalandtheoreticalresults
A.L.Kataev
TheoreticalPhysicsDivision,CERNCH-1211Geneva,SwitzerlandandInstituteforNuclearResearchoftheAcademyofSciencesofRussia,
117312,Moscow,Russia
Abstract
Areviewofrecentexperimentalandtheoreticalstudiesofcharacteristicsofneutrinodeep-inelasticscatteringispresented.Specialattentionispaidtothedeterminationofαsand1/Q2non-perturbativeeffectsfromtheQCDfitstoxF3dataatdifferentordersofperturbationtheory,withthehelpofseveraltheoreticalmethods.
ContributedtotheProceedingsofLesRencontresdelaVall´eed’Aoste,
LaThuile,Italy,March4-10,2001
CERN-TH/2001-195July2001
1Introduction
Theneutrinodeep-inelasticscattering(DIS)iscontinuingtoserveastheclassicaltoolforprobingthenucleonstructure(forreviews,seee.g.Refs.1)2)).Inthelastfewyearssomeprogresswasmadeinmoredetailedexperimentalandtheoreticalstudiesofthebehaviourofthecross-sectionsofνNDIS,andintheextractionofthestructurefunctions(SFs)xF3andF2.AmongthedatathatarecurrentlyunderactiveanalysisaretheonesprovidedbytheCCFR/NuTeVcollaborationattheFermilabTevatron(seee.g.Refs.3)4)),theexperimentalresultsoftheJINR–IHEPNeutrinoDetectorcollaboration5)collectedsometimeagoattheIHEP(Protvino)U-70protonsynchrotron,andthepreliminarydataoftheCHORUScollaboration,obtainedrecentlyattheCERNSPS6)7).
Thekinematicalregionscurrentlyavailableforcross-sectionmeasurements
areshownontheplotofFig.1,takenfromRef.8).Thisfigureclearlyshowsthattheabove-mentionedthreeexperimentswereperformedindifferentkinematicalregions,whichoverlapinpartonly.ThustheyprovidecomplementaryinformationaboutthebehaviourofSFsindifferentregimes.Moreover,additionalmoreprecisedataforνNDIScross-sectionscanbeobtainedinthefutureatneutrinofactories.IftheenergyoftheneutrinobeamisfixedatEν=50GeV,experimentsshouldpenetrateintothephysicalregionthatwasaddedtoFig.1inRef.9).Thisregionoverlapsinpartwiththosewheretheabove-mentionedthreeexperimentalcollaborationswereworking.Therefore,thestudiesofthedataonνNDIScharacteristicsavailableatpresentcanbeimportantmilestonesintheplanningofmorepreciseDISexperimentsatneutrinofactories10).2
Discussionsofsomenewexperimentalresults
Recentinterestingexperimentalnewscamefromthemodel-independentre-extractionofthebehaviouroftheF2νNSF4)fromtheCCFR’97data3).There-analysisofRef.4),whichdoesnotaffectpreviousCCFR’97xF3results,removedthewidelydiscusseddiscrepancythatexistedatx<0.1betweenthebehaviourofCCFR’97F23)andthatobtainedbytheNMCcollaboration11)fromtheprocessofµNDIS.InadditiontothenewextractionofF2fromthedifferentialcross-sectionsofCCFR,
ν
thefirstmeasurementof∆xF3=xF3−xF
Q2 (GeV2)106
ZEUS 96+97 prel.105
4
CDF/D0 - jetsZEUS SVTX 95ZEUS BPC 95ZEUS BPT 97 prel.H1 97 prel.H1 96 ISR prel.H1 SVTX 95JLAB E97-010CCFRCHORUSJINR-IHEPNuFacty = 110
H1 94-97 prel.H103102101
-1
10
101010101010
x
Figure1:Kinematicregionsinx-Q2forcross-sectionmeasurementsindeepinelasticepscattering,νscatteringandfortripledifferentialjetcross-sectionmeasurementsinpp¯collisions(fromRef.9)).
-6-5-4-3ERA-2-1collaborationin19973).Therefore,thepreliminaryCHORUSF2datashouldshowapatternidenticaltothatfoundintheCCFR’97analysis3),i.e.exceedingby10–15%F2NMCmeasurementsatx<0.1.ThisexcessisbeyondtheexistingstatisticalandsystematicerrorsofdiscussedDISexperiments.Moreover,theinclusionofthesewrongCCFR’97F2pointsintothenext-to-leadingorder(NLO)QCDfits,performedwiththehelpoftheDGLAPmethod13),leadstotheerroneouslow-xbehaviourofthegluondistributionxG(x,9GeV2)∼xbGwithbG=0.0092±0.007314).Itisinevidentcontradictionwiththenumber,obtainedpreviouslyfromtheNLOcombinedanalysisofthedatafromHERAandtheCERNSPS15),namely
2
bG=−0.267±0.043atQ20=9GeV.TakingintoaccountnewCCFRmodel-independentextractionsofF24),itseemsworthwhiletoperformmorecarefulstudiesofthepreliminaryCHORUSresults.Moreover,itisratherinterestingtotrytoverifyfromtheCHORUSdatatheexperimentalbehaviourof∆xF3,foundinRef.4).
ItshouldbestressedthattheCHORUSexperimenthasanattractivefea-ture.Indeed,ascanbeclearlyseenfromFig.1,itprovidesinformationaboutνN
NuclearshadowingeffectforF2andF310.90.80.70.6
R2R30.0001
0.001
x
AN
Figure2:TheratiosofaheavytargettothefreenucleonSFsR2=F2/F2and
AN562
R3=F3/F3calculatedforFenucleusintheregionofsmallxandQ=10GeV2.(thefigureofKulaginfromRef.10))
0.010.1
DISSFsintheregionofratherlowQ2andlowx,whichcomplementinparttheonewheretheCCFR’97datawereextracted.Inthiskinematicaldomain,theoreticalcontributionsof1/Q2andnuclearcorrectionscanplayanimportantrole.Leavingforawhilethediscussionsofpower-suppressedterms,westressthattheCHORUScollaborationwasusingaleadtarget,whiletheCCFRtargetismadeofiron.Pos-sibilitiesarethereforereallyopentostudynucleareffectsinneutrinoDIS;asshownincalculationsreportedinRef.16),theseeffectscanbeofgreatimportance.AcomparisonoftheseeffectsforthecasesofF2andxF3neutrinoDISSFsisdepictedinFig.2,constructedforthedetailedworkofRef.10).
AnotherinterestingpossibilityofDISexperimentsistheextraction,from
theirchraracteristics,ofnon-perturbativepower-suppressedtermsandthevaluesofαs(MZ).ThisquestionwasconsideredinRefs.17)−23)intheprocessofthenext-to-next-to-leadingorder(NNLO)QCDfitstodifferentdata,andinRef.24),whileperformingNLOfitstotheexperimentalresultsforcharged-leptonsDISSFs.ThemostrecentoutcomesoftheNNLOanalysisoftheCCFR’97xF3data23)willbediscussedinthenextsection.HereitisworthwhileemphasizingthattheworksofRefs.17)−20)22)23)agreeintheirconclusionthattheinclusionoftheNNLOQCDcorrectionsintothefitshasatendencytodecreasetheextractedvaluesofnon-perturbative1/Q2-terms.Whetherthisisageneraltheoreticalfeature(seee.g.Refs.25)26))oritisrelatedtothelackofprecisionoftheanalyseddatamightbeclarifiedinthefuture,iftheideasofmoredetailedexperimentsonneutrinoDISatneutrinofactories10)arerealized.
ItshouldbenotedthatthesituationatNLOismoretransparent.Indeed,
theJacobipolynomialfitsofRefs.17)19)23)demonstratedthatitisthenpos-
Table1:Theresultsofthefit,inRef.5),oftheIRRmodeltothedatafromdifferentneutrinoexperiments.Thevalueofχ2overthenumberofpoints(np)isgiven.
ExperimentIHEP–JINR
0.36±0.22
χ2/np3/12
Q2
where
h(x)=A2
′
,(1)
1
dz
x
β0
Λ23
,(3)
whereCF=4/3andβ0=(11−2/3f)isthefirstcoefficientoftheQCDβ-function.
Fixingαs(MZ)=0.118,whichcorrespondstoitsworld-averagevalue,Λ3wasex-tractedfromthefitstotheIHEP–JINRNeutrinoDectectorandCCFR’97data5).TheresultsarepresentedinTable1,takenfromRef.5).Notethatthetwoexpres-sionsforΛ23fromTable1arecomparablewithintheerrors.Averagingthenumbers
′5)obtainedthefollowingforΛ2andtransformingthemtoA32,theauthorsofRef.
value:
′
A2=−0.130±0.056(exp)GeV2,(4)
wheretheerrorincludesbothstatisticalandsystematicexperimentaluncertainties.ItisinagreementwiththevalueextractedfromtheNLOJacobipolynomialanalysisoftheCCFR’97xF3behaviour,cutatQ2≥5GeV217)19)23).Indeed,atNLO,themostdetailedfitsofRef.23)give:
A′
2=−0.125±0.053(stat)GeV2
.(5)
Note,however,thattheerrorinEq.(5)doesnotincludethesystematicuncertainties.Therefore,Eq.(4)isthemostpreciseup-to-datevalueoftheIRRmodelparameterA′2.3
NewQCDfitstoCCFRxF3data:NNLOandbeyond
WecanstartthediscussiononthephenomenologicalapplicationofsomenewN3LOperturbativeQCDresultsonthecoefficientfunctionsofoddmomentsofxF3andontheNNLOapproximationsfortherelatedanomalousdimensions28)(whicharecomplementarytothoseobtainedinRef.29)inthecaseofevenmomentsfortheF2SFofcharged-leptonsDIS),incombinationwiththeNNLOexpressionsforthecoefficientfunctions30)thatwererecentlyconfirmedinRef.31).3.1TheapplicationoftheJacobipolynomialmethod
Itisappropriate,atthispointtorecallthebasicideasoftheJacobipolynomialmethod32)whichwasdevelopedinRefs.33)34)andwas35)previouslyusedintheanalysisoftheBCDMScharged-leptonsDISdataatNLO,andinthenon-singletapproximationatNNLO36).InthecaseoftheanalysisoftheCCFRxF3data,theJacobipolynomialmethodwasappliedatNLOinRefs.37)38)andprovedusefulforperformingfitsattheNNLO17)19)23)andapproximateN3LOlevels,19)23)withandwithouttwist-4corrections(seediscussionbelow).
ThismethodallowsthereconstructionoftheSF(sayxF3)fromthefinite
numberofMellinmoments,namely
xF3Nmax(x,Q2)
=xαN(1−x)
β
maxΘαβ(n)
n(x)
cj(α,β)MMC2
h(x)
jT+2,F3(Q)+
n=0
j
n=0
Mnucl
2
n+2
Table2:TheNNLOresultsoftheparametersA,b,cofthemodelforxF3deter-mined,inRef.23)andtheircomparisonwiththevaluesobtainedinRef.40).Thenewonesaremarkedbyboldtype.
Q205GeV2
NNLO/9
10GeV2
NNLO/9
20GeV2
NNLO/9
100GeV2
NNLO/9
4.74±0.324.49±0.25
0.63±0.02
4.14±0.09
4.21±0.35
0.65±0.03
3.89±0.06
80.0/86
3.73±0.68
0.65±0.03
3.73±0.07
77.0/86
b0.66±0.03
3.52±0.08
76.3/86χ2/np78.4/86
Thecontributionofthetwist-4termstoEq.(6)isparametrizedwiththehelpofthefunctionh(x).Itwillbeneglectedforourfirststageofdiscussions.
FixingnowthebehaviourxF3attheinitialscaleQ20as
PT2b(Q0)
xF3(x,Q2(1−x)c(Q0)(1+γ(Q20)=A(Q0)x0))
2
2
,(8)
calculatingtherelatedMellinmomentsandtransformingthemtoexperimentally
accessibleregionswiththehelpoftherenormalizationgrouptechniqueatLO,NLO,NNLOandapproximateN3LO(theexplicitformulaefortherenormalizationgroupevolutioncanbefoundinRef.23)),substitutingtherenormalization-group-TMC
improvedexpressionforMn,F(Q2)intoEq.(6),andperformingthefitstothe3
experimentaldata,itispossibletodetermine5parameters,namelyA,b,c,γand(4)Λ
MS
,obtainedinRef.23)intheprocessoftwist-4
Table3:TheQ20andNmaxdependenceofΛNmax
NNLONNLONNLONNLO
6789
303±29
(76.4)328±32(76.2)334±33(74.8)330±31(73.3)5
314±34(76.3)327±35(76.7)334±35(75.7)332±35(73.6)
(4)
10
327±36(76.9)326±36(78.1)331±35(77.4)331±37(75.8)
321±32(75.7)325±33(77.3)327±34(76.6)329±32(75.7)
50
326±35(79.5)328±35(78.7)328±35(78.5)329±35(77.8)
325±33(78.0)324±33(78.5)323±34(77.3)325±32(76.7)
thevaluesofχ2.
ItisinterestingtocomparetheNNLOresultsforb(Q20)fromTable2,
2
whicharealmostQ0-independent,withthecalculationsofthesmall-xasymptoticbehaviourofnon-singletcontributionstoF1andthespin-dependentSFg1performedinRef.41)inallordersof1-loopexpressionforαs,usinginparttheapproachdevelopedinRef.42).IntheprocessofcalculationsofRef.41)thefollowing1-loopformulaforαswasused
αs(s)=
=
4π
ln2(s/Λ2)+π24π
β0
[ln(s/Λ2)+
−iπ]iπ
(9)
MS
withrespecttochangesoftheinitialscaleQ20,anddecreases
0.50.40.30.20.1
246810
x
ω(−)
0
Q2
ConsidernowsomeotherresultsoftheworkofRef.23),andinparticular
theextractionoftwist-4contributionsandthevalueofαs(MZ)atvariousordersofperturbationtheory.Tomodelthe1/Q2-termh(x)inEq.(6),threeapproacheswereusedinRef.23).ThefirstoneistheIRRmodelofRef.27)(seeEq.(2)).
AftertakingtheMellinmomentsfromEq.(2)andapplyingtheNNLO
′
andN3LOfitstotheCCFR’97xF3data,thereductionsoftheNLOvalueofA2,presentedinEq.(5),wereobserved.AttheNNLOandN3LOtheexpressionsfor′
A2becomecomparablewithzero,withinthestatisticalerrors23),namely
NNLON3LO
::
A2=−0.013±0.051GeV2′
A2=0.038±0.051GeV2.
′
(11)
However,therelatedαs(MZ)resultsweredeterminedinRef.23)withreasonableerrors:
NLONNLON3LO
:::
αs(MZ)=0.120±0.002(stat)±0.005(syst)
.010
±0.002(thresh)+0−0.006(scale)
αs(MZ)=0.119±0.002(stat)±0.005(syst)
.004
±0.002(thresh)+0−0.002(scale)
αs(MZ)=0.119±0.002(stat)±0.005(syst)
.002
±0.002(thresh)+0−0.001(scale)
(12)
wherethefirsttheoreticaluncertaintyisduetotheambiguitiesoftakingintoaccount
(5)
thresholdeffectswhiletransformingtheresultsforΛ
kandvaryingkintheconventional
interval1/4≤k≤4.Onecannoticethedrasticreductionofthescale-dependenceuncertaintiesasaresultofaddingNNLOandN3LOperturbativeQCDcorrectionsintothefits,tabulatedinthecaseoff=4inRef.23)(notethatattheN3LOthecontributionstoexpandedanomalous-dimensiontermsweremodelledusing[1/1]Pad´eapproximants).
Theresultsforαs(MZ),presentedinEq.(12),shouldbecomparedwiththe
onesobtainedfromthetwist-4independentJacobipolynomialfitstotheCCFR’97dataatNmax=923),whichgive
MS
NLONNLON3LO
:::
αs(MZ)=0.118±0.002(stat)±0.005(syst)
.007
±0.002(thresh)+0−0.005(scale)
αs(MZ)=0.119±0.002(stat)±0.005(syst)
.004
±0.002(thresh)+0−0.002(scale)
αs(MZ)=0.119±0.002(stat)±0.005(syst)
.002
±0.002(thresh)+0−0.001(scale)
(13)
Noticethattheeffectiveminimizationofthetwist-4contributionsattheNNLOandN3LO(seeEq.(11))isleadingtoratherclosedNNLOandN3LOvaluesofαs(MZ),whichwereobtainedfromthefitswithandwithout1/Q2corrections.
ItisworthstressingthaterrorsonthescaledependenceoftheNLOand
NNLOresultsfromEq.(13)havedefinitesupport.Indeed,theyareinagreementwiththeindependentestimates
∆αs(MZ)NLO=+0−0..006
004
,
∆αs(MZ)NNLO=+0−0..0025
0015
,(14)
obtainedinRef.50),whichusethemodelconstructedinthisworkfortheNNLO
NSDGLAPkernel.
Inordertostudythesecondpossibilityofmodelling1/Q2-effectsusingthe
parametrizationofh(x)byfreeconstantshi=h(xi),wherexiarethepointsintheexperimentaldatabinning,9parametershiwereusedinRef.23).ThischoicedistinguishesnewfitsfromtheonesperformedinRefs.17)19),where16variableshiwereused.TheminimizationofthenumberoffreeparameterswasmotivatedbytheworksofRefs.24)14),whereitwasdemonstratedthatadecreaseinthenumberoffittedhigh-twistparametersdecreasesthecorrelationbetweentheirerrorsandmaketheirextractionmorereliable(theproblemsofestimatingtheoreticaluncertaintiesinthecaseofthechoiceof16freeparametershiwerealsodiscussedinRef.51)).Thechoiceofasmallernumberofhiresultsinamorereliabledescriptionofthex-shapeofh(x)forthefitstotheCCFRxF3data.AsintheprocessoftheanalogousfitsofRefs.17)19),theLOandNLOx-shapesofh(x)obtainedinRef.23)areinagreementwiththepredictionoftheIRRmodelofRef.27).ThenewNNLOandN3LOresultsofRef.23),inagreementwiththeabove-discussedtendencytoanoverallminimizationoftheextractedcontributionofh(x),revealsomenewfeature,namelyanindicationofanoscillating-typebehaviourofh(x)aroundx=0,albeitwithrathersmallamplitude.
Thethirdmodelof1/Q2-corrections,consideredinRef.23),isdirectly
expressedintermsofMellinmoments,namely
′
MnHT(Q2)=n
B2
inthesecircumstances,thattheQCDfitsofRef.52)54)toBEBC–Gargamelle53)andCDHSneutrinoDISdata,performedover20yearsago,didnotallowadis-criminationbetween1/Q2andthelogarithmicdescriptionofscalingviolationtobemade.Therefore,itispossibletoconcludethatpresentneutrinoDISdatanowhavebecomemoreprecise.Indeed,theiranalysisshiftedtheeffectofperturbativescreen-ingof1/Q2-correctionsfromLOtoNNLO.ThenextgenerationofmoredetailedtestsofQCDinneutrinoDISisnowontheagenda10).3.2TheapplicationoftheBernsteinpolynomialmethod
Inthispartofourmini-reviewthebasicstepsoftheBernsteinpolynomialapproach,proposedinRef.55)andrecentlyusedintheprocessofNNLOfitstotheCCFR’97xF3datainRef.21),willberecalled.ThebasicconstructionsofthisapproacharetheBernsteinaveragesforthexF3SF:
FnkF3(Q2)
=
1
0
dxpnk(x)xF3(x,Q2),
(16)
wherepnk(x)aretheBernsteinpolynomials,whichcanbepresented,whenk≤n,
inthefollowingform:
npnk(x)−k=p(n,k)
(−1)l
l=0
2
)
)Γ(n−k+1).
(18)
2
UsingEqs.(16)–(18),itispossibletoexpresstheBernsteinaveragesforxF3through
xF3oddMellinmomentsas:
FnkF3(Q2n)
=p(n,k)
−k(−1)l
l=0
ThefinalNNLOexpression,whichincludestheestimatesofsometheoreticaluncer-tainties,is21):
NNLO:
αs(MZ)=0.1153±0.0041(exp)±0.0061(theor),
(21)
Itisworthwhiletomentionthat,despitethequalitativeagreement,thecentralNLOvaluesofEq.(20),obtainedwiththehelpoftheBernsteinpolynomialtech-nique,arelowerthantheexistingdeterminationsofαs(MZ)fromtheCCFR’97xF3data,whichresultfromtheNLODGLAPanalysis3)14)andtheapplicationoftheJacobipolynomialtechnique17)19)23).Moreover,atNNLO,theresultofEq.(21)intersectswiththeNNLOdeterminationofαs(MZ)ofRef.23)(seeEqs.(12)and(13))withinexistingerrorsonly.ThecomparisonbetweentheresultsoftheJacobiandBernsteinpolynomialdeterminationsofαs(MZ)andoftherelatedtheoreticaluncertaintieswaspresentedinRef.23).Intheprocessofthesestudies,definitedisagreementswererevealedbetweensomeresultsoftheworksofRef.21)andRef.23).Theoriginofthesedisagreementsisunclearatpresentandstimu-latesamoredetailedanalysisoftheNNLOrealizationsoftheJacobiandBernsteinpolynomialapproaches.Note,however,thatthedefinitechoiceofthescaleparam-eterintheJacobipolynomialfitsleadstoimprovingtheagreementoftheresultsofapplicationsofthetwomethods23).Inviewofthisobservation,itispossiblethattheresultsofRef.21)containlargertheoreticaluncertaintiesduetotheneglectofscale-dependenceambiguities.Ontheotherhand,contrarytotheBernsteinpoly-nomialanalysis,theNNLOJacobipolynomialfitsofRef.23)alsousedapproximateinformationaboutthevaluesoftheNNLOcorrectionstoanomalousdimensionsofevenmomentsofxF3.Itshouldbestressedthatthisapproximationcanbeelimi-natedaftercompletingtheprogramofexplicitcalculationsofNNLOcontributionstonon-singletDGLAPkernels,whichisnowinprogress56).AstothecurrentapplicationsoftheDGLAPmethodintheconcreteNNLOfitstoDISdata,theycaninprinciplebebasedonthemachineryoftheBayesiantreatmentofsystematicerrorsofDISdata(seee.g.Ref.15))andtheapproximateNNLOmodelsofDGLAPkernels,constructedinRefs.50)57)58).
Acknowledgements
IamgratefultoG.ParenteandA.V.Sidorovforourlongandfruitfulcollaboration,whichledustoanumberofresultsdiscussedinthismini-reviewandtoA.V.Kotikovforhiscontributiontoourcommonworks.ItisapleasuretothankS.I.Alekhin,G.Altarelli,S.Catani,B.I.Ermolaev,S.A.KulaginandF.J.Yndurainformanyusefuldiscussions.ItisanhonourtoexpressmywarmgratitudetoJ.TranThanhVan,whocontributedalottoorganizingnon-formaldiscussionsbetweenexperimentalistsandtheoreticiansduringtheRencontresdeMoriondQCDsessions,whichhadanessentialinfluenceontheworksdiscussedabove,andespeciallyonthosedevotedtothestudyofneutrinoDISdataoftheCCFRcollaboration.SpecialthanksgotoM.GrecoforgivingmethepossibilitytopresentthetalkattheveryproductiveLaThuileConference.IalsowouldliketoexpressmyspecialthankstothemembersoftheTheoreticalPhysicsDivisionofCERNforcreatingapleasantscientificatmosphere.References
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