Fluxes of atmospheric muons underwater depending on the small-x gluon density

The prompt muon contribution to the deep-sea atmospheric muon flux can serve as a tool for probing into the small-x feature of the gluon density inside of a nucleon, if the muon energy threshold could be lifted to 100 TeV. The prompt muon flux underwater i

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002 ebF 02 1v3812030/hp-pe:hviXraFluxesofatmosphericmuonsunderwaterdependingonthesmall-xgluondensity

AMisaki1,TSSinegovskaya2,SISinegovsky2andNTakahashi3

1WasedaUniversity,Okubo3-4-1,Shinjuku-ku,Tokyo,169-8555Japan2IrkutskStateUniversity,664003Russia3

HirosakiUniversity,036-8561Japan

E-mail:sinegovsky@api.isu.runnet.ru

Abstract.Thepromptmuoncontributiontothedeep-seaatmosphericmuon uxcanserveasatoolforprobingintothesmall-xfeatureofthegluondensityinsideofanucleon,ifthemuonenergythresholdcouldbeliftedto100TeV.Thepromptmuon uxunderwateriscalculatedtakingintoconsiderationpredictionsofrecentcharmproductionmodelsinwhichthesmall-xbehaviourofthegluondistributionisprobed.WediscussthepossibilityofdistinguishingthePQCDmodelsofthecharmproductiondi eringinthesmall-xexponentofthegluondistribution,inmeasurementsofthemuon uxatenergies10100TeVwithneutrinotelescopes.

Submittedto:J.Phys.G:Nucl.Phys.

1.Introduction

Acorrecttreatmentofthecharmhadroproductionisimportanttotheatmosphericmuonandneutrinostudies,sinceshort-livedcharmedparticles,D±,D0,

The prompt muon contribution to the deep-sea atmospheric muon flux can serve as a tool for probing into the small-x feature of the gluon density inside of a nucleon, if the muon energy threshold could be lifted to 100 TeV. The prompt muon flux underwater i

uxesatthegroundleveldependingstronglyonprotongluondistributionsatsmallxscale,x<10 5.

ThemuonspectraunderwatercomputedwiththemodelofPasqualietal.[2],inwhichusedweretheMRSD [5]andtheCTEQ3M[6]setsofPDFs,wererecentlydiscussed[7,8,9].Inthisnote,usingpredictionsofthePQCDmodel[3,4]forthecharmproduction,wediscussthePMcontributiontothedeep-seamuon uxatdepthstypicalforoperatingandconstructingneutrinotelescopes,AMANDA[10],ANTARES[11],Baikal[12],NESTOR[13].Duetolargedetectorvolumeandefectivearea(104 105m2)andhomogeneityofsurroundingmattertheseundericeanddeep-seainstallationshaveconsiderableadvantagesoverundergrounddetectorsforprobingveryhigh-energyatmosphericmuons.

Namely,herewetrytostudyaPM uxunderwaterdependenceonthepowerλofthesmall-xgluondistributionfunction:xg(x,Q2)∝x λ.Thenatureofthesmall-xbehaviourofthegluondensityisnowunderextensivediscussion(see,forexample,[14,15,16,17,18,19]).Thesmall-xbehaviourofthePDFsisthesubjectofthedeepinterestbecauseanunderstandingoftheunderlyingdynamicsisfaryetfrombeingclear.2.PDFsandcharmproductionmodels

Duetodominantsubprocessinheavyquarkshadroproduction,gg→

c

The prompt muon contribution to the deep-sea atmospheric muon flux can serve as a tool for probing into the small-x feature of the gluon density inside of a nucleon, if the muon energy threshold could be lifted to 100 TeV. The prompt muon flux underwater i

1

-V

eG1

-rs1

-s2-mc ,)θ ,µE(µφ)V3

eG / µE(10

10

10

10108

Eµ , GeV

Figure1.Verticalsea-levelmuon uxdataandpredictions.Experiments: –

Artyomovsk[24],△–Baksan[25], –MSU[26], –Frejus[27],

–MACRO[28],–LVD[29].Thelowersolidlinestandsforconventionalmuons.Therestofcurvesrepresentthetotalmuon ux,sumofpromptmuonsandconventionalones.

Pasquali,RenoandSarcevic[2](hereafterPRS–dottedlineswithnumbers1,2,3)andmodelsbyGelmini,GondoloandVarieschi[3,4](GGV–thincurveswithnumbers0.1–0.5).Thesemodelsareusedfurtherincalculationsofthedeep-seamuon ux.LetussketchoutPQCDmodels.

2.1.ThemodelbyPasquali,RenoandSarcevic

2.1.1.PRS-1.ThePRS-1model(dottedlinesin gures1,2)(identicalwiththePQCD-1inreference[9])isbasedontheMRSD set[5].ThePDFinputparametersare

thefollowings:xg(x,Q2 0.50)～xasx→0,4-momentumtransfersquaredQ20=4GeV2

;

thesealightquarkasymmetry,

d,istakingintoconsideration;theQCDscaleintheminimalsubtractionscheme(MS

4

=0.215GeV,correspondstothee ectivecouplingattheZbosonmassscaleαs(MZ2

)=0.111.ThefactorizationscaleisµF=2mc,therenormalizationoneisµR=mc,wherethecharmquarkmass,mc,ischosentobeequal1.3.Thesea-levelpromptmuon uxhasbeenparameterizedbyauthors[2]withtheequation:

lg[E3D,Λc

1µφµ(Eµ)·(cm 2ssr 1GeV2) 1]= 5.91+0.290y+0.143y2 0.0147y3,

(1)

wherey=lg(

Eµ

The prompt muon contribution to the deep-sea atmospheric muon flux can serve as a tool for probing into the small-x feature of the gluon density inside of a nucleon, if the muon energy threshold could be lifted to 100 TeV. The prompt muon flux underwater i

2.1.2.PRS-2.InthePRS-2model(thesameasthePQCD-2inreference[9])CTEQ3Mset[6]wasused.CorrespondinginputswhichwereutilizedinthismodelareΛ

(γ0+γ1y+γ2y2+γ3y3)

1GeV

cm 2s 1sr 1GeV 1.

(4)

Intable1 vesetsoftheparameterstoequation(4)arepresentedfordi erentvaluesoftheindexλofthesmall-xgluondestribution.

Table1.Parametersofthepromptmuonspectrumatsealevel(4).

0.13.122.70 0.0951.49 0.21480.23.542.71 0.0821.12 0.02850.31.802.380.045 0.820.9110.40.972.090.160 2.571.7490.50.581.840.257 4.052.455

The prompt muon contribution to the deep-sea atmospheric muon flux can serve as a tool for probing into the small-x feature of the gluon density inside of a nucleon, if the muon energy threshold could be lifted to 100 TeV. The prompt muon flux underwater i

3.Theconventionalmuon ux

Themainsourceoftheatmosphericmuonsupto～50TeVaredecayscosmicraypionsandkaons.The ux(conventional)of(π,K)-muonsbasedonthenuclearcascademodelby[33](seealso[32,34]).partofthisspectrumfortheverticalmaybeapproximatedwiththecm 2s 1sr 1GeV 1):

3.672

14.35EµforE1<Eµ E2,π,K

φµ(Eµ,0)=.3 4

10EµforEµ>E2

ofsecondary

iscomputedHigh-energyequation(in

(5)

whereE1=1.5878×103GeV,E2=4.1625×105GeV.

Zenith-angledistributionofatmosphericmuonsatsea-levelwascomputedinthereference[35]wheredetailcomparisonbetweenthecalculatedatmosphericmuonspectraandthesea-levelexperimentaldataatdi erentzenithangleswasmade(seealso[9]).Theconventionalmuon uxcomputedfortheverticaldirectionisshownin gure1(thelowersolidline).

Eachof vethinlinesin gure1presentsthesumoftheconventionalmuon ux(5)andtheGGVpromptmuon ux(4)correspondingtotheexponentλ=0.1,0.2,0.3,0.4,0.5(numbersnearlines).DottedlinesshowthesameforPRSmodels,equations(1-3).Forcomparisontherearealsoshowncontributionsduetothequark-gluonstringmodelandtherecombinationquark-partonone[31,32](thedash-dotlineandthedashlinerespectively).Ratiosofpromptmuon uxestotheconventionaloneareshownin gure2.Asonecansee,thecrossoverenergyforthePM uxandconventionalonecoversthewideregionfrom～150TeVto～3PeV,thatismorethanoneorderofthemagnitude.

ItisworthtonotethatoldQGSMprediction[31]athighenergiesiswithinGGVpromptmuon uxesaswellthatofRQPMiswithinPRSresults( gures1,2).4.Promptmuoncomponentofthe uxunderwater

Muonenergyspectraandangledistributionsofthe uxunderwaterwascomputedwiththemethodby[36].Thecollisionintegralinthekineticequationincludestheenergylossofmuonsduetobremsstrahlung,directe+e pairproductionandphotonuclearinteractions.Theionizationenergylossandthesmall-vpartofthelossduetoe+e pairproduction(v<2·10 4,wherevisthefractionoftheenergylostbythemuon)weretreatedascontinuousones.

Inourcalculationsofunderwatermuon uxesatdi erentzenithangles,weused,asaboundaryspectra,PQCDPM uxescalculatedonlyfortheverticaldirectionatthegroundlevel,supposingtheisotropicapproximationforpromptmuonstobeareliableatleastfor104<Eµ<106GeVatzenithanglesθ 80 .

Thepromptmuonfractionofthe uxunderwater,Rpm,de nedasratioofthepromptmuonintegralspectrumtotheconventionalone,ispresentedin gure3forthe

The prompt muon contribution to the deep-sea atmospheric muon flux can serve as a tool for probing into the small-x feature of the gluon density inside of a nucleon, if the muon energy threshold could be lifted to 100 TeV. The prompt muon flux underwater i

(E)

π,K

(E) / φ

D,Λc

φ

10

10

10

10

Eµ , GeV

Figure2.Ratioofthedi erentialpromptmuonspectrumatsealeveltotheconventionalone.

. pm

Ã

L '

F

FRV q K NP Z H

5430

,

5 ,

m

PS

(m *H9

**9 0567 l

**9 0567 l 356 &7(4 0 l

Figure3.Promptmuoncontributionath=4kmw.e.vs.Eµ.

The prompt muon contribution to the deep-sea atmospheric muon flux can serve as a tool for probing into the small-x feature of the gluon density inside of a nucleon, if the muon energy threshold could be lifted to 100 TeV. The prompt muon flux underwater i

( K q

m

K,p

( K q ,

m

l l (m ! 7H9

m

L '

F

5 ,

m

PS

FRVq

NP Z H

NP

Figure4.Ratioofthepromptmuon uxunderwatertotheconventionaloneasafunctionofcosθatEµ≥100TeV.

depthof4kmofthewaterequivalent(w.e.)andforcosθ=0.2.Asisseenfromthis gure,Rpmrelatedtothegluondensityslopeλ=0.5isafactor3greaterthanthatforλ=0.1atEµ 10TeV.

Zenith-angledistributionsofthepromptmuoncontributionatdepths1-4kmw.e.,calculatedforEµ>100TeV,areshownin gure4.HereweusedpredictionsoftheGGVmodelfortwovaluesofthegluondensityexponent,λ=0.1(dash)andλ=0.5(solid).Asonecanseein gure4,Rpmincreasesfortheverticaldirectionfromabout0.2atthedepthoftheBaikalNT(1.15km)[12]toabout0.5attheNESTORdepth(～4km)[13].Forthelargerzenithangles,θ～75 ,thiscontributionbecomesapparentlysizableatdepths3 4km.Di erencesinthepredictionsowingtoachangeofλ,from0.1to0.5(seeh=2and3kmw.e.),arealsoclearlyvisible:theratioRpm(λ=0.5)/Rpm(λ=0.1)ath=2kmw.e.growsfromabout1.5toabout5ascosθchangesfrom1to0.2.

Herewesupposednodi erencesbetweenPRSandGGVcalculationsapartfromthoserelatedtothecharmproductioncrosssections.Actuallyoneneedstocomparetheprimaryspectrumandcomposition,nucleonandmesonproductioncrosssectionsandotherdetailsoftheatmosphericnuclearcascadebeingusedinabovecomputations.Thesesourcesofuncertaintieswouldbeconsideredelsewhere.

The prompt muon contribution to the deep-sea atmospheric muon flux can serve as a tool for probing into the small-x feature of the gluon density inside of a nucleon, if the muon energy threshold could be lifted to 100 TeV. The prompt muon flux underwater i

5.Summary

Inordertotestthesmall-xgluondistributione ectwehavecomputeddeep-seapromptmuon uxesusingpredictionsofcharmproductionmodelsbasedonNLOcalculationsofthePQCD[2]-[4].ThepossibilitytodiscriminatethePQCDmodels,di eringintheslopeofthegluondistribution,seemstobeachievableinmeasurementsoftheunderwatermuon uxatenergies50-100TeV.

Hardlyappearedatsealevelforenergiesupto105GeV( gures1,2),adependenceonthespectralindexλofthesmall-xgluondistributionbecomesmoredistinctatdepths3 4kmw.e.( gures3,4).Atthedepthof4kmandattheangleof～78 onecouldobservethePM uxtobeequal,forλ=0.5,totheconventionaloneevenformuonenergy～10TeV(thecrossoverenergy).Whileforλ=0.1thecrossoverenergyisabout70TeV.Forthehighenergythreshold,Eµ>100TeV,andath 3kmw.e.,theratioRpmisnearlyisotropicupto～60 .The“crossoverzenithangle”atagivendepth,θc(h),dependsapparentlyonthesmall-xexponentλofthegluondensityinsidecollidingnucleons:

cosθc|λ=0.5 0.3andcosθc|λ=0.1 0.1forh=3kmw.e.

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