Fluxes of atmospheric muons underwater depending on the smal

1、fluxes of atmospheric muons underwater depending on the smal 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

2、flux underwater i 3 002 ebf 02 1v3812030/hp-pe:hvixrafluxesofatmosphericmuonsunderwaterdependingonthesmall-xgluondensity amisaki1,tssinegovskaya2,sisinegovsky2andntakahashi3 1wasedauniversity,okubo3-4-1,shinjuku-ku,tokyo,169-8555japan2irkutskstateuniversity,664003russia3 hirosakiuniversity,036-8561j

3、apan e-mail:sinegovskyapi.isu.runnet.ru abstract.thepromptmuoncontributiontothedeep-seaatmosphericmuon uxcanserveasatoolforprobingintothesmall-xfeatureofthegluondensityinsideofanucleon,ifthemuonenergythresholdcouldbeliftedto100tev.thepromptmuon uxunderwateriscalculatedtakingintoconsiderationpredicti

4、onsofrecentcharmproductionmodelsinwhichthesmall-xbehaviourofthegluondistributionisprobed.wediscussthepossibilityofdistinguishingthepqcdmodelsofthecharmproductiondi eringinthesmall-xexponentofthegluondistribution,inmeasurementsofthemuon uxatenergies10100tevwithneutrinotelescopes. submittedto:j.phys.g

5、: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 densi

6、ty inside of a nucleon, if the muon energy threshold could be lifted to 100 tev. the prompt muon flux underwater i uxesatthegroundleveldependingstronglyonprotongluondistributionsatsmallxscale,x10 5. themuonspectraunderwatercomputedwiththemodelofpasqualietal.2,inwhichusedwerethemrsd 5andthecteq3m6set

7、sofpdfs,wererecentlydiscussed7,8,9.inthisnote,usingpredictionsofthepqcdmodel3,4forthecharmproduction,wediscussthepmcontributiontothedeep-seamuon uxatdepthstypicalforoperatingandconstructingneutrinotelescopes,amanda10,antares11,baikal12,nestor13.duetolargedetectorvolumeandefectivearea(104 105m2)andho

8、mogeneityofsurroundingmattertheseundericeanddeep-seainstallationshaveconsiderableadvantagesoverundergrounddetectorsforprobingveryhigh-energyatmosphericmuons. namely,herewetrytostudyapm uxunderwaterdependenceonthepowerofthesmall-xgluondistributionfunction:xg(x,q2)x .thenatureofthesmall-xbehaviourofth

9、egluondensityisnowunderextensivediscussion(see,forexample,14,15,16,17,18,19).thesmall-xbehaviourofthepdfsisthesubjectofthedeepinterestbecauseanunderstandingoftheunderlyingdynamicsisfaryetfrombeingclear.2.pdfsandcharmproductionmodels duetodominantsubprocessinheavyquarkshadroproduction,gg c the prompt

10、 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 ,) ,e()v3 eg / e(10 10 10 10108

11、e , gev figure1.verticalsea-levelmuon uxdataandpredictions.experiments: artyomovsk24,baksan25, msu26, frejus27, macro28,lvd29.thelowersolidlinestandsforconventionalmuons.therestofcurvesrepresentthetotalmuon ux,sumofpromptmuonsandconventionalones. pasquali,renoandsarcevic2(hereafterprsdottedlineswith

12、numbers1,2,3)andmodelsbygelmini,gondoloandvarieschi3,4(ggvthincurveswithnumbers0.10.5).thesemodelsareusedfurtherincalculationsofthedeep-seamuon ux.letussketchoutpqcdmodels. 2.1.themodelbypasquali,renoandsarcevic 2.1.1.prs-1.theprs-1model(dottedlinesin gures1,2)(identicalwiththepqcd-1inreference9)isb

13、asedonthemrsd set5.thepdfinputparametersare thefollowings:xg(x,q2 0.50)xasx0,4-momentumtransfersquaredq20=4gev2 ; thesealightquarkasymmetry, d,istakingintoconsideration;theqcdscaleintheminimalsubtractionscheme(ms 4 =0.215gev,correspondstothee ectivecouplingatthezbosonmassscales(mz2 )=0.111.thefactor

14、izationscaleisf=2mc,therenormalizationoneisr=mc,wherethecharmquarkmass,mc,ischosentobeequal1.3.thesea-levelpromptmuon uxhasbeenparameterizedbyauthors2withtheequation: lge3d,c 1(e)(cm 2ssr 1gev2) 1= 5.91+0.290y+0.143y2 0.0147y3, (1) wherey=lg( e the prompt muon contribution to the deep-sea atmospheri

15、c 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-2inreference9)cteq3mset6wasused.correspondinginputswh

16、ichwereutilizedinthismodelare (0+1y+2y2+3y3) 1gev cm 2s 1sr 1gev 1. (4) intable1 vesetsoftheparameterstoequation(4)arepresentedfordi erentvaluesoftheindexofthesmall-xgluondestribution. table1.parametersofthepromptmuonspectrumatsealevel(4). 0 0.0951.49 0.21480.23.542.71 0.0821.12 0.02850.31

17、.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. th

18、e prompt muon flux underwater i 3.theconventionalmuon ux themainsourceoftheatmosphericmuonsupto50tevaredecayscosmicraypionsandkaons.the ux(conventional)of(,k)-muonsbasedonthenuclearcascademodelby33(seealso32,34).partofthisspectrumfortheverticalmaybeapproximatedwiththecm 2s 1sr 1gev 1): 3.672 14.35ef

19、ore1e e2,k (e,0)=.3 4 10eforee2 ofsecondary iscomputedhigh-energyequation(in (5) wheree1=1.5878103gev,e2=4.1625105gev. zenith-angledistributionofatmosphericmuonsatsea-levelwascomputedinthereference35wheredetailcomparisonbetweenthecalculatedatmosphericmuonspectraandthesea-levelexperimentaldataatdi er

20、entzenithangleswasmade(seealso9).theconventionalmuon uxcomputedfortheverticaldirectionisshownin gure1(thelowersolidline). eachof vethinlinesin gure1presentsthesumoftheconventionalmuon ux(5)andtheggvpromptmuon ux(4)correspondingtotheexponent=0.1,0.2,0.3,0.4,0.5(numbersnearlines).dottedlinesshowthesam

21、eforprsmodels,equations(1-3).forcomparisontherearealsoshowncontributionsduetothequark-gluonstringmodelandtherecombinationquark-partonone31,32(thedash-dotlineandthedashlinerespectively).ratiosofpromptmuon uxestotheconventionaloneareshownin gure2.asonecansee,thecrossoverenergyforthepm uxandconventiona

22、lonecoversthewideregionfrom150tevto3pev,thatismorethanoneorderofthemagnitude. itisworthtonotethatoldqgsmprediction31athighenergiesiswithinggvpromptmuon uxesaswellthatofrqpmiswithinprsresults( gures1,2).4.promptmuoncomponentofthe uxunderwater muonenergyspectraandangledistributionsofthe uxunderwaterwa

23、scomputedwiththemethodby36.thecollisionintegralinthekineticequationincludestheenergylossofmuonsduetobremsstrahlung,directe+e pairproductionandphotonuclearinteractions.theionizationenergylossandthesmall-vpartofthelossduetoe+e pairproduction(v210 4,wherevisthefractionoftheenergylostbythemuon)weretreat

24、edascontinuousones. inourcalculationsofunderwatermuon uxesatdi erentzenithangles,weused,asaboundaryspectra,pqcdpm uxescalculatedonlyfortheverticaldirectionatthegroundlevel,supposingtheisotropicapproximationforpromptmuonstobeareliableatleastfor104e106gevatzenithangles 80 . thepromptmuonfractionofthe

25、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 thresho

26、ld could be lifted to 100 tev. the prompt muon flux underwater i (e) ,k (e) / d,c 10 10 10 10 e , gev figure2.ratioofthedi erentialpromptmuonspectrumatsealeveltotheconventionalone. . pm 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=

27、4kmw.e.vs.e. 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 !

28、 7h9 m l ' f 5 , m ps frvq np z h np figure4.ratioofthepromptmuon uxunderwatertotheconventionaloneasafunctionofcosate100tev. depthof4kmofthewaterequivalent(w.e.)andforcos=0.2.asisseenfromthis gure,rpmrelatedtothegluondensityslope=0.5isafactor3greaterthanthatfor=0.1ate 10tev. zenith-angledistribu

29、tionsofthepromptmuoncontributionatdepths1-4kmw.e.,calculatedfore100tev,areshownin gure4.hereweusedpredictionsoftheggvmodelfortwovaluesofthegluondensityexponent,=0.1(dash)and=0.5(solid).asonecanseein gure4,rpmincreasesfortheverticaldirectionfromabout0.2atthedepthofthebaikalnt(1.15km)12toabout0.5atthe

30、nestordepth(4km)13.forthelargerzenithangles,75 ,thiscontributionbecomesapparentlysizableatdepths3 4km.di erencesinthepredictionsowingtoachangeof,from0.1to0.5(seeh=2and3kmw.e.),arealsoclearlyvisible:theratiorpm(=0.5)/rpm(=0.1)ath=2kmw.e.growsfromabout1.5toabout5ascoschangesfrom1to0.2. herewesupposedn

31、odi erencesbetweenprsandggvcalculationsapartfromthoserelatedtothecharmproductioncrosssections.actuallyoneneedstocomparetheprimaryspectrumandcomposition,nucleonandmesonproductioncrosssectionsandotherdetailsoftheatmosphericnuclearcascadebeingusedinabovecomputations.thesesourcesofuncertaintieswouldbeco

32、nsideredelsewhere. 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 inordertotestt

33、hesmall-xgluondistributione ectwehavecomputeddeep-seapromptmuon uxesusingpredictionsofcharmproductionmodelsbasedonnlocalculationsofthepqcd2-4.thepossibilitytodiscriminatethepqcdmodels,di eringintheslopeofthegluondistribution,seemstobeachievableinmeasurementsoftheunderwatermuon uxatenergies50-100tev.

34、 hardlyappearedatsealevelforenergiesupto105gev( gures1,2),adependenceonthespectralindexofthesmall-xgluondistributionbecomesmoredistinctatdepths3 4kmw.e.( gures3,4).atthedepthof4kmandattheangleof78 onecouldobservethepm uxtobeequal,for=0.5,totheconventionaloneevenformuonenergy10tev(thecrossoverenergy)

35、.whilefor=0.1thecrossoverenergyisabout70tev.forthehighenergythreshold,e100tev,andath 3kmw.e.,theratiorpmisnearlyisotropicupto60 .the“crossoverzenithangle”atagivendepth,c(h),dependsapparentlyonthesmall-xexponentofthegluondensityinsidecollidingnucleons: cosc|=0.5 0.3andcosc|=0.1 0.1forh=3kmw.e. refere

36、nces 12345678910111213141516171819202122 learnedjgandmannheimk2000ann.rev.nucl.part.sci.50,679.pasqualil,renomhandsarcevici1999phys.rev.d59,034020.gelminig,gondolopandvarieschig2000phys.rev.d61,056011.gelminig,gondolopandvarieschig2021phys.rev.d63,036006.martinad,stirlingwjandrobertsrg1993phys.rev.d

37、47,867. laihletal.1995phys.rev.d51,4763;laihletal.1997phys.rev.d55,1280.misakia.etal1999proc.26icrc(saltlakecity)vol2,p139,hep-ph/9905399.naumovva,sinegovskayatsandsinegovskysi2000phys.atom.nucl.63,1923.sinegovskayatsandsinegovskysi2021phys.rev.d63,096004.andreseetal.(amandacollaboration)2000astropa

38、rt.phys.13,1.amrampetal.(antarescollaboration)2000astropart.phys.13,127.belolaptikoviaetal.(baikalcollaboration)1997astropart.phys.7,263. anassontzisegetal.(nestorcollaboration)2000nucl.phys.b(proc.suppl.)85,153.anderssonb(smallxcollaboration)2021smallxphenomenology:summaryandstatus,hep-ph/0204115.

39、ballrdandlandsho pv2000j.phys.g:nucl.part.phys.26,672.kaidalovab2021reggepolesinqcd,hep-ph/0103011.schleperp2021softhadronicinteractions,hep-ex/0102051.vogtr2000prog.part.nucl.phys.45,s105. yoshidar(onbehalfofzeusandh1collaboration)2021herasmall-xand/ordi raction,hep-ph/0102262. martinad,robertsrg,stirlingwjandtorners1999nucl.phys.b(proc.suppl.)79,ihletal2000eur.phys.j.c12,375. kuraevea,lipatovlnandfadinvs1976zh.eksp.teor.fiz.71,840;kuraevea,lipatovlnandfadinvs1977zh.eksp.teor.fiz.72,377;balitskyiiandlipatovln1978yad.fiz.yad.fiz.28,1597.brodskysjetal19