Testing Mass Varying Neutrino With Short Gamma Ray Burst

In this paper we study the possibility of probing for the absolute neutrino mass and its variation with short Gamma Ray Burst (GRB). We have calculated the flight time difference between a massive neutrino and a photon in two different approaches to mass v

TestingMassVaryingNeutrinoWithShortGammaRayBurst

InstituteofHighEnergyPhysics,ChineseAcademyofSciences,

P.O.Box918-4,Beijing100049,P.R.Chinaand

b

DepartmentofAstronomy,NanjingUniversity,Nanjing210093,P.R.China

Inthispaperwestudythepossibilityofprobingfortheabsoluteneutrinomassandits

variationwithshortGammaRayBurst(GRB).Wehavecalculatedthe ighttimedi erencebetweenamassiveneutrinoandaphotonintwodi erentapproachestomassvaryingneutrinos.Firstlyweparametrizetheneutrinomassasafunctionofredshiftinamodelindependentway,thenweconsidertwospeci cmodelswheretheneutrinomassvariesduringtheevolutionoftheQuintessence elds.Ourcalculationsshowingeneralthevalueofthetimedelayischangedsubstantiallyrelativetoaconstantneutrinomass.Furthermoreournumericalresultsshowthatthe ighttimedelayinthesemodelsisexpectedtobelargerthanthedurationtimeoftheshortGRB,whichopensapossibilityoftestingthescenarioofmassvaryingneutrinowiththeshortGRB.

a

HongLia, ZigaoDaib,andXinminZhanga

arXiv:hep-ph/0411228v1 17 Nov 2004

IntherecentyearsastronomicalobservationsshowthattheUniverseisspatiallyacceleratingatthepresenttime[1].Thesimplestaccountofthiscosmicaccelera-tionseemstobearemnantsmallcosmologicalconstant,butitsu ersfromthedi cultiesassociatedwiththe netuningandcoincidenceproblem,somanyphysicistsareattractedwiththeideathattheaccelerationisdrivenbydynamicalscalar elds,suchasQuintessence.Inmodelsofdarkenergywitharemnantsmallcosmologicalcon-4

stantor(trueorfalse)vacuumenergyρ～(2×10 3ev).Thisenergyscale～10 3evissmallerthantheenergyscalesinparticlephysics,butinterestinglyiscompara-bletotheneutrinomasses.Thisindicatesapossibleconnectionbetweentheneutrinosandthedarkenergy.Furthermore,inQuintessence-likemodelsmQ～10 33eV,whichsurprisinglyisalsoconnectedtotheneutrinomassesviaasee-sawformulamQ～m2ν/MplwithMpltheplanckmass.

Istherereallyanyconnectionsbetweentheneutrinosanddarkenergy?Giventheargumentsaboveitisquiteinterestingtomakesuchaspeculationonthisconnec-tion.Ifyes,howeverintermsofthelanguageoftheparticlephysicsitrequirestheexistenceofnewdynam-icsandnewinteractionsbetweentheneutrinoandthedarkenergysector.Recentlytherearesomestudiesintheliteratureonthepossiblerealizationofthemodelsonneutrinosanddarkenergy[2,3,4,5,6,7,8,9,10,11].Oneoftheinterestingpredictionsofthesemodels[12]isthatneutrinomassesarenotconstant,butvaryasafunc-tionoftimeduringtheevolutionoftheuniverse.Inthispaperwestudythepossibilityoftestingthisscenarioofmassvaryingneutrinoswiththegammarayburst.Therearestrongevidencesforthenon-vanishingneu-trinomassesfromtheneutrinooscillationexperiments,howevertheneutrinomassesgivenbythesolarneutri-nosandatmosphereneutrinosexperimentsarenotthe

～absolutevalues,butmasssquaredi erence: m2

2

m22

m2

(1+

z′)2H

In this paper we study the possibility of probing for the absolute neutrino mass and its variation with short Gamma Ray Burst (GRB). We have calculated the flight time difference between a massive neutrino and a photon in two different approaches to mass v

thetdonewouldbeabletodeterminetheabsolutevalueoftheneutrinomass,howeveringeneralsincethepho-tonsaretrappedinsidethe reballandarereleasedmuchlater,theneutrinoandthephotonwillnotbeemittedatthesametime.Thiswilladdansystematicerrorinthemeasurementofthetd,consequentlyanuncertaintyinthedeterminationoftheneutrinomasses.ToreducethistypeofsystematicerrorwefocusontheshortGRBwhosedurationisgenerallylessthan2secondsorevenmuchshorter,ForexampleBurstAndTransientSourceExperiment(BATSE)

hasdiscoveredaGRBwithdura-tiononly5milliseconds[22].IthasbeenwidelyarguedthatshortGRBsareproducedbythemergeroftwocom-pactobjects(e.g.,neutronstarsorblackholes)[23],andthattheirenvironmentsarelikelytobelow-densitymediabecausethemergingplaceisfarawayfromthebirthsiteofoneneutronstar.Evenso,thepredictedafterglowsfromshortGRBsappeartobedetectablewithcurrentinstrumentsintheSwiftera[24].Oncesuchafterglowsaredetected,theredshiftsofshortGRBsmaybemea-sured.

FromFig.1onecanseethatingeneralthetimedelaytdislongerthanthedurationoftheshortGRB.Forexampleform=0.6eV,z=2,p=10Mev,tdisaround400seconds;especiallyfortheGRB030329withredshiftz=0.17,tdis112seconds.Thetimedelayforthesecasesisexpectedtobedetectableinprinciple.Ifnot,thiswillputalimitontheabsolutevalueoftheneutrinomassbetterthanthecosmologicallimits.

t

p eV

FIG.1:td(unitinsecond)asafunctionofpfordi erentred-shiftsz=0.17(solid),0.5(dashed),1(dotted),2(dashdotted).

Inthefollowingwewilldiscussthepossibilityoftest-ingonthevariationoftheneutrinomasseswiththeshortGRB.Firstofallweparametrizethevariationoftheneu-trinomassinamodelindependentway,thenwewilltaketwoconcretemodelsofQuintessenceforthecalculationofmassvariationandthetimedelayoftheneutrinos.Considerageneralcasewheremνisanarbitraryfunc-tionoftheredshiftZ,i.e.m(Z),onecanexpanditintermsofredshiftz.Forsmallz,weget:

m(Z)=m0+m′Z+

1

2

mZ+...).(3)

De ningc≡

m′(0)

m202

(1+z′)2H0

p

.Notethathere

theneutrinomassin(5)variesasafunctionofredshiftz.FromFig.2onecanseethatthevariationofthe

t

z=0.3

z=0.05

m/p

FIG.2:td(unitinsecond)asafunctionof

m0

f

ννφφ+h.c.,(6)

andmν～

v2

2Mpl

1+β

Q0

M),(8)

pl

In this paper we study the possibility of probing for the absolute neutrino mass and its variation with short Gamma Ray Burst (GRB). We have calculated the flight time difference between a massive neutrino and a photon in two different approaches to mass v

wherem0andQ0aretheneutrinomassandthevalueofQuintessenceatpresenttime,andMplisthePlankscale.Nowtheformulaforthetime-delayisgivenby

t1d=p)21M2pl)

t0

(1+β

Q

tQ

˙23

(ρB+

3

inradiation-dominatedandωB=0in

thematter-dominatedUniverse.

Foranumericalstudy,weconsideramodelofQuintessencewithainversepower-lawpotential[25],

V=V0Q α.

(13)

Thistypeofmodelisshown[25,26]tohavethepropertyoftrackingbehavior.Inthecalculationwetakeα=0.5whichgivesrisetoωQ0< 0.78constrainedbytheWMAP.InFig.3weplotthenumericalvalueoftd.FromFig.3onecanseethetimedelaytdisenhanced

t

z=3

z=0.17

m/p

FIG.3:Plotoftd(unitinsecond)asafunctionof

m0

p

)2～10 7,andβ= 3,tdcan

reach1624secondsatz=3.Forcomparisonaconstantneutrinomassβ=0reducestdto1075seconds.

Forthecaseofhierarchicalneutrinomasspatter,theheavyneutrinomasssetbytheatmosphareneutrinoos-cillationism～0.05eV.Withthisvalueonecanseefrom

3

Fig.3thattdisonly0.5secondsfortheconstant-mass

neutrinoswithenergyp=12Mevandredshiftz=3.ThisvalueismuchsmallerthantheshortGRBdurationandwillbedi culttobeobservable.Iftheneutrinomassvaries,asshowninFig.3tdwillbeenhancedsubstan-tially.NumericallyFig.3showsthattdcanbeincreasedto3seconds.

CertainlytheresultsshowninFig.3dependsontheQuintessencemodel.Foranillustration,weconsideranotherQuintessencemodel

V=V0exp(

λ

p

.The

threecurvescorrespondtoparametersβ=0(solid),-2(dash),10(dot)respectivelyforagivenredshift.

Insummarywehaveinthispaperdiscussedthepossi-bilityofusingtheShortGRBtoprobefortheabsoluteneutrinomassesanditsvariation.Byadetailedcalcu-lationwehaveshownthatwiththecurrentcosmologicallimitsonthedegeneratedneutrinomassesthetimedelaytdwillbeingenerallongerthanthetimeoftheduration,furthermorethechangestothetdcausedbythevariationoftheneutrinomassesarealsoexpectedtobelargerthanthetimeoftheduration.Ourresultsindicatethepossi-bilityoftestingthescenarioofmassvaryingneutrinointhefutureneutrinotelescope.

Acknowledgments:WeareindebtedtoDrs.MinzheLi,BoFengandXiaojunBiforenlighteningdiscussionsandProfs.RobertoPeccei,SandipPakvasaandDr.HeinrichPaesforreadingthemanuscript.WethankDr.NanLiangforhelpfuldiscussionsonGRBs.ThisworkwassupportedinpartbyNationalNaturalScienceFoundationofChina(grantNos.90303004,19925523,10233010and10221001)andbyMinistryofScienceandTechnologyofChina(underGrantNo.NKBRSFG19990754).

In this paper we study the possibility of probing for the absolute neutrino mass and its variation with short Gamma Ray Burst (GRB). We have calculated the flight time difference between a massive neutrino and a photon in two different approaches to mass v

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