owingww.zzz13.com

PublishedforSISSAbySpringerReceived:March13,2014Revised:April23,2014Accepted:April23,2014Published:May20,2014HiggsphobicandfermiophobicZ'asasingledarkmattercandidateNanChen,aYingZhang,bQingWang,a,c,d,1GiacomoCacciapaglia,e,f,1AldoDeandreae,f,gandLucaPanizzihaDepartmentofPhysics,TsinghuaUniversity,Beijing100084,P.
R.
ChinabSchoolofScience,Xi'anJiaotongUniversity,Xi'an710049,P.
R.
ChinacCenterforHighEnergyPhysics,TsinghuaUniversity,Beijing100084,P.
R.
ChinadCollaborativeInnovationCenterofQuantumMatter,Beijing100084,P.
R.
ChinaeUniversitedeLyon,F-69622Lyon,FrancefUniversiteLyon1,Villeurbanne,CNRS/IN2P3,UMR5822,InstitutdePhysiqueNucleairedeLyon,F-69622VilleurbanneCedex,FrancegInstitutUniversitairedeFrance,103boulevardSaint-Michel,75005Paris,FrancehSchoolofPhysicsandAstronomy,UniversityofSouthampton,Higheld,SouthamptonSO171BJ,U.
K.
E-mail:chen-n08@mails.
tsinghua.
edu.
cn,hepzhy@mail.
xjtu.
edu.
cn,wangq@mail.
tsinghua.
edu.
cn,g.
cacciapaglia@ipnl.
in2p3.
fr,deandrea@ipnl.
in2p3.
fr,l.
panizzi@soton.
ac.
ukAbstract:Aspin-1Z'particleasasingledarkmattercandidateisinvestigatedbyas-sumingthatitdoesnotdirectlycoupletotheHiggsbosonandstandardmodelfermionsanddoesnotmixwiththephotonandZboson.
TheremainingdominantverticesarequarticZ′Z′ZZandZ′Z′W+W,whichcaninduceeectiveZ′Z′qqcouplingsthroughstandard-modelgauge-bosonloops.
Wediscussconstraintsfromthecosmologicalthermalrelicdensity,anddirectandindirect-detectionexperiments,andndthatadarkZ'canonlyexistabovetheWbosonmassthreshold,andtheeectivequarticcouplingofZ′Z′VVisboundedintheregionof103102.
Keywords:BeyondStandardModel,CosmologyofTheoriesbeyondtheSMArXivePrint:1403.
29181Correspondingauthor.
OpenAccess,cTheAuthors.
ArticlefundedbySCOAP3.
doi:10.
1007/JHEP05(2014)088Contents1Introduction12SU(2)LU(1)YU(1)theoryandZ'asaDMcandidate33RelicdensityconstraintofdarkZ'84EectivequarkvertexanddirectdetectionofdarkZ'95IndirectdetectionofdarkZ'135.
1PAMELAexperiment145.
2AMSIIexperiment145.
3FermiLATexperiment156CombinedresultandotherDMrelatedissues177Summary20AListofcouplings211IntroductionAfterthediscoveryofthe125GeVHiggsboson,thestandardmodel(SM)ofparticlephysicshasbecomeacompletetheory;withintheSM,theremainingtaskistheprecisionmeasurementsofvariousHiggsproperties,inparticularitscouplings,andtofurthernarrowdownthepossibleparameterspaceofnewphysics.
Althoughthehierarchyandmeta-stablevacuumproblemsremainfortheSMHiggstheoretically,fornewphysicsbeyondtheSMtheabsenceofnewphysicssignalsattheLHCtodateimpliesthatextensionsoftheSMstillonlyneedtorelyonthetraditionalparticlephysicsfactsofnon-zeroneutrinomassesandbaryonasymmetry.
Giventhesecircumstances,thepresenceofdarkmatter(DM)inourUniversebecomesanevenmoreimportantleadingempiricalevidencefortheexistenceofnewphysics,becausenoSMparticlecanaccountforDM.
Cosmologyandastrophysicstellusthatalmost85%ofmatterinouruniverseisdark,i.
e.
,neutral,non-luminousandnon-baryonic.
ThefactthattheabundanceofDMiscomparabletothatofordinaryvisiblematterseemstoimplythatDMmayhavethesameorsimilaroriginsandpropertiesasordinarymatter.
Ifweaccepttheconclusionofquantumeldtheory(QFT)thatallmattershouldbemadeofparticles,thenanunambiguous,non-gravitationalsignalofDMmustappearinparticlephysicsexperiments.
ThishasdriventheparticlephysicscommunitytotryhardertounravelDM'sstillenigmaticproperties.
–1–BecausedetailsoftheparticlepropertiesofDMarelacking,thebestinvestigativestrategyfortheoristsistotrytocoverasmuchgroundaspossible.
ConsideringthatQFTclassiesparticlesaccordingtotheirspin(evenoroddhalf-integers),elementaryparticlesdiscoveredsofarallhavelowspins.
MostDMcandidatesdiscussedsofarintheliteraturehavebeenassumedtobespin-0scalars[1–7]orspin-1/2spinors[8,9](forspin-1/2DM,therearesomanypapers,hereweonlycitetwoeectiveeldanalysispapers).
WhereasascalarDMhasrelativelysimplestructureandprovidespossibleintimateinterplaywiththe125GeVHiggs,aspinorDMextendsthetraditionalobservationthatmatteriscomposedofspin-1/2particles.
Theheavysterileneutrinos[10]andthelightestneutralino[11]insupersymmetricmodelsareDMcandidatesbelongingtothistype.
ApartfromscalarandspinorDM,thenextlevelofhigher-spincandidateparticlescomprisespin-1vectors.
Ifwelimitourselvestothesimplestvectorparticlescenarioinparticlephysics,asingleextraneutralvectorparticle,usuallydenotedbyZ',issucient.
Weshalldiscussthispossibilityinthepresentpaper.
Ahigherspincase,spin-3/2DM,hasbeendiscussedinref.
[12].
AvectorparticleZ'canbeviewedasagaugebosonthatmediatesanextraU(1)gaugeforcebeyondtheconventionalSMstrongSU(3)cforceandelectroweakSU(2)LU(1)Yforces.
Forasyetunknownreasons,thisadditionalU(1)gaugesymmetryisspontaneouslybroken,thusyieldingamassiveZ'.
SMplusZ'isaminimalandwellmotivatedgenerali-sationoftheSM;manynew-physicsmodelshavesuchaZ′boson(fordetailsseereviewrefs.
[13–16])asanecessaryconstituentandremnantfornew-physicsinteractions.
BeforeJuly4,2012,theHiggswasthesuperstarofparticlephysicssearchesandaZ'onlyplayedasupportingrole.
Withthediscoveryofthe125GeVHiggs,aZ'nowbecomesoneofthehotnew-physicscandidateparticlesandtheLHCisactivelysearchingforitinvariouschan-nels,withthemodel-dependentlowermassboundalreadyreachingtheTeV-energyregiondependingonthenalstateitisassumedtodecayinto.
Now,ifwefurthertaketheZ'asaDMcandidatethuschangingtheZ'fromvisibletoinvisible,theinteractionsbetweentheinvisibleZ'andSMparticleswillbestronglyreduced,andthecorrespondingsearchstrategies(suchasdirectdetection,indirectdetection,andcolliderexperiment)willchangewithrespecttothoseforvisibleZ'.
Variousconstraintsmustthereforebere-examined.
Intheliterature,theinvisibleZ'hasbeenintensivelydiscussedasamessengerbe-tweenthevisiblesector(whichcontainstheSMparticles)andahiddensector(towhichDMbelongs)[17–28]:insuchascenario,theSMparticlescanbeeitherchargedundertheadditionalgaugesymmetryornot.
IntheeventthatSMparticlesareneutralwithrespecttotheextraU(1)symmetry,theinteractionoccursviaeectiveoperatorsconnectingdi-rectlyZ'totheSMsector.
ThesimplestcaseisthekineticmixingtermsbetweentheSMhyperchargeeldstrengthandthenewAbelianeldstrength[29].
TheunderlyingreasoninadoptingZ'asaportaltothehiddensectorstemsfromthetraditionalmediatingroleofgaugebosons.
InthistypeofDMmodels,therearetoomanyunknownsconcerningthehiddensector,asituationthatisnothelpfulinDMsearches.
Inthispaper,weconsideranalternativesimpleapproachbyignoringtheconventionalmessengerroleofZ',andinsteadtreatitaspurematter.
Thisapproachissimilartotheminimaldarkonmodel[1–6]whereSMisminimallyexpandedwiththeadditionofadarkscalar(SM+D),exceptnowwereplacethescalardarkonDwithasinglevectorDMcandidateZ'.
Thechangefromthe–2–traditionalZ'portalmodeltoourpresentsingledarkZ'approachissimilartothatfromtheHiggsportalmodel(whereascalaristakenasamessengerbetweenthevisibleandhiddensectors)[30,31]tothedarkonmodel.
Afterthereduction,becauseoftheuniquechoiceofDMcandidate,wecanignoretheuncertaintiesarisingfromthearbitraryhiddensectorinthetraditionalZ'orHiggsportalmodels.
ThedierenceinthepresentapproachwithrespecttoscalarDMisthatourdarkZ'isavectorparticle,whichbehavesnotlikeascalarorHiggsboson,butverymuchlikeaZbosonofSMandwillhaverelativelycomplexinteractionstructureowingtoitspolarisation.
Aspin-1darkmattercandidateappearsinmodelswithoneextradimensions[32]andhasbeenwidelystudiedinthiscontext[33,34].
Notethatthisisnotagenericpredictionofextradimensions,asinhigherthen5dimensionsthecandidateisascalar[35–37],andascalarisagainfoundinmodelsofpseudo-GoldstoneHiggsinwarpedspace[38]andtechnicolor[39].
Thispaperisorganisedasfollows.
InsectionII,intermsofthemodel-independentextendedelectroweakchiralLagrangianandthesixassumptionsneededtokeepZ'dark,wedeterminethenecessaryoperatorsthatcoupleourdarkZ'toSMparticles.
InsectionIII,wecalculatetherelicdensityproducedfromoursingledarkZ',deriveaconstraintontheeectivecouplingofthedarkZ'pairtoWorZpairs.
SectionIVlooksatthedirect-detectionconstraint,wherewecomputetheSMgauge-boson-loop-inducedZ′Z′qqvertexanddiscussdirectdetection.
SectionVexaminesindirect-detectionconstraintsandincludesdiscussionsofthePamela,AMS02,andFermiLATexperiments.
InsectionVI,wediscussthecombinedresultsandsomeotherpossibleDMrelatedissues.
SectionVIIpresentsasummary.
SomenecessaryresultsforsectionIIaretobefoundinappendixA.
2SU(2)LU(1)YU(1)theoryandZ'asaDMcandidateTomakeourinvestigationgeneral,westartfromamodel-independenteectiveextendedelectroweakchiralLagrangian(EEWCL)proposedinref.
[40],LEEWCL=L2+L42.
1)whereL2nforn=1,2,3,.
.
.
isthep2n-orderofEEWCLwithZ'andallSMgaugeelds,plusfournecessaryGoldstonebosonsdescribedbyatwo-by-twounitarymatrixeldU.
ThesymmetryoftheLagrangianisSU(2)LU(1)YU(1),whichwillbespontaneouslybrokentoU(1)em.
TheSMHiggseldandthefermioneldsarenotincludedintheaboveLagrangian:infact,couplingstotheHiggsarenotrequiredbythesymmetryofthemodeland,ifpresent,wouldreproduceamodelofHiggsportalDM.
Thisimpliesthatweignorethepossibledirect(ortree-order)couplingsbetweenZ'andHiggs(adarkZ'withdirectcouplingtotheHiggshasbeendiscussedinref.
[41])orbetweenZ'andSMfermions(adarkZ'couplingdirectlytoSMfermionshasbeendiscussedinref.
[42]).
ThesearethersttwoassumptionsweadoptforourdarkZ'.
ThesehiggsphobicandfermiophobicdarkZ'assumptionssimplifyourtheorysignicantly,andwetakeitastherststepinourinvestigation.
AlthoughattreelevelwecanignoretheexplicitHiggsandfermioncouplings,loopscanstillinduceeectivecouplings.
WeshallcarefullydiscusstheseloopsinsectionIV.
–3–Thep2-orderLagrangianL2is[40]L2=14f2tr[VV]+14β1f2tr[TV]tr[TV]+14β2f2tr[V]tr[TV]+14β3f2tr[V]tr[V],(2.
2)whereT≡Uτ3U,V≡(DU)U,andτ3isthePaulimatrix.
ThecovariantderivativeisDU=U+igWUiUτ32g′BiU(g′B+g′′X)I,(2.
3)whereW≡τi2Wi,B,andXaretheSU(2)L,U(1)Y,andU(1)gaugeelds,respectively,andg,g′,g′′arethecorrespondingcouplingconstants;g′isaspecialStueckelbergcoupling.
W,B,andXaregaugeeigenstates,andtheZ'discussedinthispaperisthephysicalstateafterdiagonalization.
InL2,β1andβ2aremassmixingparameters.
Thep4-orderLagrangianL4iscomposedofthreeterms[40]L4=LK+LB+LA,(2.
4)forwhichthekinetictermLKisLK=14BνBν12tr[WνWν]14XνXν.
(2.
5)ThenormaltermLBisLB=12α1gg′Bνtr[TWν]+i2α2g′Bνtr[T[V,Vν]]+iα3gtr[Wν[V,Vν]]+α4tr[VVν]tr[VVν]+α5tr[VV]tr[VνVν]+α6tr[VVν]tr[TV]tr[TVν]+α7tr[VV]tr[TVν]tr[TVν]+14α8g2tr[TWν]tr[TWν]+i2α9gtr[TWν]tr[T[V,Vν]]+12α10tr[TV]tr[TVν]tr[TV]tr[TVν]+α11gνρλtr[TV]tr[VνWρλ]+α12gtr[TV]tr[VνWν]+α13gg′νρλBνtr[TWρλ]+α14g2νρλtr[TWν]tr[TWρλ]+α15tr[V]tr[TV]tr[TVν]tr[TVν]+α16tr[V]tr[TV]tr[VνVν]+α17tr[V]tr[TVν]tr[VVν]+α18tr[V]tr[Vν]tr[TV]tr[TVν]+α19tr[V]tr[Vν]tr[VVν]+α20tr[V]tr[V]tr[TVν]tr[TVν]+α21tr[V]tr[V]tr[VνVν]+α22tr[V]tr[V]tr[Vν]tr[TVν]+α23tr[V]tr[Vν]tr[V]tr[Vν]+gg′′α24Xνtr[TWν]+g′g′′α25BνXν+α26νρλtr[V]tr[TVν]tr[T[Vρ,Vλ]]+ig′α27νρλtr[V]tr[TVν]Bρλ+igα28νρλtr[V]tr[TVν]tr[TWρλ]+gα29νρλtr[V]tr[VνWρλ]+ig′′α30νρλXνtr[T[Vρ,Vλ]]+ig′′α31Xνtr[T[V,Vν]]+g′′α32νρλtr[V]tr[TVν]Xρλ+α33tr[V]tr[TVν]tr[T[V,Vν]]+g′g′′α34νρλBνXρλ+gg′′α35νρλXνtr[TWρλ]+ig′α36tr[V]tr[TVν]Bν+igα37tr[V]tr[TVν]tr[TWν]+gα38tr[V]tr[VνWν]+g′′α39tr[V]tr[TVν]Xν+igα40tr[V]tr[TVνWν](2.
6)amongwhichα1,α8,α24,α25arekineticmixingparameters;α12α14,α30,α33α40areassociatedwithCP-violationterms.
TheanomaloustermLAisLA=α42g2νρλtr[WνWρλ]+α43g′2νρλBνBρλ+g′′2α44νρλXνXρλ(2.
7)WiththeexceptionofthekinetictermLKinL4,theαi(i=1,14)correspondtotermsappearingintheconventionalelectroweakchiralLagrangian(EWCL)[43]withoutZ',αj(j=15,44)correspondtotermsintheEEWCLinvolvingZ'.
Intherestofthe–4–paper,weshallignoretheCP-violationtermsandtheanomalyterms.
Thisconstitutesourthirdandfourthassumptions.
OurfthassumptionistoforbidpossiblemixingbetweentheZ'andtheelectroweakbosonsγ,ZinordertokeeptheZ'dark.
ThisimpliesthatthegaugeeigenstateXcanbeidentiedwiththephysicalstateZ′.
Nomassmixingrequiresβ2=0,andnokineticmixingleadstoα24=α25=0.
Furthermore,weneedtosettozerotheStueckelbergcouplingg′=0.
AtthispointitisimportanttostressthattherearemixingparameterswhichdonotinvolvetheZ':β1isamassmixingtermamongSMgaugebosons,thusitwillinduceacorrectiontotheρparameter(Tparameter),whileα1andα8inducekineticmixings,thusgeneratingacontributiontotheSparameter.
Theremainingβ3generatesacontributiontothemassoftheZ',whichisgivenbyMZ′=g′′f12β3.
(2.
8)Withtheaboveveassumptions,thep4-orderEEWCL(2.
4)intheunitarygaugegivesthefollowingLagrangianuptoquarticcouplings:L=14VνVν12W+νWν+iCV+VνW+Wν+iC+V(W+νWVνWνW+Vν)+iCV1V2V3Vν1V2V3ν+D++W+W+WνWν+D++W+WWνW+ν+D+V1V2W+WV1νV2ν+D+V1V2W+V1WνVν2+DV1V2V3V4V1V2V3νVν4.
(2.
9)HereVidenotestheneutralgaugebosonsZ,γ,andZ′,andthevariousCandDcoecientsintermsoftheαiandβicoecientsaregiveninappendixA.
Notethat,withourfthassumptionofnomassmixing,CV1V2V3vanishes.
InordertokeeptheZ'stable,weneedtoimposethevanishingofverticesthatarelinearintheZ'eld;thisconstitutesoursixthassumption.
ForthetriplecouplingsCZ′+andC+Z′tovanish,weneedtosetα31=0;thereisthennotriplecouplinginvolvingZ′(notethatwithoutthetriplecouplingCZ′VVwithV=γ,Z,W±,thelongitudinalWandZscatteringwillnotinvolveZ′attreelevelandthatimposesnounitarityconstraintontheZ'couplings).
LeftwithfourZ'-independenttriplecouplings,oneC+γhasxedcoecients,whichonlydependonSMcouplings,whereastheotherthreeCγ+,CZ+,andC+Zarefree,correspondingtoindependentcoecientsα2,α3,andα9.
Forquarticcouplings,D+ZZ′=0leadstoα16=0,D+ZZ′=D+Z′Z=0leadstoα17=0,andDZ′ZZZ=g3Zg′′(2α15+α16+α17)=0furtherleadstoα15=0.
(NotethatiftheStueckel-bergcouplingg′doesnotvanish,itwillalsogeneratenonzeroD+ZZ′,D+ZZ′,D+AZ′,D+AZ′.
)Wearenallyleftwith16nonzeroquarticcouplings.
Amongthem,D+γγandD+γγalsohavexedcoecientsandarenotfree.
TheotherfourD+γZ,D+γZandD+Zγ=D+γZonlyrelyonα3andthenarerelatedtothetriplevertex.
There-maining11nonzeroquarticcouplingsD++,D++,D+ZZ,D+Z′Z′,D+ZZ,D+Z′Z′,DZZZZ,DZ′Z′ZZ,DZ′ZZ′Z,DZ′Z′Z′Z,DZ′Z′Z′Z′arefree,correspondingtothe11indepen-dentcoecientsα4toα7,α10,andα18toα23.
IntableI,welistdetailsofallthetripleandquarticcouplings.
Giventhesixassumptionsstatedabove:(i)nodirectcouplingtotheHiggs;(ii)nodirectcouplingtofermions;–5–couplingsexistinSM(modiedby)independentcontrolbyvanishingconditionCγ+yes(α2,3,9)---CZ+yes(α2,3,9)---CZ′+noyesα31&g′α31=g′=0C+γyes(notmodied)---C+Zyes(α3)---C+Z′no-g′g′=0CV1V2V3no--alwaysD++yes(α3,4,9)---D++yes(α3,4,5,9)---D+γγyes(notmodied)---D+ZZyes(α3,5,7)---D+Z′Z′noyesα21-D+γZyes(α3)---D+γZ′no-g′g′=0D+ZZ′noyesα16&g′α16=g′=0D+γγyes(notmodied)---D+ZZyes(α3,4,6)---D+Z′Z′noyesα19-D+γZyes(α3)---D+γZ′=D+Z′γno-g′g′=0D+Zγ=D+γZyes---D+ZZ′=D+Z′Znoyesα17&g′α17=g′=0DZZZZnoyesα10-DZ′ZZZnoyesα152α15+α16+α17=0DZ′Z′ZZnoyesα20-DZ′ZZ′Znoyesα18-DZ′Z′Z′Znoyesα22α22=0DZ′Z′Z′Z′noyesα23-Table1.
Listoftripleandquarticcouplings.
(iii)noCPviolatingtermsintheEEWCL;(iv)noanomaloustermsintheEEWCL;(v)nokineticnormassmixingterms;(vi)nosingleZ'couplings;wearenallyleftwithfourZ′-dependentquarticverticeseachinvolvingtwoZ′elds,D+Z′Z′≡g1=4g2g′′2(α5+α21)(2.
10)–6–D+Z′Z′≡g2=4g2g′′2(α4+α19)(2.
11)DZ′Z′ZZ≡g3=g2Zg′′2(α5+2α7+4α20+2α21)(2.
12)DZ′ZZ′Z≡g4=4g2Zg′′2(α4+α6+2α18+α19)(2.
13)whereg2Z=g2+g′2.
Herewelimitourselvestoverticeswithupto4particles:theEEWCLdoescontainZ'verticeswithmoregaugeboson,howevertheirphysicaleectissubleadingandwewillnotconsiderthemanyfurther.
Theabovecouplingsdocontributetoelectroweakprecisiontestsatlooplevel:suchcontributionsarelog-divergent,andtheircontributioncanbeabsorbedinthetree-levelcontributionsβ1andα1,8,whosevaluesarethereforestronglyconstrained.
Inthefollowingwewillthereforenotconsiderboundsfromprecisiontests,astheydonotgiveuniqueindicationonthesizeofthequarticcouplingslistedabove.
Uptonow,wehavenotsuccessfullybuiltthedetailunderlyingmodelwhichsatisfyabovesixconstraintsandhavenonzerofourpointverticesof(10)-(13),weleavethismodelconstructiontofutureinvestigations.
Tokeepthenumberoffreeparameterstoaminimum,weshalltakeinouranalysisauniquecouplingconstantg0andconsidervedierentarrangementsofcouplingconstantsasfollows:universalcase:g1=g2=4g2g0,g3=38g4=32g2Zg0,orα4=α5=α19=α21≡g02g′′2,α6=α7=α18=α19=α20=0case1:g1=4g2g0,g3=32g2Zg0,g2=g4=0,orα5=α21≡g02g′′2,α4=α6=α7=α18=α19=α20=0case2:g2=4g2g0,g4=4g2Zg0,g1=g3=0,orα4=α19≡g02g′′2,α5=α6=α7=α18=α20=α21=0case3:g3=3g2Zg0,g1=g2=g4=0,orα7=α20≡g02g′′2,α4=α5=α6=α18=α19=α21=0case4:g4=6g2Zg0,g1=g2=g3=0,orα6=α18≡g02g′′2,α4=α5=α7=α19=α20=α21=0.
Thestrategyoftakingthesechoicesisthat:rst,wetrytokeeptheSMfactorsex-plicitandthentheboundsareshownong′′2αi(whereg′′isrelatedtothemassoftheZ',thusaphysicalconnection).
Second,foruniversalcase,wetrytobalancealloffourverticescouplingsbyrequiringg1=g2andg3=3g4/8.
Third,forremainingfourcases,wefocusonαicoecientsappearedintheoriginalEEWCL.
Consideringthatthereareeightcouplingsα4,α5,α6,α7,α18,α19,α20,α21enterintoeectivefourpointvertices,wetakethecorrespondingfourcouplingsg1,g2,g3,g4aregeneratedbyspecicoperatorpairs,soitwouldmakesensetofollowthepatternssuggestedbyeachtypeofoperatorpairs,asfollows:α5,α21generatecase1;α4,α19generatecase2;α7,α20generatecase3;α6,α18generatecase4.
Inthefollowingsections,weshallmainlyfocusontheabovevecasesandderiveinformationontheuniquecouplingg0whendiscussingpossibleconstraintsfromvariousexperiments.
Somepossibleexceptionsarealsodiscussed.
–7–Figure1.
TwoannihilationprocessesofZ′toSMweakgaugebosons.
3RelicdensityconstraintofdarkZ'InthestandardCosmologypicture(ΛCDM),itisassumedthattheDMparticlesareinthermalequilibriumwiththeotherSMparticlesviavariousfundamentalprocessessuchasZ′Z′→PPwherePisanySMparticle.
Inthehigh-temperatureEarlyUniverse,DMparticleswerekeptinthermalequilibriumaslongasthereactionrate,scaledbythetemperature,wasfasterthantheexpansionrateH(theHubbleparameter)oftheUniverse.
TheUniversecooleddownasitcontinuedtoexpand.
AttemperaturesaroundwhichthereactionratefellbelowtheexpansionrateH,theDMparticlesbegantodecouplefromthethermalbath.
TheDMparticleswillcontinuetoannihilateintoSMparticlesupuntilthepointwhentheynolongerencounteroneanother.
TheremainingDMparticleswillthenbecometherelicsthatwecanobservetoday.
ThetwopossibleannihilationprocessesforourdarkZ'areshowningure1.
TheirannihilationratestoSMweakgaugebosonsareσWv=11r2W9*64πmZ′[(224r4W+112r2W+136)g21(160r4W+80r2W+176)g1g2+(152r4W+128r2W+96)g22]+(88r4W40r2W+56)g213043r4W+32r2W+323g1g2+24r4W+1123r2W43g22v2(3.
1)σZv=11r2Z18*64πmZ′[(224r4Z+112r2Z+136)g23(160r4Z+80r2Z+176)g3g4+(152r4Z+128r2Z+96)g24]+(88r4Z40r2Z+56)g233043r4Z+32r2Z+323g3g4+24r4Z+1123r2Z43g24v2,(3.
2)inwhichrW=mZ′/mWandrZ=mZ′/mZ,andvistherelativevelocityofthecollidingDMparticles.
Inaboveresult,duetothenon-relativisticcharacteristicsofourdarkZ',wehavetakenexpansionintermsofpowersofvuptotheorderofv2.
Notethatfortheseannihila-tionstoarise,themassofZ'shouldbeheavierthanthemassoftheSMweakgaugebosons.
IftheZ'massislighterthantheSMweakbosonmass,weinsteadusetheloop-inducedeectiveZ′Z′qqverticesgiveninthenextsection.
Thecorrespondingannihilationrateisσv=v2616πfKV,f√22cf1m2fM2Z′M2Z′2+m2fM2Z′+13πfKVA,f√22cf1m2fM2Z′3/2M2Z′,(3.
3)whereKV,fandKVA,faretheeectivecouplingsintroducedinthenextsectionineq.
(4.
1),andcfisthecolorindex,whichis1forleptonsand3forquarks.
–8–TherelicdensityiscalculatedbysolvingtheBoltzmannequationinthestandardapproximationprocedure[44],WIMPh2=1.
07*109mplxFGeV1√gS1a+3b/xF,(3.
4)wherehisthescaledHubbleconstant,xF=mZ′/TFwithTFthefreezingtemperature,mpl=1.
22*1019GeV,gSthetotalnumberofeectivelyrelativisticdegreesoffreedomatfreeze-outtemperature,andaandbareparametersintheexpansionσv=a+bv2+O(v4).
Thefreeze-outtemperatureparameterxFcanbeevaluatedbynumericallysolvingthefollowingequation:xF=lnc(c+2)458gmZ′mpl(a+6b/xF)2π3√gSx1/2F,(3.
5)whereg=3isthenumberofdegreesoffreedomfortheZ′DM,andcisanumericalconstantusuallytakenequalto1/2.
WithDMmassrangingfromGeVtoTeV,xF≈25andremainsessentiallyconstant.
Inournumericalanalysis,wedemandthattheresultingrelicdensitybelessthanthemeasuredvaluefromPLANCKWIMPh2=0.
1199±0.
0027at68%CL[45],whichleadstoconstraintsfortheeectivecouplingconstantsgi,i=1,2,3,4andthedarkZ'massMZ′.
Theresultofthevedierentcouplingarrangementsintro-ducedattheendofsectionIIforM′Z>100GeVisshowningure2,wherewehaveusedresult(3.
1)and(3.
2)toperformourcalculation.
Notethatbecausetheordinateislogarithmicallyscaled,thediagramwouldbeunabletoshowclearlythepossibledeviationsofseveralpercentfromtheexperiment,andhencearenotplotted.
FordarkZ′massesbelowtheWmassthreshold,weinsteaduseeq.
(3.
3)toperformourestimation;there-sultsareshowningure3.
NotethatintheintermediateregionmW/2100GeV,fromtherelicdensityanddirectandindirect-detectionexperi-mentswhereeectiveZ′Z′qqcouplingsareinducedfromSMgauge-bosonloops,weproducevedierentcouplingscenariosthatareintheregion103–102(fortheuniversalcase,theresultisgiveningure14).
Thisrangeofcouplingcanberelaxedbeyondthevecasesana-lyzedfordirect-detectionexperiments,butcannotbechangedforindirect-detectionexper-iments.
ToimproveFermiLATγ-rayspectrabyourdarkZ',werequireaboostfactorfrom300to8000.
WecheckedthatevenifourdarkZ'masslieswithinthelow-energyregion,itcannotreducetensionsamongtheobservedpossibleDMsignalswithothernull-resultexperiments.
Theboundsweextractedarethereforeratherrobustandmodel-independent.
AcknowledgmentsThisworkwassupportedbytheNationalScienceFoundationofChina(NSFC)underGrantNo.
11075085,NationalBasicResearchProgramofChina(973Program)underGrantNo.
2010CB833000,theSpecializedResearchFundfortheDoctoralProgramofHighEducation–20–ofChinaNo.
20110002110010,FranceChinaParticlePhysicsLaboratory(FCPPL)nan-cialsupport,andtheTsinghuaUniversityInitiativeScienticResearchProgram.
GCandADalsoacknowledgepartialsupportfromtheLabex-LIO(LyonInstituteofOrigins)undergrantANR-10-LABX-66.
ADispartiallysupportedbyInstitutUniversitairedeFrance.
AListofcouplingsCγ+=g3g′gZ1g2+α2+α3+α9CZ+=g2gZ[1g′2α2+g2(α3+α9)]CZ′+=2g2g′′α31C+γ=gg′gZC+Z=g2gZg2gZα3C+Z′=0CV1V2V3=0(A.
1)Here,gZ≡g20+g21.
D++=g22+g4α3+α42+α9D++=g22+g4α3+α42+α5α9D+γγ=g2g′2g2ZD+ZZ=g4g2Z2g4α3+g2g2Z(α5+α7)D+Z′Z′=4g2g′′2(α5+α21)D+γZ=2g3g′g2Z2g3g′α3D+γZ′=0D+ZZ′=2g2gZg′′α16D+γγ=g2g′2g2ZD+ZZ=g4g2Z+2g4α3+g2g2Z(α4+α6)D+Z′Z′=4g2g′′2(α4+α19)D+γZ=g3g′g2Z+g3g′α3D+γZ′=D+Z′γ=0–21–D+Zγ=D+γZ=g3g′g2Z+g3g′α3D+ZZ′=D+Z′Z=g2gZg′′α17DZZZZ=14g4Z(α4+α5+2α6+2α7+2α10)DZ′ZZZ=g3Zg′′(2α15+α16+α17)DZ′Z′ZZ=g′′2g2Z(α5+2α7+4α20+2α21)DZ′ZZ′Z=4g′′2g2Z(α4+α6+2α18+α19)DZ′Z′Z′Z=4g′′3gZ(α16+α17+2α22)DZ′Z′Z′Z′=4g′′4(α4+α5+2α19+2α21+4α23)OpenAccess.
ThisarticleisdistributedunderthetermsoftheCreativeCommonsAttributionLicense(CC-BY4.
0),whichpermitsanyuse,distributionandreproductioninanymedium,providedtheoriginalauthor(s)andsourcearecredited.
References[1]V.
SilveiraandA.
Zee,Scalarphantoms,Phys.
Lett.
B161(1985)136[INSPIRE].
[2]C.
P.
Burgess,M.
PospelovandT.
terVeldhuis,Theminimalmodelofnonbaryonicdarkmatter:asingletscalar,Nucl.
Phys.
B619(2001)709[hep-ph/0011335][INSPIRE].
[3]W.
-L.
Guo,L.
-M.
Wang,Y.
-L.
Wu,Y.
-F.
ZhouandC.
Zhuang,Gauge-singletdarkmatterinaleft-rightsymmetricmodelwithspontaneousCP-violation,Phys.
Rev.
D79(2009)055015[arXiv:0811.
2556][INSPIRE].
[4]X.
-G.
He,S.
-Y.
Ho,J.
TandeanandH.
-C.
Tsai,ScalardarkmatterandStandardModelwithfourgenerations,Phys.
Rev.
D82(2010)035016[arXiv:1004.
3464][INSPIRE].
[5]X.
-G.
HeandJ.
Tandean,HiddenHiggsbosonattheLHCandlightdarkmattersearches,Phys.
Rev.
D84(2011)075018[arXiv:1109.
1277][INSPIRE].
[6]K.
Cheung,Y.
-L.
S.
Tsai,P.
-Y.
Tseng,T.
-C.
YuanandA.
Zee,Globalstudyofthesimplestscalarphantomdarkmattermodel,JCAP10(2012)042[arXiv:1207.
4930][INSPIRE].
[7]J.
M.
Cline,K.
Kainulainen,P.
ScottandC.
Weniger,Updateonscalarsingletdarkmatter,Phys.
Rev.
D88(2013)055025[arXiv:1306.
4710][INSPIRE].
[8]J.
-M.
Zhengetal.
,Constrainingtheinteractionstrengthbetweendarkmatterandvisiblematter:I.
Fermionicdarkmatter,Nucl.
Phys.
B854(2012)350[arXiv:1012.
2022][INSPIRE].
[9]K.
Cheung,P.
-Y.
Tseng,Y.
-L.
S.
TsaiandT.
-C.
Yuan,Globalconstraintsoneectivedarkmatterinteractions:relicdensity,directdetection,indirectdetectionandcollider,JCAP05(2012)001[arXiv:1201.
3402][INSPIRE].
[10]B.
W.
LeeandS.
Weinberg,Cosmologicallowerboundonheavyneutrinomasses,Phys.
Rev.
Lett.
39(1977)165[INSPIRE].
[11]J.
S.
Hagelin,G.
L.
KaneandS.
Raby,Perhapsscalarneutrinosarethelightestsupersymmetricpartners,Nucl.
Phys.
B241(1984)638[INSPIRE].
–22–[12]R.
DingandY.
Liao,Spin3/2particleasadarkmattercandidate:aneectiveeldtheoryapproach,JHEP04(2012)054[arXiv:1201.
0506][INSPIRE].
[13]P.
Langacker,ThephysicsofheavyZ′gaugebosons,Rev.
Mod.
Phys.
81(2009)1199[arXiv:0801.
1345][INSPIRE].
[14]P.
Langacker,G.
Paz,L.
-T.
WangandI.
Yavin,Z′-mediatedsupersymmetrybreaking,Phys.
Rev.
Lett.
100(2008)041802[arXiv:0710.
1632][INSPIRE].
[15]P.
Langacker,ThephysicsofheavyZ′gaugebosons,Rev.
Mod.
Phys.
81(2009)1199[arXiv:0801.
1345][INSPIRE].
[16]E.
Salvioni,G.
VilladoroandF.
Zwirner,MinimalZ′models:presentboundsandearlyLHCreach,JHEP11(2009)068[arXiv:0909.
1320][INSPIRE].
[17]J.
KumarandJ.
D.
Wells,CERNLHCandILCprobesofhidden-sectorgaugebosons,Phys.
Rev.
D74(2006)115017[hep-ph/0606183][INSPIRE].
[18]M.
Pospelov,A.
RitzandM.
B.
Voloshin,SecludedWIMPdarkmatter,Phys.
Lett.
B662(2008)53[arXiv:0711.
4866][INSPIRE].
[19]M.
Pospelov,SecludedU(1)belowtheweakscale,Phys.
Rev.
D80(2009)095002[arXiv:0811.
1030][INSPIRE].
[20]E.
Dudas,Y.
Mambrini,S.
PokorskiandA.
Romagnoni,(In)visibleZ′anddarkmatter,JHEP08(2009)014[arXiv:0904.
1745][INSPIRE].
[21]S.
Khalil,H.
-S.
LeeandE.
Ma,BoundonZ′massfromCDMSIIinthedarkleft-rightgaugemodelII,Phys.
Rev.
D81(2010)051702[arXiv:1002.
0692][INSPIRE].
[22]B.
Batell,Darkdiscretegaugesymmetries,Phys.
Rev.
D83(2011)035006[arXiv:1007.
0045][INSPIRE].
[23]M.
R.
Buckley,D.
HooperandJ.
L.
Rosner,AleptophobicZ′anddarkmatterfromgrandunication,Phys.
Lett.
B703(2011)343[arXiv:1106.
3583][INSPIRE].
[24]M.
T.
Frandsen,F.
Kahlhoefer,S.
SarkarandK.
Schmidt-Hoberg,DirectdetectionofdarkmatterinmodelswithalightZ′,JHEP09(2011)128[arXiv:1107.
2118][INSPIRE].
[25]P.
Gondolo,P.
KoandY.
Omura,LightdarkmatterinleptophobicZ′models,Phys.
Rev.
D85(2012)035022[arXiv:1106.
0885][INSPIRE].
[26]V.
Barger,D.
MarfatiaandA.
Peterson,LHCanddarkmattersignalsofZ′bosons,Phys.
Rev.
D87(2013)015026[arXiv:1206.
6649][INSPIRE].
[27]A.
Alves,S.
ProfumoandF.
S.
Queiroz,ThedarkZ′portal:direct,indirectandcollidersearches,JHEP04(2014)063[arXiv:1312.
5281][INSPIRE].
[28]G.
Arcadi,Y.
Mambrini,M.
H.
G.
TytgatandB.
Zaldivar,InvisibleZ′anddarkmatter:LHCvsLUXconstraints,JHEP03(2014)134[arXiv:1401.
0221][INSPIRE].
[29]B.
Holdom,TwoU(1)'sandchargeshifts,Phys.
Lett.
B166(1986)196[INSPIRE].
[30]B.
PattandF.
Wilczek,Higgs-eldportalintohiddensectors,hep-ph/0605188[INSPIRE].
[31]T.
Hambye,Hiddenvectordarkmatter,JHEP01(2009)028[arXiv:0811.
0172][INSPIRE].
[32]G.
ServantandT.
M.
P.
Tait,IsthelightestKaluza-Kleinparticleaviabledarkmattercandidate,Nucl.
Phys.
B650(2003)391[hep-ph/0206071][INSPIRE].
–23–[33]H.
-C.
Cheng,J.
L.
FengandK.
T.
Matchev,Kaluza-Kleindarkmatter,Phys.
Rev.
Lett.
89(2002)211301[hep-ph/0207125][INSPIRE].
[34]G.
Belanger,M.
KakizakiandA.
Pukhov,DarkmatterinUED:theroleofthesecondKKlevel,JCAP02(2011)009[arXiv:1012.
2577][INSPIRE].
[35]B.
A.
Dobrescu,D.
Hooper,K.
KongandR.
Mahbubani,Spinlessphotondarkmatterfromtwouniversalextradimensions,JCAP10(2007)012[arXiv:0706.
3409][INSPIRE].
[36]G.
Cacciapaglia,A.
DeandreaandJ.
Llodra-Perez,AdarkmattercandidatefromLorentzinvariancein6D,JHEP03(2010)083[arXiv:0907.
4993][INSPIRE].
[37]A.
Arbey,G.
Cacciapaglia,A.
DeandreaandB.
Kubik,Darkmatterinatwistedbottle,JHEP01(2013)147[arXiv:1210.
0384][INSPIRE].
[38]M.
Frigerio,A.
Pomarol,F.
RivaandA.
Urbano,Compositescalardarkmatter,JHEP07(2012)015[arXiv:1204.
2808][INSPIRE].
[39]T.
A.
RyttovandF.
Sannino,UltraminimaltechnicoloranditsdarkmatterTIMP,Phys.
Rev.
D78(2008)115010[arXiv:0809.
0713][INSPIRE].
[40]Y.
Zhang,S.
-Z.
WangandQ.
Wang,St¨uckelbergmechanismandchiralLagrangianforZ′boson,JHEP03(2008)047[arXiv:0803.
1275][INSPIRE].
[41]Q.
-H.
Cao,C.
-R.
Chen,C.
S.
LiandH.
Zhang,Eectivedarkmattermodel:relicdensity,CDMSII,FermiLATandLHC,JHEP08(2011)018[arXiv:0912.
4511][INSPIRE].
[42]Z.
-H.
Yuetal.
,Constrainingtheinteractionstrengthbetweendarkmatterandvisiblematter:II.
scalar,vectorandspin-3/2darkmatter,Nucl.
Phys.
B860(2012)115[arXiv:1112.
6052][INSPIRE].
[43]T.
AppelquistandG.
-H.
Wu,TheelectroweakchiralLagrangianandnewprecisionmeasurements,Phys.
Rev.
D48(1993)3235[hep-ph/9304240][INSPIRE].
[44]E.
W.
KolbandM.
S.
Turner,Theearlyuniverse,Nature294(1981)521[INSPIRE].
[45]Planckcollaboration,P.
A.
R.
Adeetal.
,Planck2013results.
XVI.
Cosmologicalparameters,arXiv:1303.
5076[INSPIRE].
[46]XENON100collaboration,E.
Aprileetal.
,Darkmatterresultsfrom225livedaysofXENON100data,Phys.
Rev.
Lett.
109(2012)181301[arXiv:1207.
5988][INSPIRE].
[47]LUXcollaboration,D.
S.
Akeribetal.
,FirstresultsfromtheLUXdarkmatterexperimentattheSanfordUndergroundResearchFacility,Phys.
Rev.
Lett.
112(2014)091303[arXiv:1310.
8214][INSPIRE].
[48]SuperCDMScollaboration,R.
Agneseetal.
,Searchforlow-massWIMPswithSuperCDMS,arXiv:1402.
7137[INSPIRE].
[49]PAMELAcollaboration,O.
Adrianietal.
,PAMELAresultsonthecosmic-rayantiprotonuxfrom60MeVto180GeVinkineticenergy,Phys.
Rev.
Lett.
105(2010)121101[arXiv:1007.
0821][INSPIRE].
[50]N.
Fornengo,L.
MaccioneandA.
Vittino,Constraintsonparticledarkmatterfromcosmic-rayantiprotons,JCAP04(2014)003[arXiv:1312.
3579][INSPIRE].
[51]AMScollaboration,M.
Aguilaretal.
,FirstresultfromthealphamagneticspectrometerontheInternationalSpaceStation:precisionmeasurementofthepositronfractioninprimarycosmicraysof0.
5–350GeV,Phys.
Rev.
Lett.
110(2013)141102[INSPIRE].
–24–[52]J.
Kopp,ConstraintsondarkmatterannihilationfromAMS-02results,Phys.
Rev.
D88(2013)076013[arXiv:1304.
1184][INSPIRE].
[53]J.
F.
Navarro,C.
S.
FrenkandS.
D.
M.
White,Auniversaldensityprolefromhierarchicalclustering,Astrophys.
J.
490(1997)493[astro-ph/9611107][INSPIRE].
[54]L.
Bergstrom,P.
UllioandJ.
H.
Buckley,Observabilityofgamma-raysfromdarkmatterneutralinoannihilationsintheMilkyWayhalo,Astropart.
Phys.
9(1998)137[astro-ph/9712318][INSPIRE].
[55]Fermi-LATcollaboration,M.
Ackermannetal.
,Darkmatterconstraintsfromobservationsof25milkywaysatellitegalaxieswiththeFermiLargeAreaTelescope,Phys.
Rev.
D89(2014)042001[arXiv:1310.
0828][INSPIRE].
[56]J.
L.
Feng,J.
Kumar,D.
MarfatiaandD.
Sanford,Isospin-violatingdarkmatter,Phys.
Lett.
B703(2011)124[arXiv:1102.
4331][INSPIRE].
[57]M.
I.
GreshamandK.
M.
Zurek,LightdarkmatteranomaliesafterLUX,Phys.
Rev.
D89(2014)016017[arXiv:1311.
2082][INSPIRE].
[58]V.
Cirigliano,M.
L.
Graesser,G.
OvanesyanandI.
M.
Shoemaker,ShiningLUXonisospin-violatingdarkmatterbeyondleadingorder,arXiv:1311.
5886[INSPIRE].
[59]E.
DelNobile,G.
B.
Gelmini,P.
GondoloandJ.
-H.
Huh,UpdateonlightWIMPlimits:LUX,liteandlight,JCAP03(2014)014[arXiv:1311.
4247][INSPIRE].
[60]CDMScollaboration,R.
Agneseetal.
,SilicondetectordarkmatterresultsfromthenalexposureofCDMSII,Phys.
Rev.
Lett.
111(2013)251301[arXiv:1304.
4279][INSPIRE].
[61]M.
T.
FrandsenandI.
M.
Shoemaker,Theup-shotofinelasticdown-scatteringatCDMS-Si,arXiv:1401.
0624[INSPIRE].
[62]S.
-Z.
Wang,S.
-Z.
Jiang,F.
-J.
GeandQ.
Wang,ElectroweakchiralLagrangianforW′boson,JHEP06(2008)107[arXiv:0805.
0643][INSPIRE].
[63]E.
J.
Chun,J.
-C.
ParkandS.
Scopel,Darkmatterandanewgaugebosonthroughkineticmixing,JHEP02(2011)100[arXiv:1011.
3300][INSPIRE].
–25–