Mirror World and its Cosmological Consequences

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

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aInternationalJournalofModernPhysicsA cWorldScienti cPublishingCompanyMIRRORWORLDANDITSCOSMOLOGICALCONSEQUENCESZURABBEREZHIANIDipartimentodiFisica,Universit`adiL’Aquila,67010Coppito,L’Aquila,andINFN,LaboratoriNazionalidelGranSasso,67010Assergi,L’Aquila,Italy;e-mail:berezhiani@fe.infn.itWebrie yreviewtheconceptofaparallel‘mirror’worldwhichhasthesameparticlephysicsastheobservableworldandcouplestothelatterbygravityandperhapsotherveryweakforces.Thenucleosynthesisboundsdemandthatthemirrorworldshouldhaveasmallertemperaturethantheordinaryone.Bythisreasonitsevolutionshouldsubstan-tiallydeviatefromthestandardcosmologyasfarasthecrucialepochslikebaryogenesis,nucleosynthesisetc.areconcerned.Inparticular,weshowthatinthecontextofcertainbaryogenesisscenarios,thebaryonasymmetryinthemirrorworldshouldbelargerthanintheobservableone.Moreover,weshowthatmirrorbaryonscouldnaturallyconsti-tutethedominantdarkmattercomponentoftheUniverse,anddiscussitscosmologicalimplications.Keywords:Extensionsofthestandardmodel;baryogenesis;darkmatter1.IntroductionTheoldideathattherecanexistahiddenmirrorsectorofparticlesandinteractionswhichistheexactduplicateofourvisibleworldhasattractedasigni cantinterestoverthelastyears.Thebasicconceptistohaveatheorygivenbytheproduct

G×G′oftwoidenticalgaugefactorswiththeidenticalparticlecontents,whichcouldnaturallyemergee.g.inthecontextofE8′

Inparticular,onecanconsideraminimalsymmetry×E8superstring.

GSM

SU(2)×U(1)standsforthestandardmodelofobservable×G′SM,whereGSM=SU(3)×particles:threefamiliesofquarksandleptonsandtheHiggs,whileG′SM=[SU(3)×SU(2)×U(1)]′isitsmirrorgaugecounterpartwithanalogousparticlecontent:threefamiliesofmirrorquarksandleptonsandthemirrorHiggs.(Fromnowonall eldsandquantitiesofthemirror(M)sectorwillbemarkedby′todistinguishfromtheonesbelongingtotheobservableorordinary(O)world.)TheM-particlesaresingletsofGSMandviceversa,theO-particlesaresingletsofG′SM.Besidesthegravity,thetwosectorscouldcommunicatebyothermeans.Inparticular,ordinaryphotonscouldhavekineticmixingwithmirrorphotons2,34,ordinary(active)neutrinoscouldmixwithmirror(sterile)7neutrinosortwosectorscouldhaveacommongaugesymmetryof avour.

AdiscretesymmetryP(G G′)interchangingcorresponding eldsofGandG′,socalledmirrorparity,guaranteesthatbothparticlesectorsaredescribedby

1

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

2ZurabBerezhiani

thesameLagrangians,withallcouplingconstants(gauge,Yukawa,Higgs)havingthesamepattern,andthustheirmicrophysicsisthesame.

Ifthemirrorsectorexists,thentheUniversealongwiththeordinaryphotons,neutrinos,baryons,etc.shouldcontaintheirmirrorpartners.OnecouldnaivelythinkthatduetomirrorparitytheordinaryandmirrorparticlesshouldhavethesamecosmologicalabundancesandhencetheO-andM-sectorsshouldhavethesamecosmologicalevolution.However,thiswouldbeintheimmediatecon ictwiththeBigBangnucleosynthesis(BBN)boundsonthee ectivenumberofextralightneutrinos,sincethemirrorphotons,electronsandneutrinoswouldgiveacontribu-tiontotheHubbleexpansionrateequivalentto Nν

UniversetheM-systemshouldhavealowertemperature 6.14.thanTherefore,ordinaryinparticles.theearlyThissituationisplausibleifthefollowingconditionsaresatis ed:

A.AftertheBigBangthetwosystemsarebornwithdi erenttemperatures,namelythepost-in ationaryreheatingtemperatureintheM-sectorislowerthaninthevisibleone,TR′<TR.Thiscanbenaturallyachievedincertainmodels8,9,10.

B.Thetwosystemsinteractveryweakly,sothattheydonotcomeintothermalequilibriumwitheachotherafterreheating.Thisconditionisautomaticallyful lledifthetwoworldscommunicateonlyviagravity.Iftherearesomeothere ectivecouplingsbetweentheO-andM-particles,theyhavetobeproperlysuppressed.

C.Bothsystemsexpandadiabatically,withoutsigni cantentropyproduction.Ifthetwosectorshavedi erentreheatingtemperatures,duringtheexpansionoftheUniversetheyevolveindependentlyandtheirtemperaturesremaindi erentatlaterstages,T′<T,thenthepresenceoftheM-sectorwouldnota ectprimordialnucleosynthesisintheordinaryworld.

Atpresent,thetemperatureofordinaryrelicphotonsisT≈2.75K,andthemassdensityofordinarybaryonsconstitutesabout5%ofthecriticaldensity.MirrorphotonsshouldhavesmallertemperatureT′<T,sotheirnumberdensityisn′γ=x3nγ,wherex=T′/T.ThisratioisakeyparameterinourfurtherconsiderationssinceitremainsnearlyinvariantduringtheexpansionoftheUniverse.boundon Nνimpliestheupperboundx<0.64 Nν1/4TheBBN.Asformirrorbaryons,adhoctheirnumberdensityn′bcanbelargerthannb,andiftheratioβ=n′b/nbisabout5orso,theycouldconstitutethedarkmatteroftheUniverse.

Inthispaperwestudythecosmologyofthemirrorsectoranddiscussthecom-parativetimehistoryofthetwosectorsintheearlyUniverse.Weshowthatduetothetemperaturedi erence,inthemirrorsectorallkeyepochsasthebaryoge-nesis,nucleosynthesis,etc.proceedatsomewhatdi erentconditionsthanintheobservableUniverse.Inparticular,weshowthatincertainbaryogenesisscenariostheM-worldgetsalarger11baryonasymmetrythantheO-sector,anditisprettyplausiblethatβ>1.ThissituationemergesinaparticularlyappealingwayintheleptogenesisscenarioduetotheleptonnumberleakingfromtheO-totheM-sectorwhichleadston′b≥nb,andcanthusexplainthenearcoincidenceofvisibleanddarkcomponentsinarathernaturalway12,13.

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...3

2.Mirrorworldandmirrorsymmetry

2.1.Particlesandcouplingsintheordinaryworld

NowadaysalmosteveryparticlephysicistknowsthatparticlephysicsisdescribedbytheStandardModel(SM)basedonthegaugesymmetryGSM=SU(3)×SU(2)×U(1),whichhasachiralfermionpattern:fermionsarerepresentedasWeylspinors,sothattheleft-handed(L)quarksandleptonsψL=qL,lLandright-handed(R)onesψR=qR,lRtransformdi erentlyundertheSU(2)×U(1)gaugefactor.Moreprecisely,thefermioncontentisthefollowing:

lL= νL

qL= eL uL

dL ～(1,2, 1);lR= NR～(1,1,0)(?)～(3,2,1/3);qR= eRu～(1,1, 2)R

dR～～(3(3,,11,,4 /23)/3),(1)

wherethebracketsexplicitlyindicatetheSU(3)andSU(2)contentofthemultipletsandtheirU(1)hypercharges.Inaddition,oneprescribesagloballeptonchargeL=1totheleptonslL,lRandabaryonchargeB=1/3toquarksqL,qR,sothatbaryonsconsistingofthreequarkshaveB=1.

TheSU(2)×U(1)symmetryisspontaneouslybrokenatthescalev=174GeVandW±,Zgaugebosonsbecomemassive.AtthesametimechargedfermionsgetmassesviatheYukawacouplings(i,j=1,2,3arethefermiongenerationindexes)

LYuk=YijudRqLjφd+Yije

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

4ZurabBerezhiani

However,onecansimplyrede nethenotionofparticles,namely,tocallψL, LasL-particlesandψ ψ

R,ψR,asRantiparticles

L fermions:ψL,ψ L;R antifermions:ψ R,ψR(5)

Hence,thestandardmodel elds,includingHiggses,canberecastedas:a

L set:(q,l,u ,d, e ,N )L,φu,φd;R set:( q, l,u,d,e,N)R,φ u,φ d(6)whereφ u,d=φ u,d,andthetheYukawaLagrangian(2)canberewrittenas

LYuk=u TYuqφu+d TYdqφd+e TYelφd+h.c.(7)

wheretheC-matrixaswellasthefamilyindicesareomittedforsimplicity.

Intheabsenceofright-handedsingletsNtherearenorenormalizableYukawacouplingswhichcouldgenerateneutrinomasses.However,oncethehigherordertermsareallowed,theneutrinoscouldgetMajoranamassesviatheD=5operators:

Aij

ItisconvenienttoparametrizetheirmassmatrixasM2(MijNiNj+Mij N iN j).

ij=GijM,whereMisa

typicalmassscaleandGisamatrixofdimensionless

theotherhand,theN Yukawa-likeconstants.On

statescancoupletolviaYukawatermsanalogousto(7),

andthusthewholesetofYukawatermsobtainthepattern:

lTYNφ u+M

10.InSO(10)

modelallLfermionsin(5)sitinonerepresentationL～16,whiletheRanti-fermionssitinR ～

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...5

symmetricwithrespecttoexchangeoftheLparticlestotheRones.Inparticular,thegaugebosonsofSU(2)coupletotheψL eldsbutdonotcoupletoψR.Infact,inthelimitofunbrokenSU(2)×U(1)symmetry,ψLandψRareessentiallyindependentparticleswithdi erentquantumnumbers.Theonlyreasonwhywecalle.g.stateseL lLandeRrespectivelytheleft-andright-handedelectrons,isthataftertheelectroweakbreakingdowntoU(1)emthesetwohavethesameelectricchargesandformamassiveDiracfermionψe=eL+eR.

Nowwhatwecallparticles(1),theweakinteractionsareleft-handed(V A)sinceonlytheL-statescoupletotheSU(2)gaugebosons.Intermsofantiparticles

(3),theweakinteractionswouldberight-handed(V+A),sincenowonlyRstatescoupletotheSU(2)bosons.Clearly,onecouldalwaysrede nethenotionofparticlesandantiparticles,torenameparticlesasantiparticlesandviceversa.Clearly,thenaturalchoiceforwhattocallparticlesisgivenbythecontentofmatterinourUniverse.Matter,atleastinourgalaxyanditsneighbourhoods,consistsofbaryonsqwhileantibaryonsq canbemetonlyinacceleratorsorperhapsincosmicrays.However,ifbychancewewouldliveintheantibaryonicislandoftheUniverse,wewouldclaimthatourweakinteractionsareright-handed.

InthecontextoftheSMoritsgranduni edextensions,theonlygoodsymmetrybetweentheleftandrightcouldbetheCPsymmetrybetweenL-particlesandR-antiparticles.E.g.,theYukawacouplings(7)inexplicitformread

L=( uTYuqφu+d TYdqφd+e TYelφd)L+(uTYu q φ u+dTYd q φ d+eTYe lφ d)R(10)However,althoughthese

theL-particlesandR termsarewritteninansymmetricmannerintermsof

-antiparticles,theyarenotinvariantunderL→R dueto

irremovablecomplexphasesintheYukawacouplingmatrices.Hence,Naturedoesnotrespectthesymmetrybetweenparticlesandantiparticles,butratherappliestheprinciplethat”theonlygooddiscretesymmetryisabrokensymmetry”.

Itisaphilosophicalquestion,whoandhowhaspreparedourUniverseattheinitialstatetoprovideanexcessofbaryonsoverantibaryons,andtherefore xedapriorityoftheV AformoftheweakinteractionsovertheV+Aone.ItisappealingtothinkthatthebaryonasymmetryitselfemergesduetotheCP-violatingfeaturesintheparticleinteractions,anditisrelatedtosomefundamentalphysicsbeyondtheStandardModelwhichisresponsiblefortheprimordialbaryogenesis.

2.2.ParticlesandcouplingsintheO-andM-worlds

Letusassumenowthatthereexistsamirrorsectorwhichhasthesamegaugegroupandthesameparticlecontentastheordinaryone.Intheminimalversion,whentheO-sectorisdescribedbythegaugesymmetryGSM=SU(3)×SU(2)×U(1)withtheobservablefermionsandHiggses(6),theM-sectorwouldbegivenbythegaugegroupG′SM=SU(3)′×SU(2)′×U(1)′withtheanalogousparticlecontent:L′ set:(q′,l′,u ′,d ′,e ′,N ′)L,φ′u,φ′d;R ′ set:( q′, l′,u′,d′,e′,N′)R,φ ′u,φ ′d(11)

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

6ZurabBerezhiani

Inmoregeneralview,onecanconsiderasupersymmetrictheorywithagaugesymmetryG×G′basedongranduni cationgroupsasSU(5)×SU(5)′,SO(10)×SO(10)′,etc.ThegaugefactorGoftheO-sectorcontainsthevectorgaugesuper eldsV,andleftchiralmatter(fermionandHiggs)super eldsLaincertainrepresentationsofG,whileG′standsfortheM-sectorwiththegaugesu-per eldsV′,andleftchiralmattersuper eldsL′a～rainanalogousrepresentationsofG′.

TheLagrangianhasaformLgauge+Lmat.ThematterLagrangianisdeterminedbytheformofthesuperpotentialwhichisaholomorphicfunctionofLsuper eldsandingeneralitcancontainanygaugeinvariantcombinationofthelatter:

W=(MabLaLb+gabcLaLbLc)+(Mab′L′aL′b+g′abcL′aL′bL′c)+...(12)ly,wehave

Lmat= dθ2W(L)+h.c.(13)

or,inexplicitform,

Lmat= dθ2(MabLaLb+gabcLaLbLc+Mab′L′aL′b+g′abcL′aL′bL′c)

+ d

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...7

Eithertypeofparityimpliesthatthetwosectorshavethesameparticlephysics.cIfthetwosectorsareseparateanddonotinteractbyforcesotherthangravity,thedi erencebetweenDandPparitiesisrathersymbolicanddoesnothaveanyprofoundimplications.However,inscenarioswithsomeparticlemessengersbetweenthetwosectorsthisdi erencecanbeimportantandcanhavedynamicalconsequences.

2.3.CouplingsbetweenO-andM-particles

Nowwediscuss,whatcommonforcescouldexistbetweentheO-andM-particles,includingmatter eldsandgauge elds.

GSM ×KineticG′mixingtermbetweentheO-andM-photons2,3,4.InthecontextofSM,thegeneralLagrangiancancontainthegaugeinvariantterm

L= χBµνB′µν,(19)

whereBµν= µBν νBµ,andanalogouslyforB′µν,whereBµandB′µaregauge eldsoftheabeliangaugefactorsU(1)andU(1)′.Obviously,aftertheelectroweaksymmetrybreaking,thistermgivesrisetoakineticmixingtermbetweenthe eld-strengthtensorsoftheO-andM-photons:

L= εFµνFµν′(20)

withε=ξcos2θW.Thereisnosymmetryreasonforsuppressingthisterm,andgenerallytheconstantεcouldbeoforder1.

Ontheotherhand,experimentallimitsontheorthopositroniumannihilationimplyastrongupperboundonε.Thisisbecauseonehastodiagonalize rstthekinetictermsoftheAµandA′µstatesandidentifythephysicalphotonasacertainlinearcombinationofthelatter.Onehastonoticethatafterthekinetictermsarebroughttocanonicalformbydiagonalizationandscalingofthe elds,(A,A′)→(A1,A2),anyorthonormalcombinationofstatesA1andA2becomesgoodtodescribethephysicalbasis.Inparticular,A2canbechosenasa”sterile”statewhichdoesnotcoupletoO-particlesbutonlytoM-particles.Then,theorthogonalcombinationA1couplesnotonlytoO-particles,butalsowithM-particleswithasmallcharge∝2ε–inotherwords,mirror14matterbecomes”milicharged”withrespecttothephysicalordinaryphoton2,.Inthisway,theterm(20)shouldinducetheprocesse+e →.e′+e′ ,withanamplitudejust2εtimesthes-channelamplitudefore+e →e+e Bythisdiagram,orthopositroniumwouldoscillateintoitsmirrorcounterpart,whichwouldbeseenasaninvisibledecaymodeexceedingexperimentallimitsunlessε<5×10 7orso3.

cThemirrorparitycouldbespontaneouslybrokenandtheweakinteractionscales φ =vand φ′ =v′couldbedi erent,whichleadstosomewhatdi erentparticlephysicsinthemirrorsector.Themodelswithspontaneoulsybrokenparityandtheirimplicationswereconsideredinrefs.6,9,10.Inthispaperwemostlyconcentrateonthecasewithexactmirrorparity.

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

8ZurabBerezhiani

Forexplainingnaturallythesmallnessofthekineticmixingterm(20)one

needs

to

invoketheconcept

of

grand

uni cation.

Obviously,

the

term

(19)isforbiddenifGSM×G′SMisembeddedinGUTslikeSU(5)×SU(5)′orSO(10)×SO(10)′whichdonotcontainabelianfactors.However,giventhatbothSU(5)andSU(5)′symmetriesarebrokendowntotheirSU(3)×SU(2)×U(1)subgroupsbytheHiggs24-pletsΦandΦ′,itcouldemergefromthehigherordere ectiveoperator

L= ζ

ij

2M(liφ)(ljφ)+A′

(liφ)(l′jφ′

M)+h.c.,(22)

The rstoperatorineq.(22),duetotheordinaryHiggsvacuumVEV φ =v～100GeV,theninducesthesmallMajoranamassesoftheordinary(active)neutrinos.SincethemirrorHiggsφ′alsohasanon-zeroVEV φ′ =v′,thesecondopera-torthenprovidesthemassesoftheM-neutrinos(whichinfactaresterilefortheordinaryobserver),and nally,thethirdoperatorinducesthemixingmassterms

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...9

betweentheactiveandsterileneutrinos.Thetotalmassmatrixofneutrinosν landν′ l′readsas6:

Mν= mνmνν′

mTν′ =1νν′m

2MabNaNb+h.c.(24)

Inthisway,Nplaytheroleofmessengersbetweenordinaryandmirrorparti-cles.Afterintegratingouttheseheavystates,theoperators(22)areinducedwithA=YG 1YT,A′=Y′G 1Y′TandD=YG 1Y′T.InthenextsectionweshowthatinadditiontheNstatescanmediateLandCPviolatingscatteringprocessesbetweentheO-andM-sectorswhichcouldprovideanewmechanismforprimordialleptogenesis.

ItisconvenienttopresenttheheavyneutrinomassmatrixasMab=GabM,MbeingtheoverallmassscaleandGabsometypicalYukawaconstants.Withoutlossofgenerality,Gabcanbetakendiagonalandreal.UnderPorDparities,ingeneralsomeofthestatesNawouldhavepositiveparity,whileotherscouldhaveanegativeone.

Onetheotherhand,theYukawamatricesingeneralremainnon-diagonalandcomplex.ThenDparitywouldimplythatY′=Y,whilePparityimplesY′=Y (c.f.(16)and(18)).

InteractiontermbetweentheO-quarticandM-interactionHiggses.IntermthebetweencontextoftheGSM

O-and×G′SM,

thegaugesymmetryallowsalsoaM-Higgsdoubletsφandφ′:

λ(φ φ)(φ′ φ′)(25)

Thistermiscosmologicallydangerous,sinceitwould

equilibriumintheearlyUniverseviainteractionsφφ¯bringthetwosectorsinto

→φ¯′φ′unlessλisunnaturally

small,λ<10 8.9

However,thistermcanbeproperlysuppressedbysupersymmetry.InthiscasestandardHiggsesφu,dbecomechiralsuper eldsaswellastheirmirrorpartnersφ′u,d,andsotheminimalgaugeinvarianttermbetweentheO-andM-Higgsesinthesuperpotentialhasdimension5:(1/M)(φuφd)(φ′uφ′d),whereMissomebigcuto

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

10ZurabBerezhiani

mass,e.g.oftheorderoftheGUTscaleorPlanckscale.Therefore,thegeneralHiggsLagrangiantakes

L= theform:

d2θ[µφuφd+µφ′uφ′1

d+

M d2θz(φuφd)(φ′uφ′d)+h.c.,wherez=mSθ2beingthesupersymmetrybreak-ingspurionwithmS～100GeV,givesrisetoaquarticscalarterm

λ(φuφd)(φ′uφ′d)+h.c.(28)

withλ～mS/M 1.Thusforµ,mS

stronglysuppressed,andhencearesafe.～100GeV,allthesequarticconstantsare

with MixedO-particlesmultipletsbeingbetweensingletstheofmirrortwosectors.gaugeUntilfactornowG′andwediscussedviceversa,theM-particlessituationbeingsingletsofordinarygaugegroupG.However,inprincipletherecouldbealsosome eldsinmixedrepresentationsofG×G′.Such eldsusuallyemergeifthetwogaugefactorsGandG′areembeddedintoabiggergranduni cationgroupG.Forexample,consideragaugetheorySU(5)×SU(5)′wheretheO-andM-fermionsrespectivelyareinthefollowingleft-chiralmultiplets:L～(¯5+10,1)andL′～(1,5+10,

3πM2Tr[(YY′)2](29)

exactlywhatweexpectedfromthee ectiveoperator

(21).Hence,theheavymixedmultipletsinfactdonotdecoupleandinducetheO-andM-photonkineticmixing

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...11

termproportionaltothesquareoftypicalmasssplittingsinthesemultiplets(～ Φ 2),analogouslytothefamiliarsituationforthephotontoZ-bosonmixinginthestandardmodel.

particles Interactionshavecommonviacommonforcesmediatedgaugebosons.bytheItisgaugeprettybosonspossibleofsomethatO-andadditionalM-symmetrygroupH.Inotherwords,onecanconsideratheorywithagaugegroupG×G′

correspondingly×H,wheretheirO-particlesantiparticlesareareininsomeantirepresentations,representationsofR H,La～ra,and

a

particles,viceversa,wetakeL′～r¯a.AsforM-a

Gpatterniscompatiblewiththe～r¯a,andsoR ′aofmirrorparity(17).～ra.OnlysuchaprescriptionInaddition,inthiscaseH

symmetryautomaticallybecomesvector-likeandsoitwouldhavenoproblemswithaxialanomalieseveniftheparticlecontentsofO-andM-sectorsseparatelyarenotanomaly-freewithrespecttoH.

Letusconsiderthefollowingexample.Thehorizontal avoursymmetrySU(3)Hbetweenthequark-leptonfamiliesseemstobeverypromisingforunderstandingthefermionmassandmixingpattern21,22.Inaddition,itcanbeusefulforcontrol-lingthe avour-changingphenomenainthecontextofsupersymmetry7.Onecanconsidere.g.aGUTwithSU(5)×SU(3)HsymmetrywhereL-fermionsin(6)aretripletsofSU(3)H.SoSU(3)Hhasachiralcharacteranditisnotanomaly-freeunlesssomeextrastatesareintroducedfortheanomalycancellation21.

However,theconceptofmirrorsectormakesthethingseasier.Considere.g.SU(5)×SU(5)′×SU(3)HtheorywithL-fermionsin(6)beingtripletsofSU(3)HwhileL′-fermionsin(11)areanti-triplets.Hence,inthiscasetheSU(3)Hanomaliesoftheordinaryparticlesarecancelledbytheirmirrorpartners.AnotheradvantageisthatinasupersymmetrictheorythegaugeD-termsofSU(3)Hareperfectlycancelledbetweenthetwosectorsandhencetheydonotgiverisetodangerous avour-changingphenomena7.

TheimmediateimplicationofsuchatheorywouldbethemixingofneutralO-bosonstotheirM-partners,ly,oscil-lationsπ0→π′0orK0→K′0becomepossibleandperhapsevendetectableifthehorizontalsymmetrybreakingscaleisnottoohigh.

Anotherexampleisacommonleptonnumber(orB L)symmetrybetweenthetwosectors.LetusassumethatordinaryleptonslhaveleptonchargesL=1underthissymmetrywhilemirroronesl′haveL= 1.Obviously,thissymmetrywouldforbidthe rsttwocouplingsin(22),A,A′=0,whilethethirdtermisallowed–D=0.Hence,‘Majorana’masstermswouldbeabsentbothforO-andM-neutrinosin(23)andsoneutrinoswouldbeDiracparticleshavingnaturally smallmasses,withleftcomponentsνL landrightcomponentsbeingν ′R l.

ThemodelwithcommonPeccei-QuinnsymmetrybetweentheO-andM-sectorswasconsideredin23.

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

12ZurabBerezhiani

3.TheexpansionoftheUniverseandthermodynamicsoftheO-andM-sectors

Letusassume,thatafterin ationended,theO-andM-systemsreceiveddi erentreheatingtemperatures,namelyTR>TR′.ThisiscertainlypossibledespitethefactthattwosectorshaveidenticalLagrangians,andcanbenaturallyachievedincertainmodelsofin ation8,9,10.d

Ifthetwosystemsweredecoupledalreadyafterreheating,atlatertimesttheywillhavedi erenttemperaturesT(t)andT′(t),andsodi erentenergyandentropydensities:

ρ(t)=π2

g′ (T′)T′4

30,(30)

s(t)=2π2

g′s(T′)T′3

45.(31)

Thefactorsg ,gsandg′ ,g′saccountingforthee ectivenumberofthedegreesoffreedominthetwosystemscaningeneralbedi erentfromeachother.LetusassumethatduringtheexpansionoftheUniversethetwosectorsevolvewithseparatelyconservedentropies.Thentheratiox≡(s′/s)1/3istimeindependentwhiletheratioofthetemperaturesinthetwosectorsissimplygivenby:

T′(t)

g′s(T′) 1/3.(32)

The Hubbleexpansionrateisdeterminedbythetotalenergydensityρ¯=ρ+ρ′,H=

2t=1.66

ofO-andM-temperatures M=1.66PlT(t)andT′(t), M(33)Plintermswhere

g¯ (T)=g (T)(1+x4),g¯′ (T′)=g′ (T′)(1+x 4).(34)

Inparticular,wehavex=T0′/T0,whereT0,T0′arethepresenttemperaturesoftheO-andM-relicphotons.Infact,xistheonlyfreeparameterinourmodelanditisconstrainedbytheBBNbounds.

Theobservedabundancesoflightelementsareingoodagreementwiththestan-dardnucleosynthesispredictions.AtT～1MeVwehaveg =10.75asitissatu-ratedbyphotonsγ,electronseandthreeneutrinospeciesνe,µ,τ.Thecontributionofmirrorparticles(γ′,e′andν′e,µ,τ)wouldchangeittog¯ =g (1+x4).Deviationsfromg =10.75areusuallyparametrizedintermsofthee ectivenumberofextraneutrinospecies, g=g¯ 10.75=1.75 Nν.Thuswehave:

Nν=6.14·x4.(35)

dForanalogy,twoharmonicoscillatorswiththesamefrequency(e.g.twospringswiththesamematerialandthesamelength)arenotobligedtooscillatewiththesameamplitudes.

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...13

Thislimitveryweaklydependson Nν.Namely,theconservativebound Nν<1impliesx<0.64.Inviewofthepresentobservationalsituation,confrontingtheWMAPresultstotheBBNanalysis,theboundseemstobestronger.However,e.g.x=0.3impliesacompletelynegligiblecontribution Nν=0.05.

Asfarasx4

dominatedbythe O-matter1,inarelativisticdensityandepochthepresencetheHubbleoftheexpansionM-sectorratepractically(33)isdoesnota ectthestandardcosmologyoftheearlyordinaryUniverse.However,evenifthetwosectorshavethesamemicrophysics,thecosmologyoftheearlymirrorworldcanbeverydi erentfromthestandardoneasfarasthecrucialepochslikebaryogenesis,nuclesosynthesis,etc.areconcerned.AnyoftheseepochsisrelatedtoaninstantwhentherateoftherelevantparticleprocessΓ(T),whichisgenericallyafunctionofthetemperature,becomesequaltotheHubbleexpansionrateH(T).Obviously,intheM-sectortheseeventstakeplaceearlierthanintheO-sector,andasarule,therelevantprocessesintheformerfreezeoutatlargertemperaturesthaninthelatter.

Inthematterdominationepochthesituationbecomesdi erent.Inparticular,weknowthatordinarybaryonsprovideonlyasmallfractionofthepresentmatterdensity,whereastheobservationaldataindicatethepresenceofdarkmatterwithabout5timeslargerdensity.ItisinterestingtoquestionwhetherthemissingmatterdensityoftheUniversecouldbeduetomirrorbaryons?Inthenextsectionweshowthatthiscouldoccurinaprettynaturalmanner.

ItcanalsobeshownthattheBBNepochinthemirrorworldproceedsdi er-entlyfrom11theordinaryone,anditpredictsdi ly,mirrorheliumabundancecanbeintherangeY4′=0.6 0.8,considerablylargerthantheobservableY4 0.24.

4.BaryogenesisinM-sectorandmirrorbaryonsasdarkmatter

4.1.VisibleanddarkmatterintheUniverse

Thepresentcosmologicalobservationsstronglysupportthemainpredictionsofthein ationaryscenario: rst,theUniverseis at,withtheenergydensityveryclosetothecritical =1,andsecond,primoridaldensityperturbationshavenearly atspectrum,withthespectralindexns

asmallfractionofthepresentenergy≈density,1.Theaboutnon-relativistic m

attributedtothevacuumenergy(cosmologicalterm) Λ 00..7327,matter

24whilegives

.Thethefactrestonlythatis

mand Λareofthesameorder,givesrisetosocalledcosmologicalcoincidenceproblem:whyweliveinanepochwhenρm～ρλ,ifintheearlyUniverseonehadρm ρΛandinthelateUniverseonewouldexpectρm ρΛ?Theanswercanbeonlyrelatedtoanantrophicprinciple:thematterandvacuumenergydensitiesscaledi erentlywiththeexpansionoftheUniverseρm

theyhavetocoincideatsomemoment,andweare∝justa 3andρΛ

hencehappyto∝beconst.,here.Moreover,forsubstantiallylargerρΛnogalaxiescouldbeformedandthustherewouldnotbeanyonetoaskthisthisquestion.

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

14ZurabBerezhiani

Ontheotherhand,thematterintheUniversehastwocomponents,visibleanddark: m= b+ d.Thevisiblematterconsistsofbaryonswith b

thedarkmatterwith d 0.044

veryweaklyinteractingwith 0the.22observableisconstitutedmatter.bysomeItishypotheticatantalizingparticlequestion,specieswhilewhythevisibleanddarkcomponentshavesocloseenergydensities?Clearly,theratio

β=ρd

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...15

theO-baryondensitynb.Thus,onecouldquestionwhethertheratioβ=n′b/nbcouldbenaturallyorder1orsomewhatbigger.

ThevisiblematterintheUniverseconsistsofbaryons,whiletheabundanceofantibaryonsisvanishinglysmall.IntheearlyUniverse,attempreaturesT 1GeV,thebaryonsandantibaryonshadpracticallythesamedensities,nb

nbslightlyexceedingn¯thatthebaryonnumberdensitywassmall,≈nBn=¯bwithb,so

nb n¯b nb.Iftherewasnosigni cantentropyproductionafterthebaryogenesisepoch,thebaryonnumberdensitytoentropydensityratiohadtobethesameastoday,B=nB/s≈8×10 11.e

Onecanquestion,whoandhowhaspreparedtheinitialUniversewithsuchasmallexcessofbaryonsoverantibaryons.IntheFriedmanUniversetheinitialbaryonasymmetrycouldbearrangedapriori,intermsofnon-vanishingchemicalpotentialofbaryons.However,thein ationaryparadigmgivesanothertwisttothisquestion,sincein ationdilutesanypreexistingbaryonnumberoftheUniversetozero.Therefore,afterin atondecayandthe(re-)heatingoftheUniverse,thebaryonasymmetryhastobecreatedbysomecosmologicalmechanism.

ThereareseveralrelativelyhonestbaryogenesismechanismsasareGUT25baryo-genesis,leptogenesis,electroweakbaryogenesis,etc.(forareview,seee.g.).TheyareallbasedongeneralprinciplessuggestedlongtimeagobySakharov26:anon-zerobaryonasymmetrycanbeproducedintheinitiallybaryonsymmetricUniverseifthreeconditionsareful lled:B-violation,C-andCP-violationanddeparturefromthermalequilibrium.IntheGUTbaryogenesisorleptogenesisscenariostheseconditionscanbesatis edinthedecaysofheavyparticles.

Atpresentitisnotpossibletosayde nitelywhichoftheknownmechanismsisresponsiblefortheobservedbaryonasymmetryintheordinaryworld.However,itismostlikelythatthebaryonasymmetryinthemirrorworldisproducedbythesamemechanismandmoreover,thepropertiesoftheBandCPviolationprocessesareparametricallythesameinbothcases.Butthemirrorsectorhasalowertemperaturethanordinaryone,andsoatepochsrelevantforbaryogenesistheout-of-equilibriumconditionsshouldbeeasierful lledfortheM-sector.

4.2.BaryogenesisintheO-andM-worlds

Letusconsiderthedi erencebetweentheordinaryandmirrorbaryonasymme-triesontheexampleoftheGUTbaryogenesismechanism.Itistypicallybasedon‘slow’B-andCP-violatingdecaysofasuperheavybosonXintoquarksandleptons,whereslowmeansthatatT<MtheHubbleparameterH(T)isgreaterthanthedecayrateΓ～αM,αbeingthecouplingstrengthofXtofermionsandMitsmass.Theotherreactionratesarealsoofrelevance:inversedecay:ΓI～Γ(M/T)3/2exp( M/T)forT<MX,andtheXbosonmediatedscatteringeInthefollowingweuseB=nB/swhichisrelatedwiththefamiliarη=nB/nγasB≈0.14η.However,BismoreadoptedforfeaturingthebaryonasymmetrysinceitdoesnotdependontimeiftheentropyoftheUniverseisconserved.

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

16ZurabBerezhiani

processes:ΓS～nXσ～Aα2T5/M4,wherethefactorAamountsforthepossiblereactionchannels.

The nalBAdependsonthetemperatureatwhichXbosonsgooutfromequilibrium.Onecanintroduceaparameterwhichmeasuresthee ectivenessofdecayattheepochT～M:k=(Γ/H)T=M=0.3¯g 1/2the(αMPl/M).Fork 1theout-of-equilibriumconditionisstronglysatis ed,andperdecayofoneXparticleonegeneratesthebaryonnumberproportionaltotheCP-violatingasymmetryε.Thus,wehaveB=ε/g ,g isanumberofe ectivedegreesoffreedomatT<M.Thelargerkis,thelongerequilibriumismaintainedandthefreeze-outabundanceofXbosonbecomessmaller.Hence,theresultingbaryonnumbertoentropyratiobecomesB=(ε/g )D(k),wherethedampingfactorD(k)isadecreasingfunctionofk.Inparticular,D(k)=1fork 1,whileforkexceedingsomecriticalvaluekc,thedampingisexponential.

Thepresenceofthemirrorsectorpracticallydoesnotaltertheordinarybaryo-genesis.Thee ectiveparticlenumberisg¯ (T)=g (T)(1+x4)andthusthecon-tributionofM-particlestotheHubbleconstantatT～Missuppressedbyasmallfactorx4.

Inthemirrorsectoreverythingshouldoccurinasimilarway,apartfromthefactthatnowatT′

speciesbutbyordinary～Mones:theg¯Hubble′ (T′) constantg′ (T′)(1is+notx 4dominated).Asaconsequence,bythemirrorwehavek′=(Γ/H)|T′=M=kx2.Therefore,thedampingfactorformirrorbaryonasymmetrycanbesimplyobtainedbyreplacingk→k′=kx2inD(k).Inotherwords,thebaryonnumberdensitytoentropydensityratiointheM-worldbecomesB′=n′B/s′ ( /g )D(kx2).SinceD(k)isadecreasingfunctionofk,thenforx<1wehaveD(kx2)>D(k)andthusweconcludethatthemirrorworldalwaysgetsalargerbaryonasymmetrythanthevisibleone,B′>B.fNamely,fork>1thebaryonasymmetryintheO-sectorisdampedbysomefactor–wehaveB (ε/g )D(k)<ε/g ,whileifx2<k 1,thedampingwouldbeirrelevantfortheM-sectorandhenceB′

thisdoesnot ε/g .

However,apriorimeanthat ′bwillbelargerthan b.Sincetheentropydensitiesarerelatedass′/s=x3,fortheratioβ= ′b/ bwehave:

β(x)=n′BD(kx2)

Bs=x3

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...17

longerthaninthemirrorone,andthusordinaryBAcanbesuppressedbyanexponentialBoltzmannfactorwhilethemirrorBAcouldbeproducedstillintheregimek′=kx2 1,whenbaryogenesisD(k′)≈1.However,theGUTpicturehasthefollowinggenericproblem.Inscenariosbasedongranduni cationmodelslikeSU(5),theheavygaugeorHiggsbosondecaysviolateseparatelyBandL,butconserveB L,andso nallyB L=0.Ontheotherhand,thenon-perturbativesphaleronprocesses,whichviolateB+LbutconserveB L,aree ectiveattemperaturesfromabout1012GeVdownto100GeV27.Therefore,ifB+Liserasedbysphalerontransitions,the nalBandLbothwillvanish.

Hence,inarealisticscenariooneactuallyhastoproduceanon-zeroB Lratherthanjustanon-zeroB,afactthatstronglyfavoursthesocalledleptogenesisscenario28.Theseesawmechanismforneutrinomasseso ersanelegantpossibilityofgeneratingnon-zeroB LinCP-violatingdecaysofheavyMajorananeutrinosNintoleptonsandHiggses.ThesedecaysviolateLbutobviouslydonotchangeBandsotheycouldcreateanon-zeroB L= ly,

constants,thedecayratesΓ(N→lφ)andΓ(N→ lφ duetocomplexYukawa

)canbedi erentfromeach

other,sothattheleptonslandanti-leptons lareproducedindi erentamounts.

Whensphaleronsareinequilibrium,theyviolateB+Landsoredistributenon-zeroB ly,the nalvaluesofBandB LarerelatedasB=a(B L),whereaisorder1coe cient,namelya 1/3intheSMandinitssupersymmetricextension25.Hence,theobservedbaryontoentropydensityratio,B≈8×10 11,needstoproduceB L～2×10 10.

However,thecomparativeanalysispresentedabovefortheGUTbaryogenesisintheO-andM-worlds,isessentiallytruealsofortheleptogenesisscenario.Theout-of-equilibriumdecaysofheavyNneutrinosoftheO-sectorwouldproduceanon-zeroB LwhichbeingreprocessedbysphaleronswouldgiveanobservablebaryonasymmetryB=a(B L).Ontheotherhand,thesamedecaysofheavyN′neutrinosoftheM-sectorwouldgivenon-zero(B′ L′)andthusthemirrorbaryonasymmetryB′=a(B′ L′).InordertothermallyproduceheavyneutrinosinbothO-andM-sectors,thelightestofthemshouldhaveamasssmallerthanthereheatingtemperatureTR′intheM-sector,i.e.MN<TR′,TR.ThesituationMN>TR′wouldpreventthermalproductionofN′states,andsonoB′

begeneratedinM-sector.However,onecanconsideralsoscenarioswhen L′wouldbothNandN′statesarenon-thermallyproducedinin atondecays,butwithdi erentamounts.ThenthereheatingofbothsectorsaswellasB LnumbergenerationcanberelatedtothedecaysoftheheavyneutrinosofbothsectorsandhencethesituationTR′<TRcanbenaturallyaccompaniedbyB′>B.

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

18ZurabBerezhiani

4.3.BaryogenesisviaOrdinary-Mirrorparticleexchange

Analternativemechanismofleptogenesisbasedonscatteringprocessesratherthanondecaywassuggestedinref.12.Themainideaconsistsinthefollowing.Thereexistssomehidden(shadow)sectorofnewparticleswhicharenotinthermalequi-libriumwiththeordinaryparticleworldasfarasthetwosystemsinteractveryweakly,e.g.iftheyonlycommunicateviagravity.However,othermessengersmaywellexist,namely,superheavygaugesingletslikeright-handedneutrinoswhichcanmediateveryweake ectiveinteractionsbetweentheordinaryandhiddenleptons.Then,anetB LcouldemergeintheUniverseasaresultofCP-violatinge ectsintheunbalancedproductionofhiddenparticlesfromordinaryparticlecollisions.Hereweconsiderthecasewhenthehiddensectorisamirrorone.Asfarastheleptogenesisisconcerned,weconcentrateonlyontheleptonsectorofbothOandMworlds.Thereforeweconsiderthestandardmodel,andamongotherparticlesspecies,concentrateontheleptondoubletsli=(ν,e)i(i=1,2,3isthefamilyindex)andtheHiggsdoubletφfortheO-sector,andontheirmirrorpartnersl′i=(ν′,e′)iandφ′.Theircouplingstotheheavysingletneutrinosaregivenby(24).

Letusdiscussnowtheleptogenesismechanisminourscenario.AcrucialroleinourconsiderationsisplayedbythereheatingtemperatureTR,atwhichthein atondecayandentropyproductionoftheUniverseisover,andafterwhichtheUniverseisdominatedbyarelativisticplasmaofordinaryparticlespecies.Aswediscussedabove,weassumethatafterthepostin ationaryreheating,di erenttemperaturesareestablishedinthetwosectors:TR′<TR,i.e.themirrorsectoriscoolerthanthevisibleone,orultimately,evencompletely“empty”.

Inaddition,thetwoparticlesystemsshouldinteractveryweaklysothattheydonotcomeinthermalequilibriumwitheachotherafterreheating.WeassumethattheheavyneutrinomassesarelargerthanTRandthuscannotbethermallyproduced.Asaresult,theusualleptogenesismechanismviaN→lφdecaysisine ective.

Now,theimportantroleisplayedbyleptonnumberviolatingscatteringsme-diatedbytheheavyneutrinosN.The“cooler”mirrorworldstartstobe“slowly”occupiedduetotheentropytransferfromtheordinarysectorthroughthe L=1reactionsliφ→¯l′kφ¯′,¯liφ¯→l′kφ′.IngeneraltheseprocessesviolateCPduetocomplexYukawacouplingsineq.(24),andsothecross-sectionswithleptonsandanti-leptonsintheinitialstatearedi erentfromeachother.Asaresult,leptonsleaktothemirrorsectormore(orless)e ectivelythanantileptonsandanon-zeroB LisproducedintheUniverse.

Itisimportanttostressthatthismechanismwouldgeneratethebaryonasym-metrynotonlyintheobservablesector,butalsointhemirrorsector.Infact,thetwosectorsarecompletelysimilar,andhavesimilarCP-violatingproperties.Wehavescatteringprocesseswhichtransformtheordinaryparticlesintotheirmirrorpartners,andCP-violatione ectsinthisscatteringowingtothecomplexcouplingconstants.TheseexchangeprocessesareactiveatsomeearlyepochoftheUniverse,

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...19

andtheyareoutofequilibrium.Inthiscase,ahypotheticalOobservershouldde-tectduringthecontactepochthat(i)matterslowly(incomparisontotheUniverseexpansionrate)disappearsfromthethermalbathofourworld,and,inaddition,(ii)particlesandantiparticlesdisappearwithdi erentrates,sothatafterthecontactepochendsup,heobservesthathisworldisleftwithanon-zerobaryonnumberevenifinitiallyitwasbaryonsymmetric.

Ontheotherhand,hismirrorcolleague,Mobserver,wouldseethat(i)mattercreationtakesplaceinhisworld,and(ii)particlesandantiparticlesemergewithdi erentrates.Therefore,afterthecontactepoch,healsowouldobserveanon-zerobaryonnumberinhisworld.

OnewouldnaivelyexpectthatinthiscasethebaryonasymmetriesintheOandMsectorsshouldbeliterallyequal,giventhattheCP-violatingfactorsarethesameforbothsectors.However,weshowthatinreality,theBAintheMsector,sinceitiscolder,canbeaboutanorderofmagnitudebiggerthanintheOsector,asfaraswashingoute ectsaretakenintoaccount.Indeed,thise ectsshouldbemoree cientforthehotterOsectorwhiletheycanbenegligibleforthecolderMsector,whichcouldprovidereasonabledi erencesbetweenthetwoworldsincasetheexchangeprocessisnottoofarfromequilibrium.Thepossiblemarriagebetweendarkmatterandtheleptobaryogenesismechanismiscertainlyanattractivefeatureofourscheme.

ThefastreactionsrelevantfortheO-sector′φ¯′,andthe L=2oneslikelφ→¯lφ¯arethe L=1oneliφ→¯lk,ll→φφetc.Theirtotalratesarecorrespondingly

ΓQn1=1eq

;Q2=Tr(A A)=Tr[(Y Y) G 1(Y Y)G 1

4πM2],(38)

whereneq (1.2/π2)T3isanequilibriumdensityper(bosonic)degreeoffreedom,andthesumistakenoverall avourandisospinindicesofinitialand nalstates.Itisessentialthattheseprocessesstayoutofequilibrium,which

ratesshouldnotexceedmuchtheHubbleparameterH=1.66g1 /2meansthattheirT2/MPlfortem-peraturesT≤TR,whereg isthee ectivenumberofparticledegreesoffreedom,namelyg 100intheSM.Inotherwords,thedimensionlessparameters

k1= Γ1

1/2

k g M2Γ2=2

g1 /2M2(39)

shouldnotbemuchlargerthan1.

LetusnowturntoCP-violation.In L=1processestheCP-oddleptonnum-berasymmetryemergesfromtheinterferencebetweenthetree-levelandone-loopdiagramsof g.1.However,CP-violationtakesalsoplacein L=2processes(see g.2).Thisisaconsequenceoftheveryexistenceofthemirrorsector,namely,it

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

20ZurabBerezhiani

Fig.1.Tree-levelandone-loopdiagramscontributingtotheCP-asymmetriesinlφ→¯l′φ¯′(leftcolumn)andlφ→l′φ′(rightcolumn).

comesfromthecontributionofthemirrorparticlestotheone-loopdiagramsof g.2.Thedirectcalculationgives:g

σ(lφ→¯l′φ¯′) σ(¯lφ¯→l′φ′)=( σ σ′)/2,

σ(lφ→l′φ′) σ(¯lφ¯→¯l′φ¯′)=( σ+ σ′)/2,

σ(lφ→¯lφ¯) σ(¯lφ¯→lφ)= σ;(40)

σ=3JS

32π2M4,(41)

whereSisthec.m.energysquare,J=ImTr[(Y Y) G 1(Y′ Y′)G 2(Y Y)G 1]istheCP-violationparameterandJ′isobtainedfromJbyexchangingY

Y′.Thecontributionsyieldingasymmetries σ′respectivelyforlφ→¯l′φ¯with

′and

lφ→l′φ′channelsemergefromthediagramswithl′φ′insidetheloops,notshownin g.1.

ThisisinperfectagreementwithCPTinvariancethatrequiresthattheto-talcrosssectionsfor

σ¯particleandanti-particlescatteringsareequaltoeachother:¯→X).Indeed,takingintoaccountthatσ(lφ→lφ)=σ

lφ¯(lφ→X)=σ(¯lφ(¯lφ¯→)byCPT,weseethatCPasymmetriesinthe L=1and L=2processesgItisinterestingtonotethatthetree-levelamplitudeforthedominantchannellφ→¯l′φ¯′goesas1/Mandtheradiativecorrectionsas1/M3.Forthechannellφ→l′φ′instead,bothtree-levelandone-loopamplitudesgoas1/M2

forbothlφ→¯l′φ¯.Asaresult,thecrosssectionCPasymmetriesarecomparable

′andlφ→l′φ′channels.

We briefly review the concept of a parallel `mirror' world which has the same particle physics as the observable world and couples to the latter by gravity and perhaps other very weak forces. The nucleosynthesis bounds demand that the mirror world should h

MirrorWorld...21

Fig.2.Tree-levelandone-loopdiagramscontributingtotheCP-asymmetryoflφ→¯lφ¯.Theexternalleglabelsidentifytheinitialand nalstateparticles.

shouldberelatedas

σ(lφ→X′) σ(¯lφ¯→X′)= [σ(lφ→¯lφ¯) σ(¯lφ¯→lφ)]= σ,(42)whereX′arethemirrorsector nalstates,¯l′φ¯′andl′φ′.Thatis,the L=1and L=2reactionshaveCPasymmetrieswithequalintensitiesbutoppositesigns.

But,asLvariesineachcasebyadi erentamount,anetleptonnumberdecreaseisproduced,orbetter,anetincreaseofB L∝ σ(recallthattheleptonnumberLisviolatedbythesphaleronprocesses,whileB Lischangedsolelybytheaboveprocesses).

Asfarasweassumethatthemirrorsectoriscoolerandthusdepletedofpar-ticles,theonlyrelevantreactionsaretheoneswithordinaryparticlesintheinitialstate.Hence,theevolutionoftheB LnumberdensityisdeterminedbytheCPasymmetriesshownineqs.(40)andobeystheequation

dnB L σn2eq=1.8×10 3T8

4

g3 /2M4.(44)

hObservethatthemagnitudeoftheproducedB Lstronglydependsonthetemperature,namely,largerB Lshouldbeproducedinthepatcheswheretheplasmaishotter.Inthecosmologicalcontext,thiswouldleadtoasituationwhere,apartfromtheadiabaticdensity/temperatureper-turbations,therealsoemergecorrelatedisocurvature uctuationswithvariableBandLwhichcouldbetestedwiththefuturedataontheCMBanisotropiesandlargescalestructure.

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