L Composite Higgs models, Dark Matter and J. Lorenzo Diaz Cruz FacultaddeCienciasFísico-Matemáticas,BeneméritaUniversidadAutónomadePuebla,Puebla, Pue.,CP:72570,México 9 0 Abstract. We suggestthatdarkmattercanbeidentifiedwith astable compositefermionX0,that 0 arises within the holographic AdS/CFT models, where the Higgs boson emerges as a composite 2 pseudo-goldstone boson. The predicted properties of X0 satisfies the cosmological bounds, with n m 4p f O(TeV). Thus, through a deeper understanding of the mechanism of electroweak X0 ∼ ≃ a symmetrybreaking,aresolutionoftheDarkMatterenigmaisfound.Furthermore,byproposinga J discretestructureoftheHiggsvacuum,onecangetadistinctapproachtothecosmologicalconstant 8 problem. Keywords: Electroweaksymmetrybreaking,Higgsmodels,darkmatter ] h PACS: 12.60.-i,11.10.Kk,95.35.+d p - p INTRODUCTION e h [ The notion of spontaneous symmetry breaking (SSB) [1] has been an important ingre- 1 dientfor thedevelopmentofmodern particlephysics,with applicationsthat range from v 6 the description of chiral symmetry breaking in the strong interactions [2], to the gener- 2 ation of masses in the electroweak model [3, 4], including as well the development of 1 inflationarymodels[5].However,theeffectivedescriptionoftheHiggsmechanismstill 1 . lacks afundamental understanding.Thus, explainingthe natureofelectroweak symme- 1 try breaking (EWSB) is one of the most important questions in particle physics today. 0 9 Within the standard model (SM), electroweak precision tests (EWPT) prefer a Higgs 0 bosonmassoftheorderoftheelectroweak (EW)scalev 175GeV [6]. Plenty ofphe- : ≃ v nomenological studies have provided us with an understanding of the expected proper- i X tiesoftheHiggsboson(mass,decay rates and productioncross sections),whichshould r betested soonat theLHC. a Ontheotherhand,fromthecosmologyside,therearealsobigproblems,oneofthem beingtheenigmaofdarkmatter(DM)[7].Plentyofastrophysicalandcosmologicaldata requirestheexistenceofaDMcomponent,thataccountsforabout10-20%ofthematter- energycontentofouruniverse[8].Aweakly-interactingmassiveparticle(WIMP),with a mass also of the order of theEW scale, seems a most viableoption for theDM. What is the nature of DM and how does it fit into our current understanding of elementary particles,ishowevernotknown. Given the similar requirements on masses and interactions for both particles, Higgs bosonandDM,onecannaturallyaskwhethertheycouldshareacommonorigin.Within the minimal SUSY SM [9], which has become one of the most popular extensions of the SM, there are several WIMP candidates (neutralino, sneutrino, gravitino) [10]. Amongthem,theneutralinohasbeenmostwidelystudied;itisacombinationsofSUSY partners of the Higgs and gauge bosons, the Higgsinos and gauginos. Thus, in SUSY models the fermion-boson symmetry provides a connection between the Higgs boson and DM. However, many new models have been proposed more recently [11], which provide alternative theoretical foundation to stabilize the Higgs mechanism. Some of these models, which have been originally motivated by the studies of extra dimensions [12], include new DM candidates, such as the lightest T-odd particle (LTP) within little Higgs models [13] or the lightest KK particle (LKP) in models with universal extra- dimensions[14]. Here, we summarize the results of our search for possible dark matter candidates, within the Holographic Higgs models [15]. In these constructions, EWSB is triggered byalightcompositeHiggsboson,whichemergesasapseudo-goldstoneboson[16,17]. Within this class of models, we propose that a stable composite “Baryon”, tightly bounded by the new strong interactions, can account for the DM. This picture, where stronginteractionsproducealightpseudo-goldstonebosonandaheavierstablefermion, isnotstrangeatallinnature.Thisispreciselywhathappensinordinaryhadronphysics, wherethepionandtheprotonplaysuchroles.Inthispaperweshalldiscussmodelsthat produceasimilarpatternfortheHiggsandDM,butatahigherenergyscale,andwitha stableneutralstateinsteadofacharged one. However,eveniftheHiggsbosonsisfoundattheLHC,andevenifonecouldidentify the Dark matter candidate, there will be some issues left open. One of them, probably the most difficult one, is the cosmological constant problem. Namely, we would like to understand why the Higgs vacuum does not produce the large curvature that one would expectw with naive estimates. Many efforts have been devoted to this problem, but so farno solutionhasbeen found.Thisissuewouldprobablyneed anunderstandingofthe structureofspace-time[18]. Here, we also present our discrete model of the Higgs vacuum [19], which departs fromtheusualcontinuummodel.NamelyweshallassumethattheHiggsvacuumhasan structure, and it consists of small size regions (droplets) where the vacuum expectation value is different from zero, while in the true empty regions it vanishes. For simplicity we shall consider that these regions form spherical droplets, and it will be shown that this model allows to solve the cosmological constant problem, for a certain relation between the density and size of the sherical droplets. The model is not distinguishable from the SM at the energies of current accelerators, however interesting deviations can beexpected tooccur atthecomingLHCorhigherenergies. HOLOGRAPHIC HIGGS MODELS AND DARK MATTER TheHolographicHiggsmodelsofourinterest,admitadualAdS/CFTdescription,how- ever,weshalldiscussitsfeaturesmainlyfromthe4Dpointofview,usingfirstageneric effectivelagrangianapproach,andthenpresentingspecificrealizationswithintheknown Holographic Higgs models [16, 17]. From the 4D perspective, the effective lagrangian thatdescribesthesemodels[20,21],includestwosectors:i)TheSMsectorthatcontains thegaugebosonsandmostofthequarksandleptons,whichischaracterizedbyageneric couplingg (gaugeorYukawa),andii)Anewstronglyinteractingsector,characterized sm by another coupling g and an scale M . This scale can be associated with the mass of R the lowest compositer∗esonance, which in the dual AdS/CFT picture corresponds to the lightestKK mode; in ordinary QCD M can be taken as themass of therho meson (r ). R The couplings are choosen here to satisfy g g 4p , and as a result of the dynam- sm ics of the strongly interacting sector, a compo∼site∗H∼iggs boson emerges. It behaves as an exactly massless goldstone boson because of the global symmetries that hold in the limitg 0. SM interactionsthen produceadeformation ofthetheory,and theHiggs sm → bosonbecomesapsudo-Goldstoneboson.RadiativeeffectsinduceaHiggsmass,which can bewrittenas:m (gsm)M . h ≃ 4p R Simultaneous to the Higgs appearance, a whole tower of fermionic composite states X0,X ,X ...shouldalsoappear.Ourdarkmattercandidateisidentifiedwiththelight- ± ±± estneutralstate(X0)withinthisfermionictower,andwecallitthelightestHolographic fermionic particle (LHP for short). Similarly to what happens in ordinary QCD, where the proton is stable because of Baryon number conservation, we also assume that X0 is stable because a new conserved quantum number, that we call “Dark Number” (D ). N Thus, the SM particles and the “Mesonic” states, like the Higgs boson, will have zero Dark number (D (SM) = 0), while the “baryonic” states like X0, will have +1 dark N number(D (X0)=+1).Theformationofsuch“baryonic”states,includingaconserved N numberoftopologicalorigin,hasbeenderivedrecentlyusingtheSkyrmionmodelinthe RS geometry [22]. For a strongly interacting sector that corresponds to a deformed s type model, the mass of X0 satisfies: M 4p f, where f is the analogue of the pion X0 ∼ decayconstant,thusm M .InanalogywithordinaryQCD,itisusualyassumedthat X0 R lightestresonancecorresp≃ondstoavectormeson,howeverX0itselfcouldbethelightest state. In any case, the natural value for M will be in the TeV range, somehow heavier X0 than theSUSY candidates for DM. It is importantto stress that because L M , then H R ≃ the EWPT analysis can be reinterpreted as an indirect method to obtain constraints on thedark matterscale. There are severalalternativesto accomodateourproposed LHPcandidate, withinthe HolographicHiggsmodelsproposedsofar[16],anditisoneofthepurposesofourwork toidentifythemostfavorablemodels.Fromthe4Dperspective,eachmodelisdefinedby impossingaglobal symmetryG on thenew stronglyinteracting sector, then a subgroup H of G will be gauged; here we shall consider the case when the SM group is gauged, i.e. H =SU(2) U(1) . Furthermore, in order to fix the LHP quantum numbers, one L Y × needs to specify a particular representation (G-multiplet)that will contain it. Then, this G-multiplet can be decomposed in terms of an H-multiplet plus some extra states. We callActiveDM thosecases when theLHPbelongstotheH-multiplet,whileSterileDM willbeused formodelswheretheLHPisaSM singlet. Let us consider first the models based on the group G=SU(3) U(1) [16].U(1) X X × is needed in order to get the correct SM hypercharges. Under SU(3) U(1) the SM X × doublets(Q)andd-typesinglets(D)areincludedinSU(3)triplets,i.e.Q 3 ,D 3 . ≡ ∗1/3 ≡ 0 The SM up-type singlet (U) is defined as a TeV-brane singlet field, i.e. U 1 . 1/3 ≡ T The hypercharge is obtained from: Y = 8 +X, while the electric charge arises from: √3 Q = T +Y, and T denote the diagonal generators of SU(3). Then, admiting only em 3 3,8 the lowest dimensional SU(3) representations (triplets and singlets), one can obtain the electrically neutral LHP, by requiring: X = 1/3, 2/3. Thus, for an SU(3) anti- tripletwithX =1/3:Y =(N0,C+,N0)T,therea±retwo±optionsfortheLHP:i)Model1 1 1 1 2 (active):theLHPbelongstoaSMdoublety =(N0,C+),i.e.X0=N0,andii)Model2 1 1 1 1 (sterile):theLHPisaSMsinglet,i.e.X0=N0.SimilarpatternisobtainedforX = 1/3. 2 − Choosing instead a SU(3)triplet with X = 2/3, i.e. Y =(N0,C+,C+)T, only allows ± 2 3 2 3 theLHP to be X0 =N0 (Model 3). Allowingthe inclusions of SU(3) octets leads to the 3 possibilityofhavingLHPcandidatesthatbelongtoSMtripletswithY =0, 1(Models ± 4,5). Ontheotherhand,LHPcandidatescanalsoarisewithintheminimalcompositeHiggs model (MCHM) with global symmetry G=SO(5) U(1) [23], which incorporates a X custodial symmetry. The SM hypercharge is define×d now by Y = X +TR, where TR 3 3 denotes the R-isospin obtained from the breaking chain: SO(5) U(1) SO(4) X × → × U(1) SU(2) U(1) ,andwithSO(4) SU(2) SU(2) .InthemodelMCHM , X L Y L R 5 → × ≃ × the SM quarks and leptons are accomodated in the fundamental representations (5) of SO(5), while in the option named MCHM , the SM matter is grouped in the anti- 10 symmetric (10-dimensional) representation of SO(5). For the DM candidates one can use either of these possibilities. DM models using the 5 of SO(5), can accomodate the LHP in SM doublets or singlets, similar to the pattern obtained for the SU(3) models. On the other hand, in models that employ the 10 representation of SO(5), the LHP can alsoappearinSMtriplets.Forinstance,takingX =0,allowsX0 tofitinaY =0triplet, whiletheoptionX = 1,offerstheposibilityofhavinganLHPwithinaY = 1triplet. ± ± TheeffectivelagrangiandescriptionofboththeHiggsand DM, isgivenby: a L =LH +L +(cid:229) i O , (1) H sm DM (L )n 4 in H − where LH denotes the SM Higgs lagrangian. The higher-dimensional operators O sm in (n 6) can induce corrections to the SM Higgs properties; meassuring these effects ≥ at future colliders (LHC,ILC), could provide information on the DM scale. The co- efficient a and the scale L will depend on the nature of each operator. The lead- i H ing operators are: OW = i(H†s iDm H)(Dn Wmn )i, OB = i(H†Dm H)(¶ n Bmn ), OHW = i(Dm H)†s i(Dn H)Wmni , OHB =i(Dm H)†(Dn H)Bmn , OT =i(H†Dm H)(H†Dm H), OH = i¶ m (H†H)¶ m (H†H) [20]. At LHC it will be possibleto meassure the corrections to the Higgscouplings,withaprecisionthatwilltranslateintoaboundL 5 7 TeV,while H ≥ − at ILC it will extend up to about 30 TeV [20]. These operators can also modify the SM boundsontheHiggsmassobtainedfromEWPT.Inparticular,O canincreassethelimit T ontheHiggsmassabove300GeV, fora =O(1)andL 1TeV. i H TherenormalizableinteractionsofX0withtheSM,arefi≃xedbyitsquantumnumbers, whilethecompleteeffectivelagrangianincludeshigher-dimensionaloperators,namely: a LDM =X¯0(g m Dm −MX)X0+(cid:229) (L )in 4Oin (2) X − wonheereexDpemct=s t¶hma−t L igxTiWf,miw−hgi′xleY2Bfomr.oFpoerrtahtoorssetohpaetrraetosursltthfraotmdetshceribinetecgormatpioonsitoefetfhfeecGts-, X partners of X0, one e≃xpects L M > M . Similarly, the coupling a should be of X R X X orderO(1)(b /16p 2)foroperato≃rs inducedat tree- (loop-)level. i We are interested in constraining the LHP models, using both cosmology (relic den- sity)andtheexperimentalsearchesforDM.Weshallconsiderthethreetypesofmodels: i) Active LHP models with Y = 0, ii) Active LHP models with Y = 0, and iii) Sterile 6 LHPmodels.LetusdiscussfirsttheactiveLHPmodels.Thecorrespondingrelicdensity can bewritteninterms ofthethermalaveraged cross-section<s v>as follows: 2.57 10 10 2.57 10 10M2 W h2 = × − = × − X (3) X <s v> C T,Y where C depends on the isospin (T) and hypercharge (Y) of the LHP. Numerical T,Y values ofC for the lowest-dimensional representations are:C =0.004,C = T,Y 1/2,1/2 1,0 0.01, C = 0.011. Then, in order to have agreement with current data, i.e. W h2 = 1,1 X 0.11 0.066[24],models1,3requireM =1.3TeV,whilemodel4(5)requireM =2.1 X X ± (M =2.2)TeV,respectively.Itisquiteremarkablethatthesevaluesarepreciselyofthe X rightorderexpectedin thestronglyinteractingHiggsmodel!. In order to discuss the relic density constraint for the sterile LHP DM (model 2), we notice that the couplings of X0 with the SM gauge and Higgs bosons, come from the higher-dimensional operators, which include i) 4-fermion opera- tors: O1 = 1(F¯g m F)(X¯g m X), O1 = 1(f¯g m f)(X¯g m X), OV = 1(F¯g m X)(X¯g m F), FX 2 fX 2 FX 2 OV = 1(f¯g m X)(X¯g m f), OS = 1(F¯X)(X¯F), OS = 1(f¯X)(X¯ f), ii) fermion- fX 2 FX 2 fX 2 scalar operator: OXf = (F †F )(X¯X), and iii) Fermion-vector-scalar operator: ODX = (F †Dm F )(X¯g m X). where F(f) denote the SM fermion doublet (singlet). The full analysis should include all these operators, which depends on many parame- ters, however, to obtain a simplified estimate, we shall only consider the operator O . DX This operator induces an effective vertex ZX0X0 of the form: G = g hg m , ZXX 2c W with h = 2c gc v2/M2, and c being the coefficient of O . Then, requiring x w R x DX W h2 W h2 = 0.11 0.006 [24], implies: M 0.8h TeV. Thus, for M of X DM X X order T≃eV, one would ne±ed to have h 1, which co≃uld be satisfied in some region of parameter space, although one usuall≥y expects h 1 within a strongly interacting ≤ scenario. Constraints on the LHP models can also be derived from the direct experimental search for DM, such as the one based on the nucleon-LHP elastic scattering [25]. The corresponding cross section can be expressed as: s = G2F f Y2, where f depends T,Y 2p N N on the type of nucleus used in the reaction. As it was discussed in ref. [26], vector- like dark matter with Y = 1 is severely constrained by the direct searches, unless its couplingwiththeZ bosonissuppressedwithrespectto theSM strength.A suppression of this type can be realized in a natural manner for Holographic DM models. Namely, following ref. [21], we notice that by admitting a mixing between the composite LHP and a set of elementary fields with the same quantum numbers, then the vertex ZXX will be suppressed by the mixing angles needed to go from the weak- to the mass- eigenstate basis. For model 1, with active DM appearing in a doublet y =(N0,C+)T, 1 1 1 one includes an elementary copy of these fields, which then allows to write the vertex ZXX as: G = h ′g2g m , with h < 1. The cross-section for DM+N DM+N can ZXX 2cW ′ → be written then as: s = G2F f h 2. Agreement with current bounds [25] requires to 2p N ′ have h 2 10 2 10 4, which seems reasonable. On the other hand, DM withY = 0 ′ − − automatic≤ally sati−sfies this bound, i.e. s (Y =0)=0. While for sterile dark matter, the correspondingnucleon-LHPcross-section,satisfiesthecurrentlimits[25],providedthat the factor h also satisfies h 2 10 2 10 4, which is in contradiction with the bound − − derivedfrom thecosmologica≤lrelic de−nsity,i.e. h 1, therefore we find that thesterile ≥ darkmattercandidate(Model2)seemsdisfavored. A DISCRETE MODEL OF THE HIGGS VACUUM Our presentation of the Higgs mechanism starts by considering an scalar field that interacts with gauge bosons and fermions. The lagrangian for the scalar and gauge sectorsiswritten as: L =(Dm F )†Dm F V(F ) (4) − wherethecovariantderivativeisgivenby:Dm F =(¶ m igTaVma)F ,andTa arethegen- eratorsfortherepresentationthatF belongsto.TheHig−gspotentialtakesthe“Sombrero Mexicano” form, which has a minimum at a value of the Higgs field <F >=v, which isassumedtoaccur everywhere. As we discussed in our paper [19], the continuum Higgs v.e.v. will be replaced by a distribution, i.e. we shall assume that the Higgs v.e.v. is different from zero only in some small regions (droplets), elsewhere the v.e.v. will be zero. Thus, we take the view that the Higgs vacuum is really a Bose condensate. Such condensates have been studied in condensed matter, where certain compounds are made of certain atoms, e.g. Helium,that favor the emergence of such phenomena. Therefore, onecould betempted to extrapolate that such “molecular” or“atomic” structures should also exist in order to explainthetruenatureoftheHiggsmechanism.Inthispaper,weshallconsiderthatthis may be a possibility, but will leave open the possibility that our “droplets” are indeed those “atoms” or a “molecule” or a larger collection of such atoms, which will define a hierarchyofscales. If the vacuum energy (v.e.v.) were spread continuosly, it would contribute to the cosmological constant, with a value of the order L 109 GeV4 = 1049 GeV/cm3. ≃ Howeverif the droplets, of finite size r and inter-distance l , are distributed uniformly d d with a density r , then their contribution to the cosmological constant would be L = d r vt , where the volume of the droplets is given by t r3. Thus, by saturating the d d d ≃ h observed value for the cosmological constant ( 10 4GeV/cm3), we obtain r t − d d 10 56. Furthermore, by considering that the dis≃tance between the droplets should b≃e − smaller than the shortest distance being tested at current colliders, i.e. l 10 15 cm, d − thentheresultingsizeofthedroplestisoftheorderofthePlancklength,i.e≤.r 10 33 d − ≃ cm. OnemaywonderwhycurrentexperimentshavenotdetectedthestructureoftheHiggs vacuum.Thereasonisthatcurrentprobes(photons,electrons,protons)haveanenergyor momentumthatcorrespondstoawave-lengththatislargethanthedistancebetweenthe spheres with v.e.v. different from zero. Thus, with current probes the vacuum “looks” continuum. However, one one gets an energy that is if the order of the inverse of the distancebetween thespheres, thevacuumwillstart toshowitsstructure. InordertoidentifypossibletestofourmodelthatcanbecarriedattheLHC,weshall focus on Higgs phenomenology. Let us consider the standard Higgs interaction with a fermiony , whichisdescribedbytheYukawalagrangian.AfterSSB wegetthemassof the fermion and its interaction with the Higgs. However, this will be valid only at low energies, but at high-energies the fermion will “see” less v.e.v., therefore the coupling will be not be given just by the fermion mass, but rather we need to include an energy- dependentfactorforthevertexand themass: m L =x(q2) hy¯ y +m(q2)y¯ y +h.c. (5) new L R L R v These effect can be probed at the LHC by looking at the Higgs production. For instancewecanstudythegluonfusionproduction,whichdependsontheHiggscoupling with the top. Now the cross-section needs to include the form factor x(q2), which will affect the shape of the p distributions. At lower momenta the result will be similar to T theSM, butat highermomenta,wewillobserveadeviationfromtheSM result. CONCLUSIONS Wehaveproposed anew DM candidates (LHP), withinthecontextof stronglyinteract- ing Holographic Higgs models. LHP candidates are identified as composite fermionic states (X0), with a mass of order m 4p f, which is made stable by assuming the X0 ∼ existence of a conserved “dark” quantum number. Thus, we suggest that there exists a connection between two of the most important problems in particles physics and cos- mology:EWSBand DM.Inthesemodels,theHiggscouplingsreceivepotentiallylarge corrections,whichcouldbetestedatthecoming(LHC)andfuturecolliders(ILC).Mea- suring these deviations, could also provide information on the dark matter scale. We have verified that the LHP relic abundance is satisfied for masses of O(TeV), which is the range expected in Holographic Higgs models. Furthermore, the current bounds on experimental searches for DM based on LHP-nucleon scattering, provides further con- straints on the possible models. Overall, we conclude that most favorable models are the active ones with Y = 0. It could be interesting to compare our model with other approaches thapredict acompositedark mattercandidate[28]. Here, we have also presented a model of the Higgs vacuum, which assumes that the Higgs vacuum has an structure, it consists of small size regions where the vacuum expectation value is different from zero, while in the true empty regions it vanishes. For simplicity we shall consider that these regions form spherical droplets, and it will be shown that this model allows to solve the cosmological constant problem, for a certain relation between the density and size of the sherical droplets. The model is not distinguishablefrom theSM at theenergies ofcurrent accelerators, howeverinteresting deviationscan beexpectedto occurat thecomingLHCorat higherenergies. ACKNOWLEDGMENTS The author thanks the Sistema Nacional de Investigadores and CONACyT (México) for financialsupport.TheseondpartofthisarticleisbasedonacollaborationwithP.Amore andA. Aranda, whichI sincerelyacknowledge. 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