aa r X i v : . [ h e p - ph ] N ov DUAL-CP-07-05October 25, 2018
Holographic Dark Matter and Higgs
J. Lorenzo D´ıaz-Cruz ∗ Fac. de Cs. F´ısico-Matem´aticas, BUAP,Apdo. Postal 1364, Puebla, Pue., C.P. 72000, M´exico,and Dual CP Institute of High Energy Physics (Dated: October 25, 2018)We identify possible dark matter candidates within the class of strongly interact-ing models where electroweak symmetry breaking is triggered by a light compositeHiggs boson. In these models, the Higgs boson emerges as a Holographic pseudo-goldstone boson, while dark matter can be identified as a fermionic composite state X , which is made stable through a conserved (“dark”) quantum number. An ef-fective lagrangian description of both the Higgs and dark matter is proposed, thatincludes higher-dimensional operators suppressed by an scale Λ i . These operatorswill induce deviations from the standard Higgs properties that could be meassured atfuture colliders (LHC,ILC), and thus provide information on the dark matter scale.The dark matter X , is expected to have a mass of order m X < ∼ πf ≃ O ( T eV ),which is in agreement with the values extracted from the cosmological bounds andthe experimental searches for dark matter.
PACS numbers: 12.60.Cn,12.60.Fr,11.30.Er ∗ Electronic address: [email protected]
Explaining the nature of electroweak symmetry breaking (EWSB)and dark matter (DM) have become two of the most important problems in modern ele-mentary particle physics and cosmology today [1, 2]. Within the standard model (SM),electroweak precision tests prefer the existence of a light Higgs boson, with a mass of orderof the electroweak (EW) scale v ≃
175 GeV, that should be tested soon at the LHC [3].Similarly, plenty of astrophysical and cosmological data points towards the existence of adark matter component, that accounts for about 12% of the matter-energy content of ouruniverse [4]. A weakly-interacting massive particle (WIMP) with a mass of the order of theEW scale, seems a most viable option for the dark matter. What is the nature of EWSBand dark matter, and how do they fit in our current understanding of elementary particles,is however not known.Given the similar requirements on masses and interactions for both particles, Higgs andDM, one can naturally ask whether they could share a common origin. Within the minimalSUSY SM (MSSM) [5], which has become one of the most popular extensions of the SM,there are several WIMP candidates (neutralino, sneutrino, gravitino) [6]. Among them,the neutralino has been most widely studied; it is a combination of the so called Higgsinosand gauginos, which are the SUSY partners of the Higgs and gauge bosons. Thus, in theSUSY case, the fermion-boson symmetry provides the connection between EWSB and DarkMatter. However, many new models have been proposed more recently [7], which providealternative theoretical foundation to stabilize the Higgs mechanism. Some of these newmodels assume that EWSB originates in a new strongly interacting sector, and have beenoriginally motivated by the studies of extra dimensions [8]. In some of them, candidates fordark matter have been proposed, such as the lightest T-odd particle (LTP) within little Higgsmodels [9] or the lightest KK particle (LKP) in models with universal extra-dimensions [10].In this letter, we are interested in searching for possible dark matter candidates, withinthe so called Holographic Higgs models [11]. Here, EWSB is triggered by a light compos-ite Higgs boson, which emerges as a pseudo-goldstone boson. Within this class of stronglyinteracting models, we shall propose that a stable composite “Baryon”, can account forthe dark matter. This particle can be made stable by impossing a conserved (Dark) quan-tum number, and it will be denoted as the lightest Holographic fermionic particle (
LHP ).The effective lagrangian description of both the Higgs and dark matter, includes higher-dimensional operators suppressed by an scale Λ i ( i = H, X ), which will induce deviationsfrom the SM predictions for the Higgs properties. Thus, meassuring these effects at futurecolliders (LHC,ILC), could also provide information on the dark matter scale. Furthermore,it is likely that because of compositeness, the Higgs boson will be heavier than in the SMcase, as it can be derived from EW precision tests. Having SM interactions, the LHP willshare similar characteristics with other WIMP candidates, however the composite natureof X will also have important implications for cosmological bounds and the experimentalsearches for dark matter.This picture, where strong interactions produce a light pseudo-goldstone boson and aheavier stable fermion, is not strange at all in nature. This is precisely what happens inordinary hadron physics, where the pion and the proton play such roles, namely they appearas two- and three-quark bound states, formed by the action of the strong QCD interaction.In this paper we shall propose models that produce a similar pattern for the Higgs and darkmatter, but at a higher energy scale, and with a stable neutral state instead of a charged one.We believe that such scenario is very attractive and unifying, and it could provide furtherunderstanding of both EWSB and DM problems. Although we shall formulate our ideasusing a generic effective lagrangian approach, we shall also discuss specific models withinthe known Holographic Higgs models [11]. We are thus interested in looking for adark matter candidate, within the context of strongly interacting models that produce alight composite Higgs boson. Although these models admit a dual AdS/CFT description,we shall discuss its main features from the 4D point of view, ocacionally relaying on thecorresponding 5D description to clarify some issues. From the 4D perspective, these modelsare formulated through an effective lagrangian [12, 13], that includes two sectors: i) TheSM sector that contains the gauge bosons and most of the quarks and leptons, which ischaracterized by a generic coupling g sm (gauge or Yukawa), and ii) A new strongly interactingsector, characterized by another coupling g ∗ and an scale M R . This scale can be associatedwith the mass of the lowest composite resonance, which corresponds to the lightes KKresonance in the dual 5D Ads description; in ordinary QCD it is the mass of the rho meson( ρ ). The couplings are choosen to satisfy g sm < ∼ g ∗ < ∼ π . As a result of the dynamics andglobal symmetries of the strongly interacting sector, a composite Higgs boson emerges asan exactly massless goldstone boson in the limit g sm →
0. SM interactions then producea deformation of the theory, and the Higgs boson becomes massive. Thus, radiative effectsinduce a Higgs mass, which can be written as: m h ≃ ( g sm π ) M R .Like the Higgs, we propose that dark matter arises as a composite states from the stronglyinteracting sector; in fact, a whole tower of fermionic states X , X ± , X ± , ± ... should appear.Similarly to what happens in ordinary QCD, where the proton is stable because of Baryonnumber conservation, we shall also assume that the lightest Holographic fermionic particle(LHP) X will be stable because a new conserved quantum number, that we call “DarkNumber” ( D N ). Thus, the SM particles and the “Mesonic” states, like the Higgs boson,will have zero Dark number ( D N ( SM ) = 0), while the “baryonic” states like X , will have+1 dark number ( D N ( X ) = +1). For a deformed σ type model of the strongly interactingsector, one can use NDA to derive a bound on the mass of X , namely m X < ∼ πf , where f is the analogue of the pion decay constant. It is usualy assumed that lightest resonance ofthe Holographic theory, corresponds to a vectorial resonance, in analogy with ordinary QCD.However, because we lack a detailed quantitative understanding of the strongly interactingtheory, we admit the possibility that X corresponds to the lightest state. Thus, the naturalvalue for M X will be in the TeV range, somehow heavier than the SUSY candidates fordark matter.The properties of the Holographic Higgs boson can be described in terms of the followingeffective lagrangian: L H = L Hsm + X α i (Λ i ) n − O in (1)where L Hsm denotes the SM Higgs lagrangian. The next term contains higher-dimensionaloperators O in ( n ≥ α i will depend on gauge/Yukawa couplings, mixing angles and possible loop factor,while the scale Λ i could be either f or M R , depending on the nature of each operator. Exam-ples of such operators include: O W = i ( H † σ i D µ H )( D ν W µν ) i , O B = i ( H † σ i D µ H )( ∂ ν B µν ), O HW = i ( D µ H ) † σ i ( D ν H ) W iµν , O HB = i ( D µ H ) † ( D ν H ) B µν , O T = i ( H † D µ H )( H † D µ H ), O H = i∂ µ ( H † H ) ∂ µ ( H † H ), as discussed in ref. [12]. These operators have been studied inthe past for the most general effective lagrangian extension of the SM, and they can induce,for instance, modifications to the SM bounds on the Higgs mass obtained from EW precisiontests (EWPT) [15]. In particular, the operator O T can help to increasse the limit on theHiggs mass above 300 GeV, for relatively natural values of parameters, i.e. with α i = O (1)and Λ H ≃ M R of the order 5-7 TeV, while at ILC this range will extend up toabout 30 TeV [12]. It is important to stress that such analysis would be re-interpreted asan alternative method to derive indirect constraints on the DM scale. Our proposed dark matter candidate (
LHP )can arise in any of the Holographic Higgs models proposed so far; we shall argue that it isa generic feature of this class of strongly interacting theories [11]. However, its specific real-ization will depend on the particular model under consideration, which will fix the quantumnumbers of the LHP. In this paper, we shall consider the simplest possibilities for the LHP,within the models discussed in [11, 14]. From the 4D perspective, each model is definedby impossing a global symmetry G on the strongly interacting sector, of which only theSM subgroup H = SU (2) × U (1) will be gauged. Thus, the DM model will be difined byspecifying a G -multiplet, which is composed of an H -multiplet that has SM gauge interac-tions, plus some extra singlets. We call Active DM those cases when the LHP belongs tothe H -multiplet, while Sterile DM will be used for models where the LHP is a SM singlet.Let us consider first the models that can be constructed with G = SU (3) × U (1) X .The extra U (1) X is needed in order to get the correct assignment for SM hypercharges.Under SU (3) × U (1) X the SM doublets and d-type singlets are included in SU (3) triplets,namely: the SM quark doublet appears in: Q ≃ ∗ / , while the d-type singlet is containedin: D ≃ . The SM up-type singlet is contained in a TeV-brane field that transforms asa singlet: U ≃ / . The SM hypercharge for the fermions is obtained from the relation: Y = T √ + X , while the electric charge arises from the usual relation: Q em = T + Y . Theadditional composite fermions (“Baryons”), which shall contain the LHP, must also appearin complete multiplets of SU (3), in order to keep under control their radiative contributionsto the Higgs mass [16]. Furthermore, admiting only the lowest dimensional representationsunder SU (3) (triplets and singlets) to accomodate an electrically neutral LHP candidate,requires: X = ± / , ± /
3, which admit SM singlets and doublets. A classification of thecorresponding active and sterile Holographic dark matter models is listed in Table 1. Itincludes, for instance, the case of an SU (3) anti-triplet with X = 1 /
3, which can be writtenas: Ψ = ( N , C +1 , N ) T . Therefore, we can have two options for the LHP: i) Model 1 (active)where the LHP is part of the SM doublet ψ = ( N , C +1 ), i.e. X = N , similar to a heavyneutrino, and ii) Model 2 (sterile) where the LHP is a SM singlet, i.e. X = N . In this case U (1) X G -multiplets H -multiplets LHP models+ ∗ : Ψ = ( N , C +1 , N ) T ∗ : ψ = ( N , C +1 ) T X = N (Active)2) X = N (Sterile) − ∗ : Ψ = ( C − , N , N ) T ∗ : ψ = ( C − , N ) T X = N (Active)4) X = N (Sterile)+ ∗ : Ψ = ( N , C +3 , C +4 ) T ∗ : ψ = ( N , C +3 ) T X = N (Active) − ∗ : Ψ = ( C − , N , C − ) T ∗ : ψ = ( C − , N ) T X = N (Active)+ x : Ψ =full octet mult. : ψ = ( C +7 , N , C − ) T X = N (Active)TABLE I: LHP candidates within the SU (3) × U (1) X Holographic Higgs models X does not have SM couplings at tree-level, but they could appear through the inclusion ofhigher-dimensional operators. Allowing the inclusions of SU(3) octets leads to the possibilityof having also SM triplets with Y = 0 (model 7). On the other hand, when one considersthe Higgs model with G = SO (5) × U (1) X [16], there is number of other posibilities for thequantum numbers of the G − and H − multiplets. In particular, one could have a DM neutralstate belonging to SM Doublet, SM triplet, SM singlet, etc. A detailed study of these DMoption will be carried in a forthcoming publication [17]. Here we shall only determine theviability of the SU (3) models listed in table 1.The couplings of X with the SM sector will include both renormalizable and effectiveinteractions. The renormalizable interactions will be fixed by the quantum numbers of X ,while the effective lagrangian will include higher-dimensional operators, which would repre-sent both the effects from the integration of heavy fermions that belong to the G − multiplet,as well as the composite nature of LHP. Thus, we write the full lagrangian for DM as follows: L DM = ¯ X ( γ µ D µ − M x ) X + X α i (Λ i ) n − O in (2)where D µ = ∂ µ − ig x T i W iµ − g ′ x Y B µ . For the case with Y = 0, we have g x ( g ′ x ) = g ( g ),while for Y = 1 it can be a different story, as it will be discussed next. We would like to discuss possible effectsfrom the dark matter LHP, including its composite nature. Namely, we are interestedin studying how to constrain the effective lagrangian (2), using both cosmology and theexperimental searches for DM. Three cases of models shown in table 1 will be analyzed here.Namely: i) Active LHP models with Y = 0, ii) Active LHP models with Y = 0, and iii)Sterile LHP models. We shall discuss first the calculation for the relic density of DM. Afterincluding the interaction with SM gauge bosons, the result for the relic density calculationcan be written in terms of the thermal averaged cross-section < σv > as:Ω X h = 2 . × − < σv > = 2 . × − M X C T,Y (3)where the constant C T,Y depends on the isospin (T) and hypercharge (Y) of the LHP can-didate. Numerical values for C T,Y for the lowes-dimensional representations are: C / , / =0 . C , = 0 .
01 In order to have agreement with current data on relic DM density, i.e.Ω X h = 0 . ± .
066 [18], model 1 requires M X = 1 . M X = 2 . σ T,Y = G F π f N Y , where f N is a factor that depends on the typeof nucleus used in the reaction. As it was discussed in ref. [20], vector-like dark matter with Y = 1 is severely constrained by the direct search, unless its coupling with the Z boson issuppressed with respect to the SM strength. A suppression of this type can be realized ina natural manner for Holographic dark matter models. For this, we follow ref. [13], andadmit a possible mixing between the composite LHP and some elementary fields havingthe same SM quantum numbers. Then, the vertex ZXX will be suppressed by the mixingangles needed to go from the weak eigenstates to the physical mass eigenstates. To discussan specific model, we shall consider model 1 from Table 1, i.e. the active DM appears ina doublet ψ = ( N , C +1 ). Including the elementary copy of these fields, allows to suppressthe vertex ZXX , which can be written as: Γ
ZXX = ηg c W γ µ , with η < DM + N → DM + N can be written as: σ = G F π f N η . Then, agreement with currentbounds [19] requires to have | η | ≤ − , which in turn translates into a bound Λ i >
10 TeV.Finally, we discuss the sterile LHP case, considering the model 2. In this case, thecouplings of the LHP with SM fields, only appear through higher-dimensional operators.The whole tower of dim-6 operators, includes, for instance, the following operator: O = ic x f ( H † D µ H ) ¯ Xγ µ X (4)where D µ denotes the SM covariant derivative, with c x parametrizing the strength of thiscontribution; this operator will induce the vertex ZX X . Inclusing the effect from thoseoperators that do not modify the Lorentz vectorial structure of the vertex ZX X , allowsus to write it as: Γ ZXX = g c W η ′ γ µ , with η ′ being a parameter that measures the strengthof those new effects associated with the whole tower of such operators; if only the operator(3) is included we have: η ′ = 2 c x gc w v /f . Then, requiring Ω X h ≃ Ω DM h = 0 . ± . η ′ ≃ O (0 . XX → γγ . Even more exotic signatures of this model, can be obtainedby considering the extra particles appearing in the G-multiplets, i.e. we can look for effectsform the G- or H-partners of the LHP. Within the Holographic SU (3) model 1, the LHPappears in a weak doublet, with an extra charged state X − . Because of EWPT, in particulartheir contribution to the ρ parameter, the mases of both particles X and X − should notdiffer by much. Thus, it should be possible to produce pairs X − X + at the LHC, which willdecay predominantly into X ± → W ± + X . Furthermore, in a strongly interacting theorythere should be resonances of these states, which could be searched at LHC too. Turningnow to Astrophysical signals, we could imagine that X − and other resonances, could beproduced at places with high concentrations of dark matter, where we would observe high-energy activity. Good candidates for such places, are the AGN. The high-energy signalsarising from the decays of X − into X + W − , would lead to the prediction of cosmic rayswith energies in the multi-TeV range. An extensive discussion of these searches, includingthe whole tower of dim-6 operators, will be presented in an extended version of this letter[17]. We have proposed new dark matter candidates, within the contextof strongly interacting Holographic Higgs models. These LHP candidates are identified ascomposite fermionic states ( X ), with a mass of order m X < ∼ πf , which is made stable byassuming the existence of a conserved “dark” quantum number. Thus, we suggest that thereexists a connection between two of the most important problems in particles physics andcosmology: EWSB and DM. In these models, the Higgs boson couplings receive potentiallylarge corrections, which could be tested at the coming (LHC) and future colliders (ILC).Measuring these deviations from SM predictions, will not only constrain the Higgs properties,but it could also provide information on the dark matter scale. In particular, LHC couldprovide indirect evidence of dark matter for masses of order 5-7 TeV, while ILC will be ableto reach masses of order 30 TeV. A correlated dark matter signal with these masses shouldbe also observed at LHC. A list of some of the models that can appear within the SU (3)Holographic Higgs model are shown in table 1.We have verified that the calculation of the LHP relic abundance, including the correctionsto its couplings, satisfies the astrophysical observations. Furthermore, the current bounds ondark matter experimental searches, such as those based on LHP-nucleon scattering, providesstringentconstraints on the parameters of the model. 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