Non–Baryonic Dark Matter V. Berezinsky1 , A. Bottino2,3 and G. Mignola3,4 1INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi (AQ), Italy 2Universit`a di Torino, via P. Giuria 1, I-10125 Torino, Italy 3INFN - Sezione di Torino, via P. Giuria 1, I-10125 Torino, Italy 4Theoretical Physics Division, CERN, CH–1211 Geneva 23, Switzerland
(presented by V. Berezinsky)
The best particle candidates for non–baryonic cold dark matter are reviewed, namely, neutralino, axion, axino and Majoron. These particles are considered in the context of cosmological models with the restrictions given by the observed mass spectrum of large scale structures, data on clusters of galaxies, age of the Universe etc.
1. Introduction (Cosmological environ- The structure formation in Universe put strong ment) restrictions to the properties of DM in Universe. Universe with HDM plus baryonic DM has a Presence of dark matter (DM) in the Universe wrong prediction for the spectrum of fluctuations is reliably established. Rotation curves in many as compared with measurements of COBE, IRAS galaxies provide evidence for large halos filled by and CfA. CDM plus baryonic matter can ex- nonluminous matter. The galaxy velocity distri- plain the spectrum of fluctuations if total density bution in clusters also show the presence of DM in Ω0 ≈ 0.3. intercluster space. IRAS and POTENT demon- There is one more form of energy density in the strate the presence of DM on the largest scale in Universe, namely the vacuum energy described the Universe. by the cosmological constant Λ. The correspond- The matter density in the Universe ρ is usually 2 ing energy density is given by ΩΛ =Λ/(3H0 ). parametrized in terms of Ω = ρ/ρc,whereρc ≈ Quasar lensing and the COBE results restrict the 1.88 · 10−29h2 g/cm3 is the critical density and vacuum energy density: in terms of ΩΛ it is less h is the dimensionless Hubble constant defined than 0.7. −1 −1 as h = H0/(100km.s .Mpc ). Different mea- Contribution of galactic halos to the total den- . h surements suggest generally 0 4 ≤ ≤ 1. The sity is estimated as Ω ∼ 0.03 − 0.1 and clusters recent measurements of extragalactic Cepheids in give Ω ≈ 0.3. Inspired mostly by theoretical mo- Virgo and Coma clusters narrowed this interval to tivation (horizon problem, flatness problem and 0.6 ≤ h ≤ 0.9. However, one should be cautious the beauty of the inflationary scenarios) Ω0 =1 about the accuracy of this interval due to uncer- is usually assumed. This value is supported by tainties involved in these difficult measuremets. IRAS data and POTENT analysis. No observa- Dark Matter can be subdivided in baryonic tional data significantly contradict this value. DM, hot DM (HDM) and cold DM (CDM). There are several cosmological models based on The density of baryonic matter found from nu- 2 the four types of DM described above (baryonic cleosynthesis is given [1] as Ωbh =0.025 ± 0.005. DM, HDM, CDM and vacuum energy). These Hot and cold DM are distinguished by velocity models predict different spectra of fluctuations to of particles at the moment when horizon crosses be compared with data of COBE, IRAS, CfA etc. the galactic scale. If particles are relativistic they They also produce different effects for cluster- are called HDM particles, if not – CDM. The nat- cluster correlations, velocity dispersion etc. The ural candidate for HDM is the heaviest neutrino, simplest and most attractive model for a cor- most naturally τ-neutrino. Many new particles rect description of all these phenomena is the were suggested as CDM candidates. so-called mixed model or cold-hot dark matter
1 model (CHDM). This model is characterized by Ωh2 ≈ 0.2 can be considered as the value common following parameters: for most models. In this paper we shall analyze several candi- ΩΛ =0,Ω0 =Ωb+ΩCDM +ΩHDM =1, dates for CDM, best motivated from the point of −1 −1 H0 ≈50 kms Mpc (h ≈ 0.5), view of elementary particle physics. The motiva-
ΩCDM :ΩHDM :Ωb ≈0.75 : 0.20 : 0.05, (1) tions are briefly described below. Neutralino is a natural lightest supersymmetric where ΩHDM ≈ 0.2 is obtained in ref.[2] from particle (LSP) in SUSY. It is stable if R-parity is damped Lyα data. Thus in the CHDM model conserved. Ωχ ∼ ΩCDM is naturally provided by the central value for the CDM density is given by annihilation cross-section in large areas of neu- tralino parameter space. Ω h2 =0.19 (2) CDM Axion gives the best known solution for strong with uncertaities within 0.1. CP-violation. Ωa ∼ ΩCDM for natural values of The best candidate for the HDM particle is τ- parameters. neutrino. In the CHDM model with Ων =0.2 Axino is a supersymmetric partner of axion. It mass of τ neutrino is mντ ≈ 4.7 eV .Thiscom- can be LSP. ponent will not be discussed further. Majoron is a Goldstone particle in spontaneously The most plausible candidate for the CDM par- broken global U(1)B−L or U(1)L. KeV mass can ticle is probably the neutralino (χ): it is massive, be naturally produced by gravitational interac- stable (when the neutralino is the lightest super- tion. symmeric particle and if R-parity is conserved) Apart from cosmological acceptance of DM and the χχ-annihilation cross-section results in particles, there can be observational confirma- 2 Ωχh ∼ 0.2 in large areas of the neutralino pa- tion of their existence. The DM particles can rameter space. be searched for in the direct and indirect experi- In the light of recent measurements of the Hub- ments. The direct search implies the interaction ble constant the CHDM model faces the age prob- of DM particles occurring inside appropriate de- lem. The lower limit on the age of Universe tectors. Indirect search is based on detection of t0 > 13 Gyr (age of globular clusters) imposes the the secondary particles produced by DM particles upper limit on the Hubble constant in the CHDM in our Galaxy or outside. As examples we can −1 −1 model H0 < 50 kms Mpc . This value is in mention production of antiprotons and positrons slight contradiction with the recent observations in our Galaxy and high energy gamma and neu- of extragalactic Cepheids, which can be summa- trino radiation due to annihilation of DM parti- −1 −1 rized as H0 > 60 kms Mpc . However, it is cles or due to their decays. too early to speak about a serious conflict tak- ing into account the many uncertainties and the 2. Axion physical possibilities (e.g. the Universe can be locally overdense - see the discussion in ref.[3]). The axion is generically a light pseudoscalar The age problem, if to take it seriously, can particle which gives natural and beautiful solu- be solved with help of another successful cosmo- tion to the CP violation in the strong interaction logical model ΛCDM. This model assumes that [4] (for a review and references see[5]). Sponta- Ω0 = 1 is provided by the vacuum energy (cos- neous breaking of the PQ-symmetry due to VEV mological constant Λ) and CDM. From the limit of the scalar field <φ>=fPQ results in the ΩΛ < 0.7 and the age of Universe one obtains production of massless Goldstone boson. Though ΩCDM ≥ 0.3andh<0.7. Thus this model fPQ is a free parameter, in practical applications 2 10 12 also predicts ΩCDMh ≈ 0.15 with uncertainties it is assumed to be large, fPQ ∼ 10 −10 GeV 0.1. Finally, we shall mention that the CDM with and therefore the PQ-phase transition occurs in Ω0 =ΩCDM =0.3andh=0.8, which fits the ob- very early Universe. At low temperature T ∼ 2 servational data, also gives Ωh ≈ 0.2. Therefore ΛQCD ∼ 0.1 GeV the chiral anomaly of QCD
2 induces the mass of the Goldstone boson ma ∼ production, (i) misalignment production and (iii) 2 ΛQCD/fPQ . This massive Goldstone particle is radiation from axionic strings. the axion. The interaction of axion is basically The relic density of thermally produced axions determined by the Yukawa interactions of field(s) is about the same as for light neutrinos and thus −2 φ with fermions. Triangular anomaly, which pro- for the mass of axion ma ∼ 10 eV this compo- vides the axion mass, results in the coupling of nent is not important as DM. the axion with two photons. Thus, the basic for The misalignment production is clearly ex- cosmology and astrophysics axion interactions are plained in ref.[5]. those with nucleons, electrons and photons. At very low temperature T ΛQCD the mas- Numerically, axion mass is given by sive axion provides the minimum of the potential
−3 10 at value θ = 0,which corresponds to conservation ma =1.9·10 (N/3)(10 GeV/fPQ) eV, (3) of CP. At very high temperatures T ΛQCD where N is a color anomaly (number of quark the axion is massless and the potential does not doublets). depend on θ. At these temperatures there is no All coupling constants of the axion are inversely reason for θ to be zero: its values are different in various casually disconnected regions of the Uni- proportional to fPQ and thus are determined by the axion mass. Therefore, the upper limits on verse. When T → ΛQCD the system tends to go emission of axions by stars result in upper limits to potential minimum (at θ =0)andasare- for the axion mass. Of course, axion fluxes can- sult oscillates around this position. The energy not be detected directly, but they produce addi- of these coherent oscillations is the axion energy tional cooling which is limited by some observa- density in the Universe. From cosmological point tions (e.g.,age of a star, duration of neutrino pulse of view axions in this regime are equivalent to in case of SN etc). In Table 1 we cite the upper CDM. The energy density of this component is limits on axion mass from ref.[5], compared with approximately [5, 7] revised limits, given recently by Raffelt [6]. 2 −5 −1.18 Ωah ≈ 2 · (ma/10 eV ) . (4)
Uncertainties of the calculations can be estimated Table 1 as 10±0.5. Astrophysical upper limits on axion mass Axions can be also produced by radiation of axionic strings [5],[8]. Axionic string is a one- 1990 [5] 1995 [6] dimension vacuum defect <φPQ >= 0, i.e. a sun 1 eV 1 eV line of old vacuum embedded into the new one. very red giants 1 · 10−2 eV The string network includes the long strings and uncertain closed loops which radiate axions due to oscilla- hor.–branch not 0.4 eV tion. There were many uncertainties in the axion stars considered radiation by axionic strings (see ref.[5] for a re- SN 1987A 1 · 10−3 eV 1 · 10−2 eV view). Recently more detailed and accurate cal- culations were performed by Battye and Schellard [8]. They obtained for the density of axions
As one can see from the Table the strong upper 2 −5 −1.18 Ωah ≈ A(ma/10 eV ) (5) limit, given in 1990 from red giants, is replaced by the weaker limit due to the horizontal-branch with A limited between 2.7 and 15.2 and with stars. The upper limit from SN 1987A was re- uncertainties of the order 10±0.6. The overpro- 2 considered taking into account the nucleon spin duction condition Ωah > 1 imposes lower limit −5 fluctuation in N +N → N +N +a axion emission. on axion mass ma > 2.3 · 10 eV . 2 There are three known mechanisms of cosmo- Fig.1 shows the density of axions Ωah as a logical production of axions. They are (i)thermal function of the axion mass ma. The upper limits
3 Figure 1. Axion window 1995. The curves ”therm.” and ”misalign.” describe the thermal and misaligne- ment production of axions, respectively. The dash-dotted curve corresponds to the calculations by Davis [10] for string production. The recent refined calculations [8] are shown by two dashed lines for two extreme cases, respectively. The other explanations are given in the text. on axion mass from Table I are shown above the direct search for the axion in nearest-future ex- upper absciss (limits of 1990) and below lower ab- periments (see Fig.1 and refs.[9]). sciss (limits of 1995). The overproduction region 2 Ωah > 1 and the regions excluded by astrophysi- cal observations [6] are shown as the dotted areas. 3. Axino The axion window of 1995 (shown as undot- ted region) became wider and moved to the right In supersymmetric theory the PQ-solution as compared with window 1990. The horizontal for strong CP-violation should be generalized. Within this theory the PQ symmetry breaking re- strip shows ΩCDM =0.2±0.1 as it was discussed in Introduction. One can see from Fig.1 that sults in the production of the Goldstone chiral su- string and misaligment mechanisms provide the permultiplet which contains two scalar fields and axion density as required by cosmological CDM their fermionic partner – axino (˜a). The scalar fields enter the supermultiplet in the combination model, if axion mass is limited between 7·10−5 eV (f + s)exp(a/f ), where s is a scalar field, and 7 · 10−4 eV . However, in the light of un- PQ PQ certainties, mostly in the calculations of axion saxino, which describes the oscillations of the ini- production, one can expect that this ”best calcu- tial field φ around its VEV value <φ>=fPQ, and a is the axion field. This phase transition in lated” window is between 3 · 10−5 and 10−3 eV. This region is partly overlapped with a possible the Universe occurs at temperature T ∼ fPQ.As we saw in the previous section the axion is mass-
4 less at this temperature and since supersymme- decays of the neutralinos [15],[16]. The axion chi- try is not broken yet, the axino and saxino are ral supermultiplet contains two particles which massless, too. The axion acquires the mass in the can be CDM particles, namely axion and ax- usual way due to chiral anomaly at T ∼ ΛQCD, ino. In this section we are interested in the while saxino and axino obtain the masses due to case when axino gives the dominant contribu- global supersymmetry breaking. tion. In particular this can take place in the range 9 10 The saxino is not of great interest for cosmol- 2 · 10 GeV < fPQ < 2.7·10 GeV where axions ogy: it is heavy and it decays fast (mostly into are cosmologically unimportant. two gluons). The axino can be the lightest super- Since axino interacts with matter very weakly, symmetric particle and thus another candidate the decoupling temperature for the thermal pro- for DM. duction is very high [16]: How heavy the axino can be? The mass of ax- T ≈ 109 GeV (f /1011 GeV ). (7) ino has a very model dependent value. In the phe- d PQ nomenological approach, using the global super- Therefore, axinos are produced thermally at the symmetry breaking parameter MSUSY one typi- reheating phase after inflation. The relic concen- cally obtains (e.g. [11],[12]) tration of axinos can be easily evaluated for the 2 reheating temperature TR as ma˜ ∼ MSUSY /fPQ (6) 10 ma˜ 3 · 10 GeV TR For example, if global SUSY breaking occurs due Ω h2 ≈ 0.6 ( )2 (8) a˜ 100 keV f 109 GeV to VEV of auxiliary field of the goldstino su- PQ 9 permultiplet
5 for description of observed spectrum of fluctua- duced through ντ → J + νµ decays. The strong tions. interactions between Majorons reduces the relic Unfortunatelly stable axino is unobservable. In abundance of the Majorons to the cosmologically case of very weak R-parity violation, decay of ax- required value. inos can produce a diffuse X-ray radiation, with The Majoron signature in all these models is practically no signature of the axino. given by J → γ + γ decays, which result in the production of keV X-ray line in the X-ray back- 4. Majoron ground radiation.
The Majoron is a Goldstone particle associ- 5. Neutralino ated with spontaneously broken global U(1)B−L or U(1)L symmetry. The symmetry breaking oc- The neutralino is a superposition of four spin curs due to VEV of scalar field, <σ>=vs,and 1/2 neutral fields: the wino W˜ 3,binoB˜and two σsplits into two fields, ρ and J: Higgsinos H˜1 and H˜2:
σ → (vs + ρ)exp(iJ/vs). (9) χ = C1W˜ 3 + C2B˜ + C3H˜1 + C4H˜2 (10) The field J is the Majoron. A mass of ∼ keV The neutralino is a Majorana particle. With a can be obtained due to gravitational interac- unitary relation between the coefficients Ci the tion[18],[19]. The keV Majoron has the great parameter space of neutralino states is described cosmological interest since the Jeans mass associ- by three independent parameters, e.g. mass of 3 2 ated with this particle, mJeans ∼ mPl/mJ ,gives wino M2, mixing parameter of two Higgsinos µ, 12 the galactic scale M ∼ 10 M .Inallotherre- and the ratio of two vacuum expectation values spects the keV Majoron plays the role of a CDM tan β = v2/v1. particle. It is assumed usually that the Majoron In literature one can find two extreme ap- interacts directly only with some very heavy par- proaches describing the neutralino as a DM par- ticles (e.g. with the right-handed neutrino νR). ticle. It results in very weak interaction of the Majoron (i)Phenomenological approach. The allowed with the ordinary particles (leptons, quarks etc) neutralino parameter space is restricted by the and thus makes the Majoron ”invisible” in the LEP and CDF data. In particular these data accelerator experiments. put a lower limit to the neutralino mass, mχ > The cosmological production of the Majoron 20 GeV. In this approach only the usual GUT re- occurs through thermal production[19, 20] and lation between gaugino masses, M1 : M2 : M3 = radiation by the strings [19] as in case of the ax- α1 : α2 : α3, is used as an additional assumption, ion. However, under imposed observational con- where αi are the gauge coupling constants. All straints, the Majoron in models[18],[19] has to be other SUSY masses which are needed for the cal- in general unstable with lifetime much shorter culations are treated as free parameters, limited than the age of Universe. A successful model from below by accelerator data. was developed in ref.[21], where the Majoron was One can find the relevant calculations within assumed to interact with the ordinary particles this approach in refs.[23, 24] and in the review[25] through new heavy particles. For the cosmologi- (see also the references therein). There are large cal production it was considered the phase transi- areas in neutralino parameter space where the tion associated with the global U(1)B−L symme- neutralino relic density satisfies the relation (2). try breaking, when the Majorons were produced This is especially true for heavy neutralinos with both directly and through ρ → J + J decays. mχ > 100 − 1000 GeV, ref.[26]. In these areas Another interesting possibility was considered there are good prospects for indirect detection of recently in ref.[22]. In this model the Majoron neutralinos, due to high energy neutrino radia- is rather strongly coupled with νµ and ντ neu- tion from Earth and Sun (see [27, 28] and refer- trinos (Jνµντ coupling). The Majorons are pro- ences therein) as well as due to production of an-
6 tiprotons and positrons in our Galaxy. The direct 5.1. SUSY theoretical framework detection of neutralinos is possible too, though The basic element which should be used in in more restricted parameter space areas of light the analysis is a supersymmetry breaking and in- neutralinos (see review [25]). duced by it (through radiative corrections) elec- This model-independent approach is very inter- troweak symmetry breaking [38]. We shall refer esting as an extreme case: in the absence of an ex- to this restriction as to the EWSB restriction. perimentally confirmed SUSY model it gives the One starts with unbroken supersymmetric results obtained within most general framework model described by some superpotential. It is of supersymmetric theory. assumed that local supersymmetry is broken by (ii) Strongly constrained models. This approach supergravity in the hidden sector, which commu- is based on the remarkable observation that in nicates with the visible sector only gravitation- the minimal SUSY SU(5) model with fixed parti- ally. This symmetry breaking penetrates into the cle content, the three running coupling constants visible sector in the form of global supersymme- meet at one point corresponding to the GUT mass try breaking. More specifically it is assumed that MGUT . Because of the fixed particle content the symmetry breaking terms in the visible sec- of the model, its predictions are rigid and they tor are the soft breaking terms given at the GUT 2 2 strongly restrict the neutralino parameter space. scale Q ∼ MGUT by the following expression: This is especially true for the limits due to pro- 2 2 ton decay p → K+ν. As a result very little space Lsb = m0 |φa| + m1/2 λaλa + a a is left for neutralino as DM particle. Normally X X + Am0fY + Bm0µH1H2 (11) neutralinos overclose the Universe (Ωχ > 1). The relic density decreases to the allowed values in where φa are scalar fields of the model (sfermions very restricted areas where χχ-annihilation is ac- and two Higgses H1 and H2), λa are gaug- cidentally large (e.g.due to the Z0 exchange term ino fields, fY are trilinear Yukawa couplings of - see ref.[29]. Thus, this approach looks rather fermions and Higgses and the last term is an pessimistic for neutralino as DM particle. additional (relative to the superpotential term In several recent works [30]-[35] less restricted µH1H2) soft breaking mixing of two Higgses. SUSY models were considered. In particular the Here and everywhere below we specify the scale at limits due to proton decay were lifted. A GUT which an expression and parameters are defined. model was not specified or less restrictive SO(10) The soft breaking terms (11) are described by model was used [35]. Although, the neutralino 5 free parameters : m0,m1/2,A,B and µ.This can be heavy in these models, the prospects for implies the strong assumption that all scalars φa indirect detection, including the detection of high and all gauginos λa attheGUTscalehavethe energy neutrinos from the Sun and Earth, are common masses m0 and m1/2, respectively. This rather pessimistic [32]. The direct detection is assumption can be relaxed, as we shall discuss possible in many cases [33, 34, 32]. later. (iii)Relaxed restrictions. In section 5.3, follow- The soft breaking terms (11) together with su- ing refs.[36],[37], we shall analyze the restrictions persymmetric mixing give the following potential to neutralino as DM particle, imposed by basic defined at the EW scale at the tree level: properties of SUSY theory. As in many pre- 2 2 2 2 vious works, a fundamental element of analysis V = m1|H1| + m2|H1| − is the radiatively induced EW symmetry break- − Bµm0(H1H2 + hc)+ ing (EWSB)[38]. However, some mass unifica- g2 +g2 + 1 2(|H |2 −|H |2)2, (12) tion conditions at the GUT scale are relaxed (see 8 1 2 ref.[36]). The powerful restriction from the no- The mass parameters m and m at the GUT fine-tuning condition is added. 1 2 scale are equal to 2 2 2 2 m1(GUT)=m2(GUT)=µ +m0, (13)
7 with µ defined at the GUT scale,too. The term µ2 rhs of Eq(16) only by the price of accidental com- in Eqs.(12),(13) appears due to the mixing term pensation between the different terms. It is un- µH1H2 in the superpotential of unbroken SUSY. natural to expect an accidental compensation to The radiative EWSB occurs due to evolution a value less than 1% from the initial values. This of mH2 ,themassofH2, which is connected with is the no-fine-tuning condition. the upper components of the fermion doublets Naturally this condition is just the same as the and in particular with t-quark. Because of the one due to the radiative corrections to the Higgs large mass of t-quark and consequently the large mass. Yukawa coupling Y , m2 evolves from m2 at ttH2 H2 0 the GUT scale to the negative value m2 < 0at 5.2. Restrictions: the price list H2 the EW scale. At this value the potential (12) Within the theoretical framework outlined acquires its minimum and the system undergoes above one can choose the restrictions from the the EW phase transition coming to the minimum following price list: of the potential. At EW scale in the tree approx- • Soft breaking terms (11) and EWSB condi- imation the conditions of the potential minimum tions (14), (vanishing of the derivatives) give: m2 m2 tan2 β 2 • No-fine-tuning condition, 2 H1 − H2 MZ µ = 2 − tan β − 1 2 • Particle phenomenology (constraints from −2Bµ accelerator experiments and the condition sin 2β = (14) m2 + m2 +2µ2 that the neutralino is the LSP), H1 H2 With these equations we obtain one connec- • Restrictions due to b → sγ decay, tion between five free parameters describing the soft-breaking terms (11). Thus the number • Meeting of coupling constants at MGUT , of independent parameters is reduced to four, • b − τ and b − τ − t unification, e.g.m0,m1/2,A,µ(or tan β). Using the renormalization group equations • Restrictions due to p → Kν decay. (RGE) one can follow the evolution of the scalar particles (Higgses and sfermions) and spin 1/2 Using some (or all) restrictions listed above one particles (gauginos) from the masses m0 and m1/2 can start calculations for the neutralino as DM at the GUT scale to the masses at the EW particle. The regions where the neutralinos are 2 scale. Analogously, the evolution of the coupling overproduced (Ωχh > 1) must be excluded from constants can be calculated. In particlular the consideration and the allowed region should be masses of Higgses at EW scale are given by determined according to the chosen cosmological 2 2 2 2 model (e.g. Ωχh =0.2±0.1fortheCHDM m = aim + bim + Hi 0 1/2 model). For the allowed regions the signal for 2 2 + ciA m0 + diAm0m1/2, (15) direct and indirect detection can be calculated. where ai,bi,ci and di are numerical coefficients, 5.3. SUSY models with basic restrictions which depend on tan β. Accepting all restrictions listed above one ar- Equivalently, using Eqs.(14) one finds rives at a rigid SUSY model, with the neutralino 2 2 2 2 2 MZ = J1m1/2 + J2m0 + J3A m0 + parameter space being too strongly constrained. In ref.[37] the SUSY models with basic restric- + J Am m − µ2, (16) 4 0 1/2 tions were considered. These restrictions are as where Ji are also numerical coefficients. follows: Eq.(16) allows to impose the no-fine-tuning (i)Radiative EWSB, (ii) No fine-tuning stronger condition in the neutralino parameter space. In- than 1%, (iii) RGE and particle phenomenology deed, one can keep large values of masses in the (accelerator limits on the calculated masses and
8 the condition that neutralino is LSP), (iv) Lim- its from b → sγ decay taken with the uncertain- ties in the calculations of the decay rate and (v) 2 0.01 < Ωχh < 1 as the allowed relic density for neutralinos. Rather strong restrictions are im- posed by the condition (ii); in particular it limits the mass of neutralino as mχ < 200 GeV. At the same time some restrictions are lifted as being too model-dependent: (i) No restrictions are imposed due to p → Kν decay, (ii) Unifica- tion of coupling constants at the GUT point is allowed to be not exact (it is assumed that new very heavy particles can restore the unification), (iii) unification in the soft breaking terms (11) is relaxed. Following ref.[36] it is assumed that masses of Higgses at the GUT scale can deviate from the universal value m0 as
2 2 mHi (GUT)=m0(1 + δi)(i=1,2). (17)
Figure 3. Case δ1 = −0.2, δ2 =0.4andtanβ=8.
This non-universality affects rather strongly the properties of neutralino as DM particle: the allowed parameter space regions become larger and neutralino is allowed to be Higgsino- dominated, which is favorable for detection. Some results obtained in ref.[37] are illustrated by Figs. 2 - 6. The regions allowed for the neu- tralino as CDM particle are shown everywhere by small boxes. In Fig.2 the regions excluded by the LEP and CDF data are shown by dots and labelled as LEP. The regions labelled ”fine tuning” have an acci- dental compensation stronger than 1% and thus are excluded. No-fine-tuning region inside the broken-line box corresponds to a neutralino mass mχ ≤ 200 GeV. The region ”EWSB+particle phenom.” is excluded by the EWSB condition combined with particle phenomenology (neu- Figure 2. The neutralino parameter space for the tralino as LSP, limits on the masses of SUSY mass–unification case δ1 = δ2 = 0 and tan β =8. particles etc). In the region marked by rar- efied dotted lines neutralinos overclose the Uni- 2 verse (Ωχh > 1). The solid line corresponds to m0 = 0. The regions allowed for neutralino as
9 Figure 4. Mass–unification case and tan β =53 Figure 5. Case δ1 =0,δ2 =−0.2andtanβ= 53.
2 the level of the predicted rate. CDM particle (0.01 < Ωχh < 1) are shown by small boxes. As one can see in most regions the neutralinos are overproduced. The allowed re- 6. Conclusions gions correspond to large χχ annihilation cross- section (e.g. due to Z0-pole). 1. The density of CDM needed for most cosmo- 2 Fig. 3 and Fig. 2 differ only by universality: in logical models is given by ΩCDMh =0.2±0.1. Fig. 3 δ1 = δ2 = 0 (mass–unification), while in There are four candidates for CDM, best moti- Fig. 4 δ1 = −0.2andδ2 =0.4. The allowed region vated from point of view of elementary particle in Fig. 3 becomes much larger and is shifted into physics: neutralino, axion, axino and majoron. the Higgsino dominated region. Figs. 4 and 5 are 2. There are different approaches to study the given for tanβ = 53. This large value of tan β neutralino as DM particle in SUSY models with correspond to b − τ − t unification of the Yukawa R-parity conservation. coupling constants. Again one can notice that In the phenomenological approach,apartfrom in the mass–unification case (δ1 = δ2 =0)only the LEP-CDF limits, very few other constraints a small area is allowed for neutralino as CDM are imposed. In the Minimal Supersymmetric particle, while in non-universal case (δ1 =0,δ2 = Model only one GUT relation between gauginos −0.2) the allowed area becomes larger and shifts masses, M1 : M2 : M3 = α1 : α2 : α3,isused. into the gaugino dominated region. While the coupling constants are known, the mass In Fig. 6 the scatter plot for the rate of direct parameters needed for calculations are taken as detection with the Ge detector [39] is given for the free parameters. Many allowed configurations the non-universal case (δ1 =0,δ2 = −0.2) and in the parameter space give the neutralino as DM tan β = 53. We notice that, for some configura- particle with observable signals for direct and in- tions, the experimental sensitivity is already at direct detection (antiprotons and positrons in our
10 straints, e.g. by assuming non-universality of the scalar mass term in Eq.(11), then even in the case of 1% no-fine-tuning condition the neutralino can be the DM particle in a large area of the param- eter space and can be detected in some parts of this area in direct and indirect experiments. The neutralino is an observable particle. It can be observed directly in the underground experi- ments, or indirectly, mostly due to the products of neutralino-neutralino annihilation. 3. In the framework of supersymmetric the- ory, the PQ-mechanism for solving the problem of strong CP violation, results in a Goldstone super- multiplet which contains axion and its fermionic superpartner, axino. Both of them can be CDM particles. The most important parameter here is the scale of PQ symmetry breaking fPQ which is observationally constrained as 2 · 109 GeV < 11 fPQ < 8·10 GeV . The axion can be the CDM particle, if its mass is 10−5 −10−3 eV ( the corre- 9 sponding values of fPQ are between 2 · 10 GeV Figure 6. Rate for direct detection (δ1 =0,δ2 = 10 and 6 · 10 GeV ). For larger values of fPQ the −0.2andtanβ= 53). axino can be CDM particle if a mechanism of SUSY breaking provides its mass within the in- terval 10 − 100 keV . The axino can provide both Galaxy and high energy neutrinos from the Sun CDM and HDM components needed to fit the cos- and Earth). mological observations. The axion can be directly The other extreme case, the complete SUSY observed (e.g. in microwave cavity experiments) SU(5) model with fixed particle content, with while the axino dark matter is practically unob- meeting of coupling constants at the GUT point servable. and with the constraints due to proton decay, 4. The Majoron with keV mass can be the leaves very little space for the neutralino as DM warm DM particle which explains the galactic 12 particle. scale (M ∼ 10 M ) in the structure formation In the third option the SUSY soft breaking problem. The decay of the Majoron to two pho- terms (11) and induced EW symmetry breaking tons can produce an observable X-ray line in the are used as the general theoretical framework. cosmic background radiation. Combined with a no-fine-tuning condition this Acknowledgements framework already gives essential restrictions. In The main results presented here on the neutralino particular, fine tuning allowed at the level larger are based on a work carried out with John Ellis, than 1% results in the neutralino being lighter Nicolao Fornengo and Stefano Scopel. We wish to than 200 GeV. Here the neutralino is gaugino express our thanks to them for collaboration and dominated, which is unfavorable for direct detec- discussions. Partial financial support was pro- tion. If we employ further other restrictions, such vided by the Theoretical Astroparticle Network as exact unification of gauge coupling constants under contract No. CHRX–CT93–0120 of the Di- and soft–breaking parameters at the GUT scale, rection General of the EEC. b → sγ limit and b − τ unification, the model becomes as rigid as the one considered above. If, on the other hand, we relax some con-
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