v. berezinsky, a. bottino and g. mignola- non-baryonic dark matter

8/3/2019 V. Berezinsky, A. Bottino and G. Mignola- Non-Baryonic Dark Matter

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NonBaryonic Dark MatterV. Berezinsky 1 , A. Bottino 2 ,3 and G. Mignola 3 ,41 INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi (AQ), Italy2 Universit`a di Torino, via P. Giuria 1, I-10125 Torino, Italy3 INFN - Sezione di Torino, via P. Giuria 1, I-10125 Torino, Italy4 Theoretical Physics Division, CERN, CH1211 Geneva 23, Switzerland

(presented by V. Berezinsky)

The best particle candidates for nonbaryonic cold dark matter are reviewed, namely, neutralino, axion, axinoand Majoron. These particles are considered in the context of cosmological models with the restrictions given bythe observed mass spectrum of large scale structures, data on clusters of galaxies, age of the Universe etc.

1. Introduction (Cosmological environ-ment)

Presence of dark matter (DM) in the Universeis reliably established. Rotation curves in many

galaxies provide evidence for large halos lled bynonluminous matter. The galaxy velocity distri-bution in clusters also show the presence of DM inintercluster space. IRAS and POTENT demon-strate the presence of DM on the largest scale inthe Universe.

The matter density in the Universe is usuallyparametrized in terms of = / c , where c 1.88 10 29 h2 g/cm 3 is the critical density andh is the dimensionless Hubble constant denedas h = H 0 / (100km.s 1 .Mpc 1 ). Different mea-surements suggest generally 0 .4 h 1. Therecent measurements of extragalactic Cepheids inVirgo and Coma clusters narrowed this interval to0.6 h 0.9. However, one should be cautiousabout the accuracy of this interval due to uncer-tainties involved in these difficult measuremets.

Dark Matter can be subdivided in baryonicDM, hot DM (HDM) and cold DM (CDM).

The density of baryonic matter found from nu-cleosynthesis is given [1] as bh2 = 0 .025 0.005.

Hot and cold DM are distinguished by velocityof particles at the moment when horizon crossesthe galactic scale. If particles are relativistic theyare called HDM particles, if not CDM. The nat-ural candidate for HDM is the heaviest neutrino,most naturally -neutrino. Many new particleswere suggested as CDM candidates.

The structure formation in Universe put strongrestrictions to the properties of DM in Universe.Universe with HDM plus baryonic DM has awrong prediction for the spectrum of uctuationsas compared with measurements of COBE, IRASand CfA. CDM plus baryonic matter can ex-plain the spectrum of uctuations if total density0 0.3.

There is one more form of energy density in theUniverse, namely the vacuum energy describedby the cosmological constant . The correspond-ing energy density is given by = / (3H 20 ).Quasar lensing and the COBE results restrict thevacuum energy density: in terms of it is lessthan 0.7.

Contribution of galactic halos to the total den-sity is estimated as 0.03 0.1 and clusters

give 0.3. Inspired mostly by theoretical mo-tivation (horizon problem, atness problem andthe beauty of the inationary scenarios) 0 = 1is usually assumed. This value is supported byIRAS data and POTENT analysis. No observa-tional data signicantly contradict this value.

There are several cosmological models based onthe four types of DM described above (baryonicDM, HDM, CDM and vacuum energy). Thesemodels predict different spectra of uctuations tobe compared with data of COBE, IRAS, CfA etc.They also produce different effects for cluster-cluster correlations, velocity dispersion etc. The

simplest and most attractive model for a cor-rect description of all these phenomena is theso-called mixed model or cold-hot dark matter

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model (CHDM). This model is characterized byfollowing parameters:

= 0 , 0 = b + CDM + HDM = 1 ,H 0 50 kms 1 Mpc 1 (h 0.5),

CDM : HDM : b 0.75 : 0.20 : 0.05, (1)

where HDM 0.2 is obtained in ref.[2] fromdamped Ly data. Thus in the CHDM modelthe central value for the CDM density is given by

CDM h2 = 0 .19 (2)

with uncertaities within 0.1.The best candidate for the HDM particle is -

neutrino. In the CHDM model with = 0 .2mass of neutrino is m 4.7 eV . This com-ponent will not be discussed further.

The most plausible candidate for the CDM par-ticle is probably the neutralino ( ): it is massive,

stable (when the neutralino is the lightest super-symmeric particle and if R-parity is conserved)and the -annihilation cross-section results in h2 0.2 in large areas of the neutralino pa-rameter space.

In the light of recent measurements of the Hub-ble constant the CHDM model faces the age prob-lem . The lower limit on the age of Universet0 > 13 Gyr (age of globular clusters) imposes theupper limit on the Hubble constant in the CHDMmodel H 0 < 50 kms 1 Mpc 1 . This value is inslight contradiction with the recent observationsof extragalactic Cepheids, which can be summa-

rized as H 0 > 60 kms 1

Mpc 1

. However, it istoo early to speak about a serious conict tak-ing into account the many uncertainties and thephysical possibilities (e.g. the Universe can belocally overdense - see the discussion in ref.[3]).

The age problem, if to take it seriously, canbe solved with help of another successful cosmo-logical model CDM. This model assumes that0 = 1 is provided by the vacuum energy (cos-mological constant ) and CDM. From the limit < 0.7 and the age of Universe one obtainsCDM 0.3 and h < 0.7. Thus this modelalso predicts CDM h2 0.15 with uncertainties0.1. Finally, we shall mention that the CDM with0 = CDM = 0 .3 and h = 0 .8, which ts the ob-servational data, also gives h2 0.2. Therefore

h2 0.2 can be considered as the value commonfor most models.

In this paper we shall analyze several candi-dates for CDM, best motivated from the point of view of elementary particle physics. The motiva-tions are briey described below.Neutralino is a natural lightest supersymmetricparticle (LSP) in SUSY. It is stable if R-parity isconserved. CDM is naturally provided byannihilation cross-section in large areas of neu-tralino parameter space.Axion gives the best known solution for strongCP-violation. a CDM for natural values of parameters.Axino is a supersymmetric partner of axion. Itcan be LSP.Majoron is a Goldstone particle in spontaneouslybroken global U (1)B L or U (1)L . KeV mass canbe naturally produced by gravitational interac-

tion.Apart from cosmological acceptance of DMparticles, there can be observational conrma-tion of their existence. The DM particles canbe searched for in the direct and indirect experi-ments. The direct search implies the interactionof DM particles occurring inside appropriate de-tectors. Indirect search is based on detection of the secondary particles produced by DM particlesin our Galaxy or outside. As examples we canmention production of antiprotons and positronsin our Galaxy and high energy gamma and neu-trino radiation due to annihilation of DM parti-

cles or due to their decays.

2. Axion

The axion is generically a light pseudoscalarparticle which gives natural and beautiful solu-tion to the CP violation in the strong interaction[4] (for a review and references see[5]). Sponta-neous breaking of the PQ-symmetry due to VEVof the scalar eld < > = f PQ results in theproduction of massless Goldstone boson. Thoughf PQ is a free parameter, in practical applicationsit is assumed to be large, f PQ 1010 1012 GeV and therefore the PQ-phase transition occurs invery early Universe. At low temperature T QCD 0.1 GeV the chiral anomaly of QCD

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induces the mass of the Goldstone boson ma 2QCD /f PQ . This massive Goldstone particle isthe axion . The interaction of axion is basicallydetermined by the Yukawa interactions of eld(s) with fermions. Triangular anomaly, which pro-vides the axion mass, results in the coupling of the axion with two photons. Thus, the basic forcosmology and astrophysics axion interactions arethose with nucleons, electrons and photons.

Numerically, axion mass is given by

ma = 1 .9 10 3 (N/ 3)(10 10 GeV/f PQ ) eV, (3)

where N is a color anomaly (number of quarkdoublets).

All coupling constants of the axion are inverselyproportional to f PQ and thus are determined bythe axion mass. Therefore, the upper limits onemission of axions by stars result in upper limitsfor the axion mass. Of course, axion uxes can-

not be detected directly, but they produce addi-tional cooling which is limited by some observa-tions (e.g.,age of a star, duration of neutrino pulsein case of SN etc). In Table 1 we cite the upperlimits on axion mass from ref.[5], compared withrevised limits, given recently by Raffelt [6].

Table 1Astrophysical upper limits on axion mass

1990 [5] 1995 [6]sun 1 eV 1 eV

red giants 1 10 2 eV veryuncertainhor.branchstars

notconsidered 0.4 eV

SN 1987A 1 10 3 eV 1 10 2 eV

As one can see from the Table the strong upperlimit, given in 1990 from red giants, is replacedby the weaker limit due to the horizontal-branchstars. The upper limit from SN 1987A was re-considered taking into account the nucleon spinuctuation in N + N N + N + a axion emission.

There are three known mechanisms of cosmo-logical production of axions. They are (i)thermal

production, (i) misalignment production and (iii)radiation from axionic strings.

The relic density of thermally produced axionsis about the same as for light neutrinos and thusfor the mass of axion ma 10 2 eV this compo-nent is not important as DM.

The misalignment production is clearly ex-plained in ref.[5].

At very low temperature T QCD the mas-sive axion provides the minimum of the potentialat value = 0,which corresponds to conservationof CP. At very high temperatures T QCDthe axion is massless and the potential does notdepend on . At these temperatures there is noreason for to be zero: its values are different invarious casually disconnected regions of the Uni-verse. When T QCD the system tends to goto potential minimum (at = 0) and as a re-sult oscillates around this position. The energy

of these coherent oscillations is the axion energydensity in the Universe. From cosmological pointof view axions in this regime are equivalent toCDM. The energy density of this component isapproximately [5, 7]

a h2 2 (ma / 10 5 eV ) 1 .18 . (4)

Uncertainties of the calculations can be estimatedas 10 0 .5 .

Axions can be also produced by radiation of axionic strings [5],[8]. Axionic string is a one-dimension vacuum defect < PQ > = 0, i.e. a

line of old vacuum embedded into the new one.The string network includes the long strings andclosed loops which radiate axions due to oscilla-tion. There were many uncertainties in the axionradiation by axionic strings (see ref.[5] for a re-view). Recently more detailed and accurate cal-culations were performed by Battye and Schellard[8]. They obtained for the density of axions

a h2 A(ma / 10 5 eV ) 1 .18 (5)

with A limited between 2.7 and 15.2 and withuncertainties of the order 10 0 .6 . The overpro-duction condition a h2 > 1 imposes lower limiton axion mass ma > 2.3 10 5 eV .

Fig.1 shows the density of axions a h2 as afunction of the axion mass ma . The upper limits

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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 rened calculations [8] are shown by two dashed lines for twoextreme cases, respectively. The other explanations are given in the text.

on axion mass from Table I are shown above theupper absciss (limits of 1990) and below lower ab-sciss (limits of 1995). The overproduction regiona h2 > 1 and the regions excluded by astrophysi-cal observations [6] are shown as the dotted areas.

The axion window of 1995 (shown as undot-ted region) became wider and moved to the rightas compared with window 1990. The horizontalstrip shows CDM = 0 .2 0.1 as it was discussedin Introduction. One can see from Fig.1 thatstring and misaligment mechanisms provide theaxion density as required by cosmological CDMmodel, if axion mass is limited between 7 10 5 eV and 7 10 4 eV . However, in the light of un-certainties, mostly in the calculations of axion

production, one can expect that this best calcu-lated window is between 3 10 5 and 10 3 eV.This region is partly overlapped with a possible

direct search for the axion in nearest-future ex-periments (see Fig.1 and refs.[9]).

3. Axino

In supersymmetric theory the PQ-solutionfor strong CP-violation should be generalized.Within this theory the PQ symmetry breaking re-sults in the production of the Goldstone chiral su-permultiplet which contains two scalar elds andtheir fermionic partner axino ( a). The scalarelds enter the supermultiplet in the combination(f PQ + s)exp( a/f P Q ), where s is a scalar eld,saxino, which describes the oscillations of the ini-tial eld around its VEV value < > = f PQ ,and a is the axion eld. This phase transition inthe Universe occurs at temperature T f PQ . Aswe saw in the previous section the axion is mass-

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less at this temperature and since supersymme-try is not broken yet, the axino and saxino aremassless, too. The axion acquires the mass in theusual way due to chiral anomaly at T QCD ,while saxino and axino obtain the masses due toglobal supersymmetry breaking.

The saxino is not of great interest for cosmol-ogy: it is heavy and it decays fast (mostly intotwo gluons). The axino can be the lightest super-symmetric particle and thus another candidatefor DM.

How heavy the axino can be? The mass of ax-ino has a very model dependent value. In the phe-nomenological approach, using the global super-symmetry breaking parameter M SUSY one typi-cally obtains (e.g. [11],[12])

m a M 2SUSY /f PQ (6)

For example, if global SUSY breaking occurs due

to VEV of auxiliary eld of the goldstino su-permultiplet < F > = F g , then the axino massappears due to interaction term ( g/f PQ )a aF (Fhas a dimension M 2 ), and using < F > = F g =M 2SUSY one arrives at the value (6).

The situation is different in supergravity. Inref.[13] the general analysis of the axino mass isgiven in the framework of local supersymmetry.It was found that generically the mass of axino inthese theories is m a m3 / 2 100 GeV . Even incase when axino mass is small at tree level, theradiative corrections raise this mass to the value m3 / 2 . This result holds for the most general

form of superpotential.The global SUSY result,m a m23 / 2 /f PQ , can be reproduced in the localSUSY only if one of the superpotential couplingconstants is very small, < 10 4 , which impliesne-tuning. Thus, the axino is too heavy to be aCDM particle.

The only exceptional case was found by Gotoand Yamaguchi [14]. They demonstrated that incase of no-scale superpotential the axino massvanishes and the radiative corrections in somespecic models can result in the axino mass10 100 keV , cosmologically interesting. Thisbeautiful case gives essentially the main founda-tion for axino as CDM particle.

The cosmological production of axinos can oc-cur through thermal production [16] or due to

decays of the neutralinos [15],[16]. The axion chi-ral supermultiplet contains two particles whichcan be CDM particles, namely axion and ax-ino. In this section we are interested in thecase when axino gives the dominant contribu-tion. In particular this can take place in the range2 109 GeV < f P Q < 2.7 1010 GeV where axionsare cosmologically unimportant.

Since axino interacts with matter very weakly,the decoupling temperature for the thermal pro-duction is very high [16]:

T d 109 GeV (f PQ / 1011 GeV ). (7)

Therefore, axinos are produced thermally at thereheating phase after ination. The relic concen-tration of axinos can be easily evaluated for thereheating temperature T R as

a h2 0.6

m a100 keV

(3 1010 GeV

f PQ)2

T R109 GeV

(8)

Reheating temperature T R 109 GeV gives noproblem with the gravitino production. The relicdensity (8) provides CDM h2 0.2 for a reason-able set of parameters m a , f PQ and T R . One caneasily incorporate in these calculations the addi-tional entropy production if it occurs at EW scale[17].

If the axino is LSP and the neutralino is thesecond lightest supersymmetric particle, the ax-inos can also be produced by neutralino decays[15],[16],[17]. According to estimates of ref.[17]the axinos are produced due to a + de-

cays at the epoch with red-shift zdec 108

. Ax-inos are produced in these decays as ultrarela-tivistic particles and the free-streeming preventsthe growth of uctuations on the horizon scaleand less. At red-shift znr 104 axinos becomenon-relativistic due to adiabatic expansion (redshift). From this moment on the axinos behave asthe usual CDM and the uctuations on the scales (1 + znr )ctnr (which correspond to a masslarger than 10 15 M ) grow as in the case of stan-dard CDM. For smaller scales the uctuations,as was explained above, grow less than in CDMmodel. Therefore, as was observed in ref.[17], theaxinos produced by neutralino decay behave likeHDM. It means that axinos can provide generi-cally both components, CDM and HDM, needed

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for description of observed spectrum of uctua-tions.

Unfortunatelly stable axino is unobservable. Incase of very weak R-parity violation, decay of ax-inos can produce a diffuse X-ray radiation, withpractically no signature of the axino.

4. Majoron

The Majoron is a Goldstone particle associ-ated with spontaneously broken global U (1)B Lor U (1)L symmetry. The symmetry breaking oc-curs due to VEV of scalar eld, < > = vs , and splits into two elds, and J :

(vs + )exp( iJ/v s ). (9)

The eld J is the Majoron. A mass of keV can be obtained due to gravitational interac-tion[18],[19]. The keV Majoron has the great

cosmological interest since the Jeans mass associ-ated with this particle, m Jeans m3Pl /m2J , gives

the galactic scale M 1012 M . In all other re-spects the keV Majoron plays the role of a CDMparticle. It is assumed usually that the Majoroninteracts directly only with some very heavy par-ticles (e.g. with the right-handed neutrino R ).It results in very weak interaction of the Majoronwith the ordinary particles (leptons, quarks etc)and thus makes the Majoron invisible in theaccelerator experiments.

The cosmological production of the Majoronoccurs through thermal production[19, 20] and

radiation by the strings [19] as in case of the ax-ion. However, under imposed observational con-straints, the Majoron in models[18],[19] has to bein general unstable with lifetime much shorterthan the age of Universe. A successful modelwas developed in ref.[21], where the Majoron wasassumed to interact with the ordinary particlesthrough new heavy particles. For the cosmologi-cal production it was considered the phase transi-tion associated with the global U (1)B L symme-try breaking, when the Majorons were producedboth directly and through J + J decays.

Another interesting possibility was consideredrecently in ref.[22]. In this model the Majoronis rather strongly coupled with and neu-trinos ( J coupling). The Majorons are pro-

duced through J + decays. The stronginteractions between Majorons reduces the relicabundance of the Majorons to the cosmologicallyrequired value.

The Majoron signature in all these models isgiven by J + decays, which result in theproduction of keV X-ray line in the X-ray back-ground radiation.

5. Neutralino

The neutralino is a superposition of four spin1/2 neutral elds: the wino W 3 , bino B and twoHiggsinos H 1 and H 2 :

= C 1 W 3 + C 2 B + C 3 H 1 + C 4 H 2 (10)

The neutralino is a Majorana particle. With aunitary relation between the coefficients C i theparameter space of neutralino states is described

by three independent parameters, e.g. mass of wino M 2 , mixing parameter of two Higgsinos ,and the ratio of two vacuum expectation valuestan = v2 /v 1 .

In literature one can nd two extreme ap-proaches describing the neutralino as a DM par-ticle.

(i)Phenomenological approach . The allowedneutralino parameter space is restricted by theLEP and CDF data. In particular these dataput a lower limit to the neutralino mass, m >20 GeV. In this approach only the usual GUT re-lation between gaugino masses, M 1 : M 2 : M 3 =

1 : 2 : 3 , is used as an additional assumption,where i are the gauge coupling constants. Allother SUSY masses which are needed for the cal-culations are treated as free parameters, limitedfrom below by accelerator data.

One can nd the relevant calculations withinthis approach in refs.[23, 24] and in the review[25](see also the references therein). There are largeareas in neutralino parameter space where theneutralino relic density satises the relation (2).This is especially true for heavy neutralinos withm > 100 1000 GeV, ref.[26]. In these areasthere are good prospects for indirect detection of neutralinos, due to high energy neutrino radia-tion from Earth and Sun (see [27, 28] and refer-ences therein) as well as due to production of an-

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tiprotons and positrons in our Galaxy. The direct detection of neutralinos is possible too, thoughin more restricted parameter space areas of lightneutralinos (see review [25]).

This model-independent approach is very inter-esting as an extreme case: in the absence of an ex-perimentally conrmed SUSY model it gives theresults obtained within most general frameworkof supersymmetric theory.(ii) Strongly constrained models . This approachis based on the remarkable observation that inthe minimal SUSY SU(5) model with xed parti-cle content, the three running coupling constantsmeet at one point corresponding to the GUT massM GUT . Because of the xed particle contentof the model, its predictions are rigid and theystrongly restrict the neutralino parameter space.This is especially true for the limits due to pro-ton decay p K + . As a result very little space

is left for neutralino as DM particle. Normallyneutralinos overclose the Universe ( > 1). Therelic density decreases to the allowed values invery restricted areas where -annihilation is ac-cidentally large (e.g.due to the Z 0 exchange term- see ref.[29]. Thus, this approach looks ratherpessimistic for neutralino as DM particle.

In several recent works [30]-[35] less restrictedSUSY models were considered. In particular thelimits due to proton decay were lifted. A GUTmodel was not specied or less restrictive SO (10)model was used [35]. Although, the neutralinocan be heavy in these models, the prospects for

indirect detection, including the detection of highenergy neutrinos from the Sun and Earth, arerather pessimistic [32]. The direct detection ispossible in many cases [33, 34, 32].

(iii) Relaxed restrictions . In section 5.3, follow-ing refs.[36],[37], we shall analyze the restrictionsto neutralino as DM particle, imposed by basicproperties of SUSY theory. As in many pre-vious works, a fundamental element of analysisis the radiatively induced EW symmetry break-ing (EWSB)[38]. However, some mass unica-tion conditions at the GUT scale are relaxed (seeref.[36]). The powerful restriction from the no-ne-tuning condition is added.

5.1. SUSY theoretical frameworkThe basic element which should be used in

the analysis is a supersymmetry breaking and in-duced by it (through radiative corrections) elec-troweak symmetry breaking [38]. We shall referto this restriction as to the EWSB restriction.

One starts with unbroken supersymmetricmodel described by some superpotential. It isassumed that local supersymmetry is broken bysupergravity in the hidden sector, which commu-nicates with the visible sector only gravitation-ally. This symmetry breaking penetrates into thevisible sector in the form of global supersymme-try breaking. More specically it is assumed thatthe symmetry breaking terms in the visible sec-tor are the soft breaking terms given at the GUTscale Q2 M 2GUT by the following expression:

Lsb = m20a

|a |2 + m1 / 2a

a a +

+ Am 0 f Y + Bm 0 H 1 H 2 (11)

where a are scalar elds of the model (sfermionsand two Higgses H 1 and H 2 ), a are gaug-ino elds, f Y are trilinear Yukawa couplings of fermions and Higgses and the last term is anadditional (relative to the superpotential termH 1 H 2 ) soft breaking mixing of two Higgses.Here and everywhere below we specify the scale atwhich an expression and parameters are dened.

The soft breaking terms (11) are described by5 free parameters : m0 , m 1 / 2 ,A,B and . Thisimplies the strong assumption that all scalars aand all gauginos a at the GUT scale have thecommon masses m0 and m1 / 2 , respectively. Thisassumption can be relaxed, as we shall discusslater.

The soft breaking terms (11) together with su-persymmetric mixing give the following potentialdened at the EW scale at the tree level:

V = m21 |H 1 |2 + m22 |H 1 |

2 Bm 0 (H 1 H 2 + hc) +

+g21 + g22

8(|H 1 | 2 | H 2 |2 )2 , (12)

The mass parameters m1 and m2 at the GUTscale are equal to

m21 (GUT ) = m22 (GUT ) =

2 + m20 , (13)

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with dened at the GUT scale,too. The term 2

in Eqs.(12),(13) appears due to the mixing termH 1 H 2 in the superpotential of unbroken SUSY.

The radiative EWSB occurs due to evolutionof mH 2 , the mass of H 2 , which is connected withthe upper components of the fermion doubletsand in particular with t-quark. Because of thelarge mass of t-quark and consequently the largeYukawa coupling Y ttH 2 , m2H 2 evolves from m

20 at

the GUT scale to the negative value m2H 2 < 0 atthe EW scale. At this value the potential (12)acquires its minimum and the system undergoesthe EW phase transition coming to the minimumof the potential. At EW scale in the tree approx-imation the conditions of the potential minimum(vanishing of the derivatives) give:

2 =m2H 1 m

2H 2 tan

2 tan 2 1

M 2Z

2

sin2 = 2B

m2H 1 + m2H 2 + 2

2 (14)

With these equations we obtain one connec-tion between ve free parameters describing thesoft-breaking terms (11). Thus the numberof independent parameters is reduced to four,e.g.m0 , m 1 / 2 ,A, (or tan ).

Using the renormalization group equations(RGE) one can follow the evolution of the scalarparticles (Higgses and sfermions) and spin 1/2particles (gauginos) from the masses m0 and m1 / 2at the GUT scale to the masses at the EWscale. Analogously, the evolution of the couplingconstants can be calculated. In particlular themasses of Higgses at EW scale are given by

m2H i = a i m20 + bi m

21 / 2 +

+ ci A2 m20 + di Am 0 m1 / 2 , (15)

where a i , bi , ci and di are numerical coefficients,which depend on tan .

Equivalently, using Eqs.(14) one nds

M 2Z = J 1 m21 / 2 + J 2 m

20 + J 3 A

2 m20 +

+ J 4 Am 0 m1 / 2 2 , (16)

where J i are also numerical coefficients.Eq.(16) allows to impose the no-ne-tuning

condition in the neutralino parameter space. In-deed, one can keep large values of masses in the

rhs of Eq(16) only by the price of accidental com-pensation between the different terms. It is un-natural to expect an accidental compensation toa value less than 1% from the initial values. Thisis the no-ne-tuning condition.Naturally this condition is just the same as theone due to the radiative corrections to the Higgsmass.

5.2. Restrictions: the price listWithin the theoretical framework outlined

above one can choose the restrictions from thefollowing price list:

Soft breaking terms (11) and EWSB condi-tions (14),

No-ne-tuning condition,

Particle phenomenology (constraints fromaccelerator experiments and the conditionthat the neutralino is the LSP),

Restrictions due to b s decay,

Meeting of coupling constants at M GUT ,

b and b t unication,

Restrictions due to p K decay.

Using some (or all) restrictions listed above onecan start calculations for the neutralino as DMparticle. The regions where the neutralinos areoverproduced ( h2 > 1) must be excluded fromconsideration and the allowed region should bedetermined according to the chosen cosmologicalmodel (e.g. h2 = 0 .2 0.1 for the CHDMmodel). For the allowed regions the signal fordirect and indirect detection can be calculated.

5.3. SUSY models with basic restrictionsAccepting all restrictions listed above one ar-

rives at a rigid SUSY model, with the neutralinoparameter space being too strongly constrained.In ref.[37] the SUSY models with basic restric-tions were considered. These restrictions are asfollows:(i)Radiative EWSB, (ii) No ne-tuning strongerthan 1%, (iii) RGE and particle phenomenology(accelerator limits on the calculated masses and

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the condition that neutralino is LSP), (iv) Lim-its from b s decay taken with the uncertain-ties in the calculations of the decay rate and (v)0.01 < h2 < 1 as the allowed relic density forneutralinos. Rather strong restrictions are im-posed by the condition (ii); in particular it limitsthe mass of neutralino as m < 200 GeV.

At the same time some restrictions are liftedas being too model-dependent: (i) No restrictionsare imposed due to p K decay, (ii) Unica-tion of coupling constants at the GUT point isallowed to be not exact (it is assumed that newvery heavy particles can restore the unication),(iii) unication in the soft breaking terms (11)is relaxed. Following ref.[36] it is assumed thatmasses of Higgses at the GUT scale can deviatefrom the universal value m0 as

m2H i (GUT ) = m20 (1 + i ) (i = 1 , 2). (17)

Figure 2. The neutralino parameter space for themassunication case 1 = 2 = 0 and tan = 8.

Figure 3. Case 1 = 0.2, 2 = 0 .4 and tan = 8.

This non-universality affects rather stronglythe properties of neutralino as DM particle:the allowed parameter space regions becomelarger and neutralino is allowed to be Higgsino-dominated, which is favorable for detection.

Some results obtained in ref.[37] are illustratedby Figs. 2 - 6. The regions allowed for the neu-

tralino as CDM particle are shown everywhere bysmall boxes.

In Fig.2 the regions excluded by the LEP andCDF data are shown by dots and labelled as LEP.The regions labelled ne tuning have an acci-dental compensation stronger than 1% and thusare excluded. No-ne-tuning region inside thebroken-line box corresponds to a neutralino massm 200 GeV. The region EWSB+particlephenom. is excluded by the EWSB conditioncombined with particle phenomenology (neu-tralino as LSP, limits on the masses of SUSYparticles etc). In the region marked by rar-eed dotted lines neutralinos overclose the Uni-verse ( h2 > 1). The solid line corresponds tom0 = 0. The regions allowed for neutralino as

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Figure 4. Massunication case and tan = 53

CDM particle (0 .01 < h2 < 1) are shown bysmall boxes. As one can see in most regions theneutralinos are overproduced. The allowed re-gions correspond to large annihilation cross-section (e.g. due to Z 0 -pole).Fig. 3 and Fig. 2 differ only by universality: inFig. 3 1 = 2 = 0 (massunication), while in

Fig. 4 1 = 0.2 and 2 = 0 .4. The allowedregionin Fig. 3 becomes much larger and is shifted intothe Higgsino dominated region. Figs. 4 and 5 aregiven for tan = 53. This large value of tan correspond to b t unication of the Yukawacoupling constants. Again one can notice thatin the massunication case ( 1 = 2 = 0) onlya small area is allowed for neutralino as CDMparticle, while in non-universal case ( 1 = 0 , 2 = 0.2) the allowed area becomes larger and shiftsinto the gaugino dominated region.

In Fig. 6 the scatter plot for the rate of directdetection with the Ge detector [39] is given forthe non-universal case ( 1 = 0 , 2 = 0.2) andtan = 53. We notice that, for some congura-tions, the experimental sensitivity is already at

Figure 5. Case 1 = 0, 2 = 0.2 and tan = 53.

the level of the predicted rate.

6. Conclusions

1. The density of CDM needed for most cosmo-logical models is given by CDM h2 = 0 .2 0.1.There are four candidates for CDM, best moti-

vated from point of view of elementary particlephysics: neutralino, axion, axino and majoron.

2. There are different approaches to study theneutralino as DM particle in SUSY models withR-parity conservation.

In the phenomenological approach , apart fromthe LEP-CDF limits, very few other constraintsare imposed. In the Minimal SupersymmetricModel only one GUT relation between gauginosmasses, M 1 : M 2 : M 3 = 1 : 2 : 3 , is used.While the coupling constants are known, the massparameters needed for calculations are taken asthe free parameters. Many allowed congurationsin the parameter space give the neutralino as DMparticle with observable signals for direct and in-direct detection (antiprotons and positrons in our

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Figure 6. Rate for direct detection ( 1 = 0 , 2 = 0.2 and tan = 53).

Galaxy and high energy neutrinos from the Sunand Earth).

The other extreme case, the complete SUSYSU(5) model with xed particle content, withmeeting of coupling constants at the GUT pointand with the constraints due to proton decay,leaves very little space for the neutralino as DM

particle.In the third option the SUSY soft breaking

terms (11) and induced EW symmetry breakingare used as the general theoretical framework.Combined with a no-ne-tuning condition thisframework already gives essential restrictions. Inparticular, ne tuning allowed at the level largerthan 1% results in the neutralino being lighterthan 200 GeV. Here the neutralino is gauginodominated, which is unfavorable for direct detec-tion. If we employ further other restrictions, suchas exact unication of gauge coupling constantsand softbreaking parameters at the GUT scale,b s limit and b unication, the modelbecomes as rigid as the one considered above.

If, on the other hand, we relax some con-

straints, e.g. by assuming non-universality of thescalar mass term in Eq.(11), then even in the caseof 1% no-ne-tuning condition the neutralino canbe 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 canbe observed directly in the underground experi-ments, or indirectly, mostly due to the productsof 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 fermionicsuperpartner, axino . Both of them can be CDMparticles. The most important parameter here isthe scale of PQ symmetry breaking f PQ whichis observationally constrained as 2 109 GeV

v. berezinsky, a. bottino and g. mignola- non-baryonic dark matter

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