

DFTT 21/94 NEUTRALINOS AS DARK MATTER CANDIDATES

A. BOTTINO, N. FORNENGO, G. MIGNOLA Dipartimento di Fisica Teorica, Universit`a di Torino and INFN, Sezione di Torino, via P.Giuria 1, 10125 Torino, Italy

and S. SCOPEL Dipartimento di Fisica, Universit`a di Genova and INFN, Sezione di Genova, via Dodecaneso 33, 16146 Genova, Italy

ABSTRACT We review some properties of the neutralino as a candidate for dark matter in the Universe. After presentation of evaluations for the neutralino relic abundance, possibilities for its direct and indirect detections are discussed, with emphasis for measurements at neutrino telescopes.

1 The cold side of the dark: neutralinos

As is widely discussed in the literature, neutralino may be considered as avery natural candidate for Cold Dark Matter (CDM). This property rests on two assumptions, i.e. that: i) R-parity is conserved, ii) neutralino is the lightest su-persymmetric particle (LSP). Very convenient theoretical frameworks where dark matter neutralino phenomenology may be easily studied are provided by the Min-imal Supersymmetric Standard Model (MSSM) and by its implementation in a Supergravity theory (SUGRA) [1-2].

The neutralino (O/) is defined as the lowest-mass linear combination ofphotino, zino and higgsinos

O/ = a1~fl + a2 ~Z + a3 ~H01 + a4 ~H02 : (1) Invited talk presented by A.Bottino at the 6th International Workshop on: "Neutrino Telescopes", Venice 1994

Here ~fl and ~Z are linear combinations of the U(1) and SU(2) neutral gauginos, ~B and ~W3,

~fl = cos `W ~B + sin `W ~W3; ~Z = \Gamma sin `W ~B + cos `W ~W3; (2)

and `W is the Weinberg angle.

The neutralino mass mO/ and the coefficients ai depend on the parameters: _ (Higgs mixing parameter), M1, M2 (masses of ~B and of ~W3, respectively) andtan fi = v

u=vd (vu and vd are the v.e.v.'s which give masses to up-type and down-type quarks). It is customary to employ the standard GUT relationship between

M1 and M2: M1 = (5=3) tan2 `W M2 ' 0:5 M2. We use this assumption here.

In the following for the parameters M2 and _ we will consider the ranges: 20GeV ^

M2 ^ 6 TeV, 20 GeV ^ j_j ^ 3 TeV. tanfi will be taken at the representativevalue tanfi = 8.

For the evaluation of the neutralino relic abundance and of the event rates fordirect and indirect neutralino detections one also has to assign values to the masses of a large number of particles, namely to the Higgs bosons and to the Susy scalar partners of leptons and quarks: sleptons (~l) and squarks (~q). In the MSSM schemethese values are assigned arbitrarily: a standard procedure consists in assuming

mass degeneracy both for sleptons and for squarks. Only for stop particles ofdifferent helicities non-degeneracy is sometimes introduced, since, under certain circumstances, stop-mixing may generate sizeable effects [3]. As for the neutralHiggs bosons we recall that in the MSSM there are three neutral Higgs particles: two CP-even bosons h and H (of masses mh, mH with mH ? mh) and a CP-oddone

A (of mass mA). Once a value for one of these masses (say, mh) is assigned,the other two masses (

mA, mH) are derived through mass relationships dependingon radiative effects.

Implementation of MSSM with supergravity sets a much more constrainedphenomenological framework, since SUGRA establishes strict relations between all the masses in play and the few fundamental theoretical parameters: A and B (ap-pearing in the soft symmetry-breaking interaction terms),

m0 (common scalar massat the GUT scale), m

1=2 (common gaugino mass at the GUT scale) and _. Furthermore, other specific theoretical requirements (features of the symmetry breaking,condition that the neutralino be the LSP, ...) strongly restrict the whole parameter

space. This has important consequences; for instance, it constrains the neutralinoto compositions with dominance of the gaugino components.

In the present note we wish to discuss the event rates for neutralino searchesin a framework which is not too much constrained by theoretical requirements; we then simply adopt here a MSSM, with choices for the free parameters that are onlyrestricted by experimental bounds. Our main concern is to discuss the minimal sensitivity required in experimental devices in order to undertake a significant in-vestigation of neutralino dark matter. For this reason we present here evaluations where the unknown masses are assigned the smallest values compatible with exper-imental lower bounds; this usually provides maximal values for the signals. To be

definite, in the following we will set the sfermion masses at the value m ~f = 1:2 mO/, when mO/ ? 45 GeV, m ~f = 45 GeV otherwise. Only the mass of the stop quark is assigned a larger value of 3 TeV. Higgs mass mh is set at the value of 50 GeV. Thetop mass has been fixed at m

t = 150 GeV.

We show in the M2 \Gamma _ plots of Fig.1 the iso-mass lines and the iso-composition lines for the neutralino. Along an iso-composition line the composition parameter P , defined as the gaugino fractional weight, i.e. P = a12 + a22,is kept fixed. Shown in Fig.1 are the iso-composition lines referring to the values P = 0:01; 0:1; 0:5; 0:9; 0:99 (i.e., from a very pure higgsino composition to a verypure gaugino composition).

In the following we will characterize a neutralino state by the values of mO/and P ; we will accordingly present our results for the relic abundance and the eventrates in terms of these two parameters.

2 How many neutralinos around us?

For the computation of the direct and indirect event rates for neutralinoone has to use a specific value for the neutralino density

aeO/. Obviously, it wouldbe inappropriate to assign to the neutralino local density

aeO/ the standard value

for the total dark matter density ael = 0:3 GeV cm

\Gamma 3, unless one specifically verifies

that the neutralino relic abundance \Omega O/h2 turns out to be at the level of an (\Omega h2)minconsistent with

ael. This is why a correct evaluation of the event rates for O/ detectionalso requires a calculation of its relic abundance.

Thus we evaluate \Omega O/h2 and we determine aeO/ by adopting a standard procedure [4]: when \Omega O/h2 * (\Omega h2)min, we put aeO/ = ael; when \Omega O/h2 turns out to be less than (\Omega h2)min, we take

aeO/ = ael \Omega O/h

2

(\Omega h2)min : (3)

Here (\Omega h2)min is set equal to 0.03.

For the neutralino relic abundance \Omega O/h2 we employ the results of Ref.5. In Fig.2 we report \Omega O/h2 as a function of mO/ for three representative neutralino compositions: i) a gaugino-dominated composition (P = 0:9), ii) a composition ofmaximal gaugino-higgsino mixing (

P = 0:5), iii) a higgsino-dominated composition( P = 0:1). As expected, out of the three compositions displayed in Fig.2 the

gaugino-dominated state provides the largest values of \Omega O/h2. In order to have more substantial values of \Omega O/h2, one has to consider more pure gaugino compositions (P ?, 0:99). This is explicitly shown in Fig.3, where together with the reference value P = 0:5 also displayed are the rather extreme cases: P = 0:01 (very pure higgsinocomposition) and P = 0:99 (very pure gaugino composition). The very pronounced

dips in the plots of Fig.s 2-3 are due to the s-poles in the O/ \Gamma O/ annihilation crosssection (exchange of the Z and of the Higgs neutral scalars).

In Fig.4 we display the values of \Omega O/h2 versus mO/ in the form of a scatterplot obtained by varying M

2 and _ over a grid of constant spacing in the log-logplane of Fig.1.

By comparing Fig.4 with Fig.3 one sees that, at fixed mO/, the minimum of \Omega O/h2 is provided by the O/-configuration of maximal mixing, as is expected; furthermore, one sees that the large spread in values for \Omega O/h2, displayed in Fig.4,is due to configurations of extremely pure composition.

3 Looking down, watching at muons that come up

Let us turn now to the indirect search for neutralino dark matter which can beperformed by means of neutrino telescopes [8]. Neutralinos, if present in our galactic halo as dark matter components, would be slowed down by elastic scattering off thenuclei of the celestial bodies (Sun and Earth) and then gravitationally trapped inside them. Due to the process of neutralino capture these macroscopic bodiescould accumulate neutralinos which would subsequently annihilate in pairs. An important outcome of this O/-O/ annihilation would be a steady flux of neutrinosfrom these celestial bodies.

The differential neutrino flux at a distance d from the annihilation region isgiven by

dN*

dE* =

\Gamma A 4ssd2 XF;f B

(F ) O/f dN

f*

dE* (4)

where \Gamma A is the annihilation rate and F denotes the O/-O/ annihilation final stateswhich are: 1) fermion-antifermion pairs , 2) pairs of neutral and charged Higgs bosons, 3) one gauge boson-one Higgs boson pairs, 4) pairs of gauge bosons; B(F )O/f denotes the branching ratio into the fermion f (heavy quark or o/ lepton), in thechannel

F ; dNf* =dE* denotes the differential distribution of the neutrinos generatedby the semileptonic decays of the fermion

f . The *_'s, crossing the Earth, wouldconvert into muons and generate a signal of up-going muons inside a neutrino

telescope. Calculations of this muon flux from the original neutrino flux may beperformed using standard procedures [8].

Particular care has to be taken in the evaluation of the annihilation rate \Gamma A.This quantity is given by [9]

\Gamma A = C2 tanh2( to/

A ) (5)

where t is the age of the macroscopic body (t = 4:5 Gyr for Sun and Earth), o/A = (CCA)

\Gamma 1=2, C is the capture rate of neutralinos in the macroscopic body and

CA is the annihilation rate per effective volume of the body. The capture rate C isprovided by the formula [10]

C = aeO/v

O/ Xi

oeel;i mO/mi (MB fi)hv

2esciiXi; (6)

where aeO/ and vO/ are the neutralino local density and mean velocity, oeel;i is the cross section of the neutralino elastic scattering off the nucleus i of mass mi (for someproperties of the elastic

O/-nucleus cross section see next Sect. 4 and Ref. [11]),

MBfi is the total mass of the element i in the body of mass MB, hv2escii is thesquare escape velocity averaged over the distribution of the element

i, Xi is a factorwhich takes account of kinematical properties occurring in the neutralino-nucleus

interactions.

CA is given by [9]

CA = ! oev ?V

0 (

mO/ 20 GeV )

3=2 (7)

where oe is the neutralino-neutralino annihilation cross section and v is the relativevelocity.

V0 is defined as V0 = (3m2P lT =(2ae \Theta 10 GeV))3=2 where T and ae are thecentral temperature and the central density of the celestial body. For the Earth

(T = 6000 K, ae = 13 g \Delta cm

\Gamma 3) V0 = 2:3 \Theta 1025cm3, for the Sun (T = 1:4 \Theta 107 K,

ae = 150 g \Delta cm

\Gamma 3) V0 = 6:6 \Theta 1028 cm3.

For the computation of the capture rate (and then also of o/A) one has touse a specific value for the neutralino density

aeO/. Here, for any point of the modelparameter space, we evaluate ae

O/ as explained in Sect.2.

In Fig.5 we give the results of our calculations for tanh2(t=o/A) and for \Gamma A,which include the rescaling for

aeO/. This figure refers to the Earth for the case of theusual three representative O/-compositions. For simplicity, in this figure, as well asin the following ones, only the results concerning positive values of the parameter

_ are shown; similar results hold for negative values of _. One clearly sees thatequilibrium is not reached for

mO/ ?, mW , because of the substantial suppression

introduced in \Gamma A by the factor tanh2(t=o/A). In Fig.6 we show tanh2(t=o/A) and for\Gamma

A for the Sun; here the equilibrium between capture and annihilation is reachedover the whole m

O/ range.

Now we report some of our results about the flux of the up-going muonsin the case of

O/-O/ annihilation in the Earth. In Fig.s 7-8 we show the fluxes ofthe up-going muons as functions of

mO/ for a number of values of the neutralinocomposition P , for O/-O/ annihilation in the Earth. The threshold for the muon

energy is Eth_ = 2 GeV. We recall here that the present experimental upper bound

for signals coming from the Earth is 4:0 \Delta 10

\Gamma 14cm\Gamma 2s\Gamma 1 (90 % C.L.) [12]. By

comparing this upper limit with our fluxes we see that the regions explored byKamiokande (at our representative point: tan

fi = 8; mh = 50 GeV) concern themass range 50 GeV !

, mO/ !, 65 GeV. These regions are illustrated in Fig.9 in a_-M 2 plot. In this figure we also display the regions which could be explored by aneutrino telescope with an improvement factor of 10 (and of 100) in sensitivity.

The location and the shape of the most easily explorable regions in the

M2 \Gamma _ plane depend on the Earth chemical composition and on the neutralinocomposition in terms of the gaugino, higgsino components. In fact the capture rate of neutralinos is more effective when neutralino mass matches the mass of someof the main components of the Earth and when the neutralino is a large gaugino- higgsino mixture. Because of these two properties the signal is maximal along iso-mass lines in the range 50-65 GeV, with elongations along iso-composition lines of sizeable mixing.

In Fig.10 we report the fluxes for up-going muons as functions of mO/ dueto O/-O/ annihilation in the Sun. As before the threshold for the muon energy is

Eth_ = 2 GeV. The evaluated fluxes are below the present experimental upper limit of Kamiokande: 6:6 \Delta 10

\Gamma 14cm\Gamma 2s\Gamma 1 (90 % C.L.) [12].

From these results it can be concluded that neutrino telescopes with an area above 105 m2 are very powerful tools for investigating neutralino dark matter inlarge regions of the parameter space. It also emerges from the previous results that

the signals from the Earth and from the Sun somewhat complement each other toallow an exploration about DM neutralino over a wide range of m

O/. Taking into ac-count the appropriate on-source duty factor for the Sun, it turns out that the signal

from the Sun starts overcoming the one from the Earth at about mO/ , 200 GeV.This occurs since, even if the Sun is mainly composed of very light elements, its gravitational field is much more effective compared to the Earth's one in capturingneutralinos.

4 Patiently waiting that relic neutralinos hit our detector

Another way to search for dark matter neutralinos is the direct detectionwhich relies on the measurement of the recoil energy of nuclei of a detector, due to elastic scattering of O/'s. The relevant quantities to calculate are the differentialrate (in the nuclear recoil energy

Er):

dR dEr = NT

aeO/ mO/ Z

vmax vmin(Er) dvf (v)v

doeel

dEr (v; Er) (8)

and the integrated rate Rint, which is the integral of Eq. (8) from the threshold energy Ethr , which is a characteristic feature of the detector, up to a maximal energy Emaxr . In Eq.(8) NT denotes the number of target nuclei, doeel=dEr is the differentialelastic cross section and

f (v) is the distribution of O/ velocities in the Galaxy. Itis important to note again that the local density

aeO/ is evaluated here according tothe procedure discussed in Sect. 2. In general, the

O/-nucleus cross section has two contributions: a coherent contribution, depending on A2 (A is the mass numberof the nucleus) which is due to Higgs and ~

q exchange diagrams; a spin-dependent

contribution, arising from Z and ~q exchange, proportional to *2J (J + 1).

By way of example, let us remind the expression of the coherent cross sectiondue to the Higgs-exchange [14]:

Figure 10. Fluxes of the up-going muons as functions of mO/ for O/-O/ annihilation in the Sun, for the three representative neutralino compositions P = 0:1 (dotted line), 0:5 (short-dashed line), 0:9 (long-dashed line). The threshold for the muon energy is Eth_ = 2 GeV.

oeel;H = 8G

2F

ss ff

2HA2 m2Z

m4h

m2i m2O/ (m2i + mO/)2 (9)

where ffH is a quantity depending on the neutralino-Higgs and the neutralino-quarks couplings. It is worth mentioning that ff

H depends rather sensitively on theO/-composition and on a number of parameters, such as tan fi and the Higgs masses.

Except for very special points in the parameter space, the coherent contribu-tion to elastic cross section strongly dominates over the spin-dependent one. For a detailed analysis on the calculation of the direct event rates see [15] and referencesquoted therein. For an experimental overview about dark matter detectors see Ref. [16].

Here, as an example, we simply report in Fig. 11 the event rates Rintfor a Germanium detector as a function of

mO/ for neutralino compositions P =

0:1; 0:5; 0:9. Rates are calculated by integrating the differential rate of Eq.(8) overthe electron-equivalent energy range (2-4) KeV. Fig. 12 shows R

int for more pureneutralino compositions. In Fig. 13 we show the regions of the M

2 \Gamma _ param-eter space which can be explored with an improvement of one and two orders of

magnitude in the sensitivity of the detector.

As for the shape of these regions we refer to the comments presented above,in Sect. 3, in connection with Fig. 9. Again, the signals are higher along the iso-

mass line with an mO/ matching the mass of any element composing the detector.Thus, using detectors of different compositions allows explorations of the

M2 \Gamma _plane over a wide range in mO/. For instance, investigation of regions with small mO/

values (of order of 10 GeV) with very low threshold detectors [17] would be veryinteresting. In fact this

mO/ range (which is excluded by accelerator data only under a number of assumptions) is out of reach for the indirect detection discussed in theprevious Section.

5 An epilogue: how to save stationery and be happy

The procedure for getting reliable results about neutralino relic abundanceand detection event rates requires rather elaborate calculations, demanding great care and accuracy. This is because in sizeable regions of the parameter space manydifferent channels and final states are competing, and the coupling constants of the various processes are very sensitive to the values of the free parameters. Furthermoreinterplay between calculations of different quantities has to be taken into account, if one really cares about realistic evaluations; for instance, this is the case when arescaled neutralino local density has to be used in the evaluation of the event rates.

It is clear that under simplifying assumptions and in special circumstances,for example when clear dominance of a particular process accidentally occurs, order- of-magnitude calculations may give reasonably good estimates. When this is thecase, one may go through the straightforward exercise of feeding a few numbers in simple formulae (as the one in Eq.(9)) cheaply available in the market, and workout a reasonable (order of magnitude) result. It is obvious (but nevertheless worth mentioning) that these back-of-the-envelope estimates, very enjoyable for pedagog-ical purposes, may in no way be interpreted as substitute for realistic calculations. The inadequacy of back-of-the-envelope evaluations in this field may be provedquite easily, even on the back of a stamp if saving stationery and not caring about accuracy in calculations is the main point of the game!

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