On the neutralino as dark matter candidate. II. Direct detection

Astroparticle Physics 2 (1994) 77-90 North-Holland

Astroparticle

Physics

On the neutralino as dark matter candidate. II. Direct detection

A. Bottino, V.

de Alfaro, N. Fornengo, G. Mignola and S. Scopel

Dipartimento di FL~ica Teorica dell'Unit,ersit?~ di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Turin, Italy Received 31 August 1993

Evaluations of the event rates relevant to direct search for dark matter neutralino are presented for a wide range of neutralino masses and for various detector materials of preeminent interest. Differential and total rates are appropriately weighted over the local neutralino density expected on theoretical grounds.

1. I n t r o d u c t i o n

T h e most natural way of searching for neu- tralino dark m a t t e r is provided by direct experi- ments where the effects induced in a p p r o p r i a t e detectors by n e u t r a l i n o - n u c l e u s elastic scattering may be measured. T h e aim of this p a p e r is to present the evaluation of the event rates relevant to these direct m e a s u r e m e n t s and to discuss the experimental sensitivities required in this search for it to be successful.

O u r present analysis refers to a wide range for the neutralino masses, 20 G e V _ < m x < 1 TeV, and examines various detector materials of pre- sent experimental interest: Ge, NaI, Xe, C a F 2. The theoretical framework is provided by the Minimal SuSy Standard Model (MSSM), imple- mented with a general G U T assumption, as dis- cussed in the companion p a p e r [1] (this reference will h e r e a f t e r be referred to as p a p e r I). The model depends on a n u m b e r of free p a r a m e t e r s which, apart from the masses of a few particles which come into play, are chosen here to be M2, /.t, tan ft. T h e definition of these p a r a m e t e r s is recalled in p a p e r I.

Correspondence to: A. Bottino, Dipartimento di Fisica Teorica dell'Universit~ di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Turin, Italy.

In the following sections the differential and the integrated rates will be presented and dis- cussed also in connection with the properties of the neutralino relic density.

2. Event rates

For the nuclear recoil spectrum we use the expression dR _ Z R°" Fi2(ER)I(ER) (2.1) d E R -7" ( E ~ a x ) where Ro., = Nx PX tri(t;). (2.2) m x

T h e various quantities in eqs. (2.1), (2.2) are defined as follows: E a is the nuclear recoil en- ergy, i.e. E R = m2,edt,2(l -- COS O*)/m u (0" is the scattering angle defined in the n e u t r a l i n o - n u c l e u s center of mass frame, m N is the nucleus mass, mre o is the n e u t r a l i n o - n u c l e u s reduced mass and

max 2 ,2

t' is the relative velocity), E R = 2mre J, / m u , F ( E R) denotes the nuclear form factor, tr is the n e u t r a l i n o - n u c l e u s cross section. The subscript i refers to the two cases of coherent and spin-de- pendent effective interactions. N v is the n u m b e r density of the target nuclei and Px is the local

0927-6505/94/$07.0t) © 1994 - Elsevier Science B.V. All rights reserved S S D I 0 9 2 7 - 6 5 0 5 ( 9 3 ) E 0 0 2 0 - O

78

A. Bottino et al. / The neutralino as dark matter candidate. H neutralino matter density. Finally, I ( E R) is given

by

I ( E R ) - (V) 'cm,,(ER) dt V " (2.3)

In eq. (2.3) f ( v ) is the velocity distribution of neutralinos in the Galaxy, measured in the Earth rest frame,

Umin(E

R)

is given by

/)min(ER )=

(mNER/(2m~ed)) 1/2. T h e averages appearing in

eqs. (2.2), (2.3) denote averages over the velocity distribution in the Earth rest frame. An explicit formula for I ( E R) in the case of a maxwellian velocity distribution may be found in ref. [2].

The differential rates that will be presented in section 6 will be expressed in terms of the elec- tron-equivalent energy Ec~ rather than in terms of E R. T h e s e two variables are simply propor- tional: Ee~ = QE R, where Q is defined as quench- ing factor.

In section 6 we will also discuss the total rates, obtained by integrating d R / d E , ¢ over appropri- ate ranges of Ece above the experimental energy threshold.

3. N e u t r a l i n o - n u c l e u s elastic cross s e c t i o n s

T h e total cross sections for n e u t r a l i n o - n u c l e u s elastic scattering have been evaluated following the p r o c e d u r e discussed in refs. [2,3].

H e r e we only summarize some of the main properties. N e u t r a l i n o - q u a r k scattering is de- scribed by the amplitudes with Higgs boson ex- changes and Z boson exchange in the t-channel and by the amplitudes with squark exchanges in the s- and u-channels. T h e neutral Higgs bosons considered here are the two CP-even bosons: h, H (of masses rn h, m H with m H > m h) and the CP-odd one: A (of mass rnA).

T h e relevant properties for these amplitudes are: (1) Higgs boson exchanges contribute a co- herent cross section which is only vanishing when there is no zino-higgsino mixture in the neu- tralino composition [4]; (2) Z boson exchange provides a spin-dependent cross section which

takes contribution only from the higgsino compo- nents of X; (3) squark exchanges contribute a coherent cross section (due to zino-higgsino mix- ing) as well as a spin-dependent cross section (due mainly to the gaugino c o m p o n e n t s of X) [5]. In the evaluations of the cross sections pre- sented here, we have adopted for the masses of the Higgs bosons and of the sfermions the various sets of values indicated in p a p e r I. In particular we have relaxed here the assumption, used in previous papers [2,3], that m / > 150 GeV, since this stringent lower bound, previously reported by the C D F Collaboration [6], is significantly weak- ened by more refined analyses [7,8]. Thus the following two cases have been considered for rn/: (1) m f = 1 . 2 m~, when m x > 4 5 GeV, m / = 4 5 G e V otherwise, except for the mass of the top scalar partners (the only one relevant to radiative corrections) which has been taken rh = 3 T e V (hereafter this choice for m f will simply be re- ferred to as: r n f = 1.2 rex); (2) m / = 3 T e V for all sfermions. T h e Higgs boson mass m h (taken here as a free p a r a m e t e r ) is set equal to values close to its experimental lower bound ~ 50 G e V [9].

Let us recall that, whereas the coherent cross section is proportional to the square of the mass n u m b e r A of the nucleus, the non-coherent cross section brings a spin-dependence that is com- monly represented by a factor AzJ(J+ 1). T h e values used here for this last quantity are taken from [10]; other evaluations may be found in ref. [111.

It is worth noticing that, in the model we are considering here, the event rates for neutralino detection are largely dominated by coherent ef- fects in most regions of the p a r a m e t e r space. In the small domains where spin-dependent effects dominate over the coherent ones the total rates are usually too low to allow detection. T h e n in the present p a p e r we do not consider explicitly materials enriched in high-spin isotopes, even if these materials are very interesting for a search of hypothetical dark matter particles which inter- act with m a t t e r via substantial spin-dependent interactions. In particular for G e and Xe we consider only natural isotopic compositions (en- richment in some isotope would not appreciably modify our results).

A. Bottino et aL / The neutralino as dark matter candidate. 11 79

4. Nuclear form factors

Let us turn now to the E R - d e p e n d e n c e intro- d u c e d in the nuclear recoil s p e c t r u m by the form factors. T h e s e f o r m factors d e p e n d sensitively on the n a t u r e o f the effective interaction involved in the n e u t r a l i n o - n u c l e u s scattering. F o r the c o h e r -

ent case, we simply e m p l o y the s t a n d a r d

p a r a m e t r i z a t i o n [ 12]

3 j l ( q r o )

F ( E R ) - - e x p ( - ½sEq2), (4.1)

qro

where q2 = ]q12 = 2 r u N E R is the s q u a r e d three- m o m e n t u m transfer, s = 1 fm is the thickness p a r a m e t e r for the nucleus surface, r 0 = ( r 2 -

5S2) I/2, r = 1.2 A 1/3 fm and jl(qro) is the spheri-

cal Bessel function o f index 1.

T h e form factor o f eq. (4.1) i n t r o d u c e s sub- stantial suppression in the recoil s p e c t r u m unless

qr o << 1. A noticeable r e d u c t i o n in d R ~ d E R may

already o c c u r at threshold E R = E ~ _{= E e e / Q} th

w h e n r0v ~ is not small c o m p a r e d to unity.

T h e actual o c c u r r e n c e o f this f e a t u r e d e p e n d s on a few p a r a m e t e r s of the d e t e c t o r material: nu- clear radius, q u e n c h i n g factor, threshold e n e r g y

E~ TM. T h e values of these p a r a m e t e r s for the nuclei

c o n s i d e r e d in this p a p e r are r e p o r t e d in table 1 [13]. Also the values of F E ( E th) calculated from eq. (4.1) are given in this table. W e see that the effect o f the form factor is very m o d e r a t e for 19F, 23Na and 76Ge. In 76Ge, as c o m p a r e d to 19F and

Table l

Quenching factors Q, threshold energies Eete h, squared values

of the form factor (4.1) at the threshold F2(E~) and the values k,'~)~ where the first zero of eq. (4.1) occurs are listed for the nuclei considered in our analysis. [The values of Q and E~ h are taken from refs. [14,15] (GeL refs. [16,2] (NAIL ref. [17] (Xe) and ref. [13] (CaF2).]

Nucleus Q E~ (keV) F 2(E~) E¢~ (keV) o

19F 0.09 8.5 0.84 378 23Na 0.23 4 0.96 630 4°Ca 0.07 8.5 0.44 62 76Ge 0.25 2 0.86 61 127I 0.07 4 0.05 7.4 131Xe 0.80 20 0.28 80

23Na, this occurs since the larger nuclear exten- sion is c o m p e n s a t e d by a smaller value of E th and a larger Q. T h e effect o f the form factor on the shape o f the differential s p e c t r u m as a func- tion of Eee d e p e n d s mainly on how close to the

threshold e n e r g y Et~ h the first z e r o o f the function

j , ( q r o ) / q r 0 occurs (this zero is located at E~)e---- 1 0 Q / r a N r2. C o m p a r i n g the values of E th and of

E~)~ r e p o r t e d in table 1, we see that only for lz7I (and marginally for 131Xe) this effect is expected to be noticeable.

In general for the s p i n - d e p e n d e n t case there are no analytic expressions for the form factors. However, numerical analyses have b e e n per- f o r m e d on 131Xe [12], ~3Nb [18] and 73Ge [19]. T h e general feature is that these form factors have a m u c h milder d e p e n d e n c e o n E R as com- p a r e d to the c o h e r e n t ones, b e c a u s e only a few nucleons participate in the n e u t r a l i n o - n u c l e u s scattering in this case. In o u r evaluations we use the results of refs. [12,19] for 131Xe and 73Ge. For 12Vl we assume that the form factor has the same b e h a v i o u r as in the case o f 13~Xe. F o r 23Na and I°F we e m p l o y the analytic expression (4.1), since for these light nuclei the suppression due to nuclear size is small anyway.

5. Neutralino local density

As for the value o f the local neutralino den- sit)', Px, to bc used in the rate of eq. (2.2), for any point o f the model p a r a m e t e r space wc take into a c c o u n t the relevant value of the neutralino relic a b u n d a n c e . W h e n f2xh2 is larger than a minimal (,Oh2)min consistent with the s t a n d a r d value for the local dark m a t t e r density P¢ = 0.3 G e V cm -a, then we simply put Px = P t . W h e n ,Qxh 2 turns o u r to be less t h a n (~'~hE)min, we take

p~: = pr,Oxh2//( ,OhZ)min - p ( ~ . (5.1) H e r e ($'~h2)min is set equal to 0.05. T h e effect o f this p r o c e d u r e o f rescaling the local dark mat- ter density a c c o r d i n g to the actual neutralino relic a b u n d a n c e is manifest in some characteristic features of o u r results, as is shown in the next section.

80 A . B o t t i n o et al. / The neutralino as d a r k m a t t e r candidate. !I 6 . R e s u l t s

6.1. G e r m a n i u m

We start the presentation of our results by discussing the event rates expected for Ge-detec- tors *. In fig. 1 we report the integrated rate R ( 2 K e V < E ~ < 4 K e V )

f4

~eV d E ~ d R / d E ~

= ~2 KeV

at the representative point: t a n / 3 = 20, m h = 50 GeV, m / = 1.2 m r, in the form of a scatter plot obtained by varying the M 2 , / z p a r a m e t e r s in the ranges: 20 G e V _ < M 2 _< 6 T e V and 20 G e V _< I/z[ _< 3 TeV. T h e range of the neutralino mass

m x considered here is 20 G e V _ < m r _ < l TeV; however, to simplify the presentation of our re- sults, in fig. 1 and in the following plots for R, only the range 20 G e V < m x < 300 G e V is shown

102 101 100 10-1 1 0 - 2 1 0 - 3 ' ' 1 . . . . I . . . . I . . . . I . . . . I . . . . t ~ ,I . . . . I . . . . I . . . . I , , , I . . . . 50 100 150 200 ~50 300 m x (GeV)

Fig. 1. Scatter plot of the integrated rate R(2 KcV _< Eee _< 4 KeV) for n e u t r a l i n o - G e interaction as a function of neu- tralino mass m r , at the representative point: t a n / 3 = 20, m , = 50 GeV and m / = 1.2 m~. The parameters M 2 and /.t are varied in the ranges 20 GeV _< M2 _< 6 TeV and 20 GeV < I/zl~ 3 TeV. The horizontal line denotes the present exper- imental limit Re~ ~. Filled circles represent the R values for neutralino compositions where gaugino and higgsino compo-

nents are maximally mixed (0.45 ~ a~ + a 2 < 0.55).

* For an up-to-date review of the present achievements and the new perspectives for dark matter detectors using Ge as well as the other materials discussed in the present work, see the experimental papers in ref. [20].

(at higher m x the plots retain the same smooth decrease as m r increases).

To illustrate some of the features of the scat- ter plot in fig. 1, let us recall two main points: ( i ) at fixed rn x the event rate R depends on the model p a r a m e t e r s only through the product pxo-i (see eqs. (2.1), (2.2)), (ii) this product is given by p X o ' i = p g o " i when ~ h 2 >_ ( f ~ h E ) m i n , (6.1)

px ori=£pecrio~( g2xh2)o.,.cx ari

(O'annU)

when ~2xh 2 < (~h2)min (6.2)

( ~ . , and c being the X - X annihilation cross section and the relative velocity; see eq. (3.1) of p a p e r I).

In fig. 1 the wide spread in the values of R at fixed rn x is due to the very large variations of p~o" i as we scan the M 2,/.t plane along an isomass curve. A detailed analysis of our numerical re- sults shows that the maximal value of R, at each m r, usually corresponds to a point in the M 2, plane w h e r e .Qxh 2 is substantially b e l o w (.Qh2)min. In other words, the maximal expected signal is obtained for neutralino configurations which im- ply use of eq. (6.2) for px, with values for the rescaling p a r a m e t e r ~: which may also be very small. T h e nature of the neutralino composition that at fixed rn x provides the maximum R de- pends on a very crucial balance between ~, and tTa. . in the expression (6.2). Indicatively, the largest event rates in the plot of fig. 1 entail the following values for ~:: ~: ~ 0 . 0 1 - 0 . 0 2 for m r = 40-60 G e V (in this case the gaugino-higgsino mixing is substantial), ¢ ~ 0.1 for rnx = 60-70 G e V (here the composition is mainly of the higg- sino type), ¢ ~ 0.05 for m x = 7 0 - 9 0 G e V (large gaugino-higgsino mixing); above m x = 90 G e V the neutralino composition is mainly of the gaug- ino type with ~: which reaches values of order of a few tenths. These properties are depicted in figs. 2 and 3. Figure 2 s h o w s ~'~xh 2 as a function of rnx; the diamonds denote the values of the relic abundances for those configurations that provide the maximal values of R. T h e composi- tions of these neutralino configurations are shown in the M2-/z plane of fig. 3.

A. Bottino et al. / The neutralino as dark matter candidate. II 81 I01 I0 0 l O - I 1 0 - 2 1 0 - 3 1 0 - 4 1 0 - 5 1 0 - 6 1 0 - 7 ' ' I ' ''I . . . . I . . . . I . . . . I . . . .

; , ; . . . , -

i!ltit:-'(~(:~. • .. : " 7 . " . _ i m i • . . . . ] . . . . I . . . . ] . . . . ] . . . . 50 100 150 200 250 300 m:t (CeV)

Fig. 2. Scatter plot of the neutralino relic abundance

~Ixh2

as a function of ncutralino mass mx. T h e relic abundancc is calculated in thc s a m c representative point as in fig. I and also with the samc variations over M2, ~. T h e horizontal linc denotes thc value (.f]h2)min = 0.05. D i a m o n d s reprcscnt the values of rclic abundance which givc thc maximal value of R for a given neutralino mass m ~ (they correspond to the

m a x i m a of R in fig. l).

In view o f these results we wish to e m p h a s i z e that, by disregarding the n e u t r a l i n o c o m p o s i t i o n s that give low values of O x h 2 (say, below O h 2 ~ 0.05) on the base that these c o m p o s i t i o n s are not cosmologically interesting, o n e would miss config- urations that potentially g e n e r a t e the most siz- able signals in the direct searches.

It is w o r t h noticing that the c o n f i g u r a t i o n s o f maximal g a u g i n o - h i g g s i n o mixing (defined in pa- per I) provide large values of R ( d e n o t e d by circles in fig. 1) over the whole m x range, even if the rescaling effect is always substantial for these compositions; in fact in this case the elastic X - nucleus c o h e r e n t cross sections are very large and c o m p e n s a t e for the large values of Olan n in eq. (6.2).

Because o f the p r o p e r t i e s that we have just discussed, the g r a p h o f the event rate R (fig. 1) exhibits some o f the salient features o f the plot o f the relic a b u n d a n c e /2xh 2 as a function of mx (see fig. 2). In particular we recognize the pro-

n o u n c e d dips at m x ~ 2 5 G e V and at m x ~ 4 5

G e V , i.e. at the values w h e r e o'a, . has an A ( h ) - pole a n d a Z-pole, respectively.

Finally we wish to stress that in the r e p r e s e n - tative point u n d e r discussion the maximal ex- p e c t e d signal: R ~ 6 e v e n t s / k g day, which occurs at rn x ~ 40 G e V , differs only by a factor 1.5 from

the present experimental sensitivity R~xo= 10 e v e n t s / k g day [14] ( d e n o t e d by a horizontal line in o u r plot o f fig. 1). In o r d e r to illustrate the discovery potential of the direct neutralino search by G e d e t e c t o r s we show in fig. 4 the regions in the M2, I~ plane that could be experimentally investigated in case o f i m p r o v e m e n t s in sensitivity by a factor o f 10 (heavy-dotted regions) or by a factor 100 (light-dotted regions).

In fig. 5 we give a sample o f the differential

rates d R / d E ~ c o r r e s p o n d i n g to the neutralino

compositions that, at a given m x , provide a maxi- mal value for the integrated rate R, previously defined. T h e values m~ = 40, 80, 150 G e V arc

8 v 103 102 101 _

~ / ~ . . . - /

- - _ ,

¢o.,.f-_l__,__ 'o_otl

I I I 3o , I,,_J,, ,I , l 0 50 100 500 1000 (CeV) i I i J ~ / - - io 3 _ ~ . . ~ / , / I O O O c ~ -3oo 0.1 . . . J io, , I,,

i

,x , 10 50 100 500 1000 -#z (GeV)

Fig. 3. lsomass curves and composition lines for neutralino in the M2- ~ plane for tan/3 = 20. Dashed lines are lines of constant neutralino mass (ma = 30 GeV, 100 GeV, 3(X) GeV and 1 TeV). Solid lines refer to constant gaugino fraction f~ in the neutralino composition (fg = a~ + a22): f~ = 0.99, 0.9, 0.5, 0.1 and 0.01. Squares represent the points in the M2- # parameter space which give the maximal integrated rate R for a given neutralino mass m x (they correspond to the maxima

82 A . Bottino et aL / The neutralino as dark m a t t e r candidate. H 10 3 [_5 10 2 I ! : I : I I : I . . . . ° . . . i i : : : ::!!:~: . . . iii~ii~i: / -.:2 :Xl;;; i!: iiiih~g: I ? : i i i i i i i i i i : i ?. ~ i ~ : i l i I i ~ : ! i ! i ! ! ~ i i i i i ! i i !

' :i: ii ! iiiii!iiii!i!i,

. . . . ??i?iii: . : : : : . ) i : :~!:i': i i : t : i :

i i

~: !iii!iiii !::iii : ~: ! : ~ ¸ :-'"i:": : . ' ~ -- -

i ',i iiiii. iiiiiiiHii

,)i!ii

i

\ : . , 103 101 I ... l " I . . . L I01 1000 500 I00 50 10 l0 50 I00 500 1000 - / z ( G e V ) ~ ( G e V ) /o <: v 10 2 Fig. 4. E x p l o r a b l e r e g i o n s in t h e M 2 - / z p a r a m e t e r s p a c e f o r n e u t r a l i n o - G e i n t e r a c t i o n . P a r a m e t e r s a r e : 2 0 G e V < M 2 < 6 T e V , 2 0 G e V < I ~ 1 < 3 T e V , t a n / 3 = 20, m h = 5 0 G e V a n d m / = 1.2 m ~ . N e u t r a l i n o m a s s e s e x t e n d u p to m x = 1 T e V . H e a v y d o t s d e n o t e r e g i o n s w h e r e t h e i n t e g r a t e d r a t e R is in t h e r a n g e 0.1 R ~ p < R_< Re~ p, b e i n g Rex p t h e p r e s e n t e x p e r i m e n t a l b o u n d . In l i g h t - d o t t e d r e g i o n s 0.01 Rex o ~ R ~ 0.1 Rex p o c c u r s . L i n e s d e n o t e c u r v e s o f c o n s t a n t n e u t r a l i n o m a s s : d a s h e d lines s t a n d f o r

m g = 2 0 G e V , solid l i n e s f o r m x = 100 G e V , d o t - d a s h e d lines f o r m x = 3 0 0 G e V a n d d o t t e d lines f o r m ~ = 1 T e V .

d i s p l a y e d in t h e f i g u r e t o g e t h e r w i t h t h e p r e s e n t e x p e r i m e n t a l p o i n t s [14].

In fig. 6 w e give t h e e v e n t rate R at t h e s a m e v a l u e s o f tan /3 a n d rn h as a b o v e , but for t h e

s e c o n d set o f v a l u e s for t h e s f e r m i o n m a s s e s , i.e. m [ = 3 T e V . W e n o t i c e that in this c a s e t h e e x p e c t e d m a x i m a l rate is e v e n larger t h a n in t h e p r e v i o u s e x a m p l e for m x > 70 G e V . T h i s i n c r e a s e "o 10.0 5.0 1 0 0.5 ' ' I . . . . I . . . . I . . . . l . . . . _ + - ~ ~ + + +% +++ + ~ 4`-i -+ + ÷ ++ 0 t . . . . I . . . . I . . . . I . . . . I . . . . 2 4 6 B 10 E.. (KEY) Fig. 5. N u c l e a r r e c o i l s p e c t r u m d R / d Ecc f o r n e u t r a l i n o - G e i n t e r a c t i o n as a f u n c t i o n o f t h e e l e c t r o n - e q u i v a l e n t e n e r g y E t c . L i n e s d e n o t e t h e n u c l e a r r e c o i l s p e c t r a w h i c h give t h e m a x i m a l i n t e g r a t e d r a t e R f o r a fixed n e u t r a l i n o m a s s : solid line is c a l c u l a t e d f o r m ~ = 4 0 G e V , d a s h e d line f o r m x = 80 G e V a n d d o t - d a s h e d line f o r m x = 150 G e V . M 2 a n d ~. p a r a m e t e r s a r e v a r i e d in t h e r a n g e s 2 0 G e V _< M 2 _< 6 T e V a n d 20 G e V < I/zl < 3 T e V . T h e o t h e r p a r a m e t e r s a r e : t a n / 3 = 20, m h = 5 0 G e V a n d m / = 1.2 m x . C r o s s e s

A. B o t t i n o et al. / The neutralino as dark m a t t e r candidate. I1 83 ! ' ' ! . . . . I . . . . I . . . . I . . . . ! . . . . 102 101 100 1 0 - 1 1 0 - 2 1 0 - 3 :: :i::: ... • ~ , ~'...;)~::!~1 : ~ i ! ~' " : ' ~ ' .i " " • ~ : : '. ! , , i . . . . I . . . . 1 . . . . I . . . 50 100 150 200 250 300 m x (GeV)

Fig. 6. S a m e as in fig. 1, e x c e p t for m / = 3 T e V i n s t e a d o f ms= = 1.2 rex. 102 i o 1 I ° 0 1 0 _ l t0-2 1 0 - 3 50 tO0 150 200 250 300 rn x (GeV)

Fig. 8. S a m e as in fig. I; here the representative point is: tan fl = 8, m h = 50 G c V and m / = 1.2 rex.

is d u e to c o m p o s i t i o n s which are m o r e markedly o f the g a u g i n o type (rescaling effect is small here, d u e to the suppression i n d u c e d by the high values

o f r n / on the annihilation cross section). T h e

relevant explorable regions in the M 2 - # plane are shown in fig. 7.

N o w let us see how the theoretical rates vary

as we c h a n g e the values for tan /3 and for m h. In

o u r evaluations t h e H i g g s - e x c h a n g e g r a p h s have a

substantial role; then, increasing rn h has a sup-

pression effect on cross sections, and in particular on the elastic one. This is mainly due to the

d e p e n d e n c e on the Higgs masses t h r o u g h the

p r o p a g a t o r s ( ~ 1 / m 4 ) . tan /3 enters in some o f

the coupling constants; it generally occurs that cross sections d e c r e a s e as tan /3 decreases. T o illustrate the size o f these effects, we show in figs. 8 and 9 the i n t e g r a t c d cvent rates for tan /3 = 8,

m h = 50 G e V for the two sets of values for m / a n d in figs. 10 and 11 the relevant explorable regions in the M2, p. plane. In figs. 12 and 13 R is displayed at the representative point t a n / 3 = 8,

m h = 80 G e V for m / = 1.2 m x and m / = 3 T e V 103 10 2 ' . . . ' " i " : : : I" ... i : i ! !;; i!i: I i~:~i !!l:ii!ii!!: - . : 2 ! . ~ i ; d ! : i ! ! i ! _{: . : : : : : . : : : : : : :} ~ ! : i i ! :: : : : : : " l i i ~ : i i i i l H i ~ i i i ! i i . i i ~ i i : : i iii:iii: i i i i i i ~ i i i ~ i i i i i ! i ! i : I

!i!!!!!!i!iiiiiiiiiiii!:i!i:

i! ! !!! ! ~ ' ! ' ~ ' ~ ! i ! . : i / : : : : : : : : : : : / I :i!i!i'~i ~ i i i i : i i l i ~ i i ! ; . i l i : ~ iiiiiiH " ~ . : : : 1 1 i : i f :

iii:iiiii iiiiii,i:'

x~i!iiiii~:::::

iii;ili!ii!!i

i:i

i~!!ii!iiiiii~g!!!!g:i:::i:: \ iiii!!iiiiiiiiiiiiiii:'ii : \ i:ii:'"~'~'"~ i : 103 102 1 0 1 J . . . i . . . k . . . , , I 1 0 i l o o o 5 0 0 l O O 5 0 1 o t o 5 0 l O O 5 o o ~ o o o - p , ( G e V ) /z ( G e V ) t o

Fig. 7. S a m e as in fig. 4, e x c e p t for m / = 3 T e V i n s t e a d of m ~ = 1.2 m x. D a s h e d line 20 G e V ; solid line 100 G e V ; d o t - d a s h e d line 300 G e V ; d o t t e d line 1 T e V .

84 A. Bottino et al. / T h e neutralino as d a r k m a t t e r candidate, l i 1 ° 2 ~ ' ( . . . . I . . . . I . . . . ! . . . . I . . . . i e,,, ~ : ~ i ; I i i : ,, I , , , ;i,}, i,i ,!,

,

:, ,,

,:

50 100 150 200 250 300 rn~ (GeV)

Fig. 9. Same as in fig. 1, here the representative point is: t a n / 3 = 8, m h = 50 G e V and m [ = 3 TeV. l01 100 1 0 - i 1 0 - 2 1 0 - 3

respectively. As c o m p a r e d to the previous case tan /3 = 8, m h = 5 0 G e V , the present event rate R is sizably lower, mainly because of the propaga- tor effect ~ 1linCh . T h e regions in the M 2 , /a,

plane which are experimentally explorable, by increasing the sensitivity appropriately, are shown in figs. 14 and 15.

We do not report here the results for other representative points. We only quote that for instance at t a n / 3 = 2 , as is expected, the ex- plorahle regions are smaller than the previous

ones. However, a 2-order-of-magnitude improve- m e n t in sensitivity would still m a k e feasible an exploration in the range 20 G e V _< m x _< 100 GeV. Then, as far as the G e detectors are con- cerned, we can conclude that improvements of about 2 orders of magnitude in experimental sen- sitivities would allow investigation of the neu- tralino dark m a t t e r in large regions of the model p a r a m e t e r space. It is extremely encouraging that this level of sensitivities a p p e a r s to be within reach in the near future [20].

Let us turn now to a brief discussion of the other materials mentioned before. T h e theoreti- cal analysis has been p e r f o r m e d along the same lines as in the case of Ge. For sake of brevity, here we only report the differential and the inte- grated rates for the two representative points: t a n / 3 = 20, m h = 50 GeV, m / = 1.2 mx and tan/3

= 2 0 , m h = 5 0 GeV, m [ = 3 TeV.

6.2. S o d i u m i o d i d e

Theoretical expectations for this diatomic ma- terial were already presented and c o m p a r e d with experimental values in ref. [2] for neutralino masses in the range 20 G e V < m x < 8 0 GeV.

;> v 10 3 10 2 ' . . . i f L ii::iiii::} ~ , iiiii!i ! il

i' i :ii: i

~ i ~ i I { i F ! ! i i i i I i i!~i!!!!! t ~:iii~!i!!i I ~:i: <'!iiiiiiii i! i Z_ _ ~!iJiiii iii i i ! i i !

iiii!F iiii'! iii'i

'i!;!iiii i{ i

\ ~!g~!ii!~ii:i;ii!:i: g!iH!!!iiii~! ~: , , ~ i ! i : : 1 0 1 ~ . . . I . . . - . . . . ' . . . t t O 0 0 5 0 0 t O 0 5 0 1 0 t O 5 0 1 0 0 5 0 0 t O 0 0 -~z (GeV) ~ (GeV) 1 0 3 102 1 0 1 ¢o t ~

Fig. 10. Same as in fig. 4, here the representative point is: t a n / 3 = 8, m h = 50 G e V and m / = 1.2 m x. D a s h e d line 20 GeV; solid line 100 GeV; dot-dashed line 300 GeV; dotted line 1 TeV.

A. B o t t i n o et al. / T h e n e u t r a / m o as d a r k m a t t e r c a n d i d a t e . I1 85 :> v 103 10 2 k i i ) i L o • ' . . . . . . i . . . i L i iii:ilill i

, ii {!ii; i,lit;: i;i

\ ? ; : i i ! i i i i ! i i : ; iT \ ::lTi!~i::ii: T!7 10 3 101 I . . . i , , - . . . i . . . , 101 1 0 0 0 5 0 0 tO0 50 t O tO 50 I 0 0 500 I 0 0 0 - p , ( C e V ) /z ( G e V ) ix) 10 2

Fig. 11. S a m e as in fig. 4, here the representative point is: t a n / 3 = 8, m h = 50 G e V and my: = 3 TeV. D a s h e d line 20 G e V ; solid line If)() G e V ; d o t - d a s h e d line 300 G e V ; dotted line 1 TeV.

H e r e the analysis is e x t e n d e d up to m x = 1 T e V and to m u c h w i d e r regions o f the p a r a m e t e r space. T h e results o f the integrated rate R(4.5 K e V < E ~ < 5 K e V ) are s h o w n in figs. 16 and 17. For the c a s e o f tan /3 = 20, rn h = 50 G e V , my: = 1.2 rn x w e also give in fig. 18 the m a x i m a l differential rate at three values o f rex. W e notice that the peculiar b e h a v i o u r o f d R / d E ~ e as a function o f m x is due to the d i a t o m i c nature o f NaI. T h e r e s p o n s e o f a N a I d e t e c t o r to the neu-

tralino scattering is d o m i n a t e d by iodine up to E~e ~ 6 k e V and by S o d i u m at higher Eel. This c i r c u m s t a n c e d e p e n d s on the very different co- h e r e n t f o r m factor effects in the two nuclei as is s h o w n by the values r e p o r t e d in table 1. F r o m the threshold up to Eee ~ 6 K e V in iodine the large s u p p r e s s i o n due to F Z ( E ~ ) is c o m p e n s a t e d by a large value o f its c o h e r e n t pointlike cross section ( A ~ effect). A t higher values o f E ~ the iodine c o h e r e n t form factor has a fast fall-off, b e c a u s e

10 2 1 0 l "o i 0 0 t o - 1 10 - 2 r,- 1 0 - 3 ' ' 1 . . . . 1 . . . . I . . . . I . . . . I . . . .

[

~ ~ i

•

I: ~

!

: " : "'" ;

1

[

' . - ° q ' i l l . o - . ° l o

~Jitiii i i::i~, :i, ,~l:V,;i~, It,, i ,i[~ ,, ~"

5 0 100 150 2 0 0 2 5 0 3 0 0

m ~ (CeV)

Fig. 12. S a m e as in fig. 1, here the representative point is: t a n / 3 = 8, m h = 8(} G e V and m i = 1.2 m r. a:: 102 101 1 0 0 I 0 - I 1 0 - 2 ) ' ' [ . . . . I . . . . I . . . . [ . . . . I . . . . ~ k ~ l l ~ O 10-3 ,, i i i, i, il/,i, 50 t O 0 1 5 0 2 0 0 2 5 0 3 0 0 rax (GeV)

Fig. 13. S a m e as in fig. 1, here the representative point is: t a n / 3 = 8, m h = 80 G e V and m ~ = 3 TeV.

86 A. Bottino et al. / The neutralino as d a r k m a t t e r candidate. I1 o 103 10 ~ i . . . ! I . . .

'ii iii! ,

i!i~ iiii! iliili ~ ~iii~ i ! ! / / I ... I i I I i \i: :i!::i! : '\:?!. 10 3 102 1 0 1 I . l . . . , . . . . . . . . . . . . . . . . J . . . 1 1 0 1

1000 500 I00 50 tO tO 50 tO0 500 tO00

-/~ (GeV) /~ (GeV)

O

t ~

.<

Fig. 14. Same as in fig. 4, here the representative point is: tan/3 = 8, m h = 80 GeV and m / = 1.2 m x . Dashed line 20 GeV; solid line 100 GeV; dot-dashed line 300 GeV; dotted line 1 TeV.

of the zero at E e e = 7 . 4 keV, whereas in the whole range of Eee shown in fig. 18 the sodium form factor has a rather flat behaviour.

As discussed in ref. [2] an improvement of two orders of magnitude in the NaI detectors sensitiv- ity is foreseeable in the next few years. This

v 103 102 I . . . [ I I I I .J ~: I :iil : : : : . : , . : : : : i : i i i ! ! ' : iJ • i ~ . ? ~ i ~ . . . I . . . I l I i i : i : : : ; : 10 3 102 1 0 1 I . . . - - l . . . , . . . I 1 0 1 t O 0 0 5 0 0 1 0 0 50 10 10 50 tO0 500 t O 0 0 -/~ (GeV) /~ (GeV) t ~ .< v

Fig. 15. Same as in fig. 4, here the representative point is: tan/3 = 8, m^ = 80 GeV and m / = 3 TeV. Dashed line 20 GeV; solid line 100 GeV; dot-dashed line 300 GeV; dotted line l TeV.

A. Bottino et al. / The neutralino as dark matter candidate. 11 87 io o 1 0 - 1 1 0 _ 2 1 0 _ 3 a ) r w 1 0 - 4 50 100 150 200 250 300 m x (GeV)

Fig. 16. S c a t t e r plot of t h e i n t e g r a t e d r a t e R(4.5 k e V _< E¢~ _< 5 k e V ) for n e u t r a l i n o - N a l i n t e r a c t i o n . T h e r e p r e s e n t a t i v e p o i n t is: t a n / 3 = 20, m h = 50 G e V a n d m~ = 1.2 m ~ ; t h e p a r a m e t e r s M z a n d ~ a r e v a r i e d in t h e r a n g e s 20 G e V _< M z _< 6 T e V a n d 20 G e V _< ]/z ] _< 3 TeV. F i l l e d - c i r c l e s r e p r e s e n t the R v a l u e s for n e u t r a l i n o c o m l ~ ) s i t i o n s w h e r e g a u g i n o a n d h i g g s i n o com- p o n e n t s a r e m a x i m a l l y m i x e d (0.45 _< a~ + a~ _< 0.55).

would allow to start the exploration of the physics o f dark m a t t e r neutra]inos with these detectors.

6.3. X e n o n

M u c h experimental activity is now being devel- o p e d in dark m a t t e r d e t e c t o r s which e m p l o y Xe.

100 ! ' ' l . . . . i . . . . i . . . . i . . . . ~ . . . . 10 - 2 • . g : . : : : : : : 1 0 _ 3 ! ' _. ! i i i i i i i l ! - 1 0 - 4 , I . . . . I , , , I . . . . I . . . . I , , 50 100 150 200 250 300 m x (GeV)

Fig. 17. S a m e as in fig. 16, e x c e p t for m f = 3 T e V i n s t e a d of m / = 1.2 m~.

W e r e p o r t in figs. 19 and 20 the integrated rate R(20 k e V < E¢c _< 22 keV) in the representative

points: tan /3 = 20, m h = 50 GeV, m r = 1.2 m x

and tan /3 = 20, m h = 50 G e V , m r = 3 TeV. For

the first o f these points also the differential rate is given (fig. 21). W e notice that for Xe the suppression due to the c o h e r e n t form factor is sizeable (see table 1). H o w e v e r the use o f X e for detecting dark m a t t e r neutralinos a p p e a r to be

10 3 102 1o 1 1o 0 l o _ l 1 0 - 2 r ~

~

..' ' ' ' I . . . .

I +' ' ' ' I . . . .

+ + + + + + + + + + 1

I . . . .

lO-3 . . . . I . . . . I . . . . I . . . . i . . . . 0 2 4 6 8 tO E~ (KEY)

Fig. 18. N u c l e a r recoil s p e c t r u m d R / d E e e for n e u t r a l i n o - N a ] i n t e r a c t i o n as a f u n c t i o n of t h e e l e c t r o n - e q u i v a l e n t e n e r g y Eee. L i n e s d e n o t e t h e n u c l e a r recoil s p e c t r a w h i c h give t h e m a x i m a l i n t e g r a t e d r a t e for a fixed n e u t r a l i n o mass: solid line is c a l c u l a t e d for m x = 4(} G e V , d a s h e d line for m X = 80 G e V a n d d o t - d a s h e d line for m x = 150 G e V . M e a n d tz p a r a m e t e r s are v a r i e d in the r a n g e s 20 G e V < M 2 < 6 T e V a n d 2(} G e V :~ ]~-I -< 3 TeV. T h e o t h e r p a r a m e t e r s are: t a n / 3 = 20, m h = 50 G e V a n d m f = 1.2 m x .

88 A. Bottino et al. / The neutralino as dark matter candidate. I1

I01 ~ :

I00

g !?!i!ii i!iiii?ii '!i, ii ;i

1o-Z L ~ - ~ ' . ~ : ] : ~ i . : ; i ! : : : ~ ! ! ! ! ~: "I : "1 • ',. :: i l ! ~ : [

10-4

50 I 0 0 150 200 250 300

m x (CeV)

Fig. 19. Scatter p l o t o f t h e integrated rate R(20 keV < E ~ < 22 keV) for neutralino-Xe interaction. The values f o r t h e model

p a r a m e t e r s a n d t h e n o t a t i o n s are as in fig. 16.

very promising in view o f the experimental fea- tures o f these detectors.

6 . 4 .

Calcium fluoride

Finally in figs. 22 and 23 w e display the inte- grated rate R (8.5 k e V < E~e _< 9 k e V ) for CaF2 in the usual representative points: tan /3 = 20, m h = 50 G e V , m / = 1.2 m x and tan /3 = 20, m h = 50 G e V , m / = 3 T e V . In fig. 24 also the differ-

"o v e, 0) v t~ 101 100 i 0 - 1 10-2 10-3 " l . . . . D . . . . , . . . . I . . . . i . . . . ! t 0 - 4 , I , , i . . . . i . . . . i , , i . . . . 50 tO0 150 200 250 300 m x (GeV)

Fig. 20. Same as in fig. 19, except for m { = 3 TeV instead of m / = 1.2 rex.

ential rate for the first representative point is s h o w n .

7. C o n c l u s i o n s

W e can c o n c l u d e that the use o f different materials and the a c h i e v e m e n t s o f improved ex- p e r i m e n t a l p e r f o r m a n c e s are offering very excit- ing perspectives for a direct search o f neutralino as a dark matter c o m p o n e n t . 1 o ~ | . . . . ) . . . . i . . . . ) . . . . )

....

I01 ~

1oO

--T-~.

i0_1 'o ~. 10 - 2 i0_3 t: 1 0 - 4 I 0 - 5 - , , , , I . . . . I . . . . I , , , 1 ° - 8

\ I , ~ , ,

0 2 0 4 0 6 0 8 0 100 E . (KEY)

Fig. 21. Nuclear recoil spectrum dR/dKee for n e u t r a l i n o - X e interaction as a function of the electron-equivalent energy Eee. The values for the model parameters and the notations are as in fig. 18. Here no experimental data are available.

A. Bottino et al. / The neutralino as dark matter candidate. II 8 9 ~°° ~ ' ' 1 .... I .... I .... I .... ) . . . . 1 0 - 1 ,., 1 0 _ 2 . ~ , , : ~ : ~ ' ~ I~,iIQ 41 ~1 " I !B~ I • 410000 . ~ ! :~ ; i : ~ ' ~ i i : : : : : ! : :i : ~ ! . i . 1 o - 4 ~ T , ,'~, ) . . . . i . . . ; . . . . .50 100 150 2 0 0 2.50 3 0 0 r n x (GeV)

Fig. 22. Scatter plot of the integrated rate R ( 8 . 5 k e V _< E ~ _< 9

keV) for neutralino-CaF 2 interaction. The values for the model parameters and the notations are as in fig. 16.

100 .. . . i ' " m . . . . i . . . . m . . . . tO - 1 ~'-- b~ 10 - 2 ~'~ : ~ , ~ ~. ,~ ~ : ~ •

i

io-~

. . .

i

. . . . . .

.

i

,50 100 1.50 2 0 0 2 5 0 3 0 0 m x (GeV)

Fig. 23. Same as in fig. 22, except for m / = 3 T e V instead of

m / = 1.2 m x .

The theoretical evaluations presented here aim at setting the level o f the experimental sensitivi- ties required to start the exploration o f the physics o f dark matter neutralinos. The quantitative rela- tions between the experimental sensitivity of a particular detector and its discovery potential are easily readable in our plots.

S o m e final c o m m e n t s on the main theoretical uncertainties are in order here:

(i) major improvements in theoretical predic- tive power require some experimental informa-

tions about the masses of particles to be discov- ered at accelerators: Higgs bosons, SuSy parti- cles;

(ii) the evaluation of the elastic coherent cross section suffers from an uncertainty in the esti- mate o f the H i g g s - n u c l e o n strength. In ref. [3] we have analyzed the relevance of this effect for the direct neutralino search, by comparing different estimates for the H i g g s - n u c l e o n strength [21,22]. A new recent study of this fundamental coupling has shown that the problem needs further investi- gation [23]. In the present paper we have em-

102 I01 ~" io 0 ~ 10 - 1 ~ ~o -2 "~ 1 0 - 3 1 0 - 4 I . . . . I . . . . I . . . . I . . . . I , ,"~ , 5 10 E,. (KeV) , t 15 2 0

Fig. 24. Nuclear recoil spectrum d R / d E~¢ for neutralino-CaF 2 interaction as a function of the electron-equivalent energy E ~ .

90 A. Bottino et al. / The neutralino as dark matter candidate. H

ployed for the Higgs-nucleon coupling the esti- mate of ref. [21]. Use of the estimate in ref. [22] would reduce our rates by a factor of 3 approxi- mately;

(iii) the value of ($'~h2)min t o be employed for rescaling the local neutralino density is obviously somewhat arbitrary. In the present paper we have conservatively taken (-Oha)min = 0.05. In view of the small value of ~ which appears to be implied by the estimated amount of dark matter in our galaxy, our values f o r (.(-2h2)min could quite well be reduced by a factor of 2. This reduction in (.Oh2)min would increase the maximal rate R by a factor of 2, since, as we have seen above, maximal signals occur in general for neutralino composi- tions where the rescaling effect is large.

Acknowledgement

This work was supported in part by Research Funds of the Ministero deli'Universitb, e della Ricerca Scientifica e Technologica.

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