Neutralino Dark Matter in Supersymmetric Models with Non--Universal Scalar Mass Terms

Home , Neutralino

CERN{TH 95{206

DFTT 47/95

JHU{TIPAC 95020

LNGS{95/51

GEF{Th{7/95

August 1995

Neutralino dark matter in sup ersymmetric mo dels

with non{universal scalar mass terms.

a b,c d e,c b,c

V. Berezinsky , A. Bottino , J. Ellis ,N.Fornengo , G. Mignola

f,g

and S. Scop el

INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi (AQ), Italy

Dipartimento di FisicaTeorica, UniversitadiTorino, Via P. Giuria 1, 10125 Torino, Italy

INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy

Theoretical Physics Division, CERN, CH{1211 Geneva 23, Switzerland

Department of Physics and Astronomy, The Johns Hopkins University,

Baltimore, Maryland 21218, USA.

Dipartimento di Fisica, Universita di Genova, Via Dodecaneso 33, 16146 Genova, Italy

INFN, Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy

Abstract

Neutralino dark matter is studied in the context of a sup ergravityscheme

where the scalar mass terms are not constrained by universality conditions

at the grand uni cation scale. We analyse in detail the consequences of the

relaxation of this universality assumption on the sup ersymmetric parameter

space, on the neutralino relic abundance and on the event rate for the direct

detection of relic neutralinos.

E{mail: b [email protected], b [email protected], [email protected],

[email protected], [email protected], scop [email protected] 1

I. INTRODUCTION

The phenomenology of neutralino dark matter has b een studied extensively in the Mini-

mal Sup ersymmetric extension of the Standard Mo del (MSSM) [1]. This mo del incorp orates

the same gauge group as the Standard Mo del and the sup ersymmetric extension of its parti-

cle content. The Higgs sector is slightly mo di ed as compared to that of the Standard Mo del:

the MSSM requires two Higgs doublets H and H in order to give mass b oth to down{ and

1 2

up{typ e quarks and to cancel anomalies. After Electro{Weak Symmetry Breaking (EWSB),

the physical Higgs elds consist of twocharged particles and three neutral ones: two scalar

elds (h and H ) and one pseudoscalar (A). The Higgs sector is sp eci ed at the tree level by

two indep endent parameters: the mass of one of the physical Higgs elds and the ratio of

the twovacuum exp ectation values, usually de ned as tan = v =v =.

2 1 2 1

The sup ersymmetric sector of the mo del intro duces some other free parameters: the mass

parameters M , M and M for the sup ersymmetric partners of gauge elds (gauginos),

1 2 3

the Higgs{mixing parameter and, in general, all the masses of the scalar partners of the

fermions (sfermions).

In the MSSM it is generally assumed that the gaugino masses are equal at the grand

uni cation scale M : M (M ) m and hence are related at lower scales by

GU T i GU T 1=2

M : M : M = : : (1)

1 2 3 1 2 3

where the (i=1,2,3) are the coupling constants of the three Standard Mo del gauge groups.

The neutralinos are mass{eigenstate linear sup erp ositions of the two neutral gauginos ( ~ and

~ ~ ~

Z ) and the two neutral higgsinos (H and H )

1 2

~ ~ ~

= a ~ + a Z + a H + a H : (2)

1 2 3 1 4 2

The neutralino sector dep ends, at the tree{level, on the following (low{energy) parameters:

M =(5=3) tan M , M ' 0:8 m , and tan . Neutralino prop erties are naturally

1 W 2 2 1=2

discussed in the (m , ) plane, for a xed value of tan . As an example, in Fig.1 the lines

1=2

of constant mass for the lightest neutralino (m ) and constant gaugino fractional weight

2 2

(P a + a ) are plotted in the (m , ) plane for tan =8.We observe that the mass of

1=2

1 2

the lightest neutralino increases from the b ottom left to the top right, while the neutralino

comp osition changes from higgsino dominance in the top{left region of the plane to gaugino

dominance in the b ottom{right. The regions forbidden by accelerator data are also displayed

in Fig.1.

The low{energy MSSM scheme is a purely phenomenological approach, whose basic idea

is to imp ose as few mo del{dep endent restrictions as p ossible. In this approach the lightest

neutralino is a favourite candidate for cold dark matter. This scheme has b een employed ex-

tensively in the analysis of the size and the relevance of various p ossible signals of neutralino

dark matter: direct detection [2{4], signals due to neutralino annihilation in celestial b o dies, 2

namely the Earth and the Sun [5,6], and signals from neutralino annihilation in the galactic

halo [7]. The MSSM provides a useful framework in which neutralino phenomenology may

b e analysed without strong theoretical prejudices which could, aposteriori, turn out to b e

incorrect. This scheme is also frequently employed in analyses of the discovery p otential of

future accelerators [8].

At a more fundamental level, it is natural to implement this phenomenological scheme

within the sup ergravity framework [9{ 11]. One attractive feature of the ensuing mo del is the

connection b etween soft sup ersymmetry breaking and EWSB, whichwould then b e induced

radiatively. The essential elements of the mo del are describ ed byaYang{Mills Lagrangian,

the sup erp otential, which contains all the Yukawainteractions b etween the standard and

sup ersymmetric elds, and by the soft{breaking Lagrangian, which mo dels the breaking of

sup ersymmetry. Here we only recall the soft sup ersymmetry breaking terms

2 2

L = m j j

sof t i

nh i o

l l d d u u

~ ~ ~ ~ ~ ~

+ A h L H R + A h Q H D + A h Q H U + h.c. BH H + h.c.

a 1 b a 1 b a 2 b 1 2

ab ab ab ab ab ab

M ( + ) (3) +

i i i i i

where the are the scalar elds, the are the gaugino elds, H and H are the two Higgs

i i 1 2

~ ~ ~ ~ ~

elds, Q and L are the doublet squark and slepton elds, resp ectively, and U , D and R

denote the SU (2){singlet elds for the up{squarks, down{squarks and sleptons. In Eq.(3),

m and M are the mass parameters of the scalar and gaugino elds, resp ectively, and A

i i

and B denote trilinear and bilinear sup ersymmetry breaking parameters, resp ectively. The

Yukawainteractions are describ ed by the parameters h, which are related to the masses of

the standard fermions by the usual expressions, e.g., m = h v .

t 2

The sup ergravity framework is usually implemented with a numb er of restrictive assump-

tions ab out uni cation at M :

GU T

i) Uni cation of the gaugino masses: M (M ) m ,

i GU T 1=2

ii) Universality of the scalar masses with a common mass denoted by m : m (M )

0 i GU T

m ,

l d u

iii) Universality of the trilinear scalar couplings: A (M )=A (M )=A (M )

GU T GU T GU T

A m .

0 0

These conditions have strong consequences for low{energy sup ersymmetry phenomenology,

and in particular for the prop erties of the neutralino as dark matter particle. Typically, the

lightest neutralino is constrained to regions of gaugino dominance, that entail a large relic

abundance (in wide regions of the parameter space h exceeds the cosmological upp er

b ound) and a small direct detection rate for neutralino dark matter. Indirect signals from 3

the neutralino, such as high{energy neutrinos from the Earth and Sun, and the pro ducts of

annihilatio n in the halo, are practically undetectable [11].

The ab ove assumptions, particularly ii) and iii), are not very solid, since universality

may o ccur at a scale higher than M , i.e., the Planck scale or string scale [12], in which

GU T

case renormalization ab ove M mayweak universality in the m , e.g.,between scalars in

GU T i

5 and 10 representations of SU (5) [13]. Moreover, in many string mo dels the m 's are not

universal even at the string scale.

In a numb er of recentworks [14,15], deviations from some of the uni cation conditions

have b een considered. In particular, in Ref. [14] phenomenological consequences for neu-

tralinos of a relaxation of assumption ii) have b een analysed in the regime of large values

of tan . It has b een shown that deviations from condition ii) mayentail a changeover in

neutralino comp osition from a gaugino{like state to a higgsino{li ke state (or at least to a

higgsino{ga ugi no mixed state), with imp ortant consequences for neutralino phenomenology.

In this pap er, we rst explore, over the full range of tan , the various scenarios which

may o ccur when condition ii) is relaxed, with an approach which is similar to the one adopted

in the large{tan analysis of Ref. [14]. We then discuss in detail the ensuing consequences

for neutralino dark matter, with particular emphasis for its direct detection.

In the following, we rst discuss which constraints can b e applied to the parameters

when sp eci c physical requirements are imp osed. In Sect.I I, we summarize the conditions

implied by radiative EWSB and de ne the typ e of departure from universality examined in

this pap er. Then, in Sect.I I I we establish some upp er b ounds on the sup ergravity param-

eters by requiring that radiative EWSB do es not o ccur with excessive ne tuning among

di erent terms. In Sect.IV we analyse in detail the constraints due to the requirement that

EWSB takes place radiatively. Subsequently, in Sect.V cosmological constraints, derived

from the evaluation of the neutralino relic abundance, are discussed. Other constraints,

from exp erimental data on b ! s pro cesses and on the mass of the b ottom quark m , are

applied in Sect.VI. In Sect.VI I the e ects of these various constraints are rst displayed in

the (m , m ) plane for xed tan and A , and then shown in the (m , ) plane, which

0 0

1=2 1=2

provides the most useful representation for discussing neutralino phenomenology.We recall

some sp eci c prop erties of the neutral Higgs b osons in Sect.VI I I. Finally, in Sect.IX event

rates for direct detection of neutralino dark matter are discussed. Conclusions are presented

in the last Section.

I I. RADIATIVE EWSB

We recall that the tree{level scalar p otential for the neutral Higgs elds may b e written

in the form

2 2 2 2 2 2

V =(M + )jH j +(M + )jH j B(H H + h.c.) + quartic D terms. (4)

0 1 2 1 2

H H

1 2 4

The parameters of this p otential must ob ey the following physical conditions:

sin 2 = (5)

2 2

M + M +2

H H

1 2

2 2 2

M M tan

H H

2 2

1 2

M =2 2 (6)

tan 1

2 2 2 2

M = M + M +2 > 0 : (7)

A H H

1 2

Here M is the mass of the CP{o dd neutral Higgs b oson (see Sect.VI I I b elow), and eq.(7)

must in fact b e strengthened to M (M ) , where (M ) is the exp erimental lower

A A lb A lb

b ound [16]. For instance, for tan 3, (M ) ' 45 GeV. Notice that the sign of is

A lb

de ned according to the convention of reference [1]. We remark that although Eqs.(4{7) are

expressed at the tree level, in our actual calculations 1{lo op corrections to V [17] have b een

included. The M 's (as well as the sfermion and the gaugino masses and the parameters A,

B and )evolve from the M scale down to the M scale according to the Renormalization

GU T Z

Group Equations (RGE's). This is how Eq.(6) may b e satis ed, even if M and M are

H H

1 2

equal at M .

GU T

In this work we consider deviations from universality in the scalar masses at M ,

GU T

which split M from M . This e ect is parameterized as

H H

1 2

2 2

M (M )=m (1 + ) : (8)

GU T i

H 0

The parameters which quantify the departure from universality for the M will b e varied

in the range (1,+1), but are taken to b e indep endent of the sup ersymmetry parameters.

This is an Ansatz, since, when evolving the scalar masses from the uni cation scale (Planck

scale or string scale) to the GUT scale M , the deviation parameters are in general

GU T

functions of all the sup ersymmetry parameters [18].

Following a common pro cedure, Eq.(5) is used to replace the parameter B by tan .Thus

the set of indep endent parameters b ecomes m , m , A , tan , and is given in terms of

0 0

1=2

these parameters by Eq.(6), suitably corrected by 1{lo op e ects: only the sign of remains

undetermined. Obviously,values of are accepted only if they exceed the exp erimental

lower b ound , which is derived from the lower limit on the chargino mass [16]: j j' 45

GeV.

Wehave solved the RGE's using the 1{lo op b eta functions including the whole sup er-

symmetric particle sp ectrum from the GUT scale down to M , neglecting the p ossible e ects

of intermediate thresholds. Two{lo op and threshold e ects on the running of the gauge and

Yukawa couplings are known not to exceed 10% of the nal result [19]. While this is of

crucial imp ortance as far as gauge coupling uni cation is concerned [19], it is a second{order 5

e ect on the evolution of the soft masses. Since neutralino prop erties are studied over a wide

range of variation for the high scale parameters, such a degree of re nement is not required

here.

In order to sp ecify the sup ersymmetry phenomenology, b oundary conditions for the gauge

and Yukawa couplings have to b e sp eci ed. Low{scale values for the gauge couplings and

for the top{quark and the tau{lepton Yukawa couplings are xed using present exp erimental

results. In particular, we assign for the top mass the value m = 178 GeV [20]. In addition,

we require the uni cation of the b ottom and tau Yukawa couplings at the GUT scale, as

would b e suggested by a unifying group that includes an SU (5){like structure [21].

The values of M and M at the M scale, obtained from the RGE's, may b e param-

H H Z

1 2

eterized in the following way:

2 2 2 2 2

M = a m + b m + c A m + d A m m : (9)

i i i i 0 0

1=2

H 1=2 0 0 0

(Notice that, in our notation, all running quantities written without any further sp eci cation

are meant to denote their values at M .) The co ecients in the expression (9) are functions

of tan and of the 's. They are displayed in Fig.2 (a,b) for the case of universal scalar

masses, (i.e., = 0). The co ecients for M turn out to b e very stable as functions of

tan , except for small tan . More precisely, a 2:5 for tan 4 with all the other

co ecients much smaller (of order 0.1). As far as M is concerned, whereas c and d are

1 1

again very stable (of order 0.1), a and b vary rapidly as functions of tan . This prop erty

1 1

of a and b is due to the very fast increase of h for increasing tan .

1 1

When a departure from m universalityisintro duced, the co ecients in Eq.(9), except

for a and a , b ecome functions of the parameters : b , c and d dep end on and b , c

1 2 i 1 1 1 1 2 2

and d on . Whereas the b 's are rapidly{increasing functions of the 's, the c 's and the

2 2 i i i

d 's are rather insensitive to these parameters.

Stringent constraints on the parameters m ;m ;A and tan follow from the request

0 0

1=2

2 2

that the M 's, evaluated from Eq.(9), satisfy Eqs.(6{7). Explicitly,we require that and

M , given by the expressions

2 2 2 2 2

= f(a a tan )m +(b b tan )m +

1 2 1 2

1=2 0

tan 1

2 2 2 2

(c c tan )A m +(d d tan )A m m g

1 2 1 2 0 0

1=2

0 0

2 2 2 2 Z

J m +J m +J A m +J A m m (10)

1 2 3 4 0 0

1=2

1=2 0 0 0

2 2 2

M =(a +a +2J )m +(b +b +2J )m +

1 2 1 1 2 2

A 1=2 0

2 2 2

+(d +d +2J )A m m M m (c +c +2J )A

1 2 4 0 0 1 2 3

1=2

Z 0 0

2 2 2 2 2

K m +K m +K A m +K A m m M (11)

1 2 3 4 0 0

1=2

1=2 0 0 0 Z 6

2 2

satisfy the conditions: , M (M ) mentioned earlier.

A A lb

The co ecients J and K in Eqs.(10,11) are plotted as functions of tan in Fig.2 (c,d)

i i

for the case of m universality. In Fig.2c we notice that all the J 's are p ositive, with J

0 i 1

dominating over the others: for tan 4, one has J ' 2:4. As far as the co ecients K are

1 i

concerned, we see in Fig.2d that only two of them are sizeable: K and K . They are b oth

1 2

decreasing functions of tan , with K >K .Atvery large tan these co ecients b ecome

1 2

very small, and K even b ecomes negative (but still small in magnitude) at tan 50.

In the case of non{universality, the co ecients J and K , except for J and K , b ecome

i i 1 1

functions of the parameters .We will see in Sect.IV that many imp ortant features of the

sup ersymmetry parameter space dep end on the signs of the two co ecients J and K .We

2 2

show in Figs.3 and 4 how their signs dep end on the values of the 's. In Fig.3 the lines

J = 0 are plotted in the ( , ) plane for a few values of tan : for eachvalue of tan ,

2 2 1

J is negative in the region ab ove the relevant J = 0 line and p ositive b elow. Similarly,in

2 2

Fig.4 the K = 0 lines are displayed in the same ( , ) plane at xed tan : K is negative

2 2 1 2

ab ove the K = 0 lines, and p ositive b elow.

Wenow make a few comments related to Eq.(11), since the value of M playsavery

crucial role in a numb er of imp ortant neutralino prop erties. This is due to the fact that

manyphysical pro cesses involving neutralinos are mediated by the neutral Higgs b osons.

Thus the value of M determines the size of the relevant cross sections b oth through M {

dep endence in propagators and, in an implicit way, through the couplings of the h and H

b osons to quarks and to the lightest neutralino (see Sect.VI I I). As a consequence, a small

value of M has the e ect of enhancing the magnitude of the relevant cross sections.

What values of M do we obtain from Eq.(11)? Because of the prop erties of the co e-

cients K previously analysed, M turns out to b e a rapidly{decreasing function of tan .In

i A

Fig.5, M is displayed at the representative p oint m = 50 GeV, m = 200 GeV (1{lo op

A 0 1=2

corrections to M have b een included in the calculation). One notices that M is O (M )

A A Z

for tan 45. This feature provides one of the most app ealing scenarios for neutralino

phenomenology.

I I I. CONSTRAINTS DUE TO THE ABSENCE OF FINE TUNING

2 2

Before we exploit fully the two constraints , M (M ) to restrict the pa-

A A lb

rameter space, we apply the general criterion that the expression (10) is satis ed without

excessive tuning among the various terms [22,10]. In radiative EWSB the physical value of

M , which sets the EW scale, may b e written as

2 2 2 2 2 2

M =2(J m +J m +J A m +J A m m ) : (12)

1 2 3 4 0 0

1=2

Z 1=2 0 0 0 7

Accidental comp ensation ( ne tuning) among di erent terms in Eq.(12) may o ccur. We

explicitly require the absence of to o{strong ne tuning, i.e., cancellations among exceedingly

large values of the parameters m , m , A and . Denoting by a parameter which

0 0 f

1=2

quanti es the degree of ne tuning, we require [22] that

2 2

x M

i Z

< (13)

2 2

M x

where x denotes any of the previous parameters. For instance, for A = 0, Eq.(13) provides

i 0

the following conditions

f f f

2 2 2 2 2 2 2

m < M ; m < M ; < M ' (640 GeV) (14)

1=2 Z 0 Z Z

2jJ j 2jJ j 2

1 2

where in the last approximate equalitywehave taken = 100, which means that we allow

an accidental comp ensation at the 1% level. The upp er b ound on m dep ends on tan and

the 's, whereas that on m varies only with tan (b ecause of the nature of the Ansatz

1=2

(8): see the comment after Eq.(8)).

For the sake of illustration , we give some numerical examples, taking again = 100.

< <

For tan =8,wehave, for = =0,m 400 GeV, m 1:5TeV. For two other pairs

1 2 0

1=2

of values of the 's, which will b e discussed later on, we obtain m 2:4TeV for = 0:2,

i 0 1

< <

=0:4 and m 3:0TeV for = 0:8, =0:2. At tan =53wehavem 415 GeV

2 0 1 2

1=2

and m (1:7 1:9) TeV, dep ending on the values for the 's. These inequalities imply for

0 i

the neutralino mass m 170 GeV.

In the following, when graphical representations for the parameter space are shown,

we display no{ ne{tuning upp er b ounds obtained from the general expression (13) with

= 100. These upp er b ounds are denoted by dashed lines in Figs.9{14.

IV. CONSTRAINTS DUE TO RADIATIVE EWSB

The EWSB constraints are given by the set of Eqs.(5{7), or equivalently by Eqs.(10{11),

2 2

together with the conditions and M (M ) .From these equations the values

A A lb

of m and m (or and m ) are constrained and thus some domains in the (m , m )

0 0

1=2 1=2 1=2

or (m , ) planes can b e excluded. Let us start this discussion by analyzing the condition

1=2

M (M ) , with M given by Eq.(11). For the sake of simplicity,we put A = 0 for the

A A lb A 0

moment. To discuss the role of M (M ) in placing b ounds on m and m ,we rst

A A lb 0

1=2

rewrite it explicitly as

2 2 2 2

K m + K m M +(M ) : (15)

1 2 A

1=2 0 Z lb 8

The nature of this quadratic form in the (m , m ) plane obviously dep ends on the signs

1=2 0

of the two co ecients K and K .Aswehave seen in Sect.I I, it turns out that, whereas

1 2

K is always p ositive, the sign of K dep ends on the values of tan and of the 's. Two

1 2 i

di erent situations may o ccur, dep ending on the sign of K . In the case K > 0 the region

2 2

allowed by (15) is the one ab ove an elliptical branch centered in the origin of the (m , m )

1=2

plane. Therefore, b oth parameters m and m are b ounded from b elow. When K < 0,

0 2

1=2

the region allowed by Eq.(15) is the one b etween the m axis and an upward{moving

1=2

hyp erb olic branch. Thus, whereas m is still b ounded from b elow, m is now constrained

1=2

from ab ove. The upp er b ound on m is particularly stringent when K is large and negative

0 2

and K is not large. This o ccurs, for instance, at very large values of tan in the case of

m universality.

This discussion may b e extended straightforwardly to the case A 6= 0. In this case the

constraint M (M ) may b e written explicitly as

A A lb

2 2 2 2 2 2

K m + K m + K A m + K A m m M +(M ) : (16)

1 2 3 4 0 0 1=2 A

1=2 0 0 0 Z lb

The nature of this quadratic form dep ends on the sign of its determinant. When this

determinant is p ositive, an elliptical branch in the (m , m ) plane provides lower b ounds

1=2 0

on the twovariables. On the other hand, a negative determinantentails an upward{moving

hyp erb olic branch which places an upp er b ound on m . These branches are part of conics

whose axes are somewhat tilted with resp ect to the (m , m ) axes.

1=2

2 2

Similar implications follow from the constraint , whichmay b e written explicitly

as (for A =0)

2 2 2

J m +J m + : (17)

1 2

1=2 0 lb

This quadratic form may b e discussed in much the same way as the one in Eq.(15). From

the prop erties seen in Sect.I I it turns out that the co ecient J is always p ositive, whereas

the co ecient J is p ositive in the universal case, but may b e negative when deviations

2 2

from m universality are intro duced. Thus it follows that the condition puts lower

b ounds on m and either lower or upp er b ounds on m , dep ending on the sign of J (due

0 2

1=2

to analytic prop erties identical to those discussed previously b elow Eq.(15)). The condition

2 2

sets a very stringent upp er b ound on m , whenever J is negative and large in

0 2

magnitude. The extension to the case A 6=0 may b e rep eated here in a way similar to the

ab ove discussion for Eq.(16).

2 2

Thus wehave seen that two imp ortant constraints, and M (M ) , are at

A A lb

work in b ounding m and m , when EWSB is required to o ccur radiatively. When J and

0 2

1=2

K are p ositive, the two conditions place lower b ounds on m and m . Similar constraints

2 0

1=2

are established by the requirements that also the sfermion masses and m satisfy the relevant

exp erimental b ounds. These last conditions are not explicitly discussed here, but they are

taken into account in our evaluations. 9

2 2

It is worth emphasizing that the most dramatic impact of the conditions and

M (M ) over the parameter space o ccurs when either J or K (or b oth of them) are

A A lb 2 2

2 2

negative. Under these circumstances, as wehave seen ab ove, and M (M ) may

A A lb

place stringent upp er limits on m , b ounding the neutralino parameter space considerably.

Which of the two conditions prevails over the other dep ends on the sp eci c regions of the full

parameter space and on the values of the 's. In Sect.VI I we will illustrate the implications

of these constraints in a few sp eci c examples.

V. COSMOLOGICAL CONSTRAINT

Let us turn now to the evaluation of the neutralino relic abundance h and to the

requirement that the lightest neutralino is not overpro duced, i.e., h 1.

The neutralino relic abundance h is evaluated following the standard pro cedure

2 2

[23{ 26], according to which h is essentially given by h /< v > , where

ann

int

< v> is the thermally{averaged annihilation cross section, integrated from the freeze{

ann int

out temp erature to the present temp erature. The standard expansion < v>=a+bx + :::

ann

may b e employed, with x = T=m , except at s{channel resonances (Z; A; H; h), where a more

precise treatment has to b e used for the thermal average [24]. In the evaluation of < v>

ann

the full set of annihilatio n nal states (f f pairs, gauge{b oson pairs, Higgs{b oson pairs and

Higgs{gauge b oson pairs), as well as the complete set of Born diagrams are taken into ac-

count [26]. We recall that one of the largest contributions to the annihilation cross section is

provided by diagrams with the exchange of the pseudoscalar Higgs b oson A. (More relevant

prop erties of the Higgs b osons are discussed in Section VI I I.) We note that the constraint

h 1isvery e ective for small and intermediate values of tan , but is not restrictive for

large values of tan . The strong restriction in the former case comes from the large value

of M implied by small and intermediate values of tan (see Fig.5) (also the couplings of

A to and fermions are small for these values of tan ).

We show in Figs.6{8 a few examples where h is given as a function of m in the

form of scatter plots. These scatter plots have b een obtained byvarying the parameters

m and m on a equally{spaced linear grid over the ranges 10 GeV m 2TeV,

0 1=2 0

45 GeV m 500 GeV.Furthermore, we remark that all evaluations presented in this

1=2

pap er are for p ositivevalues of , since negativevalues of are disfavoured by the constraints

due to m and b ! s pro cesses (see Sect.VI). The con gurations shown in Figs.6{8 satisfy

the constraints due to radiative EWSB, discussed previously.

In Fig.6 is shown the case tan = 8 and =0. Here, as exp ected b ecause of the

intermediate value of tan , many neutralino con gurations provide h > 1, whilst only

a few give h 1. (Also, M is large here b ecause of sizeable values of K (see Fig.2d),

A 2

which helps increase h .) An exception o ccurs when m ' M =2, since in this case the

annihilatio n cross section is greatly enhanced due to the Z{p ole contribution. 10

In Fig.7 we display h in a case of non{universality( =0:2; =0:4, for de nite-

1 2

ness). It is easier to nd h 1 in this case, since here the departure from m universality

implies a changeover of the neutralino comp osition from the gaugino dominance of the pre-

vious example to higgsino dominance (this p oint will b e elucidated in Sect.VI I). This implies

a larger { annihilatio n cross section and consequently a smaller relic abundance. Thus

only a few neutralino con gurations are excluded by the h 1 condition.

An example for h in the case of large tan and = 0 is shown in Fig.8. We see that

h 1 imp oses no constraint since, for this very large value of tan , annihilation cross

sections are very large.

VI. CONSTRAINTS FROM b ! s AND m

In the evaluation of the b ! s decay rate wehave included the sup ersymmetric con-

tributions arising from the charged Higgs lo ops and chargino lo ops given in Ref. [27]. The

Higgs term always adds to the Standard Mo del value and usually entails to o large a value for

the rate. On the other hand, the chargino contribution gives rise to a destructiveinterference

for >0 (in our convention for the sign of ). At large tan sup ersymmetric contributions

may b e sizeable: unless the destructiveinterference protects the decay rate, it can very

easily b e driven out of the present exp erimental b ounds. In the light of this prop erty, the

p ositive scenario app ears to b e the favourite one and, as already remarked, in this pap er

we only show results for this case. In comparing our predictions with observations wehave

taken into account that, as discussed in Ref. [28], large theoretical uncertainties are present,

mainly due to QCD e ects. In particular, predictions dep end very strongly on the choice

of the renormalization scale, leading to an inaccuracy of order 25%. To account for this

e ect wehave relaxed the exp erimental b ounds of Ref. [29] by the same amount, keeping

the renormalization scale xed at the representativevalue of 5 GeV. Thus, our requirement

4 4

is that the rate of b ! s decay falls into the range 0:8 10 BR(b ! s ) 5:3 10 :

The sup ersymmetric corrections to the b ottom mass include contributions from b ottom{

squark{gluino lo ops and from top{squark{chargino lo ops [30]. In the present analysis, the

b ottom mass is computed as a function of the other parameters and required to b e compatible

with the present exp erimental b ounds. Theoretical uncertainties in the evaluations of m

arise b oth from the running of the RGE's and from assumptions ab out Yukawa uni cation.

Since our choice is to solveRGE's at the 1{lo op level and without thresholds, we estimate

an uncertainty of the order of 10% in our prediction for m . In addition, a relatively

small departure (see Ref. [31]) from b ottom{ Yukawa uni cation at the GUT scale may

signi cantly change the b ottom mass result. To takeinto account such uncertainties wehave

chosen to weaken the b ounds on m given in [32] by an amount of 10%. Thus we require m

b b

to fall into the range 2:7 GeV m (M ) 3:4 GeV.

b Z 11

VI I. ALLOWED REGIONS IN NEUTRALINO PARAMETER SPACE

We discuss now in a few examples how the various constraints analysed in the previous

Sections complement each other in shaping the allowed regions in the parameter space. We

start with the (m , m ) representation, and later display our results in the (m , ) plane

1=2 1=2

which provides the most useful representation for neutralino phenomenology.

Let us rst clarify a few graphical conventions adopted in our (m , m ) and (m , )

1=2 1=2

plots. Regions are left empty when at least one of the following constraints is not satis ed:

i) exp erimental b ounds on Higgs, neutralino and sfermion masses [16,33], ii) the is the

2 2

Lightest Sup ersymmetric Particle (LSP), iii) radiative EWSB and , M (M ) .

A A lb

Regions forbidden by the cosmological constraint( h 1) are explicitly denoted by dots

and those disallowed by the b ! s , m constraints (but not by the previous ones) are

denoted by crosses (crosses are displayed only in the (m , m ) plane, but not in the (m ,

1=2 1=2

) plane, to simplify these plots). The allowed domains are denoted by squares when they

satisfy h > 0:01, or by diamonds otherwise in the (m , m ) plots. They are denoted by

1=2

squares in the (m , ) plots, indep endently of the h value. To simplify the discussion,

1=2

we rst take A =0.We comment on the A 6= 0 case at the end of this Section.

0 0

As a rst example, let us consider the representative p oint tan =8.For this intermedi-

ate value of tan , the cosmological constraint is exp ected to b e very e ective in view of the

arguments discussed in Sect.V. This is actually the case for universal m , when b oth K and

0 2

J are p ositive (see Fig.2), so that the conditions of radiative EWSB do not set any upp er

limit on m (Fig.9a). The empty region in the lower part of these gures is forbidden by

the exp erimental b ound on m . As shown in this gure, in wide regions (denoted by dots)

h > 1. Thus the cosmological constraint places a very stringent upp er b ound on m for

m 150 GeV. However, for smaller values of m , an allowed horizontal region extends

1=2 1=2

up to m ' 2TeV. In fact, along this strip, m ' M =2 and then h 1 is satis ed (see

0 Z

the discussion in Sect.V).

Moving away from the universal p ointtowards a region where J is negative, we exp ect

2 2

to b e e ective in placing a stringent upp er b ound on m . This is actually the case

in the example shown in Fig.10a, which refers to the representative p oint = 0:2, =0:4

1 2

2 2

(J = 0:07). Here it is the b ound which provides the most stringent constraint

in disallowi ng the large (empty) domain on the right side. Nevertheless, h 1 is still

e ective in excluding an internal region that would otherwise b e allowed (see the discussion

b elow).

Keeping tan =8,we complete our discussion by considering the representative p oint

= 0:8; =0:2 shown in Fig.11a, which gives an example where J is very small. The

1 2 2

p eculiarity of this example will b ecome clear when we discuss the relevant situation in the

(m , ) plane, to whichwenow turn.

1=2

The shap e and general prop erties of the physical region in the (m , ) plane are dictated

1=2 12

by the constraints previously derived, and they are determined most notably by J .Itis

convenient to distinguish the two cases i) J > 0 and ii) J < 0. For case (i) at xed m ,

2 2 1=2

increases for increasing m with the consequence that the allowed physical region extends

to the right of the m = 0 line in the (m , ) plane, allowing for the neutralino only

1=2

a gaugino{domina ted region. In the case (ii) (J < 0), starting from the m = 0 line and

2 0

increasing m at xed m , one moves to the left and then one may reach regions of sizeable

1=2

higgsino{ga ugi no mixing or even of higgsino dominance. Case i) applies in particular to the

case of m universality( = 0) for anyvalue of tan . This is clear from Fig.3, which shows

0 i

that in the ( , ) plane the origin is b elowanyJ = 0 line. An example of this situation

2 1 2

is displayed in Fig.9b (for tan = 8).

However, as wehave seen in Sect.I I, when the assumption of m universality is relaxed,

then J , which in the universal case is p ositive and small, mayvery easily b ecome negative

and sizeable. In this case a changeover in neutralino comp osition from an originally gaugino{

like state into a higgsino{l ike one o ccurs. This remarkable prop erty, discussed in Ref. [14]

for large tan , is in fact valid over the whole range of tan , if the degree of non{universality

is increased for decreasing tan . An example of case ii) (J < 0) is shown in Fig.10b, where

the allowed region extends widely into the higgsino region. It is instructive to compare Fig.9

with Fig.10. Lo oking at sections a) of these gures, we notice that changing the values

of the 's from the set = 0 to the set = 0:2, =0:4 relaxes substantially the

i i 1 2

cosmological constraint. Parts b) of these gures provide the explanations for this feature.

In fact, whereas in the former case the neutralino is mainly a gaugino, in the latter case

is higgsino{l ike or mixed. As we already remarked, this implies an increase of the {

annihilatio n cross section and a reduction of the relic abundance. The physical region also

displays an extension to the right, in the example of Fig.11b, but here the e ect is very tiny,

due to a very small J and to the severe upp er b ound on m for m 180 GeV. This is

2 0

1=2

the rst case to showavery marked (m , ) correlation.

1=2

Nowwe turn to the case of large tan , where new features app ear. First, the M

(M ) condition is no longer protected by large values of K , and may b ecome e ectivein

A lb 1

restricting the parameter space. Secondly, the m and b ! s conditions are now rather

stringentover large domains and not only o ccasionally relevant as in the smaller tan

cases. Thirdly, the cosmological constraint is usually overwhelmed by the other conditions.

In Figs.12a, 13a, 14a wehave, for tan = 53, the following sequence of examples. i)

=0; = 0 (Fig.12a): here K < 0, J > 0, and since K is negative and sizeable in

1 2 2 2 2

magnitude, the constraint M (M ) sets an extremely stringent upp er b ound on m

A A lb 0

and thus forbids the wide (empty) region on the right. ii) =0; = 0:2 (Fig.13a): here

1 2

one still has K < 0, J > 0, but jK j is smaller than in the previous case, so the constraint

2 2 2

M (M ) is still very e ective but less comp elling than in the case (i). Also, the role of

A A lb

the m and the b ! s conditions is more signi cant here. iii) =0:7; =0:4 (Fig.14a):

b 1 2

2 2

here K > 0, J < 0, M (M ) givesalower b ound on m and the condition

2 2 A A lb

1=2

provides the frontier of the empty domain on the right.

The (m , ) representations for large tan and for the representative p oints discussed

1=2

ab ove are displayed in Figs.12b{14b. We start from the universal case of Fig.12b. Here we 13

exp ect gaugino{domina ted con gurations. However, b ecause the values of m are strongly

limited from ab ove (see Fig.12a), wehave the extremely correlated states shown in Fig.12b.

In the case of Fig.13b one has J > 0, and gaugino{domina ted states o ccur. No strong

(m , ) correlation shows up in this case. The opp osite case, J < 0, is shown in Fig.14b,

1=2

where higgsino{domi nated con gurations app ear.

It is worth adding a few comments ab out the examples of Figs.11 and 12, where the

physical regions in the (m , ) plane showavery pronounced correlation in the two

1=2

2 2

variables. This feature o ccurs whenever jJ jm J m , i.e., whenever m is severely

2 1 0

1=2

b ounded from ab ove and/or jJ j is very close to zero. As far as the values of jJ j=J are

2 2 1

concerned, we notice that in the universal case (see Fig.2c), except for small values of tan ,

J =J ' 0:04 (in fact, for tan 4, J 'a '2:5, J 'b '0:1). Thus for =0 a

2 1 1 2 2 2 i

strong (m , ) correlation o ccurs whenever m O (m ). This happ ens in the example

1=2 0 1=2

of Fig.12, where m is severely b ounded by the M (M ) condition, and in the case of

0 A A lb

Fig.11, where the correlation is enforced bya very small value of J : J =0:06. A (m , )

2 2

1=2

correlation is also exhibited in Fig.9b for the range m 150 GeV, where m is b ounded

1=2

by the cosmological constraint.

In general, we do not consider these physical regions with a strong (m , ) correlation

1=2

as unnatural, since they are usually realized without much tuning. We recall that the size

of the co ecients J and J is dictated by the RGE's with their intrinsic cancellations,

1 2

and that one naturally has J = O (a few), J = O (0:1 0:01). As wehave seen, these

1 2

prop erties, combined with severe upp er b ounds on m , are sucient to generate the (m ,

0 1=2

) correlation.

We turn now to the A 6= 0 case. First we recall that A is constrained in the range

0 0

jA j 3 from the absence of charge and color breaking [34]. Thus, allowing A 6= 0 do es

0 0

not change essentially the general picture previously discussed. The previous scenarios still

o ccur, but at di erent p oints in the parameter space. Two sp eci c comments are in order

here: i) indep endently of its sign, A disfavours the changeover from gaugino dominance

to higgsino dominance in the neutralino comp osition, ii) a negative A reduces the value

of M as compared to the A = 0 case, and so either provides a light A b oson (and hence

A 0

interesting phenomenology) or enforces a more stringent constraint on the parameter space.

VI I I. NEUTRAL HIGGS BOSONS

Neutralino direct detection, to b e discussed in the next Section, is based on neutralino{

nucleus scattering. In this pro cess, exchanges of neutral Higgs b osons play a dominant role,

provided the Higgs masses are not to o heavy. Itisconvenient to recall here some relevant

prop erties of the couplings of with matter via Higgs exchange. As was already mentioned

in the Intro duction, the two Higgs iso doublets H , H yield 3 neutral Higgs mass eigenstates:

1 2

one CP{o dd (A) state, whose mass M is given by expression (11) and two CP{even states

A 14

0 0

(of masses M , M , M

h H h H

1 2

angle

0 0

H = cos H + sin H

1 2

0 0

h = sin H + cos H : (18)

1 2

It is imp ortant to notice here that dep ends very sensitively on M , b eing very close to

> <

zero for tan 4 and rising very fast to =2 for M O (M ) (see Fig.15).

A Z

The angle plays a crucial role in determining the size of the neutral h; H {quark cou-

plings. Here, as we are interested in {nucleus scattering, we discuss explicitly only the

couplings involving the CP{even states, since h; H are dominant compared to A. The low{

energy neutralino{quark e ective Lagrangian generated by Higgs exchange may b e written

as follows [35]

L = 2G F k m qq : (19)

e F h;H q q

h;H

Here F is the ratio of the Higgs{neutralino coupling to the SU (2) gauge coupling, which

h;H

dep ends on the comp osition of

F = a (a sin + a cos )

h 2 3 4

F = a (a cos a sin ) (20)

H 2 3 4

and the k are given, for the up{typ e quarks and the down{typ e quarks resp ectively,by

H h

k sin = sin cos = sin

k cos = cos sin = cos : (21)

Note that, in general, since tan >1, the strength of the coupling to the down{typ e quarks

is bigger than the one to the up{typ e quarks, and L usually gets a sizeable contribution

when the h b oson is exchanged (h is lighter than H and is therefore favored b ecause of the

propagator denominator in Eq.(19)) and when ' =2, i.e., when M O (M ). When

A Z

this regime do es not apply, the size of L is much suppressed.

The cross section for elastic neutralino{nucleus scattering which follows from the e ective

Lagrangian (19) will b e given in Sect.IX.B.

IX. DIRECT DETECTION

Much exp erimental activity is under way in the direct search for neutralino dark matter

and the p ersp ectives for signi cant improvements in exp erimental sensitivities are encourag- 15

ing [36]. In this class of exp eriments, a relic neutralino would b e detected by the amountof

energy released by its elastic scattering o nuclei in an appropriate apparatus. A signature

would b e provided bya yearly mo dulation of the signal, whose observations would require

high statistics and extremely go o d stability in the detector resp onse. Here weevaluate the

event rates for this pro cess extending previous analyses to the non{universal 6= 0 case.

Various materials are b eing used in the current exp eriments and others are under investi-

gation for future detectors. In this pap er we analyse two of the most interesting materials:

129

Ge (in its natural comp osition) [37{ 39] and Xe [40].

A. Di erential rates

The nuclear recoil sp ectrum maybeevaluated from the expression

dR R

0;i

= F (E )I (E ) (22)

R R

max

where

R = N : (23)

0;i T i

In Eqs.(22){(23) we use the following notations: the subscript i refers to the two cases of

coherent and spin{dep endent e ectiveinteractions, N is the numb er of the target nuclei p er

unit of mass, is the lo cal neutralino matter density, and E is the nuclear recoil energy

2 2

given by E = m v (1 cos )=m , where is the scattering angle in the neutralino{

R N

red

nucleus center{of{mass frame, m is the nuclear mass, m is the neutralino{nucleus re-

N red

max 2 2

duced mass and v is the relativevelo city. The maximum value of E is E =2m v =m .

R N

R red

Returning to (22{23), F (E ) denotes the nuclear form factor, and is the (coherent/spin{

R i

dep endent) neutralino{nucleus cross section. The factor I (E ) is given by

max

f (v )

I (E )= (24) dv

v (E )

min

where f (v ) is the velo city distribution of neutralinos in the Galaxy, as measured in the

2 1=2

Earth's rest frame, and v (E ) is given by v (E )=(m E =(2m )) . The averages

min R min R N R

red

app earing in Eqs.(22){(24) denote averages over the velo city distribution in the Earth's rest

frame. An explicit formula for I (E ) in the case of a Maxwellian velo city distribution may

b e found in Ref. [4].

The di erential rates to b e discussed b elow will b e expressed in terms of the electron{

equivalent energy E rather than in terms of E . These twovariables are prop ortional:

ee R

E = QE where Q is called the quenching factor: typical values of Q will b e discussed

ee R

shortly. 16

B. Neutralino{nucleus elastic cross sections

The total cross sections for neutralino{nucleus elastic scattering have b een evaluated fol-

lowing standard pro cedures [3,4,35,41,42]. Here we only summarize some of the main prop-

erties. Neutralino{quark scattering is describ ed by amplitudes with Higgs{b oson exchanges

and Z {b oson exchange in the t{channel, and by amplitudes with squark exchanges in the

s{ and u{channels. The neutral Higgs b osons considered here are the two CP{even b osons:

h; H and the CP{o dd one: A, whose couplings were previously discussed in Sect.VI I I.

The relevant prop erties for these amplitudes are: 1) Higgs{b oson exchanges contribute

a coherent cross section whichvanishes only when there is no zino{higgsino mixture in the

neutralino comp osition [35], 2) Z {b oson exchange provides a spin{dep endent cross section

which receives contributions only from the higgsino comp onents of , 3) squark exchanges

contribute a coherent cross section (due to zino{higgsino mixing) as well as a spin{dep endent

cross section (due mainly to the gaugino comp onents of ) [41]. As examples we recall here

only the expressions for the coherent cross section due to the exchange of a Higgs b oson (h

or H ) and the spin{dep endent one due to Z exchange.

The former cross section is easily evaluated from the e ective Lagrangian of Eq.(19) [35]

2 2

8G m

F Z 2 2 2

= m A (25)

h;H red

h;H

where A is the nuclear mass numb er and is given by

h;H

= F I ; I = k m hN jqq jN i: (26)

h;H h;H q q

The quantity I may b e expressed conveniently in terms of the N sigma{term and of

a parameter a which is related to the strange{quark content of the nucleon y by

: (27) a = y (m =(m + m )) ; y =2

s u d

One has

I ' k g + k g (28)

u u d d

where

4 19

g = m + a

u N N N

27 8

2 23 25

g = m + + a : (29)

d N N N

27 4 2 17

Unfortunately, the values of b oth the quantities y and are somewhat uncertain. Here,

for y we use the central value of the most recentevaluation: y =0:33 0:09, obtained from a

lattice calculation [43]. For , which is derived by phase{shift analysis and disp ersion re-

lation techniques from low{energy pion{nucleon scattering cross{sections [44,45], we employ

the value of Ref. [45]: = 45 MeV. We then nd the results: g = 123 MeV, g = 288

N u d

MeV (we use 2(m =(m + m )) = 29 [46]). We note that these values further reinforce the

s u d

role of the down{typ e quarks as compared to the up{typ e ones.

We p oint out that the Higgs{nucleon couplings for nucleons b ound in a nucleus maybe

renormalized by the nuclear medium. As a consequence, the strength of I might in principle

b e reduced to some extent [47]. However, this e ect is neglected here.

Now let us turn to the spin{dep endent cross section due to Z exchange. This maybe

cast into the usual form [41,42]

F 2 2 2 2 2 2

= (a a ) m ( T q ) J (J +1) : (30)

SD 3L;q

3 4 red

In this pap er we use this formula for Ge (this isotop e is present at the level of 7.8 % in

129 2

the natural comp osition of Ge) and to Xe. For these nuclei we employ the values of

obtained in the o dd{group mo del [3], where only the o dd nuclear sp ecies in o dd{even nuclei

are explicitly taken into account. The q 's in Eq.(30) denote the fractions of the nucleon

spin carried by the quarks q in the nucleon of the o dd sp ecies, and the T 's stand for the

3L;q

third comp onents of the quark weak isospin. The values for the q 's are taken from Ref.

[48].

It is worth noticing that the event rates for neutralino direct detection with the materials

considered here are largely dominated by coherent e ects in most regions of the parame-

ter space. In the small domains where spin{dep endent e ects dominate over the coherent

ones the total rates are usually to o small to allow detection. The exp erimental strategy of

employing materials enriched in heavy isotop es of high spin is interesting for a search for hy-

p othetical dark matter particles whichinteract with matter via substantial spin{dep endent

interactions. However, this approach do es not app ear to b e very fruitful for neutralinos.

One more ingredient whichenters the event rate in Eq.(22) is the nuclear form fac-

tor, which dep ends sensitively on the nature of the e ectiveinteraction involved in the

neutralino{nucleus scattering. For the coherent case, we simply employ the standard pa-

rameterization [49]

j (qr )

1 0

2 2

F (E )= 3 s q (31) e

2 2

where q j ~q j =2m E is the squared three{momentum transfer, s ' 1 fm is the thickness

N R

2 2 1=2 1=3

parameter for the nucleus surface, r =(r 5s) ,r =1:2A fm and j (qr ) is the

0 1 0

spherical Bessel function of index 1. 18

TABLE I. Characteristics of some current exp eriments. In the second column is rep orted

the quenching factor Q, in the third column the electron{equivalent energy at threshold, in the

fourth the square of the form factor at threshold, and in the last column the present exp erimental

sensitivity.

th 2 th

Nucleus Q E (KeV ) F (E ) evts/(Kg d KeV)

Ge [38] 0.25 2 0.87 3.0

Ge [39] 0.25 12 0.41 0.2

Xe [40] 0.80 40 0.07 0.8

The form factor in Eq.(31) intro duces a substantial suppression in the recoil sp ectrum

unless qr 1. A noticeable reduction in dR=dE may already o ccur at threshold E =

0 R R

th th

E = E =Q when r 2m E is not small compared to unity. The actual o ccurrence

0 N

R ee R

of this feature dep ends on parameters of the detector material: nuclear radius, quenching

factor, threshold energy E . The values of these parameters for the nuclei considered in

2 th

this pap er are rep orted in Table I [36,38{ 40], and the values of F (E ) calculated from

Eq.(31) are given in this same Table. Since we consider in this pap er mainly the value of

2 th

the di erential rate near threshold, F (E ) is the most relevant quantity.We see from the

values in the Table that the reduction intro duced by the form factor is mo derate in Ge, but

129

quite substantial in Xe.

In general, for the spin{dep endent case there are no analytic expressions for the form

factors. However, numerical analyses have b een p erformed for a number of nuclei. The

general feature is that these form factors havea much milder dep endence on E as compared

to the coherent ones, b ecause only a few nucleons participate in the neutralino{nucleus

131

scattering in this case. In our evaluations we use the results of Refs. [49,50] for Xe and

Ge resp ectively.

C. Lo cal Neutralino Density

We denote the lo cal halo densityby, for whichwe use the estimate =0:5 GeV

l l

[51]. For the value of the lo cal neutralino density to b e used in the rate of Eq.(23), cm

for each p oint of the mo del parameter space we takeinto account the relevantvalue of the

2 2

cosmological neutralino relic density. When h is larger than a minimal ( h ) required

min

by observational data and by large scale structure calculations we simply put = . When

2 2

h turns out less than ( h ) , the neutralino may only provide a fractional contribution

min

2 2 2

h =( h ) to h ; in this case we take = . The value to b e assigned to

min l

( h ) is somewhat arbitrary. Here we set it equal to 0.1.

min

It is worth remarking here that, due to this scaling pro cedure, for the direct detection

2 2

rate one has: i) R / for h ( h ) and ii) R / / =< v> for

0;i l i min 0;i l i l i ann int 19

2 2

h < ( h ) .Thus the rate R is large in the regions of the parameter space where

min 0;i i

is large. This is trivial in case i), but it is also true in case ii), since when is large also

increases but in sucha way that usually the ratio = increases to o. Because of the

ann i ann

relation h /< v> it follows that R is large for neutralino con gurations with

ann 0;i

int

mo dest values of the relic abundance, and vice versa.

D. Results for detection rates

The most signi cant quantity in comparing exp erimental data and theoretical evaluations

for direct detection is the di erential rate dR=dE =(dR=dE )=Q (with dR=dE de ned

ee R R

in Eq.(22)) rather than the total rates, obtained byintegration over wide ranges of E .

By using the di erential rate instead of the integrated ones, one obtains the b est signal{

to{background ratio. Note that the exp erimental sp ectra, apart from an energy interval

around threshold, usually showavery at b ehaviour, whereas signals for light neutralinos

are decreasing functions of the nuclear recoil energy.

A complete pro cedure would then b e to compare the exp erimental and theoretical rates

over the whole E range. However, to simplify the presentation here, we give our results

in terms of the rate integrated over a narrow range of 1 KeV at a sp eci c value of E , the

one which app ears the most appropriate for each exp eriment: typically it corresp onds to a

p oint close to the exp erimental threshold. To b e de nite we consider the following cases:

i) Ge (natural comp osition). Among the various running exp eriments [36], we select

the two which, at present, app ear to provide the most stringent limits: a) Caltech{PSI{

Neuchatel [38] with E = 2 KeV, di erential rate ' 3events/(Kg day KeV); b) Heidelb erg{

Moscow [39] with E = 12 KeV, di erential rate ' 0:2events/(Kg day KeV). Corresp ond-

ingly, for Ge wehaveevaluated our rate byintegrating dR=dE over the range (2{3) KeV

for exp eriment a) and over (12{13) KeV for exp eriment b). It turns out that the case b)

provides the most stringent b ound also for light neutralinos.

129

ii) Xe. In this case, taking into account the features of the DAMA exp eriment [40], we

have considered the rate R integrated over the range (40{41) KeV.

Our results are shown in Figs.16{19. Figs.16{18 rep ort the rate for a Ge detector for the

regions of the parameter space which are depicted in Figs.12{14, resp ectively. In parts (a)

and (b) of each gure, R is displayed in the form of a scatter plot, in terms of m and of

the relic abundance, resp ectively. The horizontal line denotes the present level of sensitivity

in the Heidelb erg{Moscow exp eriment. We notice that, in all cases shown in these gures,

the exp erimental sensitivity is already, for some con gurations, at the level of the predicted

rate. Some p oints of the sup ersymmetric parameter space, denoted by lled squares in

Figs.12{14, are even already excluded by present data. The exploration p otential of this

class of exp eriments as the sensitivity is improved is apparent from these gures. Fig.19 20

129

shows the rate R for Xe for the region of the parameter space displayed in Fig.13: again

the horizontal line gives the present exp erimental sensitivity. A comparison of Fig.19 with

Fig.17 shows that the Ge exp eriments are currently more e ective. However, it has to b e

noticed that exp eriments with liquid Xe may b ecome extremely comp etitive in the future

[40].

A few more remarks are in order here:

i) The cases displayed in Figs.16{18 present the common feature of providing fair chances

for direct detection. This is not a surprise, since these representative p oints all b elong to the

category of con gurations with small values of M .Aswas stressed b efore, once wemove

away from these app ealing physical regions of the neutralino parameter space, the rates for

direct detection may fall far b elow(by many orders of magnitude) the detection sensitivities

(present or future). This unfortunate situation o ccurs, for instance, typically as wemove

towards smaller values of tan .However, one should keep in mind that the regime of very

large tan , where signals may b e sizeable, represents a very interesting scenario, deserving

much attention and exploration. In fact this is one of the two options, very small or very

large tan , which seem to t low{energy phenomenology at the b est [52].

ii) The scatter plots in parts b) of Figs.16{19 show explicitly a prop erty previously men-

tioned in Sect.IX.C, namely that the scaling pro cedure adopted to evaluate the neutralino

lo cal density implies a R{ h correlation. Con gurations which provide a measurable R

usually entail a low and viceversa. Only in a few cases the neutralino may b e detectable

by direct detection and also provide a sizeable contribution to .

X. CONCLUSIONS

In the present pap er wehave discussed some p ossible scenarios for neutralino dark matter

which originate from the relaxation of the assumption of strict universality for soft scalar

masses at M .

GU T

This approach derives from the general consideration that many crucial theoretical p oints

entering not only grand uni ed and sup ersymmetric theories, not to mention the Standard

Mo del, are far from b eing understo o d and/or veri ed. For this reason, any new theoretical

assumption has to b e fully scrutinized. This is even more imp ortant b ecause new assump-

tions in sup ersymmetric mo dels are often intro duced not b ecause of solid arguments, but

rather for the sake of simplicity and for the need to reduce the large numb er of free param-

eters that would otherwise preventany rm prediction.

In our work wehave discussed di erent scenarios, by considering various physical con-

straints in a sort of hierarchical order, giving top priority to the requirement of radiative

EWSB, implemented with a no{ ne{tuning criterion, and to the cosmological relic neu- 21

tralino density constraint. Some other assumptions, often intro duced in the literature, have

b een relaxed in our work. This is in particular the case for universality in the soft scalar

masses. However, it has to b e remarked that the typ e of departure from universality that

wehave considered in our pap er is far from b eing the most general one, as was noticed in

Sect.I I. In particular, it only refers to the Higgs masses, and not to the sfermion masses.

The implications of the various scenarios on neutralino relic abundances and rates for

detection rates have b een analysed, and the impact of a non{universalityinm has b een dis-

cussed for the whole range of tan .Wehave shown that the departure from m universality

is particularly interesting in two resp ects:

i) Small values of M are allowed: this has in itself the dramatic consequence for direct

detection of generating a large value for the angle and large couplings to matter of the

lightest neutralino .

ii) Higgsino or mixed higgsino{ga ugi no con gurations app ear for all tan : this contrasts

with the pure gaugino con gurations favoured by strict m universality.

Consequences of such a departure from universality on the size of the neutralino relic abun-

dance have b een analysed for b oth large and small values of tan . It has b een shown that,

b ecause of the previous prop erties, deviations from universalitymay reduce the value of

h .

The predicted rates for direct detection has b een analysed in detail and compared with

current and foreseen exp erimental sensitivities. The role of the previous prop erties in op ening

interesting p ersp ectives for this kind of search has b een elucidated. We nd that presently{

running exp eriments are already impacting interesting regions of the neutralino parameters

space in some of the non{universal scenarios discussed here.

Acknowledgements. We wish to express our thanks to Uri Sarid for interesting dis-

cussions and for contributions in the very early stages of our work. We also gratefully ac-

knowledge very useful conversations with Marek Olechowski and informative discussions on

exp erimental asp ects of direct detection with Pierluigi Belli, Rita Bernab ei and Luigi Mosca.

NF wishes to express his gratitude to the Fondazione A. Della Riccia for a fellowship. This

work was supp orted in part by the Ministero della Ricerca Scienti ca e Tecnologica (Italy). 22

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Figure Captions

Figure 1 { The (m , ) plane for tan = 8. The lines of constant m = 30 GeV, 60

1=2

GeV, 90 GeV are displayed as dashed lines. The lines of constant P = 0.1, 0.5, 0.9, 0.99 are

shown as solid lines. The dotted region denotes the domain forbidden by present LEP data.

Figure 2 { Co ecients of the p olynomial expressions (9{11) as functions of tan :

a) co ecients of M : a solid line, b dashed line, c dot{dashed line and d dotted line,

1 1 1 1

b) co ecients of M : a solid line, b dashed line, c dot{dashed line and d dotted line,

2 2 2 2

c) co ecients of : J solid line, J dashed line, J dot{dashed line and J dotted line,

1 2 3 4

d) co ecients of M : K solid line, K dashed line, K dot{dashed line and K dotted line.

1 2 3 4

Figure 3 { In the ( , ) plane, the lines where J = 0 at xed tan are displayed:

2 1 2

tan = 53, solid line; tan = 8, dashed line; tan = 3, dot{dashed line; tan = 2, dotted

line.

Figure 4 { In the ( , ) plane, the lines where K = 0 at xed tan are displayed:

2 1 2

tan = 53, solid line; tan = 40, dashed line; tan = 8, dot{dashed line.

Figure 5 { Graph of M as a function of tan at the representative p oint m =50

A 0

GeV, m = 200 GeV.

1=2

Figure 6 { Scatter plot representing h as a function of m for tan =8, = 0 and

=0.Parameters are varied on a linear equally{spaced grid over the ranges: 10 GeV

m 2TeV, 45 GeV m 500 GeV.

0 1=2

Figure 7 { The same as in Figure 6, but with = 0:2 and =0:4.

1 2 29

Figure 8 { Scatter plot representing h as a function of m for tan = 53, =0

and =0.Parameters ranges are as in Figure 6.

Figure 9 {a)Parameter space in (m , m ) plane for tan =8, = 0 and =0.

0 1 2

1=2

Empty regions are excluded by: i) accelerator constraints, ii) radiative EWSB conditions,

iii) neutralino is not the LSP. Dots represent the region where h > 1. Regions with

crosses are excluded by b ! s and m constraints. In the regions denoted by squares,

0:01 < h 1. The region without ne{tuning is inside the b ox b ounded by dashed

lines.

b) Parameter space represented in the (m , ) plane. Solid lines corresp ond to the extreme

1=2

values of m . Notations are the same as in a), but crosses are omitted here.

Figure 10 { a) The same as in Figure 9, but with = 0:2 and =0:4. In the regions

1 2

denoted by diamonds h 0:01.

b) Notations are the same as in a). Note that the domain where the neutralino is the dark

matter particle with 0:01 < h 1 has shifted to the higgsino{ domina ted region. Crosses

are omitted here.

Figure 11 { The same as in Figure 9, but with = 0:8 and =0:2.

1 2

Figure 12 {a)Parameter space in the (m , m ) plane for tan = 53, = 0 and

0 1

1=2

= 0. Filled squares denote con gurations excluded by direct detection with a Ge detector

[39]. Other notations are as in Figure 9.

b) Parameter space represented in the (m , ) plane. Notations are the same as in a).

1=2

Figure 13 { The same as in Figure 12, but with = 0 and = 0:2.

1 2

Figure 14 { The same as in Figure 12, but with =0:7 and =0:4.

1 2

Figure 15 { a) Graphs of M and M as functions of tan .

A h

b) Mixing angle as a function of tan . 30

Here the representative p ointism = 100 GeV, m = 200 GeV.

0 1=2

Figure 16 { Scatter plot of the rate for direct detection with a Ge [39] detector for

tan = 53, = 0 and = 0, as a function of m (a) and as a function of h (b).

1 2

Parameters are varied on a linear equally{spaced grid over the ranges: 10 GeV m

2TeV, 45 GeV m 500 GeV.

1=2

Figure 17 { The same as in Figure 16, but with = 0 and = 0:2.

1 2

Figure 18 { The same as in Figure 16, but with =0:7 and =0:4.

1 2

Figure 19 { Scatter plot of the rate for direct detection with a Xe [40] detector for

tan = 53, = 0 and = 0:2, as a function of m (a) and as a function of h (b).

1 2

Parameters are varied on a linear equally{spaced grid over the ranges: 10 GeV m

2TeV, 45 GeV m 500 GeV.

1=2 31