

 1 Dec 94

IASSNS-HEP-94/76,

SU-4240-593 CfP A-94-TH-59,

CU-TP-653, UCSD/PTH-94-23

Mo del-Indep

enden tComparison

of Direct

vs. Indirect

Detection of Sup

ersymmetric

Dark Matter

Marc Kamionk

owski

a;b z, Kim

Griest

c\Lambda , Gerard

Jungman

dy, and

Bernard

Sadoulet

efi

aScho ol of Natur

al Scienc

es, Institute

for Advanc

ed Study,

Princ eton, NJ 08540

bDep artment

of Physics,

Columbia

University,

New York,

NY 10027

cDep artment

of Physics,

University

of California,

San Die go,

La Jol la, CA

92093

dDep artment

of Physics,

Syr acuse

University,

Syr acuse,

NY 13244

eCenter for Particle

Astr ophysics,

University

of California,

Berkeley, CA 94720

ABSTRA CT

We compare

the rate

for elastic

scattering

of neutralinos

from various

nuclei with the flux

of up ward

muons

induced by energetic

neutrinos from neutralino

annihilation

in the

Sun

and Earth.

We consider

both scalar

and

axial-v ector interactions

of neutralinos

with nuclei.

We find

that the even

trate

in akg

of germanium

isroughly

equiv alen tto that

in a10

5- to 10

7-m

2m uon

detector

for aneutralino

with primarily

scalar coupling

to nuclei.

For an axially

coupled

neutralino,

the even

trate

in a50-gram

hydrogen detector is roughly

the same

as that

in a10-

to 500-m

2m uon

detector.

Exp ected

exp erimen

tal bac kgrounds

fav or forthcoming

elastic-scattering

detectors for scalar

couplings

while the neutrino

detectors ha ve

the

adv an tage

for axial-v

ector couplings.

1Decem ber 1994

z kamion@ph

ys.colum bia.edu

\Lambda kgriest@ucsd.edu y jungman@npac.syr.edu

fi sadoulet@lbl.go

v

An ever-increasing

bo dy

of evidence

mak es itfeasible

that the dark

matter

in our

Galaxy

is comp

osed of weakly-in

teracting massiv eparticles

(WIMPs).

The most

promising

candidate WIMP isprobably

the ligh test

sup ersymmetric

particle, whic h in

most

cases is the

neutralino,

alinear com bination

of the

sup erpartners

of the

photon,

Z boson,

and Higgs

bosons

[1][2].

Sev eral

tec hniques

are curren

tly being

pursued

in an

effort

to disco

ver suc h

dark-matter particles. The first,

direct

detection

(DD), seeks to observ

ethe

O(k eV )energy

dep osited

in alo

w-bac

kground

detector when aWIMP

elastically scatters

from an ucleus

therein

[3]. The second,

indirect detection

(ID),

inv olv es asearc

hfor energetic

neutrinos pro duced

by annihilation

of WIMPs

that ha ve

been

captured

in the

cen ter

of the

Sun

and/or

Earth [4].

Numerous calculations

of even

trates

for both

detection

schemes for av ariet y of

candidate

WIMPs ha ve

been

performed.

Although sup ersymmetry

(SUSY) is well-motiv

ated and theoretically

highly dev elop

ed, there

are man

y

undetermined parameters, even in the

minim

alSUSY

extension

of the

standard

mo del

(MSSM),

so the

results

of an ysp

ecific

calculation

often dep end

on a

num ber of assumptions.

Consequen tly ,it

isdifficult

to compare

the constrain

ts

placed on WIMPs

from DD exp erimen

ts with

those

from ID exp erimen

ts.

In this

Letter,

we pro vide

acomparison

of rates

for DD

and ID whic

his

largely mo del

indep

enden t. The

rates

for both

tec hniques

dep end

primarily

on the

coupling

of WIMPs

to nuclei.

WIMPs couple both to the

mass

of a

nucleus through ascalar interaction

and to the

spin

through

an axial-v

ector

(spin) interaction.

Here we focus

only on WIMPs

with either

scalar or spin

interactions; the results

for ageneral

WIMP should fall in bet

ween.

In the

end,

we obtain

the ratio

of the

rate

per kg for

elastic

scattering

from av ariet

yof

nuclei versus the flux

of up ward

muons

per square

meter induced

by energetic

neutrinos from annihilation

in the

Sun

and Earth.

\Lambda

We accoun

tfor the most

significan

tmo del dep endence

quan titativ

ely ,and

we discuss

some residual

mo del

dep endence

whic hcannot

be considered

in a

\Lambda A

similar,

though less comprehensiv

e, comparison

has been

done by Ric hand

Tao [5].

1

general fashion.

The final results

are to be

tak en as appro

ximate

and general

results for abroad

class of realistic

WIMP candidates.

A more

precise

comparison can, of course,

be made

for an ysp

ecific

mo del.

It is also

easy to imagine

mo dels

with

DD/ID

ratios whic hfall

far from

our estimates.

We ha ve

chec

ked

our results,

ho wev

er, by explicitly

calculating

and comparing

DD and ID rates

for thousands

of allo

wed

parameter

choices in the

MSSM

[2].

This analysis

should be helpful

in assessing

the relativ

eeffectiv

eness of

the tw ometho

ds of probing

WIMPs with various

couplings

over awide

mass

range, and for comparing

various materials

for low-bac

kground

detectors.

We

caution that there

are numerous

exp erimen

tal factors

whic hw eaddress

only

briefly that must

be considered

to prop

erly weigh

the relativ

esensitivities

of

the exp erimen

ts to WIMPs.

Although we ha ve

neutralinos

in mind,

the results

for other

WIMP

candidates,

suc has

hea vy Dirac

or Ma

jorana

neutrinos,

should

be similar.W

eb egin

with particles

with scalar

interactions

and note

that the cross

section for neutralino

scattering from nucleus

ivia ascalar

interaction

can be

written oesci =

4m ~O/2m

4i

ss(m ~O/+

m i)

2j

hL sci

j 2;

(1)

where m ~O/is

the neutralino

mass, m iis

the

nuclear

mass, and hL sci

isthe

nucleon

matrix elemen t(scaled

by the

nuclear

mass) of the

effectiv

eLagrangian

for the

scalar neutralino-n

ucleus interaction.

The imp ortan

tthing

to note

isthat

all the

information needed ab out

an ysp

ecific

MSSM

(e.g., the neutralino

comp osition,

the masses

and couplings

of all

the

sup erpartners,

etc.) for the

scalar

neutralinonucleus interaction

isenco ded in hL sci

,and

hL sci

is indep

enden tof the nuclear

mass. To agreat

exten t, both

detection

schemes pro vide

constrain

ts on

hL sci

.

Moreo ver, theoretical

uncertain ties, suc has

the strange-quark

scalar densit yin

the nucleon,

are absorb

ed in hL sci

.These

uncertain

ties affect

both rates in the

same wa y, so

they

do not

affect

the comparison

bet ween

DD and ID.

Giv en the

cross-section

for elastic

scattering

of WIMPs

from an ucleus,

it

is straigh

tforw ard to compute

the DD even trate.

With Eq. (1), the rate

for

DD of scalar-coupled

WIMPs is giv

en by Eq.

(18) in [6].

The even trate

for

2

10 100 1000

0.0001 0.001

0.01 0.1

1

A=1 A=4 A=16 A=28 A=40 A=76 A=93 A=131

FIG. 1. Ev ent

rate

(per kg of detector)

for scalar-coupled

WIMPs in adetector

comp osed of nuclei

with mass num ber A scaled

by the

rate

in a

76Ge

detector

as a

function of WIMP

mass m ~O/.

scalar-coupled WIMPs in adetector

comp osed of nuclei

with mass num ber A,

scaled by the

rate

in a

76Ge

detector,

is sho

wn

in Fig.

1as afunction

of the

WIMP mass. Energetic

neutrinos from WIMP

annihilation

in the

Sun

or Earth

are poten tially

detectable

via observ

ation of up ward

muons.

The flux of suc

hm

uons

from WIMP

annihilation

in the

Sun

or Earth

can be written

\Gamma = dtanh

2(t=o/ )ae

0:3O/

f(m

~O/),

(m ~O/)(

m ~O/=GeV

)2( hL sci

=GeV

\Gamma 3 )2;

(2)

where dfi = 2:9

\Theta 10

8m \Gamma 2 yr

\Gamma 1

,and

d\Phi = 1:5

\Theta 10

8m \Gamma 2 yr

\Gamma 1

,and

itshould

be

noted that the quan

tities

f(m ~O/),

,(m

~O/),

and

t=o/ are differen

tfor annihilation

in the

Earth

than they are for the

Sun.

The capture

rates in the

Sun

and Earth

are C fi=

(2: 1\Theta

10

37

sec

\Gamma 1 )ae

0:3O/

ffi (m

~O/)(

hL sci

=GeV

\Gamma 3 )2 and

C \Phi =

(2: 1\Theta 10

28

sec

\Gamma 1 )ae

0:3O/ f\Phi (m

~O/)(

hL sci

=GeV

\Gamma 3 )2.

3

The neutralino-mass

dep endence

isdescrib ed by the

function

f(m ~O/)

= X

i

fiOE

iS i(m

~O/)F i(m

~O/)m

3im ~O/=(

m ~O/+

m i)

2;

(3)

where the sum

iso ver

nuclei

in the

Sun

or Earth,

the quan

tities

fi and

OEi are

mass fractions

and mean

scaled poten tials, and Si( m ~O/)

and

Fi( m ~O/)

describ

e

resonance effects and form-factor

suppression (see [2] and

[7]).

The functions

,(m ~O/)

enco

de information

ab out

the neutrino

spectra and

the pro duction

of muons,

and can be written,

,(m ~O/)

= X

F

B F[3

:47

\Omega N z2ff F;*

(m ~O/)

+2

:08

\Omega N z2ff F; _*(

m ~O/)]

;

(4)

where the sum

iso ver

all annihilation

channels F available

to the

WIMP

,and

B Fis

the branc

hing ratio for annihilation

into F. The

\Omega N z2ff

are scaled

second

momen tsof the neutrino

(and an tineutrino)

energy distribution

from final state

F for

agiv

en neutralino

mass [8]. Neutrinos

are absorb

ed in the

Sun

but not

the Earth,

so the

\Omega N z2ff

are differen

tfor annihilation

in the

Sun

than

they are

for annihilation

in the

Earth.

The most

significan

tmo del dep endence

in our

calculation

comes in Eq.

(4). There

are man

yannihilation

channels available to an ysp

ecific

neutralino

candidate (and the num

ber islarger

for larger

neutralino

masses), and the B F

ma ydep

end on av ariet

yof couplings

and particle

masses. This leads

to arange

of values

of ,(m

~O/)

for

an ygiv

en neutralino

mass. Ho wev

er, the

function

,(m ~O/)

will be brac

keted

ab ove

(belo

w) by the

value

obtained

from the annihilation

channel whic hgiv

es the

largest

(smallest)

,(m ~O/).

For

neutralinos

hea vier

than

the top quark,

the upp er (lo wer)

limit to ,(m

~O/)

comes

from annihilation

into

top quarks

(gauge bosons).

If the

WIMP

is less

massiv

ethan the W boson,

then the upp er (lo wer)

limit comes

from annihilation

into o/_o/ (b_b )pairs.

If

m W

! m ~O/!

m t,

then

the upp er (lo wer)

limit comes

annihilation

into gauge

bosons (o/_o/ pairs).

The factor,

tanh

2(t=o/

),in Eq. (2) describ

es the

suppression

of WIMP

annihilation in the

Sun

or Earth

relativ eto capture.

Here, tfi ' t\Phi

' 4:5

Gyr

isthe

age of the

solar

system,

and the o/are

equilibration

time scales

giv en by (t=o/

)=

4

r \Theta ae0

:3O/f

(m ~O/)(

oeA v) 26 \Lambda

1= 2(

m ~O/=

GeV

)3

=4(

hL sci

=GeV

\Gamma 3 ), where

(oe Av

)26

is the

total annihilation

cross section

times relativ ev elo cit yin

the limit

v! 0in

units of 10

\Gamma 26

cm

3sec

\Gamma 1 . Here,

rfi = 2:9

\Theta 10

7and

r\Phi = 5:2

\Theta 10

4.

The

analogous expression for the

Earth

is obtained

by making

the replacemen

ts

2:9 \Theta 10

7! 5:2 \Theta 10

4and

fi !

\Phi .

IfhL

sci

islarge,

then t=o/ AE 1, tanh(

t=o/ )'

1,

annihilation and capture

are in equilibrium,

and the signal

is at

full

strength.

In this

case,

\Gamma / hL sci

2, as

isthe

DD rate.

On the other

hand, ifhL sci

issmall,

then t=o/ o/ 1, annihilation

has not had time

to equilibrate

with accretion

and

the neutrino

signal issuppressed.

In this

case,

\Gamma / hL sci

4( oeA

v).

To pro ceed,

we mak

ethe

simplest

and most

attractiv

eassumption

that if

neutralinos exist, their abundance

issuitable for accoun

ting for aflat

Univ erse.

Then, \Omega O/h

2'

0:25

whic hfixes

(oe Av

)26

' 1.

We

then

consider

mo dels

whic h

giv eneutrino

fluxes in the

range

10

\Gamma 2

?, \Gamma = (m

\Gamma 2 yr

\Gamma 1

)?,

10

\Gamma 4

only

.Mo

dels with

larger

fluxes would ha ve

been

observ

ed already

,and the low er limit

is roughly

the sensitivit

yattainable

with next-generation

O( km

2) detectors

(accoun ting for the

irreducible

bac kground

of atmospheric

neutrinos). We can

then sho w that

if\Gamma ( oeA v) 26(

m ~O/=

GeV

)\Gamma

1= 2r

2?,

d,( m ~O/),

then

the signal

is at

full strength.

Taking \Gamma fi?

10

\Gamma 4

m

\Gamma 2

yr

\Gamma 1

,and

,(m ~O/)

!, 0:25

(the maxim

um

value of ,for

an yannihilation

branc his 0.25),

we find

that the neutrino

signal

from the Sun

is at

full

strength

unless m ~O/?,

10 TeV.

On the other

hand, the

signal from the Earth

isp oten

tially

suppressed

for an ym

~O/?,

10 GeV.

The results

for the

comparison

of the

rate

for elastic

scattering

from Ge

with the flux

of up ward

muons

for scalar-coupled

WIMPs are sho wn

in Fig.

2

as afunction

of the

WIMP

mass. We consider

the sum

of muons

from neutrinos

from WIMP

annihilation

both in the

Earth

and in the

Sun.

The solid

(dashed)

curv es are

the ratios

(including

equilibration

prop erly)

for the

upp er (lo wer)

limit for ,(m

~O/).

The

upp er (lo wer)

pair of these

curv es are

for WIMPs

that

giv e\Gamma

= 10

\Gamma 4

(10

\Gamma 2 )m

\Gamma 2 yr

\Gamma 1

.In

the low er-limit

curv es, the

neutrino

signal

from the Earth

isessen

tially at full

strength

and iscomparable

to (for

m ~O/?,

80

GeV) or greater

than (for m ~O/!,

80 GeV)

the signal

from the Sun.

The mo deldep enden

tuncertain

ties are indicated

by the

range

of values

bet ween

the highest

5

FIG. 2. Direct

vs indirect

detection

of scalarand spin-coupled

WIMPs. For scalarcoupled WIMPs, we plot

the ratio

of the

rate

for elastic

scattering

from Ge in a

lab oratory

detector to the

flux

of up ward

muons

induced

by neutrinos

from annihilation in the

Sun

and Earth

as afunction

of WIMP

mass. The solid

(dashed)

curv es

are the ratios

for the

upp er (lo wer)

limit for ,(m

~O/),

the

neutrino

fluxes. The upp er

(lo wer)

pair of these

curv es are

for mo dels

that giv e\Gamma

= 10

\Gamma 4(10

\Gamma 2)

m

\Gamma 2yr

\Gamma 1,

and

the mo del

dep enden

tuncertain

ties are indicated

by the

range

of values

bet ween

the

highest and low est

curv

es. For scalar-coupled

WIMPs, the ratios

for detectors

with

differen tcomp osition can be obtained

using the scalings

plotted in Fig.

1. For

WIMPs

with axial-v

ector couplings

to nuclei,

we plot

ratios

of the

rate

for elastic

scattering

from hydrogen

in alab

oratory

detector to the

flux

of up ward

muons.

The upp er

(lo wer)

dotted

curv eis the ratio

for the

upp er (lo wer)

limit to the

neutrino

fluxes for

spin-coupled WIMPs, and the mo del-dep

enden tuncertain

ty isindicated

by the

range

of values

bet ween

these curv es. In both

cases,

we neglect

detector

thresholds

and

bac kgrounds

and assume

efficiencies

of order

unit y.

and low est curv

es in this

plot.

The ratios

for DD

with

other

nuclei can be found

using the scaling

in Fig.

1.

Our results

sho w that

for scalar

couplings,

the exp ected

rate in 1kg

of Ge

isroughly equiv alen tto that

in 10

5\Gamma

10

7m

2of muon

detector.

This isthe

main

conclusion of this

Letter.

To put

itin

persp ectiv e, we

should

briefly commen

t

on the

exp erimen

tal situation.

At least

one DD exp erimen

t[9] using

1kg Ge

6

and activ ebac kground

rejection tec hniques

[10] should

be op erational

within

the coming

year with arejection

factor of 99%.

It will

ha ve

abac

kground

of roughly

300 kg

\Gamma 1

yr

\Gamma 1

with

an efficiency

close to 100%

for large

enough

masses.

ySimilarly

the Dumand

II [11]

and AMAND

A [12]

collab

orations

are

installing within similar time scales

muon detectors

with area of the

order

of

10

4m

2. The

atmospheric-neutrino

bac kground

will be roughly

300 even ts per

year, coinciden

tally the same

num ber of bac

kground

even ts as

for

akg

of Ge.

We exp ect therefore

similar bac kground

rates for the

coming

generation

of DD

(1 kg of Ge)

and ID (10

4m

2) exp

erimen

ts. Ho wev

er, for scalar

coupling,

we

ha ve

just

sho wn

that

the rates

are exp ected

to be 10 to 1000

times

larger for a

1-kg Ge exp erimen

tthan for a10

4-m 2neutrino

detector, so the

sensitivit

yof

the DD exp erimen

tis muc hgreater

in this

case.

We no wturn

to WIMPs

with only aspin

coupling.

These WIMPs

are captured in the

Sun

via scattering

from hydrogen

(H), but they

are not captured

in the

Earth,

and they

ma yb edetected

directly through scattering

only from

nuclei with spin.

The spin coupling

to protons

differs from that to neutrons,

and the spin

in hea

vier

nuclei

isgenerally

carried at least

in part

by an unpaired

neutron [13]. Furthermore,

there ma yb ea

significan

tam biguit

yin the relation

bet ween

coupling

to neutrons

and protons

due to uncertain

ties in the

measured

spin con ten tof

the nucleon

[13][14]. Therefore,

amo del-indep

enden tcomparison bet ween

ID and

DD via scattering

from an arbitrary

nucleus cannot be

made. We can still,

ho wev

er, mak

ea mo del-indep

enden tcomparison

of rates

for

ID with

rates for DD

in adetector

made of H [15].

With

Eqs. (10) and (25) in

Ref. [7] and

Eq. (18) in Ref.

[6], we find,

[Direct (H )=Indirect]

= 1:1

\Theta 10

5m ~O/\Gamma

2[,

(m ~O/)S i(m

~O/;m H)]

\Gamma 1 (kg

\Gamma 1 =m

\Gamma 2 ):(5)

yW euse

here the Ge exp erimen

tonly

for the

purp

ose of illustrati

on. A similar comparison

can be made

for the

NaI

exp erimen

ts being

constructed

and for the

sapphire and LiF cry ogenic

detectors

being dev elop

ed.

7

These results,

for the

allo wed

range

of ,(m

~O/),

are

sho wn

in Fig.

2. Capture

of

spin-coupled WIMPs in the

Sun

occurs

only via scattering

from H. Therefore,

there are no uncertain

ties from

nuclear

ph ysics

or nucleon

spin structure

that

enter into this calculation.

Moreo ver, if aWIMP

has some

scalar

coupling

as well,

there will be additional

capture in the

Sun

and Earth

by scattering

from hea vier

elemen

ts. Therefore,

Eq. (5) giv es afairly

robust upp er bound

to the

DD/ID

ratio for spin-coupled

WIMPs. It app

ears

possible

to ha ve

in

the coming

years, ab out

50 gof

H in alo

w pressure

time-pro jection cham ber

[15] whic hshould

ha ve

negligible

bac kground.

The corresp

onding limit on the

rate would

be after

1y ear,

ab out

50 kg

\Gamma 1

yr

\Gamma 1

whic

his therefore

equiv alen tto

am uon-flux

limit of 0.005

(at low

mass)

to 0.25

(at large

masses)

m

\Gamma 2

yr

\Gamma 1

.

This has already

or will

soon

be reac

hed

by the

curren

tneutrino

exp erimen

ts.

The indirect

metho dhas therefore

aclear adv an tage

in this

case.

Let us assume

for the

momen

tthat the spin

coupling

to neutrons

is the

same as that

for protons.

There isno strong

reason to believ

ethis istrue

in a

particular mo del,

but this will allo w us

to scale

appro ximately

the dep endence

of the

rate

on the

nucleus.

Then, the scaling

of the

rate

for DD

of spin-coupled

WIMPs for other

nuclei

with mass m iis

roughly

fspin jc( m ~O/;m

i)m i=(

m ~O/+

m i)

2,

where fspin isthe

mass fraction

of the

giv en isotop

ein the detector.

The function

jC is the

DD form-factor

suppression giv en in Eq.

(19) in [6],

although

we

caution that the functional

form for the

spin

interaction

ma ydiffer

somewhat

[16]. The Ge exp erimen

tquoted

ab ove

intends

to use

500 gof isotopically

pure

73Ge. Assuming

the same

bac kgrounds

as before

the Ge exp erimen

tapp ears

roughly equiv alen tto

a10

4-m 2indirect

exp erimen

t. Ho wev

er, this

is highly

dep enden

ton details

of the

spin

con ten tof

the nucleon,

and usual

estimates

[13] lead

to asubstan

tial DD rate

reduction

compared to this.

To conclude,

we ha ve

found

that for scalar-coupled

WIMPs, the even

trate

for DD

in akg

of Ge

isroughly

the same

as the

rate

in aneutrino

telescop eof

area 10

5to

10

7m

2for

equal

exp osure

times.

For spin-coupled

WIMPs, the DD

even trates

in a50-g

H detector

isroughly

equiv alen tto that

in a10-

to 500-m

2

detector. We ha ve

also

sho wn

ho w

these

results

can be scaled

to DD

rates

8

for other

target

nuclei, although

the scaling

ma yb equite

mo del

dep enden

tfor

spin-coupled WIMPs. Taking into accoun

texp ected

exp erimen

tal bac kgrounds,

the forthcoming

1-kg Ge exp erimen

twith activ ebac kground

rejection should ha ve

an

adv

an tage

over the 10

4-m

2detectors

under construction

in case

of dominan

tscalar interactions. Ho wev

er, in the

case

of dominan

tspin interactions,

the ab ove

neutrino

exp erimen

ts should

ha ve

the

adv an tage:

the isotopically

enric hed

73Ge

exp erimen tis exp ected

to ha ve

only

comparable

sensitivit yat best to that

of the

ab ove

neutrino

detectors,

and the sensitivities

of 50-g

H detectors

will be even

low er.In

realistic

SUSY mo dels,

DD via scalar

interactions

tends to dominate

DD

via spin

interactions

by factors

of thousands

over muc hof parameter

space. On

the other

hand, the spin

interaction

pla ys am

uch more

imp ortan

trole in capture

of WIMPs

in the

Sun.

Therefore,

for ageneral

WIMP the ratio

of DD

versus

ID rates

will fall somewhere

bet ween

the results

for apurely

scalar-coupled

and

apurely spin-coupled

WIMP .Ho wev er, by explicit

evaluation

of the

DD/ID

ratio in sev eral

thousand

realistic SUSY mo dels,

we find

that in most

regions

of parameter

space, the DD/ID

ratio seems

to be

well

describ

ed by the

curv es

for purely

scalar-coupled

WIMPs sho wn

in Fig.

2. This

do es

not

mean

that

there are not regions

of SUSY

parameter

space where spin coupling

dominates,

but it is lik ely

that

ageneric

neutralino

will be primarily

scalar coupled,

and

so the

upp er curv

es in Fig.

2will

apply .It should

also be noted

that there

ma y

be other

(perhaps

non-SUSY)

WIMP candidates

(suc has

Ma jorana

neutrinos)

that ha ve

only

spin couplings.

It isa

pleasure

to thank

F. Halzen,

C. Tao,

and J. Engel

for useful

discussions. B.S. thanks

M. Run

yan for assistance.

M.K. ackno wledges

the hospitalit

y

of the

CERN

Theory Group where part of this

work

was completed.

This work

was supp

orted

in part

(B.S.

and K.G.)

by the

Cen ter for Particle

Astroph ysics,

aNSF Science

and Tec hnology

Cen ter op erated

by the

U. of California

under

Co op erativ

eAgreemen

tNo. .

M.K. was supp orted

by the

W.

M. Kec kF oundation

at the

I.A.S.

and by the

D.O.E.

at Colum

bia under

contract DEF G02-92-ER

40699. G.J. was supp orted

by the

D.O.E.

under con tract

9

DEF G02-85-ER

40231. K.G. was supp orted

in part

by aD.O.E.

OJI aw ard

and the Alfred

P. Sloan

Foundation.

References

[1] H. E. Hab

er and

G. L. Kane,

Ph ys.

Rep.

117 ,75

(1985).

[2] G. Jungman,

K. Griest,

and M. Kamionk

owski, in preparation

(1994).

[3] M. W. Go odman

and E. Witten,

Ph ys.

Rev.

D 31

,3059

(1985);

I.W asserman, Ph ys.

Rev.

D 33 ,2071

(1986).

[4] J. Silk,

K. Oliv

e, and

M. Srednic

ki, Ph ys.

Rev.

Lett. 55 ,257

(1985);

L. Krauss,

M. Srednic

ki, and

F. Wilczek,

Ph ys.

Rev.

D 33 ,2079

(1986);

K. Freese,

Ph ys.

Lett.

B 167

,295

(1986);

[5] J. Ric

hand

C. Tao,

priv ate comm

unication.

[6] K. Griest,

Ph ys.

Rev.

D 38 ,2357

(1988);

FERMILAB-Pub-89/139-A

(E).

[7] M. Kamionk

owski, Ph ys.

Rev.

D 44 ,3021

(1991).

[8] G. Jungman

and M. Kamionk

owski, IASSNS-HEP-94/45.

To app ear

in

Ph ys.

Rev.

D (1994).

[9] P. D.

Barnes

et al.,

J. Lo w T.

Ph ys.

93 ,79

(1993).

[10] T. Sh utt

et al.,

Ph ys.

Rev.

Lett. 69 ,3452

(1992).

[11] DUMAND

Collab oration:

K. K. Young

et al.,

UWSEA-PUB-93-16.

Contribution to Int.

Europh

ysics Conf.

on High

Energy

Ph ysics,

Marseille,

France, Jul 22-28,

1993.

[12] D. M.

Lo wder

et al.,

Nature

353 ,331

(1991).

[13] M. T. Ressell

et al.,

Ph ys.

Rev.

D 48 ,5519

(1993).

[14] M. Kamionk

owski, L. M.

Krauss,

and M. T. Ressell,

IASSNS-HEP-94/14,

unpublished. [15] K. N. Buc

kland,

M. J. Lehner,

G. E. Masek,

and M. Mo jav er, Ph ys.

Rev.

Lett. 73, 1067

(1994).

[16] J. Engel,

S. Pittel,

and P. Vogel,

Int. J. Mo

d. Ph ys.

E 1,

1(1992).

10

