

 24 Jan 95

SU-4240-591IASSNS-HEP-94/89CU-TP-654

Gamma Ra ys From

Neutralino

Annihilation

Gerard Jungman

a yand

Marc Kamionk

owski

b;c z

aDep artment

of Physics,

Syr acuse

University,

Syr acuse,

NY 13244.

bScho ol of Natur

al Scienc

es, Institute

for Advanc

ed Study,

Princ eton, NJ 08540

cDep artment

of Physics,

Columbia

University,

New York,

NY 10027

ABSTRA CT

We calculate

the flux

of cosmic

gamma rays exp ected

from the annihilation

of neutralinos

in the

Galactic

halo. Our calculation

of the

annihilation

cross section

to tw ophotons

impro ves the existing

calculations

by

inclusion of exact

one-lo op diagrams

for the

amplitudes

inv olving

Higgs boson and chargino

states as well

as

those inv olving

fermion

and sfermion

states. A surv

ey of sup

ersymmetric

parameter space sho ws that

numerous

mo dels

would

be observ

able at the

3 oelev

el with

an air Cerenk

ov telescop

ewith an exp

osure

of 10

4m

2yr .

25 Octob

er 1994

y jungman@npac.syr.edu z kamion@ph

ys.colum bia.edu

1. INTR

ODUCTION For quite

some time, ithas been kno wn that

only ab out

aten

th of the

mass

of most

galaxies,

including our own,

isluminous,

and that

the rest

iscomp

osed

of some

sort of dark

matter

[1]. The nature

of this

nonluminous

material is

unkno wn, although

there are quite

con vincing

argumen ts that

it must

be nonbary onic.

One of the

most

promising

of man

ycandidates

for the

dark

matter

isthe neutralino

[2][3], alinear

com bination

of the

sup ersymmetric

partners of

the photon,

Z

0,

and

Higgs

bosons.

It has

been

suggested

by numerous

authors

that ifneutralinos

populate the Galactic

halo, then mono energetic

gamma rays

pro duced

by neutralino

annihilation

in the

halo

could

pro vide

aplausible

aven ue

tow ard

disco

very of suc

hdark-matter

particles [4]. In this

pap er we

re-examine

this prop osal. The goal of this

work

is to

pro

vide

results

for the

cross

section

for annihilation of neutralinos

to tw ophotons

whic hinclude

all of the

con tributions

at one

loop

to the

amplitude

for aneutralino

in an ygiv

en minimal

sup ersymmetric extension

of the

standard

mo del.

We generalize

previous calculations

of

the amplitude

for annihilation

through quark-squark

loops [5][6][7]

and Higgschargino loops [5] to arbitrary

neutralino

and squark

masses and comp

ositions.

We also

include

the recen

tappro

ximate calculation

of Bergstrom

and Kaplan

[8]

of annihilation

through W

\Sigma -b

oson-c

hargino

loops and impro

ve itb

yincluding

subleading logarithmic terms.

We then

use these

expressions

in asurv

ey of sup

ersymmetric

parameter

space to assess

the possibilit

yof disco vering

dark-matter

neutralinos in the

Galaxy via observ

ation of cosmic

gamma rays pro duced

by neutralino

annihilation. In the

follo wing,

we will

presen

tour result

for the

cross

section,

discussing

the imp ortance

of the

various

con tributions.

Then we will

giv ean

estimate

of

the signal

rates whic hare

implied.

1

Ghost Diagrams +

A Z W

f f Z A

W

~

~

c

c

H

H +-

+-

+-

+- f

f

f f

c c

c c

+- +-

+- +- +-

+-

a)b)c) FIG. 1. Feynman

diagrams for neutralino

annihilation

to photons.

2. CR

OSS

SECTION

The Feynman

diagrams for annihilation

to tw ophotons

are sho wn in Fig.

1.

The diagrams

fall into

three

categories.

The similarit

yb etw een

the diagrams

of

classes a) and

b) indicates

that the corresp

onding amplitudes

can be written

in

terms of the

same

basic functions

arising from the loop

integrations.

Furthermore, these functions

are precisely

those whic happ

ear in the

calculation

of the

cross section

for neutralino

annihilation

to tw ogluons

presen ted in Ref.

[9]. The

third class of diagrams

are those

with W bosons

in the

loop,

and those

ghost

diagrams whic hare

related

to them;

this gauge

inv arian

tset of diagrams

has

been discussed

in Ref.

[8]. By ac hoice

of non-linear

gauge, the calculation

was

reduced significan tly .In

the limit

m O/

\Sigma ?,

m O/

0(where

m O/

\Sigma and

m O/

0are

masses

of the

chargino

O/

\Sigma

and

neutralino

O/

0,

resp

ectiv

ely), whic his alw ays

appropriate when

considering

the neutralino

as the

ligh test

sup ersymmetric

particle, the

amplitude reduces to asingle

three-p oin tin

tegral,

as giv en in Ref.

[8].

2

Neutralinos in the

halo

mo ve with

velo cities

negligible

compared with the

speed of ligh

t,so

annihilation

occurs in the

sw ave

only .Therefore,

the amplitudes dep end

only

on the

outgoing

photon momen ta and

polarizations,

and the

amplitudes can be written

in the

form

A =

e24ss 2ffl(

k1; k2; ffl1; ffl2)

~A;

(2: 1)

where ki and

ffli are

the momen

ta and

polarizations

of the

outgoing

photon pair.

The total

amplitude

will be asum

of three

parts to be discussed

in turn

belo w,

A =A

f

~f+A

HC

+A

W.

Giv en this

amplitude,

the cross

section

is

oefl flv

=

ff

2m

2O/

16 ss

3j

~Aj

2:

(2: 2)

First, consider

the amplitudes

related to the

tw ogluon

amplitudes,

i.e.,

those coming

from fermion-sfermion

loop diagrams

sho wn

in class

a) in Fig.

1.

Define the follo wing

functions,

arising from the loop

integrals:

FI (a;

b;S;

D) =

1 1+ a\Gamma b `

12 'h

Sb +D

pab i

;

(2: 3)

T( c;^

A;

^Z

)=

2 p c

^A gAO/O/

4\Gamma m

2O/=m

2A +

c m

2O/m2Z ^ Zg ZO/O/

;

(2: 4)

where m A

and

m Z

are

the masses

of the

pseudoscalar-Higgs

and Z bosons,

gAO/O/ and gZ O/O/

are

the couplings

of neutralinos

to the

pseudoscalar-Higgs

and

Z bosons

[9], and

F( a;b;

S; D)

=\Gamma

12 Z

10 dx (

Sx ln fifififi x2 a+

x(b\Gamma

1\Gamma a) +1

\Gamma x

2a +x

(b\Gamma

1+ a) +1 fifififi

+

Sb

+D

pab

1+ a\Gamma b `

11\Gamma x +

11+x '

ln fifififi

x

2a

+x

(b\Gamma

a\Gamma 1) +1

b+ a(1\Gamma

x2 ) fifififi

+

1 1\Gamma b+ xa "

Sb `

1x +

11\Gamma x ' +D

pab1\Gamma x #

ln fifififi

b x

2a\Gamma

x(a +b\Gamma

1)\Gamma 1 fifififi

+

1 b\Gamma 1+ ax "

Sb `

1x \Gamma

11+x ' \Gamma D

pabx+ 1 #

ln fifififi

b x

2a

+x

(b\Gamma

1\Gamma a) +1 fifififi )

: (2: 5)

3

Note that this expression

for F( a;b;

S; D)

corrects

at yp ographical

error in

Ref. [9].The amplitude

in terms

of these

functions

is

Re

~Af ~f= X

f

cfQ

2f ( 2I i

m

2fm2O/ j

1m2O/ T i

m

2fm2O/ ;

gAf

f;

g cos

`W j

+ X ~f

1m2O/ F i

m

2O/m2~f ;

m

2fm2~f ;

Sf

~f;D

f

~f j)

;

Im

~Af ~f=\Gamma

ss X f

cfQ

2f` (m

2O/\Gamma

m

2f)

ln

1+

fif 1\Gamma fif (

\Gamma T i

m

2fm2O/ ;

gAf

f;

g cos

`W j

+

1m2O/ X

~f F

I i

m

2O/m2~f ;

m

2fm2~f ;

Sf

~f;D

f

~f j)

;

(2: 6)

where the sum

on fis

over quarks

and leptons,

and the sum

on

~fis over

the

squarks and sleptons.

Here, gAf fare

the pseudoscalar-Higgs-fermion

couplings

[9], fif = (1\Gamma

m

2f=m

2O/) 1= 2,Q

fis

the electric

charge of f, cf isa

color

factor

whic h

equals 3for quarks

and 1for

leptons,

and `(x )is

the Hea viside

step function.

The function

I(x )is

giv en

in Eq.

(2.14)

of Ref.

[9]. The couplings

of the

fermions and sfermions

are defined

by Sf ~f=

A

2f ~f+

B

2f ~fand

D f

~f=

A

2f ~f\Gamma

B

2f ~f,

with A f

~f=

12 i X

0f~f 0+

W

0f~f 0 j

;

B f

~f=

12 i X

0f~f 0\Gamma

W

0f~f 0 j

:

(2: 7)

The fundamen

tal couplings

X

0and

W

0are

the couplings

of left-handed

and

righ t-handed

fermions, resp ectiv

ely ,to

sfermions

and neutralinos,

as defined

in

Refs. [9] and

[10].

Next consider

the amplitude

from diagrams

inv olving

intermediate

chargedHiggs bosons

H

\Sigma and

charginos,

i.e., those

in class

b). We find

Re

~AH C= X

O/

\Sigma m (

2I i

m

2O/ \Sigma m

m

2O/ j

T i

m

2O/ \Sigma m

m

2O/ ;

hAO/

\Sigma mO/ \Sigma m; ffim j

+F i

m

2O/

m

2H \Sigma ;

m

2O/ \Sigma m

m

2H \Sigma ;

Sm

;D m j)

;

(2: 8)

4

where the sum

iso ver

the tw oc

harginos.

Here ffim = (g=

cos `W )(O

0Lmm\Gamma

O

0Rmm

)

is the

coupling

of the

Z to ac hargino

pair, and hAO/

\Sigma mO/ \Sigma m = g(sin

fiQ mm

+

cos fiS mm

)is

the coupling

of the

A boson

to ac hargino

pair. The couplings

of

the charginos

to neutralinos

are defined

by

Sm =

12 h\Gamma

Q

0L0m \Delta

2+\Gamma

Q

0R0m \Delta

2i ;

D m

= Q

0L0m

Q

0R0m

:

(2: 9)

Definitions and more

details

of the

couplings

O

0and

Q

0can

be found

in Refs.

[2] and

[10]. Next

consider

the amplitude

from diagrams

with intermediate

W bosons,

i.e., those

in class

c). In Ref.

[8], the imaginary

part of this

amplitude

was

calculated exactly ,Im~A W

= ss X O/

\Sigma m `

(m

2O/\Gamma

m

2W)

Cm

fi

2W

ln

1+

fiW 1\Gamma fiW

;

(2: 10)

where fiW = (1\Gamma

m

2W=m

2O/) 1= 2.

The

coupling

is giv

en by

Cm =

g22 p2

4m2O/ \Sigma m h\Gamma

O

L0m \Delta

2+\Gamma

O

R0m \Delta

2i ;

(2: 11)

where O

L0m

and

O

R0m

are

giv en in Refs.

[2] and

[10].

The real part

of the

amplitude was evaluated

in aleading-logarithm

limit in Ref.

[8], using

disp ersion

relations, giving aresult

prop ortional

to ln

2(a

). As

poin

ted out there,

it is an

excellen tappro ximation

to con

tract

the chargino

propagator

to ap

oin t, corresp onding

to the

limit

m O/

\Sigma ?,

m O/.

Ev aluating

the real

part

of the

subsequen

t

three-p oin tamplitude,

we find

Re

~AW =\Gamma

2 X O/

+m C

m (

\Gamma 34 \Gamma

12 ln

m

2W4m2O/

+

12 ln

4m

2O/_2+

14 B 1(

m

2W4m2O/

)\Gamma

B 2(

m

2W4m2O/

): )

;

(2: 12)

where B 1(a

)= Z

10 dxx ^ 1+

a\Gamma x(1\Gamma

x) x

ln fifififi

a\Gamma

x(1\Gamma

x) a

fifififi * ;

B 2(a

)= Z

10 dxx [a +x

(1\Gamma

x)] ln fifififi

a\Gamma

x(1\Gamma

x) a

fifififi;

(2: 13)

5

and _is the renormalization

poin t;note

that _app

ears because

of the

con traction of the

chargino

propagator.

We cho ose

_=

m W,

specifying

the running

coupling at the

electro

weak scale; this giv es asubleading

logarithmic con tribution to the

amplitude.

It is very

useful

to ha ve

an asymptotic

expansion

for this

result,

extending

the leading-logarithm

expansion of Ref.

[8] into

the

subleading terms. We find B 1(a

), ae

\Gamma

12ln

2a +1 +

ss

23

a!

0,

1+

16a

a! 1,

B 2(a

), ae

\Gamma

12ln

a\Gamma 1 a!

0,

\Gamma

12\Gamma

18a

a! 1.

(2: 14)

The leading

logarithmic

beha viour

of this

amplitude

agrees with that obtained

in Ref.

[8]. It should

be poin

ted out that

these

expressions

are not strictly

valid

for m O/!

m W.

Ho wev

er, since

we are

interested

in high

energy

gamma rays,

this case isnot

of interest

to us;

we henceforth

assume that m O/?

m W.

No w we

can

consider

the beha

viours

of eac

hof

these

terms.

The size of

the fermion/sfermion

loop con tribution

is sensitiv

eto the gaugino

fraction of

the neutralino

and to the

masses

of the

sfermions;

asufficien tly Higgsino-lik

e

comp osition

of the

neutralino

will suppress

these con tributions.

As poin

ted out

by Bergstrom

and Kaplan

[8], annihilation

through loops con taining

W bosons

[class c)] ma yb esignifican

t.

The Higgs-c

hargino

loop con tribution

can be imp

ortan

tfor sev eral

reasons.

First note that the couplings

whic happ

ear in these

diagrams

are all naturally

of the

order

of the

gauge

couplings,

so that

there

is no

special

suppression,

as can

sometimes

occur in couplings

of neutralinos

to fermions

and sfermions.

Furthermore, the masses

of the

intermediate

particles are comparable,

and this

matc hing of scales

pro vides

some enhancemen

tof the loop

integral.

In fact,

it is imp

ortan

tto note that the amplitude

as written

con tains

ap ole at the

poin tb = 1+

a, or m

2O/+

m

2H \Sigma

=

m

2O/ \Sigma .

This

div ergence

occurs because

we

ha ve

ignored

the widths

of the

intermediate

particles. In actualit

ythese widths

are large;

for chargino

masses greater than 100 GeV

the width

of the

chargino

is appro

ximately

\Gamma O/

\Sigma '

0:1 m O/

\Sigma .

So the

pole

is spurious,

and the amplitude

6

must be mollified

in the

region

b' 1+ a. Note

that previous

treatmen ts of this

con tribution

[5] will

not be reliable

in the

common

case that the masses

of the

particles are comparable.

The W -lo op

con tribution

is almost

alw ays

imp ortan

tin the case

that the

neutralino is hea

vier

than

the W and

primarily

Higgsino [8]. This

is because

this con tribution

dep ends

most strongly

on the

ligh test-c

hargino

mass and this

isusually not too muc hlarger

than the neutralino

mass.

3. SIGNAL

RA TES

We follo w Urban

et al.

[11]

and consider

the signal

from poin ted observation of the

Galactic

cen ter

with

an atmospheric

Cerenk ov telescop

e(A CT).

This is poten

tially apromising

metho dfor observ

ation of high-energy

gamma

rays from

neutralino

annihilation.

The flux of gamma

rays from

neutralino

annihilation,

from awindo

w of

solid angle

\Delta \Omega aimed

at the

Galactic

cen ter,

ma yb ewritten

[4],

OEfl' 2\Theta 10

\Gamma 11

cm

\Gamma 2sec

\Gamma 1(ae

0: 4O/)

2f

halo

[(oe flflv

)=10

\Gamma 30

cm

3sec

\Gamma 1]

(m O/=

10

GeV

)2

(\Delta \Omega =sr );(3:1)

where ae0

:4O/is

the local

halo densit

yin units

of 0:4

GeV

cm

\Gamma 3,and

0:3

!, fhalo

!, 2

is aparameter

whic hreflects

uncertain ty in mo deling

the galactic

halo. Eq.

(3.1) is obtained

assuming an isothermal

halo with adensit

yprofile

suitable

for accoun

ting for the

observ

ed rotation

curv e. If the

cosmological

neutralino

densit yis too small

to accoun

tfor the halo

dark matter,

neutralinos

should still

gather in galactic

halos, and the annihilation

cross section

in this

case should

generally be larger

than that in the

case

where

neutralinos

are the dark

matter.

Therefore, there ma yb ean

observ

able gamma-ra

ysignature

of neutralinos

even

ifthey exist but are too few to accoun

tfor all the

halo

dark matter.

To accoun

t

for this,

we tak ethe

halo densit

y(for \Omega O/h

2!,

1as

required

by the

age-of-theUniv erse constrain

t) to

be \Omega O/h

2=0

:25,

where

\Omega O/is

the cosmological

neutralino

7

Required Exposure (m2 - sec) mc (G

eV)

100 200 300 400 500

10

6 108

10

10 1012 1014

FIG. 2. Minim

um exp osure

required

for a3 oe detection

of gamma

rays from

neutralino annihilation

in the

Galactic

cen ter,

versus

mass of the

neutralino,

for the

surv ey of sup

ersymmetric

parameter space discussed

in the

text.

The sym bols

indicate whic hof the three

amplitude

con tributions

dominates the cross

section;

triangles

indicate that the fermion-sfermion

diagrams dominate, diamonds indicate that the W

diagrams dominate, and circles

indicate

that the Higgs-c

hargino

diagrams

dominate.

abundance, and his the Hubble

parameter

in units

of 100

km

sec

\Gamma 1Mp

c\Gamma

1.

We note

that our results

will not dep end

sensitiv

ely on this

prescription.

The natural

width of the

tw o-photon

peak is small,

and the bac kground

belo w the

peak

is con

trolled

by the

instrumen

tal resolution.

The most

imp ortan tbac

kground

for energies

belo w 1T

eV comes

from misiden

tified charged

particles [11]; the bac kground

from diffuse

cosmic gamma rays [6] con tributes

only sligh tly to the

total

bac kground

in this

regime.

Follo wing

Ref. [11], we

8

find abac

kground

flux of

OEb = (1:

2\Theta

10

\Gamma 5sec

\Gamma 1m

\Gamma 2)

\Delta E 100

GeV ^

\Delta \Omega 10\Gamma 3sr *h

m O/

100 GeVi

\Gamma p ;

(3: 2)

where \Delta E isthe

energy

resolution

on the

peak,

and p' 3:3 isthe

spectral

index

for the

bac kground,

in this

case dominated

by misiden

tified electrons

[11].

In Fig.

2w eplot

the exp osure

required

for a3 oedetection

of gamma

rays

from neutralino

annihilation

in the

Galactic

cen ter for av ariet

yof mo dels.

The

parameter ranges whic hgenerated

these mo dels

were

tak en to be

100

GeV

!

M 2!

800 GeV ,200

GeV ! _!

800 GeV ,2 ! tan

fi!

20, 300

GeV

! m A!

600 GeV ,and

200 GeV

! m ~q!

800 GeV . Here

M 2and

_ are

the gaugino

mass parameters,

m Ais

the mass

of the

pseudoscalar

Higgs particle,

and m ~qis

acommon squark mass parameter

(whic hdiffers

from the actual

squark masses

due to mixing

terms, whic hw ere included).

The grand

unification

condition

was assumed.

Mo dels

were

cut from

the plot

ifthey

violated

kno wn bounds

from e+ e\Gamma

ph ysics,

ifthey

gav eHiggs

masses in volation

of curren

tlimits,

or if

they were inconsisten

tas mo dels

for neutralino

dark matter

(for example,

we

ob viously

require that the ligh test

sup ersymmetric

particle isa neutralino).

Triangles, circles, and diamonds

indicate mo dels

where

the fermionsfermion, Higgs-c hargino,

and W -boson

diagrams,

resp ectiv

ely ,dominate

the

amplitude. According to Fig.

2, the

fermion-sfermion

diagrams are most

often

imp ortan

t, but

there

are indeed

regions of parameter

space where the Higgschargino and W loops

are dominan

t. These

results indicate

that numerous

sup ersymmetric

mo dels

could

be prob

ed by an atmospheric

Cerenk ov detector

with an area

O(10

4m 2).

4. A CKNO

WLEDGMENTS G.J. thanks

J. Buc

kley

for con versations

ab out

high-energy

gamma rays.

M.K. ackno wledges

the hospitalit

yof the Theory

Group at CERN

where part of

this work

was completed.

M.K. was supp

orted

at the

I.A.S.

by the

W. M. Kec

k

Foundation and at Colum

bia Univ

ersit yb ythe

U.S. Departmen

tof Energy

under con tract

DE-F G02-92ER40699.

G.J. was supp orted

by the

U.S.

D.O.E.

under con tract

DE-F G02-85ER40231.

9

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