

 21 Jul 94

IASSNS-HEP-94/45SU-HEP-4240-584

Neutrinos From Particle

Deca yin the Sun

and Earth

Gerard Jungman

ay and

Marc

Kamionk

owski

bz

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

ABSTRA CT

Weakly interacting

massiv eparticles

(WIMPs) ma yb eindirectly

detected by observ

ation of up ward

muons

induced by energetic

neutrinos from annihilation

of WIMPs

that ha ve

accum

ulated in the

Sun

and/or

Earth.

Energetic muon neutrinos

come from the deca

ys of o/leptons,

c, b,and

tquarks,

gauge bosons,

and Higgs

bosons

pro duced

by WIMP

annihilation.

We pro vide

analytic

expressions,

suitable for computing

the flux

of up ward

muons, for the

neutrino

energy spectra from deca ys of all

these

particles

in the

cen ter

of the

Sun

and Earth.

These analytic

expressions

should ob viate

the need

for Mon

te Carlo

calculations

of the

up ward-m

uon flux.

We

inv estigate

the effects

of polarization

of the

gauge

bosons

on the

neutrino

spectra, and find that they are small.

We also

presen

tsimple

expressions

for the

second

momen ts of the

neutrino

distributions

whic hcan

be used

to

estimate the rates

for observ

ation of neutrino-induced

muons from WIMP

annihilation.

June 1994 y jungman@npac.syr.edu z kamion@guinness.ias.edu

Address after Jan uary

1, 1995:

Dept. of Ph ysics,

Colum

bia Univ

ersit y, New

York,

NY 10027

1. In tro

duction

Weakly-in

teracting massiv eparticles

(WIMPs) with masses

in the

GeVTeV range

are among

the leading

candidates

for the

dark

matter

in our

Galactic

halo [1]. One

of the

most

promising

metho ds for

disco

very of WIMPs

in the

halo is via

observ

ation of energetic

neutrinos from annihilation

of WIMPs

in

the Sun

and/or

Earth [2][3][4]

[5] .WIMPs

in the

halo

can be captured

in the

Sun and the Earth

by elastic

scattering

from nuclei

therein.

WIMPs that ha ve

accum ulated

in the

Sun

or Earth

will annihilate

into ordinary

particles, suc h

as quarks,

leptons, and ifhea

vy enough,

gauge bosons,

Higgs bosons,

and top

quarks. Man yof

the leading

WIMP candidates--suc

h as

the

neutralino

in

sup ersymmetric

theories--are Ma jorana

particles,

so they

do not

annihilate

directly into neutrinos

[6]. Neutrinos

with energies

in the

multi-GeV

range are

then pro duced

by deca

yof

the annihilation

pro ducts,

and suc hneutrinos

are

poten tially observ able in man

yastroph

ysical neutrino

detectors.

The differen

tial energy

flux of high-energy

neutrinos of typ

ei

(e.g.,

i=

*_ ,_* _,

etc.)

from WIMP

annihilation

in the

Sun

(Earth)

is

`dOEdE ' i =

\Gamma A4ssR 2 X F

B F `

dNdE '

F;i

;

(1: 1)

where \Gamma A is the

rate

of WIMP

annihilation

in the

Sun

(Earth),

and R is the

distance of the

Earth

from the Sun

(or the radius

of the

Earth).

The sum is

over all annihilation

channels F (e.g.,

pairs of gauge

or Higgs

bosons

or fermionan tifermion

pairs), B Fis

the annihilation

branc hfor channel

F, and

(dN =dE )F ;i

isthe differen

tial energy

spectrum

of neutrino

typ ei at the

surfac

eof the Sun

exp ected

from injection

of the

particles

in channel

F in the

cor eof

the Sun.

The

total num ber of neutrinos

of typ

ei from

final state

F isN

F;i

=R

(dN =dE )F ;idE

.

The spectrum

(dN =dE )F ;iis

afunction

of the

energy

of the

neutrino

and of the

energy of the

injected

particles.

As we poin

tout

in this

pap er, the

(dN =dE )F ;i

ma yalso

dep end

on the

polarization

of the

annihilation

pro ducts.

This pap er con

tin ues

the work

of Gaisser,

Steigman,

and Tila v[4]

and

Ritz and Sec kel

[3].

The dN =dE

from the o/-lepton

and band

c-quark

final

1

states were first computed

analytically

using the spectator-quark

appro ximation in Ref.

[4], but stopping

of hea

vy hadrons

and stopping

and absorption

of neutrinos

in the

Sun

were

neglected.

A Mon

te Carlo

calculation

whic hincluded interactions

with the solar

medium

was performed

by Ritz

and Sec kel

[3] (hereafter

RS). They

also considered

deca ys of the

top quark,

but those

results, obtained

assuming atop-quark

mass of 60

GeV,

are no longer

valid.

Some of the

recen

texp erimen

tal collab

orations

ha ve

follo

wed

the example

of

RS and

used

Mon te Carlo

sim ulations

to obtain

the neutrino

energy spectra

[7][8]. Ho wev

er, Mon

te Carlo

calculations

can be cum

bersome,

esp ecially

when

numerous WIMP candidates

are to be

explored;

analytic expressions

are muc h

more con venien

t. It is also

easier

to isolate

the various

interaction

effects and

theoretical assumptions

that enter

the calculation

with analytic

expressions.

In

this pap er, we deriv

eanalytic

expressions

for the

neutrino

energy distributions

from o/-lepton,

c-, b-, and

t-quark,

and gaugeand Higgs-b

oson deca yin the Sun

and Earth

that can be used

in place

of Mon

te Carlo

results.

Our results

for band

cquarks

and o/leptons

are deriv

ed from

the results

of [3]

and

[4]. Our

results

for dN =dE

from gauge-b

oson and Higgs-b

oson final

states follo wthe

work of Refs.

[9] and

[5], but our discussion

of the

top-quark

final state

is entirely

new. If the

top quark

is hea

vier

than

the W

\Sigma boson,

then it deca

ys almost

exclusiv ely into

W bosons

and bquarks;

therefore, the

calculation of the

neutrino

spectrum must be redone.

Also note that in man

y

cases where

the neutralino

ishea vier than

the top quark,

neutralinos

annihilate

predominan tly to the

t_t final

state [10][11][12]

.Therefore,

evaluation of the

neutrino spectrum for the

t_t final

state is crucial

to calculation

of energeticneutrino even trates.

Of all our

results,

the neutrino

spectra from injected

band cquarks

are

sub ject

to the

largest

theoretical

uncertain ty. In particular,

hea vy hadrons

will be slo wed

before

they deca yas

they pass through

the Sun

[3], and

little

is kno

wn

ab out

interactions

of hea

vy hadrons

with asurrounding

medium.

Ho wev

er, this

uncertain

ty enters

the Mon

te Carlo

calculations

as well,

so little

accuracy is lost

by using

analytic

results instead.

In addition,

the effect

of

hea vy-quark

stopping becomes small at low

injection

energies, so the

resulting

2

uncertain ty issmall

at low

energies.

Once the neutralino

mass has become

large

enough that the top-quark

annihilation

channel op ens

up, the branc

hing ratio

for neutralino

annihilation

into bquarks

becomes negligible.

So although

the

b-quark neutrino

spectrum ma yb equite

uncertain

at high

injection

energies,

the resulting

uncertain ty in the

total

neutrino

even t-rate

remains

small. Note

that in all

cases,

the errors

in the

up ward-m

uon flux from

the neutrino

spectra

are dw arfed

by underlying

uncertain ty in the

local

halo densit

y.

When the WIMP

is hea

vy enough

that it annihilates

primarily to vector

bosons and/or top quarks,

then the theoretical

calculation simplifies greatly .

These particles

(as well

as o/leptons)

do not

hadronize

and are not slo wed

before they deca y, so

relativ

ely simple

analytic

expressions

are obtained

for the

neutrino spectra from the Earth.

The only poin tat whic hthe

analysis

becomes

somewhat complicated

is when

considering

the stopping

of the

bquark

from

top quark

deca yin

the Sun.

Ho wev

er, this

b-quark

con tribution

pro vides

a

small fraction

of the

total

neutrino

signal from top deca y, so

the

uncertain

ty

intro duced

from mo deling

of the

hadronic

stopping effect is negligible.

In the

end, we arriv

eat expressions

for dN =dE

from

the Earth

whic hgiv

eresults

for

the up ward-m

uon flux estimated

to be

accurate

to afew

percen tand results

for

dN =dE

from

the Sun

whic hgiv

eresults

accurate to within

5%.

We also

calculate

the neutrino

spectra from polarized

gauge bosons.

In Ref.

[5] itw

as assumed

that the gauge

bosons

pro duced

by neutralino

annihilation

were unp olarized.

In fact,

they are indeed

polarized

[13]. In this

work,

we

quan tify the effect

of polarization

of gauge

bosons

from neutralino

annihilation

on the

up ward-m

uon flux from

deca ys of these

particles

and sho w that

itleads

to no

more

than a10%

correction

to the

results

of Ref.

[5].

In most

neutrino

detectors,

the existence

of the

neutrino

isinferred

by observ ation

of an

up ward

muon

pro duced

by ac harged-curren

tin teraction

in the

roc kb

elo w the

detector.

The cross

section

for pro duction

of am

uon

is proportional to the

neutrino

energy ,and the range

of the

muon

in roc

kis

roughly

prop ortional

to the

muon

energy .Therefore,

the rate

for observ

ation of energetic neutrinos

isessen tially prop ortional

to the

second

momen tof the neutrino

energy spectrum.

Inserting the numerical

values for the

charged-curren

tcross

3

section and the effectiv

erange

of the

muon,

and ignoring

detector thresholds,

the rate

per unit

detector

area for neutrino-induced

throughgoing-m uon even ts

ma yb ewritten

[3][5]

\Gamma detect = (1:

27 \Theta 10

\Gamma 29 m

\Gamma 2

yr

\Gamma 1

)

C 1sec

\Gamma 1 i

m ~O/

1 GeVj

2X i

aib

i X F

B F \Omega

Nz

2ff F;i

;

(1: 2)

for neutrinos

from the Sun;

the same

expression

multipli ed by 5:6 \Theta 10

8(the

square of the

ratio

of the

Earth-Sun

distance to the

Earth's

radius) giv es the

rate for neutrino

even ts from

the Earth.

Here, C isthe

WIMP

capture rate in

the Sun

or Earth,

B F

is the

branc

hing fraction

for WIMP

annihilation

to the

final state

F, the

sum

on iis

over

muon

neutrinos

and muon

an ti-neutrinos,

the ai are

neutrino

scattering

coefficien ts, a* = 6:8

and

a_* = 3:1,

and the bi are

muon range coefficien

ts, b* = 0:51

and b_* = 0:67.

The quan tit y

\Omega N z2ff F;i

(E in)

j

1E2in Z

`dNdE ' F;i (

E *;E

in)

E

2*dE

*;

(1: 3)

is the

second

momen tof the spectrum

of neutrino

typ ei

from

final state F

scaled by the

square

of the

injection

energy E iof

the annihilation

pro ducts.

The quan tit yz

= E *=E

in

isthe

neutrino

energy scaled by the

injection

energy .

Strictly speaking,

neutrino telescop es observ

eneutrinos

only with energies ab ove

agiv

en threshold.

Therefore, the even trate

is prop

ortional

to

the con tribution

to the

second

momen tfrom neutrinos

with energies

ab ove

threshold--that is, there

is alo

wer

bound

to the

integral

in Eq.

(1.3).

To

obtain the most

accurate

exp erimen

tal information,

adetailed calculation of

the neutrino

spectra should be folded

in with

the detector

resp onse.

For example, an energetic-neutrino

signal from the Sun

must

be distinguishable

from

the atmospheric-neutrino

bac kground.

The direction

of the

neutrino-induced

muon is correlated

with the paren

t-neutrino

direction only within

an angular

windo w of roughly

pGeV =E *radians.

Therefore, aprop er determination

of

the signal-to-noise

ratio requires

kno wledge

of the

neutrino

energy spectrum.

On the other

hand, in cases

where

the WIMP

is quite

massiv

e, most

of

the neutrinos

ha ve

energies

large enough

to pro

duce

am uon

with

an energy

4

muc hhigher

than the detector

threshold.

For example,

ifa 100-GeV

WIMP

annihilates to gauge

bosons,

the typical

neutrino

energy isroughly

half that,

and

the typical

muon energy

typically

half that,

ab out

25 GeV.

Typical

thresholds

for curren

tand next-generation

detectors are no more

than10

GeV, so the

vast

ma jorit

yof the neutrinos

in this

example

are ab ove

threshold.

In addition,

even

ifa non-negligible

fraction of the

neutrinos

ha ve

energies

belo w the

detector

threshold, the total

second

momen tof the energy

distribution

isstill primarily

determined by the

higher-energy

neutrinos. The con tribution

of neutrinos

with

energies belo wthreshold

to the

up ward-m

uon flux will be O( E

3thresh

=m ~O/3)

where

E thresh

is the

threshold

energy . Therefore,

in cases

where

the WIMP

mass

is significan

tly larger

than the detector

threshold,

the expression

for the

rate

for neutrino-induced

up ward

muons,

Eq. (1.2),

together

with the results

for

\Omega N z2ff

presen

ted here

will pro vide

ago od theoretical

estimate to compare

with

exp erimen

tal determinations

of the

flux

of up ward

muons

from the Sun

and/or

Earth. The most

promising

metho dof detection

of the

energetic

neutrinos,

especially for higher-mass

WIMPs, is the

up ward-m

uon signal.

Therefore,

we

calculate the energy

spectrum

of muon

neutrinos

and an tineutrinos

only .

In the

next

Section,

we discuss

some general

results on neutrino

spectra.

We relate

the neutrino

spectrum from deca yof amo

ving particle

to the

neutrino

spectrum from deca yof

the particle

at rest,

and we sho w ho w

the

neutrino

spectra from particle

deca ys in the

Sun

are obtained

from those

from deca y

in the

Earth.

We discuss

ho w hadronization

and stopping

of band

cquarks

affect the neutrino

spectra. We also

discuss

ho w neutrino

spectra from Higgs

bosons are obtained

from the neutrino

spectra from the Higgs-b

oson deca y

pro ducts.

Muc hof

this isreview,

but useful

for the

discussion

in the

Sections

that follo w. In Section

3, we

calculate

the neutrino

spectra from deca ys of b

and cquarks

and o/leptons.

In Section

4, we

discuss

the neutrino

spectra from

deca ys of gauge

bosons

and quan tify the effect

of gauge-b

oson polarization

on

the neutrino

spectra. In Section

5, we

calculate

the neutrino

spectra from

deca ys of top

quarks,

and we summarize

our results

in Section

6.

5

2. Preliminar

ies For all of the

particles

considered,

calculation of the

neutrino

distribution

in the

rest

frame

of the

deca

ying

particle,

(dN =dE )rest (E *),

as afunction

of the

neutrino energy E *,

isa

standard

exercise. Once the rest-frame

distribution

is

kno wn,

the energy

distribution

of aparticle

mo ving

with an energy

E d,v

elo cit y

fi, and

fl= (1 \Gamma fi2 )\Gamma

1= 2=

E d=m

d(where

m dis

the deca

ying-particle

mass) is

related to the

rest-frame

distribution

by

`dNdE ' \Phi (

E d;E

*)

=

12 Z

E +(

E)

E \Gamma (

E)

dfflffl 1flfi `

dNdE '

rest (ffl) ;

(2: 1)

where E \Sigma (

E)

=

E *

fl(1 \Upsilon fi)

:

(2: 2)

This assumes

that there

isno prefered

direction in the

rest

frame

of the

paren

t

particle. In particular,

itapplies to deca

yof scalar

or unp

olarized

particles.

If

the deca ying

particle

is polarized

and the neutrino

direction iscorrelated

with

the spin

of the

deca ying

particle,

Eq. (2.1)

do es not

apply

.In almost

all cases

we consider,

the deca ying

particles

are unp olarized,

and we sho w that

in the

few cases

where

they are polarized,

the effect

of polarization

issmall or can

be

accoun ted for relativ

ely easily

.

The Earth

is thin

enough

that stopping

of hea

vy hadrons

and stopping

and absorption

of neutrinos

as they

pass through

the Earth

can be neglected.

Therefore, ifthe rest-frame

distribution

from aparticular

deca ying

particle

is

kno wn,

Eq.

(2.1) giv es the

correct

dN =dE

for particle

deca yin

the Earth;

hence the sup erscript

\Phi in Eq.

(2.1).

This also implies

that the neutrino

and

an tineutrino

spectra from injection

of agiv

en particle-an

tiparticle pair in the

cen ter

of the

Sun

are the same.

Also, ifthe scaled

second momen tin the rest

frame of the

deca ying

particle,

\Omega N z2ff

rest ,is

kno wn,

then

the scaled

second

momen tfor aparticle

that deca ys with

av elo cit yfi

in the

Earth

issimply

\Omega N z2ff

(fi) =\Omega Nz

2ff

rest

(1 +

fi23 ):

(2: 3)

6

On the other

hand, energetic

neutrinos will lose energy

via neutral-curren

t

interactions with the solar

medium

and become

absorb ed via

charged-curren

t

interactions as they

pass through

the Sun.

To accoun

tfor these

effects,

we use

the calculation

of RS

that

aneutrino

injected with an energy

E lea ves

the Sun

with energy

E f=

E 1+

Eo/ i ;

(2: 4)

where o/* = 1:01

\Theta 10

\Gamma 3 GeV

\Gamma 1 and

o/_* = 3:8

\Theta 10

\Gamma 4 GeV

\Gamma 1 ,and

with probabilit

y

Pf = `

1 1+

Eo/ i '

ff i;

(2: 5)

where ff* = 5:1

and

ff_* = 9:0

As aresult,

the neutrino

spectrum for aparticle

deca ying

with energy

E din

the Sun,

(dN =dE )fi ,is

related

to the

neutrino

spectrum for aparticle

deca ying

with energy

E din

the Earth,

(dN =dE )\Phi ,b y

`dNdE ' fii (

E d;E

*)

= (1 \Gamma E *o/

i)

ff i\Gamma

2 `

dNdE '

\Phi ( E d;E

m)

;

(2: 6)

where E m

= E *=(1

\Gamma E *o/ i)

isthe

energy

aneutrino

had at the

core

of the

Sun

ifit exits

with energy

E *.

Note

that stopping

and absorption

are differen

tfor

neutrinos and an tineutrinos,

so the

spectrum

of neutrinos

from WIMP

annihilation in the

Sun

isdifferen

tthan the spectrum

of an tineutrinos.

The spectrum

of neutrinos

isthe same as that

for an tineutrinos

for deca

ys in the

Earth,

so we

neglect the subscript

ion neutrino

spectra from the Earth.

Ifa bor cquark

isinjected

into the cen ter of the

Earth,

itwill lose energy

during hadronization.

As aresult,

the energy

at whic

hthe

hadron

deca ys in the

Earth, E d,

isrelated

to the

energy

E iat

whic hit isinjected

by E d=

zfE i.

We

are interested

primarily in the

second

momen tof the neutrino

distributions,

so

we use

the square

root of the

second

momen ts of the

fragmen

tation functions

computed by RS--that

is, we

tak ez f=

0:58 for cquarks

and zf = 0:73

for b

quarks. Actually ,the distribution

of zf

isdescrib

ed by afragmen

tation function

[14], but this fragmen

tation function

ishighly localized around zf. Comparison

of higher

order momen

ts of zf

suggests

that no more

than O(5%)

accuracy

is

lost by using

the cen tral

value

of the

fragmen

tation function.

7

In addition,

the core

of the

Sun

(but

not the Earth)

isdense

enough that

band

c-quark

hadrons will interact

with the solar

medium

and be slo wed

appreciably before they deca y[3].

Ifa hadron

initially has an energy

E 0=

zf E i

(after hadronization

of aquark

injected with energy

E i),

then

itwill

deca ywith

an energy

E dpic

ked from

adeca

ydistribution

[3],

` 1N dNdE d '

hadron

(E 0;E

d)

=

E cE2d exp

^E c `

1Ec \Gamma

1Ed '*

;

(2: 7)

where E c=

250 GeV

for cquarks,

E c=

470 GeV

for bquarks.

The nth momen

tof this distribution,

for agiv

en initial

energy

E 0,

is

hE

ndi (E 0)

= Z

E 0

0 `

1N dNdE d '

hadron

(E 0;E

)E

ndE

;

(2: 8)

so in particular,

the average

energy at whic

ha hadron

deca ys is

hE di(

E 0)

= E cexp`

E cE0 '

E 1 `

E cE0 '

;

(2: 9)

where E 1(x

)=R

1x( e\Gamma

y=y

)dy

[15],

and the rms

value

of the

deca

yenergy

is

E

rmsd

(E 0)

= q

hE

2di = p

E c(E

0\Gamma

hE di)

:

(2: 10)

These quan tities

will come

in handy

in the

follo wing

Section.

Of all the

factors

that enter

into the calculation

of the

neutrino

spectrum, stopping

of hea

vy hadrons

is the

most

uncertain

theoretically;

very little is kno

wn ab out

the interaction

of hea

vy-quark

hadrons with dense

matter.

The functional

form of the

deca ydistribution,

Eq. (2.7),

is ph ysically

wellmotiv ated, but the values

of E clisted

here are sub ject

to some

(perhaps

sizeable) uncertain

ty. Ho wev

er, this

theoretical

uncertain ty also

enters

the Mon

te

Carlo calculation;

in this

regard,

Mon te Carlo

sim ulations

offer no impro

vemen t

over the analytic

result. Also, the effect

of hadron

stopping

is small

at low

er

energies, so the

uncertain

ty intro

duced

into the neutrino

spectra is relativ

ely

small. Stopping

becomes muc hstronger--and

the subsequen

ttheoretical

uncertain ty muc

hlarger--at

higher injection

energies, but in most

cases where

8

the WIMP

is massiv

eenough

that it can

annihilate

into top quarks,

it annihilates almost exclusiv ely to top

quarks,

gauge bosons,

and/or Higgs bosons

[10][11][1 2]. Therefore,

the uncertain

ty in the

total

neutrino

spectrum due to

po or understanding

of hadron

stopping

isnev er very

large.

Before con tin uing,

we remind

the reader

that ligh t( u,

d, and

s) quarks

will

pro duce

long-liv

ed hadrons

that come

to rest

in the

core

of the

Sun

or Earth

before they deca y; therefore,

these final states

pro duce

no energetic

neutrinos.

The same

istrue

of neutrinos

from muon

deca y. On

the other

hand, o/leptons,

top quarks,

and gauge

and Higgs

bosons

will deca yessen

tially immediately

,so

there is no

energy

loss due to slo wing

or hadronization

before they deca y.

Finally ,consider

ac hain

deca ypro

cess, H ! l1l 2,

where

ahea vy particle,

H, deca

ys to tw oligh

ter particles,

l1 and

l2, whic

hthen

deca yto

energetic

neutrinos. For example,

the hea vy particle

could be one

of the

Higgs

bosons

whic hthen

deca ys to tw oligh

tfermions.

In the

rest

frame

of the

deca ying

H,

the energies

of the

tw oligh

tparticles

are E 1=

(m

2H+

m

21\Gamma

m

22)=

(2m

H)

and

E 2=

(m

2H + m

32\Gamma

m

21)=

(2m

H),

and their

velo cities

are fi1 = (1 \Gamma m

21=E

21) 1= 2

and fi2 = (1 \Gamma m

22=E 22) 1= 2,

where

m H,

m 1,

and

m 2are

the masses

of H,

l1, and

l2, resp

ectiv

ely .Supp

ose the neutrino

spectrum from deca yof l1 as

afunction

of neutrino

energy E *for

agiv en injection

energy E iis

(dN =dE )1( E i;E

*),

and similarly

for neutrinos

from deca ys of l2.

Then

the spectrum

from the

hea vy particle

H, whic

hdeca

ys while

mo ving

with av elo cit yfi

H

and

energy

E H

= flH

m H

is

`dNdE ' H (

E H;

E *)

= X f=1

;2

1 2fl HE

ffi ffi H Z

flH E f(1+

fiH fif )

flH E f(1

\Gamma fi Hfi f) `

dNdE ' f (

E;

E *)

dE

;

(2: 11)

where the sum

is over

the tw odeca

yparticles,

and E fand

fif are

the deca yparticle energies and velo cities

in the

rest

frame

of the

deca ying

H. Also,

if the

scaled

second momen ts of the

neutrino

distribution,

\Omega N z2ff 1(

E i)

and

\Omega N z2ff 2(E

i),

from

deca ys of l1 and

l2, resp

ectiv

ely ,are

kno wn

for injection

energies E i,then

the\Omega Nz

2ff from

deca yof H ! l1l 2is

[16]

\Omega N z2ff H(

E H)

=

1E2H X f=1

;2

1 2fl HE

ffi ffi H Z

flH E f(1+

fiH fif )

flH E f(1

\Gamma fi Hfi f)

E

2\Omega

Nz

2ff f(

E)

dE :

(2: 12)

9

These equations

can be used,

for example,

to determine

the neutrino

spectra

from Higgs-b

oson deca y. Higgs

bosons

do not

deca ydirectly

to neutrinos,

so

all the

energetic-neutrino

signal comes from the deca

yof the Higgs

deca ypro

ducts. Giv en the

neutrino

spectra from the Higgs

deca yc hannels

(e.g., quarks,

leptons, and ifhea

vy enough,

gauge bosons

and/or top quarks)

presen ted below, the neutrino

spectrum and/or

\Omega N z2ff

from

Higgs

deca yare

obtained

by

summing Eqs. (2.11)

and/or

(2.12) over all deca

yc hannels

of the

Higgs

with

the appropriate

branc hing ratios.

We will use these

results

belo wto

obtain

the

\Omega N z2ff

from

top-quark

deca y. Also,

note that these

relations,

Eqs. (2.11)

and

(2.12), are valid

for neutrino

spectra from particle

deca yin both

the Sun

and

the Earth.Giv en the

results

in this

Section,

the neutrino

spectra from eac hfinal

state

follo w from

the kinematics

of tw oand

three-b

ody deca y, and

we will

discuss

eac hfinal

state belo w.

3. Neutrinos

From o/Leptons

and band

cQuarks

3.1. Neutrinos

Fr om

o/ Leptons

We begin

with the neutrino

energy spectra from o/leptons.

Calculation

of these

spectra

is easier

than that for band

cquarks,

since o/leptons

do not

hadronize, and they

are not slo wed

before

stopping.

In addition,

we consider

deca y of

unp

olarized

o/ leptons

only . For

neutralinos,

this is valid

because

the o/leptons

pro duced

by neutralino

annihilation

are unp olarized

[13]. If for

some unforeseen

reason, the injected

o/leptons

are polarized,

the results

for the

neutrino spectra will be sligh

tly differen

t.

In the

rest

frame

of the

o/, the

energy

distribution

of the

muon

neutrino

from the deca yo/ ! __* _* o/is

[17]

` dNdE * '

rest_o/o/

=

96\Gamma

o/! _* *

m

4o/

E

2*(m

o/\Gamma

2E *);

0^ E *^

12 m o/;

(3: 1)

10

where \Gamma o/ ! _*

*'

0:18

isthe

branc hing ratio for o/deca

yto muons.

The spectrum

of neutrinos

from deca ys of o/leptons

mo ving

with an energy

E i(the

injection

energy), velo cit yfi

,and

fl= (1 \Gamma fi

2) \Gamma 1

=2

is

` dNdE * '

\Phi _o/o/ (E i;E

*)

=

48\Gamma

o/! _* *

fifl m

4o/ `

12 m o/E

2*\Gamma

23 E 3* '

min

(m o/=

2;E

+)

E \Gamma

; (3: 2)

where E \Sigma

= E *fl

\Gamma 1 (1 \Upsilon fi)

\Gamma 1

.In

the relativistic

limit (fi ! 1),

this

becomes

(dN =dE )= (2\Gamma =E i)(1

\Gamma 3 x2 +2

x3),

for 0^

E *^

E i,where

x= E *=E

i.

Giv

en

the neutrino

distribution

from o/deca

yin the Earth,

the spectrum

from o/deca

y

in the

Sun

isobtained

by using

Eq. (2.6)

to accoun

tfor stopping

and absorption

of neutrinos.Applying

the results

of Section

2, the

scaled

second

momen tfrom injection

of _o/o/

pairs

with velo cit yfi

in the

core

of the

Earth

is

\Omega N z2ff

\Phi _o/o/ =

\Gamma o/ ! _*

*

10

(1 +

fi23 ):

(3: 3)

The integral

over the distributions

from the Sun

can also be performed,

and

the result

is

\Omega N z2ff

fi_o/o/ ;i(

E i)=

\Gamma o/ ! _*

*h

o/;i(

E io/ i);

(3: 4)

where the o/i are

the neutrino

stopping coefficien ts giv

en in Section

2. For

muon

neutrinos, the function

ho/ ;i(y

)is

giv en by

ho/ ;* _(y

)=

130 4+ y

(1 +y

)4

;

(3: 5)

and for an tineutrinos,

the appropriate

function is

ho/ ;_* _(y

)=

11260 168

+354

y+ 348 y2 +190

y3 +56

y4 +7

y5

(1 +y

)8

: (3: 6)

In deriving

Eq. (3.5),

we took

ff* _=

5:0, whic his an excellen

tappro ximation

to

the RS value

of ff* _=

5:1. In addition,

the hfunctions

were obtained

using the

fi! 1limit

of (dN

=dE )\Phi _o/ o/.

Iffi

isnot

near unit y,then

E im

ust be close

to m o/,

in

whic hcase

neutrino

absorption

and stopping

will be negligible

and the dN =dE

from the Earth

can be used.

Also, in the

low-energy

limit (E io/

o/\Gamma

1 i

' TeV),

11

simple but accurate

expressions

can be obtained

from Ta ylor

expansions

of the

hfunctions. In addition

to the

energetic

muons pro duced

by direct

deca ys of the

o/, there

will be additional

indirect energetic

muon neutrinos

from the o/neutrinos.

The

o/neutrino from the o/deca

yma ypro duce an additional

o/lepton via ac hargedcurren tin teraction

with the solar

medium

as it passes

through

the Sun,

and

this o/lepton

will then

deca yand

pro duce

an additional

muon neutrino

[3]. The

energy of this

muon

neutrino

ism uch smaller

than that from direct

deca yof the

original o/lepton.

Although these neutrinos

ma ycon

tribute

non-negligibly

to

dN =dE

at low

neutrino

energies, this con tribution

to the

up ward-m

uon signal

istin y, so

itis

justifiably

ignored when considering

up ward

muons.

Therefore,

Eqs. (3.3),

(3.4), (3.5), and (3.6)

accoun

tfor the overwhelming

ma jorit

yof the

energetic neutrinos that giv erise

to up ward

muons.

In addition,

they correctly

tak ein

to accoun

tall the imp ortan

tph ysical

pro cesses:

the o/-deca

ykinematics,

and stopping

and absorption

of neutrinos

as they

pass through

the Sun.

Therefore, these

expressions

are sub ject

to little

theoretical

uncertain ty and

can be

used in place

of the

results

of Mon

te Carlo

calculations.

3.2. Neutrinos

from band

cQuarks

The three-b

ody deca yof

band

c-quark

hadrons is similar

to that

for o/

deca y,and

the treatmen

tof neutrino

stopping and absorption

in the

Sun

issimilar; ho wev

er, calculation

of the

neutrino

spectra is sligh

tly more

complicated

due to hadronization

of the

quarks

and stopping

of the

hea vy hadrons

in the

Sun. Still, we can

treat

these effects

analytically

without resorting

to Mon

te

Carlos. Hadronization

issimple: the energy

of aquark

isdegraded

from its injection energy E ito

the hadron

energy E 0=

zfE i.

Stopping

isstraigh tforw ard

as well:

we must

integrate

the neutrino

energy spectrum

from ahadron

whic h

deca ys with

energy

E do

ver

the distribution

of deca

yenergies,

Eq. (2.7).

In the

rest

frame

of the

b-quark

hadron, the energy

distribution

of the

neutrino from the deca yb ! c_*

is (neglecting

the small

correction

due to

finite c-quark

mass) similar

to that

from o/-lepton

deca y[17]:

` dNdE

* '

restb_b

=

96\Gamma

b! _* X

m

4b

E

2*(m

b\Gamma

2E *);

0^ E *^

12 m b;

(3: 7)

12

where \Gamma b ! _*

X

' 0:103

is the

branc

hing ratio for inclusiv

esemileptonic

deca y

of the

bquark

into muons

[18]. A bquark

injected with energy

E ideca

ys in the

Earth with an energy

E d=

zfE i,v

elo cit yfi

,and

fl= (1 \Gamma fi2 )\Gamma

1= 2=

E d=m

b

and the resulting

neutrino spectrum is

` dNdE * '

\Phi b_b =

48\Gamma

b! _* X

fifl m

4b `

12 m bE

2*\Gamma

23 E 3* '

min

(m b=

2;E

+)

E \Gamma

; (3: 8)

where E \Sigma

= E *fl

\Gamma 1 (1 \Upsilon fi)

\Gamma 1

.In

the relativistic

limit (fi ! 1),

this

becomes

(dN =dE )= (2\Gamma =E d)(1

\Gamma 3 x2 +2

x3),

for 0^

E *^

E d,

where

x= E *=E

d.

The neutrino

spectrum from bdeca

yin the Sun

is obtained

by applying

Eq. (2.6)

to Eq.

(3.8),

and integrating

over the distribution

of hadron

deca y

energies, Eq. (2.7):

`dNdE ' fib_b;i

(E i;E

*)

= Z

zf E i

0

` 1N dNdE d '

hadron

(E d)

\Theta (1 \Gamma E *o/

i)

ff i\Gamma

2 `

dNdE '

\Phi b_b ( E d;E

m)

dE d

=

2\Gamma b! _* X

E c

(1 \Gamma E

*o/ i)

ff i\Gamma

2exp`

E cE0 \Gamma

x '

\Theta (1 \Gamma 18 y2 +24

x2 y3

+x

+48

y3 +8

y3 x3

\Gamma 9

y2x

2

+48 y3x +2 y3x

4\Gamma 18 y2x

\Gamma 3 y2 x3)

;

(3: 9)

where E m

= E *=(1

\Gamma E *o/

i),

x=

max (E c=E

0;E c=E *),

E 0=

zfE i,

and

here,

y= E *=E

c.

The

value

of E cto

be used

here is470

GeV. Eq. (3.9)

was obtained

using the fi!

1limit

of (dN

=dE )\Phi b_ b,

so

it is not

strictly

valid for E inear

m b.

On the other

hand, ifthe injection

energy is so

low

that

fiis not near

unit y,

interactions with the solar

medium

are small

and the spectrum

from deca yin

the Earth

can be used.

In addition

to the

prompt

neutrinos

from b-quark

deca y, there

will be

additional energetic neutrinos from subsequen

tdeca ys of the

cquark.

Ho wever, the typical

energy of these

neutrinos

is roughly

1/3 the energy

of the

neutrinos from prompt

deca y, so

they

con tribute

only ab out

10%

to the

total

13

up ward-m

uon signal.

Since neutralino

annihilation

into bquarks

is alw

ays

accompanied by annihilation

into o/leptons,

cquarks, and possibly

top quarks

and gauge/Higgs

bosons, the error

intro duced

into the up ward-m

uon rate by

including only prompt

neutrinos

from bdeca

ywill nev er be

greater

than 10%.

If

greater accuracy

isdesired,

the con tribution

from cquarks

can be included

using

the c-quark

neutrino

spectra belo w and

the chain-deca

yequations

of Section

2.

The neutrino

spectra from charmed

hadrons are similar,

but the three-b

ody

deca ykinematics

issligh tly differen

t. In

the

rest

frame

of the

charmed

hadron,

the energy

distribution

of the

neutrino

from the deca yc ! s_*

is (neglecting

the small

correction

due to finite

strange-quark

mass) [17]

` dNdE * '

restc_c

=

16\Gamma

c! _* X

m

4c

E

2*(3

m c\Gamma

4E *);

0^ E *^

12 m c;

(3: 10)

where \Gamma c ! _*

X

' 0:13

isthe

branc hing ratio for inclusiv

esemileptonic

deca yof

the cquark

into muons.

(W eobtain

this value

for \Gamma c ! _*

Xb

yassuming

charged

and neutral

charmed

mesons are pro duced

with equal

probabilit

yand averaging

the branc

hing ratios

for inclusiv

esemileptonic

deca ys to muons

[18]. Our value

for this

branc

hing ratio app ears

to be

almost

twice as large

as that

used by

RS, so our

neutrino

spectra from c-quark

deca ywill

be significan

tly larger

than

those obtained

by RS.)

A cquark

injected with energy

E ideca

ys in the

Earth

with an energy

E d=

zfE i,v

elo cit yfi

,and

fl= (1 \Gamma fi2 )\Gamma

1= 2=

E d=m

cand

the

resulting neutrino spectrum is

` dNdE

* '

\Phi c_c =

8\Gamma c! _* X

fifl m

4c `

32 m cE

2*\Gamma

43 E 3* '

min

(m c=

2;E

+)

E \Gamma

; (3: 11)

where E \Sigma

= E *fl

\Gamma 1 (1 \Upsilon fi)

\Gamma 1

.In

the relativistic

limit (fi ! 1),

this

becomes

(dN =dE )= (\Gamma =E

d)[(5

=3) \Gamma 3 x2 +(4

=3) x3 ],for

0^ E *^

E d,

where

x= E *=E

d.

The deriv ation

of the

neutrino

spectrum from cdeca

yin the Sun

issimilar

to that

for bquarks

ab ove.

The result

is

`dNdE ' fic_c;i

(E i;E

*)

=\Gamma

c! _* X

3E c

(1 \Gamma E

*o/ i)

ff i\Gamma

2exp`

E cE0 \Gamma

x '

\Theta (5 \Gamma 54 y2 +48

y3x

2+ 5x +96

y3 +16

y3x

3\Gamma 27 x2 y2

+96 y3x +4 y3x

4\Gamma 54 y2x

\Gamma 96

y2x

3);

(3: 12)

14

where as before,

E m

= E *=(1

\Gamma E *o/

i),

x=

max (E c=E

0;E c=E *),

E 0=

zf E i,

and here,

y= E *=E

c.

The

value

of E cto

be used

here is250

GeV.

No w we

list

expressions

for the\Omega

Nz

2ff.

The

expression

for deca

yof

ab

quark mo ving

with velo cit yfi

in the

Earth

is

\Omega N z2ff

\Phi b_b =

z2f\Gamma

b! _* X

10

(1 +

fi23 );

(3: 13)

and that

from c-quark

deca yis\Omega Nz

2ff

\Phi c_c

=

2z

2f\Gamma c! _* X

15

(1 +

fi23 ):

(3: 14)

The second

momen ts from

band

c-quark

deca yin

the Sun

can be obtained

by integrating

the correct

expressions

for dN =dE

. Ho

wev

er, it turns

out to

be easier

to switc

hthe

order of integration--that

is, first

compute

\Omega N z2ff

for

agiv en deca

yenergy

,E d,

and

then

integrate

over the hadronic

deca y-energy

distribution. The result

is

\Omega N z2ff f;i

(E i)=

\Gamma f ! _*

X

E

2i

E

2ce

E c=E

0 Z

1Ec=E 0 dxx

2e

\Gamma x hf ;i(

E co/

i=x

)

'

hE di

2

E

2i

hf ;i `q

hE

2dio/ i '

;

(3: 15)

where the subscript

fdenotes bor cquarks,

and the subscript

irefers to neutrino

typ e( *_

or _*_),

and the mom

ents hE

ndi are

those

that were giv en in Section

2.

The integral

in the

first

line cannot

be performed

analytically ,but we ha ve

found that the appro

ximation

in the

second

line isaccurate

to afew

percen tfor

injection energies less than

afew

TeV. For bquarks,

the functions

hb;i are the

same as those

for o/leptons

giv en in the

previous

subsection:

hb;i (y)

= ho/ ;i(y

)

[c.f., Eqs. (3.5) and (3.6)].

For neutrinos,

the c-quark

functions

are,

hc;* _(y

)=

1180 32 +25

y+ 5y

2

(1 + y)

5

;

(3: 16)

and for an tineutrinos,

hc; _*_(

y) =

17560 (1344+

3186 y+ 3834

y2

+2786 y3 +1242

y4 +315

y5 +35

y6) =(1

+y )9:

(3: 17)

15

Eqs. (3.13),

(3.14), and (3.15),

with the expressions

for the

appropriate

h(y )

functions, correctly describ ethe second

momen ts of the

neutrino

distributions

from band

c-quark

deca yin

the Sun

and Earth

and include

the effects

of

hadronization and stopping

of hea

vy quarks

and stopping

and absorption

of

neutrinos in the

Sun.

They therefore

can be used

in place

of Mon

te Carlo

calculations.4. Neutrinos

From Gauge

Bosons

A W

boson

deca ys directly

to am

uon

and am uon

neutrino

10.5% of

the time,

and aZ deca ys to am

uon

neutrino-an

tineutrino pair 6.7%

of the

time. Additional

muon neutrinos

are indirectly

pro duced

by W deca

yto

b

and cquarks

and o/ leptons.

The energies

of these

neutrinos

are generally

smaller, so the

con tribution

pro vided

by these

indirect

neutrinos

to\Omega Nz

2ff isa

small fraction

of that

from the direct

neutrinos.

Recall that ligh t-fermion

final

states pro duce

only very low energy

neutrinos

since they become

thermalized

before deca ying

[3], and

their

con tributions

to the

neutrino

energy momen tsare

completely negligible. We will first

calculate

the\Omega Nz

2ff from

deca yof polarized

and unp olarized

gauge bosons

in the

Earth,

and sho w that

polarization

is nev

er more

than a

10% effect.

Giv en these

results,

we will

presen

texpressions

for dN =dE

from

deca yof unp olarized

gauge bosons

only in the

Sun

and in the

Earth,

and then

we will

pro vide

an analytic

result for\Omega Nz

2ff for

gauge-b

oson deca yin the Sun.

In the

rest

frame

of the

vector

boson, the neutrino

is emitted

with an

energy equal to half

the vector-b

oson mass;

in the

lab oratory

frame, the energy

of the

emitted

neutrino is E *=

E i(1

+ ficos

`)= 2where

`is the angle

in the

rest frame

bet ween

the neutrino

direction and the direction

of motion

of the

cen ter

of mass,

E iis

the energy

at whic

hthe

vector

boson isinjected,

and fiis

its velo

cit y.

Therefore,

for av ector

boson

injected

into the Earth

with energy

E i,the

second

momen tof the resulting

neutrino spectrum is

hE

2*i =

E

2i4 R

P(cos

`)(1 + ficos

`)

2d

(cos

`)

RP (cos

`) d(cos

`);

(4: 1)

16

where P(cos `) isthe

deca yangular

distribution

in the

rest

frame

of the

gauge

boson, and and \Gamma * isthe

fraction

of vector-b

oson deca ys whic

hpro

duce aneutrino. For unp olarized

gauge bosons,

the deca

yis isotropic,

and [5]

hE

2*i =

E

2i4 `

1+

fi23 '

:

(4: 2)

If the

gauge

boson has helicit

y h

=

\Sigma 1

(transv

ersely polarized),

then

P(cos `) / (1 \Sigma cos

`)

2,

andhE 2*i T=

E

2i4 `

1+

2fi

25 \Sigma

5fi '

;

(4: 3)

and ifthe

gauge

boson islongitudinally

polarized (h = 0),

then

P(cos

`) / sin

2`

and hE

2*i L=

E

2i4 `

1+

fi25 '

:

(4: 4)

The velo cities

of WIMPs

in the

Sun

or halo

are vo/

1, so

annihilation

proceeds only through

the low est angular-mom

entum state, the sw ave.

Although

the s-w ave

cross

sections

for annihilation

of neutralinos

into t_t [19][20][

21] ,

gauge-b oson pairs [10], and Higgs-b

oson-gauge-b

oson pairs [5] were

calculated

previously ,the helicit y-ampli

tude formalism

used by Drees

and No jiri

[13]

is

needed to describ

ethe polarization

state of the

annihilation

pro ducts.

According to the

results

of [13],

the gauge

bosons

pro duced

in the

s-w ave

annihilation

pro cesses

~O/~O/ ! W

+W

\Gamma and

~O/~O/ ! ZZ

are pro duced

with atensor

polarization; that is, the

gauge

bosons

are transv

ersely polarized,

half with

spin aligned

(h = 1)

with

the direction

of motion

and half with

spins an tialigned

(h = \Gamma 1).

Therefore, the scaled

second momen tof the neutrino

spectrum from neutralino

annihilation into W

+W

\Gamma pairs

in the

Earth

is an

average

of that

from the

tw otransv

erse-p olarization

states. The second

momen tfor neutrinos

from the

Earth isthen

giv en by\Omega N

z2ff

\Phi ~O/~O/ ! W W

= \Gamma W

! _*

14 (1

+

25 fi 2W) ;

(4: 5)

17

where \Gamma W ! _*

= 0:105.

There will be additional

muon neutrinos

from o/

leptons from W deca

y, but

their

con tribution

to\Omega Nz

2ff is small.

Only one

in fiv eo/

deca ys pro

duces

am uon

neutrino,

and the energy

of this

neutrino

is

typically only one third

the energy

of aneutrino

from direct

deca y. Therefore,

Eq. (4.5)

prop erly accoun

ts for

polarization

and should

be accurate

to better

than 3%.The expression

for ZZ

pairs

isobtained

similarly to the

ab ove.

We include

an overall

factor of tw o, whic

hcoun

ts the

Z bosons,

and arriv eat the result

\Omega N z2ff

\Phi ~O/~O/ ! ZZ

= 2\Gamma Z!

*_ _*_ 14

(1 +

25 fi 2Z);

(4: 6)

where \Gamma Z ! *_

_*_

= 0:067.

Again, there will be additional

neutrinos from Z

deca yto

b_b, and

to alesser

exten t, c_c and

_o/o/ ,and

the magnitude

of their

con tribution

to\Omega Nz

2ff iseasily

estimated.

Although the branc

hing ratio to b_b

is 2.2

times

that into *_ _*_,

am uon

neutrino

is pro

duced

in only

one in ten

b-quark deca ys. In addition,

the b-quark

energy is reduced

to 0.7

times

its

original energy during hadronization,

and the resulting

muon neutrino

carries

only athird

of that

energy

.By performing

similar estimates

for the

o/lepton

and cquark,

we find

that Eq. (4.6)

isaccurate

to better

than 4%.

Gauge bosons that are pro duced

in the

annihilation

pro cesses

~O/~O/ ! ZH

,

where H is one

of the

tw oneutral

scalar Higgs bosons,

or ~O/~O/ ! W

+H

\Gamma (or

its charge

conjugate),

where H

\Sigma is

the charged

Higgs boson,

are longitudinally

polarized [13], so Eq.

(4.4)

isthe

appropriate

result to use

for these

final states.

Th us we

ha ve

\Omega N z2ff

\Phi ~O/~O/ ! ZH

= \Gamma Z

! *_

_*_ 14

(1 +

15 fi 2Z) +Higgs

deca ycon

tribution

: (4: 7)

The Higgs-deca

ycon tribution

can be evaluated

using the results

of Section

2.

Generally ,the Higgs

con tribution

is muc

hsmaller

than that from the Z; there

is an

additional

step in the

deca yc hain,

so the

energies

of the

neutrinos

are

smaller. This isesp ecially

true ifthe

Higgs isligh tenough

that itdeca

ys only

to ligh

tfermions.Previously

,it was assumed

that the gauge

bosons

were unp olarized,

as

in Eq.

(4.2)

[5]. Giv en our

new

results,

we see that

aprop

er treatmen

tof

18

polarization changes the second

momen ts by

no more

than 5% for the

tensor

polarization, and no more

than 10% for the

longitudinal

polarization. In the

rest of this

Section,

we will

ignore

the small

effect of polarization

and treat

deca ys of unp

olarized

gauge bosons

only .

The neutrino

energy distribution

from W or Z bosons

at rest

is simply

a

delta function

with supp ort at half

the gauge-b

oson mass,

`dNdE r '

restW W ;Z

Z (

E)

= \Gamma ffi (E

\Gamma m B=

2);

(4: 8)

where m Bis

the gauge-b

oson mass and \Gamma is the branc

hing ratio for direct

deca y

into muon

neutrinos

(this expression

should be multiplied

by 2for

Z bosons).

Applying Eq. (2.1),

the mo ving-fram

edistribution

from unp olarized

W or Z

bosons mo ving

with energy

E iand

velo cit yfi

is

` dNdE

* '

\Phi W W ;Z

Z (

E i;E

*)

= ae

\Gamma ( m Bfl

fi)

\Gamma 1

for

E i2(1

\Gamma fi)

! E *!

E i2(1

+ fi),

0 otherwise.

(4: 9)

Here we ha ve

neglected

the con tribution

from other

W deca

ymo

des. Although

the con tribution

to dN =dE

from

these other mo des

ma yb esignifican

tat low

energies, these low-energy

neutrinos pro vide

little

con tribution

to the

up wardmuon flux, as discussed

ab ove,

so the

final

result

obtained

using Eq. (4.9)

should

be accurate

to better

than 4%.

The energy

distribution

of neutrinos

from gauge-b

oson deca yin

the Sun

can be obtained

by applying

Eq. (2.6).

The result

for the

scaled

second

momen t

of the

neutrino

distribution

from W deca

yin the Sun

isthen

[5]

\Omega N z2ff

fiW W ;i=

\Gamma W ! _*fi

2+

2E o/i(1

+ ffi)

+ E

2o/

2iff i(1

+ff

i)

E

3io/

3iff

i(ff

2i\Gamma

1)(1

+ Eo/

i)

ff i+1

fififififi E =E

i(1+

fi) =2

E= E i(1

\Gamma fi )=2

;(4:10)

for W 's

injected

into the core

of the

Sun

with

energy

E iand

velo cit y fi.

Once again,

the expression

for ZZ

pairs

isobtained

by replacing

\Gamma W ! _*

with

2\Gamma Z!

*_ _*_,

where

the factor

of tw ocoun

ts the

Z bosons.

This do es not

include

the con tributions

from the deca ys of the

W and

Z bosons

to hea

vy fermions,

but again,

we ha ve

already

seen that this effect

isless than ab out

4%.

19

Prop er treatmen

tof the effects

of vector-b

oson polarization

on the

scaled

second momen tfrom the Sun

leads

to av ery

small

correction

to Eq.

(4.10).

In

the low-energy

limit, the effect

of the

polarization

vanishes, as we

ha ve

seen.

In addition,

the effect

of the

polarization

must vanish

in the

high-energy

limit

as well.

This is because

the Loren

tz bo ost

factor

suppresses

the con tribution

of neutrinos

with cos `'

\Gamma 1 (an tialigned

with the gauge-b

oson direction

of

motion), but the interactions

suppress those with cos `'

1(aligned)

because

of their

higher

energy .The

net result

is that

the integrated

effect of the

polarization is nev

er more

than 2%, and is thus

negligible

for neutrinos

from

gauge-b oson deca yin the Sun.

5. Neutrinos

from Top-Quark

Deca ys

In man

ycases

where the WIMP

is hea

vier

than

the top quark,

WIMPs

annihilate predominan tly to top

quarks,

esp ecially

if the

top is as

hea vy as

recen tevidence

suggests [22][23]. Therefore,

the neutrino

spectrum from topquark deca yis needed

for aprop

er calculation

of the

flux

of energetic

neutrinos

from WIMP

annihilation

in the

Sun

or Earth.

In previous

work [3][5],

the top-quark

mass was assumed

to be

60 GeV.

In

this case,

calculation

of the

hadronization

and deca yc hannels

of the

top were

quite difficult,

and the resulting

neutrino spectrum,

calculated using Mon te

Carlo tec hniques,

was accordingly

uncertain. In addition,

the results

were

highly dep enden

ton the assumed

top-quark

mass. Giv en that

the top quark

is significan

tly hea vier

than

the W [22][23],

the calculation

must be rep eated.

The top quark

is hea

vy, so it deca

ys almost

exclusiv ely to aW

boson

and ab

quark. Hadronization

effects are unimp

ortan t,and

deca yis essen tially

afreeparticle weak deca y. Therefore,

our new

results

are less

sub ject

to theoretical

uncertain ty. The neutrino

spectrum from top quarks

follo ws from

the neutrino

spectra

of its

deca

ypro

ducts,

the W boson

and the bquark,

whic hw eha

ve already

computed. We will first

calculate

the neutrino

spectra from top-quark

deca y

in the

Earth

and Sun and assume

that the W bosons

from top-quark

deca y

20

are unp olarized.

We will then

calculate

\Omega N z2ff

from

top-quark

deca yin

the

Earth both with and without

polarization

and sho w that

polarization

of the

gauge boson is only

a3%

effect.

Also, the results

of Ref.

[13] sho w that

the

positiv eand negativ

ehelicit yamplitudes

for top-quark

pro duction

are equal.

Therefore, the top quarks

pro duced

by neutralino

annihilation

are unp olarized,

and we will

consider

unp olarized

top quarks

only . We

will then

presen

tan

analytic appro ximation

to\Omega Nz

2ff for

top-quark

deca yin the Sun.

First consider

the case

where

the top quark

deca ys at rest.

The top deca ys

t! W bwith

abranc hing ratio close to unit

y. The

spectrum

of neutrinos

from

top-quark deca yat rest isgiv

en by asum

of the

W -deca

ycon tribution

and the

b-deca ycon tribution:`dNdE '

restt_t (E )= `

dNdE ' W W

(E W;

E) + `

dNdE '

b_b (

E b;E

):

(5: 1)

In the

rest

frame

of the

top quark,

the W has

energy

E W

=

m

2t+

m

2W

2m t

;

(5: 2)

and its velo

cit yis

fiW =

m

2t\Gamma

m

2W

m

2t+

m

2W

=

E b

E W

:

(5: 3)

The W -boson

con tribution

isgiv en by Eq.

(4.9)

with injection

energy E W.

The

bquark is first

slo wed

from

its injection

energy ,E b=

(m

2t\Gamma

m

2W)

=(2 m t),

by

hadronization to an

energy

E d=

zfE b,and

has av elo cit yfi

close

to unit

y. The

b-quark con tribution

isthen giv en by the

fi!

1limit

of Eq.

(3.8).

The bo ost

form

ula, Eq. (2.1),

yields the neutrino

spectrum from unp olarized top quarks

mo ving

with velo cit yfi

tand

flt = (1 \Gamma fi

2) \Gamma 1

=2,

dNdE fififififi

\Phi t_t (E *)

=

\Gamma W ! _*

2fl tfi tE

Wfi

W

ln

min

(E +;

ffl+ )

max (E \Gamma ;

ffl\Gamma ) \Theta i

flt(1 \Gamma fit)

ffl\Gamma ! E *!

flt(1 +fi t)ffl + j

+

\Gamma b ! _*

X

2fl tfi tE

d D

b[E

\Gamma =E

d;min

(1; E +=E

d)]

\Theta i E *!

flt(1 +fi t)E d j (5: 4)

21

where ffl\Sigma = E W(1

\Sigma fiW

)=2

with

E W

and

fiW equal

to their

values

in the

topquark rest frame

[Eqs. (5.2) and (5.3)],

E \Sigma =

E *fl

\Gamma 1t (1\Upsilon

fit)

\Gamma 1 ,and

\Theta ( x) = 1if

xis true

and \Theta ( x) = 0otherwise.

Also, the b-quark

deca yenergy

is E d=

zfE b

where E bis

the energy

of the

bquark

in the

top-quark

rest frame.

The function

D bis

giv en by

D b[x;

y] =

13 h 9(x

2\Gamma

y2) +4(

x3 \Gamma y

3) +6

ln

yx i

:

(5: 5)

The kinematics

of the

deca

ydictate

that the bquarks

carry asmaller

fraction of

the energy

than the W bosons,

their energy

isfurther

degraded by hadronization, and they

undergo

three-b ody--rather

than tw o-b

ody--deca

y. Therefore,

even though

the b-quark

con tribution

to (dN

=dE )restt_t

ma yb esignifican

tat low

energies, it will

con tribute

only ab out

9% to the

final

result

for the

up wardmuon flux.The distribution

from the Sun

is obtained

by applying

Eq. (2.6)

to Eq.

(5.4). It should

be noted

that application

of Eq.

(2.6)

to Eq.

(5.4)

do es

not

tak ein

to accoun

tthe stopping

of bquarks

in the

Sun;

ho wev

er, neutrinos

from

bquarks con tribute

only 9% of the

up ward-m

uon signal,

so little

accuracy

will

be lost

by neglecting

this effect.

We no wcalculate

\Omega N z2ff

for top-quark

deca yin the Earth,

and quan

tify the

effect of gauge-b

oson polarization

on the

results

for the

up ward-m

uon signal.

Consider the case

where

the top quark

deca ys at rest.

A fraction

[24]

fL =

1 1+ 2m

2Wm2t

;

(5: 6)

of the

W bosons

are pro duced

in the

the longitudinal-helicit

ystate, and the

rest (a fraction

1\Gamma fL )are

pro duced

in the

transv

erse-helicit

y states,

with

equal probabilit

yfor positiv

eand negativ

ehelicit ystates.

The result

for\Omega Nz

2ff

for top

quarks

at rest

(E t=

m twhere

E tis

the top-quark

energy) then follo ws

from the results

of the

previous

Section. The only subtlet

yis that

the second

momen tm ust be scaled

by the

square

of the

top-quark

mass rather

than the

22

square of the

W -boson

energy .The con tribution

from the semileptonic

bdeca ys

can also be included.

Th us,

we ha ve

\Omega N z2ff

\Phi t_t( E t=

m t)=

\Gamma W ! _*

E

2W

4m

2t ^

fL ` 1+

fi2W5 '

+(1 \Gamma fL ) `

1+

2fi

2W5 '*

+\Gamma b! _* X 2

z2fE

2b

15 m

2t

=

\Gamma W

! _*

E

2W

4m

2t ^

1+

15 fi 2W(2

\Gamma fL ) * +\Gamma

b! _* X 2

z2fE

2b

15 m

2t ;

(5: 7)

For m t=

174 GeV

[22], fL ss 0:7

and

fi2W = 0:4.

If the

W bosons

were

pro duced

unp olarized,

then we would

ha ve

fL

= 0:33,

so that

consideration

of

the polarization

of the

W bosons

in the

deca

yW

! _*

results

in asmall

change

from the unp olarized

case, less than

ab out

3%.

And as discussed

ab ove,

the

con tribution

to\Omega Nz

2ff from

bquarks

issub dominan

t(ab out 9% of the

total).

The top quark

deca ys immediately

,so the scaled

second momen tfrom

deca yof atop

quark in the

Earth

mo ving

with velo cit yfi

tand

energy

E tis

\Omega N z2ff

\Phi t_t( E t)= `

1+

fi2t3 '\Omega

Nz

2ff

\Phi t_t(

E t=

m t):

(5: 8)

Therefore, the\Omega Nz

2ff for

tops

mo ving

with velo cit yfi

tis

just

(1 +fi

2t=3)

times

that for top

quarks

at rest,

as giv en in Eq.

(5.7).

Eqs. (5.7) and (5.8)

pro vide

expressions

for the

scaled

second mom ent

of the

neutrino

distribution

from top quarks

from WIMP

annihilation

in the

Earth, accurate

to at least

3%.

In order

to obtain

\Omega N z2ff

from

top-quark

deca yin

the Sun,

the stopping

and absorption

of neutrinos,

as giv en in Eqs.

(2.4) and (2.5),

and stopping

of

the b-quark

hadrons must be included.

Neutrinos come predominan

tly from

the W boson

pro duced

by the

topquark deca y, and

to alesser

exten tfrom

the bquarks.

We already

kno w the

\Omega N z2ff

from

injection

of these

particles

in the

cen ter

of the

Sun.

The only catc h

isthat, to use

Eq.

(4.10),

we must

ignore

polarization

of the

W ;the

results

of

the previous

subsection

sho w that

W polarization

in top

deca yis less than

a

3% effect.

Therefore,

the easiest

wa yto

obtain

the result

for\Omega Nz

2ff from

top

23

quarks from the Sun

isto

integrate

the\Omega Nz

2ff for

W bosons

and bquarks

over

the appropriate

injection energies.

The expression

for\Omega Nz

2ff as afunction

of the

top-quark

energy ,E t,

for

top quarks

injected

into the Sun

follo ws from

Eq. (2.12)

and is

\Omega N z2ff

fit_t( E t)=

1E2t X f= b_b;W

W

1 2fl tE ffi ffi t Z

flt E f(1+

fit fif

)

flt E f(1

\Gamma fi tfi f)

E

2\Omega

Nz

2ff

fif(

E)

dE ;(5:9)

where we ha ve

included

the sum

over both the W bosons

and bquarks

from topquark deca y. The

integral

iso ver

the injection

energy of the

deca

yparticles,

and

the momen

tsh Nz

2i

fifinclude

the effects

of interactions

with the solar

medium.

For W bosons,

\Omega N z2ff

fiis

giv en by Eq.

(4.10),

and for bquarks

by Eq.

(3.15).

The quan tities

E t,

fit,

and

flt are

the energy

,v elo cit y,

and

fl factors

of the

top quark,

and E W

and

fiW are the energy

and velo cit yof

the W boson

in

the rest

frame

of the

top,

as giv en in Eqs.

(5.2) and (5.3).

The quan tities

E b=

(m

2t\Gamma

m

2W)

=(2 m t)

and

fib ' 1are

the energy

and velo cit yof

the bquark

in the

rest

frame

of the

top.

Also,

recall that\Omega Nz

2ff for

neutrinos

from the Sun

are differen

tthan those for an tineutrinos.

The results

for\Omega Nz

2ff for

neutrinos

and an tineutrinos

from injection

of

top quarks

into the core

of the

Sun

and Earth

are sho wn

in Fig.

1. There

are

three bands

of curv

es: the top is the

result

for\Omega Nz

2ff from

the Earth

(and

is the

same

for both

neutrinos

and an tineutrinos).

The cen ter

band

is the

result for an tineutrinos

from top quarks

injected

into the core

of the

Sun,

and

the low er band

is that

for neutrinos

from top quarks

injected

into the core

of

the Sun.

The curv es for

an tineutrinos

from the Sun

are higher

than those

for

neutrinos since neutrinos

interact more strongly

with the solar

medium

than

an tineutrinos

and are therefore

more strongly

atten uated

as they

pass through

the Sun.

Comparison

of the

results

for\Omega Nz

2ff from

the Sun

and Earth

illustrates

that absorption

and stopping

of neutrinos

in the

Sun

are very

imp ortan

teffects.

In eac

hband,

the solid

curv eis the total

result

for\Omega Nz

2ff whic

haccoun

tsfor

neutrinos from both the W boson

and bquark.

In eac

hcase

we ha ve

tak

en m t=

174 GeV

[22]. The dashed

curv eis the result

obtained

ignoring the con tribution

24

100 1000

0.0001 0.001 0.01 Fig.

1. Plot

of\Omega Nz

2ff for

top

quarks

injected

into the core

of the

Sun

and

Earth as afunction

of the

injection

energy E t.

See

text

for adescription

of the

curv es. from

bquarks.

For neutrinos

from the Sun,

the b-quark

con tribution

is ab out

10% at low

energies,

and increases

to roughly

25% at energies

around 1000

GeV. The increased

imp ortance

of the

b-quark

neutrinos

from the Sun

at higher

energies isdue to the

fact

that the neutrinos

from W bosons

are more

energetic

and therefore

more strongly

absorb ed as they

tra vel

through

the Sun.

In Fig.

1, we

also

illustrate

the effect

of varying

the top-quark

mass. The

dotted curv es ab ove

and

belo wthe

solid curv es in eac

hband

are the total\Omega

Nz

2ff

25

obtained using m t=

131 GeV

(a conserv

ativ elo wer

limit

[23]) and m t=

225

GeV (the 3oe upp

er limit

suggested

by the

CDF

data [22]),

resp ectiv

ely .

We ha ve

crafted

the follo wing

analytic

fit whic

hpro

vides ago od appro

ximation (better than 10%)

to the\Omega

Nz

2ff over

the range

m t^

E t^

3000

GeV

for neutrinos

from the Sun

(the tw olo

wer

solid

curv es curv

es in Fig.

1):

log 10 h\Omega

Nz

2ff

fit_t;i i

= A i(log

10

E t)

2\Gamma

B i(log

10

E t)+

C i:

(5: 10)

For neutrinos,

A *=

\Gamma 0 :825,

B *=

\Gamma 3 :31,

and C *=

\Gamma 5 :39.

For an tineutrinos,

A _*=

\Gamma 0 :889,

B _*=

\Gamma 3 :94,

and C _*=

\Gamma 6 :40.

Of course,

ifthe top-quark

mass

differs from 174 GeV,

these coefficien

ts will

change

sligh tly .

6. SummaryIn

this pap er we

ha ve

calculated

the differen

tial energy

spectra of muon

neutrinos and an tineutrinos

from particle

deca yin

the Sun

and Earth.

These

results will be useful

for computing

the rates

for up ward-m

uon even ts in neutrino telescop

es from

annihilation

of WIMPs

that ha ve

accum

ulated in the

Sun

and/or Earth. The analytic

expressions

we ha ve

deriv

ed for

neutrino

spectra

from deca yof particles

injected in the

Sun

result

from the tw oor three-b

ody

deca ykinematics

and include

the effects

of hadronization.

The results

for particle deca yin

the Sun

tak ein

to accoun

tthe additional

effects of stopping

of

hea vy hadrons

in the

Sun

and stopping

and absorption

of neutrinos

as they

pass through

the Sun.

Neutrino

spectra are giv en for

all the

WIMP-annihilation

channels that will pro duce

energetic

neutrinos.

These expressions

will giv eresults for the

flux

of neutrino-induced

up ward

muons

accurate

to better

than

O(10%) and can be used

in place

of Mon

te Carlo

calculations.

In particular,

the energy

spectra from injection

of o/_o/ ,c _c, and

b_b, pairs

in

the Earth

are giv en by Eqs.

(3.2),

(3.8), and (3.11),

resp ectiv

ely .The

neutrino

spectra from injection

of W

+W

\Gamma and

ZZ pairs

in the

Earth

are giv en by Eq.

(4.9), and the spectrum

from injection

of top

quarks

in the

Earth

is giv

en by

Eq. (5.4).

The an tineutrino

spectra from particle

deca ys in the

Earth

are

the same

as the

neutrino

spectra. The neutrino

spectra from injection

of o/_o/ ,

26

W

+W

\Gamma , and

t_t pairs

in the

Sun

are obtained

by applying

Eq. (2.6)

to Eqs.

(3.2), (4.9), and (5.4),

resp ectiv

ely ,and

the neutrino

spectra from injection

of b_b and

c_c pairs

in the

Sun

are giv en by Eqs.

(3.9) and (3.12),

resp ectiv

ely .

Recall that interactions

of an tineutrinos

with the solar

medium

differ than those

of neutrinos,

so the

an tineutrino

and neutrino

spectra from particle

deca yin

the Sun

differ.

The neutrino

spectra from deca ys of Higgs

bosons

in the

Sun

and Earth

can be obtained

from the spectra

of the

Higgs

deca ys pro

ducts

by

applying Eq. (2.11).

The probabilit

yof detecting

aneutrino-induced

up ward

muon

is prop

ortional to the

square

of the

neutrino

energy; one po wer

comes

from the chargedcurren tcross

section for muon

pro duction,

and the other

comes

from the muon

range. We argued

that in most

cases,

the detector

thresholds

are small

enough

compared to the

WIMP

mass that the thresholds

can be ignored.

If so,

then

the up ward-m

uon flux from

WIMP

annihilation

is prop

ortional

to the

second

momen tof the neutrino

energy distribution,

Eq. (1.2)

will pro vide

an accurate

estimate of the

muon

flux from

agiv en WIMP

candidate,

and detailed

information ab out

the shap

eof the neutrino

energy distribution

isnot required.

For this

reason, we ha ve

pro vided

analytic

expressions

for the

scaled

second momen ts

of the

neutrino

distribution,

\Omega N z2ff ,for

eac hWIMP-annihilation

channel. The

scaled second momen ts from

injection

of o/_o/ ,b

_b, c_c,

W

+W

\Gamma , ZZ

,and

t_t pairs

in the

Earth

are giv en by Eqs.

(3.3), (3.13),

(3.14), (4.5), (4.6), and (5.8),

resp ectiv

ely .The

scaled second

momen ts from

injection

of o/_o/

and

t_t pairs

in

the Sun

are giv en by Eqs.

(3.4) and (5.10),

resp ectiv

ely .Those

from injection

of b_b and

c_c pairs

in the

Sun

are giv en by Eq.

(3.15),

and those

from injection

of W

+W

\Gamma and

ZZ pairs

are giv en by (4.10).

The scaled

second momen ts from

Higgs-b oson deca yin

the Sun

and Earth

can be obtained

from those

of the

Higgs deca ypro

ducts

using Eq. (2.12).

The results

for the

second

momen ts, m ~O/2\Omega

Nz

2ff,

for neutrinos

and antineutrinos from injection

of particle-an

tiparticle pairs in the

Sun

with

energy

equal to the

WIMP

mass, m ~O/,

are

plotted

in Fig.

2and

Fig. 3, resp

ectiv

ely .

The strongest

signals are from

those particles

that deca yimmedia

tely and directly to neutrinos,

the o/_o/ and

gauge-b

oson final states,

and the signal

from top

27

mc (G

eV)

mc2 <Nz2> (GeV2)

Neu trin o S ign al

0 200

400 600 800 100 0

0 500

100 0

150 0

200 0

Fig. 2. The

second

momen t,m ~O/2\Omega

Nz

2ff,

of the

neutrino

energy distribution

from injection

of particles

with energy

equal to the

WIMP

mass m ~O/in

the Sun.

The solid

curv eis for t_t pairs,

the upp er (lo wer)

dashed

curv eis for W

+W

\Gamma

(Z Z)

pairs,

and the upp er dot-dash

curv eis for o/_o/ pairs.

At the

bottom

are

the b_b and

c_c curv

es, and

the b_b curv

eis sligh

tly higher

than the c_c curv

e.

quarks isonly sligh tly weak

er. The

turno

ver in the

gauge-b

oson signal

strength

isdue to absorption

of high-energy

neutrinos in the

Sun;

the signals

from other

particles will turn

over similarly

at larger

energies.

The Figures

illustrates

the

imp ortance

of hadronization

and stopping

of hea

vy hadrons:

the signals

from

band cquarks

are significan

tly smaller

than those

from particles

that do not

from hadrons.The results

listed here con tain

the effects

of hadronization

of hea

vy quarks

and interactions

of hea

vy hadrons

and neutrinos

with the solar

medium.

We

ha ve

included

only the neutrinos

from prompt

particle deca y. In some

cases,

there will be additional

neutrinos from secondary

deca ys, but

their

con tribution

28

mc (G

eV)

mc2 <Nz2> (GeV2)

An tine

utri no Sig nal

0 200

400 600 800 100 0

0

100 0

200 0

300 0

400 0

Fig. 3. Same

as Fig.

2, but

for an tineutrinos.

to the

up ward-m

uon flux will be negligible.

For example,

there will be neutrinos

pro duced

by gauge-b

oson deca yto

o/leptons

and band

cquarks

in addition

to the

prompt

neutrinos

from gauge-b

oson deca y, but

we estimate

that their

con tribution

to the

up ward-m

uon flux will be less

than

4% of the

total.

Also,

there will be additional

neutrinos from the cquarks

pro duced

by bdeca

y, but

we estimate

that these

will con tribute

roughly 10% of the

total

up ward-m

uon

flux. The additional

con tribution

to the

up ward-m

uon flux from

deca ys of o/

leptons pro duced

by charged-curren

tin teraction

of o/neutrinos

as they

pass

through the Sun

are negligible.

If our

neutrino

spectra are used

to compute

even trates

for detection

tec hniques

whic hare

prop ortional

to the

first

mom ent

of the

neutrino

energy distribution

(suc has

searc hes for con tained

even ts), then

the con tribution

from the neutrinos

from secondary

deca ys whic

hw eneglect

ma yb esignifican

tand the even

trates

will be underestimated.

29

The neutrino

energy spectrum

from deca yof agiv en particle

dep ends

on

the energy

of the

deca

ying

particle,

and itma

ydep end on the

polarization

state

as well.

We ha ve

evaluated

the effect

of polarization

on the

up ward-m

uon flux;

we found

that itis nev er more

than a10%

effect and can typically

be accoun

ted

for easilyThe

largest

theoretical

uncertain ty inv

olv es mo

deling

the effect

of stopping

of hea

vy hadrons

in the

Sun.

This can be traced

to the

uncertain

ty in the

values

adopted for the

stopping

coefficien t, E c[c.f.

Eq. (2.7)].

Ho wev

er, neutralino

annihilation into band

cquarks

is alw

ays

accompanied

by annihilation

into

o/leptons (with abranc

hing ratio enhanced

by radiativ

ecorrections

[25]). If

hea vy enough,

WIMP annihilation

is predominan

tly to t_t pairs

and possibly

gauge-b oson final states,

so the

uncertain

ty in the

total

up ward-m

uon flux due

to uncertain

ty in the

stopping

effect is diluted.

For all annihilation

channels,

there isalso

the uncertain

ty in the

absorption

and stopping

length for neutrinos

and an tineutrinos

in the

Sun,

but this is small

at energies

near aT eV,

and

is virtually

negligible at low

er energies.

These uncertain

ties do not

effect

the

neutrino spectra from the Earth.

Note that these

theoretical

uncertain ties must

also enter

Mon te Carlo

calculations

and are not intro

duced

by our

analytic

treatmen t. 7. Ac kno

wledgmen

ts

M.K. was supp

orted

by the

U.S.

Departmen

tof Energy

under con tract

DEFG02-90ER40542, and G.J.

was supp orted

by DE-F

G02-85ER40231

at Syracuse Univ ersit y.

30

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32

