The Suppression of Neutralino Annihilation into Zh

arXiv:hep-ph/0602111v5 14 Nov 2006

(willbeinsertedbytheeditor)

The Suppression of Neutralino Annihilation into

Zh

LabonneBenjamin 1,2 ,NezriEmmanuel 3 ,OrloJean 2 1

CentredePhysiqueThéorique,CNRSLuminy, ase907,F-13288MarseilleCedex9,Fran e 2

LaboratoiredePhysiqueCorpus ulaire,IN2P3-CNRS,UniversitéBlaisePas al,F-63177AubièreCedex,Fran e 3

Servi edePhysiqueThéorique,UniversitéLibredeBruxelles,CampusdelaPlaineCP225,Belgium February,therst,2006

Abstra t TheIndire tDete tionofneutralinoDarkMatterismostpromisingthroughannihilation hannels produ ing a hard energy spe trum for the dete ted parti les, su h as the neutralino annihilation into

Zh

. A an ellation however makes this parti ular annihilation hannel generi ally subdominant inthe hugeparameterspa e ofsupersymmetri models.This an ellation requiresnon-trivialrelations between neutralinomixingsandmasses,whi hwederivefromgaugeindependen eandunitarityoftheMSSM.To showhowthe an ellationovershootsleavingonlyasubdominantresult,weuseaperturbativeexpansion inpowersoftheele troweak/supersymmetrybreakingratio

m

Z

/m

χ

.

1Introdu tion and motivations

Thereisnodoubtthatstandardmodelsofparti lephysi s and osmology alone annot des ribe the full wealth of observationaldata re ently olle tedonawidevarietyof length s ales. Ad-ho as it may seem, the Dark Matter (DM)hypothesis[1,2℄isprobablypartoftheminimalset of extraingredients needed to a ount for the in reased gravitational self-attra tion of matter on s ales ranging from galaxies tothe fullvisible universe. Moreexoti in-gredients likerepulsiveDark Energy, ormodi ationsof gravity itself, might also be ome ne essaryto ope with theapparenta elerationof theuniverse. Intheabsen e of a onvin ingunied theoreti alsolutionto these both issues, experimental sear hes are the only way to prove thevalidityofhypothesesliketheexisten eofaDM par-ti le.Forinstan e,therstissuewouldbesettledifanew parti lewerefoundanditsnon-gravitationalintera tions measured to be ompatible with the Cold Dark Matter reli density required by Cosmi Mi rowaveBa kground and larges alestru ture formation[3,4℄.However,these intera tions are then by denition weak, and we should notbesurprised that their eviden esare extremelyhard toobtain,mu hlikeforPauli'sneutrino.Debateslikethe one aroundDAMA's laim [512℄ for dire t dete tion of DMareillustrativeofthisdi ulty.Thisiswhyitis ru- ialtobeableto ross- he kandunderstandresultsinas manydierentwaysaspossible,forwhi h adeniteand wellmotivatedDMtheoreti alframeworkisne essary.In this work,weshallkeepwiththewell-studied supersym-metri lightestneutralino(

χ)

ofmSugraorMSSMmodels. Corresponden e to: labonne lermont.in2p3.fr, nezriin2p3.fr,orloin2p3.fr

Aparti ularly ru ial ross- he kwouldbetheIndire t Dete tion of theneutralino annihilation produ ts, whi h wouldshowthatsu hannihilationdidindeedo urinthe past,froze outat somepointand re-startedin hotspots likethegala ti enterorthesolar ore,wheredark mat-ter later a umulated. However, to identify an Indire t DMsignalandpossiblydeterminetheneutralinomassby looking at uxes of e.g. photons or neutrinos from su h hot spots, it is essential to be able to distinguish that signalfromthestandardbut poorlyknownastrophysi al ba kground.Thesebeing hara terizedbyenergyspe tra with fairly universal power laws, Indire t Dete tion will bemostsu essfulwhenneutralinoannihilationpro eeds through primary hannels whi h provide se ondary pho-tons orneutrinos with the hardest possible spe tra,and asharpenergy ut-o aroundtheneutralinomass.From g.1(dis ussedin AppendixA), themostpromising an-nihilation hannels are into a

τ

+

_τ

₋

pair, into twogauge bosons(

χχ → W

+

_W

−,

or

ZZ

whi hhasthesameshape) orintoonegaugebosonandaHiggs-Englert-Brout(HEB) boson[13,14℄(

χχ → Zh

).

However,thesefairlyuniversalspe traneedtobe weigh-ted by the a tual model-dependent bran hing ratios to givethenal indire t DM dete tionsignal.It wasnoted long ago [15℄ that the

Zh

hannel is then suppressed, whi h an be numeri ally he ked

1

using the DarkSusy (3.14.02version)[16℄andtheSuspe t(2.003version) ode [17℄: the top plot of g. 2 typi ally shows asuppression by three orders of magnitude for various mSugra mod-els with

tan β = 10

. To qualify this suppression, let us startby ontributionsweexpe t tobedominant,namely

1

Temporarily failing su h he k after an update of the DarkSusy-Suspe tinterfa ewasa tuallythestartingpointof thiswork.

2 LabonneBenjamin 1,2 ,NezriEmmanuel 3 ,OrloJean 2

:TheSuppressionofNeutralinoAnnihilationinto

Zh

PSfragrepla ements

x = E

ν

/m

χ

d

N

/

d

x

m

χ

= 1

TeV 10 10 10 10 10 10 1 1 -1 -1 -2 2 3

Zh

W

+

_W

−

b¯b

t¯

t

τ

+

_τ

−

dN/dx

m

χ

= 1

TeV 10 1 -1 -2 2 3

Zh

W

+

_W

−

b¯b

t¯

t

τ

+

_τ

−

x = E

γ

/m

χ

d

N

/

d

x

m

χ

= 1

TeV 10 10 10 10 10 10 1 1 -1 -1 -2 2 3

Zh

W

+

W

−

b¯b

t¯

t

τ

+

_τ

−

Figure1.Theshapeofdierentialneutrino(top)andgamma (bottom) uxes oming from neutralino annihilations into various primary hannels. The normalisations (depending on bran hing ratios) have been arbitrarily res aled to ompare theshapeofall hannels;thedependen eon

m

χ

(

1

TeVhere) isweak.

thosefromSMparti leex hanges(inour asean

s

- hannel

Z

)sin esuperpartnersarene essarilyheavier.Asseenin the bottom g. 2, this ontribution not only dominates, but even overwhelms the total ross-se tion. We there-fore need to understand a double suppression, that rst an elsthislarge ontributionandse ondbringsthetotal

Zh

annihilationbelowother hannels.The an elling on-tribution ne essarilyinvolvesnon-SMparti leex hanges, whi hseems ontradi torywiththefa tthatong.2,the an ellationgetsbetterwithin reasing

m

0

and

m

1/2

,i.e. formaximallybrokensupersymmetry.

To be asgeneralaspossible, this an ellation should be he kedinasupersymmetrybreakingindependentway. This is done in g. 3 in the more general MSSM, while keeping the usual GUT relation

M

1

=

5

3

tan θ

W

M

2

≃

0.5M

2

. A an ellation up to three orders of magnitude thusseemsageneri propertyof everybroken supersym-metrytheory.

Tobemore on rete,letusnowfo usontheparti ular mSugra model with

m

0

= 3000 GeV

,

m

1/2

= 800 GeV

,

A

0

= 0

,

tan (β) = 10

and

µ > 0

,markedbythebla kstar in g.2.Theannihilation ross-se tionsat restare:

vσ cm

3

_/s

χχ → tt

χχ → Zh

Z

ex hange

1.83 × 10

−28

4.72 × 10

−28

Alldiagrams

1.03 × 10

−28

_{1.48 × 10}

−31

vσ (χχ → all) = 1.05 × 10

−28

0

500

1000

1500

2000

2500

0.005

0.002

0.001

0.0005

0.0002

0.0001

PSfragrepla ements

m

0

(GeV )

m

1/2

(

GeV

)

σ(χχ→Zh)

Z+χ

σ(χχ→All)

0

500

1000

1500

2000

2500

505

605

765

895

1025

1155

1285

1415

1545

1675

1805

1935

2

5

10

20

50

PSfragrepla ements

σ(χχ→Zh)

_Z

σ(χχ→All)

m

0

(GeV)

m

1/2

(

G

eV )

Figure 2. Neutralinoannihilationbran hingratiosto

Zh

for various

m

0

and

m

1/2

values:

Z

-ex hangeonly(bottom),

Z

and

χ

ex hanges(top)

2000

1500

1000

500

100

150 200

300

400

500

600

700

800

900

PSfrag repla ements

µ

(GeV)

M

1

(

G

eV )

10

−

5

10

−

4

10

−

3

10

−

2

10

−

1

1

Figure 3.Contour plotof

σ(χχ→Zh)

Z+χ

σ(χχ→Zh)

Z

intheMSSM;white dots orrespondtomSugramodelsofg.2.

LabonneBenjamin 1,2 ,NezriEmmanuel 3 ,OrloJean 2

:TheSuppressionofNeutralinoAnnihilationinto

Zh

3

Theannihilationisdominatedbythe

t¯

t

hannel,3 or-dersof magnitudelargerthan the

Zh

one[18℄.However, whenrestri tingtothe

Z

ex hangediagram,theyare om-parable,with

Zh

slightlylarger.Thisiseasilyunderstood by exhibiting the ouplingsand kinemati fa tors in the amplitudes:

Z

χ

χ

t

t

Z

χ

χ

Z

h

A χχ → tt

Z

∝

−g

2

m

t

O

Z

11

cos

2

_θ

W

A (χχ → Zh)

Z

∝

−g

2

m

Z

O

Z

11

cos

2

_θ

W

where the

χχZ

oupling

O

Z

11

is dened in terms of the neutralinomixingmatrix

2

N

(seeAppendixD.1):

O

ij

Z

= (N

i4

N

j4

− N

i3

N

j3

)

(1)

It remains to understand why other ex hanges an- el the

Zh

hannel and not the

tt

one. For the latter, the

t

- hannel sfermion ex hange anbe madearbitrarily smallbytakinglargeenough

m

0

,sothatthemain ontri-bution pro eeds viaa(SM)

Z

boson ex hange, onform-ing to naive expe tations. However, two other diagram are involved in the

Zh

annihilation hannel: (1) an

s

- hannelpseudos alar

A

ex hange,whi h anbenegle ted forlarge

m

0

(inwhat followswewillalwaysworkin this pseudos alarde ouplinglimit(47)),and(2)

t

- hannel ex- hanges of the4neutralinos,whi h annot bede oupled be auseof stronglinks betweenthe ouplingsinvolvedin ea hdiagram.Theselinks appearin theexpressionofthe annihilation amplitudes [19,20℄ derived in the next se -tion:

A (χχ → Zh)

Z

=

−ig

2

√

₂

cos

2

_θ

W

m

2

χ

m

2

Z

β

Zh

× O

11

Z

(2)

A (χχ → Zh)

χ

=

ig

2

√

₂

cos

2

_θ

W

m

2

χ

m

2

Z

β

Zh

×

(3)

×

4

X

i=1

2O

Z

1i

O

h

1i

(m

χ

i

− m

χ

) m

Z

2m

2

χ

+ 2m

2

χ

i

− m

2

h

− m

2

Z

wherethe

χ

i

χ

j

h

oupling

O

h

ij

isdenedinAppendix(49) as (wherethede oupling ondition(47)hasbeenused):

O

h

ij

= (c

W

N

i2

− s

W

N

i1

) (s

β

N

j4

− c

β

N

j3

) + (i ↔ j)

(4)

Itis learthat these ond amplitude(3) annoteasilybe negle ted and might turn out omparable with the rst one(2). However,it is less learwhybothshould an el

2

Inwhatfollow,wewillalwaysworkwiththehypothesisof CPviolatingphasesabsen e,inorderthe

N

matrixtobereal (seeAppendixC)

with high pre ision, espe ially given as dierent-looking ouplingsas(1)and(4).

A toy example of the above an ellation is provided by theannihilation of aspin singlet

t¯

t

pair into

Zh

in a StandardModelwithout

SU (2)

.Onemayrstbepuzzled to get a an ellation between an

s

- hannel

Z

-ex hange, whi honlyinvolvesgauge ouplings,and a

t

- hannel top ex hange,whi hinvolvesanaprioriindependentYukawa oupling.However,onesoonrealizesthattheexisten eof the

s

- hannel requires both spontaneous breakingof the gaugesymmetryforthe

ZZh

vertex,andanaxial oupling for the

t¯

tZ

vertex. This last ouplingex ludes ontribu-tionstothetopmassotherthanthegaugebreaking

yhhi

one. The two hannels

g

1

m

2

Z

gm

z

∝ g

1

m

t

y

are then both proportional to

g/hhi

and may an el. In ontrast with this simple ase, the neutralino annihilation studied be-low is ompli ated by the presen e of another sour e of mass, namely the SUSY breaking Majorana mass terms forthegauginos.

To analysethis an ellation, westart in se tion 2 by derivingtherelevantamplitudes.Inse tion3,gauge inde-penden eisusedtodrawarstlinkbetweenthe ouplings, whi hisshowntofollowfromthegaugeinvarian eofthe massmatrix.Unitarityathighenergyisthenusedin se -tion4toderivease ondrelation,whi his ombinedwith the rst one to show that a an ellation is possible. In se tion 5,thestru ture of Rayleigh-S hroedinger pertur-bationtheoryis thenused toshowthat this an ellation isstrongerthanexpe ted.

2Amplitudes for Neutralino Annihilationat Rest

Letusstartbyderivingthepolarizedamplitude

A (Zh)

Z

for neutralino annihilation into

Zh

via

Z

ex hange. By onstru tion, neutralinos are their own antiparti les, so that an initial pair of lightest neutralinosat rest is ne -essarily in an antisymmetri spinsinglet state. Thenal state ontainingaHEBs alar,theoutgoing

Z

boson po-larizationneedsto belongitudinal, and anbe hosenas

z

-axis. It is then more onvenient to use heli ity ampli-tudes [15,19℄ than unpolarized ross-se tions [20℄. The Feynman diagram and rules dened in the Appendix D givetheamplitude

Z, q

χ, p

χ, p

′

Z, k

h, k

′

A (χχ → Zh)

Z

=

−iC

A

11

C

Z

q

2

_{− m}

2

Z

(χ (p

′

) γ

µ

γ

5

χ (p))

(5)

×



g

µν

−

q

µ

q

ν

m

2

Z



ε

ν

(k)

where

χ = u

isthein omingneutralino,

χ = v

the outgo-ingoneinanarbitrary hoi eofarrowsdire tions.

u

and

v

areexternalDira spinorsatrestinthe hiralbasis(see Appendix B),namely:

χ (p

0

) = √m

χ

 ξ

_ξ

s

s



; with ξ

+

1

2

=

 −1

0



or ξ

₋

1

2

=

 0

1



for aspinupordownalong the

z

-axis,and similarlyfor theantiparti le:

χ (p

′

0

) = √m

χ

(−η

s

′

, η

s

′

)

Thanks to the Majorana ondition (40), des riptions of the same neutralino as a parti le with

ξ

s

′

or asan an-tiparti le with

η

s

′

areequivalentprovided:

η

s

′

= −iσ

2

ξ

s

′

(6)

Thepolarizationofthelongitudinal

Z

bosonbeing

ε

ν

_{(k) =}

(k

z

_{, 0, 0, k}

0

_)/m

Z

,andtheinitialmomentumatrestsimply

q

µ

_{= (2m}

χ

, 0)

,thepolarizedamplitude(5)is:

A (χ

_↑

χ

_↓

→ Zh)

Z

=

iC

A

11

C

Z

4m

2

χ

− m

2

Z

m

χ

m

Z

(σ

µ

11

+ ¯

σ

µ

11

)

(7)

×



g

µν

−

q

µ

q

ν

m

2

Z



k

z

, 0, 0, k

0

= −2iC

11

A

C

Z

m

χ

m

3

Z

k

z

(8)

Noti ethetime-likestru tureoftheinitialstateve tor

(σ

11

µ

+ ¯

σ

µ

11

) = (2, 0, 0, 0)

:apurelyaxial ouplingtalksonly with thes alarpart ofthe two spinsat rest. Noti ealso thedisappearingofthe

Z

poleat

m

χ

= m

Z

/2

.

Expressing

k

z

_{= m}

χ

β

Zh

in terms ofthe onventional kinemati fa tor

β

Zh

=

s

1 −

(m

h

+ m

Z

)

2

4m

2

χ

s

1 −

(m

h

− m

Z

)

2

4m

2

χ

_m

_χ

_{≫ m}

≈

_Z

1

and using the denitions (43) and (46) of ouplings in termsofneutralinomixings givenintheAppendix D,we nallynd:

A (χ

_↑

χ

_↓

→ Zh)

Z

=

−ig

2

_O

Z

11

cos

2

_θ

W

m

2

χ

m

2

Z

β

Zh

(9)

Togetthe amplitudewithreversed heli ities,wejust need to repla e (1,1) omponents of the Pauli matri es in (7)by(2,2) omponents,andtaketheextrasignfrom theMajorana ondition(6) into a ount.Thanksto this sign, only an antisymmetri initial state an ontribute, andgiveswiththe orre tnormalizationfa tor:

A (χχ → Zh)

Z

=

−ig

2

√

₂

cos

2

_θ

W

m

2

χ

m

2

Z

β

Zh

O

11

Z

(10)

in agreementwith (2)andexistingresults[19,20℄. Thisamplitude(10)hastobe omparedwiththeresult ofasimilar omputationfortheannihilationinto

t¯

t

:

A χχ → tt

Z

=

−ig

2

√

₂

cos

2

_θ

W

m

χ

m

t

m

2

Z

β

tt

T

3

O

Z

11

(11) where

β

t¯

t

=

q

1 − m

2

t

/m

2

χ

and

T

3

=1istheweakisospinof thetopquark.Noti ethedierentpowerof

m

χ

,favouring the

Zh

hannelforlargemasses.

Wehaveseenin the introdu tion that

t

- hannel neu-tralinosex hangesshould reversethis on lusion. Follow-ingthesamepath,their ontributionis

χ

i

, Q

χ, p

χ, p

′

Z, k

h, k

′

A (χχ → Zh)

χ

i

=

−iC

A

1i

C

1i

h

Q

2

_{− m}

2

χ

i

× (χ (p

′

_{) ( /}

_{Q + m}

χ

i

) γ

µ

_γ

5

χ (p)) ε

µ

(k)

with

Q

µ

_{= p}

µ

_{− k}

µ

.Aftersomealgebraonthe

t

- hannel, and adding the

u

- ontribution (same diagram as before with

p

and

p

′

inter hanged),onends:

A (χχ → Zh)

χ

i

=

ig

2

√

₂

cos

2

_θ

W

m

2

χ

m

2

Z

β

Zh

×

(12)

×

2O

Z

1i

O

h

1i

(m

χ

i

− m

χ

) m

Z

2m

2

χ

+ 2m

2

χ

i

− m

2

Z

− m

2

h

The problem is now to understand how the amplitudes (12) and (10) an el with the

10

−3

pre ision shown in theintrodu tion.This anhappenonlyifthesumof se -ond lines of (12) whi h ontain four powersof the neu-tralino mixing matrix

N

somehowredu es to

O

Z

11

∼ N

2

asa onsequen eofsomesymmetry. We alreadynoti ed that supersymmetryhadto be maximallybrokenforthe an ellation to take pla e. We are thus left with gauge invarian ewhi h isinvestigatedin thenextse tion.

3Gauge Independen e and Gauge Invarian e of the Mass Matrix

Theprevious omputationswereperformedintheunitary gauge,wheremassivegaugeeldshave ompletelyeaten aGoldstoneboson.Onewaytoobtainnon-trivialrelations ofthekindweseekistoworkinthe

R

ξ

-gaugefamilyof't Hooft[21℄,andrequireindependen eof theresultonthe gauge-xingparameter

ξ

.

Let us rst noti e that the neutralino ex hange dia-grams are gauge independent by themselves. Indeed, in

LabonneBenjamin 1,2 ,NezriEmmanuel 3 ,OrloJean 2

:TheSuppressionofNeutralinoAnnihilationinto

Zh

5

a ve tor supermultiplet, only the bosoni gauge eld is gaugedependent,and nottheasso iatedgaugino. More-over,higgsinosareasso iatedwiththerealpartof omplex s alars,whi harealsogaugeindependent.

We anthus on entrateonthe

Z

ex hangediagram. Whengoingfromunitaryto

R

ξ

-gauges,this ontribution splits in two

ξ

-dependent diagrams:one with Goldstone bosonex hange,andanotherwiththe

Z

-bosonex hange. To exhibit the an ellation of their

ξ

dependen e, it is onvenientinthe

Z

propagator

−i

q

2

_{− m}

2

Z



g

µν

−

(1 − ξ) q

µ

q

ν

q

2

_{− ξm}

2

Z



tode omposethelongitudinalpartas:

m

2

Z

(1 − ξ)

(q

2

_{− m}

2

Z

) (q

2

− ξm

2

Z

)

=

1

(q

2

_{− m}

2

Z

)

−

1

(q

2

_{− ξm}

2

Z

)

.

(13)

The rst term is nothing but the longitudinal unitary gaugepropagator,andthese ond exhibitsafakepole at the Goldstone mass

q

2

_{= ξm}

2

Z

, with a wrong sign [22℄. The

ξ

-dependent

Z

ex hange

A

Zξ

= A

Z

+ A

ZG

orre-spondingly de omposes into the previously obtained

A

Z

(5)andaGoldstone-likeamplitudewithgauge ouplings:

A (χχ → Zh)

ZG

= −i

C

A

C

Z

m

2

Z

(χ (p

′

) /

qγ

5

χ (p))

q.ε (k)

q

2

_{− ξm}

2

Z

whi h,using themass-shell onditionfortheinitial state

¯

χ/

qγ

5

χ = 2m

χ

χγ

¯

5

χ

, be omes:

A (χχ → Zh)

ZG

=

−ig

2

_O

Z

11

2 cos

2

_θ

W

m

χ

m

Z

χγ

5

χ

q.ε (k)

q

2

_{− ξm}

2

Z

(14) WenowturntothegenuineGoldstonebosonex hange. The

ZZh

ouplingisrepla edbythe

GZh

one:

G

χ

χ

Z

h

C

Z

Gh

= iC

G

(k − k

′

)

µ

where C

G

= g/2 cos θ

W

(15) and the s alar propagator is simply

i/(q

2

_{− ξm}

2

Z

)

. But thingsaremoresubtleforthe

Gχχ

vertex:evenifthe neu-tralGoldstonebosonispartofthe

Z

bosonintheunitary gauge,itdoesnothavethesame ouplingto neutralinos:

C

χ

G

i

χ

j

=

igO

G

ij

2 cos θ

W

(16) with

O

ij

G

= (N

i2

c

W

− N

i1

s

W

) (c

β

N

j3

+ s

β

N

j4

) +

(i↔j)

(17)

Forthe lightest neutralinoannihilation,

i = j = 1

, and sin ethe ouplingisimaginary,theGoldstoneonly ouples totheaxialpartoftheneutralino.Theamplitudeforthe Goldstoneex hangediagramisthus:

A (χχ → Zh)

G

= −C

χχ

G

C

G

χ (p

′

) γ

5

χ (p)

(18)

×

(q + k

′

₎

µ

q

2

_{− ξm}

2

Z

ε

µ

(k)

Re alling kinemati s

(q = k + k

′

₎

, the polarization on-dition

(k.ε (k) = 0)

, and ouplings denitions, we nally have:

A (χχ → Zh)

G

= i

g

2

_O

G

11

2c

2

W

χγ

5

χ

q.ε (k)

q

2

_{− ξm}

2

Z

(19)

Now, omparing (14) and (19), gauge independen e re-quiresarelationbetween ouplings:

O

Z

11

m

χ

m

Z

= −

1

2

O

G

11

whi h by (1) and (17) an be expressed in terms of the neutralinomixingsandmasses:

N

14

2

− N

13

2

m

χ

m

Z

(20)

= − (N

12

c

W

− N

11

s

W

) (s

β

N

14

+ c

β

N

13

)

This relation an be extended for theannihilation of an arbitrarypairofneutralinos

χ

i

− χ

j

in thesame hannel:

O

Z

ij

m

χ

i

+ m

χ

j

m

Z

= −O

G

ij

whi hisashort versionoftherathernon-trivialidentity:

(N

i4

N

j4

− N

i3

N

j3

)

m

χ

i

+ m

χ

j

m

Z

(21)

= − (N

i2

c

W

− N

i1

s

W

) (s

β

N

j4

+ c

β

N

j3

) −

(i↔j)

Althoughnot ompletelyidenti al,this relationbears similarities with the ombination of masses and mixings involved in (3). It is therefore interesting to noti e that it only involves the neutralino mass matrix, and an be derivedin the followingway.Bydenition of themixing matrix

N

(42),wehave:

(N M )

ij

= m

i

N

ij

Thenforanymatrix

P

,thefollowingidentitieshold:

Asaparti ular ase,ifwetakefor

P

theisospinoperator that ips the rst higgsino sign ompared to the se ond one:

P = T

3

=

 0 0

0 −σ

3



=







0 0 0 0

0 0 0 0

0 0 −1 0

0 0 0 1







were overthegaugeindependen erelation(21).The par-ti ular form of the right hand side of this relation then followsfrom the spe ial stru ture of thesymmetri neu-tralinomassmatrix:

M =



A C

C

T

_B



where

A

is a

2 × 2

diagonal matrix ree tingthe Majo-rananatureofgauginos,

B

isa

2 × 2

anti-diagonalmatrix ree tingtheDira nature ofthe

SU (2)

- hargedhiggsino pair, and

C ∼ m

Z

is a

2 × 2

matrix that vanisheswhen

SU (2)

is unbroken and with null determinant to other-wiseensuremasslessnessofthephoton.These onditions make

P M + M P

o-diagonal, explainingthe non-trivial vanishingoftheleft-handsideof(22)with

m

Z

.

Theseremarksstressthe entral roleplayedbygauge invarian e in the stru ture of the mass matrix. The ap-pearan e of

P = T

3

is no surprise, asall

χχZ

ouplings ndtheir rootin thegaugeinvariant

˜

h˜

hZ

:higgsinos an oupletothe

Z

bosonthankstotheir

U

Y

(1)

hargewhi h di tatesaDira behaviour.Only thespontaneous break-ing of

SU (2)

and

U

Y

(1)

arriedby

C

an thensplit this degenerateDira systemintoapairofMajoranaparti les.

4High Energy Unitarity

Despite vague similarities, the gauge independen e rela-tions (21) do notyet explain the an ellation of (2) and (3), and in parti ular their dierent powers of mixings

N

ij

. Toget further, weneed another relation. Using the pin hte hnique[23℄,wethereforeturntothehighenergy behaviouroftheamplitude.Indeed,fromrotation invari-an e,theoutgoing

Z

bosonmusthaveapurelongitudinal polarization, and it is well known that this an lead to oni ts with the perturbative unitarity onstraint that s attering amplitudes be bounded by a onstant

A(s →

∞) < K

.

Having he ked gaugeindependen e,we an for sim-pli ity use the Feynman gauge

ξ = 1

, to have a purely transverse

Z

propagatorfreefromhighenergydivergen es. Thedangerous diagramswhi hmust an elare then the

s

- hannelGoldstoneand

t

- hannelneutralinosex hanges. A light-like ve torbeing orthogonal to itself, the po-larizationve torat highenergiesis approximately

ε

µ

(k) ≃

k

µ

m

Z

.

(23)

Usingthisandthekinemati identity

2q.k = q

2

_+m

2

Z

−m

2

h

, theamplitude forGoldstoneex hange (19)in this gauge be omes:

A (χχ → Zh)

G

=

ig

2

2 cos

2

_θ

W

O

G

11

2m

Z

χγ

5

χ

(24)

×



1 +

2m

2

Z

− m

2

h

q

2

_{− m}

2

Z



Therst onta ttermintheparenthesisgivesa on-tribution

A ≃

√

s =

pq

2

, whi hisdivergentandviolates unitarityinthehighenergylimit,whilethese ondtermis betterbehavedthankstotheappearan eofapropagator denominator.From adiagrammati point ofview,thisis expressed by splitting thediagram into a pin hed part andarest:

G

χ

χ

Z

h

=



χ

χ

Z

h

+

Rest

Fortheneutralinoex hange hannel,wehave

A (χχ → Zh)

χ

= −2i

4

X

i=1

C

A

1i

C

1i

h

Q

2

_{− m}

2

χ

i

×χ (p

′

) ( /

Q + m

χ

i

) γ

µ

_γ

5

χ (p)

k

µ

m

Z

Expressing

k = /

/

p− /

Q = (/p−m

χ

)−( /

Q−m

χ

i

)+(m

χ

−m

χ

i

)

, the rsttermvanishes on-shell, whilethe se ond an els thepropagatorpoleto givea onta ttermandarest:

A (Zh)

_χ

=

−ig

2

2c

2

W

4

X

i=1

O

Z

1i

O

h

1i

m

Z

× {χγ

5

χ

(25)

−

m

χ

− m

χ

i

Q

2

_{− m}

2

χ

i

χ ( /

Q + m

χ

i

) γ

5

χ}

whi h anbediagrammati allyrepresentedby:

χ

χ

χ

Z

h

=



χ

χ

Z

h

+

_Rest

′

Can ellation of the onta t terms in (24) and (25), requiresthefollowingidentity:

1

2

O

G

11

=

X

i

O

1i

Z

O

1i

h

or, using (1), (4) and (17), thefollowingrelation among mixings

N

:

(N

12

c

W

− N

11

s

W

) (s

β

N

14

+ c

β

N

13

)

(26)

=

4

X

i=1

(N

14

N

i4

− N

13

N

i3

) ×

{(c

W

N

12

− s

W

N

11

) (N

i4

s

β

− N

i3

c

β

)

+ (c

W

N

i2

− s

W

N

i1

) (N

14

s

β

− N

13

c

β

)}

LabonneBenjamin 1,2 ,NezriEmmanuel 3 ,OrloJean 2

:TheSuppressionofNeutralinoAnnihilationinto

Zh

7

As ompli ated asit may seen, this equationsimply fol-lowsfromtheorthogonality ondition

N

ij

N

kj

= δ

ik

,with

i, k = 3, 4

.Wehavethereforeshownthathighenergy per-turbativeunitarityoftheamplitudeisguaranteedbythe unitarityof themixing matrix.

By ombining(26)and(20),anewnon-trivialidentity among ouplingsisobtained:

m

χ

m

Z

O

Z

11

= −

X

i

O

Z

1i

O

h

1i

whi hisevenlesstrivialwhenusingthedenitionsof

O

Z

ij

and

O

h

ij

:

m

χ

m

Z

N

2

i4

− N

i3

2

= − P

4

i=1

(N

14

N

i4

− N

13

N

i3

)

(27)

×{(c

W

N

12

− s

W

N

11

) (N

i4

s

β

− N

i3

c

β

)

+ (c

W

N

i2

− s

W

N

i1

) (N

14

s

β

− N

13

c

β

)}

Thisidentityrelatesthe ouplingsappearingin

Z

-ex hange (2)and

χ

-ex hange(3),andsuggeststorewritethese ond as:

A (χχ → Zh)

χ

= 2i

√

2β

Zh

m

χ

m

Z

g

2

cos

2

_θ

W

×

m

χ

m

Z

O

Z

11

+

4

X

i=1

R

i

!

(28)

where the rst term exa tly an els the

s

- hannel

Z

ex- hange,andthese ondissubdominantinthehighenergy limit

s ≫ m

2

Z

, m

2

χ

. It is howevernot lear why those re-maindersshouldbesubdominantatrest,be auseevenfor large

s = 4m

2

χ

, there is onfusion between dynami ally suppressed ontributions

∼ m

2

Z

/s ≪ 1

and neutralino mixing suppressed ones

∼ m

2

Z

/m

2

χ

≪ 1

. Moreover, the denition

R

i

= O

Z

1i

O

h

1i

2m

χ

i

m

χ

− 4m

2

χ

− 2m

2

χ

i

+ m

2

h

+ m

2

Z

2m

2

χ

+ 2m

2

χ

i

− m

2

h

− m

2

Z

(29)

implies that for

m

Z

≪ m

χ

,

R

3

≈ −R

4

are omparable with the rst term in (28), and only their sum be omes negligible.Toanalyzethis nal an ellation,a perturba-tiveexpansionin

m

Z

/m

χ

isthereforeneeded.

5Perturbation theory

In the SUSY-de oupling limit (

m

Z

≪ M

1

, M

2

, µ

), the neutralinomassinAppendixCisnaturallysplit[24℄into

M = M

0

+ W

,withaleading ontribution

M

0

=







M

1

0

0

0

0 M

2

0

0

0

0

0 −µ

0

0 −µ 0







andaperturbation

W = m

Z







0

0

−s

W

c

β

s

W

s

β

0

0

c

W

c

β

−c

W

s

β

−s

W

c

β

c

W

c

β

0

0

s

W

s

β

−c

W

s

β

0

0







triggeredbyEW-symmetry breaking.Followingstandard Rayleigh-S hroedingerperturbationexpansionwestartby solvingtheunperturbedeigensystem

N

0

M

0

N

0 T

= m

0

whoseeigenvaluesare

m

0

_{= diag (M}

1

, M

2

, −µ, µ)

andwhose eigenve tors

ϕ

0

n

formthemixingmatrix

N

0 T

= (ϕ

0

1

, ϕ

0

2

, ϕ

0

3

, ϕ

0

4

) =









1 0 0

0

0 1 0

0

0 0

√

1

2

√

−1

2

0 0

√

1

2

1

√

2









Therstorder orre tionstoeigenvalues

m

1

n

=

ϕ

0

n





W



ϕ

0

n



aresimplythediagonalelementsof

W

0

_{= N}

0

_{W N}

0 T

:

W

0

_{= m}

Z











0

0

s

W

_√

s

−

2

s

W

s

+

√

2

0

0

−

c

W

_√

s

−

2

−

c

W

s

+

√

2

s

W

_√

s

−

2

−

c

W

_√

s

−

2

0

0

s

W

s

+

√

2

−

c

W

s

+

√

2

0

0











(30) where

s

±

= s

β

±c

β

.Fromthestru tureoftheperturbation

W

, these diagonalelements learly vanish. Furthermore, rstorder orre tionstoeigenve tors



ϕ

1

i

 =

X

j6=i

W

0

ji

m

0

i

− m

0

j



ϕ

0

j



anberegroupedinto

N

1 T

_{= (ϕ}

1

1

, ϕ

1

2

, ϕ

1

3

, ϕ

1

4

)

with:

N

1

= m

Z











0

0

−

s

W

C

1

M

2

1

−µ

2

s

W

S

1

M

2

1

−µ

2

0

0

c

W

C

2

M

2

2

−µ

2

−

c

W

S

2

M

2

2

−µ

2

−

_√

s

W

s

−

2(M

1

+µ)

c

W

s

−

√

2(M

2

+µ)

0

0

s

W

s

+

√

2(µ−M

1

)

c

W

s

+

√

2(M

2

−µ)

0

0











and

C

1,2

= (µs

β

+ M

1,2

c

β

)

,

S

1,2

= (M

1,2

s

β

+ µc

β

)

. Forabino-likeneutralino(

M

1

< M

2

, µ)

,the

Zχχ

ver-texdoesnotexistwithoutele troweaksymmetrybreaking andknowingthediagonalizationmatrix

N = N

0

_{+ N}

1

up to rst order in

m

Z

allows in fa t to ompute the rst non-trivial ontributionto

Z

-ex hange(2)whi happears atse ond order:

A (χχ → Zh)

Z

= −

√

2β

Zh

M

2

1

m

2

Z

g

2

_c

2β

t

2

W

m

2

Z

(M

2

1

− µ

2

)

(31) Atthesameorder,therstnon-trivial ontributionto

χ

-ex hange (3)requires onlyoneperturbation ofthe

Zχχ

i

vertex,andnoperturbationofthe

hχχ

i

vertex,whi hdoes exist in the absen e of ele troweak symmetry breaking. Furtherexpandingthepropagatorinpowersof

m

Z

would givethesamestru tureas(28):

A (χ → Zh)

χ

=

√

2β

Zh

M

2

1

m

2

Z

g

2

cos 2β tan

2

θ

W

(32)

×

m

2

Z

(M

2

1

− µ

2

)



1 +

1

2

m

2

Z

+ m

2

h

M

2

1

+ µ

2



,

suggesting that the remainder is

O(m

4

Z

)

, i.e. twoorders lowerthantheleadingterm.However,tormlyestablish this on lusionrequires a justi ationfor the absen eof terms

O(m

3

Z

)

, whi h weshall nowndbyexamining the generalstru tureoftheperturbativeexpansion.

Whensolvingtheeigensystem

(M

o

+ W )

_ij

N

jl

T

= N

il

T

m

l

(33)

bypowerexpansionsin

m

Z

foreigenvaluesand eigenve -tors:

m

i

= m

0

i

+ m

1

i

+ ...

(34)

N = N

0

+ N

1

+ ...

(35)

we antoallorders hoosethe orre tiontoaneigenve tor in

N − N

0

orthogonalto the orrespondingunperturbed ve torin

N

0

: the only pri e is to end up with non-unit ve torsin

N

.This hoi ehoweversimpliesthere urren e relationforthesolutionatorder

q

to:

m

q

i

= N

0

W N

q−1 T

ii

(36)

N

0

N

q T

ji

=

N

0

_{W N}

q−1 T

ji

m

0

i

− m

0

j

−

q−1

X

p=1

m

p

_i

N

0

_N

q−p T

ji

m

0

i

− m

0

j

The expressions for

q = 0

, and

q = 1

were given above. We saw that

M

0

is

2 × 2

blo k-diagonal, and so is

N

0

, whereas

W

andthus

N

1

areblo ko-diagonal.Following there urren e, this an begeneralized,to showthat

N

q

must be blo kdiagonal for

q

evenandblo ko-diagonal for

q

odd. Be ause of this stru ture,

m

q

i

will vanish for

q

odd, so that

m

i

(m

Z

)

is holomorphi in

m

2

Z

. Ina sim-ilar way, diagonalblo ksof

N

have apurely even power expansionin

m

Z

, whereaso-diagonalones only ontain oddpowers.

These results an be extended to the amplitudes

A

Z

and

A

χ

whi h then ontainonlyeven powersof

m

Z

: the lowest order

O(m

−2

Z

)

vanishes for both as it should to allow for a smooth

m

Z

→ 0

limit, the next

O(m

0

Z

)

is equal and opposite for

s

- and

t

- hannel ex hanges, and the remainder

O(m

2

Z

)

di tates the amplitude of the

Zh

annihilation hannel at rest to belowerthan the

t¯

t

pair hannel. This anbelooselyexpressedas:

A (Zh) ∝

m

2

Z

m

2

χ

≪ A tt
∝

m

t

m

χ

≪ A (Zh)

Z

∝ 1

Theorderofmagnitudeandthepowerofthesuppression

σ (χχ → Zh)

Z+χ

σ (χχ → Zh)

Z

∝

m

4

Z

M

4

1

thenagreewiththosedisplayedings.2,3.

6Con lusion

In this work, we have given a quantitative understand-ingofwhyneutralinosat rest annotannihilate predomi-nantlyinto

Zh

.Thepossiblerelevan eofthispro essfor indire tDMdete tionhasbeenshownintheIntrodu tion, before pointing out similarities and dieren es between the

tt

and

Zh

annihilation hannels.Wealsostressedhow mu hanaiveestimateofthislast hannel anfailby ignor-ing thesubtlebut tightlinks between ouplingsimposed by symmetries, espe ially broken ones.Be ause of these links,ane an ellationdoeso urwhi hrequiresa loser look. Having noti ed that this an ellation was getting nerwithin reasinglybrokensupersymmetry,weshowed thatbrokengaugesymmetryhadtobeinvestigated.This wasdone both at thelevelof gauge independen e in

R

ξ

gauges,andofunitarityathighenergy,knowntobe deli- ateforlongitudinalgaugebosons.Both onstraintsledto non-trivialrelations among ouplingswhi h showed that indeed,theSMparti lesex hanges anbe an elledby su-perpartnersex hanges, asheavyasthese mightbe. How-ever, to quantitatively estimate the importan e of what remains after this an ellation required to show that a perturbativeexpansionoftheamplitudes in

m

Z

/m

χ

on-tained only even powers. Whether su h an ellation an beextended from large

s

to large

m

χ

=

√

_s/2

at restfor alldiagramssueringfromlarge

s

unitarityproblems, re-mains anopenquestion.

It is a pleasure to a knowledge fruitful dis ussions with A. Djouadi, P. Gay, R. Grimm, J.-L. Kneur, Y. Mambrini, V. Morenas, G. Moultaka and J. Papavassiliou. This work was partiallyfundedbytheGdR2305SupersymétrieoftheFren h CNRS.

A Indire t Dete tion Energy Spe tra

The neutrino and photon dierential energy spe tra of gure 1 were extra ted from a PYTHIA simulation of

10

6

eventsfor ea h hannel, shown as afun tion of

x =

E

ν, γ

/m

χ

.Forthehard(anti-)neutrinosfromthe

Z

-boson in the

Zh

hannel that on ernedus most,wehavebeen areful to orre t the unpolarized PYTHIA results by a fa tor

∝ x(1 − x)

translatingthepurely longitudinal po-larizationof the

Z

-boson, whi h suppressesforward neu-trinos with respe t to the

W W

hannel. In spite of this fa tor,the

Zh

hannelprodu esthenext-to-hardest neu-trinospe trum.

Forphotons,thehard(at) omponentofthe

Zh

han-nel around

x ≈ 1

omes from

h

loop-de aying into two

LabonneBenjamin 1,2 ,NezriEmmanuel 3 ,OrloJean 2

:TheSuppressionofNeutralinoAnnihilationinto

Zh

9

photonsand therefore onlyappearsat the largestvalues of

x

.Thisleavesonlyasmallnumberofevents(~10/bin) and large statisti al errors whi h do not appear in the t shownin the bottom gure1.There are of ourseno eventsfor

x > 1

, but the pre ise shapeof this vanishing (probably similar to that of neutrinos from

W W

above, as shown)ishiddenbythese errors.

B Dira and Majorana fermion onventions TorepresenttheCliordalgebraofDira matri es

{γ

µ

, γ

ν

} = 2η

µν

weusethe hiralbasis:

γ

µ

₌



0 σ

µ

σ

µ

0



γ

5

=

 −1 0

_{0 1}



where

σ

µ

_{= (1, σ) σ}

µ

_{= (1, −σ)}

are4-dextensionsofthePaulimatri es

σ

= σ

1

_{, σ}

2

_{, σ}

3

. ForMajoranaspinors,weuse[25℄theusualDira F eyn-manrulesafter having hosenanarbitraryorientationof thespinorlines.TheMajorana onditionis

χ = χ

c

_{= Cχ}

T

(37) andtheplanewaveexpansionofaMajoranaeldoperator isthus[26℄:

χ (x) =

Z

_d

3

_p

(2π)

3/2

X

s=±

a

s

(p) u

s

(p) e

−ip.x

(38)

+a

†

s

(p) v

s

(p) e

ip.x

ImplementingtheMajorana ondition(37)in(38),wend therelations

u

s

= Cγ

0

v

s

∗

(39)

v

s

= Cγ

0

u

∗

s

(40)

whi h allow to ip the orientation of external Majorana lines.

CNeutralino Mass Matrix

The neutralino mass eigenstatesare linear ombinations ofgauginoandhiggsinoelds

 ˜

B, ˜

W

3

, ˜

H

b

, ˜

H

t



χ

i

= N

i1

B + N

˜

i2

W

˜

3

+ N

i3

H

˜

b

+ N

i4

H

˜

t

whi hdiagonalizethemassmatrix:

M =







M

1

0

0 M

2

m

Z

× C

m

Z

× C

T

0 −µ

−µ 0







(41)

C

is the

2 × 2

ele troweak-breaking ontribution to this neutralinomassmatrix:

C =

 −s

_c

W

c

β

s

W

s

β

W

c

β

−c

W

s

β



with

s

W

= sin θ

W

, c

W

= cos θ

W

s

β

= sin β,

c

β

= cos β

The normalized eigenve tors an be olle ted into a unitarymatrix

N

satisfying:

N

⋆

_{M N}

−1

_{= M}

D

(42)

where

M

D

isadiagonalmatrix ontainingneutralinomasses. Inthe absen e of CPviolating phases, thematrix

N

anbe hosenasarealmatrix,andatleastoneneutralino massisthennegative.

DLagrangian terms for relevant ouplings D.1

Z − χ − χ

oupling

L =

1

2

4

X

i,j=1

χ

i

γ

µ

C

ij

V

− C

ij

A

γ

5

χ

j

Z

µ

with

C

ij

V

=

g

4 cos θ

W

O

Z

ij

− O

Z∗

ij

(43)

C

ij

A

=

g

4 cos θ

W



O

Z

ij

+ O

Z

∗

ij



(44) and

O

Z

ij

= N

i4

N

j4

∗

− N

i3

N

j3

∗

Intheabsen eofCP violation,thereisabasissu h that

C

V

ij

= 0

. D.2

h

-

Z

-

Z

oupling

L =

1

2

C

Z

hZ

µ

Z

µ

(45) where

C

Z

= −

gm

Z

cos θ

W

sin (α − β)

(46) In pra ti e, we will always work in the de oupling limit

m

A

≫ m

Z

,whi himplies:

α = β −

π

2

(47)

D.3

h − χ − χ

oupling

L =

g

₂

hχ

i

s

α

L

ij

∗

P

L

+ L

ij

P

R

(48)

+c

α

K

ij

∗

P

L

+ K

ij

P

R

χ

j

with

K

ij

=

1

2

N

i4

(N

j2

− tan θ

W

N

j1

) + (i ↔ j)

L

ij

=

1

2

N

i3

(N

j2

− tan θ

W

N

j1

) + (i ↔ j)

Thesymmetri formof

K

and

L

omesfrom thefa t weareworkingwithMajoranaparti les:

χ

i

(1 ± γ

5

) χ

j

= χ

j

(1 ± γ

5

) χ

i

Forreal

N

,thisLagrangiansimpliesto:

L =

1

₂

4

X

i,j=1

C

ij

h

hχ

i

χ

j

where

C

ij

h

=

g

2 cos θ

W

O

h

ij

(49) and

O

h

ij

= (c

W

N

i2

− s

W

N

i1

) (s

α

N

j3

+ c

α

N

j4

) + (i ↔ j)

D.4

Z

-fermion-fermion oupling

L =

X

f

f γ

µ

C

f f

ZV

− C

f f

ZA

γ

5

f Z

µ

where

C

f f

ZV

= −

g

2 cos θ

W

T

3f

− 2 sin

2

θ

W

Q

f

C

f f

ZA

= −

g

2 cos θ

W

T

3f

Q

f

and

T

3f

arethe hargeandthirdweakisospin ompo-nent,withtheusualnormalization:

T

3top

= 1

,and

Q

top

=

2

3

.

Referen es

1. G. Bertone, D. Hooper, J. Silk, Phys. Rept. 405, 279 (2005).

2. K. Griest, M. Kamionkowski, Phys. Rev. Lett. 64, 615 (1990).

3. M. Tegmark, M. Zaldarriaga, Phys. Rev.Lett. 85, 2240 (2000).

4. L.Verde,Int.J.Mod.Phys.A19,1121 (2004).

5. R.Bernabeietal.,Phys.Lett.B389,757(1996);424,195 (1998);450,448(1999);480,23(2000);509,283(2001); Riv. NuovoCim. 26,1(2003);Int.J.Mod.Phys.D 13, 2127(2004);P.Bellietal.,Phys.Rev.D61,023512(2000); Phys.Rev.D66,043503(2002).

6. D. S. Akerib et al., Phys. Rev. D68, 082002 (2003);D. Abramsetal.,Phys.Rev.D66,122003(2002).

7. D. S. Akerib et al. [CDMS Coll.℄, Phys. Rev. Lett. 93, 211301(2004).

8. V. Sanglard [the EDELWEISS Coll.℄, astro-ph/0306233 (2003).

9. V. Sanglard [the EDELWEISS Coll.℄, Phys. Rev. D71, 122002(2005).

10. J.Jo humetal.,Nu l.Phys.Pro .Suppl.124,189(2003); G.Angloheretal.,Phys.Atom.Nu l.66,494(2003)[Yad. Fiz.66(2003)521℄.

11. L. Stodolsky, F. Probst, talk at The Dark Side of the Universe, AnnArbor,May2004.

12. M. Brhlik and L. Roszkowski, Phys. Lett. B464, 303 (1999); C.J. Copi and L.M. Krauss, Phys. Rev. D67, 103507 (2003); R.Foot,Phys.Rev.D69,036001 (2004); Mod.Phys.Lett. A 19, 1841 (2004); A.M.Green, Phys. Rev.D63,043005(2001);Phys.Rev.D66,083003(2002); Phys. Rev. D68, 023004 (2003) Erratum ibid. D69, 109902 (2004); A. Kurylov and M. Kamionkowski, Phys. Rev. D69, 063503 (2004); G. Prézeau, A. Kurylov, M. Kamionkowski,andP.Vogel,Phys.Rev.Lett.91,231301 (2003);C.Savage,P.Gondolo,and K.Freese,Phys.Rev. D70,123513 (2004); D.Smithand N. Weiner, Phys.Rev. D64, 043502 (2001); P.Ullio, M. Kamionkowski, and P. Vogel,JHEP0107,044(2001),P.GondoloandG.Gelmini, Phys.Rev.D71,123520(2005)

13. F.Englert,R.Brout,Phys.Rev.Lett.13,321(1964). 14. P.W.Higgs,Phys.Rev.Lett.13,508(1964).

15. M.Drees, M.M.Nojiri,Phys.Rev.D47,376(1993). 16. P.Gondolo, J.Edsjö, L. Bergström,P.Ullio, M. S helke

andT.Baltz,JCAP0407,008(2004).

17. A.Djouadi,J.-L.Kneur,G.Moultaka,hep-ph/0211331. 18. V. Bertin, E. Nezri, J. Orlo, Eur. Phys. .J. C26, 111

(2002).

19. G.Jungman,M.KamionkowskiandK.Griest,Phys.Rept. 267,195(1996).

20. T. Nihei, L. Roszkowski and R. Ruiz de Austri, JHEP 0203, 031(2002).

21. J.Rosiek,Phys.Rev.D41,3464(1989).

22. J. M. Cornwall, D. N. Levinand G. Tiktopoulos, Phys. Rev.D10,1145(1974).

23. J.M.Cornwall,Phys.Rev.D26,1453(1981);J.M. Corn-wall, J.Papavassiliou, Phys.Rev. D40, 3474 (1989); D. Binosi,J.PhysG30,1021(2004).

24. J. F. Gunion and H. E. Haber, Phys. Rev. D37, 2515 (1988).

25. H.E.HaberandG.L.Kane,Phys.Rept.117,75(1985). 26. R. N. Mohapatra and P. B. Pal, Massive Neutrinos in