The Inert Doublet Model: an Archetype for Dark Matter

1.8 MB·English

by  Laura Lopez Honorez

ULB-TH/06-27
The Inert Doublet Model: an Ar
hetype for Dark Matter
Laura Lopez Honorez, Emmanuel Nezri, Josep F. Oliver, Mi
hel H.G. Tytgat
Servi
e de Physique Théorique, Université Libre de Bruxelles,
CP225, Bld du Triomphe, 1050 Brussels, Belgium
7
0
0
2 Abstra
t

n The Inert Doublet Model (IDM), a two Higgs extension of the Standard Model with an unbroken
Z
a 2 symmetry, is a simple and yet ri
h model of dark matter. We present a systemati
analysis of the
J
darkmatterabundan
e and investigate the potentialities fordire
t and gamma indire
t dete
tion. We

9
showthat the modelshouldbe within the rangeoffuture experiments, likeGLAST and ZEPLIN. The

lightest stable s
alar in the IDM is a perfe
t example, or ar
hetype of a weakly intera
ting massive
2
v parti
le.
5
7
2 1 Introdu
tion
2
1 Contemporary
osmologi
alobservations
on
urtoindi
atethatthemajorityofmatterintheuniverse
6
not only does not shine but is not even made of ordinary atoms [1, 2℄. De
iphering the nature of this
0
so-
alled Dark Matter has be
ome one of the most important issue at the frontier of parti
le physi
s,
/
h astrophysi
s and
osmology. A profusion of dark parti
les have been proposed over the years and it
p
is mu
h hoped that present and forth
oming experiments will throw some light on the matter. For a
-
p review, see for instan
e [3, 4℄.
e In the present arti
le weinvestigate further arather mundane, although in our opinion quite inter-
h
esting, form of dark matter. The parti
le
andidate is a weakly intera
ting massive s
alar and it has
: H1 H2
v been advo
ated re
ently in [5℄ and [6, 7℄. The framework is that of a two Higgs doublets, and ,
Z
Xi version of the Standard Model with a 2 symmetry su
h that
H H H H .
r 1 → 1 and 2 →− 2
a
Z
2
All the (cid:28)elds of the Standard Model are even under . This model has been (cid:28)rst dis
ussed by
Z
2
Deshpande and Ma in [8℄. Following [6℄, we will assume that is not spontaneously broken, i.e.
H
2
does not develop an expe
tation value. Among other things, the dis
rete symmetry prevents the
H 0
2
appearan
e of (cid:29)avour
hanging neutral
urrents. The model is de(cid:28)nitely not the h i → limit of a
generi
two Higgs model like, for instan
e, the Higgs se
tor of the MSSM.
The most general, albeit renormalizable, potential of the model
an be written as
λ
V =µ2 H 2+µ2 H 2+λ H 4+λ H 4+λ H 2 H 2+λ H†H 2+ 5 (H†H )2+h.c. .
1| 1| 2| 2| 1| 1| 2| 2| 3| 1| | 2| 4| 1 2| 2 h 1 2 i (1)
U(1) λ = 0
5
There is an Pe

ei-Quinn global symmetry if . This limit is however not favored by dark
matter dire
t dete
tion experiments (se
tion 2.3).
SU(2) U(1) H
1
The × symmetry is broken by the va
uum expe
tation value of ,
v
H =
1
h i √2
v = µ2/λ =248 µ2 >0
with − 1 1 GeV while, assuming 2 ,
H =0.
2
h i
1
h
The mass of the Brout-Englert-Higgs parti
le ( or Higgs for short) is
M2 = 2µ2 2λ v2
h − 1 ≡ 1 (2)
H+ H A H
0 0 2
while the mass of the
harged, , and two neutral, and ,
omponents of the (cid:28)eld aregiven
by
M2 = µ2+λ v2/2
H+ 2 3
M2 = µ2+(λ +λ +λ )v2/2
H0 2 3 4 5
M2 = µ2+(λ +λ λ )v2/2.
A0 2 3 4− 5 (3)
H A H
0 0 2
Forappropriatequarti

ouplings, or isthelightest
omponentofthe doublet. Intheabsen
e
Z
2
of any other lighter -odd (cid:28)eld, either one is a
andidate for dark matter. For de(cid:28)niteness we
hoose
H A
0 0
. All our
on
lusions are un
hanged if the dark matter
andidate is instead. Following [6℄ we
H λ = (λ +λ +λ )/2
0 L 3 4 5
parameterize the
ontribution from symmetry breaking to the mass of by ,
h H
0
whi
h is also the
oupling
onstant between the Higgs (cid:28)eld and our dark matter
andidate . Of
v
the seven parameters of the potential (1), one is known, , and four
an be related to the mass of the
µ λ
2 2
s
alar parti
les. In the sequel, we take and as the last two independent parameters. The later
a
tually plays little role for the question of dark matter.
H
0
The abundan
e of as the dark matter of the universe as well as the
onstraints from dire
t
dete
tion experiments have been tou
hed upon in the re
ent literature [6, 7℄ (see also [9℄). In the
H
0
presentarti
leweinvestigatethe potentialitiesforindire
tdete
tionof usinggammarayteles
opes
and dete
tors. Forthe sakeof
ompleteness, wealsopresenta systemati
(albeit tree-level)analysisof
H
0
the abundan
e of the in the light of WMAP data, as well as the prospe
t for dire
t dete
tion by
underground dete
tors.
Before doing so, we would like to brie(cid:29)y emphasize some of the virtues of this, so-
alled, Inert
Doublet Model (IDM), both in general and for the issue of dark matter in parti
ular. True, the model
is ad ho
. Also, it does not address any deep issue, say the hierar
hy problem. Yet it is a very simple
extension of the Standard Model and its phenomenology is nevertheless very ri
h [5, 6, 10, 11℄. In
referen
e [6℄, Barbieri et al
onsidered this model as a way of pushing the mass of the Higgs parti
le
towardtheTeVs
ale,without
ontradi
tingLEPpre
isionmeasurements(seealso[10℄,andse
tion3.1
ofthepresentpaper). Also,inthesimplestversiondis
ussedheretherearenoYukawa
ouplingstothe
H
2
doublet, but it is somewhat natural to imagine
oupling it to (odd) right-handed neutrinos. This
opensthepossibilityofgeneratingthemassoftheSMneutrinosthroughloop
orre
tions,ame
hanism
introdu
ed by Ma in [5℄ and further addressed in a series of papers [12, 13, 14℄.
H
0
As a dark matter
andidate, is not without interest either. It belongs to the family of Weakly
Intera
tingMassiveParti
les(WIMP) andhasbeen proposedasadarkmatter
andidatebyMa. The
most a

laimed member of this family is
ertainly the neutralino, the lightest supersymmetri
parti
le
(LSP).Thesupersymmetri
extensionsoftheSMarewithoutdoubtveryinterestingandwellmotivated
but they involve many new parameters and their phenomenology is, to say the least,
omplex. It is
perhaps far fet
hed to
ompare the respe
tive advantages and disadvantages of the IDM and of the
H
0
MSSM but it is interesting that the shares many similarities with the LSP (weak intera
tions,
similar mass s
ale, et
) while being
onsiderably mu
h simpler to analyze. Interestingly, if we take
H
0
seriouslythe ideathat isthedominantformofdarkmatter, theparameterspa
eoftheIDMmodel
is quite
onstrained. It is also, to some extent,
omplementary to that of the MSSM, as we show in
H
0
the
on
lusions. And,
herry on the top, the phenomenologyof the as a
andidate for dark matter
isneatly intertwined with thatof the Higgsparti
le. In ouropinion,it is this
onjun
tion ofsimpli
ity
and ri
hness that makes the lightest stable s
alar a perfe
t example, or ar
hetype, of dark matter.
The plan of the paper is as follow. In the next se
tion (2) we give a summary of our methodology
forthe
al
ulationsofthedarkmatterabundan
e,thephoton(cid:29)uxforindire
tdete
tionbygammaray
teles
opesandthe
ross-se
tionsfordire
tdete
tion. Wethendis
ussinsomedetailstheresultsofour
analysis, in the light of WMAP data, and study the prospe
ts for both indire
t and dire
t dete
tion
by existing and forth
oming experiments. We give our
on
lusions in the last se
tion, together with

omparison of the Model with that of the MSSM, and the prospe
ts for future analysis.
2
2 Dark matter aspe
ts
2.1 Dark Matter abundan
e
H
0
The abundan
e of has been estimated in [6℄ and in [7℄ for two di(cid:27)erent mass ranges. As in these
papers, we
onsider only thHe standard freeze-out me
hanismW. ThereZare then esseMntially>tw80oGreeVgimes,
depending on the mass of 0 with respe
t to that of the and bosons. If H0 , the
H M >M ∼
0 annihilate essentially into Z and W boson pairs (see Figure (1)). For H0 h, the annihilation
h H
0

hannel into pairs opens (see Figure (2)). Otherwise, the annihilate essentially through an
H
0
intermediate Higgs, provided the Higgs itself is not too heavy. Some amount of
oannihilation of
A H+ Z W+
0
with the next-to-lightest s
alar parti
le (resp. ) through a (resp. ) boson
an be present
[15℄, a feature that substantially
ompli
ates the determination of the dark matter abundan
e (see
Figure (3)).
H
0
We have
omputed the reli
abundan
e of using mi
rOMEGAs2.0, a new and versatilepa
kage
for the numeri
al
al
ulation of Dark Matter abundan
e from thermal freeze-out [16℄. The latest
implementation of this
ode, whi
h was originally developed to study supersymmetri
models, allows
onetoenteranymodel
ontainingadis
retesymmetrythatguaranteesthestabilityofthedarkmatter
parti
le. The
ode takesadvantageofthe fa
tthatea
hodd(i.e. nonstandard)parti
lesofthemodel
H
0
will eventually de
ay into the lightest odd parti
le, in our
ase the . One
an then sum the system
of Boltzmann equations of all odd spe
ies and the reli
density of dark matter is
al
ulated by solving
the Boltzmann equation for the lightest parti
le only [17℄:
dY πg (T)
= ∗ M σv (Y2(T) Y2(T)),
dT r 45 Plh i − eq (4)
Y n /s
DM
where ≡ is the
omovingdensity of dark matter. All the pro
essesof the model enterin the
σv T = T =T
0
thermally averaged
ross-se
tion h i. Integrating Eq.(4) from ∞ to the reli
abundan
e
is given by M
Ω h2 =2.72 108 DMY(T ).
DM 0
× GeV (5)
Mi
rOMEGAs2.0 itself is build upon CALCHEP, a
ode for
omputing tree-level
ross-se
tions. We
have used both Mi
rOMEGAs2.0 and CALCHEP to
ompare with our analyti
al
al
ulations and to
performanumber∗of
ross-
he
ksthathavestrengthenedourfaith in thenumeri
alresultsreportedin
the present paper.
2.2 Indire
t dete
tion
Themeasurementofse
ondaryparti
les
omingfromdarkmatterannihilationinthehaloofthegalaxy
isapromisingwayofde
ipheringthenatureofdarkmatter. Thispossibilitydependshowevernotonly
on the properties of the dark matter parti
le, through its annihilation
ross-se
tions, but also on the
astrophysi
alassumptionsmade
on
erningthedistributionofdarkmatterinthehalothatsupposedly
surroundsourgalaxy. Thegala
ti

enter(GC)regionispotentiallyaveryattra
tivetargetforindire
t
dete
tion of dark matter, in parti
ular through gamma rays. The produ
ed gamma ray (cid:29)ux from the
annihilation of dark matter parti
les
an be expressed as
Φ dNi 1
γ = γ σ v ρ2 dl,
dΩdE dE h i i4πm2 Z (6)
Xi γ DM l.o.s.
m ρ σ v dNi/dE
where DM is the darkmatter parti
le mass, is the dark matterdensity pro(cid:28)le, h i i and γ γ
v
are,respe
tively,thethermallyaveragedannihilation
rossse
tiontimestherelativevelo
ity andthe
di(cid:27)erential gamma spe
trum per annihilation
oming from the de
ay of annihilation produ
ts of (cid:28)nal
i
state . The integral is taken along the line of sight. The pro
esses involved are shown in the Figure
(3), (cid:28)rst diagram and Figures (1) and (2).
∗
The Feynman rules and de(cid:28)nition (cid:28)les of the IDM to be used with mi
rOMEGAs
an be obtained upon request to the
authors of thepresentarti
le and will be in
ludedin thenextavailable version of the
ode.
3
10−8
The mcmod−e2ls.sw−1ill be
onstrainedbythe existingEGRET [18℄ experimentallim10i−t1o0n th1e0(cid:29)−u11x,∼
pcmho−t2o.nss−1 ∆Ω,=a1n0d−3the for1t0h−
5oming GLAST [19℄ expe
ted sensitivity, ∼ − photons
for and srad respe
tively.
Our purpose for the time being is merely to prospe
t the IDM with regard to gamma ray indire
t
dete
tion. In this paper, we fo
us on the parti
le physi
s parameter dependen
e of the model and
work withinRa=(cid:28)x8e.5d astrophysi
al framework, the popular Navarro-Frankρ-W=h0it.e3(GNeFVW.cm) −ha3lo pro(cid:28)le
0 0
[20℄. With kp
equaltothedistan
efromtoSuntotheGCand thedark
matter density in our neighborhood, the NFW pro(cid:28)le density
an be parameterized as
[1+(R /a)α](β−γ)/α R γ
0 0
ρ(r)=ρ ,
0 [1+(r/a)α](β−γ)/α (cid:18) r (cid:19) (7)
(a,α,β,γ) = (20,1,3,1)
with . It should be emphasized that the (dark) matter distribution in the
γ
innermost region of the galaxy is poorly known so that is not very
onstrained, a freedom that
an
give rise to very di(cid:27)erent values of the gamma ray (cid:29)ux. Typi
ally a suppression or an enhan
ement
of two orders of magnitude
an be obtained if one
onsiders respe
tively a halo with a (cid:29)at
ore (e.g.
γ = 0 γ 1.5
isothermal, ) or a deeper
usp ( ∼ ) [21℄
oming for instan
e from baryoni
infall. Hen
e,
dependingontheastrophysi
sassumptions,theestimationofgammaraysignal
anvaryquitestrongly
(see e.g. [23, 22℄).
1GeV
To
a∆l
Ωul=ate10t−h3e (cid:29)ux we have integrat∆edΩ(=6)10fo−r5a NFW pro(cid:28)le above and around a solid
angleof sradforEGRET and sradforGLAST.Thedi(cid:27)erentialspe
traofea
h

hannel are given by mi
rOMEGAs (we have
he
ked the agreement with Pythia simulations of [23℄)
σv
as well as h i at rest and we have integrated the square of the dark matter density along the line of
sight.
2.3 Dire
t dete
tion
A lo
al distribution of weakly intera
ting dark matter
ould be dete
ted [24℄ by measuring the energy
deposited in a low ba
kground dete
tor by the s
attering of a dark matter parti
le with a nu
lei of
H
0
the dete
tor. One distinguishes spin dependent and spin independent intera
tions. For intera
ting
Z
H q A q
0 0
with the quarks of the nu
lei, there are two spin independent pro
esses at tree level, −→
h
H q H q
0 0
and −→ (see Figure (4)). The experiments have rea
hed su
h a level of sensitivity that the
Z Z
ex
hange
ontribution is ex
luded by the
urrent experimental limits. Consequently, to forbid
A H
0 0
ex
hange by kinemati
s, the mass of the parti
le must be higher than the mass of by a few
100keV λ 0
5
(we will thus disregard the → limit). We assume that the main pro
ess here is the one
h
with Higgs parti
le ex
hange . The
ross se
tion at tree-level is [6℄
m2 λ 2
σh = r L f2m2,
H0−p 4π (cid:18)M M2(cid:19) p (8)
H0 h
f f 0.3 m
r
where is a form fa
tor estimated in the literature to be ∼ and is the redu
ed mass of the
M < M
system. [25℄M. In a>ddMition, for H0 W, we have negle
ted the oσne-loop ex4
.h6a1n0g−e13opfbtwo gauge
bosons. For H0 W, we in
lude this pro
ess with, following [7℄, 1H00−−6p ≃ . We take
into a

ount the
onstraints given by the CDMS [26℄ experiment (∼ pb) and the model is also

ompared to the next generation experiments like EDELWEISS II [2σ7℄ or10a−t8on-sizσe ex1p0e−ri1m0ent like
ZEPLIN [28℄ with the valley of their sensitivities respe
tively around ∼ and ∼ pb.
3 Analysis
σv
The pro
esses driving the (
o)annihilation
ross se
tion h i relevant for the reli
density from freeze-
H
0
out and indire
t dete
tion of gamma rays are shown in the Figures (1), (2) and (3). The -proton
intera
tions relevant for dire
t dete
tion are shown in Figure (4).
H W
0
There are two qualitatively distin
t regimes, depending on whether the is lighter than the
Z
and and/or the Higgs boson, in whi
h
ase annihilation pro
eeds through the diagrams of Figure
4
(3). At low mass, the se
ond diagram of (3) may
ontribute to the dark matter abundan
e if there is
A
0
a substantial number of at the time of freeze-out (i.e.
oannihilation).
H
0
In the plots of Figure (5), the reli
abundan
e of dark matter parti
les, the gamma ray (cid:29)ux
H
0
due to annihilation at the gala
ti

enter (indire
t dete
tion) and, (cid:28)nally, their
ross-se
tion for
σ
elasti
s
attering o(cid:27) a proton through Higgs boson H0−p (dire
t dete
tion) ex
hange are shown, for
M =120 200 M ,µ
twoparti
ularHiggsmasses( h GeV and GeV) in the ( H0 2) plane. We refer to this as
H H M =120
0 0 h
thelow massregime. Forhigher masses,similarplotsaredisplayedin Figure(6)(for
GeV), respe
tively for the darkmatterabundan
e, the gamma ray (cid:29)ux and the
ross-se
tionfor dire
t
dloegt(eΩ
tionh.2)Inloge(aΦ
h(of t−h2es−e1)t)hree
laosge(sσ, the (
olo)r) gradients
orrespond respe
tively to gradients in
H0 , γ
m s and H0−p pb .
M ,µ λ =0
Sin
e weplot ourresultsin the ( H0 2)plane,thediagonalline
orrespondsto L , i.e. to no
H λ λ <0 λ >0
0 L L L

ouplingbetween andtheHiggsboson. Awayfromthisline, in
reases,with (resp. )
∆MA =M M ∆MH =M M
above (resp. below) the diagonal. Also, we write 0 A0 − H0 and c H+ − H0.
In the plots of the dark matter abundan
e, the areas between the two dark lines
orrespond to
0.094 < Ω h2 < 0.129
DM
regions of the parameter spa
e su
h that , the range of dark matter energy
densities
onsistentwithWMAPdata. Inthe
rossse
tionandgammaray(cid:29)uxplots,thelinesindi
ate
the areasof the parameter spa
e within rea
h of the various experiments we take in
onsideration.
3.1 Constraints
The shaded areas in the plots of Figures (5) and (6)
orrespond to regions that are ex
luded by the
following
onstraints:
• Va
uum stability:
Va
uum stability (at tree level) demands that
λ > 0,
1,2
λ ,λ +λ λ > 2 λ λ .
3 3 4 5 1 2
−| | − (9)
p
λ < 0
L
As a result, negative
ouplings, and among others, are largely ex
luded. This is the
µ >M
shaded area in the domains 2 H0 of Figures (5) and (6).
• Perturbativity:
λ > 4π 1 < λ < 4π
i i
Strong
ouplings | | are ex
luded but intermediate
ouplings, | | , might
be tolerated. The latter
orrespond to the regions with horizontal lines. Noti
e that the mass
µ
2
splitting relative to the s
ale is smaller in the high mass regime (see Eq.(3)). For this reason,
goingawayfrom the diagonalline in the plotsof Figure(6), werunfasterinto the large
oupling
regime than in those of Figure (5). The regions ex
luded by the strong
oupling
onstraint is of
µ =M

ourse symmetri
with respe
t to the 2 H0 axis.
• Charged Higgs s
alar:
H+
The mass of the
harged s
alar is
onstrained by LEP data to be larger than 79.3 GeV
[M30℄. A<s3w0e (cid:28)x the mass di(cid:27)eren
es in our plots, this
onstraint translates in the ex
luded region
H0 GeV in the low mass regime abundan
e plots.
∼
• Ele
troweak Pre
ision Tests (EWPT):
H
2
New physi
s
an a(cid:27)e
t ele
troweak pre
ision measurements. The impa
t of the new doublet
S T

an be des
ribed in term of the , ele
troweak pre
ision parameters. As pointed out in [6℄,
H
2
with appropriate mass splittings between its
omponents, an
ould s
reen the
ontribution
T M 500 S
h
to the parameter of a large Higgs masses, ∼ GeV. The e(cid:27)e
ts on the parameter is
parametri
ally smaller for the region of parameter spa
e satisfying the previous
onstraints. In
T
all the
ase we have studied, we have
he
ked that the
ontribution to the parameterresulting
H
2
from the Higgs mass and the
omponents are
Monsi<ste2n0t0with EWPT. Unlike Barbieri et al,
h
weonly
onsideredsmallvaluesoftheHiggsmass, GeV.Givenour
hoi
eoftheHiggs
M H ∼ H
h 0 2
mass and of mass splittings between the parti
les and the other
omponents of , the

onstraints from EWPT are easily satis(cid:28)ed.
H
0
We now turn to the analysis of the model, starting with the low mass regime.
5
W+(Z)
H0 H0 W+(Z) H0 W+(Z)
h
PSfrag repla
ements PHS−fr(aAg0)repla
ements
PSfrag repla
ements
H0 W−(Z) H0 W−(Z) H0 W−(Z)
g2 g2 λ g
L
Figure 1: Annihilation
hannels into gauge bosons (cid:28)nal state with
orresponding
ouplings.
H
0 h
H h
0 H h
0
h
H
0
PSfrag repla
ementsHP0Sfrag repla
ehmentsH0 PSfrag repla
emhentsH0 h
λ λ2 λ λ
L L L 1
Figure 2: Annihilation
hannels into Higgs (cid:28)nal state.
H f H f(′)
0 hPSfrag repla
ements 0 Z(W+)
PSfrag repla
ements
f¯ A (H+) f¯
H 0
0
λ y g2
L f
Figure 3: (Co)Annihilation
hannels into fermion anti-fermion (cid:28)nal state.
6
H0 A H0 H0
PSfrag repla
ements
PSfrag repla
ements Z h
q q q q
λ y g2
L f
σ
Figure 4: Leading
hannels
ontributing to H0−p (dire
t dete
tion).
M
H
3.2 Low mass regime, around 100 GeV
W Z
The results of this se
tion are shown in Figure (5). Two pro
esses are relevant below the , or
h H H A Z
0 0 0
threshold: annihilation through the Higgs and
oannihilatibo¯bn with througHh ex
hange.
0
Both give fermion-antifermion pairs, the former predominantly into . As the mass of goes above
W Z h H WW ZZ hh
0
, or threshold, annihilation into , and be
ome in
reasingly e(cid:30)
ient, an e(cid:27)e
t
H
0
whi
h strongly suppressesZthe reli
density. ∆MA † ∆M
0
CoanTnihilatMionin/t2o5a mayo

Hurprovided isnottooimb¯bportant , roughly mustbeof
orderof fo ∼ H0 . Otherwise 0 annihilate dominantly into . This regime is quite interesting:
the intera
tions are either
ompletely known (i.e. ele
troweak intera
tions), or highly sensitive to the
M H h λ
h 0 L
Higgs boson mass and the e(cid:27)e
tive
oupling !
M =120 M =200
h h
Forthesakeofillustration,westudytwo
ases, GeVand GeVtoshowhowthe
predi
tions
hange asa fun
tionof the Higgsboson mass. A few generallessons
anbe extra
ted from
∆MA ∆MH
0 c
these two spe
i(cid:28)
examples. Note that the mass di(cid:27)eren
es and are (cid:28)xed respe
tively
∆MA
0
to10and50GeVwhile islargeenoughtoavoidthe
onstraintsfromdire
tdete
tionandsmall
Z
enough to have a some amount of
oannihilation into a .
We may distinguish (cid:28)ve di(cid:27)erent regions (see Figure (5), top left):
Regions 1 and 2 are ex
luded respe
tively by
harged Higgs produ
tion and va
uum stability
on-
λ
1
straints. The latter involves the self
oupling of the Higgs, and thus depends on the Higgs mass
M
h
(seeEq.(2)): region2isbroaderforsmallerHiggsmass . Althoughregion1isexperimentallyruled
H
0
out, it isinteresting to understandthetrends. In this region,thereli
abundan
e of de
reaseswith
M µ =M M =120 M =200
in
reasing H0 asonegoesalong 2 H0. Comparingthe h GeV and h GeV plots
M2 /M4
we observe that the gradients depend, again, on the Higgs mass. This is simply due to the H0 h
M
dependen
eofthe annihilation
rossse
tionthroughtheHiggs,i.e. smallerabundan
eforhigher H0
M µ = M
and/or lower h. In region 2, the abundan
e is smaller theλfurther oMne de<viaMtes from 2M >H0M, re-
(cid:29)e
ting the dependen
e of the annihilation
ross se
tion on | L|. For H0 W (resp. H0 h),
H H ¯bb H H hh ∼ M < M <∼ M
0 0 → (resp. 0 0 → ) is the dominant pro
ess. Otherwise, for W H0 h,
H H WW,ZZ ∼ ∼
0 0
→ dominate.
µ >InMregio,nM3, the
ouplings areratherlarge,but nevertheless
onsistentwiHthvHa
uumWs+taWbi−lity. Sin
e
2 H0 W, the dominant
ontribution to the
ross se
tion is given by 0 0 → pro
ess,
large enough to bring the reli
abundan
e far below WMAP data.
W
Region4isbelowthe threshold. Itistheonlyregion
onsistentwiththedarkmatterabundan
e
predi
tedby WMAP datain the lowmassregime. The pro
ess thatdetermines the reli
abundan
eof
H
0
is again the annihilation through the Higgs. Coannihilation pro
esses be
ome dominant near the
M +M M H M 40
resonan
eat A0 H0 ≈ Z. Thisistheoriginofthedipinthe 0reli
abundan
earound H0 ≈
M 50
GeV (
orresponding to A0 ≈ GeV for our
hoi
e of mass splittings). Similarly, annihilation near
†H H+ ∆MH
0 c

oannihilation is suppressedfor our
hoi
e of .
7
M M /2
the Higgs resonan
e generates a se
ond dip around H0 ≈ h . There is an island
onsistent with
M 80 WW
WMAPdata,whi
hextendsupto H0 ∼ GeV,atwhi
hpoint annihilationbe
omesimportant
and suppresses the reli
abundan
e.
M > M
Region 5, (cid:28)nally,
orresponds to H0 W. Annihilation into a gauge boson pair is dominant,
M <M
at least as long as H0 h. Gauge boson pair produ
tion dominates above the Higgs threshold for
µ <M µ >M
2 H0 while Higgs pair produ
tion is dominantMif 2 <60H00. In all instan
es, the reli
abundan
e
is too suppressed to be
onsistent with WMAP for H0 GeV.
∼
As expe
ted, and as revealedby visualinspe
tion, the photon (cid:29)ux plots shares some of the
hara
-
teristi
softheabundan
eplots. Theonlynewsalientfeatureistheabsen
eofadipattheZresonan
e.
A
0
This is of
ourse be
ause today, in
ontrast with the early universe, all are gone. Existing experi-
ments are not very
onstraining but future gamma ray dete
tion experiments, su
h as GLAST, might
severely
hallenge the model. As usual we should bear in mind the astrophysi
al un
ertainties: by

hanging the gala
ti
dark matter density pro(cid:28)le, we
an get higher (or smaller) (cid:29)uxes.
Theplotofthe
rossse
tionfordire
tdete
tionisprettytransparent. Su(cid:30)
estonoti
e(seese
tion
σ λ2
2.3)that H0−p at treelevelisproportionalto L. The dominant
ontributiontotheelasti
s
attering
µ =M µ M

ross se
tion is thus zero on the 2 H0 axis while it in
reases for larger values of | 2− H0|.
σ
The dependen
e on the Higgs mass (Eq. 8)
learly appears when
omparing the H0−p plots for
M = 120 M = 200 M > M
h GeV and h GeV. For H0 W, the one-loop
ontribution to the
ross se
tion
W
from10−13ex
hangeistakenintoa

ount. HoweveritdHoesnotaA(cid:27)e
tmu
htheresultsasit onlyamounts
0 0
for pb. Unless we suppose that the mass of and are nearly degenerate, existing dire
t
dete
tion experiments are not very
onstraining. Forth
oming experiments however might put a dent
on some of the solutions
onsistent with WMAP.
M M
H W
3.3 The high mass regime, ≫
The results of this se
tion are shown in Figure (6), left
olumn. No new annihilation
hannel opens if
M
H0 is heavierthan the Higgs orgaugebosons. There arethen essentiallytwosort of pro
esseswhi
h

ontrolboththeabundan
eandthegammaray(cid:29)ux: theannihilationintotwogaugebosons,dominant
µ <M µ >M
if 2 H0, and the annihilation into twoHiggs,whi
hdominatesif 2 H0. Coannihilationplays
little role. It a(cid:27)e
ts a bit the reli
abundan
e alongthe diagonalbut, even so, it is not the key pro
ess
to get the WMAP abundan
e.
The abundan
e of dark matter is suppressed over most of the area of the plot be
ause of large
quarti

oupling e(cid:27)e
ts on the
ross se
tions. Strong
ouplings are ex
luded on a physi
al basis, but
we found this limit nevertheless useful to understand the interplay between the various pro
esses. In
thepresentsubse
tion,wewill arguethatit ispossibletorea
hagreementwithWMAP data,but only
to the pri
e of some (cid:28)ne tuning between the di(cid:27)erent annihilation
hannels.
Consider(cid:28)rst theannihilationinto twoHiggsbosons(the dashedline in thegraphsFigure(7)). Its
λ =0 µ <M λ >0

ross se
tion is vanishing for L . For 2 H0 (
orresponding to L ), there is a destru
tive
µ > M
interferen
e between the diagrams of Figure (2), whi
h is absent if 2 H0. The annihilation
into gauge bosons depends on the quarti

ouplings between the s
alars (see Figure (1)). Indeed, the
λ
L
annihilation through an intermediate Higgs is
ontrolled by . Also, the t and u
hannels ex
hange
H
2
diagramsaresensitivetothemassdi(cid:27)eren
esbetweenthe
omponentsofthe doublet,whi
hdepend
λ Z λ +λ W
5 4 5
respe
tivelyon fortheannihilationinto bosonsandon fortheannihilationinto bosons.
λ 0
L
If ≃ , thereisno orlittleannihilationinto aHiggs. The relevantdiagramsarethenthequarti
A H+ λ =0
0 5
vertex with two gauge bosonsand the t and u
hannels with (resp. ) ex
hange. If (resp.
λ +λ = 0 Z W α2 /M2
4 5 ) the
ross se
tion into a (resp. ) boson pair is minimal and s
ales like W H0.
λ = 0 Z
5
If, for instan
e, 6 the annihilation into a boson pair re
eives a
ontribution whi
h grows like
α2 (M M )2/M4 W
W A0 − H0 Z. A similar result holds for the annihilation into a boson pair provided
λ +λ = 0 α2 (M M )2/M4 λ2/M2
4 5 6 . Noti
e that W A0 − H0 Z ∼ 5 H0. The e(cid:27)e
ts of weak isospin breaking
H λ
2 1
betweenthe
omponentsof isreminis
entofwhat happensfortheHiggsin the regimeof strong

oupling (the so-
alled Goldstone boson equivalen
e theorem [31℄).
8
To have an abundan
e of dark matter in agreement with WMAP, the mass splittings between the
H
2

omponents of must be kept relatively small. First be
ause large mass splittings
orrespond to
large
ouplings and se
ond be
ause the di(cid:27)erent
ontributions to the annihilation
ross se
tion must
λ = 0 M µ M M
be suppressed at the same lo
ation, around L (ie H0 ≃ ≃ A0 ≃ H+ in this
ase). This
is illustrated in Figure (7) for two di(cid:27)erent mass splittings. The (cid:28)rst plot is for small mass di(cid:27)eren
es
∆MA =5 ∆MH =10 ∆MA =10
0 c 0
( GeVand GeV), these
ondonefor(relatively)largersplittings (
∆MH =50
c
GeV and GeV). In the se
ond
ase, the
ross se
tion is too large to obtain an abundan
e
σv pb
WMAP

onsistent with WMAP, h i ∼ .
λ =0
L
In theMlimit>of60sm0 all masssplittings and vanishing wehavetherightamountof darkmatter
provided H0 GeV.Thisregime
orrespondstothenarrowregionaroundthediagonalin Figure
∼ M
(6). The abundan
e in
reases for in
reasing H0, but this
an be somewhat
ompensated by playing
with∆tMheAm=ass5splittings,∆wMhi
Hh, =as1d0is
ussed above, tend to in
rease the
ross sMe
tio>n.8F0o0r instan
e,
for 0 GeV and c GeV, we get the right reli
abundan
e for H0 GeV.
∼
There is however a limit to this trend. Indeed, the total annihilation
ross se
tion of a s
alar
H
0
parti
le, like our , is
onstrained by unitarity to be smaller than
4π x
σv unit f
h iv→0 ≈ M2 r π (10)
H0
aresultwhi
h,inthe
ontextofdarkmatterreli
sfromfreeze-out,hasbeen(cid:28)rstputforwardbyGriest
andKamionkowski[32℄. Inthepresentmodel,in
reasingthemasssplittingdrivestheannihilation
ross
se
tion toward the strong quarti

oupling regime. A similar result holds, for instan
e, for a weakly
intera
ting massiveneutrino
andidate of dark matter. In this
ase, both the neutrino and its
harged
1
partner should be kept lighter than, say TeV, sin
e their mass
omes from Yukawa
ouplings to the
Higgs (cid:28)eld. (See [33℄ and the dis
ussion in [32℄). In our
ase, the bulk of the mass of the
omponents
H µ
2 2
of the doublet
omes from the mass s
ale , whi
h is a prioriMarbit<rar1y3.0However, the unitarity
limit and WMAP data
onstraints, translate into the upper bound H0 TeV [32℄.
∼
Unfortunately neither dire
t, nor indire
t dete
tion experiments are sensitive to the large mass
region dis
ussed in this se
tion. Forth
oming dire
t dete
tion experiments might do better, but as the
plots show rather
learly, other forms of dark matter would then have to be introdu
ed in order to
explain the amount of dark matter that is
urrently observed. However, if the pro(cid:28)le of dark matter
in the galaxy is as assumed in the present paper (i.e. NFW), future gamma dete
tors (GLAST) will
probe most of the parameter spa
e
onsidered in this se
tion, keeping in mind that we had to (cid:28)nely
tune the mass splittings to obtain the right abundan
e.
4 Con
lusions
The dark matter
andidate of the Inert Doublet Model stands (cid:28)er
ely by the neutralino. The lightest
stable s
alar is a weakly intera
ting massive parti
le with a ri
h, yet simple, phenomenology and it
has a true potential for being
onstrained by existing and forth
oming experiments looking for dark
matter. For the sake of
omparison, we show in the right part of Figure (6) a fair sample of models,
(M ,Ω h2)
DM DM
both for the IDM and the MSSM. In the plane, we
learly see the two regimes (low
mass and high mass) of the IDM that may give rise to a relevant reli
density (i.e near WMAP). The
(100
MSSM models have a more
ontinuous behavior, with O GeV) dark matter masses. For indire
t
dete
tion,theIDM darkmatter
andidateshavetypi
allyhigherdete
tionratesthantheneutralinoin
SUSY models, espe
ially at high mass. It is in parti
ular interesting that the IDM
an give the right
reli
abundan
e in a range of parameters whi
h will be probed by GLAST. Dire
t dete
tion is also
promising but only at low masses, where the models are within rea
h of future ton-size experiments.
The high mass models, however, have too small
ross se
tions in the regions
onsistent with WMAP.
The phenomenology of the IDM
andidate for dark matter is intertwined with that of the Brout-
h h
Englert-Higgsparti
le, . It
ouldbe ofsomeinterestto investigatetheprospe
tfordete
tionofthe
H
2
and
omponentsattheLHC,in thelightofthe
onstraintsfordarkmatterdis
ussedinthepresent
paper.
9
log10 [W h2] : mh=120 GeV ; l2=10-1 ; D MA0= 10 GeV ; D MHc= 50 GeV log10 [W h2] : mh=200 GeV ; l2=10-1 ; D MA0= 10 GeV ; D MHc= 50 GeV
200 200
2 2
180 2 180
Excluded Excluded
160 160 0
0
140 140
-2
GeV) 120 5 3 -2 GeV) 120 WMAP
(2100 (2100 -4
m m
80 4 WMAP -4 80
-6
60 1 < l i < 4 p 60 1 < l i < 4 p
-6
40 1 40 -8
20 20
-8
20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200
mH0 (GeV) mH0 (GeV)
log10 [flux1 g (cm-2s-1)] : mh=120 GeV ; l2=10-1 ; D MA0= 10 GeV ; D MHc= 50 GeV log10 [flux1 g (cm-2s-1)] : mh=200 GeV ; l2=10-1 ; D MA0= 10 GeV ; D MHc= 50 GeV
NFW NFW
200 200
-4
180 180
-4
160 160 EGRET
EGRET -6
140 EGRET 140 EGRET -6
V) 120 V) 120
e -8 e
G G
(2100 (2100 -8
m m
80 EGRET -10 80 GLAST EGRET
-10
60 GLAST 60
GLAST
40 -12 40 -12
GLAST
20 20
20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200
mH0 (GeV) mH0 (GeV)
log10 [s ] : mh=120 GeV ; l2=10-1 ; D MA0= 10 GeV ; D MHc= 50 GeV log10 [s ] : mh=200 GeV ; l2=10-1 ; D MA0= 10 GeV ; D MHc= 50 GeV
DM-p DM-p
200 200
CDMS EDELWEISS II -6
180 EDELWEISS II 180
-6
-7
160 160
-8
140 -8 140
-9
V) 120 V) 120
e e
G G -10
(2100 -10 (2100
m m -11
80 80
ZEPLIN -12
-12
60 60
ZEPLIN -13
40 EDELWEISS II 40 ZEPLIN
ZEPLIN -14 -14
20 20
-15
20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200
mH0 (GeV) mH0 (GeV)
10
Figure 5: From top to bottom: Reli
density, gamma indire
t dete
tion and dire
t dete
tion
ontours in
(M ,µ ) m = 120 m = 200
the H0 2 plane. Left: h GeV. Right: h GeV.