cosmic rays through the higgs portal
TRANSCRIPT
arX
iv:0
803.
1444
v3 [
astr
o-ph
] 1
6 Ju
l 200
8
Cosmic rays through the Higgs portal
Rainer Dick ∗,a,b , Robert B. Mann a,c , Kai E. Wunderle b
aPerimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo,Ontario, Canada N2L 2Y5
bDepartment of Physics and Engineering Physics, University of Saskatchewan,116 Science Place, Saskatoon, Saskatchewan, Canada S7N 5E2cDepartment of Physics and Astronomy, University of Waterloo,
200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1
Abstract
We consider electroweak singlet dark matter with a mass comparable to the Higgs mass. Thesinglet is assumed to couple to standard matter through a perturbative coupling to the Higgsparticle. The annihilation of a singlet in the mass range mS ∼ mh is dominated by proximity tothe W , Z and Higgs peaks in the annihilation cross section. We find that the continuous photonspectrum from annihilation of perturbatively coupled singlets in the galactic halo can reach alevel of several per mil of the EGRET diffuse γ ray flux.
Key words: Cosmic rays, electroweak singlet, dark matterPACS: 12.60.Fr, 95.30.Cq, 95.35.+d, 96.50.S-, 98.70.Sa
1 Introduction
Revealing the distribution and nature ofdark matter is one of the most interestingcurrent challenges of particle physics andastrophysics. Numerous candidates havebeen proposed and studied, including ax-ions [1], neutralinos [2], Kaluza-Klein pho-tons [3], Kaluza-Klein or string dilatons[4], and superheavy dark matter either di-
∗ Corresponding authorEmail addresses: [email protected]
(Rainer Dick),[email protected] (Robert B.Mann), [email protected] (Kai E.Wunderle).
rectly from inflationary expansion [5,6] orfrom preheating after inflation [7]. Unsta-ble particles like axions and dilatons haveto be very light, in the sub GeV range,to survive long enough to serve as darkmatter. However, neutralinos are usuallyassumed to have masses beyond 100 GeV,although lower mass limits strongly de-pend on supersymmetric models [8,9,10].
Another very interesting model for darkmatter which can have a large mass is a sta-ble electroweak singlet S which couples tostandard model matter exclusively througha coupling to the Higgs boson H ,
HI =η
2
∫
d3~xS2H†H. (1)
Preprint submitted to Elsevier Science 13 February 2013
We focus in particular on the Z2 symmetricmodel proposed in [11,12,13,14,15], whichalso allows for a bare mass term for thesinglet, but does not include Higgs-singletmixing terms. The Lagrangian in the scalarsector is
L=−1
2∂µS∂µS − 1
2m2
SS2 − DµH†DµH
−η
2S2
(
H†H − v2h
2
)
−λ
4
(
H†H − v2h
2
)2
. (2)
The derivatives Dµ are the appropriateSU(2) × U(1) covariant derivatives actingon the Higgs field. We assume perturbativesinglet-Higgs coupling for our calculationsof singlet annihilation cross sections in thenon-relativistic limit. We report results inparticular for η2 = 0.1 and for η2 = 0.01.The singlet sector may also include a Z2
symmetric singlet self-interaction ∼ λSS4
if the positive coupling λS is weak enoughsuch that its loop contributions can be ne-glected in the present perturbative calcu-lation of singlet annihilation cross sections.The assumption of perturbative couplingsin the non-relativistic limit is compatiblewith the fact that the β functions are pos-itive in leading order in the couplings [14].We assume that the singlet vacuum expec-tation value vanishes, vs = 〈S〉 = 0. Other-wise the singlet-Higgs coupling would yielda singlet-Higgs mixing term ∼ ηvsvhsh.
Later on we will also allow for a set ofN singlet states with a global O(N) sym-metry and vanishing vacuum expectationvalues. This ensures mass degeneracy anduniversality of the singlet-Higgs couplingstrength η.
The Higgs-channel between dark mat-ter and the standard model was denotedas a Higgs portal in [15]. The model pro-vides a minimal renormalizable dark mat-
ter model 1 [11,12,13,14,15,16,17,18]. Ithas also been discussed as a model forquintessence in [22]. McDonald introducedthe variant with a complex singlet [12],and effects of gauging the ensuing hiddenU(1) symmetry are discussed in [23]. Thepresence of the singlet coupling obviouslymodifies the Higgs effective potential andimpacts electroweak symmetry breaking[24], eventually triggering a strong firstorder phase transition [25,26].
A light electroweak singlet could revealitself as missing energy in collider basedHiggs search experiments [27,28,29,23]or in B decays [17]. March-Russell et al.pointed out that electroweak singlets andtheir fermionic partners in supersymmet-ric theories can also be very heavy, withmasses up to 30 TeV [30].
In the present paper we will be con-cerned with the prospects of observationof intermediate mass electroweak singletsthrough their annihilation products in cos-mic rays. We will take the proposals ofan electroweak singlet coupling throughthe Higgs portal as a minimal renormal-izable dark matter model seriously, anddiscuss possible annihilation signals underthe assumption mS ∼ mh. The relevantmechanism for a sizable signal for an in-termediate mass singlet would be annihila-tion through an intermediate Higgs bosoninto W and Z bosons. Because of the im-portance of the opening of the W channelat 80 GeV, we will denote the mass range80 GeV < mS < 1 TeV as the intermediatemass range for electroweak singlets. Prox-imity of mS to the W , Z, and Higgs peaks
1 Other recently proposed classes of mini-mal dark matter models introduce heavy elec-troweak multiplets with a lightest neutralcomponent [19,20], or another scalar messen-ger between dark matter and standard matter[21].
2
in the annihilation cross section increasesthe product vσ substantially, thus poten-tially yielding a strongly enhanced fluxof annihilation products from the galactichalo even without invoking boost factorsfrom strong local dark matter overdensi-ties. However, we will see that the effectof the enhanced annihilation cross sectionis partially compensated for in standardLee-Weinberg theory for the creation ofthermal relics, because the requirementS = dm together with mS ∼ mh will re-quire N -plets of electroweak singlet states,and the net effect is a scaling of the fluxj ∝ vσ/N .
The strength of a direct galactic darkmatter annihilation signal depends on thedark matter distribution in our galactichalo. The cosmic ray flux from a Navarro-Frenk-White (NFW) dark matter halo anda cored isothermal halo will be reviewed insection 2. Model independent limits on theflux will be discussed in 3.
Annihilation cross sections which followfrom the singlet-Higgs coupling (1) are re-ported in section 4. The application of Lee-Weinberg theory for intermediate masselectroweak singlets is discussed in section5. Section 6 summarizes our conclusions.
2 The flux from the galactic halo
Annihilation of dark matter particles ofmass mS and number density n(~r) gener-ates a diffuse cosmic ray flux at our loca-tion ~r⊙ [31]
j =∫
d3~rνn2(~r)
4π|~r⊙ − ~r|2
×dN (E, 2mS)
dE
σv
4π sr. (3)
Here ν = 1/2 if the annihilating parti-cles are Majorana to avoid overcountingof collisions [32], and ν = 1/4 otherwise
(assuming in the non-Majorana case equalamounts of dark matter and anti-matterin the halo). The case of interest to us isν = 1/2. The fragmentation function
F(E, Ein) =dN (E, Ein)
dE=
1
σ
dσ
dE
gives the number of particles per energyinterval and per annihilation event foran event with initial energy Ein. Lateron we will mostly focus on the photonicpart dNγ/dE = σ−1dσ(γ)/dE, as chargedand hadronic components of dark mat-ter annihilation products are masked bya relatively larger cosmic ray backgroundthan photons or neutrinos. However, inthe present section we will also discuss thefraction of the total flux of cosmic raysfrom dark matter annihilation productscompared to the cosmic rays flux jCR fromall sources.
The factor σv in equation (3) is the prod-uct of annihilation cross section and rela-tive speed in the non-relativistic limit. Forprocesses at high redshift or annihilationof very low mass particles, thermal averag-ing 〈σvF〉 would have to be included. Forhigh redshift sources, redshifting of dσ/dEwould also have to be included and the dis-tance |~r⊙ − ~r| has to be replaced by theluminosity distance. However, for annihila-tion of heavy particles in the galactic halo,equation (3) and its corresponding line-of-sight counterparts below are perfectly ade-quate.
Equation (3) yields a cosmic ray flux av-eraged over all directions. If the detector isonly sensitive to cosmic rays from a smallsolid angle ∆Ω ≪ 1 sr, or if correspondingcuts can be applied, the observed flux perunit of solid angle is [33] (see also [34]),
j∆Ω =
∞∫
0
dx∫
∆Ω
dϑdϕ sinϑνn2(x, ϑ, ϕ)
4π
3
×dN (E, 2mS)
dE
σv
∆Ω. (4)
The vector ~x with length x and directionϑ, ϕ is related to the vector ~r in (3) through~x = ~r − ~r⊙, n(~x) ≡ n(~r).
The averaged flux (3) is recovered fromequation (4) in the following way. The ob-served flux for aperture ∆Ω → 0 is the in-tegral along the line of sight (ϑ, ϕ),
j∆Ω→0(ϑ, ϕ) =
∞∫
0
dxνn2(x, ϑ, ϕ)
4π
×dN (E, 2mS)
dEσv. (5)
Averaging j∆Ω→0(ϑ, ϕ) over all directionsyields the diffuse flux (3),
j = 〈j∆Ω→0(ϑ, ϕ)〉
=
∞∫
0
dx
π∫
0
dϑ
2π∫
0
dϕ sin ϑνn2(x, ϑ, ϕ)
4π
×dN (E, 2mS)
dE
σv
4π sr
=∫
d3~xνn2(~x)
4π|~x|2
×dN (E, 2mS)
dE
σv
4π sr. (6)
We will use equation (3) to estimate theaveraged cosmic ray flux from dark matterannihilation in the galactic halo and equa-tion (5) to estimate the diffuse cosmic rayflux along a line of sight orthogonal to thegalactic plane.
We refer to the differential fluxes (3,4) asinclusive fluxes. Stable final states includephotons, neutrinos, electrons, positrons,protons and anti-protons. The differentialphoton fluxes are found by substitutingdN (E, 2mS)/dE → dNγ(E, 2mS)/dE inequations (3-5).
The density profile is assumed as anNFW profile [35],
ρ(r) =M
r(r + rs)2, (7)
with a mass parameter M = 4.85 ×1010M⊙ = 5.41 × 1064 TeV/c2 and a scaleradius rs = 21.5 kpc. These parameterscorrespond to the fit by Klypin et al. tothe galactic halo [36], see also [37].
Equation (3) can be evaluated analyti-cally for an NFW profile (we use ν = 1/2in the following),
j =
∞∫
0
drr
r⊙ln
(
r + r⊙|r − r⊙|
)
ρ2(r)
4m2S
×dN (E, 2mS)
dE
σv
4π sr
=M2
4m2S
dN (E, 2mS)
dE
σv
4π sr
1
r⊙r4s
×[
π2
6+ L2
(
rs
rs + r⊙
)
+ L2
(
rs − r⊙rs
)
+1
2ln2
(
rs + r⊙rs
)
+4rsr⊙(2r2
s − r2⊙)
3(r2s − r2
⊙)2
− 2r⊙rs9r4
s − 8r2sr
2⊙ + 3r4
⊙
3(r2s − r2
⊙)3ln
(
rs
r⊙
)]
(8)
where L2 denotes Euler’s dilogarithmicfunction 2 [38,39].
Substitution of data for the galactic haloyields
j =1.95 × 1014 ×(
TeV
mS
)2
×dN (E, 2mS)
dE
σv
cm5 sr
= 1.95 × 1014 ×(
TeV
mS
)3
2 Note that L2(x) = f(1−x) in [39]. Althoughof no practical relevance, we point out thatthe singularity at rs = r⊙ cancels between thelast two terms in equation (8).
4
×dN (x, 2mS)
dx
σv
TeV cm5 sr, (9)
with the scaled energy variable x = E/mS.Looking only along a line of sight or-
thogonal to the galactic plane to minimizebackground effects will cost approximatelya factor 3 in observed flux from the galactichalo,
j⊥ = 6.20 × 1013 ×(
TeV
mS
)3
×dN (x, 2mS)
dx
σv
TeV cm5 sr. (10)
Cirelli et al. also compared their calcu-lations with other density profiles [20], us-ing the assumption of same local dark mat-ter density. This yields for the isothermalcored profile
=µ
r2 + r2c
, rc = 5 kpc,
a parameter µ = 6.92 × 1062 GeV/kpc =2.24 × 1041 GeV/cm.
The absence of the central cusp reducesthe flux averaged over all directions,
jisc =πµ2
8m2Src(r2
c + r2⊙)
dN (E, 2mS)
dE
σv
4π sr
= 1.20 × 1014 ×(
TeV
mS
)3
×dN (x, 2mS)
dx
σv
TeV cm5 sr, (11)
but the flux along a line of sight perpendic-ular to the galactic plane is virtually un-changed and slightly increased due to theweaker local gradient in the dark matterdistribution,
j(isc)⊥ =6.36 × 1013 ×
(
TeV
mS
)3
×dN (x, 2mS)
dx
σv
TeV cm5 sr. (12)
We will report numerical results for the fluxj from equation (9). The other fluxes canthen easily be derived from equations (10-12).
The fragmentation function into all finalstates must satisfy the energy sum rule
1∫
0
dxxdN (x)
dx= 2.
An often used parametrization is
dN (x)
dx=
2xα(1 − x)β
B(α + 2, β + 1). (13)
We will use fiducial values α = −1.5, β = 2for numerical estimates. The two slope be-havior of log(dN (x)/dx) versus x (see e.g.Fig. 17.1 in the review [40]) also suggests aphenomenological fit
dN (x)
dx= A exp(−αx) + B exp(−βx). (14)
The values found by the HRS Collabo-ration in e+e− annihilation at
√s = 29
GeV correspond to normalized valuesA = 360.5, α = 28.4, B = 97.2 andβ = 7.91 [41]. A disadvantage of these 2-temperature distributions is that they arevery small, but do not vanish at x = 1.However, they work very well between0.1 < x < 0.9.
The shape of fragmentation functions isonly weakly energy dependent between 12GeV and 202 GeV [40], and dominated bythe fragmentation properties of intermedi-ate partons. Therefore we also use (14) fornumerical work besides (13).
We will use the differential photon frag-mentation function proposed in [42],
dNγ(x)
dx=
0.42 exp(−8x)
x1.5 + 0.00014, (15)
for the differential photon spectrum. Thiscorresponds to about 11% photon energy
5
yield.
3 Model independent bounds on the
cosmic ray flux from dark matter
annihilation
Although we are primarily interested incosmic ray fluxes from the Higgs portalcoupling (1), we would also like to pointout that model independent estimates forcosmic rays from dark matter annihilationarise from unitarity limits and from limitson neutrino fluxes. Another model inde-pendent limit arises from halo stability[43]. However, this limit has been super-seded by the neutrino limit for dark mattermasses heavier than 0.1 GeV [44,45].
The unitarity limit on s wave annihila-tion cross sections σ ≤ 4π/k2 [46,47] im-plies
σv≤ 4.40 × 10−19 cm3
s
×(
TeV
mS
)2 100 km/s
v. (16)
Substitution in equation (9) yields a limiton the diffuse cosmic ray flux from galacticdark matter annihilation
jS ≤ 8.57 × 10−5
TeV cm2 s sr
dN (x)
dx
×(
TeV
mS
)5 100 km/s
v. (17)
Beacom et al. and Yuksel et al. recentlyfound that limits on the diffuse cosmic neu-trino signal and the halo signal can be usedto impose stronger limits on the dark mat-ter annihilation cross section for dark mat-ter masses below 10 TeV [44,45]. Between10 GeV and 10 TeV, the cosmic and theisotropic halo neutrino flux limits approxi-mately reduce the limit on the annihilationcross section ∼ m2. We take this into ac-
count through a correction factor
βν =
(
mS
10TeV
)2, 10 GeV ≤ mS ≤ 10 TeV,
1, mS > 10 TeV.
Comparison with the cosmic ray flux be-low 1 PeV [48]
jCR =2.582 × 10−5
TeV cm2 s sr
(
TeV
E
)2.68
=1.236 × 10−5
GeV cm2 s sr
(
100 GeV
E
)2.68
shows that a galactic dark matter annihi-lation signal could reach several per cent ofthe total differential cosmic ray flux if theannihilation cross section could get close tothe upper limits either through Sommer-feld enhancement or through resonance ef-fects. We find
jS
jCR≤βν × 3.32x2.68 dN (x)
dx
×(
TeV
mS
)2.32 100 km/s
v,
e.g. for mS = 100 GeV,
jS
jCR
≤ 6.94 × 10−2x2.68 dN (x)
dx× 100 km/s
v.
The maximum of x2.68dN (x)/dx for thefragmentation function (13) is 0.23 for x =0.37. For the fragmentation function (14)the maximum of x2.68dN (x)/dx is 0.37 forx = 0.34.
However, cosmic rays are strongly dom-inated by charged particles, while darkmatter annihilation products are expectedto contain a relatively higher neutral com-ponent of photons and neutrinos. γ ray andneutrino observatories are therefore theprimary search tools for dark matter anni-hilation products. We will use the diffusephoton background published by EGRETbetween 30 MeV and 120 GeV [49],
6
jγ,E =6.89 × 10−10
TeV cm2 s sr
(
TeV
E
)2.10
=8.68 × 10−11
GeV cm2 s sr
(
100 GeV
E
)2.10
, (18)
for benchmarking. This is the diffuse pho-ton flux observed by EGRET after subtrac-tion of conventional galactic sources. It hasbeen pointed out that improved models forstandard sources could reduce the EGRETsignal, see [50] and references there. Thesensitivity calibration beyond 1 GeV hasalso been called into question [51]. The en-ergy range between 1 GeV and 100 GeVis particularly relevant for continuous pho-ton signals from dark matter in the massrange of interest here, 80 GeV < mS <1 TeV, and we will see that signal levelsmay be small. Reliable further subtractionsof standard sources from the diffuse γ raybackground signal, and a lower total sig-nal, would help to identify or constrain apossible dark matter signal. EGRET sets auseful benchmark until GLAST/LAT pub-lishes data on the diffuse γ ray background.
Normalizing the averaged photon fluxfollowing from equation (9) to the EGRETflux (18) yields
jγ
jγ,E= 2.83 × 1023x2.1 dNγ(x)
dx
×(
TeV
mS
)0.9 σv
cm3 s−1.
The maximum of x2.1dNγ(x)/dx is 0.05near x ∼ 0.08. Therefore any annihilationcross section of a dark matter particle witha mass below 1.5 TeV is constrained to
σv < 10−23 cm3
s×(
mS
100 GeV
)0.9
. (19)
4 Annihilation cross sections for the
electroweak singlet
The coupling (1) reduces in unitarygauge to
HSh =ηvh
2
∫
d3~xS2h +η
4
∫
d3~xS2h2. (20)
For the annihilation of the electroweak sin-glet through an intermediate Higgs, we alsoneed the couplings
Hhh =∫
d3~xm2
h
2vh
(
h3 +h4
4vh
)
, (21)
Hfh =∫
d3~x∑
f
mf
vhhf · f (22)
and
HW,Zh =∫
d3~x
(
2mW
2
vh2
W−W+ +mZ
2
vh2
Z2
)
×(
vhh +h2
2
)
. (23)
These couplings yield the followingannihilation cross sections in the non-relativistic limit,
vσSS→hh = η2
√mS
2 − mh2
16πmS3
×∣
∣
∣
∣
∣
2mS2 + mh
2
4mS2 − mh
2 + imhΓh
− 2ηvh2
2mS2 − mh
2
∣
∣
∣
∣
∣
2
,
vσSS→ff = η2Ncmf2
4πmS3
×√
mS2 − mf
23
(4mS2 − mh
2)2 + mh2Γh
2,
vσSS→WW = η2
√mS
2 − mW2
4πmS3
7
×3mW4 − 4mW
2mS2 + 4mS
4
(4mS2 − mh
2)2 + mh2Γh
2,
vσSS→ZZ = η2
√mS
2 − mZ2
8πmS3
×3mZ4 − 4mZ
2mS2 + 4mS
4
(4mS2 − mh
2)2 + mh2Γh
2,
with Nc = 3 for quarks and Nc = 1for leptons. The σSS→ff cross section issummed over final spin states. The crosssection σSS→hh contains the scattering am-plitude from the S2h2 contact vertex inequation (20), the scattering amplitudefrom the s-channel contribution with anintermediate Higgs boson from the S2hvertex in equation (20) and the h3 vertexin equation (21), and the t-channel andu-channel amplitudes with an intermedi-ate singlet from the S2h vertex. Thoseamplitudes are with the normalizationSfi = δfi − iMfiδ
4(p1 + p2 − k1 − k2),
M(1)SS→hh =
η
16π2
× 1√
ES(~k1)ES(~k2)Eh(~p1)Eh(~p2),
M(2)SS→hh =−3ηmh
2
16π2
× 1√
ES(~k1)ES(~k2)Eh(~p1)Eh(~p2)
× 1
(k1 + k2)2 + mh2 − iǫ
,
M(3)SS→hh =−η2vh
2
8π2
× 1√
ES(~k1)ES(~k2)Eh(~p1)Eh(~p2)
×(
1
(k1 − p1)2 + mS2 − iǫ
+1
(k1 − p2)2 + mS2 − iǫ
)
.
Note that due to the constraint mS2 >
mh2 for SS → hh, neither the s-channel
resonance for mS = mh/2 nor the t-channelor u-channel resonance for mS = mh/
√2
can be realized for σSS→hh. Furthermore,ΓS = 0 in our models.
The amplitude for annihilation of twosinglets with momenta ~k1 and ~k2 into mas-sive vector bosons with momenta ~p1, ~p2
and polarizations ǫ(α)(~p1), ǫ(β)(~p2) is pro-portional to mW
2,
MSS→WW = −ηmW2
8π2
× 1√
ES(~k1)ES(~k2)EW (~p1)EW (~p2)
× ǫ(α)(~p1) · ǫ(β)(~p2)
(k1 + k2)2 + mh2 − iε
(24)
However, the longitudinal polarization vec-tor for massive vector bosons comes with afactor mW
−1, e.g. the polarization vectorsfor ~p in z-direction are
ǫ(1)(~p) = (0, 1, 0, 0), ǫ(2)(~p) = (0, 0, 1, 0),
ǫ(3)(~p) =1
mW
(
|~p|, 0, 0,√
~p2 + mW2
)
.
The tensor product of massive polarizationvectors
∑
α
ǫ(α)µ(~p) ⊗ ǫ(α)ν(~p) = ηµν +pµpν
mW2
implies
∑
α,β
(
ǫ(α)(~p1) · ǫ(β)(~p2))2
=
(
ηµν +pµ
1pν1
mW2
)
(
ηνµ +p2νp2µ
mW2
)
= 2 +(p1 · p2)
2
mW4
.
Together with the amplitude (24) thisyields the cross section σSS→WW . An equiv-alent way to understand why the cross
8
sections σSS→WW and σSS→ZZ do notvanish in the limit of vanishing couplingvh
2 ∼ mW2 ∼ mZ
2 → 0 are the residualGoldstone boson modes h± and χ0 fromthe Higgs field, which would contribute tosinglet annihilation in the SU(2) × U(1)symmetric limit.
Standard model decay widths of theHiggs particle are small unless the Higgsboson is heavy enough to decay intoweak gauge bosons [52], e.g. Γh(mh =115 GeV) = 4.56 MeV, Γh(mh = 160 GeV) =6.37 MeV, Γh(mh = 165 GeV) = 247 MeV,Γh(mh = 203 GeV) = 1.54 GeV.
The energy dependent cross sectionsσv(K) in order O(η2) in all cases corre-spond to the substitution
m2S → (K + mS)2
in the equations for vσ, where K is thekinetic energy of an electroweak singlet.The non-relativistic limits for vσ are there-fore not only excellent approximations forT ≪ mS, but also provide upper limits onthe thermal averages at high temperature
〈σv〉 =
∫∞0 dK
(K+mS)√
K(K+2mS)
exp[(K+mS)/T ]−1σv(K)
∫∞0 dK
(K+mS)√
K(K+2mS)
exp[(K+mS)/T ]−1
.
The temperature dependence is veryweak. The O(η2) cross section for mS =200 GeV is only reduced from σv =1.04η2 × 10−23 cm3/s at T = 0 to 〈σv〉 =0.39η2 × 10−23 cm3/s at T = 100 GeV.
The tree-level cross sections reportedhere yield cross sections of order σv ∼η2 × 10−23 cm3/s in the mass rangemS ∼ mh, see also figures 1-5 below. In ad-dition there are also loop suppressed pho-ton lines at Eγ = mS from a γγ final state,and at Eγ = mS − (mZ
2/4mS) from a γZstate. These contributions are of order of aper cent compared to the continuous spec-trum, because the loop amplitudes contain
two vertices of electroweak strength or thedirect W+W− → γγ contact vertex. Thecontribution to vσSS→γγ through a W+W−
loop with contact vertex e.g. is approx-imately of order (neglecting logarithmicmass dependencies)
vσSS→γγ ∼32πα2η2mW
4
mS2[(4mS
2 − mh2)2 + mh
2Γh2]
∼ 10−3vσSS.
The continuous spectrum yields a con-servatively estimated photon energy yieldof order 10% (see equation (15)). Togetherwith the ratio of cross sections, this im-plies that in terms of photon energy yieldsfrom singlet annihilation, the line spectrumshould be of order of a few per cent of thecontinuous spectrum. Therefore we focuson the continuous spectrum in the presentinvestigation. However, further study of theloop induced line spectrum is also of inter-est.
4.1 Annihilation signal from a heavy elec-
troweak singlet in tree level approxi-
mation
If we assume a Higgs mass limit mh ≤203 GeV from electroweak analysis [53]and absence of a very heavy fourth gen-eration below mS, we get in leading orderthe following annihilation cross section fora heavy (mS > 1 TeV) electroweak singletcoupling through the Higgs portal,
vσSS ≃ 7η2
64πm2S
= 4.06η2 × 10−25(
TeV
mS
)2 cm3
s. (25)
This translates into a cosmic ray fluxfrom heavy electroweak singlet annihila-
9
tion through the Higgs portal at tree level,
jS ≃ 7.92η2 × 10−11
TeV cm2 s sr
(
TeV
mS
)5 dN (x)
dx.
This is very small compared to the galac-tic background flux from supernovae andtheir remnants, and again photon signalsare expected to be more sensitive. Normal-izing to the EGRET flux formally yields
jγ
jγ,E= 0.115η2x2.1 dNγ(x)
dx
(
TeV
mS
)2.9
.
Due to the energy limit of EGRET thisshould only be used for x < 0.12 TeV/mS.For mS = 1 TeV and E ≃ 76 GeV we findjγ/jγ,E ≃ 5.6η2 × 10−3.
For heavier singlets, March-Russell etal. have pointed out that Sommerfeld en-hancement due to scalar exchange betweenannihilating supersymmetric singlets canboost annihilation cross sections by factors103 to 105 for singlet masses between 1 TeVand 30 TeV [30]. Sommerfeld enhanced an-nihilation signals from heavy electroweakmultiplets have recently been discussedby Cirelli et al. [20]. In the present paper,we will instead focus on electroweak sin-glet annihilation in the intermediate massrange 80 GeV < mS < 1 TeV, when σSS
is enhanced due to proximity to the W , Zand Higgs peaks in the cross section.
4.2 Enhancement of electroweak singlet
annihilation for mS ∼ mh due to the
W , Z and Higgs peaks
Electroweak singlet annihilation throughan intermediate Higgs can be strongly en-hanced when the channels SS → WW andSS → ZZ open up. This is especially rele-vant when mS ∼ mh, which is the primarymass range of interest in our present inves-tigation.
We will primarily use Higgs mass val-ues mh = 115 GeV from the direct searchlimit [53], and mh = 160 GeV in the rangeof highest sensitivity for search for a lightStandard Model Higgs boson at the Teva-tron [54]. The value mh = 160 GeV is alsoin the preferred mass range for minimaldark matter models identified in reference[14].
Compared to σSS→WW the contributionsof light quarks and leptons to σSS are oforder 10−4 and the contributions of heavyquarks and leptons are of order 10−2 in themass range of interest. We will include c, band t quarks with masses mc = 1.25 GeV,mb = 4.20 GeV and mt = 172.5 GeV inthe calculation of vσSS. The τ lepton willbe included with mτ = 1.78 GeV.
mS/GeV70 80 90 100 110 120 130 140
sv/h
2
1
2
3
4
Fig. 1. The cross section η−2vσSS in units of10−23 cm3/s for very weak coupling η2 . 0.01.The red (initially and finally highest) curve isfor mh = 129GeV, the blue (initially and fi-nally middle) curve is for mh = 115GeV, andthe green (initially and finally lowest) curve isfor mh = 89GeV.
The effect of the t-channel plus u-channelamplitude M(3)
SS→hh is small for very weakcoupling η2 . 0.01. The O(η2) cross sec-tions for very weak coupling are displayed
10
for several Higgs mass values in figures 1and 2.
Figure 1 shows the scaled cross sec-tion η−2vσSS for mh = 115 GeV and70 GeV < mS < 140 GeV. The cross sec-tions for mh = 89 GeV and mh = 129 GeVare included for comparison. The edge at80 GeV arises from the opening of the Wchannel SS → WW , the edge at 91 GeVarises from the Z channel, and the edge atmS = mh arises from the Higgs channelSS → hh.
mS/GeV120 130 140 150 160 170 180 190 200
sv/h
2
1.5
2.0
2.5
3.0
3.5
4.0
Fig. 2. The cross section η−2vσSS in units of10−23 cm3/s for very weak coupling η2 . 0.01.The blue (initially and finally lower) curve isthe cross section for mh = 160GeV. The red(initially and finally upper) curve is the crosssection for mh = 165GeV.
The scaled cross sections η−2vσSS formh = 160 GeV and mh = 165 GeV aredisplayed in figure 2.
The cross sections for larger values of mS
approach the asymptotic limit (25), see fig-ure 3.
For electroweak strength coupling η2 =0.1, the t-channel plus u-channel amplitudeM(3)
SS→hh suppresses the Higgs threshold inthe cross sections, see figures 4 and 5.
The continuous photon spectrum for
mS/GeV200 300 400 500 600
sv/h
2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fig. 3. The cross section η−2vσSS inunits of 10−23 cm3/s in the mass range160GeV < mS < 640GeV for very weak cou-pling η2 . 0.01. The red (initially upper) curveis the asymptotic cross section (25). The blue(initially lower) curve is for mh = 115GeV.The cross section for mh = 160GeV also dif-fers by less than 10% from the aymptotic limitfor mS > 400GeV.
Eγ < 120 GeV in units of the EGRETflux is displayed in figure 6 for mS = 120GeV, mh = 115 GeV, η2 = 0.1 andvσSS = 2.03 × 10−24 cm3/s.
The corresponding flux ratio for mS =120 GeV, mh = 160 GeV, η2 = 0.1 andvσSS = 3.87 × 10−24 cm3/s is diplayed infigure 7.
The fluxes in figures 6 and 7 assume thatall cold dark matter is in a singlet state,but do not take into account correlationsbetween mS and mh from thermal creationof singlets. This topic will be addressed inthe next section.
5 Thermal creation and abundance
estimates
A thorough and beautiful application ofLee-Weinberg theory [55] for abundance
11
Fig. 4. The cross section vσSS in units of10−24 cm3/s for coupling η2 = 0.1. The red(initially and finally highest) curve is formh = 129GeV, the blue (initially and finallymiddle) curve is for mh = 115GeV, and thegreen (initially and finally lowest) curve is formh = 89GeV.
estimates of electroweak singlets has beengiven in [12]. We will revisit the subject forthe particular range of masses and crosssections of interest to us. The applica-tion of Lee-Weinberg theory in reference[12] used Ωdm = 1 while the applicationfor light electroweak singlets in [13] usedΩdm = 0.6. Here we use Ωdm = 0.2 [53].
In the absence of more detailed modelassumptions about singlet generating in-teractions behind the Higgs portal or co-annihilations, the effect of thermal cre-ation of electroweak singlets below theelectroweak phase transition is taken intoaccount through a thermal productionterm in the rate equation
d
dt(na3) = Nthermal − 〈σv〉n2a3, (26)
where a(t) is the scale factor in theRobertson-Walker metric. The thermalproduction term is determined from the
Fig. 5. The cross section vσSS in units of10−24 cm3/s for coupling η2 = 0.1. The blue(initially and finally lower) curve is the crosssection for mh = 160GeV. The red (initiallyand finally upper) curve is the cross section formh = 165GeV.
Fig. 6. The photon flux jγ in the continuousspectrum for mh = 115GeV, mS = 120GeV,η2 = 0.1 and Eγ < 120GeV in units of theEGRET flux.
equilibrium requirement d(na3)/dt = 0[55,56],
Nthermal = 〈σv〉n20a
3, (27)
12
Fig. 7. The photon flux jγ in the continuousspectrum for mh = 160GeV, mS = 120GeV,η2 = 0.1 and Eγ < 120GeV in units of theEGRET flux.
where n0 is the thermal equilibrium den-sity for temperature T . The resulting rateequations during radiation domination is
dn
dt+
3n
2t= −〈σv〉
(
n2 − n20
)
. (28)
Radiation domination also yields
t =b
T 2, (29)
and this is used to rewrite equation (28) inthe form
d
dT
n
T 3= 2b〈σv〉n
2 − n20
T 6. (30)
Equation (29) follows from the relationsfor the energy density in radiation fortinflation ≪ t < teq (teq ≃ 2.4× 1012 s is thetime of radiation-matter equality),
γ = g∗(T )π2(kBT )4
30(~c)3=
3m2P lanckc
4~t2. (31)
Here we use the reduced Planck massmP lanck = (~c/8πGN)1/2. We have g∗(T ) =91.5 for mb < T < mW . The parameter b is
b =3~mP lanckc
2
πk2B
√
5
2g∗(T )
= 2.53 × 10−7k−2B s GeV2
=3.41 × 1019 s K2.
The analytic approximation proposed byLee and Weinberg [55] uses the equilibriumdensity
n0(T ) =1
2π2(~c)3
×∞∫
0
dK(K + mc2)
√
K(K + 2mc2)
exp [(K + mc2)/kBT ] − 1
until a freeze-out temperature Tf is reachedwith
d
dT
n0(T )
T 3
∣
∣
∣
∣
∣
T=Tf
= 2b〈σv〉n20(Tf )
T 6f
. (32)
This temperature usually turns out tosatisfy Tf . 0.05mS, such that the non-relativistic limit of n0(T ),
n0(T ) =
1
~
√
mSkBT
2π
3
exp
(
−mSc2
kBT
)
,
can be used in the evaluation of the Lee-Weinberg condition (32). The solution isthen extended for T < Tf using domina-tion of the annihilation term in the rateequation (26),
dn
dt+
3n
2t= −〈σv〉n2, tf < t < teq (33)
dn
dt+
3n
a
da
dt= −〈σv〉n2, t > teq, (34)
with the initial condition n(tf ) = n0(Tf ).Here a = a(t) is the scale factor in theRobertson-Walker line element. One mightuse the following set of equations for t >teq,
dn
dt+
2n
t= −〈σv〉n2, teq < t < tΛ,
13
dn
dt+
2n
τΛ
coth(
t
τΛ
)
= −〈σv〉n2, t > tΛ,
with the time constant (for Λ = 0.76c =4.27 keV/cm3)
τΛ = 2mP lanck
√
c
3~Λ= 3.23 × 1017 s,
and tΛ following from
1 + zΛ ≡ a(t0)
a(tΛ)=
(
sinh(t0/τΛ)
sinh(tΛ/τΛ)
)2/3
=(
ΩΛ
Ωm
)1/3
= 1.47,
see e.g. the appendix to reference [57]. How-ever, we will see from the solution of equa-tion (33) that n(teq)〈σv〉teq < 10−9 ≪ 1,see equation (39) below. Therefore bothn(t)〈σv〉t ≪ 1 and n(t)〈σv〉τΛ tanh(t/τΛ) ≪1 for t ≥ teq, i.e. the expansion term dom-inates strongly over the annihilation termfor t > teq. This yields standard cold darkmatter evolution for late times,
n(t) = n(teq)
(
a(teq)
a(t)
)3
, t > teq.
We define ξ = mSc2/kBTf . The Lee-Weinberg condition (32) takes the follow-ing form
exp(ξ)=2bk2
BmSc2
(√
2π~c)3〈σv〉
√ξ
ξ − 1.5
=4.18 × 1011 〈σv〉10−24 cm3/s
× mSc2
100 GeV
√ξ
ξ − 1.5. (35)
We are interested in a perturbativeHiggs portal η2 . 0.1 and weak scalesinglet masses. Our previous results oncross sections then imply that the fac-tor (〈σv〉/10−24 cm3 s−1) × (mS/100 GeV)should be in the range between 0.1 and 10.
This will yield values for ξ between 20 and30. Thermal theories of particle creationusing equation (27) generically predictthat particles will remain thermal until thetemperature has dropped to a value wellbelow their mass threshold.
Integration of equation (33) yields
n(teq) =
1
n(tf)
(
teqtf
)3/2
+ 2〈σv〉tf
×
(
teqtf
)3/2
− teqtf
−1
≃ n(tf )
1 + 2n(tf )〈σv〉tf
(
tfteq
)3/2
, (36)
where we used that freeze-out tempera-tures following from (35) will at least be afew GeV for the parameter range of inter-est here, and therefore (teq/tf)
1/2 > 109.The relation between temperature and
time and the definition of ξ imply
tf =b
Tf2
=bkB
2
mS2c4
ξ2
=2.53 × 10−11ξ2 s ×(
100 GeV
mSc2
)2
. (37)
This yields
〈σv〉tf =2.53 × 10−35ξ2 cm3
×(
100 GeV
mSc2
)2
× 〈σv〉10−24 cm3/s
.
On the other hand, the density is with (35)
n(tf ) =
(
mSc√2πξ~
)3
exp(−ξ)
=mS
2c4
2bkB2〈σv〉
ξ − 1.5
ξ2
=1.98 × 1034 ξ − 1.5
ξ2cm−3
×(
mSc2
100 GeV
)2
× 10−24 cm3/s
〈σv〉 ,
14
and therefore
2n(tf )〈σv〉tf = ξ − 1.5. (38)
This equation also implies
n(teq)〈σv〉teq =n(tf )〈σv〉tf
1 + 2n(tf )〈σv〉tf
√
tfteq
=ξ − 1.5
2ξ − 1
√
tfteq
< 10−9, (39)
and therefore the annihilation term is muchsmaller than the expansion term for t ≥ teq,d(na3)/dt = 0,
n(t0) = n(teq)z−3eq , (40)
where
zeq ≡a(t0)
a(teq)− 1 ≃ a(t0)
a(teq).
We use zeq = 3000 for numerical work.Equations (36), (37), (38), and (40) yield
the current energy density in one singletspecies,
(1)S = n(teq)mSc2
=2ξ − 3
2ξ − 1ξ
kB
√b
2〈σv〉teq3/2zeq3
≃ 2ξ − 3
2ξ − 1ξ × 2.51 eV/cm3
〈σv〉/10−24 cm3/s. (41)
However, we can have an O(N) symmetricN -plet of electroweak singlets of mass mS,such that the current energy density in sin-glets is
S ≃ 2ξ − 3
2ξ − 1ξ × N × 2.51 eV/cm3
〈σv〉/10−24 cm3/s. (42)
The condition S = dm = 1.106 keV/cm3
then determines ξ in terms of 〈σv〉/N ,and substitution in equation (35) thenrelates mS, mh, and N . We report the re-sulting singlet masses in the mass range
mh = 115 GeV mh = 160 GeV
N = 1 mS = 879 GeV mS = 883 GeV
Tf = 35.9 GeV Tf = 36.1 GeV
xσ = 0.0531 xσ = 0.0531
N = 10 mS = 273 GeV mS = 285 GeV
Tf = 10.7 GeV Tf = 11.1 GeV
xσ = 0.557 xσ = 0.558
Table 1: Singlet masses in the mass rangemW < mS < 1TeV which satisfy the Lee-Weinberg condition (35) for η2 = 0.1, mh =115GeV or mh = 160GeV, and N = 1 orN = 10.
mh = 115 GeV mh = 160 GeV
N = 1 mS = 293 GeV mS = 304 GeV
Tf = 12.6 GeV Tf = 13.0 GeV
xσ = 0.0505 xσ = 0.0506
N = 10 n/a mS = 111 GeV
Tf = 4.47 GeV
xσ = 0.540
Table 2: Singlet masses in the mass rangemW < mS < 1TeV which satisfy the Lee-Weinberg condition (35) for η2 = 0.01, mh =115GeV or mh = 160GeV, and N = 1 orN = 10.
mW < mS < 1 TeV for η2 = 0.1 and forN = 1 or N = 10 in table 1. We also reportthe corresponding freeze out temperatures.The annihilation cross sections are givenin the form
xσ ≡ vσSS
10−24 cm3/s.
Table 2 shows solutions in the mass rangemW < mS < 1 TeV for very weak couplingη2 = 0.01.
The cross sections in table 2 for givenmh and N are similar to the cross sectionsin table 1, in spite of the weaker coupling.
15
This is due to the fact that the correspond-ing singlet masses in table 2 are smaller andmuch closer to the peaks in the cross sec-tion.
For the flux calculations for N > 1, wehave to rescale the flux (9) by a factor 1/N ,because the density factor n2 for each anni-hilating species is now suppressed ∝ N−2,but there are N annihilating species of sin-glets of mass mS. Therefore we find withequations (9) and (15)
jγ =1.95 × 1020
N×(
GeV
mS
)2
×dNγ(Eγ)
dEγ
vσSS
cm5 sr
=8.19 × 1019
N×(
GeV
mS
)2
×√
mS exp(−8Eγ/mS)
Eγ1.5 + 0.00014mS
1.5
vσSS
cm5 sr. (43)
The parameter (vσSS/N)/(10−24 cm3/s) =xσ/N varies only in the range 5.05×10−2 ≤xσ/N ≤ 5.58 × 10−2 for the solutions intables 1 and 2. However, jγ scales withmS
−1.5, and therefore the low-mass solu-tions from table 2 yield higher flux thanthe solutions from table 1. We will giveresults for jγ in the case η2 = 0.1, N = 10,mh = 160 GeV and mS = 285 GeV,and also in the case η2 = 0.01, N = 10,mh = 160 GeV and mS = 111 GeV.
The expected contribution to the photonflux below 120 GeV in units of the EGRETflux is shown in figures 8 and 9 for the twocases. The corresponding number of pho-tons per GeV · cm2 · s · sr is given in figures10 and 11, respectivley.
The integrated photon flux
Φγ(Eγ) =
∞∫
Eγ
dEjγ(E)
in the two cases is shown in figures 12 and
Fig. 8. The photon flux jγ in the contin-uous spectrum for η2 = 0.1, N = 10,mh = 160GeV, mS = 285GeV andEγ < 120GeV in units of the EGRET flux.
Fig. 9. The photon flux jγ in the contin-uous spectrum for η2 = 0.01, N = 10,mh = 160GeV, mS = 111GeV andEγ < 111GeV in units of the EGRET flux.
13.For an instrument with an effective area
of 8000 cm2 and a field of view of 2.4 sr,comparable to the Large Area Telescopeon GLAST, the photon flux jγ in figure
16
Fig. 10. The photon flux jγ in units of(GeV cm2 s sr)−1 for η2 = 0.1, N = 10,mh = 160GeV, mS = 285GeV and Eγ be-tween 15GeV and 30GeV. The maximum infigure 8 corresponds to Eγ ≃ 22GeV.
Fig. 11. The photon flux jγ in units of(GeV cm2 s sr)−1 for η2 = 0.01, N = 10,mh = 160GeV, mS = 111GeV and Eγ be-tween 5GeV and 15GeV. The maximum infigure 9 corresponds to Eγ ≃ 9GeV.
11 corresponds to an annual rate of 470photons with energies between 5 GeV <Eγ < 15 GeV from dark matter annihila-
Fig. 12. The integrated photon flux Φγ aboveenergy Eγ in units of (cm2 s sr)−1 for η2 = 0.1,N = 10, mh = 160GeV, mS = 285GeV andEγ between 100MeV and 10GeV.
Fig. 13. The integrated photon flux Φγ
above energy Eγ in units of (cm2 s sr)−1 forη2 = 0.01, N = 10, mh = 160GeV,mS = 111GeV and Eγ between 100MeV and10GeV.
tion on a diffuse astrophysical backgroundof 90,000 photons per year. The flux in fig-ure 10 corresponds to an annual rate of 48photons with energies between 15 GeV <
17
Eγ < 30 GeV from dark matter annihila-tion on a diffuse astrophysical backgroundof 22500 photons per year. There are twocharacteristic features which would help toidentify jγ as an excess due to dark matterannihilation. The excess would extend overan energy range of order 10 GeV, and itwould be correlated with the galactic halo.
6 Conclusions
We have considered perturbatively cou-pled electroweak singlet dark matter inthe intermediate mass range mW < mS <1 TeV. The leading order annihilationcross section of the singlets is enhancedand varies strongly due to proximity tothe W , Z and Higgs peaks. The productvσSS is of order η2 × 10−23 cm3/s for mS
close to mh, i.e. it can substantially exceedstandard estimates of dark matter annihi-lation cross sections even for perturbativesinglet-Higgs coupling.
For singlet masses above the SS → WWthreshold and electroweak strength cou-pling η2 ≃ 0.1, the Lee-Weinberg conditionand the requirement ΩS = Ωdm push mS
to high mass values around 900 GeV ifthere is only one singlet state. However, ifthere is an N -plet of electroweak singletsor if the coupling is weaker, η2 . 0.01,lower singlet mass values can be achieved,and the annihilation signal from the con-tinuous γ ray spectrum can reach a level ofseveral per mil of the EGRET diffuse γ raysignal for photon energies Eγ ∼ 0.08mS.This excess contribution over the expectedcosmological background would appeartypically over an energy range of order 10GeV, and its dark matter signature wouldbe its correlation with the galactic halo.
The flux reported by EGRET was thediffuse γ ray flux after subtraction of thenknown or expected galactic components.
Reduction of the diffuse “excess” γ rayflux due to subtraction of a larger com-ponent from interstellar gas and standardextragalactic sources increases the relativeimportance of an annihilation signal forany possible excess signal and improvesdetectability. The diffuse γ ray flux willbe measured with higher sensitivity andprecision in the near future by the LargeArea Telescope aboard the GLAST satel-lite. This will also cover a larger energyrange up to 300 GeV.
The minimal dark matter models con-sidered here include four basic parameters,the singlet mass mS, the number of singletstates N , the singlet-Higgs coupling η, andthe Higgs mass mh. The assumption thatelectroweak singlets of mass mS provide thedark matter in the universe relates theseparameters through Lee-Weinberg theory.Measuring the Higgs mass at the Tevatronor the LHC will reduce the number of freeparameters in this class of minimal darkmatter models to two, or maybe even toonly one free parameter if a missing energysignal can be used to constrain a combina-tion of η, N and mS. The anticipated small-ness of singlet annihilation signals in cos-mic γ rays indicates that observation of theHiggs particle at the Tevatron or the LHCmay be needed for a successful search for asinglet annihilation signal in cosmic rays.
Acknowledgement
This work was supported by NSERCCanada. RD thanks Freddy Cachazo, RobMyers and Tom Waterhouse for interestingdiscussions, and gratefully acknowledgesthe hospitality of the Perimeter Institutefor Theoretical Physics. We also thank thereferee for constructive criticism of a firstdraft of this manuscript.
18
References
[1] J. Preskill, M.B. Wise, F. Wilczek, Phys.Lett. B 120 (1983) 127; L.F. Abbott, P.Sikivie, Phys. Lett. B 120 (1983) 133;M. Dine, W. Fischler, Phys. Lett. B 120(1983) 137.
[2] S. Weinberg, Phys. Rev. Lett. 50 (1983)387; H. Goldberg, Phys. Rev. Lett. 50(1983) 1419; J.R. Ellis, J.S. Hagelin, D.V.Nanopoulos, K.A. Olive, M. Srednicki,Nucl. Phys. B 238 (1984) 453.
[3] E.W. Kolb, R. Slansky, Phys. Lett. B 135(1984) 378.
[4] Y.M. Cho, Phys. Rev. D 41 (1990) 2462;R. Dick, Mod. Phys. Lett. A 12 (1997) 47;Fortschr. Phys. 45 (1997) 537; Y.M. Cho,Y.Y. Keum, Mod. Phys. Lett. A 13 (1998)109; Class. Quantum Grav. 15 (1998)907; Y.M. Cho, J.H. Kim, Dilaton as adark matter candidate and its detection,arXiv:0711.2858 [gr-qc].
[5] D.J. Chung, E.W. Kolb, A. Riotto, Phys.Rev. Lett. 81 (1998) 4048; Phys. Rev. D59 (1999) 023501.
[6] V.A. Kuzmin, I.I. Tkachev, Phys. Rev. D59 (1999) 123006; Phys. Rep. 320 (1999)199.
[7] L. Kofman, A.D. Linde, A.A. Starobinsky,Phys. Rev. Lett. 73 (1994) 3195; Phys.Rev. D 56 (1997) 3258.
[8] G. Bertone, D. Hooper, J. Silk, Phys. Rep.405 (2005) 279.
[9] D. Hooper, T. Plehn, Phys. Lett. B 562(2003) 18.
[10] A. Bottino, F. Donato, N. Fornengo, S.Scopel, Phys. Rev. D 68 (2003) 043506.
[11] V. Silveira, A. Zee, Phys. Lett. B 161(1985) 136.
[12] J. McDonald, Phys. Rev. D 50 (1994)3637.
[13] C.P. Burgess, M. Pospelov, T. terVeldhuis, Nucl. Phys. B 619 (2001) 709.
[14] H. Davoudiasl, R. Kitano, T. Li, H.Murayama, Phys. Lett. B 609 (2005) 117.
[15] B. Patt, F. Wilczek, Higgs-field portal intohidden sectors, hep-ph/0605188.
[16] D.E. Holz, A. Zee, Phys. Lett. B 517(2001) 239.
[17] C. Bird, R. Kowalewski, M. Pospelov,Mod. Phys. Lett. A 21 (2006) 457.
[18] H. Murayama, Physics beyond theStandard Model and dark matter,arXiv:0704.2276 [hep-ph].
[19] M. Cirelli, N. Fornengo, A. Strumia, Nucl.Phys. B 753 (2006) 178; M. Cirelli, A.Strumia, M. Tamburini, Nucl. Phys. B787 (2007) 152.
[20] M. Cirelli, R. Franceschini, A. Strumia,Nucl. Phys. B 800 (2008) 204.
[21] J. McDonald, N. Sahu, JCAP 0806 (2008)026.
[22] O. Bertolami, R. Rosenfeld, The Higgsportal and a unified model for dark energyand dark matter, arXiv:0708.1784 [hep-ph].
[23] W.F. Chang, J.N. Ng, J.M.S. Wu, Phys.Rev. D 74 (2006) 095005; Phys. Rev.D 75 (2007) 115016; Phenomenologyfrom a U(1) gauged hidden sector,arXiv:0706.2345 [hep-ph].
[24] K.A. Meissner, H. Nicolai, Phys. Lett. B648 (2007) 312.
[25] J.R. Espinosa, M. Quiros, Phys. Rev. D76 (2007) 076004.
[26] S. Profumo, M.J. Ramsey-Musolf, G.Shaughnessy, JHEP 08 (2007) 010.
[27] A. Datta, A. Raychaudhuri, Phys. Rev. D57 (1998) 2940.
19
[28] H. Davoudiasl, T. Han, H.E. Logan, Phys.Rev. D 71 (2005) 115007
[29] R. Schabinger, J.D. Wells, Phys. Rev. D72 (2005) 093007.
[30] John March-Russell, Stephen M. West,Daniel Cumberbatch, Dan Hooper, Heavydark matter through the Higgs portal,arXiv:0801.3440 [hep-ph].
[31] P. Blasi, R. Dick, E.W. Kolb, Astropart.Phys. 18 (2002) 57.
[32] P. Ullio, L. Bergstrom, J. Edsjo, C. Lacey,Phys. Rev. D 66 (2002) 123502.
[33] L. Bergstrom, P. Ullio, J.H. Buckley,Astropart. Phys. 9 (1998) 137.
[34] N.W. Evans, F. Ferrer, S. Sarkar, Phys.Rev. D 69 (2004) 123501.
[35] J.F. Navarro, C.S. Frenk, S.D.M. White,Mon. Not. R. Astron. Soc. 275 (1995) 720;Astrophys. J. 462 (1996) 563.
[36] A. Klypin, H. Zhao, R.S. Somerville,Astrophys. J. 573 (2002) 597.
[37] A. Boyarski, A. Neronov, O. Ruchayskiy,M. Shaposhnikov, I. Tkachev, Phys. Rev.Lett. 97 (2006) 261302.
[38] Higher Transcendental Functions Vol. 1,ed. A. Erdelyi (McGraw-Hill, New York1953) pp. 31-32.
[39] Handbook of Mathematical Functions, eds.M. Abramowitz and I.A. Stegun (UnitedStates National Bureau of Standards,10th printing, Washington 1972) pp.1004-1005.
[40] O. Biebel, D. Milstead, P. Nason, B.R.Webber, Fragmentation functions in e+e−
annihilations and lepton-nucleon DIS, in[53].
[41] D. Bender et al. (HRS Collaboration),Phys. Rev. D 31 (1985) 1.
[42] L. Bergstrom, J. Edsjo, P. Ullio, Phys.Rev. Lett. 87 (2001) 251301.
[43] M. Kaplinghat, L. Knox, M.S. Turner,Phys. Rev. Lett. 85 (2000) 3335.
[44] J.F. Beacom, N.F. Bell, G.D. Mack, Phys.Rev. Lett. 99 (2007) 231301.
[45] H. Yuksel, S. Horiuchi, J.F. Beacom, S.Ando, Phys. Rev. D 76 (2007) 123506.
[46] S. Weinberg, The Quantum Theory ofFields Vol. 1 (Cambridge UniversityPress, Cambridge 1995) Sec. 3.7.
[47] K. Griest, M. Kamionkowski, Phys. Rev.Lett. 64 (1990) 615; L. Hui, Phys. Rev.Lett. 86 (2001) 3467.
[48] F.A. Aharonian, W. Wittek etal. (HEGRA Collaboration), Astropart.Phys. 17 (2002) 459.
[49] P. Sreekumar et al., Astrophys. J. 494(1998) 523.
[50] A.W. Strong, I.V. Moskalenko, O.Reimer, Astrophys. J. 613 (2004) 962.
[51] F.W. Stecker, S.D. Hunter, D.A. Kniffen,Astropart. Phys. 29 (2008) 25.
[52] J.F. Donoghue, E. Golowich, B.R.Holstein, Dynamics of the Standard Model(Cambridge University Press, Cambridge1992) Sec. XV.2.
[53] W.-M. Yao et al. (Particle Data Group),J. Phys. G 33 (2006) 1 and 2007 partialupdate for the 2008 edition.
[54] M.P. Sanders, Search for Low Mass SMHiggs at the Tevatron, arXiv:0805.1248[hep-ex]; A. Duperrin, Review of Searchesfor Higgs Bosons and Beyond theStandard Model Physics at the Tevatron,arXiv:0805.3624 [hep-ex].
[55] B.W. Lee, S. Weinberg, Phys. Rev. Lett.39 (1977) 165.
20
[56] G. Steigman, Ann. Rev. Nucl. Part. Sci.29 (1979) 313.
[57] M.G. Barnett, R. Dick, K.E. Wunderle,Mon. Not. R. Astron. Soc. 349 (2004)1500.
21