Dissertation

Dissertation
Dissertation
submitted to the
Combined Faculties of the Natural Sciences and Mathematics
of the Ruperto-Carola-University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Viviana Niro
Born in Venaria Reale, Italy
Oral examination: June 7th 2010
Indirect detection of Dark Matter with neutrinos
Referees: Prof. Dr. Manfred Lindner
Prof. Dr. Tilman Plehn
Zusammenfassung
In dieser Doktorarbeit wird die indirekte Detektion von Dunkler Materie mittels Neutrinos untersucht. Wir führen eine detaillierte Berechnung der Neutrino-Spektren durch,
die von Annihilationen Dunkler Materie innerhalb der Sonne und der Erde herrühren,
wobei wir alle Prozesse mit einbeziehen, die während der Propagation auftreten können:
Oszillationen und Wechselwirkung mit Materie. Wir analysieren systematisch alle
Möglichkeiten der direkten Vernichtung von Dunkler Materie in Neutrinos für die beiden Fälle von skalarer und fermionischer Dunkler Materie. Außerdem berechnen wir
die Vernichtungs-Querschnitte für Diagramme verschiedener Topologien. Hierbei identifizieren wir die vielversprechendsten Szenarien, für welche auch das Verhalten des
Wirkungsquerschnittes angegeben wird. Danach beschreiben wir die Phänomenologie
der leptophilen Dunklen Materie und zeigen auf, wie die experimentellen Limits an den
von Annihilationsprozessen in der Sonne herrührenden Neutrinofluss dieses Modell als
Erklärung der Ergebnisse des DAMA-Experiments in Bedrängnis bringen. Schließlich
wird eine detaillierte Analyse des erwarteten Neutrino-Flusses stammend von NeutralinoAnnihilationsprozessen innerhalb der Sonne und der Erde präsentiert. Hierbei berücksichtigen wir sowohl teilchenphysikalische als auch astrophysikalische Unsicherheiten und
unterteilen den Fluss in durchgehende und stoppende Myonen.
Abstract
In this doctoral thesis, we discuss indirect Dark Matter detection with neutrinos. We
perform a detailed calculation of the neutrino spectra coming from Dark Matter annihilations inside the Sun and the Earth, taking into account all the possible processes that
could occur during propagation: oscillation and interaction with matter. We examine
in a systematic way the possibilities of Dark Matter annihilation directly into neutrinos, considering the case of scalar and fermionic Dark Matter. We explicitly calculate
the annihilation cross section for different typologies of diagrams. We identify the most
favourable scenarios, for which the behaviour of the cross section is given. We then
describe the phenomenology of the leptophilic Dark Matter and show how experimental bounds on the neutrino flux coming from annihilations inside the Sun disfavour this
model as explanation of the DAMA results. Finally, a carefull analysis of the neutrino
flux expected from neutralino annihilations inside the Sun and the Earth is presented.
We consider uncertainties coming from both particle physics and astrophysics and we
divide the fluxes in through-going and stopping muons.
Contents
1 Introduction
1
2 The
2.1
2.2
2.3
Dark Matter
Evidence and observations . . . . . . . . . . . .
Density and velocity distributions . . . . . . . .
Dark Matter searches . . . . . . . . . . . . . .
2.3.1 Direct detection . . . . . . . . . . . . .
2.3.2 Indirect detection . . . . . . . . . . . . .
2.3.3 Collider experiments . . . . . . . . . . .
2.4 Dark Matter candidates . . . . . . . . . . . . .
2.4.1 WIMP candidates . . . . . . . . . . . .
2.4.2 Non-WIMP candidates . . . . . . . . . .
2.4.3 Non-standard Dark Matter interactions
3 Indirect detection with neutrinos
3.1 Neutrino flux from the Sun and the Earth
3.1.1 Capture and annihilation rates . .
3.1.2 Neutrino production . . . . . . . .
3.1.3 Neutrino propagation . . . . . . .
3.2 Neutrino flux from the galactic center . .
3.3 Muon flux . . . . . . . . . . . . . . . . . .
3.3.1 Neutrino-Muon conversion . . . . .
3.3.2 Atmospheric background . . . . .
3.3.3 Muon detection . . . . . . . . . . .
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4 Dark Matter annihilation into neutrinos
4.1 The neutrino mass terms . . . . . . . . . . . . . .
4.1.1 Dirac mass term . . . . . . . . . . . . . .
4.1.2 Majorana mass term . . . . . . . . . . . .
4.1.3 See-saw mechanisms . . . . . . . . . . . .
4.2 Production of monoenergetic neutrinos . . . . . .
4.2.1 Scalar Dark Matter . . . . . . . . . . . . .
4.2.2 Fermionic Dark Matter . . . . . . . . . .
4.3 Discussion of unsuppressed cases . . . . . . . . .
4.3.1 s-channel: the triplet scalar mediator . .
4.3.2 t-channel: the singlet fermionic and scalar
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i
5 Indirect versus direct Dark Matter detection
5.1 Leptophilic Dark Matter . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Effective Dark Matter interactions . . . . . . . . . . . .
5.1.2 Dark Matter scattering on electrons . . . . . . . . . . .
5.1.3 Signals in direct detection experiments . . . . . . . . . .
5.1.4 Loop induced interactions . . . . . . . . . . . . . . . . .
5.1.5 Discussion of Lorentz structure . . . . . . . . . . . . . .
5.1.6 Event rates . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.7 Super-Kamiokande constraints . . . . . . . . . . . . . .
5.2 Neutralino Dark Matter . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Theoretical model . . . . . . . . . . . . . . . . . . . . .
5.2.2 WIMP-nucleon cross section: hadronic uncertainties . .
5.2.3 Numerical evaluations . . . . . . . . . . . . . . . . . . .
5.2.4 Fluxes from the Earth and the Sun . . . . . . . . . . . .
5.2.5 Fluxes of stopping muons for configurations compatible
DAMA results . . . . . . . . . . . . . . . . . . . . . . .
6 Summary and conclusions
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the
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A Neutrino interactions inside the Sun
107
A.1 Neutral current interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A.2 Charged current interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 109
B Neutrino cross sections
111
B.1 Neutral current cross sections . . . . . . . . . . . . . . . . . . . . . . . . . 111
B.2 Charged current cross sections . . . . . . . . . . . . . . . . . . . . . . . . 112
C Annihilation cross sections
115
C.1 Scalar Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
C.2 Fermionic Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
C.3 Vector Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Bibliography
ii
123
1
Introduction
The roots of our current knowledge and understanding of the Universe can be traced
back to the 1929, when the Hubble’s law was presented for the first time [1]. Edwin
Hubble and Milton Humason proposed a linear proportionality between the redshifted
light emitted from galaxies and their distances. If the redshift is interpreted as Doppler
effect, related to the recession velocity of galaxies, the conclusion that the Universe
is expanding will be reached. After this discovery, the idea of a static Universe has
gradually been abandoned and the cosmological models of Big Bang began to take over.
Nowadays, after the recent data from type-Ia supernovae [2], we know that the expansion
of the Universe is accelerating.
Results from many different observations, carried out in the past decades, have provided a precise understanding of the composition of our Universe, bringing cosmology
to face its “golden age”. In particular, a lot of different experimental evidences point
towards the existence of a form of non-luminous matter, baptized with the name “Dark
Matter”, which should account for almost 23% of the total mass-energy of the Universe
and for almost 84% of its mass. Thus, by far most of the Universe is made of a kind of
matter different from ordinary one.
One of the most exciting and difficult challenges of particle physics is to understand
the real nature of Dark Matter (DM). A rich zoo of candidates for DM is present in
the literature. All these particles arise in theories beyond the Standard Model (SM) of
particle physics. However, depending on the model, the characteristics of the DM particle
can be rather different and the values of the mass and the scattering cross section can
vary within several orders of magnitude.
This ignorance might be partially attenuated by the investigation of physics at the
electroweak (EW) scale, that will be provided by the Large Hadron Collider (LHC) at
CERN. Since the end of November 2009, the LHC is operating again and its forthcoming
results will hopefully be fundamental to test the physics beyond the SM. At the same
time, it will be able to restrict the viable DM candidates among those with masses around
1
Chapter 1 Introduction
the EW scale.
Despite that, even with the detection of a new particle that could successfully act as
DM, the accelerator experiments cannot directly prove that the same particle is present
in the galactic halo. For this reason, direct detection experiments that search for scattering of DM particles off atomic nuclei inside a detector are fundamental. There are
several experiments now running and taking data, which use different materials and
detection techniques. Among them, only the DAMA experiment has searched for a
model-independent DM signature: an annual modulation in the count rate due to the
Earth’s motion with respect to the Sun. In April 2008, the DAMA collaboration has
released new data [3], where a modulated signal is detected at 8.2 σ confidence level.
These new results have received particular attention from the theoretical particle physics
community, in the attempt of reconciling them with the negative results from the other
direct detection experiments. So far, the DAMA experiment is the only one that has
claimed a detection of DM.
Another possibility to detect DM is to search for its annihilation products (such as γrays, antimatter and neutrinos) in the Milky Way galactic center and in the galactic halo,
in dwarf spheroidal galaxies and in celestial bodies, like the Earth or the Sun. Recently,
there arose an increased interest in this field, in particular due to the cosmic ray anomaly
revealed at the end of October 2008 by the satellite experiment PAMELA [4]. An excess
in the positron flux has been detected, while no excess has been found for antiprotons.
This anomaly could be caused by DM annihilation in the galactic halo or by astrophysical
objects such as pulsars.
The annihilation of DM particles can produce also high-energy neutrinos, which can
be detected through water Cherenkov detectors, like Super-Kamiokande [5], or through
neutrino telescopes, like IceCube [6], ANTARES [7] and its future extension KM3Net [8].
Being neutral, neutrinos are not deflected by magnetic fields and have only weak interactions, so they can travel unperturbed through the interstellar medium.
The role of neutrinos in physics is often compared to the one of X-rays in diagnostic
radiography, since with their detection we are able to get an “image” of regions of space
or of celestial bodies that are accessible only partially with other methods, if at all. A
remarkable example is given by the measurements of the solar neutrino flux, through
which the Standard Solar Model has been confirmed and important information on neutrinos has been derived, i.e., the resonant oscillation in matter. Now that we gained
a good knowledge of the neutrino physics and of the neutrino oscillation parameters,
it is possible to make precise predictions regarding the neutrino flux coming from DM
annihilation.
It has been shown in several papers, see e.g. Refs. [9, 10, 11, 12], that this method
represents a promising tool to detect DM, since neutrinos conserve directionality and
are the only particle that can escape from celestial bodies with energies high enough to
be detected. The common hope is that the solar neutrino example could be repeated
and that now, through the analysis of the high-energy neutrino flux, we could obtain
important information on DM properties, like branching ratios and the mass. No excess
in the neutrino flux has been detected so far, with respect to the expected background.
2
However, the Super-Kamiokande limits, derived from analyses of the data collected from
May 1996 to July 2001 [13], are able to put stringent bounds on the DM scattering
and annihilation cross section. New future data will be able to restrict the allowed DM
configurations even more or, maybe, to detect an important signal.
In this thesis, the indirect DM detection through the neutrino portal is considered. In
Chapter 2, a brief review on the physics of DM is given. We report the main evidences
and observations for DM and their main astrophysical uncertainty: the density profile
of the DM in the halo. We also discuss the DM velocity distribution and the different
detection methods. Finally, a summary of the most common candidates present in the
literature will be given.
Chapter 3 focuses on the indirect search for DM using neutrinos. We explain how to
calculate the neutrino flux from the Sun and the Earth, considering in particular the
capture rate and the neutrino propagation aspects. Also the case of the galactic center
is analyzed. We finally discuss the neutrino-muon conversion and the main present and
future detectors.
In Chapter 4, a model-independent detailed analysis of the DM annihilation directly
into neutrinos is given. We initially review the theory of massive neutrinos and the most
common neutrino mass models. For each of these scenarios, we analyze the implications
on the annihilation cross section, considering separately a scalar or a fermionic DM
particle. We then identify the generically unsuppressed cases. For the most promising
ones, we explicitly show the behaviour of the annihilation cross sections and the bounds
coming from various experiments.
In Chapter 5, the indirect detection with neutrinos is compared with the results from
direct detection experiments. In the first part, we present the phenomenology of the
leptophilic DM and we reanalyze in this context the Super-Kamiokande bounds on the
muon flux coming from the Sun. The obtained constraints are then confronted with
the DAMA annual modulation region and with the limits from other direct detection
experiments. In the second part, we consider the light neutralino as DM candidate, in
the framework of an effective MSSM model. We derive the flux of stopping and throughgoing muons expected from all the allowed supersymmetric configurations and from those
compatible with the DAMA results.
Finally, Chapter 6 contains a summary and the conclusions of our work.
The work presented in this thesis has been partially already published in Refs. [14, 15].
During my Ph.D., I have been involved also in other projects, on topics different from
the ones contained in this thesis, see Refs. [16, 17, 18] for more details.
3
Chapter 1 Introduction
Figure 1.1: Schematic mechanism for indirect Dark Matter detection with neutrinos,
in the case of annihilation inside the Sun.
4
2
The Dark Matter
In this Chapter, we review the physics of DM. In Sec. 2.1 we report the main cosmological evidences that point towards the DM postulation and in Sec. 2.2 we discuss the
DM density profile in the galactic halo and its velocity distribution. The experimental
methods to detect DM are presented in Sec. 2.3, while Sec. 2.4 contains a summary of
the most important DM candidates. Exhaustive reviews on the DM topic are given in
Refs. [19, 20, 21, 22].
2.1 Evidence and observations
To find the first origin of the DM idea, we have to go back to the 1933. Studying the
Coma cluster, F. Zwicky found a discrepancy of two orders of magnitude between the
mass inferred by dispersion velocity measurements of the galaxies in the cluster and the
one expected by the analysis of the luminous components [23]. In 1936, S. Smith also
arrived at a similar conclusion with an analysis of the Virgo cluster [24]. The subsequent
evidences for DM arrived only after more than thirty years. V. Rubin and K. Ford
measured in 1970 the velocity rotation curve of the Andromeda Nebula [25] and found
a flat behaviour at large radii. Three years later, M. Roberts and A. Rots extended the
analysis to different galaxy types [26]. A systematic study of the velocity dispersions
in spiral galaxies was presented in 1980 by V. Rubin, K. Ford and N. Thonnard [27].
These last results blew away skepticisms and conviced the astronomy community that
the presence of DM would be necessary to explain the rotation curves, if Newtonian
dynamics was valid at the scale of galaxies and galaxy clusters. Indeed, applying the
Newton’s law of gravity, the rotational velocity as a function of the distance r is given
by
r
G M (r)
vrot (r) =
,
(2.1)
r
5
Chapter 2 The Dark Matter
with G being Newton’s gravitational constant and M (r) the mass contained within a
distance r from the center. The observation of flat rotation curves implies that the mass
increases linearly with the distance from the center, in contrast to the distribution of
luminous matter. Thus, we can picture galaxies and clusters as surrounded by a DM
halo that is spherically distributed.
The analysis of rotation curves has been extended more recently to a large number
of spiral galaxies, see for instance Ref. [28]. These observations represent one of the
strongest hints for DM at the level of galaxies.
At the scale of clusters, compelling evidences for DM arise from gravitational lensing
techniques. Einstein’s theory of general relativity predicts that a massive object deforms
the space-time curvature. Since the light rays follow geodesics, they are deflected by
strong gravitational fields. The deviation from a straight-line path is proportional to the
mass of the object, that acts like an optical lens. Usually, quasars are used as sources,
since they are distant and very bright.
Two different types of gravitational lensing are manly used in DM searches: strong
and weak lensing. In the first case, the bend in the light path is clearly detected by the
presence of multiple images of the same object, arcs and Einstein rings. Studying these
effects, it is possible to infer that DM is needed also at the scale of galaxy clusters [29].
In the case of weak lensing, instead, the deformations are much smaller and cannot be
identified using only one source, since multiple images are not present. Therefore, a
large number of galaxies is usually considered and a statistical analysis is done to reveal
possible correlated distortions and elongations. From the shapes and orientations of the
galaxies, the mass of the lens can be reconstruct. It has been shown in Ref. [30] that
weak lensing represents a powerful tool to measure the presence of DM.
Two recent outstanding applications of the gravitational lensing methods are given by
the so-called “Bullet Cluster” [31] and “Baby Bullet” [32] observations. They represent
examples of collisions between two clusters of galaxies. During this process, the stars of
the galaxies and the DM halos behave as collisionless components, since they interact only
through gravity. Electromagnetic interactions, instead, affect strongly the intergalactic
gas distributions that, as result, become separated from the galaxies. This can be seen by
comparing the “image” of the colliding clusters in visible light, obtained by the Hubble
telescope, and the one in X-rays, observed by Chandra. The DM distribution is then
gathered from gravitational lensing methods and it is found to follow the luminous one.
Since the hot gas represents most of the baryonic matter presents in the clusters, the
result from gravitational lensing can be interpreted as a clear evidence for the presence
of DM. Moreover, these observations are not only a success of the DM model, but also a
robust disproval of MOND (Modified Newtonian dynamics) theories at the scale of galaxy
clusters. Indeed, if a modification of Newton’s law of gravity would be the explanation of
the flat rotational curves, the lensing would follow the distribution of the hot interstellar
medium, this being the major source of baryonic matter.
At cosmological scales, the most convincing evidence of dark matter arises from the
analysis of the Cosmic Microwave Background (CMB). This electromagnetic radiation
was first predicted by G. Gamow in 1946 [33] and later on discovered by A. Penzias
6
2.1 Evidence and observations
and R. Wilson in 1965 [34]. The CMB consists of relic photons that decoupled from
the matter in the early Universe. The detection of this radiation is the most convincing
confirmation of the Big Bang Model. For a review of the CMB physics, see Ref. [35].
The COBE satellite revealed that the CMB radiation follows a thermal black body
spectrum with temperature T = 2.728 ± 0.004 K (95% C.L.) [36]. It also detected for the
first time some small fluctuations in the CMB temperature [37]. The anisotropies δT /T
were measured with precision by the WMAP satellites to be at the 10−5 level [38].
The measurements performed by WMAP have been fundamental to determine the
geometry and composition of the Universe. They are considered as milestone for the
actual model of cosmology. After WMAP, our Universe appears as flat and dominated
by an unknown form of energy, called “Dark Energy”, which is usually denoted by the
Greek letter “Λ”. The CMB observations are also a key ingredient to obtain with good
accuracy the actual amount of DM, which amounts to about 23% of the total mass-energy
of the Universe.
To define the content of the Universe, the density parameter Ωi = ρi /ρc is usually
introduced. The critical density ρc is the density at which the Universe has a vanishing
spatial curvature:
3H02
ρc =
≃ 1.9 × 10−29 h2 g cm−3 ,
(2.2)
8πG
where H0 is the Hubble constant at the present time, which is commonly rewritten as
H0 = 100 h km Mpc−1 s−1 . From the five-years WMAP data, the following parameters
at 1σ confidence level are found [39]:
ΩΛ = 0.742 ± 0.030 ,
(2.4)
= 0.1099 ± 0.0062 ,
(2.5)
h =
2
ΩDM h
2
Ωb h
(2.3)
0.719+0.026
−0.027 ,
= 0.02273 ± 0.00062 ,
(2.6)
where with Ωb we have denoted the density fraction of baryons present in the Universe.
It is remarkable that the value of Ωb h2 obtained with WMAP is in good agreement with
the one obtained through Big Bang Nucleosynthesis [40]. The current dominance of Dark
Energy (DE) has also been confirmed by observations of type-Ia supernovae [2]. The data
from galaxy clusters, CMB and supernovae are combined all together to derive with high
precision the matter and energy contents of the Universe. The complementarity of these
observations is clearly visible in the left panel of Fig. 2.1.
The presence of DM is fundamental to explain the formation processes of stars, galaxies
and clusters. After recombination, the baryons collapse in structure, because they fall
in the gravitational potential wells created by the DM. Depending on the type of DM,
different structure formation scenarios are present. The DM is divided in two main
categories: Hot Dark Matter (HDM), if the particles are relativistic when they decouple
from the primordial plasma, and Cold Dark Matter (CDM), if they are non-relativistic.
The first case will lead to a “top-down” structure formation, in which only clusters and
superclusters of galaxies can initially form. All the structures at small scales are indeed
7
Chapter 2 The Dark Matter
Figure 2.1: Concordance model of the Universe (left panel) and evidences for DM at
different astrophysical scales (right panel).
washed out by the pressure of Hot Dark Matter. In the second case, instead, a “bottomup” formation can be realized. Small structures collapse first and then merge together
to form larger objects.
The Sloan Digital Sky Survey (SDSS) [41] and the 2dF Galaxy Redshift Survey (2dFGRS) [42] are two galaxy surveys that are mapping portions of the sky. They have
detected, respectively, over 800 000 and over 200 000 galaxies at different redshift. This
information has allowed to study the large-scale structure of the Universe, revealing the
presence of voids, filaments and walls.
The N-body computer simulations show that the hierarchical model of structure formation, driven by CDM, can reproduce the observed structure present in the Universe.
These results are robust evidences that most of the DM should be present in the form of
CDM.
We wish to recall that the bottom-up formation model is also supported by observations
of the so-called Lyman-α Forest. This is a collection of Lyman-α absorption lines, caused
by the presence of intergalactic gas, in high redshift (z ≃ 2 − 4) spectra of quasars. This
set of data provides information on the distribution of neutral hydrogen, which is then
compared to numerical simulations. The outcome is that the presence of CDM is essential
to correctly reproduce the observed distributions, see e.g. Ref. [43].
All the observations described have been crucial to test the DM hypothesis at different
astrophysical scales. A summary of all these evidences is given in the right panel of
Fig. 2.1. The concordance between the different measurements has led to consider as the
Standard Model of Big Bang Cosmology the one in which the Universe is constituted by
8
2.2 Density and velocity distributions
manly DE and CDM. This is commonly referred as “ΛCDM model”.
2.2 Density and velocity distributions
The presence of DM at different astrophysical scales is confirmed by many observations,
which we have summarized in the previous Section. On the other hand, the actual DM
density distribution in the galaxies is still not known precisely. From the measurements
of rotational curves, we know that the DM profile should decrease as ρχ ∝ r−2 , at
large radii from the galactic center, but information on the innermost part is difficult to
obtain from the data. For this reason, the N-body simulations represent the common
tool used to derive DM profiles. This numerical method suffers, however, from numerous
complications. Most notably, the fact that baryons are not included in the simulations
makes it difficult to explore the very central region (r . 1 kpc) of the galaxies.
Moreover, the presence of a Supermassive Black Hole (SBH) in the inner part of a
galaxy could change the DM distribution at very small scales. The models of adiabatic
growth of a SBH predict the presence of a spike around the SBH [44]. In this case, the
DM mass distribution would follow a power-law and the annihilation signals from the
galactic center would be significantly increased.
However, it was pointed out in Ref. [45] that the merger history of the galaxy and the
SBH can also influence the presence of the spike and that core scouring effects of merging
black holes can actually reduced the DM density in the central region of the galaxies,
see also Ref. [46].
Using N-body simulations results, the DM density profile ρ(r) is usually parameterized
as follows:
r γ 1 + (r /r )α (β−γ)/α
0
0 s
,
(2.7)
ρ(r) = ρ0
r
1 + (r/rs )α
where the “scale radius” rs is the distance at which ρ ∝ r−2 , r0 = 8.5 kpc is the distance
of the Solar System from the galactic center and ρ0 is the local DM density. The values
of the parameters α, β, γ, rs for the Isothermal [47], Navarro-Frenk-White [48] and
Moore [49] profiles are reported in Tab. 2.1. Recent numerical simulations prefer the
Einasto profile [50]:
α
2
r
ρ(r) = ρs exp −
−1
,
(2.8)
α
rs
with α, ρs and rs fixed to the values in Tab. 2.1. In Fig. 2.2 the different behaviours of
the density profiles in the central region is clearly visible. We explicitly show also the
extrapolation to the very small scale r . 1 kpc.
Beyond the density profiles described before, other scenarios are possible. In Ref. [51]
these possibilities have been analyzed extensively. The DM density distributions were
classified into the following categories: spherically symmetric matter density ρ with
isotropic velocity dispersion, spherically symmetric matter density with non-isotropic velocity dispersion, axisymmetric models and triaxial models. For each model, the allowed
ranges for the local DM density ρ0 are derived. Assuming maximal or minimal non-halo
9
Chapter 2 The Dark Matter
Halo model
Isothermal
NFW
Moore
Halo model
Einasto
α
2
1
1.5
α
0.17
β
2
3
3
ρs [GeV/cm3 ]
0.06
γ
0
1
1.5
-
rs [kpc]
5
20
28
rs [kpc]
20
Table 2.1: Parameters for the Isothermal, NFW, Moore and Einasto Dark Matter density profiles.
105
104
Ρ@GeVcm3 D
103
102
Moore
NFW
r0 = 8.5 @kpcD
Ρ0 = 0.3 @GeVcm3 D
Einasto
101
100
Isothermal
10-1
10-2
10-3 -3
10
10-2
100
10-1
101
102
[email protected]
Figure 2.2: The different DM density profiles, as predicted by N-body simulations.
The local DM density ρ0 has been fixed to the default value of 0.3 GeV cm−3 .
components in the Galaxy, the intervals are determined using constraints on the local rotational velocity v0 from the galactic rotational curve: 170 km s−1 ≤ v0 ≤ 270 km s−1 [52]
at 90 % C.L.. A standard value of ρ0 , commonly used in the literature, is 0.3 GeV cm−3 .
If the density profile is an essential quantity for the estimation of the DM annihilation
signal, the velocity distribution function fgal (v) of DM particles at the Earth’s location
enters the calculation of even rates in direct detection experiments.
Once the density distribution ρ and the gravitational potential Φ are fixed, the sixdimensional phase-space distribution function Fgal (r, v) can be determined using the
method of Eddington. The velocity distribution function fgal (v) is then given by Fgal (r0 , v),
where r0 = (r0 , 0, 0) is the Earth’s location in the Galaxy, with r0 ≃ 8.5 kpc.
For spherically symmetric models with isotropic velocity dispersion, the local velocity
distribution can be approximated by a Maxwell-Boltzmann distribution, which is usually
truncated at a maximal escape velocity vesc , since DM particles with high kinetic energy
10
2.3 Dark Matter searches
can escape the gravitational field of the Galaxy:
2
2
2
2
fgal (v) = N e−v /v0 − e−vesc /v0 ,
for v < vesc ,
(2.9)
with N being a normalization factor. The common default values, compatible with
vesc = 650 km s−1 [51]. The root mean
rotational data, are v0 = 220 km s−1 [52] and p
square velocity of the DM is thus given by v̄ = 3/2 v0 ≃ 270 km s−1 .
The DM velocity distribution in the Earth’s reference frame f (v) is obtained from the
halo distribution function fgal through a Galilean velocity transformation:
f (v) = fgal (v + w⊕ (t)) ,
(2.10)
where w⊕ (t) is the velocity of the Earth in the galactic reference frame, whose modulus
is given by
w⊕ (t) = v⊙ + v⊕ cos γ cos [ω(t − t0 )] ,
(2.11)
where v⊙ ≃ (220 + 12) km s−1 is the Sun’s velocity with respect to the galactic frame, including the local Keplerian velocity as well as the Sun’s peculiar velocity, and v⊕ ≃ 30 km s−1
is the velocity of the Earth relative to the Sun. The angle γ ≃ π/3 is the inclination
of the ecliptic with respect to the galactic plane, ω = 2π/T with T = 1 year and
t0 ≃ 2nd of June is the time of the year when the Earth’s and Sun’s velocities are aligned
in the same direction. As we will discuss in Sect. 2.3.1, the motion of the Earth with
respect to the Sun gives rise to a modulated even rate with a phase of 1 year, which can
be revealed by direct detection experiments.
2.3 Dark Matter searches
In this Section we discuss the different DM searches. Sect. 2.3.1 is devoted to direct
detection techniques, while in Sect. 2.3.2 the indirect detection methods are summarized.
Finally, in Sect. 2.3.3, we present the actual limits provided by collider experiments. A
review on DM searches is given in Ref. [53].
2.3.1 Direct detection
The presence of DM particles in the galactic halo could be incontrovertibly proved by
the observation of their scatterings with the nuclei of a target material. To reveal the
rare DM interactions, detectors must have high target mass, a precise control of the
background and a low energy threshold. The nuclear recoil due to DM scattering off
a nucleus can induce different signals inside a detector: heat deposition, ionization and
scintillation. Most of the existing experiments are hybrid detectors that profit from the
simultaneous measurement of two signals. We report in Tab. 2.2 the most important
ones, divided according to their detection techniques.
11
Chapter 2 The Dark Matter
Techniques
Scintillation
Scintillation & Heat
Scintillation & Ionization
Ionization & Heat
Bubble chamber
Experiments
DAMA, KIMS, ANAIS
CRESST
ZEPLIN, XENON, WARP, ArDM
CDMS, EDELWEISS
COUPP, PICASSO
Table 2.2: Direct detection experiments divided with respect to their detection techniques.
The differential event rate dR/dER for DM scattering, in units of counts per energy
per kg detector mass per day, is given by the following expression:
Z
dR
η ρχ
dσN
=
d3 vf (v) v
(v, ER ) ,
(2.12)
dER
ρdet mχ v≥vmin
dER
where ER = Eχ − Eχ′ is the recoil energy, i.e. the energy deposited in the detector, η is
the number density of target particles, ρdet is the mass density of the detector, ρχ is the
local density of the DM particle χ and mχ is its mass. The astrophysical uncertainties
are contained in the local DM velocity distribution in the rest frame of the detector
f (v). We have denoted by vmin the minimal DM velocity that can lead to a recoil energy
ER (note that v = |v|). For example, in the case of elastic scattering χN → χN ,
vmin = [mN ER /(2µ2N )]1/2 , with µN = mχ mN /(mχ + mN ) being the DM-nucleus reduced
mass. Note that, if the target contains different elements (like in the case of NaI crystals),
the sum over the corresponding counting rates is implied.
The differential cross section dσN /dER encodes all particle and nuclear physics factors
and is given by the sum of the spin-independent (SI) and the spin-dependent (SD) cross
sections. Using the assumption of isotropy, the differential cross section can be rewritten
at low energy as [54]
dσN
1
SI 2
SD S(ER )
,
(2.13)
(v, ER ) = max σN F (ER ) + σN
dER
ER
S(0)
SI,SD
max is the
is the zero momentum DM-nucleus effective cross-section and ER
where σN
2
2
maximum recoil energy that, for elastic scattering, is equal to 2µN v /mN . The functions F (ER ) and S(ER ) are, respectively, the SI and SD nuclear form factors. Different
parameterizations for the SI form factor are used in the literature. The simplest one is
given by an exponential function,
F (ER ) = exp(−ER /(2 q0 )) ,
(2.14)
with q0 = 3~2 /(2mχ R02 ) and R0 being the nuclear radius
R0 = [0.91 (mN /GeV)1/3 + 0.3] × 10−15 m .
12
(2.15)
2.3 Dark Matter searches
Another common parameterization is represented by the Helm form factor, defined as
F (ER ) = 3 e−κ
2 s2 /2
sin(κr) − κr cos(κr)
,
(κr)3
(2.16)
√
√
with κ = 2 mN ER , s = 1 fm, r = R2 − 5s2 and R = 1.2 fm A1/3 .
The SD form factor is instead cast in the form:
S(ER ) = (ap + an )2 S00 (ER ) + (ap − an )2 S11 (ER ) + (ap + an )(ap − an )S01 (ER ) , (2.17)
with ap and an being the DM couplings to protons and neutrons. The functions S00 and
S11 are, respectively, the isoscalar and isovector spin-dependent form factors and S01 is
the interference term.
SI,SD
, are then related to the ones on
The cross section on protons and neutrons, σp,n
SI,SD
, by the simple relations
nuclei, σN
SI
σN
=
SD
σN
=
[Zfp + (A − Z)fn ]2 µ2N SI
σ ,
2
fp,n
µ2p p,n
(2.18)
µ2 SD
4 2
λp,n J(J + 1) N2 σp,n
,
3
µp
(2.19)
where µp is the reduced DM-proton mass, A is the atomic mass number, Z is the atomic
number and fp,n are the couplings of the DM to protons and neutrons. We have denoted
by J the total angular momentum of the nucleus and
λp,n =
ap hSp i + an hSn i
,
ap,n J
(2.20)
with hSp i and hSn i being the averaged spin expectaction values of the proton and the
neutron inside the nucleus.
Note that, from the experimental measurements of the even rates dR/dER , it is possible
to derive information on the quantity ρχ σ SI,SD only, for a fixed velocity distribution
function f (v). Indeed, in general, the DM could consist of different components, with
densities much smaller than the standard value 0.3 GeV cm−3 .
Since the velocity distribution function f (v) is a function of time, as discussed in
Sect. 2.2, also the DM rate is expected to vary during the year, due to the motion of the
Earth around the Sun. Therefore, in each energy bin k, the number of signal events is
given by
Z
dR
Sk =
≃ S0,k + Sm,k cos [ω(t − t0 )] ,
(2.21)
dER
dE
R
Ek
where S0,k is the average signal, Sm,k is the modulation amplitude, ω = 2π/T with
T = 1 year, and t0 = 152.5 days (corresponding to the 2nd of June). The function Sm,k
can be approximated as
Z
Z
dR
dR
1
dER
(June 2) −
(December 2) .
(2.22)
dER
Sm,k ≃
2
dER
dER
Ek
Ek
13
Chapter 2 The Dark Matter
Residuals (cpd/kg/keV)
2-6 keV
DAMA/NaI (0.29 ton×yr)
(target mass = 87.3 kg)
DAMA/LIBRA (0.53 ton×yr)
(target mass = 232.8 kg)
Time (day)
Figure 2.3:
The time-dependent residual rate in the DAMA/NaI and in the
DAMA/LIBRA annual modulation experiments. Figure taken from Ref. [3].
A seasonal effect in the count rate is identified as being due to the DM scatterings, in
case the following requirements are fullfilled: it is modulated as a cosine function with a
period of one year, a peak around the 2nd of June is present and a modulation amplitude
≤ 7% is observed. Moreover, since the modulation is due to DM induced recoils, it must
be present only in the low energy bins and only in the single hit events. It is really
difficult that systematic effects can fulfil all these requirements.
The investigation of the annual modulation signature has been carried out by the
DAMA collaboration, with the use of scintillation light from NaI(Tl) crystals as detection
technique. They have collected data with the DAMA/NaI detector [53], over 7 annual
cycles, and with the DAMA/LIBRA detector [3], over 4 annual cycles, corresponding to
a total exposure of 0.82 tons yr. The combined results show a modulation signal with
8.2 σ significance, as can be clearly seen in Fig. 2.3.
The interpretation of this seasonal variation as caused by DM elastic scatterings on
nuclei is tightly constrained by bounds coming from other direct detection experiments.
In particular, in the case of the spin-independent cross section, only a light DM particle
with mass of the order mχ . 10 GeV might be marginally compatible with the limits from
CDMS [55] and XENON10 [56], see Refs. [57, 58, 59, 60, 61, 62, 63, 64] for recent works.
In the case of the spin-dependent cross section, instead, the DAMA annual modulation
region is not in conflict with CDMS and XENON10 limits, but strong constraints from
the COUPP [65], KIMS [66] and PICASSO [67] experiments apply [63]. Note that here
and in the following we use the acronym “DAMA” to denote the combined DAMA/NaI
and DAMA/LIBRA data.
An important and still not completely clarified aspect of the direct detection search
is represented by the channeling effect [68]. The scattered nucleus loses its energy by
electromagnetic and nuclear interactions, but only the first kind of interaction leads to
a scintillation signal in the detector. Therefore, in general, just a fraction q of the total
nuclear recoil energy ER is measured. The event energy is measured in equivalent electron
14
2.3 Dark Matter searches
energy (in keVee), defined by q × ER for the total nuclear recoil energy ER in keV. The
parameter q si called “quenching factor” and for the nuclei in the DAMA detector, one
has that qN a ≃ 0.3 and qI ≃ 0.085. In Refs. [68, 69] it has been pointed out that particles
travelling along crystal planes lose all their energy electronically and thus q ∼ 1. These
are called “channeled events”. So far this effect has not been confirmed experimentally
in the relevant energy range [70]. However, if present, the channeling effect could play an
important role in the analysis of the DAMA data, since it could sizably shift the annual
modulation region [68].
If more complicated scenarios than elastic scattering are considered, the formulae for
the event rates might change and the partial discrepancy between the DAMA result and
other experiments might be attenuated. For example, this is the case for inelastic DM
scattering off nuclei, see Sect. 2.4.2 for more details. In this scenario, the expression for
vmin is modified and the DAMA allowed region, derived for a spin-dependent interaction,
would be in agreement with all the experimental data [71]. A lot of other models have
been proposed to reconcile all the results of direct detection experiments. These include
mirror world DM [72], DM with electric or magnetic dipole moments [73] and leptophilic
DM [14].
The CDMS collaboration has recently released new results [74], in which two events
survive after background reduction. These could be due to DM interactions inside the
detector at 90% confidence level.
Finally, we want to remember that the velocity distribution function of DM particles
can influence the DM event rate in direct detection experiments. This effect has been
recently analyzed in Ref. [75].
2.3.2 Indirect detection
In this Section we summarize the main indirect DM searches. A detailed review on this
topic is given in Ref. [76].
Monochromatic photons with energy Eγ ≃ mχ would represent a clear DM signature.
Unfortunately, they can only be produced at one-loop level, since the DM particle is
electrically neutral. Thus, the branching ratio for this channel is usually suppressed.
There are, however, four other processes through which γ-rays can be produced by DM
annihilation: i) bremsstrahlung emission by charged particles; ii) decays of hadrons, like
π 0 , coming from quark hadronization; iii) annihilation into three-body final states, one
of which is a photon; iv) synchrotron radiation due to e± propagation in the galactic
magnetic field. Searches for γ-rays from DM annihilation in the galactic center, in the
galactic ridge and in dwarf spheroidal satellite galaxies are carried out by the HESS
telescope [77] and by the Fermi space satellite [78].
The PAMELA (Payload for Anti-Matter Exploration and Light-nuclei) satellite [4] and
the balloon experiments ATIC [79] and PPB-BETS [80] have recently obtained important
results on positron and antiproton searches.
The PAMELA satellite revealed an excess in the positron fraction, starting from energies of 10 GeV, while no excess with respect to the background estimation was reported
15
Chapter 2 The Dark Matter
in the antiproton flux, see the upper panel of Fig. 2.4. In Ref. [81], the authors have systematically studied the possibilities to fit these experimental data under the hypothesis
that the PAMELA anomaly in the cosmic ray flux is due to DM annihilations. They
identified two different scenarios in which a satisfactory fit could be obtained: i) the
DM particle annihilates predominantly into leptons and has a mass above a few hundred
GeV; ii) the DM particle annihilates into W, Z or Higgses and has a mass greater than
10 TeV. These characteristics are rather exotic, since most of the theoretical models
predicts a DM particle with a mass lower than O(TeV) and with a negligible branching
ratio into leptons. A possibility to explain the PAMELA data with DM is represented
by a leptophilic DM, that we will briefly discuss in Sect. 2.4.3.
The DM framework is not the only possibility to explain the PAMELA anomaly.
Indeed, an astrophysical nearby electron source, like a pulsar or a supernova remnant,
could account for the excess in the positron fraction [82]. Moreover, the claim of DM
evidence from these data is, to some extend, model-dependent, since the estimation of
the astrophysical background flux suffers from big uncertainties [83].
A rise in the total flux of positrons and electrons has been measured for the first time by
the PPB-BETS experiment. More recently, the ATIC balloon has detected the presence
of an abrupt peak at energies of about 400-500 GeV. This last result has catalyzed a
lot of attention, since for its particular spectral feature it could be interpreted as a DM
annihilation signal. However, the HESS [84] and Fermi [85] data do not confirm the
presence of this peak. They, instead, report a more smooth behaviour, but an excess
is still present respect to the conventional expected background. These experimental
results are reported in the lower panel of Fig. 2.4.
Severe constraints on the DM interpretation of the PAMELA/Fermi anomaly arise
from the analysis of the photon flux produced by charged particles. While propagating
in the Galaxy, the e± can undergo inverse Compton scattering with the photons of
the starlight, of the infrared light or of the CMB. The production of γ-rays through
this mechanism should not exceed the existing limits provided by HESS and FERMI.
Moreover, the synchrotron radiation bounds coming from radio observations represent
another strong constraint on the DM annihilation scenario. Both γ-rays and synchrotron
emission prefer a cored isothermal DM density profile. Indeed, a steep DM density
profiles, like NFW, Moore and Einasto, would easily violate the experimental bounds,
see e.g. Ref. [86].
The DM particles could also be indirectly detected by two other annihilation products: antideuterons and neutrinos. The GAPS [87] and AMS-02 [88] experiments search
for antideuterons D̄ from DM annihilations in the galactic center and in the galactic
halo. The indirect detection technique through neutrinos, will be presented in detail in
Chapter 3.
2.3.3 Collider experiments
The bound from the Z-boson decay width is one of the strongest constraints imposed by
collider experiments on a light DM particle χ. From the analysis of the data from the
16
2.3 Dark Matter searches
×10
0.3
0.35
0.2
0.3
Donato 2001 (D, φ =500MV)
Simon 1998 (LBM, φ =500MV)
Ptuskin 2006 (PD, φ =550MV)
PAMELA
+
+
-
Positron fraction φ(e ) / (φ(e )+ φ(e ))
-3
0.4
0.1
p/p
0.25
0.2
0.15
0.1
0.02
0.05
PAMELA
0.01
1
10
100
Energy (GeV)
0
1
10
kinetic energy (GeV)
102
Figure 2.4: Data from the PAMELA satellite on the positron fraction and on the
antiproton flux (upper panel), figures taken from Ref. [4]. The spectrum of electron plus
positron, provided by the Fermi satellite (lower panel), figure taken from Ref. [85].
17
Chapter 2 The Dark Matter
e+ e− collider experiment LEP2, the decay width ΓZ→χχ is required to be less than 4.2
MeV [89].
Existing limits on the mass of DM candidates are, unfortunately, strictly dependent
on the specific model considered. Indeed, experimental bounds have usually been placed
on new electrically charged particles. From LEP2 data, their masses are now forced to
be greater than 100 GeV. Once a particular model is fixed, these limits can be translated
to limits on the DM mass. The most common example is represented by supersymmetric
models with gaugino mass unification at the GUT scale (we refer to Sect. 2.4.1 for more
details on Supersymmetry and its DM candidates). In this specific framework, the mass
of the lightest neutralino is set to be equal to half the mass of the charginos. Thus, in
this case, the allowed range on the chargino mass, mχ̃± & 103 GeV, implies a lower limit
1
of about 50 GeV on the lightest neutralino. However, we want to stress that no bound
on the lightest neutralino mass is predicted by collider experiments, in the case that no
gaugino mass unification is assumed.
Important constraints on new particles and on new physics models are provided by
electroweak precision measurements, carried on by the LEP2 and Tevatron experiments.
The precision data are commonly expressed using the Peskin-Takeuchi parameters: S,
T and U . The best-fit value on the S parameter disfavours new chiral fermions beyond
the SM ones, while the T parameter sets a limit on the vacuum expectaction value
of new non-singlet scalars [89]. The U parameter, instead, is defined as (SW − SZ ),
with SW (SZ ) given by the difference between the W -boson (Z-boson) self-energy at
2 (M 2 ) and Q2 = 0. Other robust limits can be inferred from the following
Q2 = MW
Z
experimental searches: measurements of the b → s + γ decay process, with an actual
limit of 2.89 ≤ B(b → s + γ) × 10−4 ≤ 4.21 [90]; measurements of the muon anomalous
magnetic moment aµ ≡ (gµ − 2)/2, whose deviation ∆aµ from the theoretical evaluation
within the SM is equal to (−98 ≤ ∆aµ × 1011 ≤ 565) [91]; the upper bound on the
branching ratio BR(Bs0 → µ− + µ+ ), that is set to BR(Bs0 → µ− + µ+ ) < 1.2 × 10−7 [92].
All these constraints have to be taken into account when a specific DM candidate is
considered. For an exhaustive explanation of the different experimental searches at
colliders, regarding Supersymmetry and New Physics in general, we refer to Ref. [89].
Through the future LHC proton-proton collisions at 14 TeV center-of-mass energy, new
important results on physics beyond the SM can be achieved. These will provide further
strong hints for a specific DM candidate. However, in most of the models present in the
literature, the DM particle can, in general, be produced at colliders only after a long decay
chain, which makes the extrapolation of its properties rather involved. For instance, in
the case of supersymmetric models, we could have q̃ → χ̃02 q → ˜llq → χ̃01 llq, where we have
denoted the lightest neutralino by χ̃01 . The precise value of its mass could be extracted by
an analysis of the mass distribution endpoints or “edges”. For a dedicated description
of this method, we refer to Ref. [93]. Moreover, to disentangle different models with
DM candidates, for example Supersymmetry and Extra Dimensions, it is fundamentally
important to measure the spin of the lightest neutral particle produced in the decay
chain. This possibility has been vastly analyzed in Ref. [94].
18
2.4 Dark Matter candidates
Figure 2.5: Summary of the most common Dark Matter particles as a function of their
masses and cross sections, taken from Ref. [97].
2.4 Dark Matter candidates
Astrophysics provides us with compelling evidences of DM. Unfortunately, on the particle physics side, several models predict candidates with rather different characteristics:
the DM mass can range from about 10−15 GeV to 1015 GeV and the scattering cross
section can span several order of magnitude, from around 10−35 pb to 1 pb. The various possibilities for DM candidates fulfill, however, some common properties: the DM
particle is stable or at least very long-lived and neutral under electric charge and colour
charge.
In Sect. 2.4.1, we focus on Weakly Interacting Massive Particles (WIMPs) as DM
candidates, while in Sect. 2.4.2, we summarize the more common non-WIMP candidates.
The different characteristics for each model can be seen in Fig. 2.5. In Sect. 2.4.3, we
also report examples of DM particles with more exotic interactions. For recent reviews
on DM candidates we refer to Refs. [95, 96].
2.4.1 WIMP candidates
The WIMPs are the more common DM candidates considered in the literature. These
are particles created thermally in the Early Universe, with a weak cross section and with
a mass around the EW scale. The main motivation for the WIMP hypothesis is that
its characteristics are sufficient to obtain a relic density in agreement with the WMAP
data, reported in Eq. (2.5). This prediction is the so-called “WIMP miracle”.
The complete relic abundance calculation for a thermal relic χ is reported in Ref. [98],
to which we refer for more details. Here, we just report the final expression that can be
19
Chapter 2 The Dark Matter
cast as
Ωχ h2 =
3.3 × 10−38 cm2
.
1/2
hσann viint
g∗s (Tf )
xf
(2.23)
In the previous formula, we have defined the variable x as mχ /T , with mχ being the mass
of the thermal relic and T the temperature of the Universe. Its value at the freeze-out
temperature Tf is denoted with xf . The function g∗s encodes the number of relativistic
degrees of freedom:
3
3
X
Ti
7 X
Ti
gi
+
,
(2.24)
gi
g∗s =
T
8
T
i=bosons
i=fermions
where the factor 7/8 for fermions arises from Fermi-Dirac statistics, on the contrary to
the Bose-Einstein one. Finally, hσann viint is the thermally-averaged annihilation cross
section, integrated with weight 1/x2 from the freeze-out till today:
Z ∞
1
b
hσann viint =
dx 2 hσann vix ≃ a +
,
(2.25)
x
2xf
xf
where we have used the low velocity expansion hσann vix ≃ a + b/x, for the DM being a
non-relativistic particle.
The approximate freeze-out temperature can be found using the following equation [98]:
hσann vixf
mχ
1
10
xf + ln (xf g∗ (Tf )) = ln 9 × 10 g
,
(2.26)
2
100 GeV 5 × 10−37 cm2
where g is the number of degrees of freedom of the DM particle and
4
4
X
Ti
Ti
7 X
gi
gi
g∗ =
+
.
T
8
T
i=bosons
(2.27)
i=fermions
Considering mχ ≃ 100 GeV, g = 2 and hσann vixf ≃ 5 × 10−37 cm2 , Eq. (2.26) leads to
xf ≃ 20 and, correspondingly, g∗ (Tf ) ≃ 80. Using these values with Eq. (2.23), we find
Ωχ h2 ≃ 0.15, in agreement with the 1σ range allowed by WMAP data.
The main WIMP candidates arise from New Physics models at the EW scale. These
proposed high-energy theories attempt to solve some of the problems of the SM, in
particular the one-loop quadratically divergent quantum corrections to scalar masses,
also called the “hierarchy problem” [99].
In the following, we briefly discuss the WIMP candidates present in three extensions
of the SM: Supersymmetry, Extra Dimensions and Little Higgs theories. In each of these
different models, the WIMP candidate is stable, because it is protected by a conserved
quantum number: R-parity, K-parity and T -parity, respectively.
Supersymmetric particles
Supersymmetric theories are based on a symmetry between fermions and bosons [99, 100].
20
2.4 Dark Matter candidates
Minimal Supersymmetric Standard Model
Interaction eigenstates
q̃L , q̃R
squark
˜lL , ˜lR
slepton
ν̃L
sneutrino
g̃
gluino
W̃ ±
H̃ ±
B̃
W̃ 3
H̃u0
H̃d0
wino
higgsino
bino
wino
higgsino
higgsino
Mass Eigenstates
q̃1 , q̃2
˜l1 , ˜l2
ν̃L
g̃
χ̃±
1,2
charginos
χ̃01,2,3,4
neutralinos
Table 2.3: Particle content of the MSSM. We report explicitly the interaction and mass
eigenstates.
The Minimal Supersymmetric Standard Model (MSSM) is the minimally supersymmetric
version of the SM, in which to each fermion of the SM there is an associated spin-0 particle
and to each Higgs or gauge boson there is a spin-1/2 particle. We wish to recall that
two Higgs doublets are necessary in the MSSM to avoid gauge anomalies. In Table 2.3
we list all the particles present in the MSSM and their corresponding names.
No experimental observations have been found so far for the existence of superpartners.
These particles, if they exist, are thus forced to be heavier than their SM companions
and Supersymmetry (SUSY) is expected to be a broken symmetry. The simplest way to
break SUSY is through the introduction of “soft terms” in the MSSM Lagrangian [99].
These are terms that explicitly break SUSY, without introducing ultraviolet divergences.
If we wrote in the superpotential W all the possible gauge invariant and renormalizable
terms, we would obtain a theory that violates both baryon number B and lepton number
L. This would lead to extremely fast proton decay, in contrast to the experimental data
that set the proton lifetime to be O(> 1033 ) years. This problem is easily overcome with
the postulation of a Z2 symmetry called “R-parity”, defined as
PR = (−1)3(B−L)+2s ,
(2.28)
where s is the spin of the particle. For all the SM particles R = 1, while for the
superpartners R = −1. If R-parity is conserved, the interaction terms that lead to proton
decay are forbidden. Moreover, the lightest supersymmetric particle (LSP) represents a
well motivated DM candidate, in case it is neutral. Indeed, the LSP has to be stable,
since it cannot decay to SM particles without violating R-parity. A review on SUSY DM
21
Chapter 2 The Dark Matter
is given in Ref. [101] and its collider, direct and indirect detection has been analyzed in
Ref. [102].
In most of the SUSY models, the LSP is the lightest one of the neutralinos χ̃01 . Neutralinos are four mass eigenstates, given by linear combinations of the bino B̃, the neutral
wino W̃ 3 and of the two Higgsino states H̃d0 , H̃u0 :
(i)
(i)
(i)
(i)
χ̃0i ≡ a1 B̃ + a2 W̃ 3 + a3 H̃d0 + a4 H̃u0
(i = 1, 2, 3, 4) .
These states are eigenstates of the following mass matrix:

M1
0
−mZ cβ sθW mZ sβ sθW


0
M2
mZ cβ cθW −mZ sβ cθW

Mχ̃0 = 
 −mZ cβ sθ
mZ cβ cθW
0
−µ
W

mZ sβ sθW −mZ sβ cθW
−µ
0
(2.29)




,


(2.30)
where we have used the notation cα ≡ cos α and sα ≡ sin α. The angle β is related to
the ratio of the Higgs vacuum expectaction values vu = hHu0 i and vd = hHd0 i:
tan β = vu /vd ,
(2.31)
with vu2 + vd2 ∼ (174GeV)2 . The angle θW is the Weinberg angle, whose value at the
Z-boson mass scale is sin2 θW ≃ 0.23120.
The µ parameter in the neutralino mass matrix comes from the Higgs mixing mass
term present in the superpotential of the MSSM:
WMSSM ⊃ µ(Hu )α (Hd )β ǫαβ = µ(Hu+ Hd− − Hu0 Hd0 ) ,
(2.32)
where we have denoted by Hu = (Hu+ , Hu0 ) and Hd = (Hd0 , Hd− ) the chiral superfields
and with α, β = 1, 2 the weak isospin indices. The parameters M1 and M2 come from
the bino B̃ and winos W̃ i mass terms present in the soft SUSY breaking Lagrangian:
1
i
i
Lsoft
MSSM ⊃ − (M1 B̃ B̃ + M2 W̃ W̃ + M3 g̃g̃ + c.c.) ,
2
(2.33)
where, for completeness, we have also reported the gluino g̃ soft breaking mass term.
The gaugino mass parameters Mi (i = 1, 2, 3) are in principle free parameters, but
they are usually assumed to unify to a common value called m1/2 at the Grand Unified
(GUT) scale MGUT ∼ 2 × 1016 GeV, where the gauge couplings of the MSSM unify [99].
Using the renormalization group equations, it is possible to derive the following relation
between M1 and M2 :
5
(2.34)
M1 = tan2 θW M2 ,
3
valid at each energy scale. In particular, at the EW scale, the following relation holds:
M1 ≃ 0.5 M2 .
22
(2.35)
2.4 Dark Matter candidates
If the gaugino soft breaking parameters, M1 and M2 , and the Higgs mixing parameter
µ are much greater than the electroweak scale, i.e. M1 , M2 , µ ≫ mZ , the neutralino
eigenstates assume the following approximate compositions:
1
1
χ̃0i ≃ {B̃, W̃ 3 , √ (H̃d0 − H̃u0 ), √ (H̃d0 + H̃u0 )} ,
2
2
(2.36)
with mass values given by
mχ̃0 ≃ {M1 , M2 , |µ|, |µ|} .
(2.37)
i
From the previous expressions, we can trace the two asymptotic behaviours of the lightest
neutralino. For large values of the soft breaking parameters M1 and M2 , χ̃01 is “higgsinolike”, with a mass determined mainly by the µ value: mχ̃01 ≃ µ; on the contrary, for large
values of µ, the lightest neutralino is “bino-like”, with mχ̃01 ≃ M1 ≃ 0.5 M2 .
−
The LEP2 experiment has searched for charginos through the channel e+ e− → χ̃+
1 χ̃1 ,
where χ̃±
& 103 GeV
1 denotes the lightest chargino. The experimental lower bound of mχ̃±
1
can be considered as a lower bound also on the M2 and µ parameters (M2 , µ & 103 GeV),
since the chargino mass matrix in the basis (W̃ + , H̃u+ , W̃ − , H̃d− ) assumes the form


0 XT
,
Mχ̃± = 
(2.38)
X 0
with

X= √
M2
2 mZ cβ cθW
√
2 mZ sβ cθW
µ

.
(2.39)
Therefore, the bound on the gaugino mass parameter M2 implies a bound on the lightest
neutralino of about 50 GeV. We want, however, to stress that mχ̃01 might be smaller
than this value, in an effective MSSM model in which the gaugino mass unification of
Eq. (2.35) does not hold [58, 103]. In this case, the lowest possible values of the lightest
neutralino mass are determined by the M1 parameter and χ̃01 ≃ B̃. A theoretical model
with M1 = R M2 and 0.01 ≤ R ≤ 0.5 will be employed in Chapter 5 to analyze the muon
fluxes expected by the neutralino configurations compatible with the DAMA annual
modulation region.
The neutralino couplings to the Z-boson are proportional to (a23 − a24 ), with a3 and
a4 the two higgsino fractions. Remember, indeed, that trilinear couplings between the
Z-boson and two W 3 -bosons or two B-bosons do not exist in the SM. Therefore, the couplings between the Z-boson and two winos W̃ 3 or binos B̃ are absent in the MSSM. Very
light neutralinos, with mχ̃01 < mZ /2, are almost in a pure bino configuration (a1 ≃ 1).
Therefore, most of the light neutralino configurations will survived to the constraints
imposed by Z-boson decay width. The χ̃01 can have a sizable mixing with H̃d0 only for
small µ values.
Besides the neutralino, also the sneutrino ν̃, the supersymmetric partner of the SM
neutrino, can be a viable LSP. In the framework of the simple MSSM, in which only the
23
Chapter 2 The Dark Matter
left-handed neutrinos exists, the sneutrino has been tightly constrainted by direct DM
detection experiments [104], since its large coupling to the Z-boson induces a scattering
cross section off nucleons higher than the experimental limits. Because of the same
coupling, also the annihilation cross section results really large, implying a very small
sneutrino relic density [105].
Sneutrinos have been reanalyzed in extended model in which right-handed neutrinos
are added to the MSSM and lepton-number violating terms are included [106]. They
have also been studied in the framework of the Next-to-Minimal Supersymmetric Model
in which, beyond the right-handed neutrinos, also a singlet scalar is added [107]. In these
non-minimal models the sneutrino turns out to still be a viable DM candidate.
Another SUSY DM candidate is represented by the gravitino, the superpartners of
the graviton. This particle does, however, not belong to the WIMP category, having
a cross section that is much lower than the standard weak interaction one, as can be
seen from Fig. 2.5. The gravitino may or may not be thermally produced and it is
usually assumed to decay, as otherwise its relic density would be much higher than the
one derived from the WMAP data. Strong constraints on this type of DM come from
Big Bang Nucleosynthesis, since the gravitino decay products could alter the primordial
abundances of light elements. This is also known as “cosmological gravitino problem”. A
working scenario in which the gravitino represents a viable DM candidate, in agreement
with cosmological data, is represented by a SUSY model with small R-parity violating
terms [108]. Unfortunately, it is extremely difficult to detect the gravitinos, since they
have only gravitational interactions.
Kaluza-Klein particles
Kaluza-Klein particles are a kind of excitations of the SM fields. They appear in Extra
Dimensional models, in which space-time is considered to have more dimensions than the
standard four. To reconcile these theories with the observed four-dimensional Universe,
the Extra Dimensions are compactified. It is possible that standing waves are present
in the extra compactified dimensions. Their existence would predict an infinite number
of states, the so-called “Kaluza-Klein tower”, with energy values given by E = n h c/R,
where R is the radius of the extra dimension, n is an integer, h is the Planck’s constant
and c is the speed of light. This prediction is a peculiar characteristic of this type of
models.
In models of Universal Extra Dimensions [109], a discrete symmetry arises from momentum conservation in the Extra Dimension. This is called “K-parity” and, in analogy
to R-parity, it ensures that the lightest Kaluza-Klein particle (LKP) is stable and thus
a good DM candidate [110]. In many Extra Dimensions models, the LKP results to be
the first Kaluza-Klein excitation of the photon. The DM particle in this case is a boson,
in contrast to the SUSY neutralino.
Little Higgs particles
Little Higgs models have been proposed as a solution of the hierarchy problem, alterna-
24
2.4 Dark Matter candidates
tively to Supersymmetry and Extra Dimensions. In these models, the Higgs boson is a
pseudo-Nambu-Goldstone boson arising from a global symmetry, broken spontaneously
at the TeV scale. As a consequence, the Higgs mass is stable, with respect to one-loop
corrections up to an energy of about 10 TeV. More details on Little Higgs models and
on their phenomenology can be found in Ref. [111].
The Little Higgs models can provide a good DM candidate if a discrete symmetry,
called “T -parity”, is implemented. The SM particles are even under this symmetry,
while all the heavy particles predicted by the model are odd. The lightest T -odd particle
(LTP) is stable and in most of the models it is the heavy photon [112].
2.4.2 Non-WIMP candidates
In this Section we summarize some of the most common non-WIMP DM candidates:
neutrinos, axions and axinos, and wimpzilla particles.
Neutrinos
The SM neutrinos have been considered in the past as possible DM candidates. The Big
Bang model, indeed, predicts the existence of a cosmic neutrino background, with a relic
density given by
P
mi
2
,
(2.40)
Ων h = i
94 eV
where we have denoted by mi the i-th neutrino mass. Considering the limit on the
electron neutrino mass from the tritium β-decay experiments [89], the sum of the neutrino
masses can be at most equal to 6 eV, implying that Ων h2 ≤ 0.06. Therefore, SM
neutrinos can only be a subdominant DM component.
More stringent constraints on the contribution of SM neutrinos to the DM come from
the analysis of the WMAP data on CMB anisotropies, combined with supernovae and
large scale structure observations [113]. We wish to recall that neutrinos would act
as HDM and would induce a top-down scenario in structure formation, which is not
supported by data from the SDSS and the 2dFGRS. The limit at 95% C.L. on the sum
of the neutrino masses is about 0.67 eV, which translates into Ων h2 ≤ 0.007.
Hypothetical neutrinos without SM interactions, besides mixing with SM neutrinos,
could also behave as DM particles. They are called “sterile neutrinos” and they usually
act as Warm Dark Matter, i.e. a type of DM with intermediate characteristics between
CDM and HDM. Their mass range is tightly constrained by X-ray bounds and constraints
on the DM relic abundance. The actual allowed region depends on the sterile neutrino
production mechanism: for non-resonant production, a lower bound of 1.8 keV and an
upper bound of 4 keV is set, while for resonant production the corresponding bounds are
weaker, with a lower bound of 1 keV and an upper bound of 50 keV, see Ref. [114] for a
review.
In principle, other SU (2)L doublets, containing heavy neutrinos, could be added to the
SM. Currently, the mass of heavy neutrinos is constrained to be heavier than 45 GeV,
25
Chapter 2 The Dark Matter
from data on the invisible Z-boson decay width. An upper bound on the mass is set
to 300 GeV, a value above which the heavy neutrinos would overclose the Universe.
The allowed mass window for heavy neutrinos could be reduced even more, in case an
asymmetry between neutrinos and anti-neutrinos is present, see e.g. Ref. [115].
Axions and axinos
The Peccei-Quinn theory [116], which aims to solve the strong-CP problem of QCD,
predicts the existence of a neutral particle, called “axion”, that could form the DM.
Direct searches and astrophysical constraints, from globular clusters and from the supernova 1987A, restrict the axion to have an extremely small mass, between 10−6 eV and
10−3 eV, and a very low cross section with the SM particles.
The CERN Axion Solar Telescope (CAST) is a strong magnet pointing towards the
Sun, searching for axions. If these particles exist, they could be produced in the Sun by
photon scatterings with protons and electrons, in the presence of strong electric fields.
The CAST experiment, using an intense magnetic field, would then convert the axions
back to X-ray photons. By now, no experiment has detected axions. The only evidence
was reported by the PVLAS collaboration in 2005 [117], but this result has been ruled
out by new data, obtained after an upgrade of the experiment [118].
In SUSY models, also the fermionic partner of the axion, the axino, could be a good
DM candidate [119].
Wimpzilla particles
Superheavy DM particles with a mass mχ > 1010 GeV are usually called “wimpzillas”.
These particles are no thermal relics of the early Universe, but arise from non-thermal
processes like, for example, gravitational production at the end of inflation [120]. They
can play an important role also in astrophysics, since they could explain the observed
cosmic rays, at energies above the GZK cutoff, as due to annihilations or decays of
wimpzilla particles [121].
2.4.3 Non-standard Dark Matter interactions
In Sect. 2.3.1, we presented the physics of DM direct detection, assuming that the DM
particle scatters elastically off nuclei inside a detector. However, different models beyond
this common scenario have been proposed as well. In this Section we briefly discuss two
models in which the DM has “non-standard interactions”: inelastic DM and leptophilic
DM.
Inelastic Dark Matter
The inelastic Dark Matter (iDM) scenario has first been proposed in Ref. [122] to explain
the results obtained by the DAMA/NaI experiment. Recently, this model has been reconsidered in several papers, see for instance Ref. [123], in the light of the DAMA/LIBRA an-
26
2.4 Dark Matter candidates
nual modulation data. The two main hypotheses of iDM are: i) the existence of an excited
state χ∗ of the DM particle χ with a small mass splitting δ = m∗χ − mχ ≃ 100 keV, ii) a
suppressed elastic scattering χN → χN , with respect to the inelastic process χN → χ∗ N .
In this framework, the minimal DM velocity necessary to deposit an energy ER in the
detector is
mN ER
1
vmin = √
+δ .
(2.41)
µN
2mN ER
The different kinematics of iDM leads to several consequences: scattering on heavy nuclei
is favoured over that on light ones, as can be deduced from the expression of vmin , the
annual modulation signal is enhanced and the low-energy events are suppressed [124].
The analyses reported in Ref. [123] have shown that iDM with spin-independent cross
section is a viable DM candidate, consistent at the same time with the DAMA data
and with the limits from the CDMS [55], XENON10 [56], KIMS [66], ZEPLIN [125]
and CRESST [126] experiments. The iDM with dominantly spin-dependent cross section has been studied in Ref. [71]. This scenario is able to explain the DAMA results
and to fulfill the strong constraints on spin-dependent cross sections coming from the
COUPP [65], KIMS [66] and PICASSO [67] experiments. Theoretical models for iDM
have been presented, for example, in Refs. [124, 127].
Leptophilic Dark Matter
In leptophilic models, the DM particle couples mainly to leptons rather than to quarks.
They have been introduced for two main reasons. First, they could explain the excess
in cosmic rays, detected by PAMELA and ATIC (see Sect. 2.3.2), in terms of a DM
scenario. Second, they could reconcile the DAMA results with the other direct detection
experiments. Indeed, electronic events can contribute to the scintillation light signal in
the DAMA detector, but are rejected by most of the other DM experiments, like CDMS
and XENON10.
A simple model of leptophilic DM has been presented in Ref. [128]. In this constext,
a Dark Sector (DS) is added to the SM and the DM χ is a Dirac fermion charged under
a new Abelian gauge symmetry U (1)DS . All the SM particles are odd under a discrete
DS-parity, while the DM is even.
The Lagrangian of the DS is given by
1 ′2
+ χ̄γ µ Dµ χ + |Dµ φ|2 − Mχ χ̄χ − VDS (φ) ,
LDS = − Fµν
4
(2.42)
where φ is a scalar Higgs field that breaks the gauge group U (1)DS and F ′ is the field
strength of the new gauge boson U . The latter is supposed to be leptophilic and to
mediate the coupling between the SM and the DS (at least some of the SM leptons
must be charged under the new gauge group). Tight constraints from measurements of
lepton magnetic dipole moments and from different low-energy leptonic cross sections
force the U boson to have a small coupling to the electron and the muon [128]. The
correct annihilation cross section to explain the PAMELA/ATIC data is provided by
27
Chapter 2 The Dark Matter
the Sommerfeld enhancement [129] for a DM mass of O(800 GeV) and for a U boson of
mass 1-10 GeV. Moreover, within this model, the authors identify a region of the allowed
parameter space, where the DAMA results are consistent with the lack of detection by
the other experiments.
In Chapter 5 we will analysis the lepthophilic DM scenario using a model independent
formalism and, in particular, we will show how constraints coming from indirect detection
with neutrinos can provide strong bounds.
28
3
Indirect detection with neutrinos
In this Chapter, the method of indirect DM detection with neutrinos is presented in
detail. The basic ingredients for the calculation of the neutrino flux coming from celestial
bodies, like the Sun and the Earth, are given in Sect. 3.1. In Sect. 3.2, instead, the
neutrino flux from the galactic center is considered. The calculation of the muon flux is
presented in Sect. 3.3.
3.1 Neutrino flux from the Sun and the Earth
In Sect. 3.1.1, we summarize the main formulae which we employed to evaluate the capture rates of DM particles by celestial bodies. The process of neutrino production is
discussed in Sect. 3.1.2, while the neutrino propagation aspects are treated in Sect. 3.1.3.
3.1.1 Capture and annihilation rates
If DM particles exist in the galactic halo, they have a finite probability to scatter with
the nuclei present in the Sun or the Earth. Through subsequent scatterings, they lose
energy and once their velocity is less than the escape velocity of the body, they become
gravitationally bound. Being captured, the DM particles will continue to cross the celestial body and scatter with its nuclei. In this way, their velocities will gradually decrease
and they will sink into the central part of the body, where they accumulate. In Ref. [130],
the authors found that this process always occurs for the standard WIMP cross section.
The calculation of the DM capture by the Sun and the Earth has been firstly carried
on by Gould, in Ref. [131]. Considering a spherically symmetric shell of material, the
DM capture rate per unit shell volume may be written as
Z ∞
dC
f¯(v)
=
dv
u Ω(u) ,
(3.1)
dV
v
0
29
Chapter 3 Indirect detection with neutrinos
where v is the velocitypof the DM at infinity (far from the gravitational potential of
the body), while u = v 2 + u2esc is the one at the DM-nucleus interaction point and
uesc is the escape velocity at that particular point of the celestial body. The velocity
distribution f¯(v) is defined as
f¯(v) = v 2 2π
Z
1
d cos θ f (v) ,
(3.2)
−1
where f (v) is the Maxwell-Boltzmann distribution of the DM particles, as seen by an
observer that moves with a velocity v⊙ relative to the DM halo,
nχ
(v + v⊙ )2
f (v) =
,
(3.3)
exp −
v02
(πv02 )3/2
with nχ = ρχ /mχ being the local Dark Matter number density that we fixed to the
standard value of 0.3 GeV cm−3 . For the calculation of the capture rate, we assume
v⊙ = v0 = 220 km s−1 .
The function Ω(u) denotes the rate of DM scatterings from a velocity v to a velocity
less that uesc . If the DM elastic scattering cross section σN is isotropic and velocity
independent, and if the temperature of the shell can be neglected, the following simple
relation holds [131]:
Ω(u) = σN nN u P ,
(3.4)
where nN is the number density of nuclei with mass mN in the celestial body and P is
the probability that the DM scatters at a velocity less than uesc ,
p
1
v2
2
P= 2
β
u
−
v
,
(3.5)
u
−
θ
− esc
esc
v + u2esc
β−
with β− = 4mχ mN /(mχ − mN )2 .
If the DM particles scatter on elements heavier than hydrogen, the differential cross
section should also contain a form factor, see Eq. (2.13). Using the exponential parameterization given in Eq. (2.14), the expression of the scattering probability P is modified
to
Z E max
p
R
1
ER
β− uesc − v ,
(3.6)
dER max exp −
P=
θ
ER
q0
mχ v 2 /2
where ER = mχ ∆u2 /2, with ∆u2 = u2 − u2f , uf being the DM velocity after the scattermax is equal to 2µ2 u2 /m .
ing. The quantity ER
N
N
The total capture rate C is then obtained by integrating Eq. (3.1) over the radius of
the body and by summing over the different elements i present in the body. The final
expression can be cast in the form [131, 132]
C=
2
X 8 1/2 ρχ
3ũesc
Mi
σN,i
v̄
hφi
i ξ(∞)Si ,
3π
mχ
mN,i
2v̄ 2
i
30
(3.7)
3.1 Neutrino flux from the Sun and the Earth
where ũ2esc is the escape velocity at the surface of the body (ũ2esc ≃ 618 km s−1 for the
Sun and ũ2esc ≃ 11.2 km s−1 for the Earth), Mi is the total mass of the element i in the
body and hφii is the reduced gravitational potential, φ(r) = u2esc (r)/ũ2esc , averaged over
the mass distribution of the element i. The factor ξ(∞) ≃ 0.75 is a suppression factor
due to the motion of the solar system with respect to the halo. The function Si takes
into account the kinematical properties occurring in the DM-nucleus interactions. Its
analytic expression can be found in Ref. [131].
In Fig. 3.1, we present the capture rate for some of the most abundant elements present
in the Sun. The solar composition is taken from the solar model BS2005-AGS,OP [133]
for light elements, up to 16 O, and from Ref. [134] for the heavier elements. We neglect
the effect of DM evaporation [135], that can be important only for DM masses lower
than 10 GeV, and the gravitational effects from planets like Jupiter, recently studied in
Refs. [130, 136]. In the left panel of Fig. 3.2, instead, the capture rate of the Earth is displayed. The different peakes are due to resonant capture of DM on oxygen, magnesium,
silicon and iron.
The DM particles could be capture by celestial bodies also through inelastic scatterings
on nuclei and through elastic scatterings on electrons. For the analysis of the former case,
we refer to Ref. [137], while we describe in the following the scenario of DM capture by
the Sun, due to interactions with electrons. This possibility will then be applied to the
study of leptophilic DM, carried out in Chapter 5.
In the calculation of the capture rate that we have described above, the DM particles
are assumed to interact with material at zero temperature, neglecting the solar temperature of about 1.5 × 107 K in the center and 8.1 × 104 K at the surface. Although this
is a reasonable assumption for DM candidates interacting with hydrogen and the other
nuclei, it fails for the case of DM scattering on the free electrons in the Sun. Indeed, the
effect of a non-zero temperature on the capture rate depends on the ratio of the thermal
velocity of the target to the DM velocity. The thermal kinetic
penergy kB T is independent
of the mass, but the thermal velocity is larger by a factor mp /me ≃ 45 for electrons
compared to hydrogen.
We calculate the rate for DM capture by a body at finite temperature following
Ref. [131] and considering the temperature distribution for the electrons inside the Sun
as predicted by the solar model BS2005-AGS,OP [133]. In the right panel of Fig. 3.2, we
show the effect of the non-zero temperature on the capture rate for electrons, hydrogen
and all other nuclei in the Sun. We find that the capture rate on electrons is enhanced
by about one order of magnitude, while the effect is hardly visible at the scale of the plot
for hydrogen. The temperature effect can be neglected for scattering off heavier nuclei.
The annihilation rate Γ is expressed in terms of the capture rate by the formula [138]
t
C
2
,
(3.8)
Γ = tanh
2
τA
where t is the age of the macroscopic body (t = 4.5 Gyr for Sun and Earth), τA =
(CCA )−1/2 , and CA depends on the DM annihilation cross section and on the effective
31
Chapter 3 Indirect detection with neutrinos
1030
1030
4
H @solidD, He @dottedD
1029
C @solidD,
1029
N @dottedD
0 @solidD, Ne @dottedD
CŸ @s-1 D
CŸ @s-1 D
1027
1026
1026
1025
1025
Σ p = Σn = 1 pb
200
S @dottedD
Na @solidD, Cr @dottedD
1028
1027
1024
10
Si @dottedD
Mg @solidD,
Ar @solidD, Ni @dottedD
Ca @solidD, Al @dottedD
1028
Fe @solidD,
Σ p = Σn = 1 pb
400
600
m Χ @GeVD
800
1024
10
1000
200
400
600
m Χ @GeVD
800
1000
Figure 3.1: Capture rate C⊙ in the Sun as a function of the Dark Matter mass,
assuming scattering off the different nuclei inside the Sun, with a scattering cross section
of 10−36 cm2 .
1030
1020
1029
Σe = Σ p = 1 pb
1019
Nuclei
Σe = Σ p = 1 pb
1028
1018
56
1017
28
CŸ @s-1 D
CÅ @s-1 D
1027
Fe
H
1026
1025
1024
Si
1023
10
24
16
16
1015
10
Mg
e-
1022
O
1021
20
50
100
200
m Χ @GeVD
500
1000
1020
10
200
400
600
m Χ @GeVD
800
1000
Figure 3.2: Left panel: capture rate C⊕ in the Earth as a function of the Dark Matter
mass, assuming a scattering cross section of 10−36 cm2 . Right panel: capture rate C⊙
in the Sun as a function of the Dark Matter mass, assuming scattering off electrons,
hydrogen, and all other nuclei in the Sun, with a scattering cross section of 10−36 cm2 .
The solid curves correspond to scattering off particles at zero temperature, whereas the
dotted curves show the effect of the actual temperature distribution inside the Sun for
electrons and hydrogen.
32
3.1 Neutrino flux from the Sun and the Earth
volume V0 of the confining region in which the DM particles are trapped:
CA =
hσann vi mχ 3/2
.
V0
20 GeV
(3.9)
We denote by hσann vi the thermally averaged total annihilation cross section times the
relative velocity, at the present time. The volume of the confining region is explicitly
given by
3/2
3m2P l T
V0 =
,
(3.10)
2ρ × (10 GeV)
where T and ρ are the central temperature and the central density of the celestial body.
For the Earth V0 = 2.3 × 1025 cm3 (T = 6000 K, ρ = 13 g cm−3 ) and for the Sun
V0 = 6.6 × 1028 cm3 (T = 1.4 × 107 K, ρ = 150 g cm−3 ).
We recall that, according to Eq. (3.8), in a given macroscopic body the equilibrium
between capture and annihilation (i.e. Γ ≃ C/2) will be established only if t & τA .
The expression for the annihilation rate given above refers to a macroscopic body as
a whole. This is certainly enough for the Sun which appears to us as a point source. On
the contrary, in the case of the Earth, one also has to define an annihilation rate referred
to a unit volume at point r from the Earth center:
1
Γ(r) = hσann vin2 (r) ,
2
(3.11)
where n(r) is the DM spatial density, which may be written as [138]
2
n(r) = n0 e−α̃ mχ r .
Here, α̃ = 2πGρ/(3T ) and n0 is a normalization such that
Z
1
Γ = hσann vi d3 r n2 (r) .
2
(3.12)
(3.13)
Concerning the annihilation into neutrinos, more exotic scenarios has been studied as
well, in which for example high energy electrons resulting from DM annihilations in the
Sun could escape the Sun in case the DM annihilates into long-lived states [139]. For
our study we will neglect this situation.
3.1.2 Neutrino production
Once the DM particles are accumulated in the center of the Sun or the Earth, they can
annihilate, producing directly neutrinos with energies Eν ≃ mχ , where mχ is the DM
mass. In the framework of the SUSY neutralino, the branching ratio for this annihilation
channel is proportional to the neutrino mass, and thus negligible. However, depending on
the nature of the DM particle and on the particular channel through which the annihilation occurs, there might be cases where the direct neutrino production is unsuppressed.
33
Chapter 3 Indirect detection with neutrinos
In Chapter 4 we will systematically classify all the different possibilities, reporting for
each of them the associate annihilation cross sections.
The DM particles can annihilate also into charged leptons, quarks, gauge and Higgs
bosons, which can then decay or hadronize producing neutrinos. In Ref. [140], the authors have used a PYTHIA Monte Carlo simulation to calculate the spectra of neutrinos,
coming from DM annihilation in the Sun and in the Earth, for the following channels:
bb̄, τ τ̄ , cc̄, q q̄, gg (with q = u, d, s quarks). Three main differences and improvements
have been implemented in Ref. [140] with respect to previous calculations. The first one
is the prediction of the neutrino spectra for the different neutrino flavours: νe , νµ and ντ
(not only νµ , like in previous works). The second main improvement consists of an appropriate implementation of the energy loss that hadrons and leptons can experience before
decaying. Finally, the third difference is represented by the calculation of the neutrino
spectra for light quarks u, d, s, that were usually neglected in previous calculations.
We will use the initial neutrino spectra of Ref. [140] for the analyses of the leptophilic
and the light neutralino DM, reported in Chapter 5.
3.1.3 Neutrino propagation
For a precise estimate of the neutrino flux at the detector site, it is important to take
into account the main processes that can occur during the neutrino propagation: the
oscillation and the incoherent interaction with matter. These effects have been vastly
analyzed in Refs. [140, 141], and have then been applied to specific model-dependent
studies, see e.g. Refs. [142, 143].
The equations that describe the evolution of the neutrino spectra can be formally
written using the density matrix formalism:
dρ
dρ dρ +
.
(3.14)
= −i [H, ρ] +
dr
dr N C
dr CC
The first term describes the oscillations of neutrinos in matter, with the total Hamiltonian
given by the sum of the vacuum one and of the Wolfenstein potential:
Hw =
Mw √
± 2 GF Ne diag(1, 0, 0) ,
2E
(3.15)
where GF = 1.66 × 10−5 GeV−2 is the Fermi constant and Ne is the matter electron density. The minus (plus) sign holds for (anti-)neutrinos and Mw = U diag(m21 , m22 , m23 )U †
is the mass matrix in the weak basis (νe , νµ , ντ ). The matrix U is the Pontecorvo-MakiNakagawa-Sakata (PMNS) matrix, often parameterized as


c12 c13
s12 c13
s13 eiδCP
c13 s23  ,
U = diag(1, eiα , eiβ )· −s12 c23 − c12 s13 s23 eiδCP c12 c23 − s12 s13 s23 eiδCP
iδ
iδ
CP
CP
s12 s23 − c12 s13 c23 e
−c12 s23 − s12 s13 c23 e
c13 c23
(3.16)
34
3.1 Neutrino flux from the Sun and the Earth
where cij ≡ cos θij and sij ≡ sin θij . The parameter δ is the Dirac CP-violating phase,
while α and β are the two Majorana phases, absent in the case of Dirac neutrinos. The
Majorana phases do not appear in the oscillation probability formulae for neutrinos and,
therefore, oscillation experiments do not provide any information on their values. The
possibility of distinguishing Dirac from Majorana neutrinos is given by other kinds of
experiments, like neutrinoless-double-β decay.
In Ref. [144], the authors made a three flavours global fit for the neutrino oscillation
parameters, using data from solar, atmospheric, reactor (KamLAND and CHOOZ) and
accelerator (K2K and MINOS) experiments. They found the following best-fit values,
with 1σ errors, for the mixing angle parameters:
sin2 θ12 = 0.304+0.022
−0.016 ,
sin2 θ23 = 0.50+0.07
−0.06 ,
2
sin θ13 =
(3.17)
0.01+0.016
−0.011 ,
and for the mass squared differences
−5
∆m221 = 7.65+0.23
eV2 ,
−0.20 × 10
−3
eV2 .
|∆m231 | = 2.40+0.12
−0.11 × 10
(3.18)
The sign of ∆m231 is still unknown. It can be positive for normal mass ordering (m1 <
m2 < m3 ) or negative for inverted mass ordering (m3 < m1 < m2 ). No experimental
information on the value of δCP is present at the moment.
In our study, the neutrino mixing angles θ12 and θ23 and the squared mass differences
are fixed to their best-fit values reported in Eqs. (3.17)-(3.18). We consider the case
of normal mass ordering and we set the oscillation parameter θ13 to zero. A different
choice of θ13 would marginally affect the prediction on the neutrino flux, as reported in
Refs. [140, 141].
The second term in Eq. (3.14) takes into account the neutrino energy loss and their
reinjection due to neutral current interactions. The last term, instead, represents the
neutrino absorption and the ντ regeneration through charged current interactions. The
explicit expressions of these terms are reported in Appendix A.
For DM annihilation inside the Sun, the integro-differential equation (3.14) for the
density matrix has been solved numerically by a Fortran program. In Fig. 3.3 and
Fig. 3.4 we report our results for the propagated neutrino spectra at one astronomical
unit, in the case of the ν ν̄ and of the τ τ̄ annihilation channel. These spectra will be used
in Chapter 5 in the context of a leptophilic DM candidate. Our results of Fig. 3.3 and
Fig. 3.4 match very well with the ones given in Refs. [140, 141]. Notice how the effects
of incoherent neutrino interactions are clearly visible from the propagated spectra of the
ν ν̄ annihilation channel.
In the case of annihilations inside the Earth’s core, the calculation of the neutrino
spectra can be further simplified. Indeed, the interactions with matter can be neglected,
since the mean free paths of neutrinos, in the core and in the mantle, are much bigger
35
Chapter 3 Indirect detection with neutrinos
than the Earth’s radius R⊕ for Eν . 10 TeV (for anti-neutrinos, the mean free paths are
almost a factor two greater, due to the difference in the cross-sections):
R⊕
1
,
≃ 3.6 × 104
core
σ ν Ne
(Eν /GeV)
1
R⊕
.
(Eν /GeV)
(3.19)
Therefore, for the propagation inside the Earth, only the oscillation effects can be taken
into account. Moreover, for Eν & 1 GeV, the dependence on the “solar” parameters ∆m221
and θ12 is extremely weak and can be neglected. Since in our analyses we are considering
vanishing θ13 , Earth’s matter effects are negligible and neutrino oscillations are driven by
the “atmospheric” parameters ∆m231 and θ23 . In this case, the main oscillation channel
is νµ ↔ ντ and the value of the oscillation and the survival probability Pαβ is simply
given by the vacuum two-flavors formula:
(∆m231 /eV2 )(r/km)
2
2
Pαβ (r, Eν ) = δαβ − ǫαβ sin (2θ) sin 1.27
,
(3.20)
(Eν /GeV)
λcore =
λmantle =
σν Nemantle
≃ 9.2 × 104
where the parameter ǫαβ is equal to 1 (−1) for α = β (α 6= β).
In the case of the Earth, the differential muon-neutrino flux at the detector, from the
annihilation channel f and as a function of the zenith angle θz , can be written as:
!
dφνµ
dNνfµ
Γ⊕
dNνfτ
Gµµ (θz , Eν )
,
(3.21)
=
+ Gµτ (θz , Eν )
2 BRf
dEν d cos θz
dEν
dEν
4πR⊕
where the function Gαβ (θz , Eν ) encodes the dependence on the oscillation probability
and on the DM distribution inside the Earth. Using Eq. (3.12), we find the following
expression:
Z
2 (2 mχ β̃)3/2 y
2
Gαβ (θz , Eν ) =
dr exp −2 mχ α̃ r2 + R⊕
− ry Pαβ (r, Eν ) , (3.22)
1/2
π R⊕
0
2 . The differential muon anti-neutrino
with y ≡ 2 R⊕ cos θn , θn ≡ π − θz and β̃ = α̃ R⊕
flux at the detector can be obtained by a formula analogous to Eq. (3.21). In Fig. 3.5 we
report our results for the neutrino spectra at the detector site, in the case of the bb̄ and
of the τ τ̄ annihilation channel. We will use these spectra in Chapter 5, considering the
light neutralino as DM particle. The spectra of Fig. 3.5 have been compared with the
ones of Refs. [140, 141], finding a very good agreement. Note the oscillatory behaviour
of the spectra due to the neutrino propagation along the radius of the Earth.
The neutrino flux at the detector, from the annihilation channel f with branching ratio
BRf , is given by
dφfν
Γ dNνf
= BRf
,
(3.23)
dEν
4πd2 dEν
with dNνf /dEν being the neutrino spectrum after propagation and d being the distance
between the source and the detector (the Sun-Earth distance or the Earth’s radius).
36
3.1 Neutrino flux from the Sun and the Earth

ΝΜ from Νe Ν e channel at 1 AU


Ν Μ from Νe Ν e channel at 1 AU
10
10
m Χ @GeVD
m Χ @GeVD
10
10
100
1
100
1
500
500
1000
dNΝ dx
dNΝ dx
1000
0.1
0.01
0.001
0.0
0.1
0.01
0.2
0.4
0.6
x = EΝ m Χ
0.8
0.001
0.0
1.0
0.2

ΝΜ from ΝΜ Ν Μ channel at 1 AU
0.4
0.6
x = EΝ m Χ
10
m Χ @GeVD
m Χ @GeVD
10
10
100
1
100
1
500
500
1000
dNΝ dx
dNΝ dx
1000
0.1
0.01
0.1
0.01
0.2
0.4
0.6
x = EΝ m Χ
0.8
0.001
0.0
1.0
0.2

ΝΜ from ΝΤ Ν Τ channel at 1 AU
0.4
0.6
x = EΝ m Χ
1.0
10
m Χ @GeVD
m Χ @GeVD
10
10
100
1
100
1
500
500
1000
dNΝ dx
1000
dNΝ dx
0.8


Ν Μ from ΝΤ Ν Τ channel at 1 AU
10
0.1
0.01
0.001
0.0
1.0


Ν Μ from ΝΜ Ν Μ channel at 1 AU
10
0.001
0.0
0.8
0.1
0.01
0.2
0.4
0.6
x = EΝ m Χ
0.8
1.0
0.001
0.0
0.2
0.4
0.6
x = EΝ m Χ
0.8
1.0
Figure 3.3: Spectra of muon (anti-)neutrinos at 1 AU, for DM pair-annihilation inside
the Sun into νe ν̄e , νµ ν̄µ and ντ ν̄τ . The spectra at the production are given by a δ
function centered at Eν = mχ .
37
Chapter 3 Indirect detection with neutrinos


Ν Μ from ΤΤ channel at 1 AU
10
1
1
dNΝ dx
dNΝ dx

ΝΜ from ΤΤ channel at 1 AU
10
0.1
0.1
m Χ @GeVD
0.01
m Χ @GeVD
10
0.01
100
500
500
1000
0.001
0.0
10
100
0.2
1000
0.4
0.6
x = EΝ m Χ
0.8
0.001
0.0
1.0
0.2
0.4
0.6
x = EΝ m Χ
0.8
1.0
Figure 3.4: Spectra of muon (anti-)neutrinos at 1 AU, for DM pair-annihilation inside
the Sun into τ τ̄ .

ΝΜ from bb channel at RÅ

ΝΜ from ΤΤ channel at RÅ
10
10
m Χ @GeVD
10
35
1
1
55
dNΝ dx
dNΝ dx
80
0.1
0.1
m Χ @GeVD
0.01
0.01
10
35
55
80
0.001
0.0
0.2
0.4
0.6
x = EΝ m Χ
0.8
1.0
0.001
0.0
0.2
0.4
0.6
x = EΝ m Χ
0.8
1.0
Figure 3.5: Spectra of muon neutrinos at R⊕ , for DM pair-annihilation inside the Earth
into bb̄ (left panel) and τ τ̄ (right panel). The spectra of anti-neutrinos are equivalent.
38
3.2 Neutrino flux from the galactic center
3.2 Neutrino flux from the galactic center
The galactic center (GC) region represents another site to look for neutrino signal coming
from DM annihilations. The great advantage of the GC signal, with respect to signals
in neutrinos coming from celestial bodies (like the Earth and the Sun), is represented by
its direct proportionality to the DM annihilation cross section. Indeed, no dependence
on the scattering cross section is present.
Suppose that a pair of DM particles with mass mχ annihilates near the center of the
Milky Way into να ν̄β , with α and β flavour indices. The flux of muon neutrinos, arriving
at the Earth from a solid angle ∆Ω will be then given by (see e.g. Ref. [145])
dφνµ
J∆Ω r0 ρ20
dNνα
=
P(να → νµ ) ,
(σann v)αβ
2
dEν
4π 2 mχ
dEν
(3.24)
where σann v is the annihilation cross section times the relative velocity between the two
DM particles, r0 is the distance of the Earth from the center of the Galaxy
P and ρ20 is the
local DM density. The oscillation probability P(να → νµ ) is given by i |Uαi | |Uµi |2 ,
with U being the neutrino mixing matrix of Eq. (3.16). The function J is defined as [146]
Z smax
Z
Z ψ
Z
Z 2π
1
ds ρ2 (r)
ds ρ2 (r)
1
=
,
(3.25)
sin
ψ
dψ
dΩ
dϕ
J=
2
∆Ω ∆Ω
∆Ω 0
r0 ρ20
l.o.s r0 ρ0
0
0
p
with ρ(r) being the DM density profileqand r = s2 + r02 − 2r0 s cos ψ. The upper limit
2
− sin2 ψ r02 ) + r0 cos ψ, where rhalo & tens
of the integration is given by smax = (rhalo
of kpc is the size of the DM halo. An equation analogous to Eq. (3.24) can be written
for the antineutrino flux.
It has been shown in Refs. [147, 148, 149, 150] that the neutrino signal coming from
the GC can be used to set a limit on the total DM annihilation cross section. However,
as we have discussed in Sect. 2.2, the DM density profile is not well known close to the
center of the Milky Way, since the presence of the baryons, that constitute the dominant
matter component in the central region of the Galaxy, is not included in the numerical
N-body simulations. For this reason, the neutrino flux coming from a small angular
region around the GC can suffer from large astrophysical uncertainties, which can be
partially reduced if a large angular region is considered, for definiteness a cone-half angle
of about 30◦ around the GC. In any case, to be conservative, it is always preferable to
calculate the neutrino flux from the GC considering the DM density profile that provides
the smallest signal, i.e. the isothermal profile, for which we report the values of J∆Ω in
Table 3.1. For comparison, also the values in the case of a NFW profile are given.
We also want to add that the presence of a Supermassive Black Hole, in the center of
the Milky Way, could influence the central part of the DM density profile and alter the
slope of the extrapolated profiles of Fig. 2.2. In particular, a spike or a trough toward
the GC could be present, depending on the merging history of the Galaxy.
It has been recently shown in Refs. [151, 152] that the neutrino flux from the GC can
be clearly observable at neutrino telescopes if the DM particles annihilate mainly into
39
Chapter 3 Indirect detection with neutrinos
J∆Ω
Isothermal
NFW
5◦
0.3
5.9
10◦
1.2
10.5
20◦
4.1
17.2
30◦
7.3
21.9
Table 3.1: Values of the angular factor J∆Ω in the case of an isothermal and a NFW
DM density profile.
neutrinos and if the annihilation cross section is roughly two orders of magnitude greater
that the one expected from a standard WIMP: σann v ≃ 6 × 10−24 cm3 s−1 . For smaller
cross sections, instead, the atmospheric neutrino background dominates over the DM
signal.
3.3 Muon flux
The neutrinos coming from DM annihilations can undergo charge current interactions
with the nucleons present in the rock or in the ice below the detector. Among the charged
leptons produced, muons can be easily detected by their Cherenkov light emission. In
Sect. 3.3.1, the neutrino-muon conversion process is summarized. The background to the
DM annihilation signal in neutrinos is discussed in Sect. 3.3.2, while in Sect. 3.3.3 the
main muon detectors are described.
3.3.1 Neutrino-Muon conversion
For the calculation of upward-going muons, i.e. muons coming from below the detector,
we follow the formalism described in Refs. [153, 154], which we briefly summarize in this
Section. The double energy differential muon flux is defined as:
d2 φµ
= NA
d cos θz dEµ dEν
Z
∞
Z
dX
0
Eν
Eµ
dEµ′ g(Eµ , Eµ′ ; X) S(Eν , Eµ′ ) ,
(3.26)
where NA is Avogadro’s number and
S(Eν , Eµ′ )
dφν
=
d cos θz dEν
dσνp
dσνn
′
′
Np ′ (Eν , Eµ ) + Nn ′ (Eν , Eµ )
dEµ
dEµ
(3.27)
is the product of the differential neutrino flux and the differential Charged Current (CC)
cross section, which is mainly due to Deep Inelastic Scattering (DIS) for the energy
Eν > 1 GeV. The expressions for the differential cross sections dσνp,n /dEµ are given in
Appendix B. The parameters Np and Nn are the fractional numbers of protons and
neutrons at the point of muon production. If the interaction can be assumed to occur in
standard rock, like for the Super-Kamiokande detector, we have Np ≃ Nn ≃ 0.5, since
the number of protons is almost equal to the number of neutrons (Z ≃ 11, A ≃ 22); if
the interaction occurs inside the ice, like for IceCube, we have Np ≃ 5/9 and Nn ≃ 4/9.
40
3.3 Muon flux
The function g(Eµ , Eµ′ ; X) represents the probability that a muon with energy Eµ′ will
have an energy Eµ after a distance X, due to energy-loss processes. The average rate of
muon energy loss can be written as [155]:
−
dE
= a(E) + b(E) E ,
dx
(3.28)
where a(E) represents the energy loss due to ionization and b(E) the one due to radiative
effects. For muon energies in the GeV-TeV range, the dependence of the parameters a
and b on the energy can be neglected and the function g(Eµ , Eµ′ ; X) can be approximated
by the following analytic expression [153]:
δ(X − X0 )
,
a + bEµ
(3.29)
1 a + bEµ′
ln
,
b a + bEµ
(3.30)
g(Eµ , Eµ′ ; X) =
where X0 is the mean muon range in matter,
X0 (Eµ′ , Eµ ) =
with Eµ′ and Eµ being, respectively, the initial and the final energy of the muon. Standard
values for the quantities a and b in rock are
a ≃ 2.2 × 10−3 GeV/ g cm−2 ,
(3.31)
−6
−2
b ≃ 4.4 × 10 / g cm
.
(3.32)
Substituting the expression for g(Eµ , Eµ′ ; X), the differential muon flux acquires the form
dφµ
1
= NA
d cos θz dEµ
a + bEµ
Z
mχ
dEν
Eµ
Z
Eν
Eµ
dEµ′ S(Eν , Eµ′ ) ,
(3.33)
with mχ being the DM mass.
The total muon flux can then be divided in through-going muons (muons that pass
through the detector) and stopping muons (muons that stop inside the detector) by using
the formula
Z ∞
dφµ
1
S,T
Φµ (cos θz ) =
dEµ
AS,T (L(Eµ ), θz ) ,
(3.34)
A(Lmin , θz ) Eµth
d cos θz dEµ
where Eµth is the energy threshold of the detector for upward-going muons, L the mean
muon stopping range in water L(Eµ ) ≡ X0 (Eµ , mµ ) and Lmin ≡ L(Eµth ). The functions
AS,T (L, θz ) represent the detector effective areas for stopping and through-going muons,
while A(Lmin , θz ) is the total effective area of the detector, i.e. the projected area that
corresponds to internal path-lengths longer than Lmin , for a fixed value of the zenith
angle θz .
41
Chapter 3 Indirect detection with neutrinos
The classification of upward-going muons into the two subcategories reported above is
strictly detector-dependent, since it depends on the shape and the size of the detector.
For a detector with cylindrical geometry (with radius R and height H), it has been shown
in Ref. [156] that the function A(L, θz ) acquires the form:
h
i
p
p
A(L, θz ) = 2RH sin θz 1 − x2 + 2R2 | cos θz | cos−1 x − 3x 1 − x2 Θ(Lmax (θz ) − L) ,
(3.35)
with x = L sin θz /2R and Lmax (θz ) = min [2R/ sin θz , H/| cos θz |]. The effective area for
stopping muons AS (L, θz ) is given by Eq. (3.35) and the one for through-going muons is
AT (L, θz ) ≡ [A(Lmin , θz ) − A(L, θz )].
In the case of contained events (the neutrino-muon conversion takes place inside the
detector volume), the energy differential muon flux is given by [151, 152]
dΦµ
=
dEµ
Z
D
dz
Z
mχ
dEν
Eµ
0
dPcont dφµ
,
dzdEµ dEν
(3.36)
where D is the size of the detector and
dσνp
dσ n
dPcont
= np
(Eν , Eµ ) + nn ν (Eν , Eµ ) ,
dzdEµ
dEµ
dEµ
(3.37)
where np and nn are the number densities of protons and neutrons in the detector:
np,n = NA Np,n ρ, with ρ being the density of the detector material.
3.3.2 Atmospheric background
For the indirect DM search with neutrinos, the background to a possible signal is represented by atmospheric neutrinos with GeV-TeV energies. Solar neutrinos, instead, do
not contribute to the background, since they have MeV energies.
The interactions of primary cosmic rays with the nuclei in the Earth’s atmosphere
produce π and K mesons, which then generate atmospheric muons and neutrinos through
the decay chains
π±, K ± →
µ± + νµ (ν̄µ )
֒→ e± + ν̄e (νe ) + ν̄µ (νµ ) .
Atmospheric muons coming from above the detector reach the Earth’s surface before the
decay processes µ± → e± + ν̄e (νe ) + ν̄µ (νµ ) can occur. To avoid the big background of
atmospheric muons, the DM search is done looking only for upward-going muons. In this
way, only atmospheric neutrinos will contribute to the background.
In our studies, we will use the atmospheric neutrino fluxes as calculated by Honda et
al. [157]. In the literature, two other commonly used results are the FLUKA fluxes by
Battistoni et al. [158] and the Bartol fluxes by Barr et al. [159]. These different predictions
lead to an uncertainty of order 10% in the flux estimations.
42
3.3 Muon flux
3.3.3 Muon detection
As mentioned before, the muons are detected using the Cherenkov radiation emitted
when a charged particle moves faster than the speed of light in water. This radiation has
a typical form of a ring and can be recorded using photomultipliers. From the brightness
of the ring, the energy of the charged particle can be inferred and, from measurements
of its shape, it is possible to distinguish muons from electrons. Electrons produce fuzzy
rings, since they generate electromagnetic showers inside the detector. Muons, instead,
do not suffer from multiple scatterings and are more easy to detect and identify, since
they produce rings with sharp edges.
In the following, we discuss the main characteristics of the Super-Kamiokande underground detector and we also briefly present other existing and planned neutrino telescopes, present in the Northern (ANTARES, NEMO, NESTOR, KM3Net) and in the
Southern hemisphere (IceCube).
Super-Kamiokande
The Super-Kamiokande (SK) detector is an underground water Cherenkov detector, located in Japan in the city of Hida, that now includes the old Kamioka town. It is a
cylindrical stainless steel tank, with a diameter of 39.3 m and an height of 41.4 m, filled
with 50 000 tons of ultra-pure water. An internal stainless steel structure divide the
detector volume into an outer and inner detector. The latter has a radius of R = 16.9 m
and an height of H = 36.2 m.
This neutrino observatory has been in operation since 1996 and it is expected to
run for another ten years. By measuring the solar neutrino flux, it has provided the
first detection of neutrino oscillations in 1998 [160]. In the same year, it also detected
fundamental evidences for the discovery of atmospheric neutrino oscillations [161]. The
SK detector has also been used to search for proton decay, for supernovae neutrinos and
for neutrinos from DM annihilation.
The SK collaboration classifies the data into three main categories: fully-contained
events (FC), partially contained events (PC) and upward-going muon events. The last
ones are then divided in two subcategories: upward stopping muon events and upward
through-going muon events [13]. Usually, only through-going muons are used for the
analysis of neutrinos coming from DM annihilations, since for mχ & 18 GeV almost 90%
of the muons produced fall in this category [5]. However, for a smaller DM mass, a great
part of the upward-going muon signal would be in stopping muons, rather than throughgoing. For this reason, we will consider both subcategories of upward going-muons in
the study of the light neutralino signal, reported in Chapter 5.
The SK detector has an energy threshold Eµth = 1.6 GeV, that corresponds to a pathlength cut Lmin = 7 m applied on upward-going muons, see the left panel of Fig. 3.6. In
the right panel of Fig. 3.6 we report the effective area for the SK detector, obtained using
Eq. (3.35) and the size of the inner detector.
The upward-going muons can be divided into through-going and stopping muons,
applying Eq. (3.34). Using the tabulated values of Ref. [163] for the muon energy loss in
43
1000
1400
100
1200
10
Effective Area @m2 D
X0 @mD
Chapter 3 Indirect detection with neutrinos
7m
1
0.1
800
1.6 GeV
1
1000
10
EΜ @GeVD
100
1000
600
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
cos ΘZ
Figure 3.6: Left panel: muon range in water. Right panel: zenith angle dependence
of the effective area for the Super-Kamiokande detector, obtained using the analytic
expression reported in Eq. (3.35). The agreement with the Monte Carlo simulation of
Ref. [162] is excellent.
rock and in water, we calculate the expected muons background coming from atmospheric
neutrinos. Our results, shown in Fig. 3.7, reproduce with great accuracy the zenith angle
distributions as predicted by the SK collaboration.
The limits on the muon fluxes coming from DM annihilations in the Sun and in the
Earth can be defined as [5]
Φµ (θz ; 90% C.L.) =
N90
,
A(Lmin , θz ) × T
(3.38)
where N90 is the upper Poissonian limit at 90% C.L., given the measured events and the
muon background from atmospheric neutrinos, and T is the detector lifetime. We do not
consider the detector efficiency, since it is almost equal to 100% for upward-going muons.
Using Eq. (3.38) and the SK data collected from May 1996 to July 2001 [13], we find
the following limits on through-going (ΦTµ ) and stopping (ΦSµ ) muons at 90% confidence
level:
ΦTµ,Sun . 1.2 × 10−14 cm−2 s−1 ,
(3.39)
ΦSµ,Sun . 0.5 × 10−14 cm−2 s−1 ,
(3.40)
ΦTµ,Earth . 0.8 × 10−14 cm−2 s−1 ,
(3.41)
ΦSµ,Earth . 0.5 × 10−14 cm−2 s−1 .
(3.42)
The limits for the Earth are obtained considering a cone half-angle θ ≃ 25◦ around the
Earth’s center, while the values reported for the Sun are valid for a cone half-angle θ ≃ 20◦
44
3.3 Muon flux
STOPPING MUONS
THROUGH-GOING MUONS
1.2
4.0
Super-K 2001 HStat. err. only - 90% CLL
3.5
Super-K 2001 HStat. err. only - 90% CLL
1.0
FΜ @10-13 cm-2 s-1 sr-1 D
FΜ @10-13 cm-2 s-1 sr-1 D
3.0
0.8
0.6
0.4
2.5
2.0
1.5
1.0
0.2
0.5
0.0
-1.0
-0.8
-0.6
-0.4
cos ΘZ
-0.2
0.0
0.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
cos ΘZ
Figure 3.7: Upward stopping and through-going atmospheric muons at the SuperKamiokande detector. The solid blue (dashed red) lines represent the expected muon
fluxes without (with) oscillations. The data points are taken from [13]. We report only
the statistical error at 90% confidence level. The agreement with the Monte Carlo simulations of Ref. [162] is extremely good.
around the direction of the Sun. The values for through-going muons are consistent with
the ones provided by the SK collaboration [5].
Note that in the calculation of the muon flux, we have neglected the kinematical angle
between the neutrino and muon direction, which can be relatively large for muons close
to threshold. In any case, the average deflection angle is at most of the same order of the
angular bin over which we integrate our signal, for the stopping and for the through-going
muons. Considering also the detector resolution, neglecting the kinematical angle does
not affect our results in a relevant way. This is confirmed by the quite good agreement we
obtain in our calculation of the atmospheric neutrino events with the SK evaluation [162].
The values reported in Eqs. (3.39)÷(3.42) can be compared with the muon flux induced
by DM annihilations and can be used to set limits on the DM scattering cross section.
We will apply these limits in the study of the muon flux for leptophilic and the light
neutralino DM, see Chapter 5.
In Fig. 3.8, we display the ratios between the muon fluxes at the Super-Kamiokande
detector, arising from DM pair annihilations into bb̄ and τ τ̄ , and the DM annihilation
rates, in the case of the Sun and the Earth. The muons have been divided into stopping and through-going events. Since the light neutralinos annihilates manly into bb̄ for
mχ . 30 GeV, it is clear from the plots that the category of stopping muons will provide
the dominant signal for low values of the neutralino mass.
45
Chapter 3 Indirect detection with neutrinos
SUN
EARTH
10-19
10-10

ΤΤ
FΜ GÅ @km-2 y-1 sD
FΜ GŸ @km-2 y-1 sD

bb
-21
10
-22
10
10-12

bb
10-13
10-14
solid lines: through-going
10-23
20
30
40
50
mΧ
60
solid lines: through-going
10-15
dashed lines: stopping
10-24
10

ΤΤ
10-11
10-20
dashed lines: stopping
70
80
10-16
10
20
30
40
50
60
70
80
mΧ
Figure 3.8: Ratio between the muon flux Φµ and the annihilation rate Γ, for DM
pair-annihilation inside the Sun and the Earth, as a function of the DM mass mχ . We
consider the τ τ̄ and the bb̄ annihilation channels and we divide between upward stopping
(dashed lines) and through-going muons (solid lines), considering the geometry of the
Super-Kamiokande detector.
Northern hemisphere
The ANTARES detector is a water Cherenkov detector in the deep Mediterranean Sea,
offshore from Toulon in France [7]. It consists of 12 strings, covering an area of 0.1 km2 ,
with a height of 350 m and is by now the largest neutrino telescope on the Northern
hemisphere.
The construction of two other small neutrino telescopes in the Mediterranean Sea is
ongoing: NEMO in the Sicilian Sea [164] and NESTOR [165] near Pylos, in Greece.
ANTARES, NEMO and NESTOR are three pilot projects for the future construction of
a cubic kilometer telescopes in the Northern hemisphere: the KM3Net project [8]. This
future neutrino telescope, with a planned energy threshold of about 50-100 GeV, will be
complementary to Ice-Cube, which is located at the South Pole. It will be particularly
important for DM searches, as it will be able to look at the galactic center that is hardly
visible with a neutrino telescope at the South Pole. In Fig. 3.9, we report the bounds on
the DM annihilation cross section σann v that could be set using contained muon events in
a KM3Net-like detector. We have considered a cone half-angle of 30◦ around the galactic
center and we have fixed the muon energy threshold to Eµ = 100 GeV. The exposure is
set to one year. The curves are derived assuming that the DM particles annihilate 100%
into neutrinos with a flavour-blind branching ratio. We show the limits for the isothermal
and the NFW density profiles. The Halo Angular bound has been derived in Ref. [150]
comparing the energy spectrum produced by DM pair-annihilation into neutrinos with
the atmospheric neutrino background and considering a cone half-angle of about 30◦
46
3.3 Muon flux
10-22
Halo Angular
-23
10
Σann v @cm3 s-1 D
Isothermal
10-24
NFW
10-25
Natural Scale
10-26
100
150
200
300
m Χ @GeVD
500
700
1000
Figure 3.9: Limits (at 3σ level) from contained muon events on the total annihilation
cross section σann v, after one year of exposure and for a cone half-angle of 30◦ around
the galactic center. The energy threshold has been fixed to Eµ =100 GeV and we have
considered an energy independent effective area equal to Aeff =1 km2 (KM3Net-like
detector). We have taken the isothermal and the NFW density profiles. The gray solid
line indicates the standard value of σann v for a thermal relic (natural scale). The Halo
Angular line represents the bound from neutrino searches, see text for more details.
around the GC with a value of J ≃ 25.
Southern hemisphere
The IceCube detector [166] is a neutrino telescope under construction at the South Pole,
that replaced the old AMANDA detector [167]. It will be completed in January 2011
and it will consist of 80 strings, covering one km3 of volume and 6 additional strings
concentrated in the central part, which will form the Deep Core sub-detector. Each
string carries 60 Digital Optical Modules (DOM) to detect Cherenkov light.
The recent results of IceCube with 22-strings [6, 168] improved the SK bound on the
muon flux Φµ in the high mass region: mχ & 200 GeV for the hard annihilation channel
(W + W − ) and mχ & 500 GeV for the soft channel (bb̄). The IceCube detector, provided
with the Deep Core arrays [169], will also constrain the parameter space at lower masses,
improving significantly the SK bound for the mass region mχ & 40 GeV [136]. However,
it will not be able to put bounds on really light DM particles, for which the SK bounds
will still remain to be the strongest ones.
47
4
Dark Matter annihilation into neutrinos
The most interesting signal to look for at neutrino telescopes is represented by a monochromatic neutrino signal, which can be produced by DM pair-annihilations directly into ν ν̄
or into νν (ν̄ ν̄), if we allow for lepton number violating processes (LNV). The neutrino
energy spectra produced in these annihilation channels are constituted of a soft part and
a line at energy Eν ≃ mχ . The detection of this kind of signal would provide a clear and
distinct hint for a DM annihilation origin.
The main scope of this Chapter is to carefully analyze the DM pair-annihilation into
neutrino final states. We restrict our study to the two-body direct production, since
this is the golden channel for DM discovery at neutrino telescopes. In our analysis, we
distinguish between Dirac and Majorana neutrinos and, in the latter case, we consider
explicitly different neutrino mass mechanisms, since this can be fundamental to correctly
relate the physical neutrino mass to the neutrinos Yukawa couplings with, e.g., a scalar
particle. A brief review of the possible neutrino mass terms is presented in Sect. 4.1.
We report in Sect. 4.2 the various possibilities for monoenergetic neutrino production,
considering explicitly scalar and fermionic DM, as well as the corresponding s, t, and u
channels. For simplicity, we do not extend the Standard Model (SM) gauge group, but
we contemplate different SU (2)L representations for the DM and the mediator particles.
This kind of systematic analysis was not presented before in the literature. A summary
of all the unsuppressed scenarios is given in Sect. 4.3. We explicitly show the behaviour
of the annihilation cross sections for a promising s-channel and t-channel diagram, considering both the case of scalar and fermionic DM. Our results are then compared to
experimental limits on µ and τ decays and on lepton flavour violating processes. The
constraints coming from neutrino searches are also considered. For specific models in
which the DM particles annihilate mainly in neutrinos see e.g. Refs. [170, 171].
49
Chapter 4 Dark Matter annihilation into neutrinos
4.1 The neutrino mass terms
Throughout our work we consider the SM gauge group, SU (3)c × SU (2)L × U (1)Y . In
this framework, the left-handed components of the neutrinos and the charged leptons
form a doublet under SU (2)L , while the right-handed components of the neutrinos, if
they are present, are total singlets:
να
∼ (1, 2, −1) ,
να R ∼ (1, 1, 0) ,
(4.1)
Lα L =
α L
where α is the generation index (α = e, µ, τ ). Depending on the nature of the neutrinos,
different mass terms can be present in the Lagrangian. If the neutrinos are Dirac particles,
they will be associated with a Dirac mass term, see Sect. 4.1.1, while, if they are Majorana
particles, a Majorana mass term will be allowed, see Sect. 4.1.2. The most satisfactory
ways to explain the smallness of the neutrino mass are represented by the type I and
type II see-saw mechanisms, which we briefly review in Sect. 4.1.3. For more details on
the physics of massive neutrinos, we refer to Refs. [115, 172]
4.1.1 Dirac mass term
If the neutrinos are Dirac particles, they will get their mass by the SM Higgs mechanism:
Lmass = −YDαβ LαL H̃ νβR + h.c. → −vH YDαβ ναL νβR + h.c. ,
(4.2)
where H is the SM Higgs and vH = 174 GeV is its vev. We have used the notation
H̃ ≡ iσ2 H ∗ , with σ2 being the second Pauli matrix. The neutrino mass matrix in
the flavour basis Mν = vH YD is then related to the diagonal neutrino mass matrix
Dν = diag(m1 , m2 , m3 ) by
M ν = U Dν U † ,
(4.3)
where U is the leptonic mixing (PMNS) matrix. Note that, since the neutrino masses
are small, the Yukawa couplings must be tiny, of the order of YD ∼ 10−12 .
4.1.2 Majorana mass term
In case the neutrinos are Majorana particles, beyond the standard mass term of Eq. (4.2),
also terms of the form (νL )C νL or (νR )C νR may be allowed. In the first case, a scalar
triplet field T , with
− √
T0√
T / 2
,
(4.4)
T =
T −−
−T − / 2
is needed to obtain a term that is gauge invariant under the SM gauge group:
1
1
Lmass = − YLαβ (LαL )C (iσ2 T ) LβL + h.c. → − mαβ
(ναL )C νβL + h.c. ,
2
2 L
50
(4.5)
4.1 The neutrino mass terms
αβ
where mαβ
L = vT YL , with vT being the vev of the neutral component of the scalar
triplet.
If right-handed neutrinos are present, a Majorana mass term can be written without
extending the SM scalar sector, since
1
Lmass = − MRαβ (ναR )C νβR + h.c. ,
2
(4.6)
is invariant under the SM gauge group. The Majorana mass term could arise from
a high energy theory beyond the SM, whose symmetries might be broken at the grand
unification scale. In this case, we would expect MR to be of the order of 1014 − 1016 GeV.
Moreover, since MR is not related to the vev vH , there is no a priori reason for it to be
at the electroweak scale.
4.1.3 See-saw mechanisms
The combination of the Dirac and Majorana mass terms leads to the so-called see-saw
mechanisms, where the smallness of the neutrino mass is a consequence of the heavy
right-handed neutrino fields.
In the case of a type I see-saw mechanism we have, additionally to Eq. (4.2), also a pure
Majorana mass term for the right-handed neutrinos, which we denote in this Section by
NR . The complete neutrino mass term, after symmetry breaking, is given by
1
Lmass = −νL mD NR − (NR )C MR NR + h.c. =
2
1
0
mD
(νL )C
C
= − (νL , (NR ) )
+ h.c.,
NR
mTD MR
2
(4.7)
where mD = vH YD is the Dirac mass matrix and MR is the Majorana mass matrix
for the right-handed neutrinos. The former is connected to the electroweak scale vH ,
while the latter can have a much larger value, since it is a singlet under the SM gauge
group. The above Lagrangian is not yet written in the mass basis. Before a complete
diagonalization, it is useful to bring the matrix to a block diagonal form, which will then
separate the neutrino states into heavy and light ones. Denoting the rotated states by
ν ′ and N ′ , we can write the Lagrangian as
′ C
1 ′
−mD MR−1 mTD
0
(νL )
′
C
Lmass ≈ − (νL , (NR ) )
+ h.c. ,
(4.8)
NR′
0
M
2
R
where we have neglected the small corrections to the heavy neutrino masses. The rotation
required to bring the matrix into this form is only a very tiny one. In the case of only one
generation of fermions, the corresponding mixing angle between heavy and light states is
mD
∼ 10−14 − 10−12 . As a consequence, the interaction eigenstate νL is essentially a
θ≈M
R
mD
light mass eigenstate, while NR has only a small fraction M
of the light mass eigenstate.
R
In the flavour basis, the corresponding light neutrino mass matrix is given by
Mν ≡ −mD MR−1 mTD = U Dν U T .
(4.9)
51
Chapter 4 Dark Matter annihilation into neutrinos
If beyond the terms of Eq. (4.8) a left-handed Majorana mass term mL is present, a
type II see-saw mechanism will be induced. In this case, we would have
1
mL mD
(νL )C
C
+ h.c. ,
(4.10)
Lmass = − (νL , (NR ) )
NR
mTD MR
2
with mL = vT YL . After a tiny rotation, the above mass matrix assumes the following
block diagonal form:
′ C
−1 T
1 ′
m
−
m
M
m
0
(νL )
L
D
′
D
R
Lmass ≈ − (νL , (NR )C )
+ h.c.
(4.11)
NR′
0
MR
2
Considering only one generation of fermions, the rotation angle is given by
θ∼
mD
mD
≈
,
MR − mL
MR
(4.12)
where we have used the fact that mL ≪ MR , since the correction to the ρ-parameter
forces the vev vT to be . O(1 GeV). In the case of see-saw type II, Eq. (4.9) is thus
modified to
Mν ≡ mL − mD MR−1 mTD = U Dν U T ,
(4.13)
where Mν is the light neutrino mass matrix in flavour space.
4.2 Production of monoenergetic neutrinos
Depending on the gauge quantum numbers assigned to the DM particle and to the
neutrino, specific annihilation processes will be allowed in the s, t, u channels. In our
analysis, we consider the DM particle χ and the mediator particle φ to be a singlet,
doublet or triplet representation of SU (2)L . In general, for a scalar ψs and for a fermion
ψf we use the following definitions:
ψs;1 ∼ (1, 1, 0) ,
ψ+
∼ (1, 2, 1) ,
ψs;2 =
ψ0
+ √
ψ / 2
ψ ++√
=
∼ (1, 3, 2) ,
ψ0
−ψ + / 2
ψs;3
ψf ;1 ∼ (1, 1, 0)
0
ψ
∼ (1, 2, −1) ,
ψf ;2 =
ψ−
− √
ψ0 √
ψ / 2
ψf ;3 =
∼ (1, 3, −2) .
ψ −−
−ψ − / 2
We will comment later on the possibility of having an SU (2)L triplet with a null hypercharge.
We present a model independent analysis of all the possible production channels, extending the work presented in Ref. [173], in which the authors restrict themselves to
the case of Dirac DM annihilating through an s channel diagram. In Sect. 4.2.1 and
Sect. 4.2.2, we present the results for direct neutrino production in the case of scalar and
fermionic DM, respectively. To be exhaustive, we explicitly divide the results for four
52
4.2 Production of monoenergetic neutrinos
different neutrino scenarios: Dirac neutrinos, for which the left-handend and the righthanded neutrinos are both present and independent; Majorana neutrinos, in which case
the singlet neutrinos ναR are not present and the right-handed neutrinos are simply given
by the (ναL )C ; Majorana neutrinos with see-saw type I or type II, if the right-handed
neutrinos ναR are present and acquire generally a heavy mass.
In our study, we do not consider explicitly the case of vector DM. It is known, indeed,
that a spin one DM particle can have a sizable branching ratio into neutrinos, see e.g.
Ref. [171]. The main aim of our analysis is, instead, to show that also in the framework
of a scalar and a fermionic DM the direct neutrino production can be relevant. However,
for completeness, we report in Sect. C.3 the explicit expressions for the annihilation cross
sections in the case of a vector DM.
We wish to recall that the neutrino production through DM annihilations into three
body final states has also been vastly discussed in the literature. For instance, in
Ref. [174] the authors analyzed the electroweak bremsstrahlung processes χχ → ν ν̄Z
and χχ → νeW . The hadronic decays of the weak bosons can lead to the production
of photons, which can then be used to further constraint the annihilation cross section
value, see e.g. Ref. [175] for the Z-strahlung process. Moreover, the DM annihilation
into neutrinos will induce, at loop level, electromagnetic final states, for which the synchrotron radiation bounds of Ref. [176] can be imposed, see Ref. [177] for an exhaustive
discussion on this aspect.
4.2.1 Scalar Dark Matter
This Section summarizes the results that we obtained for the case of scalar DM, considering singlet, doublet and triplet representations of SU (2)L . The basic assumptions are
that the scalar DM has a null vev hχi = 0 and that it is stable, because being odd under
some Z2 -parity, while all the other SM particles are even.
Scalar mediator, s-channel
In the case of a singlet, doublet or triplet scalar mediator, the following Yukawa interactions with the neutrinos are allowed:
LYν;1
αβ
= −Yν;1
(ναR )C φs;1 νβR + h.c. ,
LYν;2
=
LYν;3
=
αβ
−Yν;2
αβ
−Yν;3
(4.14)
LαL φ̃s;2 νβR + h.c. ,
(4.15)
(LαL )C (iσ2 φs;3 ) LβL + h.c. ,
(4.16)
where α and β are flavour indices. We have defined φ̃ ≡ iσ2 φ∗ , with σ2 being the second
Pauli matrix. Note that the entries of the Yukawa coupling matrices are in general
complex numbers and that a triplet scalar mediator with zero hypercharge φs;3 ∼ (1, 3, 0)
does not couple to neutrinos in a s channel diagram.
The singlet scalar mediator φs;1 can couple to a pair of scalar DM particles, which
transform under SU (2)L as a singlet, doublet or triplet. However, it will always produce
53
Chapter 4 Dark Matter annihilation into neutrinos
a physical right-handed (light) neutrino as well as a left-handed (light) anti-neutrino.
Both these particles are sterile and making them interacting would require a coupling to
the Higgs field (or, equivalently, a helicity flip), which is proportional to mν . This would
lead to a negligible muon flux at neutrino telescopes. Notice also that the coupling of the
singlet scalar to neutrinos, Eq. (4.14), could, in general, generate a violation of lepton
number L and is hence connected to Majorana neutrinos. Indeed, the interaction term
(νR )C φs;1 νR either directly violates lepton number or it forces the singlet scalar to carry
lepton number. In the latter case, the coupling of φs;1 to the SM Higgs, H † Hφs;1 , will
be problematic. However, if such a coupling is forbidden in certain specific models, one
might still be able to conserve lepton number.
If the doublet scalar mediator φs;2 does not get a vev, the entries in the Yukawa
coupling matrix Yν;2 could be large, as they do not contribute to the neutrino mass.
However, a fundamental problem arises from the coupling to the scalar DM. Since we
consider only the cases for which the scalar DM particle does not get a vev, the corresponding vertex must arise from a fundamental 3-scalar coupling in the Higgs potential.
In SU (2)L , such a fundamental 3-scalar coupling is impossible, since we have 2⊗1⊗1 = 2,
2 ⊗ 2 ⊗ 2 = 2 ⊕ 2 ⊕ 4, 2 ⊗ 3 ⊗ 3 = 2 ⊕ 2 ⊕ 4 ⊕ 4 ⊕ 6. This problem could be overcome
if one allowed for a non-vanishing vev hφs;2 i =
6 0. However, in this way the Yukawa
coupling Yν;2 becomes directly proportional to the light neutrino mass for the case of
Dirac neutrinos. In the presence of a see-saw situation, the Yukawa coupling could be
in principle sizable, since it is not directly related to the neutrino mass. Despite that,
since the light mass eigenstate of νR must be produced, this possibility is suppressed
by
the mixing angle θ between the heavy and light neutrinos. This is of O mD MR−1 and
hence very small, for the standard value of MR = O(1016 GeV).
The interaction of the triplet scalar mediator φs;3 with neutrinos, Eq. (4.16), induces
a violation of lepton number (in analogy to the singlet scalar mediator φs;1 ) and is
thus associated only to Majorana neutrinos and not to Dirac neutrinos. In case the
scalar mediator has a null vev, the neutrino coupling Yν;3 is unsuppressed, since is not
constrained by the neutrino mass scale. Furthermore, two active neutrinos are produced,
since the triplet scalar couples to (νL )C νL . This conclusion does not depend on the
particular neutrino mass model considered. Indeed, in the case of see-saw type I, the
correction factor, resulting from νL being not an exact mass eigenstate, is given by
(1 − θ)2 ≃ 1. If the neutrinos acquire a mass through a see-saw type II model, the only
difference is the presence of an additional Higgs triplet with vev, in order to have the
correct see-saw type II neutrino mass formula.
The DM vertex for the case hφs;3 i = 0 can come from a fundamental 3-scalar term in
the Higgs potential. This coupling is allowed only if χ is an SU (2)L doublet. In this
case, the important term in the Lagrangian is of the form
(2,3)
Lχφ
(2,3)
⊃ γχφ (χ†s;2 φs;3 χ̃s;2 ) + h.c.
(4.17)
If the triplet scalar mediator has a nonzero vev, hφs;3 i =
6 0, it will contribute to a
Majorana neutrino mass (νL )C νL and it will induce a see-saw type II situation. Thus,
54
4.2 Production of monoenergetic neutrinos
the light neutrino mass matrix would be given by
2
T
Mν = vT Yν;3 − vH
Yν;2 MR−1 Yν;2
,
(4.18)
where vH is the electroweak vev and vT is the triplet scalar vev. To yield physically
realistic light neutrino masses, the entries in the Yukawa coupling matrix Yν;3 of the
triplet to the neutrinos must be very small, in case the triplet contribution dominates
the physical neutrino masses.1 On the other hand, the combination of the Dirac Yukawa
coupling Yν;2 and the heavy neutrino mass matrix MR has to be tiny as well, if this part
dominates the physical neutrino mass. The only case where we can have larger values
for Yν;3 , which, in turn, could lead to larger annihilation rates, is the one where there
2 Y M −1 Y T in Eq. (4.18). For simultaneously
is a cancellation between vT Yν;3 and vH
ν;2 R
ν;2
having Yukawa couplings of O(0.1) and sub-eV neutrino masses, this cancellation would,
however, need to be at the level of 10−8 (for vT ≈ 1 GeV), which would require a strong
fine-tuning. Nevertheless, this possibility might be motivated in a certain specific model.
The corresponding couplings of the SU (2)L triplet scalar mediator φs;3 with non vanishing vev to the DM particles can arise from the following terms in the Lagrangian:
(1,3)
= λχφ (χ†s;1 χs;1 )Tr(φ†s;3 φs;3 ) ,
(2,3)
= λχφ (χ†s;2 χs;2 )Tr(φ†s;3 φs;3 ) + βχφ (χ†s;2 φ†s;3 φs;3 χs;2 ) +
i
h
(2,3)
+ γχφ (χ†s;2 φs;3 χ̃s;2 ) + h.c. ,
Lχφ
Lχφ
(3,3)
Lχφ
(1,3)
(2,3)
(4.19)
(2,3)
(4.20)
(3,3)
= λχφ Tr(χ†s;3 χs;3 )Tr(φ†s;3 φs;3 ) +
i
h
(3,3)
+ ξχφ Tr(χs;3 χs;3 )Tr(φ†s;3 φ†s;3 ) + h.c. .
(4.21)
The triplet scalar mediator appears as the most promising case for having a sizable
neutrino production. However, depending on the specific model, its coupling to the
leptons can be subject to constraints coming from different experiments. We postpone
the explicit discussion of these bounds to Sect. 4.3.1.
Z-boson mediator, s-channel
The coupling between the neutrinos and the Z-boson comes from the kinetic term in the
Lagrangian. We define the covariant derivative as
g′
g
Dµ = ∂µ + i (σ · Wµ ) + i Y Bµ ,
2
2
(4.22)
with σ being the Pauli matrices. The couplings g and g ′ are, respectively, the gauge
couplings of SU (2)L and U (1)Y . The corresponding gauge fields are denoted by Wµ and
1
Note that, although the vev is forced by the correction to the ρ-parameter to be vT . O(1 GeV), the
Yukawa coupling still needs to be tiny to yield sub-eV neutrino masses.
55
Chapter 4 Dark Matter annihilation into neutrinos
Bµ . Introducing the physical states
Wµ1 ∓ iWµ2
√
,
2
= Bµ cos θW + Wµ3 sin θW ,
Wµ± =
Aµ
Zµ = −Bµ sin θW +
Wµ3 cos θW
(4.23)
(4.24)
,
(4.25)
where θW is the Weinberg angle, the interaction term of the neutrinos with the Z-boson
is given by
g
µ
Lkin
νL γ µ νL Zµ .
(4.26)
L = LL iγ Dµ LL → −
2 cos θW
Only if the DM particle transforms as a doublet or triplet under SU (2)L , it can couple
to the Z-boson. The specific couplings arise from the following gauge-kinetic terms:
Lkin
χ;2
Lkin
χ;3
(Dµ χs;2 )† (Dµ χs;2 )
ig
→ −
(cos2 θW + Y sin2 θW ) ∂µ χ0,∗ χ0 Z µ + h.c. ,
2 cos θW
h
i
= Tr (Dµ χs;3 )† (Dµ χs;3 )
=
→ −
ig
(2 cos2 θW + Y sin2 θW ) ∂µ χ0,∗ χ0 Z µ + h.c. ,
2 cos θW
(4.27)
(4.28)
where the covariant derivative for χs;2 is defined analogously to Eq.(4.22), while for χs;3
it is given by
g
g′
Dµ χs;3 = ∂µ χs;3 + i [σ · Wµ , χs;3 ] + i Y Bµ χs;3 .
(4.29)
2
2
Fermionic mediator, t&u-channels
For the t and u-channel diagrams, either the scalar DM or the fermionic mediator has
to be flavoured, such as in the case of a sneutrino DM or a neutrino mediator. This
property has to be taken into account in any specific model and it will decide on the
actual existence of a t-channel diagram. For definiteness, throughout our discussion,
we suppose that the scalar DM particle carries a flavour. Our conclusions are as well
applicable to the case in which the fermionic mediator is flavoured.
We consider a fermionic mediator, whose left and right components can transform
under SU (2)L as singlets, doublets or triplets. If the fermionic mediator is an SU (2)L
singlet, [φf ;1 ]L,R , the following interaction terms are allowed:
(1,1)
Tαk
(2,1)
Tαk
Lχφν =
Lχφν =
56
(1,1)
α χk [φ
νR
s;1 f ;1 ]L + h.c. ,
(4.30)
(2,1)
LαL χ̃ks;2 [φf ;1 ]R + h.c. ,
(4.31)
4.2 Production of monoenergetic neutrinos
(i,j)
where Tαk are trilinear couplings, with α being an index in flavour space and k being
the index that denotes the lightest scalar particle. In general, indeed, different flavoured
states of the scalar particle χβs;1 can exist. The DM particle will then be identified as the
lightest particle among the mass eigenstates, χks;1 = Wkβ χβs;1 , with W being a rotation
matrix. The indices (i, j) are, respectively, the SU (2)L representations of the DM and
of the fermionic mediator. If the DM is a singlet scalar, it will only couple to sterile
neutrinos, while, if it is the neutral component of a doublet, active neutrinos can be
produced.
If the fermionic mediator is an SU (2)L doublet, [φf ;2 ]L,R , the interaction terms that
lead to a coupling between the DM particle and the neutrino are:
(1,2)
(1,2)
LαL [φf ;2 ]R χks;1 + h.c. ,
(2,2)
(2,2)
α (iσ χk )T [φ
νR
2 s;2
f ;2 ]L + h.c. ,
(3,2)
(3,2)
(LαL )C (iσ2 χks;3 ) [φf ;2 ]L + h.c.
Lχφν = Tαk
Lχφν = Tαk
Lχφν = Tαk
(4.32)
In this case a singlet and a triplet scalar DM can couple to active neutrinos, while only
sterile neutrinos will be produced if the DM is an SU (2)L doublet.
Finally, if the fermionic mediator is an SU (2)L triplet, [φf ;3 ]L,R , we can have the
following couplings:
(2,3)
(2,3)
LαL [φf ;3 ]R χks;2 + h.c. ,
n
o
(3,3)
k
α [φ
= Tαk Tr νR
]
χ
+ h.c.
f ;3 L s;3
Lχφν = Tαk
(3,3)
Lχφν
(4.33)
Active neutrinos arise from a doublet scalar DM, while a triplet scalar DM couples only
to sterile neutrinos. Moreover, if the fermion mediator is an SU (2)L triplet with Y = 0,
it can also couple to a scalar DM triplet with Y = 0 and to a right-handed neutrino.
This coupling would produce only sterile neutrinos and thus lead to a negligible flux.
As in the case of scalar DM pair-annihilations into neutrinos through a scalar exchange,
the couplings involved in the t-channel process are subject to experimental limits, coming in particular from lepton flavour violation processes (LFV). For example, if active
neutrinos are produced and if the fermionic mediator belongs to a doublet or triplet
representation of SU (2)L , the couplings involved in the t-channel diagram will also contribute to the µ → eγ decay. Another experimental constraint that could be present is
on the actual existence of the fermion particle mediating the process. We will comment
on these topics in Sect. 4.3.2.
4.2.2 Fermionic Dark Matter
In this Section we consider the DM as fermionic particle and, in analogy to the scalar
case, we allow for SU (2)L singlet [χf ;1 ]L,R ∼ (1, 1, 0), doublet [χf ;2 ]L,R ∼ (1, 2, −1) and
triplet [χf ;3 ]L,R ∼ (1, 3, −2) representations.
57
Chapter 4 Dark Matter annihilation into neutrinos
Scalar mediator, s-channel
For the s-channel, the considerations for the neutrino vertex are exactly the same as
in the scalar DM case. Therefore, in the following we will focus on the DM vertex
only. An intermediate scalar singlet φs;1 could couple to all types of fermionic DM under
consideration:
LYχ;1
(1,1)
=
(1,1)
(4.34)
(2,2)
−Yχ;1 [χf ;1 ]L [χf ;1 ]R φs;1 + h.c. ,
LYχ;1
=
(2,2)
(4.35)
(3,3)
=
−Yχ;1 [χf ;2 ]L [χf ;2 ]R φs;1 + h.c. ,
o
n
(3,3)
−Yχ;1 Tr [χf ;3 ]L [χf ;3 ]R φs;1 + h.c.
LYχ;1
(4.36)
In all the above cases, the left and right components of the DM particle belong to the
same representation of SU (2)L , i.e. the DM is a vector-like fermion. Note that another
possible expression can be obtained replacing [χf ;1 ]L with [χf ;1 ]RC . The interaction term
obtained in this way is associate with Majorana DM particles, since it can induce a
Majorana mass term. As in the scalar DM case, the problem arises at the neutrino
vertex, since only sterile neutrinos can be produced by a scalar singlet.
If the scalar mediator is an SU (2)L doublet, φs;2 , we could have the following couplings
to the DM particle:
LYχ;2
(1,2)
=
LYχ;2
(3,2)
=
(1,2)
(4.37)
(3,2)
(4.38)
−Yχ;2 [χf ;1 ]L (iσ2 φs;2 )T [χf ;2 ]R + h.c ,
−Yχ;2 φ†s;2 [χf ;3 ]L [χf ;2 ]R + h.c.
These possibilities will only be present if the left and right components of the DM particle
belong to different representations of SU (2)L , i.e. the DM is a chiral fermion. As for the
scalar DM case, the situation in which φs;2 has a nonzero vev can be neglected, since the
Yukawa couplings would be proportional to the neutrino mass or the tiny mixing angle
θ between the heavy and light neutrinos would be present.
If the scalar mediator is a triplet under SU (2)L , φs;3 , the following terms are allowed:
(2,2)
LYχ;3
(1,3)
LYχ;3
(2,2)
= −Yχ;3 [χf ;2 ]LC (iσ2 φs;3 ) [χf ;2 ]L + h.c ,
o
n
(1,3)
= −Yχ;3 Tr [χf ;1 ]L φs;3 [χf ;3 ]R + h.c ,
(4.39)
(4.40)
where the first term will be present if the DM particle is a vector-like fermion, while the
second one will be there if it is a chiral fermion. An analogous expression to Eq. (4.39) can
be obtained by exchanging the subscripts “L” with “R” and considering a new Yukawa
′(2,2)
coupling Yχ;3 . Notice that, if Eq. (4.39) holds, the triplet scalar mediator in the schannel is associated only with Majorana DM (as well as Majorana neutrinos), since it
leads to terms that violate lepton number. If the scalar triplet acquires a nonzero vev, a
see-saw type II situation is induced, in analogy to the scalar DM case, to which we refer
for more details. Remember that, in principle also a DM coupling to a triplet scalar with
zero hypercharge is possible, but this would not lead to a coupling to neutrinos.
58
4.2 Production of monoenergetic neutrinos
Z-boson mediator, s-channel
A fermionic DM particle can couple to the Z-boson, if it is a doublet or a triplet under
SU (2)L . The corresponding couplings arise from the following gauge-kinetic terms in the
Lagrangian:
Lkin
χ;2
Lkin
χ;3
[χf ;2 ]L iγ µ Dµ [χf ;2 ]L
g
→ −
(cos2 θW − Y sin2 θW )[χf ;2 ]0L γ µ [χf ;2 ]0L Zµ + h.c. ,
2 cos θW
o
n
= Tr [χf ;3 ]L iγ µ Dµ [χf ;3 ]L
g
→ −
(2 cos2 θW − Y sin2 θW )[χf ;2 ]0L γ µ [χf ;2 ]0L Zµ + h.c. ,
2 cos θW
=
(4.41)
(4.42)
and analogous expressions can be written for the right-handed components [χf ;2 ]R and
[χf ;3 ]R of the DM particle.
Scalar mediator, t&u-channels
As in the case of scalar DM, for the t and u-channel diagrams, either the fermionic DM
or the scalar mediator has to be flavoured, such as in the case of a heavy neutrino DM or
a sneutrino scalar mediator. For definiteness, throughout the discussion of this Section
we will suppose that the scalar mediator carries a flavour. Our conclusions are as well
applicable to the case in which the DM is flavoured.
If the scalar mediator is an SU (2)L singlet φs;1 , the following interaction terms are
allowed:
Lχφν =
(2,1)
Tαk
(1,1)
Tαk
Lχφν =
(2,1)
LαL [χf ;2 ]R φks;1 + h.c. ,
(4.43)
(1,1)
k
α [χ
νR
f ;1 ]L φs;1 + h.c. ,
(4.44)
(i,j)
where Tαk are trilinear couplings, with α being an index in flavour space and k being
the index that denotes the mass eigenstate of the scalar mediator. The indices (i, j) are,
respectively, the SU (2)L representations of the DM particle and of the scalar mediator. In
order not to produce only sterile neutrinos, the right-handed component of the fermionic
DM particle has to be an SU (2)L doublet.
If, instead, the scalar mediator is an SU (2)L doublet, φs;2 , the interaction terms are:
(1,2)
(1,2)
LαL φ̃ks;2 [χf ;1 ]R + h.c. ,
(2,2)
(2,2)
α (iσ φk )T [χ
νR
2 s;2
f ;2 ]L + h.c. ,
(3,2)
(3,2)
LαL [χf ;3 ]R φks;2 + h.c. ,
Lχφν = Tαk
Lχφν = Tαk
Lχφν = Tαk
(4.45)
among which only the ones involving a singlet or a triplet fermionic DM lead to active
neutrinos in the final state. One specific example falling in this category would be a slight
59
Chapter 4 Dark Matter annihilation into neutrinos
extension of the MSSM, with an additional singlet chiral superfield, whose fermionic
component acts as DM particle, while the sneutrino is the scalar mediator.
Finally, if the scalar mediator is an SU (2)L triplet, φs;3 , we have
(2,3)
(2,3)
(LαL )C (iσ2 φks;3 ) [χf ;2 ]R + h.c. ,
n
o
(3,3)
α φk [χ
= Tαk Tr νR
+ h.c. ,
f ;3 ]L
s;3
Lχφν = Tαk
(3,3)
Lχφν
(4.46)
where, only in the case of a doublet fermionic DM particle, the production of active
neutrinos is possible.
In our analysis of the t-channel diagram for fermionic DM, we have decided to neglect
the possibility that the intermediate scalar mediator acquires a nonzero vev. In this
case, a mixing between the DM particle and the neutrino is induced. The corresponding constraints on the Yukawa couplings become strongly model dependent and general
considerations will not be possible anymore.
As in the case of scalar DM pair-annihilations, the couplings involved in the t-channel
process could be subject to the experimental limits coming from LFV processes. We
refer to Sect. 4.3.2 for more details.
4.3 Discussion of unsuppressed cases
A summary of our results can be found in Table 4.1 and Table 4.2, respectively, for the
case of scalar DM and of fermionic DM. In these tables, we explicitly divide between the
s, t and u annihilation channels, and we consider different possible SU (2)L representations for the DM and the mediator particle. Moreover, different neutrino scenarios are
considered.
In Appendix C we give the explicit expressions of the annihilation cross sections for
the different cases. The results are reported in a model independent way and, therefore,
can be used for any specific model.
For a scalar DM particle, the s-channel annihilation diagram can be relevant only in
the presence of a triplet scalar mediator with zero vev. Moreover, this case is present
only for Majorana neutrinos. The explicit expression of the annihilation cross section can
be found using Eq. (C.3). Another promising situation for neutrino production is given
by a t-channel diagram with a singlet, a doublet or a triplet fermion exchange. In the
first case the DM particle should be a doublet under SU (2)L , in the second case a singlet
or a triplet and in the third case a doublet. For the t-channel diagram, the annihilation
cross section will be determined mainly by the mass of the mediator, see Eq. (C.5) and
Eq. (C.6).
For a fermionic DM particle, a triplet scalar exchange in an s-channel diagram can give
rise to a sizable neutrino production if the left-handed or right-handed components of
the DM particle transform as a doublet under SU (2)L and if the neutrinos are Majorana
particles. For a chiral fermion DM, the s-channel diagram might be relevant if the scalar
mediator is a doublet or a triplet under SU (2)L . The first case can be present if the
60
4.3 Discussion of unsuppressed cases
Annihilation
channels
s
s
s
s
s
s
t, u
t, u
t, u
t, u
t, u
t, u
Internal
mediator
Scalar 1
Scalar 3
Scalar 1 with vev
Scalar 2 with vev
Scalar 3 with vev
Z-boson
Fermion 1
Fermion 1
Fermion 2
Fermion 2
Fermion 3
Fermion 3
Dark Matter
SU (2)L -rep.
1, 2, 3
2
1, 2, 3
1, 2, 3
1, 2, 3
2, 3
1
2
1, 3
2
2
3
Dirac
neutrino
L/
L/
L/
mν
L/
4(p)
R
2
2
R
2
R
Majorana
neutrino
2
mν
4(p)
2
2
2
-
See-saw
type I
R, θ2
2
R, θ2
θ
4(p)
R, θ2
2
2
R, θ2
2
R, θ2
See-saw
type II
R, θ2
2
R, θ2
θ
f.t.
4(p)
R, θ2
2
2
R, θ2
2
R, θ2
Table 4.1: Summary table for scalar Dark Matter. 2
: potentially unsuppressed in
at least one channel; 4(p): suppressed for non-relativistic Dark Matter (p-wave term);
f.t.: fine tuning required between two couplings to get a sizable rate; L/: LNV terms are
present; - : a see-saw type I and/or type II situation is present; R: yields only right-handed
neutrinos; θn : suppressed by the n-th power of the mixing angle between heavy and light
neutrinos; mν : the Yukawa coupling involved is proportional to the light neutrino mass.
neutrinos are Dirac particles, while the second one can be there if they are Majorana
particles. The explicit expression for the annihilation cross section can be found using
Eq. (C.7). As in the case of scalar DM, another promising case for neutrino production
is given by the t-channel diagram with a singlet, doublet or triplet scalar exchange. In
the first case, the DM particle should be a doublet under SU (2)L , in the second case a
singlet or a triplet, while in the third case it must be a doublet. Other unsuppressed tchannel diagrams require explicitly a chiral fermion DM, see Table 4.2. For the t-channel
diagram, the annihilation cross section will be determined mainly by the mass of the DM
particle, see Eq. (C.10) and Eq. (C.11). Moreover, if the DM was a Dirac fermion, also the
s-channel diagram with the Z-boson exchange could lead to sizable neutrino production.
In this case, the annihilation cross section will be proportional to the mass of the DM
particle, see Eq. (C.8). However, particles with strong couplings to the Z-boson are
constrained by DM direct detection experiments [104].
For definiteness, we focus on two different typologies of unsuppressed cases: one involving an s-channel diagram, in Sect 4.3.1, and one with a t-channel diagram, in Sect. 4.3.2.
For the first possibility, we consider a triplet scalar exchange with null vev and a DM
particle that transforms as a doublet under SU (2)L . We explicitly distinguish between
the case of scalar and Majorana DM. Remember that a triplet scalar exchange in an
s-channel diagram is associated to Majorana neutrinos only. For the t-channel diagram,
61
Chapter 4 Dark Matter annihilation into neutrinos
Annihilation
channels
s
s
s
s
s
s
s
s
s
t (u)
t (u)
t (u)
t (u)
t (u)
t (u)
t (u)
t (u)
t (u)
t (u)
Internal
mediator
Scalar 1
Scalar 2
Scalar 3
Scalar 3
Scalar 1 with
Scalar 2 with
Scalar 3 with
Scalar 3 with
Z-boson
Scalar 1
Scalar 1
Scalar 1
Scalar 2
Scalar 2
Scalar 2
Scalar 2
Scalar 3
Scalar 3
Scalar 3
vev
vev
vev
vev
Dark Matter
SU (2)L -rep.
1, 2, 3
(1,2) ,(2,3)
2a
(1,3)
1, 2, 3
(1,2) ,(2,3)
2a
(1,3)
2, 3
1
2
(1,2)
1, 3
2
(1,2)
(1,3)
2
3
(2,3)
Dirac
neutrino
L/
2
L/
L/
L/
mν
L/
L/
/4(p)
2
R
2
2
2
R
2
2
2
R
2
Majorana
neutrino
2
2
mν
mν
/4(p)
2
2
2
2
2
-
See-saw
type I
R, θ2
θ
2
2
R, θ2
θ
/4(p)
2
R, θ2
2
θ
2
R, θ2
θ
2
2
R, θ2
θ
See-saw
type II
R, θ2
θ
2
2
R, θ2
θ
f.t.
f.t.
/4(p)
2
R, θ2
2
θ
2
R, θ2
θ
2
2
R, θ2
θ
Table 4.2: Summary table for chiral and vector-like fermionic Dark Matter. 2
: potentially unsuppressed in at least one channel; 4(p): suppressed for non-relativistic Dark
Matter (p-wave term); f.t.: fine tuning required between two couplings to get a sizable
rate; L/: LNV terms are present; - : a see-saw type I and/or type II situation is present;
R: yields only right-handed neutrinos; θn : suppressed by the n-th power of the mixing
angle between heavy and light neutrinos; mν : the Yukawa coupling involved is proportional to the light neutrino mass; x/y: applies for Dirac DM/applies for Majorana DM;
a the coupling is present for Majorana DM only.
we also consider a DM particle that is a doublet under SU (2)L . In the context of a
scalar DM, we focus on the possibility of a Majorana singlet mediator, while in the case
of Majorana DM we consider a scalar singlet mediator. As an example, we consider the
case of Majorana neutrinos for the t-channel diagrams.
4.3.1 s-channel: the triplet scalar mediator
The couplings involved in an s-channel diagram with a triplet scalar exchange will not
be connected to the neutrino mass, if the triplet has a null vev. However, the entries
of the Yukawa coupling matrix Yν;3 are constrained by different experimental results, in
62
4.3 Discussion of unsuppressed cases
particular by the limits on µ and τ decays, and by the values of the electron and the
muon anomalous magnetic moments. In the following, we summarize these bounds.
Experimental constraints
The singly charge triplet component φ−
s;3 might transmit a lepton number violating muon
−
−
decay with one µ -ν µ -φs;3 and one e− -ν e -φ−
s;3 vertex. Considering the experimental
uncertainty on GF of about 10−10 GeV−2 [89], obtained through µ-decay measurements,
the corresponding diagonal entries of Yν;3 are set to be:
µµ 2
ee 2
|
|Yν;3
| |Yν;3
. 0.1
10−10 m2φ
GeV2
!2
.
(4.47)
In general, also the electrically neutral component φ0s;3 of the Higgs triplet will mediate
µe
µ-decay. However, the corresponding diagram involves the LFV coupling Yν;3
, that is
constrained stronger by the experimental limit on the branching ratio for µ → 3e (see
later).
The singly charged triplet component φ−
s;3 might transmit a lepton number violating
− -ν -φ− or µ− -ν -φ− vertex. Therefore, the diand
one
e
tau decay with one τ − -ν τ -φ−
e s;3
µ s;3
s;3
agonal elements of Yν;3 receive bounds also from the experimental limit on the τ lifetime.
Taking into account that the uncertainty on Γτ is roughly 0.1% [89], we find
!
−5 m2 2
10
φ
µµ
ττ 2
ee 2
.
(4.48)
|Yν;3
| |Yν;3
| + |Yν;3 |2 . 0.1
GeV2
If the Yukawa coupling matrix Yν;3 contains off-diagonal terms, the triplet will also
have LFV couplings. In this case, the strongest constraint arises from µ → 3e decay.
Indeed, this process can be mediated at tree-level by the doubly charged component
− + −−
− − −−
of the triplet φ−−
s;3 with one µ -e -φs;3 and one e -e -φs;3 vertex. From the experiment SINDRUM I [178], we know that BR(µ → 3e) . 10−12 at 90% confidence level.
Therefore, the bound on the off-diagonal entries reads
!
−11 m2 2
10
φ
eµ 2
ee 2
.
(4.49)
| . 5.4
|Yν;3 | |Yν;3
GeV2
Note that the µ → eγ process naturally arises only at 1-loop level and is therefore
suppressed with respect to the µ → 3e decay. The branching ratio of the τ decay into
three leptons l (with l=e,µ) is, instead, constrained from the BELLE experiment [179]
to be BR(τ → lll) . (2 − 4) × 10−8 at 90% confidence level. This implies the following
limit on the off-diagonal τ Yukawa entries:
!
−9 m2 2
10
φ
lτ 2
ll 2
.
(4.50)
|Yν;3
| |Yν;3
| . 0.6
GeV2
63
Chapter 4 Dark Matter annihilation into neutrinos
ee and Y µµ are also subject to constraints coming from measureThe Yukawa entries Yν;3
ν;3
ments of the electron and the muon anomalous magnetic moments [180]:
m φ
ee
|Yν;3
| . O(10−4 )
,
(4.51)
MeV
m φ
µµ
.
(4.52)
|Yν;3
| . O(10−6 )
MeV
ee and Y µµ are of the same magnitude,
If we suppose that the diagonal elements Yν;3
ν;3
Eq. (4.47) and Eq. (4.48) imply that the only sizable diagonal Yukawa entry is given by
ττ :
the element Yν;3
!
−1 m2
10
φ
ττ 2
|Yν;3
| . min 1,
,
(4.53)
GeV2
where we have explicitly imposed that the Yukawa coupling is at most of order one. Since
τ τ |2 . 1.
in our numerical analysis we always consider mφ & 100 GeV, we have that |Yν;3
For simplicity, we neglect the contributions coming from the off-diagonal terms of the
Yukawa matrix Yν;3 .
In the case of scalar DM, the coupling between the DM particles and the scalar triplet
mediator φs;3 in the s-channel arises from a trilinear term in the potential, see Eq. (4.17).
The existence of this coupling and at the same time the possibility for the scalar triplet
to have a null vev will depend on the actual form of the scalar potential. In a particular
model, one has to check that these two conditions are fulfilled.
In the case of vector-like fermionic DM, the coupling between the DM particles and
the scalar triplet mediator φs;3 in the s-channel can arise from two different Yukawa
(2,2)
′(2,2)
couplings: Yχ;3 , which is related to the DM left-handed components, and Yχ;3 , which
is connected to the DM right-handed components. In case these two couplings result
to be of the same order, the s-wave contribution to the annihilation cross section will
vanish, see Eq. (C.7). However, there is no a priori reason for them to be of the same
magnitude. Therefore, we will suppose in our analysis that one of the Yukawa couplings,
(2,2)
′(2,2)
Yχ;3 , dominates over the other one, Yχ;3 .
We want to stress that, even though the scalar triplet can be associated also to a chiral
DM (see Table 4.2), we neglect this possibility, since strong bounds from electroweak
precision measurements apply on new chiral fermions beyond the SM ones. Indeed, a
new multiplet of degenerate fermions will contribute to the value of the S parameter in
the following way [89]:
X
S = NC
(t3L (i) − t3R (i))2 /3π ,
(4.54)
i
where t3L (i) and t3R (i) are the third components of weak isospins of the left-handed
and the right-handed components of the fermion i and NC is the number of colors.
Considering a SM Higgs mass MH = 117 GeV, the new physics contribution to the S
parameter is constrained to be . 0.06 at 95% C.L. [89].
To be consistent with direct searches at collider experiments, we consider the mass of
the triplet scalar mediator in the s-channel to be & 100 GeV [181].
64
4.3 Discussion of unsuppressed cases
The annihilation cross section
Using the Lagrangian terms of Eq. (4.16) and Eq. (4.17) and the expression of the annihilation cross section given in Eq. (C.3), we find that
(2,3) 2
τ τ |2
|Yν;3
1
σann v =
+ O(v 2 )
8π (4m2χ − m2φ )2
γχφ
for scalar DM,
(4.55)
where we have assumed for simplicity that the DM particle and the lightest neutral scalar
mediator correspond to the real components of χ0s;2 and φ0s;3 , respectively. The parameter
(2,3)
γχφ
is set to be real. Considering, instead, Eq. (4.39) and Eq. (C.7), we conclude that
τ τ |2
1 |Yχ |2 |Yν;3
m2 + O(v 2 )
σann v =
4π (4m2χ − m2φ )2 χ
with
(2,2)
Yχ = Yχ;3
for Majorana DM,
h
i
(2,2) ∗
+ Yχ;3
.
(2,2)
(4.56)
(4.57)
′(2,2)
We have assumed the Yukawa coupling Yχ;3 to dominate over Yχ;3 . If these two couplings are of the same order, instead, the first nonzero contribution to the annihilation
cross section would be given by a p-wave term. Moreover, we have considered the imaginary component of the φ0s;3 as exchanged particle. The real component would, indeed,
have a zero s-wave due to parity conservation.
The expressions reported above refer to the production of tau neutrinos. The DM
annihilation into neutrinos with different flavours would be more suppressed, because of
the bound reported in Eq. (4.47), and can therefore be neglected.
In Fig. 4.1, we show the behaviour of the annihilation cross section into tau neutrinos
for the case of scalar DM (left panel) and of Majorana DM (right panel). The annihilation
cross sections result to be of the order of the value expected for a thermal relic for a wide
range of the parameter space. From our plots, it is possible to identify which are the
values of the Yukawa couplings and of the triplet and DM mass in which the neutrino
production might be relevant. This can then be applied to specific model, in which a
triplet scalar without vev is present.
In general, we can say that the neutrino flux from the galactic center (GC), generated
by the triplet scalar exchange, might be accessible to neutrino telescopes only in the
resonant region, in which mχ ≃ mφ . Indeed a boost factor of order 100 or more is
required to overcome the atmospheric neutrino background [152]. In this case, however,
a DM production mechanism different from the thermal one is necessary. Moreover,
the CMB measurements of the WMAP satellite impose stringent limits on DM models
with very large annihilation cross section, as has been pointed out in Refs. [182, 183].
Remember that even a DM particle that annihilates mainly into neutrinos will generally
produce electromagnetic final states by loop diagrams [177]. In considering specific DM
models, the CMB bounds of Refs. [182, 183] must be imposed.
65
Chapter 4 Dark Matter annihilation into neutrinos
10-22
10-22
Halo Average
Halo Average
200
200
10-23
10-23
Halo Angular
Halo Angular
100
Σann v @cm3 s-1 D
10
BF=10
500
-25
10
Natural Scale
800
10-26
BRΝΤ =0.1
10-27
10-28
Σann v @cm3 s-1 D
100
-24
10-24
BF=10
500
10-25
Natural Scale
800
10-26
BRΝΤ =0.1
10-27
ΓH2,3L
ΧΦ =100
100
200
GeV,
2
ΤΤ 2
ÈYΝ;3
È =1
500
1000
m Χ @GeVD
ÈY Χ È = 1,
2000
5000 1 ´ 104
10-28
100
200
ΤΤ 2
ÈYΝ;3
È =
1
500
1000
2000
5000 1 ´ 104
m Χ @GeVD
Figure 4.1: Dark Matter annihilation cross sections into tau neutrinos through the
exchange of a scalar triplet with null vev, in an s-channel diagram. Left panel: scalar
Dark Matter. Right panel: Majorana Dark Matter. The numbers next to each curve
denote the different values of the scalar triplet mass (in GeV). The Halo Angular and the
Halo Average lines represent bounds from neutrino searches, see text for more details.
The gray solid line indicates the standard value of σann v for a thermal relic (natural
scale), while the gray dashed lines mark the values for a 10% branching ratio into tau
neutrinos (BRντ ) and for a boost factor (BF) equal to ten (where the natural scale is
taken as reference).
The signals from the Sun and the Earth could, instead, be detected for a wide range of
the parameters, depending on the value of the DM scattering cross section. For the Sun
a 5σ discovery, after one year of data taking with the IceCube detector, can be achieved
if σp BRν ≃ 6 × 10−7 pb for mχ ≃ 200 GeV or if σp BRν ≃ 10−5 pb for mχ ≃ 1 TeV,
where σp is assumed to be dominated by spin-dependent interactions and where BRν
is the branching ratio into neutrinos of all flavours [184, 185]. For the Earth, assuming
equilibrium between the capture and the annihilation rate, the 5σ discovery can be
reached if σpSI BRν ≃ 9 × 10−10 pb for mχ ≃ 200 GeV and if σpSI BRν ≃ 3 × 10−9 pb for
mχ ≃ 1 TeV [184, 185]
In the plots we also report the limits on the annihilation cross section σann v, as derived by the authors of Ref. [150] comparing the energy spectrum produced by DM pairannihilation into neutrinos with the atmospheric neutrino background measured by the
Super-Kamiokande, Frejus and AMANDA detectors. The Halo Angular bound corresponds to a cone half-angle of about 30◦ around the GC and to a value of the J-factor,
see Eq. (3.25), of 25. The Halo Average bound is associated, instead, to J ≃ 5, which is
an average value for the whole sky. As can be seen from the figures, these constraints
are not really strong and exclude only a small fraction of the parameter space in the
resonance region.
66
4.3 Discussion of unsuppressed cases
Note that, for the case of Majorana DM, the scalar triplet could also induce a neutrino
production through a t-channel diagram. For simplicity, we show in Fig. 4.1 only the
s-channel annihilation cross sections.
4.3.2 t-channel: the singlet fermionic and scalar mediator
The couplings involved in a t-channel diagram are subject to experimental bounds, since
they induce LFV processes at one loop. A summary of these experimental limits is given
in the following, considering for definiteness the case of a singlet vector-like fermionic
mediator φf ;1 (for a doublet scalar DM) and of a singlet scalar mediator φs;1 (for a doublet
vector-like DM). Bounds from measurements of the anomalous magnetic moments of the
electron and the muon also apply.
Experimental constraints
In the case of a scalar DM particle that is a doublet under SU (2)L , the µ → eγ process
can be mediated by the charged scalar χ−
s;2 and the fermionic singlet φf ;1 . Instead, for a
doublet vector-like DM, the µ → eγ process can be induced by the charged fermion χ−
f ;2
and the scalar singlet φs;1 . Using the limit on the BR(µ → eγ) provided by the MEGA
experiment [186], we can write
3.2 × 10
9
m2µ /GeV2
m4s /GeV4
ξ14 H 2 (t) . 1.2 × 10−11 ,
(4.58)
h
i
(2,1)
(2,1) ∗
where mµ is the muon mass. We have defined ξ12 = Tek
Tkµ
and t = m2f /m2s , with
mf and ms being, respectively, the mass of the fermion and the scalar particles involved
in the loop process. The function H(t) is given by [187]
( 2
2 ln t
2t +5t−1
− 2 t(t−1)
for scalar DM,
4
12 (t−1)3
(4.59)
H(t) =
2
t ln t
t −5t−2
+ 2 (t−1)
for fermionic DM.
4
12 (t−1)3
In analogy, we find the following constraint on the couplings involved in the τ → µγ
process:
m2 /GeV2 4 2
2.1 × 106 τ4
ξ2 H (t) . 4.5 × 10−8 ,
(4.60)
ms /GeV4
h
i
(2,1)
(2,1) ∗
where mτ is the tau mass. We have defined ξ22 = Tek
Tkτ
and we have used
the experimental limit on the BR(τ → µγ) as provided by the BELLE experiment [188].
Finally, the last bound is given by the BaBar [189] experimental limit on the BR(τ → eγ):
2.1 × 106
m2τ /GeV2 4 2
ξ3 H (t) . 1.1 × 10−7 ,
m4s /GeV4
(4.61)
67
Chapter 4 Dark Matter annihilation into neutrinos
(2,1)
where in this case we have ξ32 = Tµk
(2,1)
h
i
(2,1) ∗
Tkτ
(2,1)
. Moreover, the couplings Tek
and
Tµk are also subject to constraints coming from measurements of the electron and the
muon anomalous magnetic moments [180]:
m φ
(2,1)
,
(4.62)
|Tek | . O(10−4 )
MeV
m φ
(2,1)
.
(4.63)
|Tµk | . O(10−6 )
MeV
(2,1)
For simplicity, in our numerical examples we consider the situation in which Tek
≃
(2,1)
(2,1)
Tµk
≪ Tτ k . In this case, using Eq. (4.58) and Eq. (4.60), we find the following
constraint:
2
2
1
(2,1) 2
−4 ms /GeV
|Tτ k | . min 1, 8.7 × 10
,
(4.64)
mτ /GeV H(t)
where we have explicitly imposed that the trilinear coupling is at most of order one.
For the t-channel diagram, we restrict our analysis to a singlet Majorana mediator and
to a singlet scalar mediator with masses & 100 GeV. We use the limit of Eq. (4.64) in
the numerical evaluation, considering also that in the case of scalar DM ms ≃ O(mχ )
and mf = mφ , while in the case of fermionic DM ms = mφ and mf ≃ O(mχ ). We
finally wish to add that, in the case of a chiral mediator φf ;1 or of a chiral DM χf ;2 , the
constraints from LFV processes might be much stronger. In particular, in Eq. (4.64), we
would have the mass of the fermionic particle exchanged in the loop instead of the tau
mass. This can be related to a chirality flip in the fermionic line.
The annihilation cross section
Using the Lagrangian term of Eq. (4.31) and the expression of the annihilation cross
section given in Eq. (C.6), we find that
(2,1)
1 |Tτ k |4
m2 + O(v 2 )
σann v =
8π (m2χ + m2φ )2 φ
for scalar DM,
(4.65)
where we have assumed that the fermionic mediator is a Majorana particle and that the
DM is the real component of χ0s;2 . This can be, for example, the case of sneutrino annihilation through a neutralino exchange. Considering, instead, Eq. (4.43) and Eq. (C.11),
we conclude that
(2,1)
|Tτ k |4
1
m2 + O(v 2 )
σann v =
64π (m2χ + m2φ )2 χ
for Majorana DM,
(4.66)
where we have considered the real component of φs;1 to be the lightest scalar mediator.
Remember that, if the Majorana particle is the supersymmetric neutralino, the couplings
(2,1)
Tτ k will be proportional to the neutrino mass and thus the annihilation cross section
into neutrinos will be negligible, see the discussion after Eq. (C.14). In a more general
68
4.3 Discussion of unsuppressed cases
10-22
10-22
Halo Average
Halo Average
10-23
Halo Angular
10-24
Σann v @cm3 s-1 D
Σann v @cm3 s-1 D
10-23
BF=10
10-25
Natural Scale
10-26
BRΝΤ =0.1
10-27
H2,1L
ÈTΤk È2 =
10-28
100
200
100
1
500
1000
m Χ @GeVD
2000
100
BF=10
200
10-25
Natural Scale
10-26
500
800
BRΝΤ =0.1
10-27
500
200
10-24
Halo Angular
H2,1L
ÈTΤk È2 =
800
5000
1 ´ 104
10-28
100
200
1
500
1000
2000
5000
1 ´ 104
m Χ @GeVD
Figure 4.2: Dark Matter annihilation cross section into tau neutrinos through the
exchange of a singlet mediator, in a t-channel diagram. Left panel: scalar Dark Matter
and Majorana mediator. Right panel: Majorana Dark Matter and scalar mediator. The
numbers next to each curve denote the different values of the singlet mediator mass (in
GeV). The Halo Angular and the Halo Average lines represent bounds from neutrinos
searches, see text for more details. The gray solid line indicates the standard value of
σann v for a thermal relic (natural scale), while the gray dashed lines mark the values for
a 10% branching ratio into tau neutrinos (BRντ ) and for a boost factor (BF) equal to
ten (where the natural scale is taken as reference).
model, however, the couplings are not fixed and the neutrino production can be sizable,
even if the DM particle is Majorana. This possibility is often overlooked in the literature.
The expressions reported above refer to the production of tau neutrinos, which we have
assumed to be the dominant channel. Depending on the structure of the matrix Tαk ,
the other neutrino flavours could lead to sizable contributions. Nevertheless, the total
annihilation cross section into neutrinos would be of the same order as the one obtained
considering the tau neutrino as the dominant flavour channel.
The behaviour of the annihilation cross sections into tau neutrinos is reported in
Fig. 4.2 for the cases of scalar DM (left panel) and Majorana DM (right panel). For a
wide range of the parameter space, the annihilation cross sections can cover the order of
magnitudes expected for a standard WIMP. In our specific examples, the experimental
limits on LFV processes reported in Eq. (4.64) result to be quite weak and do not restrict
the allowed parameter space in the interesting region of σann v. However, we want to
remind that in the cases of a chiral mediator, for scalar DM, or of a chiral DM, the
bounds from LFV processes might be much stronger. In the plots we also report the
Halo Angular and Halo Average bounds [150], which partially limit the regions of the
annihilation cross section under consideration.
The neutrino signal from the GC, generated by a t-channel singlet exchange, could be
69
Chapter 4 Dark Matter annihilation into neutrinos
hardly accessible to neutrino telescopes, since a cross section of the order of & 10−24 cm3 s−1
is almost never reached. The signal from the Sun and the Earth, instead, might be detected, depending on the value of the scattering cross section, as we have explained in
Sect. 4.3.1.
70
5
Indirect versus direct Dark Matter
detection
Indirect evidence of DM particles in our halo by measurements of upward-going muons
at neutrino detectors has been the subject of many investigations in the past, see for
instance Refs. [9, 10, 11, 12].
Recently, a number of papers appeared where possible signals at neutrino telescopes
are discussed. These consider either a generic DM particle with assumed dominance of
specific annihilation channels [190, 191] or discuss specific DM candidates, like WIMPless
DM or mirror DM [191]. Also DM particles which directly annihilate into neutrinos have
been analyzed [192].
In Sect. 5.1, we present the neutrino constraints coming from DM searches at the
Super-Kamiokande detector in the framework of the leptophilic DM, while, in Sect. 5.2,
we calculate the muon fluxes expected for the neutralino DM. The constraints from
direct DM detection experiments, in particular the DAMA experiment, are implemented
for both the leptophilic and the neutralino DM.
5.1 Leptophilic Dark Matter
The leptophilic DM model has been proposed in Ref. [193] to reconcile the DAMA results
with the absence of a signal in experiments like CDMS and XENON, that search for
nuclear recoils from DM scattering, see Sect. 2.4.3. Indeed, while electronic events will
contribute to the scintillation light signal in the DAMA detectors, most of the other
DM experiments reject pure electron events by aiming at a (close to) background free
search for nuclear recoils. As shown in Ref. [193], DM scattering off electrons at rest
cannot provide enough energy to be seen in a detector. However, exploiting the tail of
the momentum distribution of electrons bound in an atom may lead to a scintillation
71
Chapter 5 Indirect versus direct Dark Matter detection
light signal in DAMA of order few keVee. The signal in direct detection experiments
from DM-electron scattering has been considered recently also in Ref. [194].
Such a framework, where DM recoils against electrons bound in atoms, might be also
motivated by recent cosmic ray anomalies [4, 79, 85] observed in electrons/positrons, but
not in anti-protons. A simple model for leptophilic DM has been presented in Ref. [128],
see in this context for example also Refs. [195, 196].
In this work, we consider the hypothesis that DM has tree-level interactions only with
leptons and has no direct couplings to quarks. We use an effective field theory approach to perform a model independent analysis. In Sect. 5.1.1 we introduce the effective
Lagrangian for DM-lepton interactions, considering all possible Lorentz structures. In
Sect. 5.1.2 we analyze the scattering on electrons in more details, while the possible experimental signatures of leptophilic DM in direct detection experiments are discussed in
Sect. 5.1.3.
Even in such a leptophilic DM scenario, in many cases a DM-quark interaction is
induced at one or two-loop level by photon exchange. In Sect. 5.1.4, we identify the
Lorentz structures for which the loop induced coupling to quarks is present. For these
cases, the DM-nucleon scattering dominates over DM-electron scattering, since the latter
is suppressed by the bound state wave function.
Sect. 5.1.5 contains a summary of the possible Lorentz structures and their relative
cross sections. We identify only one possible Lorentz structure, the axial vector type
coupling, where DM-electron scattering dominates, since the loop diagram vanishes, and
the scattering cross section is not additionally suppressed by small quantities. The
expressions for the event rates in direct detection experiments are given in Sect. 5.1.6.
The Super-Kamiokande bounds on neutrinos from leptophilic DM annihilations inside
the Sun are presented in Sect. 5.1.7. We show how the constraints on the scattering
cross section, provided by this indirect detection method, disfavour the possibility of the
leptophilic DM as viable explanation of the DAMA annual modulation signal.
5.1.1 Effective Dark Matter interactions
In this Section, we pursue a model independent analysis of the leptophilic DM candidate,
using an effective interaction description. In the case of fermionic DM, the most general
dimension six four-Fermi effective interactions are, shown pictorially also in Fig. 5.1 (right
diagram),
X
1
Leff =
G (χ̄Γiχ χ) (ℓ̄Γiℓ ℓ)
with
G= 2,
(5.1)
Λ
i
where Λ is the cut-off scale for the effective field theory description, while the sum is over
different Lorentz structures. A complete set consists of scalar (S), pseudo-scalar (P ),
vector (V ), axial-vector (A), tensor (T ), and axial-tensor (AT ) currents. The four-Fermi
72
5.1 Leptophilic Dark Matter
χ, kµ′
e− , p′µ
χ, kµ′
e− , p′µ
χ, kµ
e− , pµ
−→
φ
χ, kµ
e− , pµ
Figure 5.1: Example for generating an effective local DM-electron interaction vertex
(right diagram) as used in our analysis by the exchange of a heavy intermediate particle
φ (left diagram).
operators can thus be classified to be of
scalar-type:
vector-type:
tensor-type:
Γχ = cχS + icχP γ5 ,
Γµχ = (cχV + cχA γ5 )γ µ ,
µν
Γµν
χ = (cT + icAT γ5 )σ ,
Γℓ = cℓS + icℓP γ5 ,
Γℓµ = (cℓV + cℓA γ5 )γµ ,
Γℓµν = σµν ,
(5.2)
where σµν = 2i [γµ , γν ].1 If DM is a Majorana particle, vector and tensor like interactions
vanish, i.e., cχV = cχT = cχAT = 0.
In our work we do not rely on any specific realization of the effective interaction. The
simplest example would just be assuming that the interaction is induced by the exchange
of an intermediate particle, whose mass is much larger than the recoil momenta, that are
of order a few MeV/c. The intermediate particle can then be integrated out leaving an
effective point interaction. Let us look at the χ-lepton interaction mediated by a scalar
field φ, shown in Fig. 5.1. It gives an amplitude
igSχ (ūχ uχ )
i
ig ℓ (ūℓ uℓ )
2
q − m2φ + iǫ S
−→
i
gSχ gSℓ
(ūχ uχ )(ūℓ uℓ ) ,
m2φ
(5.3)
where on the right-hand side we have neglected the momentum transfer q 2 = (p′ − p)2 ≪
m2φ . The same amplitude is obtained from a local operator (χ̄χ) (ℓ̄ℓ) with a Wilson
coefficient gSχ gSℓ /m2φ (in the notation used in Eqs. (5.1), (5.2) we have cχS = gSχ , cℓS =
gSℓ , Λ = mφ ).
In the case of scalar DM, there is only one dimension five operator. The effective
Lagrangian is given by
1
Leff = G5 (χ† χ) ℓ̄(dS + idP γ5 )ℓ
The relation σ µν γ5 =
T ⊗ AT = AT ⊗ T .
i µναβ
ǫ
σαβ
2
with
G5 =
1
.
Λ
(5.4)
implies that the AT ⊗ AT coupling is equivalent to T ⊗ T , and
73
Chapter 5 Indirect versus direct Dark Matter detection
5.1.2 Dark Matter scattering on electrons
To simplify the discussion, we investigate the DM scattering on electrons at rest. This
will enable us to see for which types of Lorentz structures in the effective DM-lepton
Lagrangian, Eq. (5.1), this interaction is relevant. We comment in Sect. 5.1.6 on the
complications introduced by the fact that electrons are actually bound in atoms.
We consider a DM particle χ of mass mχ scattering elastically on a free electron at
rest, assuming that all the particles are non-relativistic. The scattering cross sections for
fermionic DM are then:
2
2
χ e 2 me v
χ e 2
χ e 2
0
+
scalar-type:
σ = σe (cS cS ) + (cS cP ) + (cP cS ) 2
mχ 2
(cχ ce )2 m2e 4
v
,
(5.5)
+ P P
3
m2χ
v2
vector-type:
σ = σe0 (cχV ceV )2 + 3 (cχA ceA )2 + (cχV ceA )2 + 3 (cχA ceV )2
, (5.6)
2
tensor-type:
σ = σe0 12 c2T + 6 c2AT v 2 .
(5.7)
In the above expressions there are two suppression factors, the DM velocity in our halo
v ∼ 10−3 c and the ratio me /mχ . The cross section for each Lorentz structure is given
to leading order in these expansion parameters. Up to the velocity or electron mass
suppression the typical size of the scattering cross section is
−4
G2 m2e
Λ
m2e
0
−39
2
σe ≡
.
(5.8)
=
≈ 3.1 × 10
cm
π
πΛ4
10 GeV
For scalar DM the χe scattering cross section is induced by the dimension 5 operator,
Eq. (5.4), giving
d2
0
σ = σe,5
d2S + P v 2 ,
(5.9)
2
with
0
σe,5
1 m2e
G2 m2
= 7.7 × 10−42 cm2
≡ 5 2e =
4π mχ
4πΛ2 m2χ
Λ
10 GeV
−2 mχ −2
.
100 GeV
(5.10)
Compared to fermionic DM two powers of Λ are replaced by mχ which typically is larger
than Λ. The scalar DM scattering cross section is thus further suppressed compared to
the fermionic case for given Λ. The results of Eqs. (5.5)÷(5.7) and (5.9) are summarized
in the middle column of Tab. 5.1.
5.1.3 Signals in direct detection experiments
When a leptophilic DM particle interacts in a detector, it is possible to have the following
types of signals (see also Ref. [194]):
74
5.1 Leptophilic Dark Matter
Γχ ⊗ Γℓ
S⊗S
S⊗P
P ⊗S
P ⊗P
V ⊗V
V ⊗A
A⊗V
A⊗A
T ⊗T
AT ⊗ T
Γℓ
S
P
fermionic DM
1
σ(χN → χN )/σN
σ(χe → χe)/σe0
2
1
αem
[2-loop]
2
O(v )
−
2 v2
O(re2 v 2 )
αem
[2-loop]
O(re2 v 4 )
−
1
1
[1-loop]
O(v 2 )
−
O(v 2 )
v2
[1-loop]
3
−
12
qℓ2
[1-loop]
2
2
−2
O(v )
qℓ v
[1-loop]
scalar DM
0
1
σ(χe → χe)/σe,5
σ(χN → χN )/σN,5
2
1
αem
[2-loop]
2
O(v )
−
Table 5.1: Scattering cross section suppression by small parameters for DM-electron
scattering and loop induced DM-nucleon scattering for all possible Lorentz structures.
Here, v ∼ 10−3 c is the DM velocity, re = me /mχ , and qℓ = mℓ /mN (ℓ = e, µ, τ ).
0 , σ1 , σ1
The reference cross sections σe0 , σe,5
N
N,5 are defined in Eqs. (5.8), (5.10), (5.24).
χ
ℓ
The couplings c , c , d have been set to one. The entries for χN → χN are orders of
magnitude estimates.
1. WIMP–electron scattering (WES): The whole recoil is absorbed by the electron
that is then kicked out of the atom to which it was bound.
2. WIMP–atom scattering (WAS): The electron on which the DM particle scatters
remains bound and the recoil is taken up by the whole atom. The process can
either be elastic (el-WAS) in which case the electron wave function remains the
same, or inelastic (ie-WAS), in which case the electron is excited to an outer shell.
3. Loop induced WIMP–nucleus scattering (WNS): Although per assumption DM
couples only to leptons at tree level, an interaction with quarks is induced at
loop level, by coupling a photon to virtual leptons, see Fig. 5.2. This will lead to
scattering of the DM particle off nuclei.
The WES produces a prompt electron and possibly additional Auger electrons or Xrays. This leads to a signal in scintillation detectors such as DAMA, but is rejected in
nuclear recoil experiments like CDMS and XENON. In the other two cases, instead, the
signal consists of a scattered nucleus and shows up in all direct detection experiments
searching for DM nuclear recoils. Note that quenching and channeling (see Sect. 2.3.1) is
75
Chapter 5 Indirect versus direct Dark Matter detection
relevant in DAMA in the cases of WAS and WNS , while the scattered electrons in the
case of WES produce unquenched scintillation light.
The event rate in direct detection experiments is proportional to the differential cross
section dσ/dER , where
ER = Eχ − Eχ′ ,
(5.11)
is the energy deposited by the WIMP in the detector. The DAMA annual modulation
signal is observed at ER ≃ 3 keVee. Also for other direct detection experiments typical
values are in the few to tens of keVee range. Just from kinematics the cross section is
proportional to
dσ
∝ G2 me (G2 mN ) for
dER
WES (WAS, WNS) ,
(5.12)
where G is defined in Eq. (5.1) and me (mN ) is the electron (nucleus) mass. This suppresses the WES induced event rate by a factor me /mN with respect to WAS and WNS.
In order for WES to deposit ∼ keV energy in the detector, the electron that a WIMP
scatters off has to have quite a high momentum. Indeed, the maximal detectable energy
from DM scattering on electrons at rest is 2me v 2 , with typical DM velocities of v ∼ 10−3 c.
Hence, the maximal detectable energy is of order eV, far too low to be relevant for the
DAMA signal at few keV. Therefore, one has to explore the scattering off bound electrons
with non-negligible momentum [193]. In this case, the energy transfer to the detector is
ER ∼ O(pv), and an electron momentum p ∼ MeV is required to obtain ER ∼ keV. Since
electrons are bound in the atom, there is a nonzero but small probability that it carries
such high momentum. The detailed calculations of Ref. [14] show that the suppression
factor from the wave function is given by the expression
Z
p
dp p
ǫWES = 2me (ER − EB ) (2l + 1)
|χnl (p)|2 ∼ 10−6 .
(5.13)
(2π)3
The integral is over MeV momenta, while χnl (p) is the momentum wave function of the
shell nl with the binding energy EB .
Similarly, ie-WAS is also suppressed by the overlap of atomic wave functions of the
initial and final states of the electron [14]:
X X
′
|hn′ l′ m′ |ei(k−k )x |nlmi|2 ∼ 10−19 .
(5.14)
ǫWAS =
nlm n′ l′ m′
We will show in Sect. 5.1.6 that, for the cases in which WAS is relevant, the el-WAS can
be safely neglected and only the ie-WAS should be taken into account.
Loop induced WNS does not suffer from any wave function suppression, but instead
carries a loop factor. At 1-loop the suppression is of order (αem Z/π)2 , with Z being the
charge number of the nucleus. Combining this with Eqs. (5.12), (5.13), (5.14), we obtain
the following rough estimate for the ratios of ie-WAS, WES and WNS induced event
rates (neglecting order-one factors but also possible different v dependences):
me
αem Z 2
WAS
WES
WNS
R
:R
:R
∼ ǫWAS : ǫWES
:
∼ 10−17 : 10−10 : 1 ,
(5.15)
mN
π
76
5.1 Leptophilic Dark Matter
χ, kµ′
N , p′µ
ℓ , (k − k + q)µ
−
′
γ,(k − k ′ )µ
ℓ− , qµ
N , pµ
χ, kµ
χ, kµ′
N , p′µ
χ, kµ′
N , p′µ
ℓ−
ℓ−
γ
γ
q
q
γ
γ
χ, kµ
ℓ−
ℓ−
N , pµ
χ, kµ
N , pµ
Figure 5.2: DM-nucleus interaction induced by a charged lepton loop and photon exchange at 1-loop (top) and 2-loop (bottom).
where in the last step we used mN = 100 GeV and Z = 53. We conclude that whenever
a loop induced cross section is present it will dominate the rate in direct detection
experiments. This holds for 1-loop as well as 2-loop cross sections, since the latter will
be suppressed by another factor (αem Z/π)2 ≃ 5 × 10−6 Z 2 relative to 1-loop, and hence
they are still much larger than the WES contribution.
5.1.4 Loop induced interactions
We have assumed that DM is leptophilic, so that at scale Λ only operators connecting DM
to leptons, Eqs. (5.1), (5.2), (5.4), are generated. However, even under this assumption,
at loop level one does induce model independently also couplings to quarks from photon
exchange between virtual leptons and the quarks. The diagrams that can arise at one
and two-loop order are shown in Fig. 5.2.2 The lepton running in the loop can be either
an electron or any other charged lepton to which the DM couples.
The one loop contribution involves the integral over loop momenta of the form
#
"
Z
d4 q
q/′ + mℓ µ q/ + mℓ
γ 2
,
(5.16)
Tr Γℓ ′ 2
(4π)4
q − m2ℓ
q − m2ℓ
2
Similar diagrams with a photon replaced by a Z or a Higgs boson are power suppressed by
2
(k − k′ )2 /MZ,H
and thus negligible.
77
Chapter 5 Indirect versus direct Dark Matter detection
with q ′ = k − k ′ + q and k, k ′ the incoming momenta as denoted in Fig. 5.2 and Γℓ the
Dirac structures given in Eqs. (5.2), (5.4). The one loop contribution is non-zero only for
vector and tensor lepton currents, Γℓ = γµ , σµν . For the scalar lepton current, Γℓ = 1,
the loop integral vanishes, reflecting the fact that one cannot couple a scalar current to
a vector current. The DM-quark interaction is then induced at two-loops through the
diagrams shown in Fig. 5.2. In contrast for pseudo-scalar and axial vector lepton currents,
Γℓ = γ5 , γµ γ5 , the diagrams vanish to all loop orders. One insertion of γ5 gives either zero
or a fully anti-symmetric tensor ǫαβνµ . Since there are only three independent momenta
in a 2 → 2 process, two indices need to be contracted with the same momentum, yielding
zero.
The cross section for scattering of a non-relativistic DM particle χ with mass mχ on
a nucleus at rest having a mass mN is
dσ
|M|2
=
,
dER
32πmN m2χ v 2
(5.17)
with M the matrix element for χN → χN scattering. For the explicit calculation of the
1-loop and 2-loop cross sections for all the non vanishing cases we refer to Ref. [14]. Here
we will discuss only the case of the vector DM-lepton interaction, since is the only one
important for the subsequent discussion.
For vector type interaction between leptons and DM, Lℓ = G(χ̄Γµχ χ)(ℓ̄cℓV γµ ℓ), with
Γµχ = (cχV + cχA γ5 )γ µ , the matrix element for χN → χN scattering, generated through
the one loop diagram of Fig. 5.2, is
X
(1)
Qi q̄i γµ qi |N (p)i
M = CV (µ) ū′χ Γµχ uχ hN (p′ )|
i
(5.18)
(1)
′
′ µ
= CV (µ) ūχ Γχ uχ ZF (ER ) ūN γµ uN .
The sum is over the light quarks qi with charges Qi , F (ER ) is the nuclear form factor,
(1)
and CV (µ) is the 1-loop factor calculated in the MS scheme
Z 1
h −x(1 − x)q 2 + m2 − i0 i
2αem ℓ
ℓ
dxx(1 − x) log
GcV
=
,
(5.19)
π
µ2
0
√
where q 2 ≃ −κ2 with κ = 2mN ER being the momentum transfer. In the calculation
we set µ = Λ, with Λ ∼ 10 GeV, since this corresponds roughly to the scale Λ, where
our effective theory is defined.
Even though in our numerical analysis we use the full 1-loop result, we give in the
following also analytic result for the cross section in the case of the vector DM-lepton
interaction, considering the “leading log” approximation, neglecting the remaining logarithmic dependence on momentum transfer. For mℓ ≫ κ one can neglect the momentum
transfer in the integral of Eq. (5.19), giving an approximate expression
(1)
CV (µ)
CVLL (µ) =
78
αem ℓ
GcV log m2ℓ /µ2 ,
3π
(5.20)
5.1 Leptophilic Dark Matter
which is very precise for muon and tau running in the loop. It is quite precise also for the
electron, even though me ∼ κ. The reason is that there is still a hierarchy me ≪ µ ≃ Λ.
Expanding also in the χ velocity v to first non-zero order, the differential cross section
dσ/dER results
m2 i2 1 n
h
dσ 1 h
m2N io 2
dσ
χ ℓ 2
χ ℓ 2 2
2
ℓ
F (ER ) , (5.21)
(c
c
)
+
(c
c
)
v
+
v
= N log
2
−
V
V
d
V
A
dER dER
µ2
9
µ2N
where the 1-loop cross section prefactor is
1
dσN
mN αem Z 2
G .
=
dER
2π v 2
π
(5.22)
In the previous formula, the parameters mN and Z are the nucleus mass and charge,
respectively, while µN = mN mχ /(mN + mχ ) is the reduced mass of the two-particle
system. The two
p small parameters are the χ velocity v and the velocity of the recoiled
nucleus, vd = 2ER /mN . The kinetic recoil energy of the nucleus ER in the χN → χN
scattering, cf. Eq. 5.11, has a size ER ∼ keVee.
We also report the result for the total χN → χN cross section, integrated over the
recoil energy ER . For simplicity we neglect the dependence on the nuclear form factors
and set F (ER ) = 1 for this comparison, giving
h
m2 i2 1 n
h
1 µ2N io
χ ℓ 2 2
χ ℓ 2
1
ℓ
σ =σN
log
c
)
v
c
)
+
(c
(c
,
1
+
V
V
A
V
µ2
9
2 m2N
(5.23)
1 is the integral of the differential cross section of Eq. (5.22)
where σN
1
σN
µ2 αem Z 2
G ≈ 1.9 × 10−32 cm2
= N
π
π
Λ
10 GeV
−4 µN 2
10 GeV
Z
53
2
.
(5.24)
The above result and the ones for the other interaction types (see Ref. [14] for the explicit
calculations) are summarized in Table 5.1, facilitating comparison with χ scattering on
free electrons. In Table 5.1 we took µN ∼ mN ∼ mχ , while the scaling for other values of
nucleon and DM masses is easy to obtain from above results. In Table 5.1, we also report
1 /dE and σ 1
the case of scalar DM where dσN,5
R
N,5 are given, respectively, by Eq. (5.22)
and Eq. (5.24) with G → G5 /(2mχ ).
5.1.5 Discussion of Lorentz structure
In Sect. 5.1.3, we have estimated a strong hierarchy between the three types of signals
as RWAS ≪ RWES ≪ RWNS , see Eq. (5.15). These results imply that whenever WNS at
1-loop or 2-loop is generated, it dominates the event rate in direct detection experiments.
The Lorentz structures for which this situation applies can be read off from Table 5.1.
To be specific we will use as a representative example of this class the V ⊗ V coupling.
From the table we also see that there is one case — the A ⊗ A coupling — where no χN
79
Chapter 5 Indirect versus direct Dark Matter detection
scattering is induced at loop level and moreover the WIMP-electron cross section is not
additionally v and/or me /mχ suppressed. Hence, we chose the A ⊗ A coupling as our
second representative example to quantitatively discuss the case of a WES dominated
event rate. The results from these two examples can be qualitatively extrapolated to
other Lorentz structures using Table 5.1.
In the case of axial vector like DM-lepton coupling, the signal in DAMA will be dominated by WES. Then, WAS is still irrelevant for DAMA, but since WES will not
contribute to the rate in CDMS and XENON, WAS might in principle lead to a signal
in those experiments. The χe → χe cross section in the A ⊗ A case has to be very large
(corresponding to Λ ∼ O(100 MeV)) in order to be relevant for the DAMA experiment.
For the cases in Table 5.1 where σe0 is further suppressed by small numbers, like for example S ⊗ P or P ⊗ P , the scale Λ would have to be even lower, so that the effective
field theory description would break down.
Finally, let us mention the tensor coupling T ⊗ T , where the 1-loop cross section is
suppressed by m2ℓ /m2N , while χe scattering is enhanced by a factor 12. If DM couples only
to the electron and not to µ and τ the suppression of the loop is of order m2e /m2N ∼ 10−10 ,
and hence, WES and WNS rates can be of comparable size. However, in general one
expects also a coupling to the µ and τ leptons. To be specific, in our numerical analysis
of V ⊗ V and A ⊗ A cases we will assume equal couplings to all three leptons. For the
tensor case the same choice would mean that WNS dominates.
5.1.6 Event rates
In this Section we provide the event rates in direct detection experiments. For WES
and WAS we assume A ⊗ A coupling and for WNS we take V ⊗ V . As argued above,
the A ⊗ A and V ⊗ V cases are representative enough to cover qualitatively all possible
Lorentz structures. Here we report the main important formulae, that have been used
for the numerical fits to DAMA, CDMS, and XENON data, while technical details and
supplementary information are given in Ref. [14]. In our numerical analysis we fix the
local DM density to the standard value ρχ = 0.3 GeV cm−3 and we consider a MaxwellBoltzmann distribution with velocity dispersion v0 = 220 km s−1 .
WIMP-electron scattering
To obtain an expression for the event rate in the case of WES it is necessary to take into
account the fact that electrons are bound to the atoms. The kinematics of scattering off
bound electrons has some important differences compared to scattering off free particles.
2
= p2 + m2 .
The bound electron does not obey the free-particle dispersion relation Ee(free)
Instead it has a fixed energy Ee = me − EB , determined by the binding energy of the
atomic shell, EB ≥ 0, whereas its momentum p follows a distribution which is given by
the square of the Fourier transform of the bound state wave function corresponding to
that shell. Energy conservation reads in this case Eχ + me − EB = Eχ′ + Ee′ , or
Ee′ = me + ER − EB .
80
(5.25)
5.1 Leptophilic Dark Matter
After some algebra, it is possible to arrive at the following expression for ER :
ER ≈ −
p2
− pv cos θ ,
2mχ
(5.26)
where3 cos θ = kp/kp. In the derivation we used the approximation ER ≪ me ≤ Ee ≪
mχ and v ∼ 10−3 . We see that to obtain detectable energies relevant for DAMA (ER of
few keV), electron momenta of order MeV are required.
Taking into account the peculiarities of scattering on bound electrons, the count rate
for the axial vector Dirac structure, Γχ = Γe = A, is (we also set cχA = ceA = 1 for
simplicity) [14]:
Z
η 3ρχ me G2 X q
dRWES
dp p
WES
|χnl (p)|2 I(vmin
),
≃
2me (ER − EB,nl ) (2l + 1)
dER
ρdet 4πmχ
(2π)3
nl
(5.27)
where η is the number density of the target particles, ρdet is the mass density of the
detector, and χnl (p) is the momentum wave function of the electron. The function
I(vmin ) is
Z
f (v)
θ(v − vmin ) ,
(5.28)
I(vmin ) ≡ d3 v
v
while the minimal velocity required to give detectable energy ER follows from Eq. (5.26):
WES
vmin
≈
ER
p
+
.
p
2mχ
(5.29)
For mχ & 10 GeV and p of order MeV the first term dominates.
The sum in Eq. (5.27) is over the atomic shells of both iodine and sodium with quantum numbers nl, and EB,nl is the corresponding binding energy. The electron can only
be kicked out of its atomic shell if its binding energy is smaller than the total energy
deposited in the detector (cf. Eq. (5.25)):
ER ≥ EB,nl .
(5.30)
Only the shells satisfying this requirement can contribute to the event rate in Eq. (5.27).
The dominant contribution to WES in DAMA comes from the inner s-shells of iodine
because these are largest at high p [14]. Electrons from the 1s, 2s, 2p shells do not
contribute to the DAMA signal region of ER ≃ 2 − 4 keVee since the binding energies
are too large, respectively 33.2 keV, 5.2 keV, and 4.7 keV [197]. The shell dominating
the signal in the 2-4 keVee region is the 3s shell of iodine, with a binding energy of about
1 keV. This has been overlooked in Ref. [193], while it has important consequences on
the size of the needed cross section.
3
We always denote the DM momentum with k and the electron (or nucleus) momentum with p. Bold
symbols refer to 3-vectors and k ≡ |k|, and similar for p.
81
Chapter 5 Indirect versus direct Dark Matter detection
WIMP-atom scattering
We now consider the case when the electron on which the DM particle scatters remains
bound and the recoil is taken up by the whole atom. We specialize to the case of axial
vector coupling, Γµχ = Γµe = γ µ γ 5 and set cχA = ceA = 1. We use non-relativistic spinors,
′
which is certainly justified for urχ and urχ , and also for use except, perhaps, for electrons
from the 1s shell of iodine. In this last case, relativistic corrections are of order 20%.
Let us first consider the case when the electron remains in its state, and hence the
scattering on the atom is elastic (el-WAS). Then we have s = s′ and nlm = n′ l′ m′ .
Furthermore, we have to sum coherently over all shells and electron spins, since it is
impossible in principle to identify on which electron
has scattered. It turns
P thes WIMP
µ γ 5 us vanishes. This can be
ū
γ
out that for the axial vector case the spin sum
e
s e
verified by using explicit expressions for the spinors use , and follows from the fact that
the different signs due to γ5 of right-handed and left-handed components of the electron
cancel each other in case of a coherent sum over spins.4 The elastic scattering may be
relevant for other Lorentz structures where this cancellation does not occur. However, in
Sec. 5.1.5 we have argued that the only case of practical relevance is the axial coupling,
and therefore we will not consider el-WAS further.
We are left now with the case where the electron is excited to an outer free shell
which corresponds to inelastic WIMP-atom scattering (ie-WAS). In this case the sum
over all occupied electron states nlm, over all unoccupied states n′ l′ m′ , and over WIMP
and electron spins has to be incoherent because one can distinguish in principle different
initial and final states, e.g. by x-ray spectroscopy. The corresponding expression for the
counting rate is [14]
ie-WAS
dRN
η mN 3ρχ G2 X X
′
ie-WAS
=
|hn′ l′ m′ |ei(k−k )x |nlmi|2 I(vmin
),
dER
ρdet 2π mχ
′ ′ ′
(5.31)
nlm n l m
with mN the mass of the target nucleus. The function I is defined in Eq. (5.28), and
the minimal velocity required to give detectable energy ER follows from the kinematics
2 /2, and momentum
implied by energy conservation, ER = Eχ − Eχ′ = δEB + mN vN
′
conservation, k = k + mN vN :
ie-WAS
vmin
=
ER (mχ + mN ) − mN δEB
p
,
mχ 2mN (ER − δEB )
(5.32)
where δEB is the difference of the binding energies of the initial and final shells: δEB =
EB,nlm − EB,n′ l′ m′ .
4
This argument will not hold if an unpaired valence electron is available so that we cannot sum over
spins. However, most chemically bound systems are formed in such a way that this does not happen.
Even in this case el-WAS would be suppressed since scattering on outer electrons is highly suppressed
by the smallness of the binding energy of these electrons compared to the transferred momentum.
82
5.1 Leptophilic Dark Matter
Loop induced WIMP-nucleus scattering
The event rate for loop induced DM-nucleus scattering is given by [14]
ρχ η
dσN 2
dRWNS
WNS
=
v I(vmin
).
dER
mχ ρdet dER
(5.33)
In this case the minimal velocity to produce
a detectable energy ER is given for WIMPq
WNS
nucleus elastic scattering by vmin = ER mN /2µ2N .
We now specialize to the V ⊗ V case. The event rate depends on the χ mass and the
coupling constant of the effective operator G (we set cχV = cℓV = 1 from now on). For
easier comparison with previous works, it is useful to trade G for the total χe → χe
cross section σe0 = G2 m2e /π, Eq. (5.8). For the V ⊗ V case, considering the leading log
approximation, we have
h
m2 i2
dσN 2
mN αZ 2
2
ℓ
F
(E
)
,
log
v = σe0 ×
R
dER
18m2e π
µ2
(5.34)
to be inserted in Eq. (5.33). Furthermore, we assume (somewhat arbitrarily) equal couplings to all three leptons. The logarithm in Eq. (5.34) implies then a relative contribution of e : µ : τ ≃ 30 : 7 : 1. Note that the rate is dominated by the contribution from
the electron in the loop assuming equal couplings at the scale Λ ∼ 10 GeV. Therefore,
our results are conservative, in the sense that per assumption DM has to couple to the
electron.
5.1.7 Super-Kamiokande constraints
Any DM candidate, that is considered in a theoretical model, has to fulfill the constraints
on the upward through-going muons coming from water Cherenkov detectors, like SuperKamiokande [5], and from neutrino telescopes [6, 166, 167]. Here, we reanalyze, in the
framework of leptonically interacting DM, the bound on the muon flux coming from
the DM annihilations inside the Sun, provided by the Super-Kamiokande experiment.
We do not consider the possibility of annihilation inside the Earth, since in this case
the equilibrium between capture and annihilation rate generally depends on the specific
characteristics of the model, because of the weaker gravitational field with respect to the
Sun.
For our calculations we initially assume that the capture and annihilation processes are
in equilibrium, i.e. τA ≪ t⊙ , where t⊙ is the age of the Sun. In this case, the annihilation
rate is just half the capture rate and becomes independent of the annihilation cross section
hσann vi. We will later comment on the validity of the equilibrium limit for our model.
As discussed in Sect. 3.1.1, the temperature effect on the capture rate can be neglected
for scattering off heavier nuclei, which dominates the capture in the case of loop induced
WNS, while it has to be considered when WES is dominant.
Since we are interested in annihilations into leptons, we consider the following four
channels: τ τ̄ , νe ν̄e , νµ ν̄µ and ντ ν̄τ . Note that annihilations into electrons do not provide
83
Chapter 5 Indirect versus direct Dark Matter detection
neutrinos, and muons are always stopped before decay, giving rise to neutrinos in the
MeV energy range which is below the Super-Kamiokande threshold [198]. In the case
of direct neutrino channels, the initial neutrino spectrum is simply a Dirac δ function
centered at Eν = mχ , and we assume a flavour-blind branching ratio, i.e., BRνe =
BRνµ = BRντ = 1/3. The results do not depend strongly on this assumption, since
flavours are mixed due to oscillations.5 For the τ τ̄ channel, we use the initial neutrino
spectrum given in [140].
Using the SK limit on through-going muons of Eq. (3.39), an upper bound on the DM
scattering cross section as a function of mχ can be obtained. In Fig. 5.3 we display the
bounds obtained for the case of loop induced WIMP-nucleus scattering (upper panel) and
WIMP-electron scattering (lower panel). For the calculation of loop induced scattering
we used q 2 ≃ −O(m2χ v 2 ). We show the limit for annihilations into τ τ̄ and ν ν̄ (assuming
equal branchings into the 3 flavours) starting from mχ & 10 GeV, since for lower masses
a great part of the muon signal would be in the form of stopping muons and thus a more
carefully calculation should be pursued.
In the case of WNS, annihilations into neutrinos exclude the region compatible with
DAMA, while annihilations into tau leptons might be marginally consistent with it at
3σ. In contrast, in the case of WES the neutrino bound excludes the region indicated by
DAMA by more than 6 orders of magnitude. This implies that if DM couples to electrons
with a cross section as large as necessary to explain the DAMA results through WES ,
DM annihilation into neutrinos must be very strongly suppressed.
We now wish to comment on the validity of the equilibrium assumption, between
WIMP captures and annihilations in the Sun. Let us first estimate the cut-off scale Λ
for the effective theory description of the DM-lepton coupling. For the two examples
of V ⊗ V and A ⊗ A couplings, the neutrino bounds are of order σe0 ∼ 10−43 cm2 and
10−38 cm2 , respectively, see Fig. 5.3. From Eq. (5.8) we can estimate the corresponding
cut-off scales as ΛV ∼ 100 GeV and ΛA ∼ 10 GeV, where we took coupling constants
cχi to be of order O(1). In DM annihilations the four-momentum transfer squared is of
order m2χ . For mχ ∼ 10 GeV, relevant for WNS, the WIMP annihilations may then also
be described by effective field theory. Using effective interactions in Eq. (5.1) (extending
them to neutrinos), we find
Vector: hσann vi ∼
G2 m2χ
m2χ
= σe0 2 ∼ 10−24 cm3 s−1
π
me
σe0
−43
10
cm2
mχ 2
. (5.35)
10 GeV
In the WES case, however, the effective theory typically cannot be applied since the
momentum transfer for annihilations is above the cut-off scale. Therefore, in general
we cannot make model independent statements about hσann vi without specifying the
5
There is some difference of the ντ ν̄τ -channel due to ντ regeneration effects [140, 141], which are
important for high energies. Assuming annihilations with branching ratios equal to one for each of the
three flavours we find that the muon neutrino flux at the Earth is practically the same for all three
initial flavours up to mχ ≃ 100 GeV. For mχ = 1 TeV the ratio of the muon neutrino fluxes at Earth
is roughly 1 : 3.5 : 6.4 for annihilations into νe ν̄e : νµ ν̄µ : ντ ν̄τ .
84
5.1 Leptophilic Dark Matter
H90%L
Ge H90%L
Σ0e @cm2 D
CDMS-II
10-43
XENON 10
10-42
Leptophilic DM, Vector interactions
Scattering on nuclei ž 1 loop
DAMA H90%3ΣL
no channeling
DAMA
H90%3ΣL
10-44
SK, ΧΧ ® ΤΤ H90%L
SK, ΧΧ ® ΝΝ H90%L
10-45
10-46
101
102
m Χ @GeVD
103
Leptophilic DM, Axial vector interactions
Scattering on bound electrons
-25
Σ0e @cm2 D
10
10 H90%L
CDMS-II Ge
H90%L
XENON 10
10-30
DAMA
H90%3ΣL
10-35
SK, ΧΧ ® ΤΤ H90%L
SK, ΧΧ ® ΝΝ H90%L
10-40
1
10
102
m Χ @GeVD
103
Figure 5.3: DAMA allowed region at 90% and 3σ C.L. in the case of 1-loop induced
WIMP-nucleus scattering (V ⊗ V coupling) and in the case of WIMP-electron scattering
(A ⊗ A coupling). The allowed region is shown in terms of the WIMP-electron cross
section σe0 = G2 m2e /π, with and without taking into account the channeling effect. The
bounds at 90% C.L. from CDMS-II and XENON10 are displayed. The dashed curves
show the 90% C.L. constraints from the Super-Kamiokande limit on neutrinos from the
Sun, by assuming annihilation into τ τ̄ or ν ν̄. Note how neutrino bounds are much
stronger than the ones from direct detection experiments, in the case of A ⊗ A coupling.
UV completion of the effective χℓ vertex. An order of magnitude estimate can still be
obtained from dimensional analysis as
Axial:
hσann vi ∼
−2
m
g4
χ
−21
3 −1
4
,
∼
10
cm
s
×
g
m2χ
100 GeV
(5.36)
85
Chapter 5 Indirect versus direct Dark Matter detection
with g a typical coupling constant between leptons and the dark sector.
Equilibrium of WIMP capture and annihilations is obtained if tanh2 (t⊙ /τA ) is close
to one, see Eq. (3.8). Fig. 5.4 shows the values of hσann vi for which t⊙ /τA = 1 and 5 as a
function of mχ . The values of scattering cross sections σe0 for V ⊗ V and A ⊗ A Lorentz
structures were chosen to be above (but close to) the Super-Kamiokande bounds shown in
Fig. 5.3. Since tanh2 x ≈ 1 for x & 5, WIMP capture and annihilations are in equilibrium
in the Sun for values of hσann vi above the curve for t⊙ /τA = 5. Comparing Eqs. (5.35)
and (5.36) with the ranges shown in the figure we conclude that the assumption of
equilibrium is very well justified in the cases of our interest.
In this study we have considered the hypothesis that DM has tree level couplings only
to leptons but not to quarks and within this framework we have derived the bounds on
the scattering cross sections coming from the indirect DM detection through neutrinos,
carried out by the SK experiment.
By closing the lepton legs to a loop, we obtain a coupling to the charge of the nucleus
by photon exchange. Whenever the Dirac structure of the DM-lepton coupling allows
such a diagram at 1 or 2-loop, WIMP-nucleus scattering will dominate the event rate
in direct detection experiments, since the scattering over electrons is highly suppressed
by the high momentum tail of the bound state wave function. The WIMP capture by
the Sun is also dominated by this induced coupling to nucleons and the capture rate on
electrons is negligible, cf. also Fig. 3.2. Concerning the direct detection experiments, a
DM-lepton vector like coupling leads to a situation very similar to the standard WIMP
case, implying the well-known tension between the annually modulated scintillation signal
in DAMA and the bounds from CDMS and XENON. In this case, the indirect detection
in neutrinos provides limits that are competitive with the ones from direct detection
experiments, see Fig. 5.3 .
If the DM-lepton coupling is axial vector like, no loop will be induced and hence
the scattering proceeds only by the interaction with electrons bound to the atoms of
the detector. This model is strongly disfavored by the indirect DM search with neutrinos, because the cross section required to explain the DAMA signal is ruled out by
the Super-Kamiokande constraints by many orders of magnitude, see Fig. 5.3. Moreover,
the predicted spectral shape of the modulated and/or unmodulated signal in DAMA
provides a very bad fit to the data [14].
The applicability of the neutrino bounds depends of course on the assumption that
neutrinos are produced by DM annihilations. Due to SU (2)L gauge symmetry, generically one expects that DM will couple to both, charged leptons and neutrinos, which
would open the annihilation channel into ν ν̄. If for some reason DM couples only to
charged leptons, DM would generically also annihilate into τ τ̄ , leading again to the neutrino signal. In order to evade the Super-Kamiokande constraint one has to forbid the
coupling of DM to neutrinos and to the tau lepton. Let us mention that the most generic
way to avoid coupling to neutrinos is the chiral coupling only to right-handed leptons.
Note, however, that such a chiral V + A coupling involves a vector-like coupling which
will induce DM-quark scattering via the loop diagram. In this way, the DAMA results
will be again in tension with the other direct detection experiments. Moreover, annihi-
86
5.1 Leptophilic Dark Matter
105
5.0
4
Σann v @10-32 cm3 sD
10
1.0
103
5.0
102
1.0
101
Axial Vector Type:
Σ0e
-38
= 10
cm2
0
10
10-1
10
Vector Type: Σ0e = 10-43 cm2
200
400
600
m Χ @GeVD
800
1000
Figure 5.4: Contours of t⊙ /τA = 5 and t⊙ /τA = 1. For the case of vector (axial vector)
coupling we have used a scattering cross section of σe0 = G2 m2e /π = 10−43 (10−38 ) cm2 ,
motivated by the results of the Super-Kamiokande bound. For values of hσann vi above
the curve for t⊙ /τA = 5, WIMP capture and annihilations are in equilibrium in the Sun.
lation into charged leptons generates almost model independently also annihilation into
neutrinos from W -boson exchange at 1-loop. Thus annihilation into neutrinos is typically
suppressed by a loop factor of O(10−4 ) compared to annihilation into charged leptons,
that however does not compensate the gap of more than 6 orders of magnitude between
the DAMA region and the SK bounds. This consideration rules out all leptophilic DM
models with dominant direct annihilation into leptons as an explanation of DAMA.
A possible realistic way to evade the bound from annihilations would be to assume
that DM is not self-conjugate and postulate the presence of a large χ − χ̄ asymmetry in
our halo, see e.g. Refs. [199, 200, 201].
In conclusion, we have shown that the hypothesis of DM-interactions only with leptons
does not provide a satisfactory solution to reconcile the DAMA annual modulation signal
with constraints from other direct detection experiments. In the scenario of vector like
coupling we recover the tension existing between DAMA and the other direct detection
experiments, while in the case of axial vector coupling the bounds from indirect detection
in neutrinos result extremely strong and hard to escape. However, even if the leptophilic
DM candidate does not represent a successful explanation of the DAMA results, it is
not ruled out as a whole. Indeed, it might still be a well motivated model to explain
the PAMELA anomaly in the cosmic ray flux. In this case, our work provides a carefully description of its phenomenology for both direct detection experiments and indirect
searches with neutrinos.
87
Chapter 5 Indirect versus direct Dark Matter detection
5.2 Neutralino Dark Matter
In the papers of Ref. [103] it was shown that light neutralinos with a mass in the range
7 GeV . mχ . 50 GeV are interesting Dark Matter candidates, with events rates
accessible by direct detection experiments. This population of light neutralinos arises in
the MSSM when the unification of gaugino masses at the GUT scale is not assumed [202],
see Sect. 5.2.1 for more details. In this supersymmetric framework the lower bound on
the neutralino mass of about 7 GeV is set by a cosmological bound on the neutralino
relic density [103]. This is at variance with the lower bound mχ & 50 GeV, which is
derived from the LEP2 lower limit on the chargino mass, within the MSSM with gaugino
mass unification at the GUT scale, cf. Sect. 2.4.1.
It was proved in Refs. [57, 203] that the population of light neutralinos fitted very well
the results from the DAMA/NaI experiment [53], independently of the possible presence
of channeled events. This good agreement has been further confirmed in Ref. [58], using
the DAMA/LIBRA combined data [3].
The neutrinos produced by pair-annihilations of light neutralinos captured in the Earth
and the Sun were discussed in Ref. [204]. We reconsider that analysis, by implementing
and extending it in various distinctive features.
In the calculation of the neutrino flux we include all the main processes that occur
during the neutrino propagation, i.e. neutrino oscillations and neutrino incoherent interactions with matter. The muon events are then divided in through-going and stopping
muons, using the geometry of the Super-Kamiokande detector. The last category of
events was not considered before in the literature and, actually, this turns out to be the
most promising possibility to constrain the parameter space.
In the evaluation of the signals, we take also into account the relevant particle-physics
uncertainties in hadronic quantities and astrophysics uncertainties, which affect the capture rate of relic neutralinos by the celestial bodies. These effects are discussed in details
in Sect. 5.2.2 and Sect. 5.2.3.
Our results are given in Sect. 5.2.4 for the whole population of light neutralinos, while
the final analysis of Sect. 5.2.5 is focussed on the upward muon fluxes generated by those
neutralino configurations which are able to explain the annual modulation data of the
DAMA experiment [3]. In our study, we separate the case where the channeling effect is
included from the one where this effect is neglected. For definiteness, we consider only
the results from the DAMA experiment and we do not impose constraints coming from
other DM detection experiments, like CDMS and XENON.
5.2.1 Theoretical model
The supersymmetric scheme we employ in this analysis is the one described in Refs. [58,
103]. It is an effective MSSM scheme (effMSSM) defined at the electroweak scale, with
the following independent parameters: M1 , M2 , µ, tan β, mA , mq̃ , ml̃ and A. Notations
are as follows: M1 and M2 are the U (1) and SU (2) gaugino masses (these parameters
are taken here to be positive), µ is the Higgs mixing mass parameter, tan β the ratio of
88
5.2 Neutralino Dark Matter
the two Higgs vev’s, mA the mass of the CP-odd neutral Higgs boson, mq̃ is a squark
soft mass common to all squarks, ml̃ is a slepton soft mass common to all sleptons, and
A is a common dimensionless trilinear parameter for the third family, Ab̃ = At̃ ≡ Amq̃
and Aτ̃ ≡ Aml̃ (the trilinear parameters for the other families being set equal to zero).
In our model, no gaugino mass unification at the GUT scale is assumed. The lightest
neutralino is required to be the lightest supersymmetric particle and stable, because of
R-parity conservation.
The numerical analysis is performed by a scanning of the supersymmetric parameter space, with the following ranges of the MSSM parameters: 1 ≤ tan β ≤ 50,
100 GeV ≤ |µ| ≤ 1000 GeV, 5 GeV ≤ M1 ≤ 500 GeV, 100 GeV ≤ M2 ≤ 1000 GeV,
100 GeV ≤ mq̃ , ml̃ ≤ 3000 GeV, 90 GeV ≤ mA ≤ 1000 GeV, −3 ≤ A ≤ 3.
The supersymmetric parameter space is subjected to all available constraints due to
accelerator data on supersymmetric and Higgs boson searches (CERN e+ e− collider
LEP2 [205] and Collider Detectors D0 and CDF at Fermilab [206]) and to other particlephysics precision results, as reported in Sect. 2.3.3.
Also included is the cosmological constraint that the neutralino relic abundance does
not exceed the maximal allowed value for Cold Dark Matter, i.e. Ωχ h2 ≤ (ΩCDM h2 )max ,
with (ΩCDM h2 )max = 0.122, as derived at the 2σ level from the results of Ref. [39]. We
recall that this cosmological upper bound implies on the neutralino mass the lower limit
of about 7 GeV [103].
For each neutralino configuration, we calculate the total neutrino flux summing, with
the appropriate branching ratios, the neutrino spectra coming from the following annihilation channels: bb̄, τ τ̄ , cc̄, q q̄ and gg. The annihilation of two neutralinos can also
produce two Higgs bosons or one gauge and one Higgs boson in the final state, although
these two channels (as well as the annihilation channels into tt̄ and into two gauge bosons)
are absent in our study, since we consider neutralinos with mass mχ ≤ 80 GeV.
5.2.2 WIMP-nucleon cross section: hadronic uncertainties
In Ref. [207] it is stressed that the couplings between Higgs bosons (or squarks) with
nucleons, which typically play a crucial role in the evaluation of the neutralino-nucleus
cross section, suffer of large uncertainties [208]. Actually, these couplings are conveniently
expressed in terms of three hadronic quantities: the pion-nucleon sigma term
1
¯
σπN = (mu + md ) < N |ūu + dd|N
>,
2
(5.37)
the quantity σ0 , related to the size of the SU (3) symmetry breaking,
1
¯ − 2s̄s|N > ,
σ0 ≡ (mu + md ) < N |ūu + dd
2
(5.38)
and the mass ratio r = 2ms /(mu + md ).
Because of a number of intrinsic theoretical and experimental problems, the determination of these hadronic quantities is rather poor. Conservatively, their ranges can be
89
Chapter 5 Indirect versus direct Dark Matter detection
hadronic set
σπN [MeV]
σ0 [MeV]
r
MIN
REF
MAX
41
45
73
40
30
30
25
29
25
Table 5.2: Set of values for the hadronic quantities considered in the numerical analysis.
summarized as follows (we refer to Refs. [58, 207] for details):
41 MeV . σπN . 73 MeV ,
(5.39)
σ0 = 30 ÷ 40 MeV ,
(5.40)
r = 29 ± 7 .
(5.41)
and
In the present paper, in order to display the influence of the uncertainties due to the
hadronic quantities on the signals at neutrino detectors, we will report our results for
three different sets of values for the quantities (σπN , σ0 , r) as shown in Tab. 5.2. The set
REF corresponds to the set of value referred to as reference point in Ref. [58]. The sets
MIN and MAX listed in Tab. 5.2 bracket the range of hadronic uncertainties.
In the case where the neutralino-nucleus interaction is dominated by the exchange of
Higgs bosons, it is straightforward to estimate by how much the capture rate C is affected
by the hadronic uncertainties. Indeed, in this case the dominant term in the interaction
amplitude of the neutralino-nucleus scattering is provided by coupling between the two
CP-even Higgs bosons and the down-type quarks:
2
23
23
gd =
mN +
σπN +
r (σπN − σ0 ) ,
(5.42)
27
4
5
where mN is the nucleon mass. Then:
CMIN /CREF ≃ (gd, MIN /gd, REF )2 ,
CMAX /CREF ≃ (gd, MAX /gd, REF )2 .
(5.43)
Using the values of Table 5.2 for the three sets of hadronic quantities, one finds for gd :
gd, MIN = 99 MeV, gd, REF = 290 MeV, gd, MAX = 598 MeV, respectively. We thus conclude
that, because of the hadronic uncertainties, the capture rate in the case of set MIN is
reduced by a factor ∼ 9 as compared to the capture rate evaluated with the set REF,
whereas C, evaluated with set MAX, is enhanced by a factor ∼ 4.
The consequences over the annihilation rate ΓANN is more involved, since the capture
rate C enters in ΓANN not only linearly but also through τA . When in the celestial body
capture and annihilation are in equilibrium (t & τA ), one has ΓANN ∼ C/2; then ΓANN ,
90
5.2 Neutralino Dark Matter
as a function of the hadronic quantities, rescales as C (see Eq. (5.43)); however, when
the equilibrium is not realized, the uncertainties in ΓANN can be much more pronounced.
For instance, for t ≪ τA , ΓANN is proportional to C 2 , thus the rescaling factors for ΓANN
are the squares of those in Eq. (5.43).
These estimates will be confirmed by the numerical analysis displayed in the following
section.
5.2.3 Numerical evaluations
For the velocity distribution of relic neutralinos in the galactic halo we use, for definiteness, the standard isothermal distribution parametrized in terms of the local rotational
velocity v0 (model A1 in Ref. [51]). The local rotational velocity v0 is set at three different representative values: the central value v0 = 220 km s−1 and two extreme values
v0 = 170 km s−1 and v0 = 270 km s−1 which bracket the v0 physical range, cf. Sect. 2.2.
Associated to each value of v0 we take a value of ρ0 within its physical range established
according to the procedure described in Ref. [51]. In conclusion, we will provide the numerical results of our analysis for the following three sets of astrophysical parameters:
1) v0 = 170 km s−1 , ρ0 = 0.20 GeV cm−3 ; 2) v0 = 220 km s−1 , ρ0 = 0.34 GeV cm−3 ;
3) v0 = 270 km s−1 , ρ0 = 0.62 GeV cm−3 . Note that these values of ρ0 correspond to
the case of maximal amount of non halo components to DM in the galaxy [51].
It is however to be recalled that the actual distribution function could deviate sizably
from the isothermal one [51] or even depend on non-thermalized effects [209]. Also the
possible presence of a thick disk of DM could play a relevant role in the capture of DM
by celestial bodies [210].
The density of neutralinos ρχ can be assumed equal to the local value of the total DM
density ρ0 , when the neutralino relic abundance (Ωχ h2 ) turns out to be at the level of a
minimal (ΩCDM h2 )min consistent with ρ0 . On the contrary, when (Ωχ h2 ) is smaller than
(ΩCDM h2 )min , the value to be assigned to ρχ has to be appropriately reduced. Thus we
evaluate Ωχ h2 and we determine ρχ by adopting a standard rescaling procedure [9]:
ρχ = ρ0 ,
ρχ = ρ0
Ωχ h2
,
(ΩCDM h2 )min
when
Ωχ h2 ≥ (ΩCDM h2 )min
when
Ωχ h2 < (ΩCDM h2 )min
(5.44)
Here (ΩCDM h2 )min is set to the value 0.098, as derived at 2σ level from the results of
Ref. [39].
It is worth noticing that the neutralino density ρχ , evaluated according to Eq. (5.44),
enters not only in the capture rate C but also in parameter τA (through C). Therefore
the use of a correct value for ρχ (rescaled according to Eq. (5.44), when necessary) is important also in determining whether or not the equilibrium is already set in a macroscopic
body.
Explicit calculations over the whole parameter space show that, whereas for the Earth
the equilibrium condition depends sensitively on the values of the model parameters, in
91
Chapter 5 Indirect versus direct Dark Matter detection
the case of the Sun equilibrium between capture and annihilation is typically reached
for the whole range of mχ , due to the much more efficient capture rate implied by the
stronger gravitational field [131, 132].
The left panel of Fig. 5.5 shows the scatter plots for the ratios of the capture rates
Earth (where i = set MIN, set MAX). One sees that, as anticipated in the
CiEarth /CREF
previous section, the numerical values accumulate (most significantly for light masses),
around the numerical factors shown in Eq. (5.43).
Earth
The scatter plots for the ratios ΓEarth
ANN,i /ΓANN,REF (where i = set MIN, set MAX)
are displayed in the right panel of Fig. 5.5. As expected and discussed before, these
numerical values are much larger than those of Eq. (5.43), since many supersymmetric
configurations are not able to provide a capture-annihilation equilibrium inside the Earth.
The dependence of the annihilation rate for the Sun, ΓSun
ANN on the hadronic uncertainties is shown in Fig. 5.6. Since the capture-annihilation equilibrium is realized in the Sun
Sun
for all supersymmetric configurations of our model, one has here that ΓSun
ANN,i /ΓANN,REF =
Sun , which implies that ΓSun
Sun
Sun
Sun
CiSun /CREF
ANN,MAX /ΓANN,REF . 4 and ΓANN,MIN /ΓANN,REF & 1/9.
This is at variance with the case of the Earth which we have commented before.
Moreover, one notices from Fig. 5.6 that for many supersymmetric configurations ΓSun
ANN
depends very slightly (or negligibly) on the variations in the hadronic quantities. This
is due to the fact that on many instances the capture of neutralinos from the Sun is
dominated by spin-dependent cross-sections, due to squarks exchange.
5.2.4 Fluxes from the Earth and the Sun
For the case of neutralino annihilations in the Earth, we fix the angular opening to
−1.0 ≤ cos θz ≤ −0.9, while, for the Sun, we divide the upward muons in stopping and
through-going, using the Super-Kamiokande effective area averaged over the zenith angle.
For simplicity, we neglect the ντ regeneration effect, since it provides only a negligible
correction for the WIMPs mass range of our interest: mχ ≤ 80 GeV.
The upper panel of Fig. 5.7 displays the scatter plots for the expected muon flux
integrated over the muon energy for Eµ ≥ 1.6 GeV for the upward through-going muons.
The three columns refer to the evaluation of the fluxes using in turn the three different
set of hadronic quantities defined in Sect. 5.2.2.
The various peaks for mχ . 40 GeV are due to resonant capture of neutralinos on
oxygen, magnesium and silicon; indeed, these elements are almost as abundant in Earth
as iron, which is the most relevant target nucleus for the capture of neutralinos of higher
mass. The dip at mχ ∼ 45 GeV is a consequence of a depletion of the neutralino local
density, implied by the rescaling recipe of Eq. (5.44) and a resonant effect in the (Zexchange) neutralino pair annihilation when mχ . mZ /2 (note that the neutralino relic
abundance is inversely proportional to the neutralino pair-annihilation).
The fact that the muon signal for light neutralinos (mχ . 25-30 GeV) is lower than the
one at higher masses can be understood by considerations on the neutralino annihilation
channels. Indeed, for light masses the branching ratio of the annihilation process into
the τ τ̄ final state, which is the one with the highest neutrino yield per annihilation, is
92
5.2 Neutralino Dark Matter
Figure 5.5: Ratios of capture rates (left panel) and annihilation rates (right panel), in
the case of the Earth, calculated for the hadronic sets MIN and MAX with respect to the
hadronic set REF. The local rotational velocity is set to its central value: v0 = 220 km s−1
(ρ0 = 0.34 GeV cm−3 ).
Figure 5.6: Ratios of annihilation rates, in the case of the Sun, calculated for the
hadronic sets MIN and MAX with respect to the hadronic set REF. The local rotational
velocity is set to its central value: v0 = 220 km s−1 (ρ0 = 0.34 GeV cm−3 ).
93
Chapter 5 Indirect versus direct Dark Matter detection
suppressed. This last property being in turn due to the fact that, for these masses, the
final state in bb̄ in the annihilation cross section has to be the dominant one in order to
keep the neutralino relic abundance below its cosmological upper bound [103]. Moreover,
lower mχ masses imply softer neutrino spectra, which entail fewer muons above threshold.
The comparison of the fluxes in the three columns shows how relevant can be the role of
the size of the hadronic quantities on the final outputs. The suppression (enhancement)
of the flux in the case of the set MIN (MAX) as compared to the flux for the set REF are
set by the numerical factors previously discussed for ΓEarth
ANN . This entails that, whereas
the overall muon flux is completely below the present experimental bound in the case
of the minimal set of the hadronic quantities, some part of the spectrum would emerge
sizably above the limit for neutralino masses mχ & 50 GeV for the other sets. In the case
of set MAX, the neutralino configurations with masses mχ ∼ 15 GeV or mχ ∼ 25-30 GeV
would produce a neutrino signal higher than the SK experimental bound.
Since also the dependence of the muon signals on the astrophysical parameters v0 and
ρ0 is important, in the lower panel of Fig. 5.7 we display the through-going fluxes for the
three representative values of v0 and ρ0 which we discussed in Sect. 5.2.3. The overall
increase in the fluxes in moving from left to right is essentially due to the increase in the
value of the local DM density. In these scatter plots the hadronic quantities are set to
the value REF.
The fluxes for upward stopping-muons from the Earth are given in Fig. 5.8. The scheme
of this figure is the same as the one of the previous Fig. 5.7: the dependence of the fluxes
on the hadronic quantities can be read in the upper panel, the one on the astrophysical
parameters is displayed in the lower panel.
Because of the uncertainties affecting the evaluations of the muon fluxes, mainly due
to the hadronic quantities, we cannot convert these results in terms of absolute constraints on supersymmetric configurations. However, we can conclude that the analysis
of stopping muons from the Earth can have an interesting discovery potential not only
for masses above 50 GeV, but also for light neutralinos with mχ ∼ 15 GeV or mχ ∼ 2530 GeV. Notice however that the neutralino configurations which provide the highest
values of the muon fluxes, mainly at mχ ∼ 50-70 GeV, are actually disfavored by measurements of WIMP direct detection [211] which have their maximal sensitivity in this
mass range.
The fluxes of upward through-going muons and of stopping muons from the Sun are
provided in Fig. 5.9 and in Fig. 5.10, respectively: the dependence of the fluxes on the
hadronic quantities can be read in the upper panels of Fig. 5.9 and Fig. 5.10, the one on
the astrophysical parameters is displayed in the lower panels of the same figures.
From these results one notices that through-going muons can only be relevant for
neutralinos with masses mχ & 50 GeV or mχ ∼ 35-40 GeV, whereas stopping muons can
potentially provide information also on some supersymmetric configurations with masses
down to mχ ∼ 7 GeV, in the favourable cases of high values of the hadronic quantities
and of the astrophysical parameters.
94
5.2 Neutralino Dark Matter
Figure 5.7: Upward through-going muon flux, generated by light neutralino pairannihilation inside the Earth. The upper panel shows the dependence of the muon flux on
the hadronic quantities, for fixed values of the astrophysical parameters: v0 = 220 km s−1
and ρ0 = 0.34 GeV cm−3 . The lower panel shows the dependence of the muon flux on
the local rotational velocity v0 and the total DM density ρ0 , for the hadronic set REF.
The horizontal line represents the experimental limit on through-going muons from the
Earth obtained using the SK data, see Eq. (3.41).
95
Chapter 5 Indirect versus direct Dark Matter detection
Figure 5.8: The same as Fig. 5.7, but in the case of light neutralino pair-annihilation
inside the Earth and of upward stopping muons. In this case, the horizontal line refers to
the experimental limit on stopping muons from the Earth obtained using the SK data,
see Eq. (3.42).
96
5.2 Neutralino Dark Matter
Figure 5.9: The same as Fig. 5.7, but in the case of light neutralino pair-annihilation
inside the Sun. In this case, the horizontal line refers to the experimental limit on
through-going muons from the Sun obtained using the SK data, see Eq. (3.39).
97
Chapter 5 Indirect versus direct Dark Matter detection
Figure 5.10: The same as Fig. 5.7, but in the case of upward stopping muons. In this
case, the horizontal line refers to the experimental limit on stopping muons from the Sun
obtained using the SK data, see Eq. (3.40).
98
5.2 Neutralino Dark Matter
5.2.5 Fluxes of stopping muons for configurations compatible with the
DAMA results
Now we give the expected upward muon fluxes from the Earth and the Sun which would
be produced by neutralino configurations which fit the annual modulation data of the
DAMA experiment [3]. As before, for definiteness the analysis is performed in the framework of the isothermal sphere. The selection of the supersymmetric configurations is performed on the basis of the analysis carried out in Ref. [58]: for any set of astrophysical
parameters and hadronic quantities, from the whole neutralino population are extracted
the configurations which fit the experimental annual modulation data, and the relevant
muon fluxes are evaluated. As for the yearly modulation data, we consider both outputs
of the experimental analysis of the DAMA Collaboration: those where the channelling
effect [68] is included as well as those where this effect is neglected. We recall that
the way by which the channeling effect has to be taken into account in the analysis is
still under study; thus the actual physical outputs in the analysis of the experimental
data in terms of specific DM candidates could stay mid-way, between the case defined as
channeling and the no-channeling one, respectively.
We only report the results for stopping muons, since, as we have seen above, this is the
category of events which can provide the most sizable signals. The fluxes are calculated
varying the hadronic quantities inside their allowed ranges. Fig. 5.11 displays the fluxes
for the upward stopping muons expected from the Earth in case of no-channeling (upper
panel) and in the case of channeling (lower panel). The corresponding fluxes from the
Sun are shown in Fig. 5.12.
We note that depending on the role of channeling in the extraction of the physical
supersymmetric configurations, the stopping muon fluxes can have a discovery potential
with an interesting complementarity between the signals from the two celestial bodies:
whereas the flux from the Earth cannot give insights into neutralino masses below about
15 GeV, the flux from the Sun would potentially be able to measure effects down to
mχ ∼ 7 GeV.
It is worth remarking that under favourable conditions provided by the actual values of
the involved parameters, a combination of the annual modulation data and of measurements at neutrino detectors could help in pinning down the features of the DM particle
and in restraining the ranges of the many quantities (of astrophysical and particle-physics
origins) which enter in the evaluations and still suffer from large uncertainties. In general, we can affirm that, if the channeling effect is absent, light neutralinos might have
better possibility to be discovered through measurements of stopping muons.
We stress once more that the present analysis, for definiteness, was performed only in
the standard case of a halo DM distribution function given by an isothermal sphere. Use
of different halo distributions, such as those described in Ref. [51], could modify the role
of specific supersymmetric configurations.
We wish here to recall that indirect signals of light neutralinos could also be provided
by future measurements of cosmic antideuterons in space [212] with forthcoming airborne
experiments [87, 88]. Finally, investigations at the Large Hadron Collider will hopefully
99
Chapter 5 Indirect versus direct Dark Matter detection
provide a crucial test bench for the very existence of these light supersymmetric stable
particles [213].
Figure 5.11: Upward stopping muon flux, generated by light neutralino pairannihilation inside the Earth. The configurations displayed are only the ones compatible
with the DAMA annual modulation region, obtained without including the channeling
effect (upper panel) and including the channeling effect (lower panel). The three columns
show the results for the different sets of astrophysical parameters, defined in Sect. 5.2.3.
The horizontal line represents the experimental limit on stopping muons from the Earth
obtained using the SK data, see Eq. (3.42).
100
5.2 Neutralino Dark Matter
Figure 5.12: The same as Fig. 5.11, but in the case of light neutralino pair-annihilation
inside the Sun. In this case, the horizontal line refers to the experimental limit on
stopping muons from the Sun obtained using the SK data, see Eq. (3.40).
101
6
Summary and conclusions
A long time has passed since F. Zwicky in 1933 proposed for the first time the Dark
Matter hypothesis. Several different and complementary experiments, carried out in the
past decades, have confirmed the presence of an unknown form of matter at the level
of galaxies and clusters. The most precise determination of its abundance is provided
by the analysis of the Cosmic Microwave Background: roughly 84% of the mass of the
Universe is in the form of a non-luminous unknown matter.
After compelling evidences from astrophysical and cosmological experiments, the Dark
Matter concept is now commonly accepted by the whole physics community. However,
despite these strong experimental hints, we know very little about the nature of the particle (or the particles) that constitute the Dark Matter. To overcome this poor knowledge,
different types of experimental searches are necessary. A review on our current understanding of the Universe, on the Dark Matter detection methods and on the different
Dark Matter candidates has been given in Chapter 2.
In this work, we have focused on a particular class of indirect Dark Matter detection
methods: the search for neutrinos coming from Dark Matter annihilations. We have
extensively explained in Chapter 3 how to calculate the neutrino flux in the case of
annihilation inside celestial bodies or near the galactic center.
If the Dark Matter particles annihilate directly into neutrinos, the energy spectrum
of the neutrinos will consist of a line centered at energy Eν ≃ mχ . This peculiar signal could certainly be distinguished from the background of atmospheric neutrinos. In
Chapter 4, we have systematically investigated the different annihilation cross sections
into neutrinos, identifying all the cases in which a non-negligible branching ratio might
be present. With our analysis, we shed light on the main characteristics and criteria that
have to be fullfilled to obtain a sizable neutrino production. The explicit behaviour of
the annihilation cross section has also been shown for specific examples.
Most of the different theoretical models that arise from extentions of the Standard
Model of particle physics contain a viable Dark Matter candidate. One possibility to
103
Chapter 6 Summary and conclusions
reduce this vast set of scenarios is to combine results coming from different Dark Matter
searches. In Chapter 5, we have considered the interplay between indirect Dark Matter
detection with neutrinos and the direct Dark Matter detection. In particular, we have
focused on two different candidates: the leptophilic Dark Matter and the neutralino
Dark Matter. In the first case, we have carefully described the phenomenology of the
leptophilic Dark Matter for direct detection experiments and indirect detection with
neutrinos. We have then shown how the Super-Kamiokande bounds on neutrinos from
Dark Matter annihilations inside the Sun provide a strong constraint on the leptophilic
candidate. Indeed, the cross section required to explain the DAMA data within this
scenario is excluded by many orders of magnitude by the neutrino constraints.
In the case of neutralino Dark Matter, we have calculated the fluxes in throughgoing and stopping muons, as expected at the Super-Kamiokande detector, and we have
compared them to the existing bounds. Depending on the category of events and on the
values of the various astrophysics and particle physics parameters, we have derived the
ranges of neutralino masses which could be explored at a water Cherenkov detector with
a low muon energy threshold (around 1 GeV). Moreover, we have shown how stopping
muons could be used to explore the low mass region in the allowed neutralino parameter
space. For this category of events, we have also calculated the expected fluxes for the
supersymmetric configurations selected by the DAMA annual modulation data.
We are in an important and exciting moment concerning the Dark Matter searches. A
number of experiments that use direct or indirect detection techniques are now running
and taking data, while others are under construction. Most notably, we should remember
that new future results are expected from the Super-Kamiokande detector and that
the IceCube neutrino telescope at the South Pole will soon be completed. Of strong
importance for the analysis of the neutrino flux from the galactic center is the planned
KM3Net neutrino telescope. Furthermore, the LHC collider experiment is expected to
provide important information on the nature of Dark Matter, since most of the existing
theoretical framework predicts particles with masses accessible to the energies that LHC
will reach. Future LHC data will also be fundamental to constrain the existing models
of New Physics and their Dark Matter candidates. Finally, we want to stress that
the important task of identifying the nature of Dark Matter will require a joint effort
between the astrophysics and the particle physics community, both on the theoretical
and on the experimental side. Only with combined analyses that consider data form
different experiments, we will be able to shed light on what has remained a mystery for
about eighty years.
104
Acknowledgments
First of all I would like to thank my supervisor Prof. Manfred Lindner for giving me the
possibility to come to Heidelberg and for accepting me as PhD student. Thanks for all
the enthusiasm you put in physics and for all the interesting discussions we had. Thanks
also for leaving me the freedom to find my way. I wish to thank Prof. Tilman Plehn for
agreeing to be my second referee. Thanks also to Prof. Stephanie Hansmann-Menzemer
and to Prof. Eva Grebel for accepting to take part in my exam.
A very big thank you to all the people that made this thesis possible: my collaborators.
Thanks to Joachim Kopp for all the discussions we had together and for helping me
throughout my doctorate. Thanks to Alexander Merle for constantly supervising me in
the last period of my PhD. Thanks to Manfred Lindner, Thomas Schwetz, Jure Zupan
and Thomas Underwood for the fruitful collaborations. A special thank you to the
Astroparticle Physics group of the University of Torino. In particular I would like to
express all my gratitude to Nicolao Fornengo for always finding the time to answer my
questions, for guiding me during our research and for hospitality in Torino. Thanks
also to Alessandro Bottino for all his passion towards physics and for the work we did
together. I want to thank Werner Rodejohann and Sandhya Choubey, with whom I
wrote my first paper. Thanks to Evgeny Akhmedov for giving me an interesting project
to work on. A particular thank you also to Andreas Hohenegger for having always the
solution to whatever computer-related problem. Thanks to my ex-officemate Mathias
Garny to always have had an answer to my questions.
I am also grateful to Alexander Merle and to Joachim Kopp for proofreading my thesis. I want to thank all my colleagues (and ex-colleagues) from the Division on Particle
and Astroparticle Physics for creating such an exciting working environment: Adisorn
Adulpravitchai, Evgeny Akhmedov, Fedor Bezrukov, Alexander Blum, Mathias Garny,
Claudia Hagedorn, Hans Hettmansperger, Andreas Hohenegger, Martin Holthausen,
Alexander Kartavtsev, Alexander Merle, Werner Rodejohann, Michael Schmidt, Thomas
Schwetz-Mangold, Tom Underwood and Elisa Resconi. Thanks to Anja Berneiser for
helping me out with all the bureaucracy.
These years would have not been the same without having by my side wonderful people
that made my stay in Heidelberg special. I am really thankful to Sara for all the time
we spent together, for always being there to listen to my problems. It was great that
you came here and that we shared also the PhD together. Thanks to Giulia for trying
to make me feel at home here since the first time we met and for having always a good
advice. Thanks to Claudia for being always full of ideas and for being an example for me.
Thanks to Giovanna for all her sweetness and naivety. Thanks to Isabel for believing in
friendship. Thanks also to Vivı́, Olga and Matteo. Thanks to Giovanni for being a good
105
friend (especially in the last period!) and thanks to Brian for all the good DVDs.
I would also like to thank Fede and Giulia for always being happy to see me each time
I was coming back to Torino. It is such a great feeling to know that I have you as friends.
Thanks also to Gianni, Andrea and Stefano for the funny time together.
Un grazie particolare alla mia famiglia: mamma, papá, Lavinia e Fabrizio. É stato
difficile e doloroso stare cosı́ lontano in tutti questi anni e senza tutto il vostro supporto
non sarei mai riuscita a finire questa tesi. Grazie per essermi stati vicino nei momenti
piú difficili e per aver sempre creduto in me.
A
Neutrino interactions inside the Sun
In this Appendix, we report the explicit expressions for the neutral current and charged
current terms, which appear in the evolution of the neutrino density matrix, see Eq. (3.14).
We do not discuss the oscillatory term, since it has already been written explicitly in
Eq. (3.15).
A.1 Neutral current interaction
The neutral current contribution to the density matrix equation is given by the sum of two
terms that describe, respectively, the processes of neutrino energy loss and reinjection:
Z E
dρ
dΓN C
= −
dE ′
(E)
(E, E ′ ) ρ(E) +
′
dr
dE
NC
Z0 mχ
dΓN C ′
(E , E) ρ(E ′ ) ,
(A.1)
dE ′
+
dE
E
where ΓN C is defined as
with
ΓN C (E, E ′ ) = diag ΓeN C (E, E ′ ), ΓµN C (E, E ′ ), ΓτN C (E, E ′ ) ,
(A.2)
ΓlN C (E, E ′ ) = Np (r) σ(νl p → νl′ X) + Nn (r) σ(νl n → νl′ X) .
(A.3)
Since the neutral current cross sections are identical for the different flavours, the matrix
ΓN C (E, E ′ ) is proportional to the unit matrix. The functions Np and Nn represent
the proton and neutron number densities of the medium in which the neutrinos are
propagating. In the upper panel of Fig. A.1, we report the electron and neutron number
107
Appendix A Neutrino interactions inside the Sun
density as predicted by the Standard Solar Model of Ref. [214]. As a comparison, we also
show the prediction of the approximate exponential density profile [215]:
Ne
= 245 exp(−10.54 x) cm−3 ,
NA
(A.4)
with Ne being the electron number density (Np = Ne for the Sun). The chemical composition of the ratio Np /Nn varies from a value of ∼ 7 in the outer region of the Sun to
Np /Nn ∼ 2 in the central region, as displayed in the lower panel of Fig. A.1.
100
Nn
Ne
10
NN A @cm-3 D
Ne EXPO
1
0.1
0.01
RŸ = 6.955 × 108 m
0.001
0.0
0.2
0.4
0.6
0.8
1.0
rRŸ
RŸ = 6.955 × 108 m
8
N p Nn
6
4
Solar
Radiation
Convection
Core
Zone
Zone
2
0.0
0.2
0.4
0.6
0.8
rRŸ
Figure A.1: Upper panel: neutron and electron number density inside the Sun, as
predicted by the Standard Solar Model. We report also the exponential approximation
of Eq. (A.4). Lower panel: ratio Np /Nn as a function of the radius of the Sun.
108
A.2 Charged current interaction
A.2 Charged current interaction
For the charged current interaction contributions, we have to consider both the equations for the neutrino and antineutrino density matrix, since they are coupled by the
regeneration processes due to tau decays after charged current interactions. Indeed, an
initial ντ or ν̄τ with energy E in that undergoes charged current scatterings on nucleons
can produce secondary neutrinos through the following decay chains:


X + ν̄τ
X + ντ










e+ + νe + ν̄τ
e− + ν̄e + ντ
ν̄τ → τ + →
ντ → τ − →








 +
 −
µ + νµ + ν̄τ
µ + ν̄µ + ντ
The equations for the charged current terms read
Z mχ
dρ {ΓCC , ρ}
dE in =
−
Πτ ρτ τ (E in ) ΓτCC (E in ) fτ →τ (E in , E) +
+
in
dr CC
2
E
E
+ Πe,µ ρ̄τ τ (E in ) Γ̄τCC (E in ) fτ̄ →e,µ (E in , E) ,
Γ̄CC , ρ̄
dρ̄ =−
dr CC
2
+
mχ
dE in Πτ ρ̄τ τ (E in ) Γ̄τCC (E in ) fτ̄ →τ̄ (E in , E) +
E in
E
+ Πe,µ ρτ τ (E in ) ΓτCC (E in ) fτ →ē,µ̄ (E in , E) ,
Z
where Πl is a diagonal matrix that projects onto the flavour νl , e.g. Πe = diag(1, 0, 0).
The matrix ΓCC is defined as
ΓCC (E) = diag ΓeCC (E), ΓµCC (E), ΓτCC (E) ,
(A.5)
and each component is given by:
ΓlCC (E) = Np (r) σ(νl p → lX) + Nn (r) σ(νl n → lX) .
(A.6)
For antineutrinos, the function Γ̄lCC (E) is defined analogously to the above expression
with the replacement of the neutrino cross sections by the ones of antineutrino. In
Fig. A.2, we report the energy distribution function f (E in , E) of the secondary neutrinos
produced by ντ and ν̄τ charged current scatterings on protons. The functions for the
scatterings on neutrons are nearly the same.
109
Appendix A Neutrino interactions inside the Sun
4.0
4.0
Ein =50 GeV
3.5
Ein =50 GeV
3.5
in
E =500 GeV
3.0
fΤ® Τ
2.5
Ν spectra
Ν spectra
2.5
2.0
1.5
1.0
1.0
0.0
0.0
f Τ ® e , Μ
0.2
0.5
0.4
0.6
x=EE
in
0.8
1.0
f Τ ® Τ
2.0
1.5
0.5
Ein =500 GeV
3.0
0.0
0.0
f Τ ® e, Μ
0.2
0.4
0.6
0.8
1.0
x=EEin
Figure A.2: Energy distributions of secondary neutrinos, generated by the decays of τ
or τ̄ , which are produced by ντ or ν̄τ charged current scatterings on protons. We show
the range of variability in case the initial neutrino energy E in changes from 50 GeV to
500 GeV.
110
B
Neutrino cross sections
In this Appendix, we report the neutrino cross sections that have been used in the
calculation of the neutrino spectra coming from Dark Matter annihilations as well as the
corresponding muon flux. In Sect. B.1, we discuss the neutral current interactions, while
in Sect. B.2 the charged current interactions are treated.
B.1 Neutral current cross sections
Throughout our analyses we have studied neutrinos and antineutrinos of energies of the
order of GeV-TeV. In this range, the interaction with protons and neutrons is essentially
dominated by deep inelastic scatterings.
As a good approximation, we can consider u and d as valence quarks, and ū and d¯ as
sea quarks. The differential cross section for the neutral current process νp → ν ′ X can
be obtained by summing the differential cross sections for the parton processes, weighted
by the quark distributions:
2mp G2F
dσ
′
(νp
→
ν
X)
=
dE ′
π
with:
hqi =
X
q={u,d}
Z
h
2 +
hqip gLq
E′2 2
g
E 2 Rq
2 +
+ hq̄ip gRq
E′2 2
g
E 2 Lq
+
i
,
(B.1)
1
dx x q(x) ,
0
where x is the fraction of the total momentum of the nucleon carried by the quark q and
the function q(x) is the probability that the quark q has a fraction of the total momentum
equal to x. We have indicated by E the initial energy of the neutrino, while E ′ stands
111
Appendix B Neutrino cross sections
for the final one. The coefficients gL and gR arise from the couplings between the quarks
and the Z-boson:
1 2
− sin2 θW ,
2 3
1 1
= − + sin2 θW ,
2 3
2
gRu = − sin2 θW ,
3
1
gRd = sin2 θW ,
3
gLu =
gLd
where θW is the Weinberg angle:
sin2 θW = 0.237 ± 0.006 .
In the case of antineutrinos the cross section is given by:
2mp G2F
dσ
′
(ν̄p
→
ν̄
X)
=
dE ′
π
X
q={u,d}
h
2 +
hqip gRq
2 +
+ hq̄ip gLq
E′2 2
g
E 2 Lq
E′2 2
g
E 2 Rq
+
i
.
(B.2)
If we substitute mp , hqip and hq̄ip by, respectively, mn , hqin and hq̄in , we will obtain the
cross section for the scattering on neutrons. The values that we have used in our work
are:
huip = hdin = 0.25 ,
hdip = huin = 0.15 ,
hūip = d¯ n = 0.04 ,
d¯ p = hūin = 0.06 .
(B.3)
(B.4)
(B.5)
(B.6)
We finally want to add that the expressions of the cross sections that we have reported
in this Section are valid for energies Eν ≪ MZ2 /(2 mp ) ≃ 3600 GeV, with MZ being the
mass of the Z-boson.
B.2 Charged current cross sections
In the case of charged current interaction, the intermediate boson that is exchanged
between neutrinos and partons inside the proton is a W -boson. In this case, neutrinos
and antineutrinos do not interact with all of the quarks, since the charge has to be
conserved at each vertex of a Feynman diagram. The parton cross sections are given by
the following expressions:
dσ̂
(νl d → lu) =
dy
dσ̂
¯ =
(νl ū → ld)
dy
112
G2 ŝ
dσ̂ ¯ ¯
(ν̄l d → lū) = F ,
dy
π
G2 ŝ
dσ̂
(ν̄l u → ¯ld) = F (1 − y)2 .
dy
π
(B.7)
(B.8)
B.2 Charged current cross sections
After integrating over y, with 0 ≤ y ≤ 1, we obtain the following total cross sections:
G2F ŝ
,
π
2
¯ = 1 GF ŝ ,
σ̂(ν̄l u → ¯ld) = σ̂(νl ū → ld)
3 π
σ̂(νl d → lu) = σ̂(ν̄l d¯ → ¯lū) =
(B.9)
(B.10)
where ŝ = s x is the square of the total energy of the partonic process in the center-ofmass frame, while s is the one of the (anti)neutrino-nucleon process. Integrating over x,
we can arrive at the final expressions for the cross sections:
Z 1
¯
σ(νl p → lX) =
dx dp (x) σ̂(νl d → lu) + ūp (x) σ̂(νl ū → ld)
0
G2F s
1
=⇒ σ(νl p → lX) =
hdip + hūip ,
π
3
σ(ν̄l p → ¯lX) =
Z
0
1
(B.11)
dx d¯p (x) σ̂(ν̄l d¯ → ¯lū) + up (x) σ̂(ν̄l ū → ¯ld)
G2F s ¯
1
¯
d p + huip .
=⇒ σ(ν̄l p → lX) =
π
3
(B.12)
Analogous expressions can be obtained for the scattering cross sections on neutrons.
The equations reported above refer to charged current conversions of a neutrino or
antineutrino into an electron or a muon. The cross sections for the conversion into a
tau lepton are slightly different than the ones derived before. Indeed, since we study
neutrinos with energies of the order of mτ , this will affect the range of variability of the
Bjorken variables x and y. We implemented this correction following Ref. [216]. The
behaviour of the charged current cross sections is reported in Fig. B.1.
The expressions of the cross sections that we have reported in this Section are valid up
2 /(2 m ) ≃ 3600 GeV, with M
to energies Eν ≪ MW
p
W being the mass of the W -boson.
Energy-differential charged current cross section
The energy-differential cross section for deep-inelastic scattering is given by
dσ
1 dσ
=
,
dEµ
Eν dy
(B.13)
where y = 1 − Eµ /Eν . Using Eq. (B.7) and Eq. (B.8), the explicit expressions for the
energy-differential cross sections of neutrino and antineutrino scatterings off a proton
113
Appendix B Neutrino cross sections
10-34
10-34
10-35
10-35
10
10
-37
10-37
10-38
10-38
Σ @cm2 D
Σ @cm2 D
Νe N, ΝΜ N
-36


Ν e N, Ν Μ N
10-39
10
20
50
100
EΝ @GeVD
200
500
1000
ΝΤ N
10-36
10-39

Ν ΤN
10
20
50
100
EΝ @GeVD
200
500
1000
Figure B.1: Cross sections for the charged current interactions of ν and ν̄ on a nucleon
N . Left panel: cross sections for νe,µ and ν̄e,µ . Right panel: cross sections for ντ and ν̄τ .
The effect of the τ mass is well visible at low energies.
are:
"
2 #
2 mp G2F
Eµ
hdip + hūip
≃
π
Eν
"
2 #
Eµ
≃ 0.5 + 0.1
10−38 cm2 GeV−1 ,
Eν
"
2 #
2
2
m
G
dσ
n
F
¯ p + huip Eµ
hdi
(ν̄l p → ¯lX) =
≃
dEµ
π
Eν
"
2 #
Eµ
≃ 0.2 + 0.8
10−38 cm2 GeV−1 .
Eν
dσ
(νl p → lX) =
dEµ
Analogous equations can be derived for scattering off a neutron.
114
(B.14)
(B.15)
C
Annihilation cross sections
The differential annihilation cross section of two DM particles χ into two neutrinos is
given by [217]
dσann
1 1
v
|M|2 ,
(C.1)
=
d cos θ∗
16π s
where v is the relative velocity between the two DM particles and θ∗ is the scattering
angle in the center-of-mass frame. In the previous formula we have neglected the neutrino
mass and we have denoted by |M|2 the spin-averaged matrix element:
|M|2 =
1
(2SDM +
1)2
X
spins
|M|2 ,
(C.2)
where SDM is the spin of the DM particle.
Since we do not focus on a particular model, our results are general and can be applied to the calculation of the annihilation cross section into neutrinos for a specific DM
candidate. Moreover, from our expressions it is easy to see which are the channels and
the possible cases that could lead to a sizable DM branching ratio into neutrinos. For
simplicity, throughout our analysis we consider only the Standard Model as gauge group.
For the numerical calculation we use the FeynCalc package [218].
C.1 Scalar Dark Matter
For a scalar DM, the neutrino production can occur through a scalar and a Z-boson exchange in an s-channel diagram and through a fermion exchange in a t-channel diagram.
In case the neutrinos are Majorana particles, also a u-channel diagram is present.
115
Appendix C Annihilation cross sections
Scalar mediator, s-channel
Indicating the coupling of the scalar mediator to the DM with D and the coupling to Dirac
neutrinos with NL PL +NR PR , with the projection operators defined as PL,R = (1∓γ5 )/2,
the total annihilation cross section can be written as
!
2
2 + |N |2
2m
|N
|
2−n
χ
L
R
|D|2
σann v (χs ; φs ; s) =
4n 1 −
v 2 + O(v 4 ) , (C.3)
8π
(4m2χ − m2φ )2
(4m2χ − m2φ )
where mφ is the scalar mediator mass and n = 0, 1 for Dirac and Majorana neutrinos,
respectively. In case of Majorana neutrinos, a factor 1/2 is present to avoid double
counting of identical particles in the final state and a factor 4 arises from the Feynman
rule for the effective vertex, since the Majorana neutrinos are self-conjugate particles.
Note that, for simplicity, we use the same form of the Yukawa couplings for Dirac and
Majorana neutrinos. We want to stress that, in general, this is not the case, since
Dirac and Majorana neutrinos usually couple to scalar mediators with different SU (2)L
representations, see Table 4.1.
Z-boson mediator, s-channel
Indicating the coupling of the Z-boson to the DM generically as D(k1 − k2 )µ , with k1
and k2 being the DM four momenta, and the coupling to the neutrinos with NL γ µ PL ,
the total annihilation cross section can be written as
σann v (χs ; Z; s) =
NL2
1
m2 v 2 + O(v 4 ) ,
D2
12π
(4m2χ − m2Z )2 χ
(C.4)
with D and NL being real numbers. In this case the annihilation cross section is proportional to the DM velocity, as we would naively expect from angular momentum conservation. If the neutrinos are Majorana particles, the annihilation cross section is equivalent
to the one given in Eq. (C.4). Indeed, it is well known that weak interactions mediated
by the Z-boson do not distinguish between Dirac and Majorana neutrinos [219].
Fermionic mediator, t&u-channels
Indicating the coupling of the DM particle to the fermionic mediator and the neutrino
with FL PL + FR PR at one vertex, and with GL PL + GR PR at the other vertex, the total
annihilation cross section is given by


4
mφ
1 |FL |2 |GL |2 + |FR |2 |GR |2  2
2 2
m
−
σann v (χs ; φf ; t) =

2 mχ v  +
φ
8π
(m2χ + m2φ )2
m2χ + m2φ
+
116
1 |FR |2 |GL |2 + |FL |2 |GR |2 2 2
mχ v + O v 4 ,
2
48π
m2χ + m2φ
(C.5)
C.2 Fermionic Dark Matter
where mφ is the fermionic mediator mass. Notice that, in general, GL = FR∗ and GR = FL∗
for a Dirac mediator, while also GL = FL and GR = FR are allowed for a Majorana
mediator. In the first case a pair of ν ν̄ is produced, while in the second case νν (or
ν̄ ν̄) are produced. If the DM particle is a real scalar, also a u channel is present. The
corresponding cross section is equivalent to the one in Eq. (C.5).
In the case of Majorana neutrinos both, the t-channel and the u-channel diagram,
must be considered and added together with a relative minus sign. The annihilation
cross section is hence modified to


2 3m2 + m2
m
2
2
2
2
χ
φ
φ
1 |FL | |GL | + |FR | |GR |  2
2 2
σann v (χs ; φf ; t&u) =
mφ −
2 mχ v  +
2
4π
3 m2χ + m2φ
m2χ + m2φ
+
1 |FR GL − FL GR |2 2 2
4
2 mχ v + O v .
48π
m2χ + m2φ
(C.6)
C.2 Fermionic Dark Matter
For a fermionic DM, the neutrino production can occur through a Z-boson exchange
in an s-channel diagram and through a scalar exchange in an s-channel or a t-channel
diagram. In case the DM or the neutrinos are Majorana particles, also a u-channel
diagram is present.
Scalar mediator, s-channel
Indicating the coupling of the Dirac DM particle to the scalar mediator with DL PL +
DR PR and the one of the Dirac neutrinos with NL PL + NR PR , the total annihilation
cross section can be written as
σann v (χf ; φs ; s) =
×
2−n |NL |2 + |NR |2 n m
4 4 ×
16π (4m2χ − m2φ )2
|DL − DR |2 m2χ −
m2φ
2 (4m2χ − m2φ )
(|DL |2 + |DR |2 ) m2χ v 2 +
m2χ
∗
+
(DL DR
+ c.c.) m2χ v 2
8 (4m2χ − m2φ )
!
+ O(v 4 ) .
(C.7)
where mφ is the scalar mediator mass, n=0 (n=1) for Dirac (Majorana) neutrinos and
m=0 (m=1) for Dirac (Majorana) DM. The factor 1/2 is present to avoid double counting
of identical particles in the final state, while the factors 4 come from the Feynman rules
for the effective vertex. For simplicity, we have used the same Yukawa couplings for Dirac
and Majorana neutrinos. However, they generally couple to different scalar particles, see
Table 4.2.
117
Appendix C Annihilation cross sections
Note, that in case the DM couples to the scalar mediator through a scalar coupling
(i.e. DL = DR ), the cross section will be proportional to the DM velocity v. This is
a consequence of parity conservation: a fermion-antifermion pair has a parity of (−1)
and can therefore, in an s-wave configuration, only couple to a pseudoscalar particle (i.e.
DL = −DR ).
Z-boson mediator, s-channel
Indicating the coupling of the DM particle to the Z-boson with γ µ (DL PL + DR PR ) and
the one of the neutrino with NL γ µ PL , the total annihilation cross section can be written
as
σann v (χD
f ; Z; s) =
NL2
1
×
8π (4m2χ − m2Z )2
×
(DL + DR )2 m2χ −
−
(m2Z + 2m2χ )
2
(D2 + DR
) m2χ v 2 −
3 (4m2χ − m2Z ) L
!
4 m2χ
(DL DR ) m2χ v 2
(4m2χ − m2Z )
+ O(v 4 ) .
(C.8)
with DL , DR and NL being real numbers. The cross section for Majorana neutrinos
is equivalent to Eq. (C.8), since, as we have mentioned before, the weak interactions
mediated by the Z-boson do not distinguish between Dirac and Majorana neutrinos [219].
If the DM particle is a Majorana fermion, the cross section reported above is drastically
modified. Indeed, in an s-wave annihilation, the fermions in the initial state are forced to
have opposite spins by the Pauli exclusion principle. As a consequence, since the Z-boson
has a spin of one, we expect that the first non zero contribution to the annihilation cross
section for Majorana DM is given by the p-wave term. Indeed, we find:
σann v (χM
f ; Z; s) =
NL2
1
2 2 2
4
(D
−
D
)
m
v
+
O
v
,
L
R
χ
2
12π (4 m2χ − mZ )2
(C.9)
where the same expression holds if the DM and the neutrinos are both Majorana particles.
Scalar mediator, t&u-channels
Indicating the coupling of the DM particle to the fermionic mediator and to the neutrino
at one vertex with FL PL + FR PR and at the other one with GL PL + GR PR , the total
annihilation cross section is given by
σann v (χf ; φs , t) =
×
118
1 (|FL |2 + |FR |2 )(|GL |2 + |GR |2 )
×
32π
(m2χ + m2φ )2
!
(m4φ − 3m2χ m2φ − m4χ ) 2 2
2
mχ +
mχ v + O(v 4 ) ,
3 (m2χ + m2φ )2
(C.10)
C.3 Vector Dark Matter
where mφ is the fermionic mediator mass. Note that, in general, GL = FR∗ and GR = FL∗
or GL = FL and GR = FR . In the first case a pair of ν ν̄ is produced, while in the second
case νν (or ν̄ ν̄) are produced.
A t-channel and a u-channel diagram must be considered in the case of Majorana
neutrinos and/or Majorana DM. The expression for the annihilation cross section is thus
modified to
σann v (χf ; φs ; t&u) =
−
1
1
m2 × A −
2
32π (mχ + m2φ )2 χ
1
1
2 2
4
m
v
×
B
+
O
v
,
χ
192π (m2χ + m2φ )4
(C.11)
The functions A and B are given by the following expressions in the case of Majorana
neutrinos:
Aν
Bν
= |FL |2 |GL |2 + |FR |2 |GR |2 + |FL GR − FR GL |2 ,
(C.12)
= (|FL |2 |GL |2 + |FR |2 |GR |2 ) (m4χ + 4m2χ m2φ − 3m4φ ) +
+ (|FL |2 |GR |2 + |FR |2 |GL |2 ) (m4φ − 3m2χ m2φ − m4χ ) −
− 2 (FL FR∗ G∗L GR + c.c.) (3m2φ + 2m2χ ) m2χ .
(C.13)
In the case of Majorana DM, the corresponding expressions for A and B are given by
Aχ = 2 |FL |2 |GL |2 + 2 |FR |2 |GR |2 ,
Bχ = 2 (|FL |2 |GL |2 + |FR |2 |GR |2 ) (3m4φ − 4m2χ m2φ − m4χ ) +
+ 4 (|FL |2 |GR |2 + |FR |2 |GL |2 ) (m4φ + m4χ ) .
(C.14)
(C.15)
Notice that, for Majorana DM, terms proportional to FL GR or FR GL are not present
in the s-wave. Indeed, due to the Pauli principle, two Majorana particles cannot have
parallel spins if their relative angular momentum l is zero. The only nonzero contribution
to the s-wave configuration will be present if FL 6= 0 and GL 6= 0. This situation can arise
in supersymmetric models only in the presence of a mixing between the left and right
sfermions. However, a mixing term between f˜L and f˜R is proportional to the fermion
mass. For this reason, the annihilation cross section of a neutralino pair into fermions,
through a t-channel sfermion exchange, is always proportional to the mass of the fermions
produced. This conclusion does, in general, not hold when we consider a Majorana DM
beyond a supersymmetric framework.
C.3 Vector Dark Matter
In this Section we report the annihilation cross sections for the case of vector DM, since
in specific models, for example in Extra Dimensions, new vector particles can be present,
119
Appendix C Annihilation cross sections
even without an extension of the Standard Model gauge group. The neutrino production
can then occur through a scalar exchange in an s-channel diagram and through a fermion
exchange in a t-channel diagram. In case the neutrinos are Majorana particles, also a
u-channel diagram is present.
Scalar mediator, s-channel
Indicating the coupling of the scalar mediator to the DM particles with D and the one
to the neutrinos with NL PL + NR PR , the total annihilation cross section can be written
as
!
2 + m2 )
(2m
2−n 2 NL2 + NR2
χ
φ
σann v (χv ; φs ; s) =
4n 1 −
v 2 + O(v 4 ) , (C.16)
D
24π
(4m2χ − m2φ )2
3 (4m2χ − m2φ )
with D, NL and NR being real numbers and n = 0, 1 for Dirac and Majorana neutrinos, respectively. As in the previous cases, we have considered for simplicity the same
couplings for Dirac and Majorana neutrinos.
Fermionic mediator, t&u-channels
Indicating the coupling of the DM particle to the fermionic mediator and to the neutrino
at one vertex with γ µ (FL PL + FR PR ) and at the other one with γ ν (GL PL + GR PR ), the
total annihilation cross section is given by
2
σann v (χv ; φf ; t) =
+
2
2
2
2
2
2
2
2
2
1 4mχ (FL GL + FR GR ) + 5mφ (FR GL + FL GR )
+
72π
(m2χ + m2φ )2
1
1
2
4
v
×
C
+
O
v
,
(C.17)
432π (m2χ + m2φ )4
with
C = 12 m6φ (FL2 G2R + FR2 G2L ) + 13 m6χ (FL2 G2L + FR2 G2R ) +
+ m2χ m4φ (13FL2 G2L + 13FR2 G2R + 2FL2 G2R + 2FR2 G2L ) +
+ 2 m4χ m2φ (FL2 G2L + FR2 G2R ) + 20 m4χ m2φ (FL2 G2R + FR2 G2L ) .
(C.18)
Note that, in general, GL = FL and GR = FR . For Majorana neutrinos, a t-channel and
a u-channel diagram are present. The annihilation cross section is then modified to
σann v (χv ; φf ; t&u) =
+
120
2
2
2 2
2 2
2
1 (GL + GR )(2mχ FL + 3mφ FR ) + 4mχ GL GR FL FR
+
36π
(m2χ + m2φ )2
1
1
v2 × D + O v4 ,
(C.19)
2
2
4
432π (mχ + mφ )
C.3 Vector Dark Matter
with
D = 12 m6φ (FL2 G2R + FR2 G2L ) + 13 m6χ (FL2 G2L + FR2 G2R ) +
+ 13 m2χ m4φ (FL2 G2L + FR2 G2R ) − 4 m2χ m4φ (FL2 G2R + FR2 G2L ) +
+ 2 m4χ m2φ (FL2 G2L + FR2 G2R ) + 16 m4χ m2φ (FL2 G2R + FR2 G2L ) −
− 2 FL FR GL GR (9m4χ + 10m2χ m2φ − 7m4φ ) .
(C.20)
121
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