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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 9 11 11 15 16 19 19 25 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 29 29 33 34 39 40 40 42 43 . . . . . . . . . . 49 50 50 50 51 52 53 57 60 62 67 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mediator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the . . . 71 71 72 74 74 77 79 80 83 88 88 89 91 92 99 103 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@GeVcm3 D 103 102 Moore NFW r0 = 8.5 @kpcD Ρ0 = 0.3 @GeVcm3 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 C @s-1 D C @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 C @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 G @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 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 NN A @cm-3 D Ne EXPO 1 0.1 0.01 R = 6.955 × 108 m 0.001 0.0 0.2 0.4 0.6 0.8 1.0 rR R = 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 rR 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=EE 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=EEin 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 Bibliography [1] E. Hubble, A relation between distance and radial velocity among extra-galactic nebulae, Proceedings of the National Academy of Sciences of the United States of America 15 (1929), no. 3 168–173, [http://www.pnas.org/content/15/3/168.full.pdf+html]. [2] S. Perlmutter and B. P. Schmidt, Measuring Cosmology with Supernovae, Lect. Notes Phys. 598 (2003) 195–217, [astro-ph/0303428]. [3] R. Bernabei et. al., First results from DAMA/LIBRA and the combined results with DAMA/NaI, Eur. 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