Lepton Flavour Violation and Dark Matter Phenomenology

UNIVERSIDAD AUTONOMA´ DE MADRID FACULTAD DE CIENCIAS DEPARTAMENTO DE F´ISICA TEORICA´

Memoria de Tesis Doctoral realizada por Bryan Zald´ıvar Montero presentada ante el Departamento de F´ısica Teorica´ de la Universidad Autonoma´ de Madrid para la obtencion´ del T´ıtulo de Doctor en Ciencias.

Tesis Doctoral dirigida por Alberto Casas Jesus´ Moreno del I.F.T. de la Universidad Autonoma´ de Madrid

Madrid, Febrero 2013.

AGRADECIMIENTOS

CONTENTS

1 General Introduction and Motivations 1

2 Supersymmetry and Lepton Flavour Violation 5 2.1 Elements of SUSY ...... 5 2.1.1 Neutrino masses and mixings ...... 9 2.1.2 Supersymmetric See-saw ...... 9 2.1.3 LFV processes in the supersymmetric See-saw ...... 11 2.2 Leptogenesis ...... 14

3 Dark Matter phenomenology 17 3.1 Experimental evidences of Dark Matter ...... 17 3.2 Dark Matter candidates ...... 18 3.2.1 Baryonic Dark Matter? ...... 18 3.2.2 Non-baryonic Dark Matter ...... 19 3.3 Early universe ...... 20 3.3.1 The dynamics of the universe’s evolution ...... 20 3.3.2 Thermodynamics ...... 23 3.4 Dark Matter abundance ...... 29 3.4.1 Freeze-out of cold Dark Matter ...... 31 3.5 Dark Matter Detection ...... 33 3.5.1 Indirect detection ...... 33 3.5.2 Direct Detection ...... 35 3.5.3 Collider searches ...... 36

4 Fair Scans of the Seesaw I 39

5 Collider Bounds on Dark Matter 63 6 Complementarity among different strategies of Dark Matter Searches 79

7 Conclusions 121

8 Conclusiones 125

Appendices 129

Appendix A Corrected Boltzmann equation 131

List of Figures 135 CHAPTER 1

GENERAL INTRODUCTION AND MOTIVATIONS

One of the open questions in particle physics nowadays is the so-called flavour puzzle: why there is a hierarchical structure of fermion masses and mixings, and why there are two repli- cas of the lightest fermions? The leptonic sector presents even more challenging features than the quark sector, where the mixings between flavour eigenstates show an approximately perturbative texture. Indeed, in the neutrino sector the values of the mixing angles appear ‘randomly’ distributed, and one of the mixing angles is close to maximal. On the other hand, the masses of neutrinos are believed to be below 0.3 eV, i.e. six orders of magnitude ∼ smaller than the lightest charged fermion (the electron). This should be compared with the five orders of magnitude expanded by the masses of the nine charged fermions (from the ∼ electron to the top mass), more or less equally distributed in between. Certainly, the Standard Model (SM) can be trivially extended to accommodate neutrino masses. However, the previous huge gap between the neutrinos and the rest of SM particles has posed a strong motivation to develop theoretical models that could explain it. One of the most popular theoretical frameworks to accommodate such small neutrino masses is the so- called Seesaw mechanism. The idea is that right-handed (RH) neutrinos, which are singlets under the SM gauge group, can have large Majorana masses (as they are not controlled by the electroweak-breaking scale). Then the lightest (approximately pure left-handed) neutrinos get masses suppressed by the ratio of the Higgs VEV and the right-handed Majorana mass, thus becoming extremely light. The Seesaw scenario can be easily formulated within a supersymmetric framework. Indeed, this is highly motivated by the fact that the massive right-handed neutrinos introduce large (logarithmic) corrections to the Higgs mass, which worsens the notorious Hierarchy- Problem of the Standard Model. On the other hand, the neutrino Yukawa couplings must present an off-diagonal structure (to generate the neutrino mixings), which in turn induces off-diagonal entries in the slepton matrices. The latter may potentially trigger processes which violate Lepton flavour, for example µ eγ. → 2 General Introduction and Motivations

The results of the ﬁrst part of this thesis have been worked out in this context. In particular we have derived an algorithm to scan the large parameter space of the (supersymmetric or not) Seesaw in a complete way. We have done this using the so-called R parametrisation − and show that the results are consisted with those obtained using other parametrisations. In particular, we demonstrate that if the scan is complete, the branching ratio BR(µ eγ) is → not sensitive to the angle θ13 of the neutrino (PMNS) mixing matrix. This result rectiﬁes previous claims in the literature stating that there was a strong dependence between these two quantities. We explain that the reason for those previous results was essentially that they were based on an incomplete scan of the Seesaw parameter space.

The second part of the thesis is devoted to the study of Dark Matter (DM) issues. The existence of DM is probably the strongest evidence to believe that the Standard Model is an incomplete theory, since in order to account for the assorted observations of DM one has to extend the SM with extra degrees of freedom. There are many candidates to play the role of the DM particles. One of the best-motivated ones are the so called WIMPs (weakly interacting massive particles). The reason is the following. If one assumes that the DM-genesis occurs in the early universe, in a similar fashion to the rest of the particles, then the DM- particles lived in thermal equilibrium until the global temperature decreased below a certain threshold. Then the DM particles decoupled from the thermal bath, annihilating massively until -due to the expansion of the universe- no DM particle were able to find a partner to annihilate with. After this moment the DM abundance became “frozen”. The calculation shows that for DM-masses of electroweak size (from GeV to TeV) with weak interactions (similar to neutrinos) the final DM-abundance is in the range consistent with observations. In this context, the DM particles are expected to participate not only in gravitational interactions with the rest of SM particles, but also in other kind of processes, e.g. in electroweak interactions, behaving similarly to neutrinos but with a much larger mass. There are three basic experimental strategies to search for DM nowadays. The first one is the so-called “Indirect Detection”, in which one attempts to detect the DM annihilation or decay subproducts generated in the Galactic Centre (or in Galaxy Clusters). For example, the electrons and positrons produced in processes like DM DM e+e taking place in re- → − gions where there are strong magnetic fields, will produce synchrotron radiation which can directly point towards us. Thus by measuring synchrotron fluxes from different directions in the sky, we may detect DM signals. The second experimental approach is “Direct De- tection”. The idea is that the DM particles of the galactic halo can interact, from time to time, with nucleons of a heavy material, e.g. xenon placed in containers deep underground, producing recoil energies which could be measured if the process is energetic enough. The third experimental strategy is the DM production in colliders. E.g. the LHC, the idea is that the DM particles could be directly produced in some proton-proton collisions. Once produced, the DM particles would escape the detector, as neutrinos do, contributing to the missing energy. In this way a DM signal can be identified indirectly by studying the other 3

-visible- products of the scattering. This occurs for instance in processes like qq¯ j DM → DM, in which a quark-antiquark pair annihilate into a pair of DM particles plus a jet (which can be originated by a gluon emitted by the initial states). In our results, we ﬁrst combine different bounds coming from collider searches of assorted (leptonic or hadronic) nature, as LEP and Tevatron. We show that, considering leptonic and hadronic bounds at the same time constrains much further the parameter space of generic DM models, which we study in a model-independent fashion through different classes of effective theories. We include in the analysis additional bounds coming from Di- rect Detection searches, as well as the requirement of a correct relic abundance. All this results in quite severe exclusion limits, excluding a particular class of effective theories. Finally, we perform a detailed study of possible synchrotron signals coming from DM annihilation at the Galactic Centre, comparing the corresponding bounds with those obtained from other searches, e.g. direct production in colliders (LEP, Tevatron and LHC) and other indirect-detection searches. We study the DM induced synchrotron-signal not only in the effective theory, but also in two particular ultraviolet realisations, which represent the most simple extensions of the SM to account for Dark Matter, obtaining important restrictions on their parameter-spaces.

The thesis is organised as follows. In chapter 2 I expose some introductory elements and motivations of Supersymmetry, the MSSM and the Seesaw mechanism. I also give the basis of the calculation of Lepton-Flavor violation effects and Leptogenesis in this context. In chapter 3 I introduce the Dark Matter subject, describing the ﬁrst experimental evidences, the theoretical formalism and the present experimental strategies. Finally, the original works and results are presented in chapters 4, 5 and 6, which are a compendium of the main ac- complishments of my thesis. 4 General Introduction and Motivations CHAPTER 2

SUPERSYMMETRY AND LEPTON FLAVOUR VIOLATION

2.1 Elements of SUSY

Nowadays, Supersymmetry (SUSY) is an usual ingredient in theories beyond the Standard Model (SM). There are indeed strong theoretical reasons supporting the idea that SUSY is a good symmetry of nature in some region of scales. Maybe the first motivation about its existence came from the idea of generalising the Poincare symmetry, allowing for transformations that take fermions into bosons and vice- versa. The idea was improved in the context of String Theory, at its early stages. The easiest way to introduce fermions in the original bosonic string was through supersymmetry. From a more phenomenological perspective, a supersymmetric version of the Standard Model presents very attractive properties, as the gauge coupling unification at a high scale (not far from MP) and the relaxation of the infamous hierarchy problem. The electroweak breaking requires the Higgs mass-term to be O(100 GeV). However, the quadratic contributions to this mass-term arising from radiative corrections (e.g. from a loop of tops) are much higher if the cut-off is beyond a few TeV. This translates into a strong fine-tuning between the initial tree-level mass and the size of the radiative corrections. SUSY provides an automatic cancellation of the quadratic divergences, and thus an elegant way-out to this problem. Although there may be caveats about the interpretation of the fine-tuning, the hierarchy problem is still commonly accepted by the scientific community as a strong indication that there should exist new physics at O(TeV). Unfortunately, Supersymmetry has not been observed in experiments, which requires it to be spontaneously broken. The corresponding effective theory at low-scale consists of a supersymmetric Lagrangian plus a bunch of terms which break SUSY in an explicit, but “soft”, fashion. 6 Supersymmetry and Lepton Flavour Violation

Concerning the supersymmetric Lagrangian, the minimal (N=1) supersymmetric extension of the Standard Model is the so-called Minimal Supersymmetric Standard Model (MSSM). The particle content of the MSSM is given in Table 2.1. It consists of the usual SM spectrum plus a supersymmetric partner of each SM particle. Besides, the Higgs sector 1 has to be extended, including two Higgs doublets, Hu,Hd. This is required to keep the anomaly cancellation (since the fermonic components of the Higgs superﬁelds contribute to the gauge anomalies) and also to provide masses to up- as well as down-type quarks and leptons.

Chiral super-multiplet ﬁelds

Names spin 0 spin 1/2 SU (3)C,SU (2)L,U(1)y Q (u˜L,d˜L)(uL,dL) 3, 2, 1/3 ¯ ˜ † c ¯ squarks, quarks U u¯L = u˜R u¯L =(uR) 3, 1, -4/3 ¯ ¯˜ ˜† ¯ c ¯ D dL = dR dL =(dR) 3, 1, 2/3 L (ν˜ ,e˜ )(ν ,e ) 1, 2, -1 sleptons, leptons eL L eL L ¯ ˜ † c E e¯L = e˜R e¯L =(eR) 1, 1,2 + 0 + 0 Hu (Hu ,Hu )(H˜u ,H˜u ) 1, 2,1 0 ˜ 0 ˜ Higgs, Higgsinos Hd (Hd ,Hd−)(Hd ,Hd−) 1, 2, -1 Gauge super-multiplet ﬁelds

spin 1/2 spin 1 SU (3)C,SU (2)L,U(1)y gluinos, gluonsgg ˜ 8, 1,0 0 0 winos, W bosons W˜ ±, W˜ W ±, W 1, 3,0 bino, B boson BB˜ 1, 1,0

Table 2.1: Particle content of the Minimal Supersymmetric Standard Model.

The superpotential of the MSSM is given by the most general renormalizable holomorphic function of the superﬁelds consistent with all the symmetries of the theory (including some version of R parity to prevent lepton- and baryon-number violation and thus proton − decay), W = yijU¯ Q H yijD¯ Q H yijE¯ L H + µH H , (2.1) u i j · u − d i j · d − e i j · d u · d where the dots denote SU(2)-invariant products of two doublets (colour indices are not shown), the y’s are the Yukawa couplings and µ mass-term is an additional supersymmetric parameter. The corresponding supersymmetric Lagrangian can be derived using the standard rules. In particular, this Lagrangian contains the following pieces:

1From the more formal point of view, this comes from the need of building a holomorphic superpotential. 2.1 Elements of SUSY 7

Yukawa-terms. These arise from the general expression • 2 1 ∂ W(φ) T ∑ ψnLεψmL , 2 nm ∂φn∂φm where the superpotential is expressed as a function of just the scalar components of all the superﬁelds, φn; and ψn are the corresponding fermions. This part of the Lagrangian contains the usual SM Yukawa terms plus their supersymmetric counterparts. E.g. from the u type Yukawa one gets − y uQ¯ H + y uQ˜¯ H˜ + y u¯Q˜ H˜ . u · u u · u u · u Scalar potential. It arises from the general expression • ∂W(φ) 2 2 V = + φ ∗(t )nmφm . ∑ ∂φ ∑ n A n n nm This includes terms like y 2 Q˜ H 2 + y 2 u˜¯ H 2 + y 2 u˜¯Q˜ 2 + y u˜¯Q˜ + µH 2 | u| | · u| | u| | · u| | u| | | | u d| and similar expressions for down-type quarks and leptons. Usual pure-gauge Standard Model interactions coming from the kinetic terms for φ • and ψL plus additional gaugino – gauge-boson – gaugino vertices, ¯ µ λaγ CabcAbµ λc

where a,b,c are group-labels of the corresponding adjoint representation and Cabc is the group structure-function. Matter-Gaugino interactions: They have the general structure • √ i 2 ∑ (ψ¯nL(ta)nmλa)φm + h.c., Anm including terms like (uT¯ g˜)u˜ +(dT¯ g˜)d˜+(uT¯ W˜ +)d˜+(uT¯ W˜ 0)u˜ +(uT¯ B˜0)u˜ + SU (3) SU (3) SU (2) SU (2) U(1) ··· On the other hand, the SUSY soft-breaking Lagrangian is parametrised by the soft terms, namely:

Gaugino masses for each gauge group: • 1 (M g˜a g˜a + M W˜ a W˜ a + M B˜ B˜ + h.c.), (2.2) − 2 3 · 2 · 1 ·

where Mi are assumed to be real, are all real, so they do not introduce any new CP- violation. However they can be either positive or negative. 8 Supersymmetry and Lepton Flavour Violation

Squark (mass)2 terms: • m2 Q˜† Q˜ m2 u˜¯† u˜¯ m2 d˜¯† d˜¯ ; (2.3) − Qi j i · j − ui˜¯ j Li Lj− dij˜¯ Li Lj

Slepton (mass)2 terms: • m2 L˜ † L˜ m2 e˜¯† e˜¯ ; (2.4) − Li j i · j − ei˜¯ j Li Lj Higgs (mass)2 terms: • m2 H† H m2 H† H (bH H + h.c.) (2.5) − Hu u · u − Hd d · d − u · d † + 2 0 2 + 0 0 where H H = H + H (and similarly for H ) and H H = H H− H H . u · u | u | | u | d u · d u d − u d Trilinear scalar couplings: • aiju˜¯ Q˜ H + aijd˜¯ Q˜ H + aije˜¯ L˜ H + h.c. (2.6) − u Li j · u d Li j · d e Li j · d

Admittedly, the soft terms introduce a huge amount of parameters in the theory. How- ever, extensive regions of the parameter space are in fact excluded phenomenologically. In particular, generic values of most of the new parameters trigger processes of ﬂavour changing neutral currents (FCNC) or CP-violation, at levels that are excluded by experiments. For ex- 2 2 † ˜ ample, if the matrix mL˜ in (2.4) has a non-suppressed off-diagonal term such as (mL˜ )eµ e˜LµL (in the basis where charged-lepton masses are diagonal), then µ e + γ occurs at unac- → ceptable rates. These kinds of restrictions partially justify the assumptions of the popular “constrained MSSM” (CMSSM), which amounts to start with complete universal soft terms at the uniﬁcation scale (MX ) M3 = M2 = M1 = m1/2; (2.7)

2 = 2 = 2 = 2 = 2 = m2 , (2.8) mQ˜ mu¯˜ md¯˜ mL˜ me¯˜ 01 2 = 2 = 2 mHu mHd m0; (2.9) and au = A0yu, ad = A0yd, ae = A0ye. (2.10) Of course, such full universality is not required to get consistency with the previous phenomenological restrictions, but it is a possibility that can arise in some well-motivated scenarios of SUSY breaking and transmission to the observable sector, e.g. gravity mediation in minimal supergravity or dilation-dominance in string-inspired supergravity. On the other hand, despite the initial diagonal mass matrices for squarks and leptons, they get off-diagonal entries along the RG evolution from the MX to the electroweak scale, triggered by the various (ﬂavour-violating) Yukawa couplings. 2.1 Elements of SUSY 9

2.1.1 Neutrino masses and mixings

Assuming neutrinos as Majorana particles, the ﬂavour eigenstates, ν , can be expressed as | α linear combinations of the mass eigenstates, ν | i 3 να = ∑ Uα∗i νi (2.11) | i=1 | where U is the unitary matrix (known as the PMNS matrix) that diagonalises the neutrino mass-matrix written in the ﬂavour-basis. U can be parametrized according to the following decomposition: U = V diag(eiφ1 ,eiφ2 ,1), (2.12) · where the unitary matrix V has the form

iδ c13c12 c13s12 s13e− iδ iδ V = c23s12 s23s13c12e c23c12 s23s13s12e s23c13 . (2.13)  − − iδ − iδ  s23s12 c23s13c12e s23c12 c23s13s12e c23c13  − − −  with s sinθ , c cosθ . The present experimental ranges for the three mixing angles ij ≡ ij ij ≡ ij are

sin2(2θ )=0.857 0.024, sin2(2θ ) > 0.95, sin2(2θ )=0.098 0.013 (2.14) 12 ± 23 13 ± Concerning the mass eigenvalues, the experiments of neutrino oscillations provide precise information on two squared-mass differences. Denoting m3 the most split mass-eigenvalue, these are given by

2 2 5 2 m2 m1 =(7.50 0.20) 10− eV | 2 − 2| +±0.12 × 3 2 m3 m2 =(2.32 0.08) 10− eV (2.15) | − | − × The absolute scale of the neutrino masses is not known, though we have several upper bounds coming from from tritium beta-decay, neutrino less double beta-decay and cosmological (CMB) limits. Roughly speaking, the heaviest neutrino should be at most 0.3 eV. This ∼ allows for two kinds of neutrino mass- hierarchy: m3 > m1,m2 (“normal hierarchy”) or m3 < m1,m2 (“inverted hierarchy”). The remarkable smallness of the neutrino masses cries out for a theoretical explanation. In this sense, the Seesaw mechanism has probably became the most popular one. Next we describe the Seesaw framework in the supersymmetric context.

2.1.2 Supersymmetric See-saw The Seesaw mechanism is based on the assumption that the neutrinos have standard Yukawa- like interactions, which requires to introduce one right-handed neutrino per family, plus a 10 Supersymmetry and Lepton Flavour Violation

Majorana mass terms for the latter. These are described by the following extended superpotential 1 W = W (νc)T Mνc +(νc)TY L H , (2.16) 0 − 2 R R R ν · 2 where W is the superpotential (2.1), and M is a 3 3 matrix accounting for the masses of 0 × the right-handed neutrinos2. The diagonalization of the full 6 6 neutrino mass-matrix gives × 2 2 two mass eigenstates (per generation), one with masses of order Yν v v, and the other with M mass of order3 M. Since the right-handed Majorana mass-matrix does not have electroweak origin, it can in principle take arbitrary large values, which translates in very suppressed masses for the light ( left-handed) neutrinos. ∼ Alternatively, one can integrate-out the right-handed neutrinos, thus obtaining the effective Lagrangian with the mass term for the three light left-handed neutrinos. Let us now sketch this calculation

The effective generating functional Zeff is deﬁned in this case according to the following expression: c 2 4 c [dN ] exp i d xL(N (x),φi(x)) eiZeff R R (2.17) c 2 4 L c ≡ [dNR] exp (i d x (NR(x),0)) c c where NR are the basis of right-handed neutrinos where M is diagonal, and L(NR(x),φi(x)) is the Lagrangian obtained from the superpotential (2.16). So the procedure is now to complete the square, which can be done by the following rearrangement:

1 1 d4x L = d4x (Nc)T DNc + (Nc)TY L H + h.c. −2 R R 2 R ν · 2 4 c 1 T c 1 = d x N D− (Y L H ) D N D− (Y L H ) R − ν · 2 R − ν · 2 4 T 1 d x(Y L H ) D− (Y L H ) (2.18) − ν · 2 ν · 2 where D ∂/ + M, and D 1 = ∆ (x y), with ≡ x − F − (∂/ + M)∆ (x y)= δ (4)(x y). x F − − − Now, redeﬁning a new heavy ﬁeld as

c c 1 N N D− (Y L H ) R ≡ R − ν · 2 2If one imposes conservation of lepton number, this Majorana term is forbidden, but since lepton number is a global symmetry (which is accidentally conserved in the pure SM) there is no reason why it should be respected. 3This is an estimation for the case of 1-generation, where M is the mass of the unique right-handed neutrino. 2.1 Elements of SUSY 11

c 2 c 2 we see that [dNR] = [dNR ] ; thus 1 1 [dNc]2 exp i d4x (Nc)T DNc + (Y L H )T D 1(Y L H ) R −2 R R 2 ν · 2 − ν · 2 eiZeff = 1 [dNc]2 exp i d4x (Nc)T DNc R −2 R R i 4 T 1 = exp d x(Y L H ) D− (Y L H ) 2 ν · 2 ν · 2 i = exp d4yd4x(Y (x)L(x) H (x))T ∆ (x y)(Y (y)L(y) H (y)) . 2 ν · 2 F − ν · 2 (2.19)

Finally one may obtain a local Lagrangian by noting that the heavy particle propagator is peaked at small distances. So for J(x) Y (x)L(x) H (x) we can Taylor expand ≡ ν · 2 J(y)=J(x)+(y x)µ [∂ J(y)] + ... − µ y=x and keep to a good approximation the leading term. Also we can write the propagator as

e ip (x y) e ip (x y) d4y∆ (x y)= d4yd4 p − · − = d4yd4 p − · − F − − p + M − M( + p/M) 1 so for small momenta compared to the scales in M one has 1 d4y∆ (x y) (2.20) F − ≈−M giving the result 1 4 T 1 Z = d x(Y L H ) M− (Y L H ) (2.21) eff ν · 2 ν · 2 2 The relation (2.21) is simply the Weinberg, dim=5 operator, which accounts for the light neutrino masses. Once the scalar Higgses of the MSSM acquire vevs, we have:

T 1 0 2 M = Y M− Y H . (2.22) ν ν ν 2

2.1.3 LFV processes in the supersymmetric See-saw The mixing in the neutrino sector leads, in analogy with the Kobayashi-Maskawa effect for quarks, to processes for which the lepton ﬂavour is violated (LFV processes). In the context of the SM those processes are strongly suppressed due to the tiny mass of the neutrinos. However, in the supersymmetric case, even starting with universal conditions at MX , the non-diagonal neutrino mass matrices contribute along the RG running to the appearance of off-diagonal entries in the slepton mass matrices; which triggers LFV processes. Let us focus 12 Supersymmetry and Lepton Flavour Violation

Figure 2.1: Feynman diagrams which give rise to li l jγ. ν˜X and l˜X represent the mass eigen- 0 → states of sneutrinos and charges sleptons, and χ˜ −, χ˜ the charginos and neutralinos. (Taken from “Oscillating neutrinos and µ e,γ”. J.A. Casas and A. Ibarra. Nucl.Phys. B618 (2001) 171-204). →

Figure 2.2: Dominant diagram in the process l l γ, in the mass-insertion approximation. L˜ i → j i are the slepton doublets in the basis where the gauge interactions and the charged-lepton Yukawa couplings are ﬂavour-diagonal. (Taken from “Oscillating neutrinos and µ e,γ”. J.A. Casas and A. → Ibarra. Nucl.Phys. B618 (2001) 171-204).

on the most popular ones, namely l l γ processes (i = j are family indices). At 1-loop i → j level, the relevant diagrams are shown in Figure 2.1. The expression for the amplitude of the process, in terms of effective vertices is given by:

α 2 1 1 β 2 2 T = eε ∗u¯ (p q) q γ (A P + A P )+m iσ q (A P + A P ) u (p) (2.23) i − α L L R R lj αβ L L R R j where e is the charge of the electron and muon; εα is the polarisation vector of the photon; ui(p) is the wave-function of a spinor i with momentum p; q is the momentum of the photon; PL,R are the usual helicity projectors, and (n) (c) AL,R = AL,R + AL,R (2.24) are the amplitudes of the processes. If the emitted photon is on-shell4, then the ﬁrst term

4The photon can be virtual in other types of processes, e.g. µ e conversion, where the emitted photon − interacts with the recoiling nucleus. 2.1 Elements of SUSY 13

on the RHS of (2.23) gives no contribution to the effective amplitude and so only the A2 coefﬁcients in eq. (2.23) are relevant. Hence we drop the super-index “2” in what follows. The super-indices (n) and (c) label neutralino or chargino contributions, respectively. The L(left) or R(right) labels are associated to the chirality of the interacting fermions. Now, in the chargino-neutralino basis, the A amplitudes read: −

(n) 1 1 L(l) L(l) 1 ∗ AL = 2 2 NiAx NjAx 4 (2.25) 32π m˜ 6(1 XAx) lx − (1 6X + 3X2 + 2X3 6X2 lnX ) × − Ax Ax Ax − Ax Ax M 0 L(l) R(l) χ˜A 1 2 + N N ∗ (1 X + 2X lnX ) iAx jAx m (1 X )3 − Ax Ax Ax lj − Ax

(c) 1 1 L(l) L(l) 1 ∗ AL = 2 2 CiAx CjAx 4 (2.26) −32π m 6(1 XAx) ν˜x − (2 + 3X 6X2 + X3 + 6X lnX ) × Ax − Ax Ax Ax Ax M L(l) R(l) χ˜A− 1 2 + C C ∗ ( 3 + 4X X 2lnX ) iAx jAx m (1 X )3 − Ax − Ax − Ax lj − Ax

(n) (n) AR = AL (2.27) L R ↔ (c) (c) AR = AL (2.28) L R ↔ 2 2 where XAx M 0 /m˜ (with A labelling the gaugino, and x labelling the slepton), and the C, ≡ χ˜A lx N matrices are deﬁned as:

R ν L mli ν CiAx = g2(OR)A1UX,i, CiAx = g2 (OL)A2UX,i , (2.29) − √2mW cosβ with A = 1,2 and X = 1,2,3; and

R g2 l mli l NiAx = [ (ON)A2 (ON)A1 tanθW ]UX,i + (ON)A3UX,i+3 , (2.30) −√2{ − − mW cosβ }

L g2 mli l l NiAx = (ON)A3UX,i + 2(ON)A1 tanθWUX,i+3 (2.31) −√2{mW cosβ }

with A = 1..4 and X = 1..6. The matrices OR,L (ON) are the ones that diagonalise the chargino (neutralino) mass matrix. The matrices Uν and Ul are the ones diagonalising the sneutrinos and charged sleptons, respectively. 14 Supersymmetry and Lepton Flavour Violation

In the previous expressions, the ﬂavour-violating sources are in the non-diagonal structure of the C, B matrices, which in turn comes from the off-diagonal entries in the slepton mass matrices generated along the RG running. A more intuitive way to see that is to evaluate the previous diagrams in the mass-insertion approximation, as shown in Fig. 2.2. Then it is (m2) clear that they are proportional to l˜ ij. The dominant contribution to these off-diagonal entries is usually due to the neutrino-Yukawa interactions (unless the scale of the right-handed Majorana masses is much below MX ) and has the form

2 1 2 2 † MX (m )ij (3m + A ) (Y ) ln δ (Y ) (2.32) l˜ ⊃−8π2 0 0 ν ik M kl ν lj k

where Mk is the mass of the k-th right-handed neutrino (k = 1...3).

2.2 Leptogenesis

Observations indicate that the number of baryons present in the universe is unequal to the number of anti-baryons. Indeed the stars, galaxies and clusters consist essentially of matter and there is no antimatter. There are good reasons to believe that this asymmetry has been generated dynamically instead of being caused by an asymmetry in the initial conditions; the most important being maybe the one related to inﬂation, which is nowadays quite accepted by the community. In this scheme, any primordial baryon asymmetry would have been exponentially diluted away by inﬂation. The ingredients that have to be present in order for a dynamical asymmetry to be generated are called the Sakharov conditions: 1) There has to be a baryon number (B) violation. This condition is required in order to evolve from an initial state with ∆B = 0 to a state ∆B = 0. 2) C and CP violation, because otherwise the processes creating baryons would have the same rate as those creating anti-baryons. And 3) departure from equilibrium, since in chemical equilibrium the inverse processes would erase the asymmetries generated. Actually the Standard Model have all the ingredients present but it fails in explaining an asymmetry as large as the one observed. The observed baryon asymmetry is:

nB nB¯ 10 η − =(6.19 0.15) 10− , (2.33) B ≡ n ± × γ 0 where nB,nB¯ is the number density of baryons and anti-baryons, respectively, and nγ is the number density of photons. The subscript 0 denotes values at present time. There are different ways to infer the previous value of ηB. One of them is using Big Bang Nucleosynthesis (BBN) predictions. It turns out that the primordial abundances of the light elements (D, 3He, 4 7 He, and Li) are strongly dependent on ηB, but most surprisingly, there is a range of ηB values which is consistent with all four abundances, and it is in total agreement with (2.33). 2.2 Leptogenesis 15

The second method takes into account the Cosmic Microwave Background (CMB) radiation, whose main features can be described as “acoustic” waves of the photon-baryon ﬂuid. The observed spectrum strongly depends on the baryon number density nB and thus on the asymmetry ηB. It is impressive that the requirement on nB coming from CMB is perfectly in agreement with that coming from BBN. Leptogenesis comes as one of the alternatives of New Physics for the generation of the baryon asymmetry. In this framework, the asymmetry is ﬁrst generated in the lepton sector. For example, in the Seesaw mechanism, the Yukawa interaction of the neutrinos provide the necessary source of CP violation. The rate of those interactions can be slow enough in order not to reach the equilibrium with the thermal bath. Lepton number violation comes directly from the RH Majorana mass term in the Lagrangian. Finally, the Standard Model sphaleron processes play a crucial role in partially converting the lepton asymmetry into a baryon asymmetry. Thus, in the context of Seesaw, leptogenesis is qualitatively unavoidable, and the question of whether or not it can successfully explain the baryon asymmetry is a quantitative one.

We consider leptogenesis via the decays of N1, which is the lightest RH neutrino. When the decay is into a single ﬂavour α, N L φ or L¯ φ †, one can write 5 1 → α α

nB nB¯ 135ζ(3) Y∆B = − 4 Csphal η ε , (2.34) s 4π g × × × where s is the entropy density. Here g is the number of degrees of freedom of the theory. The other factors in this equation deserve a more detailed explanation. The quantity ε is the CP asymmetry factor, which is deﬁned as

Γ(N φL) Γ(N φ †L¯) ε 1 → − 1 → . (2.35) ≡ Γ(N φL)+Γ(N φ †L¯) 1 → 1 → It is straightforward to see that ε = 0 if the above rates are computed at tree-level. However the interference between the tree-level and the 1-loop contributions induces a non-vanishing ε. More precisely, if we consider the amplitude M = c0A0 + c1A1, where ci,Ai are the coefﬁcients and amplitudes of the tree-level and 1-loop processes, respectively, ε is given by: ˜ Im(c0c∗)2 Im(A0A∗)δdΠL,φ ε = 1 1 (2.36) 2 2 ˜ c0 A0 δdΠL,φ | | | | where 3 3 ˜ 4 (4) d pL d pφ δ =(2π) δ (Pi Pf ), dΠL,φ = 3 3 − 2EL(2π) 2Eφ (2π) and Pi,Pf are, respectively, the incoming 4-momentum (i.e. PN1 ) and the outgoing 4-momentum (i.e. Pφ + PL). Notice that ε is directly dependent on the imaginary parts of the amplitudes,

5For more details see section 3.4.1 16 Supersymmetry and Lepton Flavour Violation

encoding the CP violation. For the Seesaw model the result is:

† 2 1 ∑ Im [(YY ) j] g(x j) j=2,3 { 1 } ε = † (2.37) 8π 8π(YY )11

where Y is the neutrino Yukawa matrix and x M2/M2. The g(x) function, within the j ≡ j 1 MSSM, is 2 1 + x g(x)=√x log 1 x − x − Coming back to eq.(2.34), the quantity η parametrises the out-of-equilibrium dynamics, and is called the efficiency factor. Indeed, if the processes are almost in thermal equilibrium, inverse processes can probably washout the asymmetries generated by the N1 decays, thus making the asymmetry generation less efficient. A simple criterion6 for evaluating if a process thermalises with the particles of the thermal bath, is to compare its reaction rate Γ with the expansion rate of the universe, H. In this way, processes for which Γ > H occur more rapidly than the expansion rate of the universe and those thermalise. In the same way, if Γ < H those process do not reach thermal equilibrium. However, in general, one needs to solve the full relevant Boltzmann equations in order to check if a process actually thermalises or not. The efficiency factor η comes from the solution of the Boltzmann equations for the number density of the lightest RH neutrino, nN , and the B L asymmetry, nB L, in such 1 − − a way that in the limit of an initial thermal abundance of N1 and no washout, it tends to unity. It actually smoothly interpolates between two regimes: the strong washout regime, in which the reaction rate of the N1 decay, ΓD, is larger than H, and thus the inverse decay strongly erases the asymmetry; and the weak washout regime, where ΓD < H, and the inverse processes are not efficient enough to affect the asymmetry generated by the direct ones. In the weak washout regime the final asymmetry in general depends on the initial conditions. The two main quantities involved in the computation of η are precisely related to Γ and H: 2 † 2 2 8πv (YY )11v 8πv 3 m˜ 2 ΓD = , m 2 H(T = M1) 10− eV . (2.38) ≡ M1 M1 ≡ M1 In terms of these, the η parameter can be expressed as:

1 1.16 − 2˜m 2m η = + (2.39) m m˜

Finally, the Csphal term in (2.34) has to do with the conversion of part of the L generated by the leptogenesis, into B by means of Standard Model sphalerons, which render B + L to zero, while preserving the total B L. −

6I will elaborate much more around this in the next chapter, dedicated to Dark Matter. CHAPTER 3

DARK MATTER PHENOMENOLOGY

The starting point of the subject goes back to 1933. In that year, the Swiss astrophysi- cist Fritz Zwicky discovered a mismatch between luminous and gravitational masses of the Coma galaxy cluster, and called this extra content “dark matter” (DM). Today, 80 years after that discovery, DM is still evading detection, although at present the new generation of experiments could make us feel that the discoveries are closer than ever. In the second part of this thesis, I deal with the phenomenology of DM, focusing on the implications of colliders data, as well as on the study of sub-products of DM annihilation in our Galaxy. In my works I attempted to contribute to the complementarity between these two approaches in constraining DM properties. This chapter is organised as follows. In section 1 I comment about well stablished experimental evidences of missing mass in the universe. Section 2 is devoted to the DM candidates of different nature that have been considered. In section 3, I deal with the dynamics and thermodynamics of the early universe. Section 4 is devoted to the Boltzmann formalism which is necessary to study the evolution of DM abundance. Finally, in section 5, I present the main experiments covering the different strategies to search for DM candidates.

3.1 Experimental evidences of Dark Matter

After Zwicky’s discovery, there were almost 40 years where the scientiﬁc community basically ignored the “missing mass problem”. However in the early 70’s Vera Cooper Rubin measured the velocity curves of edge-on spiral galaxies to an unprecedented accuracy. She demonstrated that most stars in spiral galaxies orbit the centre at roughly the same speed. This was consistent with an embedding of the spiral galaxies in a much larger halo of invisible mass. Different methods have been used to estimate the matter density of the universe. One 18 Dark Matter phenomenology

of the oldest is the so-called mass-to-light (M/L) ratios. Using the second Newton’s Law, one can estimate the rotation velocity vl of a luminous mass Ml(r) contained in a radius r, by computing Ml(r) as a function of the mass density which is extracted from line-of- sight radial luminosity distribution. This velocity vl = GMl(r)/r is then compared with actual observations of the kinematics of stars at different radii, i.e. the rotation curves. In the absence of non-luminous matter, the theoretical prediction should coincide with the observed rotation curves. However, it has been demonstrated that galaxies, clusters of galaxies and super-clusters have a significant fraction of this missing component. Starting from small scales (stellar sizes, say) the observations show a linear increase of M/L ratio with R. However, as more large-scale measurements are considered, that initial linearity disappears, and beyond R 1 Mpc the M/L ratios approach to a plateau, consistent ≈ with a total matter density Ω h2 0.3. m ≈ There are other methods which independently probe and quantify the presence of dark matter in the universe. One of them is the so-called “gravitational lensing” effect. This effect is a direct consequence of General Relativity, where the trajectory of photons is affected by the curvature of space-time induced by the presence of massive objects. As a consequence, light rays leaving a source in different directions are sometimes focused on the same spot (the observer here on Earth) by an intermediate galaxy or cluster producing multiple distorted images of the source. A third method is based on the measurement of the baryons-to-matter ratio, defined by f Ω /Ω . Assuming that the luminous (baryonic) matter is measured, then the aim b ≡ b m is to measure independently the ratio fb. It has been demonstrated that the visible matter in galaxies largely resides in hot ionised gas, with only a small fraction in stars. This means that baryon-to-matter is well approximated by the gas-to-matter ratio, which can be measured via: a) X-ray spectrum; by measuring the mean gas temperature from the overall shape of the X-ray spectrum, and the absolute value of the gas density from the X-ray luminosity; and b) the Sunyaev-Zeldovich effect; by which the CMB photons can Inverse-Compton scatter off the hot electrons of the gas, lowering down the intensity of CMB radiation as compared to the unscattered CMB. These independent methods, among others, provide compelling evidence that the baryon density is of order 5% of the critical density, while the total matter density is about five times larger. This clearly states that most of the matter in the universe is in the form of a non- luminous, “dark” matter.

3.2 Dark Matter candidates

3.2.1 Baryonic Dark Matter?

One possibility which is nowadays disfavoured by experiments, is the so called MACHOs (from MAssive Compact Halo Objects), which are purely baryonic objects, however very 3.2 Dark Matter candidates 19 difficult to detect. Indeed, our ability to observe stars has its limitations, because below some level of luminosity we can not detect anything. It turns out that the total mass density due to stars is well approximated by ρ m 0.35/0.35, where m is the lowest observable stellar mass. It s ≈ c− c means that the reduction of the lowest threshold of detectable stellar mass by a factor of 2 results in an increase of the stellar mass density of a factor 3.6 . Brown dwarfs are an example of a MACHO. Stars are born from self-gravitating clouds of gas. Gravitational collapse of this gas will cause the temperature to rise until nuclear burning can begin, after which the star is born. However, it can happen that before the temperature rises up to this point, the electron degeneracy prevents the collapse to continue. Electron degeneracy is a consequence of the Pauli exclusion principle, and causes the electrons in dense matter to be in a state of continuous motion, increasing the pressure of the gas. This happens for objects whose mass is order 0.08M or smaller. ⊙ White dwarfs are more popular. They form from the collapse of the stellar core once the nuclear burning ceased. Their mass is typically smaller than 1.4M , and their death as a star is followed by a helium flash which blows off the outer parts of the⊙ star, thus creating a planetary nebula. However the core continues in its own gravitational collapse until electron degeneracy is reached. They gradually cool down, becoming frequently difficult to observe 3 4 due to its low luminosity (typically 10− to 10− of the Sun’s one). Neutron stars are another example of a MACHO. They are the created after the gravitational collapse of a supernova, remaining afterwards typical masses in the range 1.4M ⊙ M 2M . In this case, the collapse of the dead star is stronger and electrons and protons can get sufficiently⊙ closely packed that neutrons are formed. Then, the collapse continues until neutron degeneracy is reached. This objects can have typical sizes of 30 km diameter. Finally, if the star’s mass is larger than 2M , the gravitational collapse is so strong that the radius of the objects shrinks under the Schwarzschild⊙ radius, thus becoming a black hole. All of these are compact sub-stellar objects that can not be directly observed, thus being a form of “dark matter”. However, even with their contributions added to the visible matter, the total matter content of the universe is significantly short. In summary, all the above points to the presence of a non-baryonic dark matter.

3.2.2 Non-baryonic Dark Matter

The strongest hint about the non-baryonic nature of DM comes from Big Bang Nucleosyn- thesis (BBN). Indeed, having extra baryonic matter would spoil the success of BBN predictions. Thus, particles not affecting BBN have to be either very rare or very weakly coupled (WIMPs). Dark Matter can be classified according to its kinetic energy at the decoupling time: Hot Dark Matter (HDM). The Standard Model’s candidate for DM, the neutrino, enters into this category. When the temperature in the universe was about 1 MeV, the rate 20 Dark Matter phenomenology of interactions such as νe νe, which keep neutrinos coupled to the rest of the plasma, ↔ dropped below the rate of the expansion of the universe. Therefore, the neutrinos were relativistic by then (thus they are “hot”). These velocities were larger at high redshifts, which resulted in major effects on the development of self-gravitating structures. In this case, the structure formation occurs with larger super-clusters forming first in flat sheets and subsequently fragmenting into smaller pieces to form clusters, galaxies and stars. The predictions of HDM model strongly disagree with observations. Cold Dark Matter (CDM). Most of the DM candidates coming from theories BSM fall into this category. Popular examples are neutralinos, coming from SUSY theories, sterile heavy neutrinos, and extra scalars coming from theories with minimal extra content. Most of WIMP models lie also in this option. Cosmological observations strongly favour this option, as it implies a bottom-up structure formation, from smaller structures forming first (stars, galaxies) to the large-scale structures that with observe today. However, there are two important discrepancies between predictions of the CDM model and observation of galaxies and their clustering: a) the cuspy halo problem (CDM predicts the central density slopes of galaxies to be much steeper than the observed ones); and b) the missing satellites problem (CDM simulations predict an excess in the number of small dwarf galaxies, those having about 1/1000 of the Milky Way mass). Warm Dark Matter (WDM). This model interpolates between HDM and CDM, and examples of these can be sterile neutrinos which are not so heavy, and gravitinos. However if the gravitino is not very heavy (order TeV or less), it can lead to the so-called gravitino problem. In the gravitino case, if it is stable, then its density turns out to be much larger than the DM density observed. If it is unstable, since its interactions are only gravitational, the decay rate is very suppressed, and its life-time is much larger than the era of BBN, thus its decay products (photons, hadrons) can have disastrous effect on BBN predictions.

3.3 Early universe

In this section I brieﬂy describe the dynamics in the early universe, from the time when inﬂation is over. At the end of the day the relic density of a Dark Matter particle will be derived from the Boltzmann equation.

3.3.1 The dynamics of the universe’s evolution

The evolution of the universe as a whole can be studied taking into account only its gravitational interactions. Let us consider the Einstein equation:

1 R g R = 8πGT , (3.1) αβ − 2 αβ αβ 3.3 Early universe 21

where Rαβ is the Riemman tensor, R stands for the scalar curvature and Tαβ is the energy- momentum tensor. Two particular useful forms of the latter are the so-called dust approximation: Tαβ = ρuα uβ , (3.2) and the perfect ﬂuid approximation:

Tαβ =(ρ + P)uα uβ + Pgαβ , (3.3) where ρ is the energy density, P the pressure, and u x˙ . Dust approximation provides a α ≡ α good description of a matter-dominated universe, whereas the perfect ﬂuid approximation is good when dealing with the radiation-dominated era. When solving (3.1) in the perfect ﬂuid approximation1, we arrive to the set of Fried- mann equations

8πG a˙2 + k = ρa2, ρ˙ + 3(ρ + P)H = 0, P = P(ρ) , (3.4) 3 where a is the scale factor in the Friedman-Lemaitre-Robertson-Walker (FLRW) metric tensor, k =+1,0, 1 for a positive, ﬂat or negative curvature of the universe, respectively; and − H stands for the Hubble parameter H a˙/a. ≡ The solution of the 2nd Friedmann equation shows how the density scales with the radius a: a) In matter domination, P = 0 and

a˙ d ρ˙ + 3ρ = 0 (ρa3)=0 a ⇒ dt 3 ρ ∝ a− (3.5)

1 b) In radiation domination, P = 3 ρ and a˙ d ρ˙ + 4ρ = 0 (ρa4)=0 a ⇒ dt 4 ρ ∝ a− (3.6)

Then, knowing ρ = ρ(a), we can solve the 1st Friedmann equation to see the evolution of the scale factor a(t), in a ﬂat, closed or open universe. The result is represented in ﬁg. 3.1. On the other hand, the 1st Friedmann equation can be rewritten as:

k 8πG ρ 2 2 = 2 ρ 1 1 Ω 1 , (3.7) a H 3H − ≡ ρc − ≡ −

1Note that the dust approximation is the pressure-less limit of the perfect ﬂuid one. 22 Dark Matter phenomenology

Figure 3.1: Evolution of the scale factor a(t) for different geometries of the universe. (Taken from “Course on Astrophysics”. Balsa Terzic. PHYS 652 -2008).

2 where we have deﬁned the critical energy-density ρc 3H /(8πG). The density ratio Ω = ≡ 2 Ωm + Ωr + ΩΛ is composed by matter, radiation and dark-energy . In terms of how the different energy densities scale with a, and considering3

ρ a n a n 3H2 a n i = 0 = (0) (0) 0 = 0 (0) 0 ( ) ρi ρc Ωi Ωi ρ 0 a ⇒ a 8πG a i we have (0) (0) (0) 2 2 Ωr Ωm Ωk (0) H = H0 4 + 3 + 2 + ΩΛ (3.8) a a a where we have normalised the scale factor today a = 1, and Ω k/a2H2. 0 k ≡− Age of the universe From eq.(3.8) we can estimate the age of the universe. Considering that it is ﬂat (k consistent with zero) we have

1/2 1 (0) (0) − 1 Ωr Ωm ( ) t = da + + a2Ω 0 . (3.9) 0 H a2 a Λ 0 0 Solving (3.9) numerically and considering that today, according to observations, we have ( ) ( ) ( ) Ω 0 0,Ω 0 0.27,Ω 0 0.73, H 0.71 km/s/Mpc, we can estimate the value of t r ≈ m ≈ Λ ≈ 0 ≈ 0 ≈ 4 1017s 13 109 years. × ≈ × Matter-radiation equality. Since we know how the energy density of radiation ρr be- haves, we can compare it to the energy density of matter ρm in order to know when both

2After including the cosmological constant Λ in the Einstein equation (3.1). 3The subindex i corresponds to matter, radiation or dark-energy, and n = 3,4,0 for matter, radiation and (0) dark-energy cases, respectively. a0 and H0 are the scale factor and Hubble parameter today, as well as ρ and Ω(0) refer to the values today. 3.3 Early universe 23

components in the universe were comparable. We know that ρr/ρm ∝ a0/a = 1 + z, where a0 is the scale factor today. Thus we can write ρ ρ ρ r = r (1 + z) 1 + z = m , (3.10) ρ ρ ⇒ eq ρ m m today r today where z is the redshift at the time of the matter-radiation equality. We can evaluate those eq densities today: 2 today 3H0 Ωm 32 2 3 ρ = 1.8 10− Ωmh kg/cm (3.11) m 8πG × 2 today 4 today π (kBTγ ) 37 3 ρr = g 7.8 10− kg/cm . ∗ 30 c5h¯ 3 × Now, evidence suggests that Ω h2 0.13, giving z 3200, or, in terms of temperature4, m ∼ eq ∼ T 0.74eV, or approximately 8700 K. In time, integrating (3.9) until a = 1/(1 + z ) eq ∼ eq eq ≈ 0.0003, it would be approximately 7 10 6secs. × − Radiation-Dark Energy equality. A similar estimation can be made in order to know when radiation and dark energy became equal: ( ) ρ Ω 1 Ω 0 4.12 10 5 r = r = r = × − 4 (0) (0) ρΛ ΩΛ a 4 2 ΩΛ a h ΩΛ aeq a ρr =1 0.1 . (3.12) ≡ | ρΛ ≈ This is translated into 5.4 108 years, redshift z 9, and a temperature of 27K approximately, × ≈ or 0.002eV. Matter-Dark Energy equality. Finally, the last equality happened “soon” ago, after which the universe became dark energy-dominated and will continue to be like this. The moment in which this occurred is ( ) ρ Ω 1 Ω 0 0.27 m = m = m = 3 (0) 3 ρΛ ΩΛ a 0.73a ΩΛ aeq a ρm =1 0.73 . (3.13) ≡ | ρΛ ≈ This is approximately 4 109 years ago, at redshift z 0.37, when the universe had a tem- × ≈ perature of 3.7 K, or 0.0003 eV.

3.3.2 Thermodynamics Although the evolution of the universe as a whole can be described by its gravitational interactions, this picture is not sufﬁcient to study the behaviour of realistic particle interactions at high temperatures. Here is where the thermodynamic description comes into play.

4 T(z)=T0(1 + z). 24 Dark Matter phenomenology

When studying the properties of the early universe one often faces the need of differ- entiating between different kind of equilibria. Thermal, kinetic and chemical equilibrium are commonly referred in the literature, but a clear definition is usually missing. This may be due to the subtle differences between them. Thermal equilibrium. It is the state where all the species of the system share the same temperature. Then, the particles A of the system can be described by the distribution function 1 fA = gA , (3.14) e(EA µA)/T 1 − ± where (+) is for Fermi-Dirac statistics and ( ) for Bose-Einstein. In the non-relativistic − limit the above expression tends to f exp[ (E µ )/T], which is called the Maxwell- A ≈ − A − A Boltzmann approximation, corresponding to the classical limit. Here µA is the corresponding chemical potential, to be defined below, EA is the particle’s energy, and gA is the spin degeneracy. The temperature T appearing in (3.14) is of course an essential ingredient in thermodynamics. For a system in thermal equilibrium, it is defined as T (∂U/∂S) , i.e. the change ≡ X of internal energy U of the system when varying the entropy S, while keeping constant the rest of macroscopic properties, here collected in X. At the classical level, the Equipartition Theorem directly relates the temperature with the average kinetic energy of the ensemble. However at the quantum level this theorem looses its validity. Still, for practical purposes, it is very useful to think about temperature as an idea of what the kinetic energy of particles are. Kinetic equilibrium. According to the tight correlation between temperature and average kinetic energy, kinetic equilibrium can be safely identified with thermal equilibrium. A system containing particles A and B is kept in kinetic (or thermal) equilibrium in elastic scatterings of the type A + B A + B. Once these reactions are not efficient enough, those → species kinetically decouple. This usually happens for very low temperatures. In the very early universe on the contrary, where the temperature is very high, all existing particles are described by a thermal bath, in which thermal equilibrium is present. Chemical equilibrium. Provided there exist interactions between different species in a system, the chemical equilibrium is present when the number density of reactants and products are constant in time. This kind of equilibrium is maintained by the detailed balance in the reaction A+A¯ B+B¯, for example, where direct and inverse processes are occurring ↔ at equal rates. In terms of chemical potentials it implies that µA + µA¯ = µB + µB¯. When, for example, the rate of the direct process becomes less than the rate of the inverse (B- annihilation) process, the particles B approaches a chemical decoupling. A rule of thumb to evaluate whether a particle A is in chemical equilibrium with the thermal bath TB, provided the reactions A + A¯ TBare present, is to compare the reaction ↔ rate Γ with the rate of expansion of the universe, given by H. Here Γ = n σv ¯ , where A AA TB n is the number density of the target particles, and σv the thermally averaged cross-section→ A of the reaction, to be defined in section (3.4). Indeed, ∆t = 1/Γ represents the mean time 3.3 Early universe 25 between two reactions to occur. On the other hand, if the universe changes its size by ∆a in precisely the time ∆t, we have ∆a/a = H∆t. Thus if the Hubble rate H Γ, the universe ∼ doubles its size in exactly the time one particle finds another target particle. If Γ >> H, then these reactions can produce and maintain the chemical equilibrium at the plasma temperature T.

The chemical potential is a thermodynamic quantity that can be defined whenever there are conserved charges in the system. A classical example is that of a box containing N particles. There, N is the conserved number in the reaction within the system, thus it makes sense to study how much the energy U of the system changes when extracting or introducing a particle. This is the standard definition of the chemical potential: µ dU/dN. At the quantum ≡ level, it is not the number of particles which can be conserved, but quantum numbers, for example, the Lepton Number L. Reactions like e+e γγ conserve L, thus it makes sense − → to define a chemical potential µL, which characterises how much the energy of the system changes, when introducing (or extracting) a lepton. However, even if not in a rigorous sense, it is possible to work with assigned chemical potentials, e.g. µL, in the case of L-violation, which were defined before L-violation occurs. In chemical equilibrium, the net chemical potential is locally conserved in a reaction A + B + ... C + D + ..., meaning that ↔

µA + µB + ... = µC + µD + ...

This can be used to relate unknown chemical potentials to each other. For example since photons are not conserved (say, in the reaction e + γ e + 2γ), in chemical equilibrium − → − µ = 0. This implies that if pair production or annihilation takes place, e.g. e+e 2γ, we γ − → have µ + = µ . e − e− Given the distribution function (3.14), the number density, energy density and pressure are given by

d3 p d3 p d3 p p 2 n = g f (p), ρ = g E (p) f (p), P = g | | f (p) . A A ( )3 A A A ( )3 A A A A ( )3 E (p) A 2π 2π 2π 3 A (3.15) From this we can see another consequence of the chemical potential. Indeed, it is straightforward to see that the particle asymmetry n n ¯ of a species A, in presence of chemical A − A equilibrium, (µ ¯ = µ ), is such that the asymmetry is zero if µ = 0, and vice-versa. This A − A A is closely related to the question of conserved charges discussed above. For example for leptons µ = µ = 0 µ + µ + µ + µ = 0 (3.16) L ∑ i 1 ¯1 2 ¯2+... i=lep ⇒ meaning that there is some lepton i which is not compensated by its ¯i counterpart, i.e. an asymmetry is present. 26 Dark Matter phenomenology

Coming back to expressions (3.15), it is useful to show their relativistic and non-relativistic limits for bosons B and fermions F. Relativistic species, small chemical potential:(m << T, µ << T)

gζ(3) 3 3 nB = T , nF = nB (3.17) π2 4 g 7 ρ = π2T 4, ρ = ρ (3.18) B 30 F 8 B ρ ρ P = B , P = F (3.19) B 3 F 3 Non-relativistic species:(m >> T) In this case the fermionic and bosonic nature of the particles is irrelevant, and

3/2 mT (m µ)/T n g e− − , ρ = mn, P nT (3.20) ≈ 2π ≈

Let us now focus on the entropy. For a system in thermal equilibrium, assuming small chemical potentials, the second Law of thermodynamics can be written as:

dU = TdS PdV Vdρ + ρdV = TdS PdV , (3.21) − ⇒ − where the internal energy U = ρV and V ∝ a3 is the volume of the system. On the other hand, the 2nd Friedmann equation can be written in terms of the volume too: dρ a˙ 1 dV = 3 (ρ + P)= (ρ + P) . (3.22) dt − a −V dt Substituting this in the time-derivative of (3.21) gives dV dV dS dV (ρ + P) + ρ = T P − dt dt dt − dt dS = 0 . (3.23) dt This means that in thermal equilibrium and in the absence of chemical potentials, the entropy is conserved. Consider then the entropy density s S/V. Substitution in (3.21) gives ≡ dV dρ Tds=(Ts ρ P) , − − − V but since in thermal equilibrium the quantities ρ and s depend only on the temperature, they do not change with the volume, so for every T: ρ + P s = . (3.24) T 3.3 Early universe 27

Let us compute the entropy density due to photons. Using (3.18) and (3.19) we have 2 s = π2g T 3 . (3.25) γ 45 γ γ In fact, from inspection of eqs.(3.17)-(3.20) we can see that the contribution to the total entropy density is strongly dominated by the relativistic species, so we can approximate the total entropy density as the contribution from just the relativistic degrees of freedom. Different species can indeed have different temperatures, if the mean energy differs from one species to other. However, in a state close to thermal equilibrium, those temperatures are expected to be quite similar. It is then useful to express the total entropy density as if it were only due to photons, but absorbing the difference between species in a global effective s degeneracy factor g , instead of the known gγ = 2: ∗ 2 2π s 3 s = g Tγ (3.26) 45 ∗ T 3 7 T 3 gs g i + g i . ≡ ∑ i T 8 ∑ i T ∗ i=bosons γ i= f ermions γ In the very same fashion, we can express the total energy density as:

2 π 4 ρ = g Tγ (3.27) 30 ∗ 4 4 Ti 7 Ti g gi + gi . ∗ ≡ ∑ T 8 ∑ T i=bosons γ i= f ermions γ

In order to see how those ratios Ti/Tγ behave, we need to study the evolution of temperatures for different particles. Since the entropy is conserved:

s 1/3 1 Tγ ∝ (g )− a− , (3.28) ∗ So in the very early universe, as long as no particle decouples from the plasma, gs is con- s ∗ stant and Tγ drops smoothly as 1/a. However when a particle decouples, g decreases, and 5 ∗ the temperature Tγ receives an extra contribution . Tγ thus decreases less slowly (as compared to 1/a) in the correspondent decoupling period. After a species is decoupled, its own temperature starts to depart from Tγ . Physical distances, dP, and comoving ones, dc, are related by dP = a(t)dc. So the momentum p1 of a particle at the time where the scale factor was a1, gets red-shifted to p = a1 p in later times, where a > a . Let us see thus how the temperature of a particle 2 a2 1 2 1 scales with a(t), after being decoupled from the plasma, assuming the universe had aD at time tD: 5This contribution comes from the annihilation of the particle in the decoupling process, which heats the plasma while ensuring entropy conservation. 28 Dark Matter phenomenology

a) in the case of relativistic particles (m << T), the kinetic energy is basically the momentum (EK = p), so the temperature of relativistic particles scales as T(t)=[aD/a(t)]TD; 2 b) in the case of non-relativistic particles (m >> T), the kinetic energy goes as EK ∝ p , such 2 that the temperature evolves as T(t)=[aD/a(t)] TD.

Neutrino decoupling Using this knowledge, we can study the neutrino decoupling. At high temperatures, neutrinos are kept in equilibrium with the plasma via weak interaction processes (e.g. νν¯ e+e me- ↔ − diated by a Z, or e ν¯ e ν¯ through a W , etc), with a cross-section which goes essentially − ↔ − − as σ G2 E2 G2 T 2. Thus the total interaction rate turns out to be Γ = n σv G2 T 5. F F F F ν F Taking into account that in this radiation-dominated epoch H 5.44 T 2/M , we obtain a Pl rough condition for neutrino decoupling as:

Γ T 3 F . (3.29) H 1MeV This means that neutrinos decouple when the temperature was around 1MeV, shortly before the temperature drops below the electron mass. When this happened, almost all the electrons and positrons annihilated into photons 6.

So before neutrino decoupling, when the universe had a scale factor a1, all the temper- 43π2 3 atures where the same and the entropy density was s(a1)= 90 T1 . After annihilation, Tν = s Tγ , and while Tν ∝ 1/a, Tγ received an extra contribution, coming from the decreasing of g . 4π2 3 3∗ The entropy density after annihilation, at time t2 with a = a2 is s(a2)= 45 (Tγ + 21/8Tν ). 3 3 But since the entropy is conserved, s(a1)a1 = s(a2)a2. This leads to 43π2 4π2 21 T 3a3 = T 3 + T 3 a3 (3.30) 90 1 1 45 γ 8 ν 2 2 2 3 43π 3 3 4π Tγ 21 3 Tν a2 = 3 + (Tν a2) 90 45 Tν 8

where in the second line we have used the fact that Tν a is constant. Solving for Tν leads to

4 1/3 T = T 0.7T . (3.31) ν 11 γ ≈ γ Hence, neutrinos are an example of a particle whose decoupling occurs while still being relativistic. Actually nowadays the temperature of the universe is smaller than the heaviest neutrino species7, so we can consider them as non-relativistic. However as they are

6 Not all, because there is an excess of electrons to have charge neutrality with protons, ne = np, but in any case np << nγ . 7Even considering that the lightest neutrino is massless, the masses of the other two species are determined by the known experimental mass differences; and it turns out that the resulting masses are much larger than 2.7K O(10 13)GeV. ∼ − 3.4 Dark Matter abundance 29 not in equilibrium since long time, its number density does not get Boltzmann-suppressed, but constant in a comoving volume. In any case, since the neutrino bath is out of equilibrium, it does not conserve entropy in general8, and consequently the global (photons plus neutrino) bath does not conserve the entropy neither. Below, we will see that in the case of thermally produced Dark Matter the decoupling occurs when the particle was already non-relativistic; so its number density is Boltzmann suppressed at the time of decoupling, and it essentially does not contribute to the entropy anymore. So in this case there is no entropy conservation issue.

3.4 Dark Matter abundance

The formalism to evaluate not only dark matter’s, but all particle’s abundances (for example, at BBN) is provided by the Boltzmann equation, which I describe in this section. The Boltzmann equation generalises the 2nd Friedmann equation which describes how an abundance of a species of particles evolves with time. For matter-domination (dust approximation, P = 0) this becomes

a˙ 3 d 3 3 d 3 ρ˙ + 3ρ = a− (ρa )=0 a− (na )=0 , (3.32) a dt ⇒ dt where n is the abundance (number density) of a species. The Boltzmann equation relates the rate of change in the abundance of a given particle to the difference between the rates for producing and eliminating the species. For a process 1 + 2 3 + 4: ↔ 4 3 3 d(n1a) d pi 4 (4) 2 a− = (2π) δ (p + p p p ) M dt ∏ ( )3 E 1 2 − 3 − 4 | | i=1 2π 2 i [ f f (1 f )(1 f ) f f (1 f )(1 f )] , (3.33) × 3 4 ± 1 ± 2 − 1 2 ± 3 ± 4 where f is the distribution function of particle i. Here +( ) refers to bosons (fermions). In i − the absence of interactions, this Boltzmann equation reduces to the 2nd Friedmann equation. In the particular case of DM analyses, (3.33) is useful to study the abundance when DM is about to chemically decouple, because before decoupling, as usual, it is a very good approximation to take n n . We have seen with the neutrinos’s example that one can DM ≈ γ have situations where the decoupling occurs when the particle is still relativistic. However, this possibility for DM is highly disfavoured by observations (see the comments about hot and warm relics in section 3.2.2). So, as a cold (non-relativistic) DM is favoured by data (although having some problems mentioned before), we can use this fact to simplify the DM distribution functions to be those of Maxwell-Boltzmann (MB):

1 (EX µX )/T fX = e− − . (3.34) e(EX µX )/T 1 ≈ − ± 8Remember that the entropy conservation was obtained by assuming a system in thermal equilibrium 30 Dark Matter phenomenology

However, for the case of particles of the rest of the thermal bath, under the assumption of a cold relic, it means that mDM > mbath, so a priori one can not approximate the distribution functions of the bath particles to follow a MB law. It is straightforward to get that the limit for which 1 + f 1 (in which case MB applies) is such that m 2T. This condition is i ≈ bath not true in general for a thermal bath made of SM particles, at the time of DM decoupling. However, as shown in the appendix (A)9, the energy conservation condition of the bb¯ χχ¯ ↔ process forces the energy of the bath particles b to be such, that the MB approximation works, even if in the large majority of cases it does not10. With this approximation, the last line in (3.33) then becomes

(E +E )/T (µ +µ )/T (µ +µ )/T e− 1 2 e 3 4 e 1 2 − where conservation of energy (E1 + E2 = E3 + E4) has been used. Now, since

3 3 d p µ /T d p E /T n g f g e i e− i , (3.35) i ≡ i ( )3 i ≈ i ( )3 2π 2π

eq eq µi/T it is useful to deﬁne the equilibrium number density ni such that ni = ni e . Then, we have

(E +E )/T (µ +µ )/T (µ +µ )/T (E +E )/T n3n4 n1n2 e− 1 2 e 3 4 e 1 2 = e− 1 2 . − neqneq − neqneq 3 4 1 2 On the other hand, the thermally averaged cross-section σv can be deﬁned as: 1 4 d3 p σv i (2π)4δ (4)(p + p p p ) f f M 2 . (3.36) ≡ eq eq ∏ ( )3 E 1 2 − 3 − 4 1 2| | n1 n2 i=1 2π 2 i which using the Maxwell approximation becomes

4 3 1 d pi 4 (4) (E +E )/T 2 σv (2π) δ (p + p p p )e− 1 2 M . (3.37) eq eq ∏ ( )3 E 1 2 − 3 − 4 | | n1 n2 i=1 2π 2 i Using (3.37), the Boltzmann equation ﬁnally simpliﬁes to

3 d(n1a) eq eq n3n4 n1n2 a− = n n σv . (3.38) dt 1 2 neqneq − neqneq 3 4 1 2 From (3.38), one can study all BBN processes, but also the abundances of a particular DM candidate. 9This result is not taken from any reference, since in all the literature at hand that approximation is made without a correct justiﬁcation. However here I have arrived to the result by a proper treatment of the equations. 10In the large majority of cases, in the regime m << T << m , we have that E T, but then nor the bath DM bath ∼ production nor the annihilation of DM occurs. 3.4 Dark Matter abundance 31

3.4.1 Freeze-out of cold Dark Matter

Had the DM been still in thermal equilibrium, its abundance would be negligible, as shown in eq.(3.20). However this is not what is observed, meaning that at some time, the DM “freezed- out”, when the reaction rate became smaller than the expansion rate of the universe, such that it became too rare for a DM particle X to ﬁnd its partner X¯ to self-annihilate. Consider a generic scenario in which two heavy WIMPs χ and χ¯ annihilate, producing two light (essentially massless) particles l. The light particles are assumed to be in thermal eq equilibrium with the cosmic plasma, so nl = nl . Then, the Boltzmann equation (3.38) simpliﬁes to: 3 3 d(nχ a ) eq 2 2 a− = σv (n ) n . (3.39) dt χ − χ The LHS of this equation is further reduced to the yield Y n /s as: χ ≡ χ 3 3 d(nχ a ) dYχ a− = s dt dt

11 3 where we have used the fact that s ∝ a− . The Boltzmann equation can be expressed just in terms of the yield: dYχ = σv s (Y eq)2 Y 2 . (3.40) dt χ − χ As a function of a new variable x m /T, this equation becomes: ≡ χ dY λ χ = (Y eq)2 Y 2 (3.41) dx x2 χ − χ 2 where λ gs 2π m3 σv /H(x = 1) is the ratio of annihilation rate to expansion rate, at the ≡ 45 χ time where the∗ temperature equals the DM mass. The solution of (3.41) is found numerically and shown in ﬁg. 3.2. There we clearly observe that DM departs from equilibrium at some x value, say xFO. The estimation of eq xFO is done as follows. We consider that around xFO there is a small deviation of Yχ , as eq Y = Yχ (1 + δ). Substituting this into (3.41), and writing

eq eq nχ gχ 45 3/2 x Y = = x e− (3.42) χ s gs 2π2 ∗ we get x gχ 45 λxe− 1 + δ s 2 3 ( 2δ) (3.43) ≈ g 2π √x x3/2 − ∗ 2 − 11 4 T, taken as the temperature of photons Tγ , is such that ργ ∝ Tγ , and on the other hand, in radiation- 4 dominated universe, ρ ∝ a− . 32 Dark Matter phenomenology

0.01

0.001

0.0001

1 10 100 1000

Figure 3.2: Evolution of the comoving number density of DM with temperature. (Taken from “TASI 2008 Lectures on Dark Matter”. Dan Hooper, FERMILAB-CONF-09-025A.) where we have neglected terms of O(δ 2), and also the derivative dδ/dx, which holds for sufﬁciently small δ. Besides, since the freeze-out starts at T m , we have x 1 and ∼ χ ≥ normally x3/2 > 1.5√x. The numerical equation for x becomes:

g 45 λ δ x log (3.44) ≈ gs π2 √x 1 + δ ∗ which is typically, for many theories, around x O(20). FO ∼ An alternative way of expressing (3.40), which is useful to extract the different solution regimes and connects with the simpler decoupling criterion used before, is given in terms of the reaction rate: eq 2 d logY Γ Y χ = 1 χ (3.45) d loga −H  − Yχ    The qualitative behaviour of the solution of (3.45) can be estimated analytically in some limits. For Γ >> H the reactions are occurring so fast that DM have enough time to thermalise, eq thus Yχ Yχ : ≈ eq n g T 3/π2 g Y eq χ = X = X . (3.46) χ ≈ T 3 T 3 π2 eq On the other hand when Γ H, as soon as T mχ , Yχ becomes rapidly Boltzmann sup- ≈ d logYχ pressed and the above equation becomes d loga O(1). In this regime since DM is semi- eq ≈− decoupled, Yχ goes down but not as fast as Yχ . Finally, at much later times when Γ << H, 3.5 Dark Matter Detection 33

d logY we have χ 0, implying that the comoving number density of DM remains approxi- d loga ≈ mately constant in time, thus getting “frozen”. The yield today can be analytically estimated as: x Y 0 FO . (3.47) χ λ 0 With the yield today, Yχ , it is straightforward to compute the relic density of DM:

0 0 2ρχ 2mχ S(T0)Yχ 2mχYχ Ω h2 = = ; (3.48) χ ρ ρ 3.6 10 9GeV crit crit × − where I have assumed there is no asymmetry in the population of χ with respect to χ¯ , and 12 S(T0) is the entropy today . From eqs.(3.47) and (3.48), one can see that the stronger the annihilation cross-section, the less amount of DM remains until today. This is in general one of the strongest constraints on theories BSM, because for large portions of the parameter space of those theories, one commonly has a too small cross-section, which leads to too much relic density. On the other hand, it may happen that a model, for a particular choice of parameters, gives a too large cross-section. In that case, the contribution of the correspondent DM candidate to the relic density is too small; although the model is not ruled-out as long as one assumes that there 2 exits other contributions which add to the previous one to obtain the correct Ωχ h . Finally, some words about the freeze-out of a possible hot relic, as a neutrino-like particle for example. In this case the freeze-out occurs while being relativistic, so when Γ H, Y Y eq, contrary to the cold relic case. Thus the comoving number density remains ≈ ≈ constant in time.

3.5 Dark Matter Detection

In this section I brieﬂy summarise the idea behind different DM detection methods, and afterwards, I describe those methods for the most important DM detection experiments.

3.5.1 Indirect detection

The Milky Way, as happen with the rest of the galaxies we observe, requires a certain amount of DM to explain observations. This Dark Matter Halo has been the subject of many studies, including complex numerical simulations which are compatible with a DM density profile ρDM decreases exponentially from the centre to negligible amounts beyond 20 kpc or so. While different profiles differ significantly on the ρ value at the very centre of the galaxy

12 2π2 s 3 s In general, S(T)= 45 g (T)T ; with g being the effective number of degrees of freedom in the thermal bath for the entropy. ∗ ∗ 34 Dark Matter phenomenology

(some profiles produce divergent densities), the agreement on the local (Earth) value is very good, and presently the central value is ρ 0.43 GeV/cm3. ⊙ The idea of indirect detection (ID) is that DM particles can annihilate or decay mostly at the Galactic Centre (GC), given that its number density is very high in this region. So even though the kinetic energy of DM particles is presently very suppressed13, the high number density at the GC allows them to possibly find each other and annihilate. In the case of and unstable DM, even if its lifetime should be very long in order to avoid BBN problems, having that high number density makes it probable for a decay to happen from time to time. In either case, being annihilation or decay, the idea is to register the DM products, which could reach the Earth in the form of cosmic rays (CR). Then, the hope is to observe a cosmic ray signal coming from the GC which could not be explained by known standard astrophysical sources, thus opening the window for a DM interpretation. The annihilation(decay) sub-products we can look for are photons (from low, radio frequencies to highly energetic gamma-rays), electrons and positrons, neutrinos, (anti)protons as well as (anti)deuterons, for example. They can of course be direct products (primary cosmic rays) of DM, or products of decays of those (secondary cosmic rays). Balloons. Launching balloons to the sky is a technique which has been used since long time ago (around 1938) and, in spite of the development of satellites as ID experiments, they continue to be used because for some tasks they are very efficient, and very cheap.The CREAM (Cosmic Ray Energetics and Mass) experiment consists of a set of particle detectors (including Cherenkov, tungsten-scintillators, etc) which are able to measure energies between 1011–1015 eV, working at a height of 40km. CREAM is presently the best experiment to measure the ratios of secondary to primary cosmic rays, thus being extremely important to constrain cosmic ray propagation. Another balloon experiment is ATIC (Advanced Thin Ionization Calorimeter), which is similar to CREAM but focused on low energies instead, in the range 10–300 GeV. Satellites. The disadvantage of using balloons is that, as they can not get much higher than some tens of km, the cosmic rays which they detect are contaminated from the interaction with the atmosphere (whose thickness is around 800 km), which introduces important uncertainties. Satellites instead work at much higher altitudes. An example is PAMELA (Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics). It is an italian particle detector installed on-board the russian Resurs-DK1 satellite. It flies at an altitude ranging from 350 to 610 km, and it is probably the best instrument looking at CR of energies lower than 1 TeV. Another satellite-like experiment is the Fermi Gamma-ray space telescope, which is the result of a worldwide collaboration. It works as a particle detector, which however is not able to distinguish particles from antiparticles, since it does not posses a magnet. Because its high altitude orbit (550 km) and its orientation, it can collect a lot of statistics without being affected by the variations of the Earth’s magnetic field. They have already released data of electron/positron fluxes in the range of 7 to 870 GeV. A third experiment is called AMS (Alpha Magnetic Spectrometer). Its AMS-02 version is operating on the

13Remember than at freeze-out they were already non-relativistic, with kinetic energies E T m /20. K ∼ ∼ χ 3.5 Dark Matter Detection 35

International Space Station since 2011, and presently is measuring CR with unprecedented accuracy. It is expected to have its first data release within the next months. Ground-based experiments. Very high-energy cosmic rays have two important compli- cations: the statistics is very poor because of the very low fluxes, and can not be efficiently studied in outer space because of size limitations. Indeed a typical spatial detector can have at maximum few squared meters, which is not sufficient to study CR beyond few TeV. The ground-based detectors are then useful: they do not observe the CR directly but their show- ering when passing through the atmosphere. As a typical CR shower is very extense, the probability for detection is high, since those experiments are not quite limited in size, being possible detections areas of several squared km. HESS (High Energy Stereoscopic Sys- tem) is an example. It consists of four 12m Cherenkov telescopes in Namibia. It has been designed to measure gamma-rays from 100GeV to 100 TeV. KASCADE (The KArlsruhe Shower Core Array DEtector) in Germany is a 700 m 700 m array made of almost 300 de- × tectors. KASCADE aims to directly detect the particles of the extended air shower (instead of their Cherenkov light). Pierre Auger cosmic ray observatory, in Argentina, works similarly as KASCADE, but it is much larger (3000 km2), and is composed by about 1600 detectors, which can jointly study CR energies of about 1018 eV, including the identification of their original direction. A final example is the LOFAR experiment (LOw Frequency ARray for radio astronomy), which is a huge array of radio antennas spanning over the Netherlands, the United Kingdom, Sweden, Germany and France. All of them are radio-interferometers listening in the 10–240 MHz frequency band.

3.5.2 Direct Detection

Direct detection (DD) experiments lie behind the idea that our solar system and our planet would be passing through a flux of DM particles. Then, assuming that DM can weakly interact with nucleons, it can deposit a measurable amount of recoil energy within an appropriate detector. This can occur through an elastic scattering between the incident WIMP and a nucleus in the fiducial volume of some detector material. The effect of an interaction would depend on the DM velocity distribution, which is centred about a few hundred km/s, and of course on the DM mass. With this information the expected energy distribution of events from interacting DM can be calculated. As a result, for an O(GeV) WIMP the typical transferred energy to a nucleus ranges in tens of keV, while the recoil energy of electrons is much more challenging instead (tens of eV). The main DD experiments searching for WIMPs are the following: XENON. It is located in the Gran Sasso mountain, in Italy. Its new version XENON100 utilises 100kg of fiducial liquid xenon target, and it is expected to register 1 event per 100kg per year. They have recently released results from the last year run, after which two events where observed, consistent with the total background expectation. A profile likelihood analysis of these data leads to the upper limit on the spin-independent elastic WIMP-nucleon scattering cross section, which is minimal (2 10 45cm2) for a DM mass of 55 GeV at 90% × − 36 Dark Matter phenomenology confidence level. It is presently the most constraining DD experiment among the existing ones. They will soon introduce more material (1ton), which is expected to reduce much more the parameter space of the WIMP models. CDMS. The name stems for Cryogenic Dark Matter Search. This experiment, located in Minnesota, aims to measure the recoil energy of a nucleus due to collisions with WIMPs by using detectors which are highly sensitive to the ionisation and phonon signals that result from a collision. Its detectors represent the state-of-the-art superconducting films deposited on 600g germanium crystals to accurately measure information about the collisions. The experiment has now increased the experimental sensitivity by a factor 10 and operates at the deeper SNOLAB facility, now operating in Canada. This location provides significantly improved shielding from cosmic rays which are an important source of background in WIMP searches. They have recently reported limits on annual modulation of the low-energy event rate, finding no evidence consistent with nuclear recoils. CoGeNT. This experiment is also located in Minnesota. It used a 440 gr high-purity germanium crystal cooled to liquid nitrogen temperatures to register nuclear recoils from WIMP collisions. The CoGeNT detector has the advantage of a very low energy threshold (0.5 keV) which allows it to search for events produced by light WIMPs, down to around 5GeV, with very good resolution. Last year, they released data of 145kg.day which show an excess of events at low recoil energies consistent with WIMP-nucleon scattering, pointing 40 2 towards a 7–8GeV particle with a cross section of about 10− cm . More importantly, they were able to measure the annual modulation of the signal. Because the velocity of the Earth with respect to the dark matter sea changes during the year due to the orbital motion around the Sun, the event rate of dark matter scattering is expected to oscillate with a peak in June and a minimum in December. DAMA. This experiment is, as XENON, located at Gran Sasso. Several low-background highly-pure scintillator materials have been used, for example, NaI (sodium iodine), with 100kg of material, 6.5kg of LXe (liquid xenon), and the new generation DAMA/LIBRA, with 250kg of NaI, in operation since 2003. Both DAMA/NaI and DAMA/LIBRA have found the presence of annual modulations, claiming a model independent evidence for the presence of DM particles in the galactic halo with 8 standard deviations. However, due to the lack of transparency in the conditions of which this experiment is made, the DAMA signal is up to now taken with some skepticism by the community.

3.5.3 Collider searches

Finally, a third branch of experiments looking for DM, but not exclusively, are the particle colliders. The idea is that proﬁting an interaction cross-section of electroweak order, WIMPs could be directly produced at those colliders by annihilation of SM particles, in quite the opposite way the DM annihilation works for ID experiments. WIMPs, being weakly interacting, of course escape from the detectors, so the hope is to observe associated ﬁnal states which one can differentiate from SM background, opening the possibility for a DM interpre- 3.5 Dark Matter Detection 37

Figure 3.3: Different bounds to DM-nucleon elastic scattering cross-section, coming from direct detection searches. (Taken from “Direct Search for Dark Matter”. Josef Jochum. Talk given at “IFT 2012 Xmas Workshop”). tation. The minimal topology one can study associated to direct WIMP production, is a final state consisting of a single photon, or a single jet. However, there is plenty of studies (mainly in the framework of supersymmetry) which analyse multi-jets or multi-lepton final states associated with DM production. There are many efforts in this direction using LEP, Tevatron and LHC, and it turns out that the level of exclusion reached by collider experiments are complementary to those of DD experiments, being able to probe the very low mass regime quite efficiently, where DD looses all the sensitivity. 38 Dark Matter phenomenology CHAPTER 4

FAIR SCANS OF THE SEESAW I 40 Fair Scans of the Seesaw I JHEP03(2011)034 ) γ ect” e, ff .This → -matrix. 13 µ Springer θ R and ). , b γ March 7, 2011 e, : February 4, 2011 February 13, 2011 ) are larger than : November 18, 2010 → : γ : µ e, is an “optical e → 13 Published Revised θ µ 10.1007/JHEP03(2011)034 Accepted Received doi: [email protected] , ) gets very insensitive to Roberto Ruiz de Austri γ b Published for SISSA by e, → µ Nuria Rius, a [email protected] , , although decreases the typical value of BR( 13 θ [email protected] , e´sM.Jesús Moreno, a 1010.5751 -parametrization”, in a complete way. This includes a very simple rule to a R Neutrino Physics, Supersymmetric Standard Model We give a straightforward procedure to scan the seesaw parameter-space, using [email protected] SISSA 2011 E-mail: [email protected] Instituto IFT-UAM/CSIC, de Teórica, F´ısica Nicolas Cabrera 13, UAM, Cantoblanco,Instituto 28049 de Madrid, Corpuscular, F´ısica Spain IFIC-UV/CSIC, Apartado de Correos 22085, 46071 Valencia, Spain b a c ArXiv ePrint: required for succesful leptogenesis). In mostthe cases experimental the upper values bound. of BR( Includingfurther (unflavoured) leptogenesis dependence does on not introduce any Keywords: As a relevant application, we showin that supersymmetric the seesaw somewhat models propagated depends beliefproduced strongly that on by BR( the incomplete value scans, of study. and does When not the hold completeholds after scan a even is careful if done, analytical the and BR( right-handed numerical neutrino masses are kept constant or under control (as is Abstract: the common “ incorporate the perturbativity requirement as a condition for the entries of the Bryan Zald´ıvar predictions on LFV processes J. Alberto Casas, Fair scans of the seesaw. Consequences for JHEP03(2011)034 6 5 3 4 9 7 4 1 3 2 18 19 11 18 15 16 .Such 2 L ; more pre- m ν ). The basic γ Y e, → µ has a strong impact on Lepton Flavour 13 θ ] and explored in later works (see, for exam- 4 –1– -diagonal entries enable LFV processes through ) ff , γ 3 e, → µ matrix − matrix and the perturbativity condition R .Thentheseo − ij ] that, in the context of the supersymmetric (SUSY) seesaw sce- ) 2 R ν , 13 1 Y θ † ν Y -diagonal entries in the (left-handed) slepton mass matrix, ( ff ∼ parametrization parametrization ij parametrization parametrization ]). 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In section 5 we give a parametrization in the present 16 → e 10 R − ν − Y µ 13 L θ 2 V H · ]. ect is apparently very important, pro- ]. L ff 12 ν 15 Y , ) on the -parametrization of the see-saw (see section γ 14 cT R R –2– matrix has plenty of freedom. This is because ν e, ered in [ ν ff + → Y † ν 1 µ Y H mixing angle increases; furthermore, such behaviour is · is very small, even negligible. L 13 e θ 13 was already noticed in refs. [ Y θ 13 cT R θ e ⊃ W ) are the leptonic doublets (charged singlets), c R are the two supersymmetric Higgs doublets. e ( 2 , 1 L H In the standard SUSY seesaw the relevant superpotential is In section 2 we present the framework and set the notation. Section 3 is devoted to The insensitivity to In this paper we will show that these results are essentially an optical e -parametrization). For related work see [ -parametrization, even when the values of the right-handed neutrino masses are kept L and From now on we use the conventions and notation of ref. [ where making use of theconstraints. mentioned Finally, scan. in section Section 8 we 7 present is our devoted main2 to conclusions. the inclusion Framework of and leptogenesis notation describe the two parameterizationssection of 4 the we show seesawconcerning and we the explain are dependence the dealingprocedure of apparent with to BR( contradiction scan in between the thisof the seesaw work. perturbativity parameter two (details space approaches, are In instudy given a in both fair appendix in way, A). incorporating an In the analytical the constraint and same numerical section and way section the 6, dependence we of BR( R constant or undermaining control freedom (as is isV scanned required in for a succesful complete leptogenesis), way (something provided non-trivial the to re- check in the will present easy rulesrating to perform the complete perturbativity explorations requirement.implemented, in the We this impact will parametrization, of show incorpo- that when such appropriaterametrization scan of is the seesawpaper, parameter see space details (called in section 3). We will show here that such insensitivity holds in the by scanning only a partresults of were the whole obtained parameter3), using space which the of is the so-called seesaw. prettyscan common More of precisely, and these the useful freedom for allowed studing by some the issues. parametrization may However, lead an to incomplete biased results. Here we strengthened by leptogenesis constraints.ducing increases The of e severalexperimental orders window. of This magnitude remarkableing behaviour in has scans BR( been upon shown theanalytical up explanation mainly seesaw for by parameters it perform- (with has been and o without leptogenesis constraints). An and neutrino mixing matrix),in the the standard seesawlow-energy scenario observables (9). there Consequently, for are givenprocesses more low-energy can observables initial the vary rate high-energy within of parameters ashifts LFV certain (18) to range. than larger The values claim as of ref. the [ JHEP03(2011)034 ) M . (3.1) (2.5) (2.6) (2.2) (2.3) (2.4) ∼    1 , are given 2 / φ i ,theneutrino MNS − U U parametrization e − 2 . Hence, for given . / R ), is determined by φ and U ) i 2 i − , 2.1 κ e ) 3 LH    κ ν ,    2 Y ( κ δ i 1 , 13 13 1 − − c c e κ 23 23 M 13 c s s T ) , δ 2 . i diag( † δ is the Majorana mass matrix of right- i e , ν U e ≡ 12 LH κ , and the mixing matrix, κ Y i 12 left-handed) neutrinos is s ν M κ 2 √ 1 s κ 13 Y 0 2 − 2 ∼ 13 ( s s 0 RD H 2 12 1 2 M 23 s . 23 c –3– T H M ν ) are valid at the seesaw scale. Besides, they are s ν 13 = ,D + √ − c Y − Y ν = D 2.4 1 12 i ≡ ¯ 12 c = MNS H = M c m ), ( · U 23 κ Y 23 s Y L , κ c 2.2 e − ective superpotential, matrices, which contain 18 independent parameters. On ff Y δ i δ MNS i e T MNS e U cT R M 12 U e 12 c ≡ c = 13 13 s U ⊃ and s κ 12 23 c 23 D s c Y ] that, for a given set of low-energy observables, ff 13 e . The standard parametrization of the MNS matrix is − c 3 − 16 W κ 12 12 s ≤ s ective mass matrix for the light ( parametrization. 23 2 23 ff c − κ s L − parametrization V ≤ −    1 R κ = , and the three mixing angles and the three phases contained in From now on, for the sake of notation clarity, we will drop labels in the neutrino mixing The neutrino mass-eigenvalues, i κ MNS 3.1 It was shown in [ Yukawa matrix (at the seesaw scale) has the form: the other hand, there∝ are 9 low-energyvalues neutrino of observables: the the low-energydimensional observables, three the parameter neutrino freedom space. masses, inother There the words, seesaw are of Lagrangian two parametrizing expands mainand our a ways the ignorance). 9- of describing We such will space call (or, them in the 3 Parameterizations of theThe seesaw high-energy seesaw Lagrangian,the given entries by of the superpotential the ( and Yukawa matrices: by with Note that the seesaw equationsobtained ( using the (reasonable) approximation ofof decoupling a at (more a unique accurate) threshold, decoupling instead in three steps (the three right-handed neutrino masses). Then, the e with in flavor space (flavor indiceshanded are neutrinos. dropped) Once and right-handeda neutrinos mass are operator decoupled is (at left the in seesaw the scale e U JHEP03(2011)034 , . L 3) )], 13 V × (3.4) (3.2) (3.3) 2.6 is has also R R V ) is general up 3.2 charged leptons of i,j , l    .Eq.( i on it. This becomes more 3 θ 3 s c , 2 21 (with parametrization, see ref. [ erent phases; and 2 ) s s Y ff 1 − γ 1 Y c , 2 L D c † ), on the MNS matrix (and in j c is a complex orthogonal (3 ∗ V l L 1 − γ , the neutrino Yukawa matrix (at Y − c V 3 U e, R 3 c → T s 1 i 1 . A usual parametrization of U s l . s 1 → R − and − κ † L i µ , and V 3 κ } 3 c Y i UD s 2 , proportional (in the leading-log approxima- † κ 2 s L j D s 1 V √ R 1 s –4– 2 = { Y s V c i 1 − . D , − s = 3 R ), namely ij 3 s ) c 1 = Y 2 1 c 2.4 c = diag L , and the three angles and three phases contained in ∗ − R i κ y V m 3 3 √ M 2 c s 2 s 2 D . D erent mixing angles and three di -diagonal entries in them. In particular, the (left) slepton mass c c 2 † , | ff ff R }    ij V i parametrization is the so-called ) 13 2 − M = θ L R √ is the corresponding diagonal matrix, which contains the three right- matrix element. On the other hand, at first order in the mass-insertion m { R ( | M ij ]). Here, the 9 independent parameters that parametrize our ignorance ) is a unitary matrix with identical structure as the MNS matrix [eq. ( D ) occurs mainly via the dependence of ( 18 Y L 13 † -diagonal entries ( V θ = diag ff Y ; ) are the sine (cosine) of the three complex angles i i ) and y parametrization M c is the diagonal matrix containing the three (real and positive) neutrino Yukawa ( matrix and the three right-handed neutrino masses are obtained by substituting √ − families) is non-vanishing and proportional to the squared of the corresponding 3.4 i is the unitary matrix that diagonalizes the symmetric matrix in the right hand − ) in the seesaw expression ( Y L D s R V D R V 3.3 Consequently, the dependence of, say BR( i, j V -diagonal entry, ]. For a given set of low-energy observables, ff particular on tion) to the ( expansion, the branching ratiothe of the LFV process o 4 The impact of As mentioned in thethe introduction, RG even running generates starting o withmatrix diagonal gets and o universal soft masses, i.e. side of ( handed masses. details see e.g. [ are the three Yukawa couplings, The eq. ( Here couplings, but, of course, with three di identical structure but with the diagonal matrix of phases acting from the left (for more 3.2 An alternative to the 17 the seesaw scale) can be written as where to reflections changing the sign of det where matrix. So, the 9handed see-saw masses parameters and that the parametrize 3 our complex ignorance angles defining are the three right- JHEP03(2011)034 . in γ e, 21 (4.3) (4.4) (4.5) (4.1) (4.2) ) any ) in the Y ,while ]). An → matrix, Y † does not D 13 13 − 12 µ ): s θ . Y R 13 θ β 3.1 2 . X are the universal tan M 0 2 A . Let us concentrate . It may happen that 21 , ) 0 R U Y m † ) and a given . So, varying Y ··· and M 13 ( , we can expand ( i θ 3 D 2 ) + κ . M using the two parametrizations † X 3.3 ∗ 13 13 U M 2 U M θ κ κ √ 33 . parametrization means moving along . on † )log − L R 2 0 13 RD L V 1 . For given Yukawa couplings ( and thus of s 21 A V M κ 2 ) Y M ) has a similar dependence on U U D γ + D Y D ∝ † † † L 2 0 e, R V R Y –5– m 21 κ ) in the . We will discuss soon the validity of the previous = → √ (3 23 Y 13 2 U 13 † θ Y τ 1 π 3 θ † matrix can be easily obtained from eq. ( UD 8 Y κ ( 2, we see that, in this approximation, Y − = Y matrix element, which is the relevant one for √ † = parametrization. From ( / 8 S Y Y 1 21 − † in the seesaw parameter space; while changing ) 3 m 21 L ) ∼ Y α 2 F L Y V | † V Y G observed in the literature (this was also noticed in [ † is quite strong, and is really the source of the dependence of 23 Y U ∼ Y | 2 13 ( 13 ) and s s γ i does not depend at all on y ∼ e, and | Y → † 13 s . The first term of such expansion is µ | Y ) on i is compatible with any choice of γ κ at all. This result seems to be in contradiction with the one obtained using the ) is almost independent of = L γ parametrization the √ e, | BR( V represents a typical supersymmetric leptonic mass, and Y µ − 13 † S R → U → Y | (and thus LFV processes) has a non-trivial dependence on m One might argue that changing Let us now turn to the Next, we analyze the dependence of ( µ τ parametrization means moving along a line of constant parametrization. Y ect † − − ff R choice of a R a line of constant expansion, and thus of these results. Clearly, now This dependence on BR( analogous argument showsBR( that BR( Since This equation tells usY that, for given right-handedfor masses the ( moment onAssuming the a ( hierarchical spectrum ofpowers neutrinos, of 4.1 A “paradox” In the Here scalar mass and the universal trilinear coupling at theof unification the scale seesaw discusseddiscuss in its section explanation. 3. We will find first a kind of “paradox”, then we will clear from the following approximatelater formula for for qualitative the discussions, branching ratio, which will be useful JHEP03(2011)034 . ) in Y Y 4.2 )we 21 (4.9) (4.7) (4.8) (4.6) ) ), it is )] goes 3.1 Y γ .The † matrix is 3.1 13 e, Y ). One can θ Y constant but → 3.4 , and write M matrix remains µ R D V Y will not change, as † 2 i Y and and y . matrix that maximizes ˆ . Let us keep fixed all † R M U L D fixed, and vary parametrization ( V parametrization the change ) tells us that an increase of ,the that maximizes ( departs necessarily from its ! So, at least we have found UU , which are known. We can − R . − κ 13 21 L Y L 4.3 θ R 21 † D V κ V 1 , Y Y [and thus of BR( − √ and matrix that appears in the expan- Y † † ). The corresponding κ 1 was “designed” to maximize this † and U ˆ − √ M Y 21 , and a R . ), it is straightforward from eq. ( Y L 3.4 Y D U M V 2 3 M 13 Y D y UD YU D θ can be obtained from eq. ( † D D † † = 1 2 L † , parametrization. Namely, in eq. ( Y R † V ˆ Y ) that corresponds to this optimal choice: R † V = U − Y ˆ R , U , R and then calculate D UU on M κ 1 R Y o –6– max 21 † − L √ V D 13 V D θ . Therefore M 1 ). Then, using the =3 Y (recall that 2 Y L − † . For each value of √ 2. Then D V ), but the Yukawa eigenvalues, = 13 3.3 13 D Y L θ θ 21 √ V ) (e.g. by varying ˆ / 3.1 R † matrix (say = parametrization. Assuming hierarchical neutrino Yukawa Y and U − M † − U in terms of ˆ =1 R R R L D Y U | † V V ( R is orthogonal indeed). If we keep now → change according to eq. ( ˆ 23 R = ) ˆ ,except ). From the point of view of the R U , it is obvious that the choice of R L ≤ 2 3 U V V Y y ( 4.2 † matrix from ( | )). So something goes wrong with the argument used to obtain matrix reads 21 ) and can only decrease in magnitude. However, from the point of view Y ). The corresponding = ) and Y | 2 2 4.5 4.6 Y y Y 4.5 M 13 † † ) D for which the impact of ). To see what, we construct the L Y Y ). Suppose we choose V ( ( 2 1 R | 4.4 y 3.3 parametrization, the approximate expression eq. ( in eq. ( keeps constant along the first line but it increases along the second one. This is ): ), ( − )is 21 21 R ) is due to a change in 4.3 4.3 4.5 Let us work first in the Y Y should reflect in an increase in the magnitude of † † Y 13 Y Y Obviously, quantity in eq.eqs. ( ( sion ( (it is funny to checkchange that the MNS matrix, that the new construct now the straightforward to derive the a choice of exactly opposite that claimed in the literature. 4.2 The reasonThe behind solution to thestart previous conflict with can a be choice found for by doing the the Yukawa above eigenvalues, steps explicit. We it is obvious from eq.in ( maximum value ( of the s the same, but given by ( This matrix can becan easily straightforwardly expressed solve inwonder the now what happens if wematrix keep will these change values of according to ( As mentioned above, thismixings value and is phases available in for any choice of ( true, but even assuming this possibility there remains acouplings, conflict, as we areeq. about ( to see. JHEP03(2011)034 , 13 θ and and ˆ (4.10) R R against 2 | 21 10 ) Y † 9 Y | 8 is not realistic, since it leads i ) are separated. Then the real θ 7 can in principle take any value 3.2 i , which is really general, and the . θ π ]. Sometimes the real and imaginary 2 j 6 κ ) are in principle as large as the first i 19 ≤ 1 κ , i 4.3 θ √ 11 5 [grad] ∝ 13 Re e matrix, correspondent the expression (4.7). Here ij 4 ≤ ]), or scanned within a range similar to the real –7– − ˆ R ˆ 20 R M 3 D † ˆ R 2 matrix? In the literature one can find several approaches. matrix and the perturbativity condition the matrix element decreases. for a given − − R 13 1 13 θ θ R vs. 0 ]). This is not general, since Im ) GeV. 2 2 | 0.25 12 , matrix is simply frozen, as in [

0.244 0.243 0.242 0.241 0.249 0.248 0.247 0.246 0.245

21 range. Of course a too-large value of Im goes opposite that naively expected. This raises the question: Is this ) | Y) |(Y −

constant, for this particular example. As expected, the maximum occurs 2 + , Y } R angles appearing in the parametrization ( † 13 11 i M θ ∞ Y θ 10 , | and for larger D , , o on 10 .( ˆ R −∞ 21 { =3 ) Thus we have constructed an explicit counter-example, where the dependence of = (10 ) taken above or it? is more general? What can we expect for a generic choice of 13 Y † θ M M M Y to non-perturbative Yukawa couplings, making the whole approach inconsistent. We will parts of the part is varied asimaginary an part ordinary is angle, fixed say topart 0 zero (as (see in e.g. [ refs.in [ the 5 Fair scans in the The final questions ofa the truly previous section generic pose scanSometimes an in the interesting the issue: how can we perform ( “wrong” behaviour a consequenceD of the specialD choice of the seesaw parameters ( The consequence is thatterm: all the terms expansion neglected iskeeping in not ( sensible.at In figure 1 we have plotted ( D Hence Figure 1 JHEP03(2011)034 ij 21 R (5.3) (5.4) (5.1) (5.2) Y † )], the matrix matrix, Y .Since, − 4.2 − Y R R D parametrization ) or ( − erent from just Intuitively, since R is scanned in all ff 3.1 1 R ]. , in contrast to the ), matrix. 23 13 θ Y 4.2 ) to the extreme, as is 5.3 , once is varied? M , D 13 θ jj . , matrix [see eqs. ( )) R Y 3 − † M 4.6 . U was constructed to maximize Y is not a usual matrix considered in the D j † tr ˆ 2 i R ˆ κ R 1 2 R i y 1 M j i either. A simple and sensible approach is to κ ect equally the magnitude of the various 3 U ff decreases for increasing matrix in the literature. It also explains the , –8– erent by orders of magnitude. This is clearly 21 2 ) , ff = − erent, but still incomplete, scans of the 2 21 | Y R =1 ff † j ij Y Y † one. Y ) translates into R † = | ( − Y )]. Recall that L Y . how does it change when 5.1 i Y V † tr θ 4.7 0 Y and tr ]. On the other hand, in some cases it has been argued that for 21 .Then 21 Y 13 † , these do not depend on the θ R Y ] or [ ). But it still corresponds to a perfectly sensible -matrix [eq. ( fixed) than in the (1) number is as good as 3 here). Now, from eq. ( 14 ˆ R i and O 4.7 M M , we can expect a global range (see eq. ( D entries. We will see that they are quite simple, but very di Y † ij Y R ]. In ocassions a truly general scan has been attempted, as in [ tr Now, we can pose the following question: For a given The perturbativity requirement has to do with the Yukawa eigenvalues, In this section we address the issue of performing a complete scan of the This is not just an academic problem, since a similar one arises typically when leptogenesis constraints 22 ≤ 1 2 3 (which can keep y are incorporated, as we will see soon. Such a task is more naturally addressed in the literature. Actually, it “exploits”clear the from perturbativity eq. ( condition ( its generality (respecting perturbativitysponding and range orthogonality for conditions), what is the corre- entries. Actually, theyin can contrast be with easilystructure typical of di scans the ofat the a certain valueusual of behaviour observed in the literature. But This condition isnot very we handy consider and athe supersymmetric perturbativity easy-to-use. requirement version does of Besides, not the it a seesaw. clearly An applies important whether remark or is that (of course, any so the perturbativity constraint ( perturbativity criterion cannot dependimpose on a constraint on the trace of Yukawa couplings, say examining the restrictionsof that the the perturbativitychoosing criterion a certain imposes range on for Im the magnitude for a given come back soon to thiscan point. be Other found di in [ certain physical problems ain particular [ (incomplete but suitable) scan was fair enough, as JHEP03(2011)034 . 6 | √ 12 / (6.1) (5.6) (5.5) U 1 | factors ∼ < , κ | , remains. , √ 13 13 . GeV θ U . | =0 11 , range. Besides, )] more closely. i 2 ∗ 1 − o = 246 GeV. The φ U GeV .Wehavestudied 4.2 i 10 10 v = = 10 κ R 12 − × 1 o β √ φ ji is only a factor , certainly not orders of =5 = = 10 R 2 tan 21 3 3 δ κ , 2, with 2 =1 i √ Y in the 0 √ † entries are imposed. But the 13 v/ Y + θ R matrices which are far below the matrix elements. In consequence, ∗ 13 − − 100 GeV U R ,m 0 R − 2 3 . A simple way to do it is explained in for a fixed vanilla κ 1 H into predictions for the branching ratio = parametrization [eq. ( √ 13 0 GeV = 3 θ − ,M 21 j 12 R R R ) ,A − Y T ) implies that typically all the γ † GeV R 10 j –9– , Y e, 5.3 4 11 M × matrices, consistent with the perturbativity condi- cannot depend much on the value of / in the π ∗ jk − 10, so that → 21 R R 21 =9 = = 10 k µ 2 2 = 250 GeV Y κ changes with parametrization. are shown in figure 2. As expected, the dependence of 23 † Y 2 θ β √ − † / 21 Y k 1 L 13 matrix considered, we scan 2 Y θ V − U Y † ,M R =1 3 Y matrices. This simple argument can be made more rigorous (and and erence that is easily compensated by the fact that ,m ), but those are statistically rare exceptions. j,k − ff ,M R R 5.3 , the first term within the brackets is the responsible for the = | GeV 13 GeV ,adi s 21 , 12 is very small, almost negligible. Note that, indeed, there are cases for | 3 = 500 GeV 6 − 10 , or how κ / 0 13 Y π 13 √ θ = is not likely to have a noticeable impact on † m ), we have to assume first a supersymmetric model. We have chosen a minimal θ = 10 = = 10 γ Y | 13 on 1 1 θ e, 13 12 ) and the orthogonality condition, θ m M U 21 | ) are compensated by the typical sizes of the → 5.3 dependence observed in the literature. However, the other two terms within the Let us now show the results of the numeric scan. First of all, we need a systematic µ Y 5.5 † − 13 Y 6 The branching ratioIn BR( oder to translateBR( these results about supergravity (mSUGRA) model defined by results of the scan in which the dependence isperturbativity stronger, limit corresponding ( to Furthermore, we have used tan procedure to scan the whole rangetion of ( appendix A. Then, for each for the numerical example we have taken the following values for the other parameters: magnitude for vanilla cumbersome) once the orthogonalitybasic result, conditions that on the the rangeThis of contradicts the commonthe lore result in obtained the using the literature, and it is much more consistent with θ brackets can be easily assmaller big than as the firstBesides, one. the Note first perturbativityin that ( condition ( changing Since But still we dothe not value know of how feasiblethis in or a natural numerical is wayexamining (more to the details reach in expression these short), of boundsThis but expression depending we can can on get be insight written into as these issues by JHEP03(2011)034 ). → ) on µ 4.1 γ ]. The e, 25 → µ erent prior for ff 10 ) can be large for an γ e, ) for the same random set ) of the mSUGRA model, 8 γ ) and the approximate for- → 6.1 e, µ 5.4 → matrices obeying the perturbativity µ − R 6 erent matrices will change. The dependence of parametrization. In this case we have to [grad] ff − − 13 e R L V 4 for di – 10 – 13 , shown in figure 2. Concerning the size of BR( θ 2 matrices in the way we did it, the number of those − 21 vs. ) R 2 | Y 2 † erent way (in Bayesian language, using a di 21 ], which uses the full one-loop expressions of ref. [ ) ff Y ( 24 Y † . Then we have calculated BR( Y | X are the universal scalar mass, gaugino mass and trilinear coupling 0 0 M 1 A 0.1 0.01

0.001 1e-05 1e-06

0.0001

| Y)

|(Y and

2 + 2 / 1 leading to branching ratios well below the experimental limit. What happens matrices is relatively very small. However, one has to keep in mind that scanning M R , ), one can check that the expected branching ratio is very large. Some comments − . Scatter plot of ( 0 R parameter space in a di m 4.1 matrices used for figure 2. The computation was made by means of our own modified − For the sake of comparison between parametrizations, we have also shown in figure − R follows closely the one of ), we see that in general is very large, quite above the experimental upper bound. This is R γ 13 3 (right panel) a similar survey using the these results on the priorthat is the out same of prior the dependence scope wasature. of implicit Let in this the us paper. scatter finally Nevertheless plots remarkthe it shown that branching in is by the ratio worth changing previous noticing changes the liter- parameters parametrically, as ( indicated in the approximate formula ( choices of is that, scanning the“good” space of the the that space), the abundance of those “good” mula ( are in order here.ordinary First, minimal although SUGRA it model, is commonvalue we of lore see the that here branching BR( that, ratio afteris is a performing typically very a quite suggesting complete above result. scan, the Second, the experimental although upper it bound, is not which visible in figure 3, there are of course of version of the SPhenoresults code are [ shown in figureθ 3 (left panel). Ase, expected, the dependence ofin BR( fact not surprising: using the approximate general range ( condition (5.3). where at the unification scale Figure 2 JHEP03(2011)034 L ). V 10 6.1 0011, . =0 8 1 3. Therefore y ). Hence, the ). This pushes matrices (or . The branching 4.1 2 − 3 6 5.6 13 y R θ parametrization. − [grad] L 13 e V 4 . Its present experimental = 1 in the latter. Note that γ 3 n y 2 ]), where non-trivial correlations parametrization the righthanded 3, which allows − 26 L erent if the set of ≤ V on LFV processes was reported to get ff 0 parametrization, eq. ( 13 Y

1e-06 1e-07 1e-08 1e-09 1e-10 1e-11 1e-12

† θ

a ! − ) ->e, BR( Y R does not depend at all on parametrization; (right) 21 − – 11 – 10 R Y † Y 8 .(left) 13 parametrization can be more than one order of magnitude parametrizations (left and right panels) we note a similar can be almost one order of magnitude bigger than in this θ − − 2 ). Furthermore, in the 6 L -parametrization) is constrained by additional considerations. R V 21 L )vs. ) 5.4 V γ [grad] to the number density of photons Y 13 e, † e and one. This is mainly due to the choice B 4 Y → n ( L − , which is one of our main results. Besides, in the ﬁgure we see that µ V R 13 θ 2 = 1. Besides, we have chosen the same mSUGRA model deﬁned in eq. ( .BR( 3 y parametrization we have imposed tr − matrices for the 0 03, R . R

1e-12 1e-07 1e-08 1e-09 1e-10 1e-11 1e-05 1e-06

V Figure 3 a ! ) e, -> BR( The last point raises a final question: what happens if leptogenesis constraints are Of course, the results of this section could be di survey, see equation ( =0 L 2 density of baryons 7 Constraints from leptogenesis The baryon asymmetry of the universe (BAU) is usually defined as the ratio of the number can indeed occur. imposed in the scenario? Actually,reinforced the once impact successful of leptogenesis iswe incorporated re-visit in the the leptogenesis analysis. issue. In the next section, downwards further the branchingtwo ratio surveys through are perfectly the consistent. log factor, see eq.and ( This is the case of GUT models (as those studied in [ larger than in the in the the upper limit of V neutrino masses are an output.less In degenerate) the than example the chosen they choice come made out for typically the bigger (and Recall that in this parametrization ratio does, but in asurveys marginal with way. the All thisinsensitivity is to apparent from figurethe 3. branching ratios Now, comparing in the the choose the values ofy the three neutrino Yukawa couplings. We have taken JHEP03(2011)034 ]. R T 34 1 1 (7.3) (7.4) (7.1) (7.2) – M M 32 ), taking 13 matrix). θ ,where Y R τ T e, µ, , 1 1 = M − i ects can be significant cannot be much larger ff ]. Note here that the rel- ] R τ T 30 36 m +˜ ∗ µ . m m 2 . 10 . − +˜ , 13 e θ η 10 2 ˜ ), contains the required lepton number m η 2 is the CP-violating contribution to the × − , is given by [ 2.1 η 10 + 15) ] sphal . × 0 16 C . ect, , and to compare the results with the previous 31 1 ff ± 1 γ 04 13 eV . – 12 – n θ n 3 ) 1 ects here, since our main goal in this paper is to ] for a recent review). Then the lepton number is 19 τ − - parametrization. This may be an indication that flavour . ff = 29 m L 10 − V γ B ] it was shown that successful leptogenesis is possible within the +˜ n ∼ =(6 n ciency factor, γ B 35 µ 1 2 ∗ 1 n n γ ffi B m n m n /M /M +˜ 2 u 1 v e 2 ciency factor, which takes into account the partial erasure of the H | ects were not considered). m 2 u i ffi ff 1 v 2( ˜ ) can be written as [ π Y | 7.2 ects and using the = =4 ff contains the sphaleron e i = is the e ∗ ˜ η m η m sphal ) are in equilibrium in the thermal bath, and flavour e C ] τ GeV. In this temperature regime, one or more charged-lepton mass-eigenstates ; is the equilibrium number density of the lightest righthanded neutrino, which GeV, to avoid the gravitino problem. In the following we will assume 3 27 ], in which a net lepton number is produced by the out-of-equilibrium decay of the 10 1 10 M n e, µ, 28 10 In the unflavored case, the e The final value for the BAU is given by: = It is worth mentioning here that in [ 2 2 ects are not going to introduce any dramatic dependence on ( R ff e SUSY seesaw for any valueinto of account the flavour still e unmeasured low energy neutrino parameters (including where because the correspondingHowever lepton we asymmetries will follow not anexamine consider the independent flavour dependence e evolution of the [ literature results (where on flavor e The thermal production of righthandedis neutrinos the is reheating suppressed temperature unless afterthan inflation. 10 On the otherT hand, asymmetry, and CP asymmetry by inverse decays andstandard scattering model processes. eq. In ( the minimal supersymmetric where is the mainM responsible for the asymmetry for hierarchical righthanded masses, (seesaw) right-handed neutrinos (see [ converted into baryon number by sphaleron-mediated processesevant Lagrangian, [ defined by the superpotentialand ( CP violating-terms (the latter are provided by appropriate phases in the value is [ Perhaps the most popular mechanismnesis to [ generate such BAU is nowadays thermal leptoge- JHEP03(2011)034 (7.6) (7.7) (7.5) matrix space. − ]: U 10 R 37 , after imposing successful ) 2 1 8 13 θ matrices that lead to strong /M − 2 j . , is given by [ , R M . ) on ( M γ g 13 √ θ 6 e, } (and thus of the branching ratio) on D matrices were precisely those selected Planck 2 ] † ) — is independent of 11 j − 21 → ) R 1 † /M ) R κ 3.1 2 µ † 1 [grad] Y † 13 M " YY RD ∗ Y ( YY g 4 M [( – 13 – √ √ { parametrization; including leptogenesis constraint. D 66 − . Im = R τ . . † µ, 13 in = θ )=1 j 2 1 in its whole parameter space. Clearly, no important 13 x +1 YY ), does not depend on θ M depend on the Yukawa couplings through the combination , x R π ) on 1 7.3 = 8 ln γ 75 . As we have seen in the previous section, there are some )vs. . T − e, = γ ( 13 parametrization — eq. ( and θ x e, H → − 0 2 η − R → µ = = 228 1

µ 1 1e-07 1e-08 1e-09 1e-10 1e-11 1e-12 1e-05 1e-06

) on

! ! ) e, -> BR( H γ x √ e, MSSM | .BR( ∗ g → )= x denoting the Hubble parameter. The CP factor, ( µ g H ,which,inthe Figure 4 shows the dependence of the BR( Note that both Still, it could happen that leptogenesis constraints strengthened the dependence † Figure 4 in the complete scenario. To analyze whether this possibility (or a more moderate is important. If (in an extreme case) those matrices for which the dependence of − 13 13 leptogenesis and scanning one) takes really place, itPerforming is partial important scans to in perform thisintroduce a space complete can unwanted be scan biases: useful of to the dependences one show of particular could BR( features, artificially but it select may of BR( R θ by successful leptogenesis, weθ would find a strong dependence of the branching ratio on YY Hence, the BAU, given by ( where with JHEP03(2011)034 ) γ on i e, erent (7.8) ˜ m ff .The ciently i ). The → ffi 7.6 µ and η matrix elements, ) and ( − 60 50 40 30 20 10 0 70 R 7.4 . both through the CP 10 ), ( − Y 0.001 10 7.3 , × τ ) that the CP asymmetry is matrices which are su 7] 7.6 − − 0.0001 e, µ, R = 1e-05 ) , and thus for larger " i ( 1 Y ,i 1e-06 16 . 1 Yukawas approximately cancels out due to the – 14 – − . Note from eq. ( 1 1e-07 η ciency factor is given by i CP-factor R ) smoothly interpolates between the weak ( ˜ ν ffi m ∗ i Yukawa couplings (that is, the second and third rows for random R-matrices, every point represents a di 7.4 m (see figure 4). 3 1e-08 η R 2 R ν . ) washout regimes. However for all the points in our scans ∗ and 13 ≈ θ are selected, thus reducing the final value of the branching ratio. m 1e-09 decreases for larger ,ineq.( and η ,thusthee η η ciency factor η 2 ∗ ffi R i m ν ˜ 1 1e-10 m 0.1 i is observed. The trend is very similar to the one with no leptogenesis

0.01

0.001 1e-05 1e-06 1e-07 1e-08 ! ) Efficiency ( Efficiency ˜ 0.0001 13 m θ i and the e ciency factor, ). So, after imposing enough BAU, only those ffi factor in the denominator. As a consequence, we find a mild dependence of . Scatter plot of 3.1 . The red and green lines denote the lower and upper BAU limits, respectively. For clarity 11 R ) † ), since the dependence on the In figure 5 we have performed a scatter plot showing the values of The e Note that when the leptogenesis constraint is imposed, the values BR( matrix. The color coordinate represents the average absolute-value of the elements in the first ) and strong ( R ∗ − YY experimental window is also shown for reference. From the figure it is clear that enough and it becomes clear that see eq. ( small in order to optimize m we find that baryon asymmetry dependsasymmetry on the neutrinomainly driven Yukawa by matrix the of ( the elements in the first row of constraints (figure 3). Thecreate) conclusion any is dependence that on leptogenesis constraints do notdecrease enhance (or at least onefigure order 3 of (left). magnitude This with behaviour respect can to be the understood case from without eqs. leptogenesis, ( R row of in the plot, we have relaxed the allowed BAU window todependence be on [5 Figure 5 JHEP03(2011)034 ) γ ma- e, erent ν ff → Y † ν µ erent way Y ff )insuper- γ e, , which changes -parametrization η → R µ does not hold after a are smaller than their 13 ) gets very insensitive to θ γ R e, matrix elements. -parametrization, this quan- ]. In the − L -matrices in a consistent way occurs through the → 13 -parametrization (section 3.2) V R R L µ 13 V θ parameter space in a di − can be reached for any value of the R ) on , which essentially disappears once the γ 13 e, θ → µ -matrices compatible with orthogonality and – 15 – R (or to any observable parameter), so BR( ). We also give (in appendix A) an straightforward 13 5.3 θ matrix. This is in contrast with the case of − R is the neutrino Yukawa matrix. In the ν ) might change. In Bayesian language, this is equivalent to use a di -parametrization (section 3.1) and the Y γ R seems to have a dependence on parametrization. e, -parametrization (which is the most common in the literature) a fair scan − ij R ) R → ν µ Y , as already mentioned. Moreover, the values of the branching ratio are typically † ν -matrix. It is given by eq. ( 13 Y BR( Once such scan is performed theθ branching ratio BR( larger than the experimental upper bound,one which is has a to very keep suggesting in result.(though mind However, that still if covering we the scanned the whole space) the relative abundance of points with large R procedure to completely scanwith the this space requirement and of theto complex orthogonality avoid possible one. biases This producedin procedure by the can incomplete scans be of very the useful seesaw parameter space togenesis), a result which is non-trivial inIn either the parametrization. implies to allowperturbativity all of the the Yukawathe couplings. complex perturbativity of We Yukawa couplings give as a a very condition in simple the rule entries of to the incorporate orthogonal tity is trivially insensitive to is. This was already( noticed in early9-dimensional papers, parameter as space ref. is [ fairlyneutrino covered. masses This are holds kept even constant if or the under right-handed control (as is required for succesful lep- the seesaw scenario.9-dimensional This space amounts or, to two incalled alternative other the ways words, of of traveling parametrizing across our this The ignorance. main potential These dependence of are BR( trix, where The somewhat propagated beliefsymmetric seesaw that models the depends branchingcareful strongly ratio analytical on BR( and the numerical value of study.native We parametrizations have of analyzed this theobservables issue 9 (neutrino using degrees masses, two of alter- mixings freedom and that, phases), besides expand the the 9 parameter low-energy space of • • • • direction, which means that the same values of − dramatically when varying the typical size of the first8 row Conclusions The main results and conclusions of this paper are the following: BAU is onlyperturbative produced bound, in if agreementreason the with for elements our the decrease previous of of analytical the the estimations. branching first ratio.first This row row is Besides, entries in the there of is the no color gradient in the JHEP03(2011)034 ) ). ) on γ A.1 γ (A.1) e, e, → → µ -parameter- µ )presentsthe R 11 -matrix entries. , condition ( R R 3 ( matrices, to cover κ 33 − . The main impact of R R 2 13 κ θ matrix in the full allowed ects) in the analysis does ff − R 1 ) on κ γ , and the perturbativity condi- 1 e, = → , decreases for large . η µ R j T κ i R 1 M ) by more than one order of magnitude. This – 16 – 2 γ | ij e, R ciency factor, | → ffi matrix µ − R matrix can reach their corresponding perturbativity limit. Nor- − R matrix elements are allowed to be larger (in absolute value) when you − R ), which for convenience we repeat here: 5.3 ), ( not introduce any furtherleptogenesis, dependence besides of remarkably BR( reducing the acceptablespace, volume is of the a decreasecomes from of the BR( fact thatHence, the successful e leptogenesis prefers smaller ones, and thus smaller BR( prior for the parameter space.the The prior dependence is of out the of typicalsame size the prior of scope dependence BR( was of implicit this in paper. the Nevertheless scatterIncluding it plots leptogenesis is shown constraints worth in (disregarding noticing previous flavour that literature. e the 5.1 In general, given a normal hierarchy among neutrinos, • mally, if one sets any entry of the first row (column), at its maximum magnitude, then the tells us that the move from bottom-right to top-left insmallest the (largest) matrix. upper Thus, bound. the element Onnot the all other the hand, elements orthogonality of implies that, in practice, The algorithm presented belowalso has leptogenesis been constraints. designed to be easily modified if one considers A General scan ofIn the this sectionall we possibilities explain compatible the withtion details ( orthogonality, of the scan made on the Generalitat Valenciana grants PROMETEO/2009/116Comunidad and de Madrid PROMETEO/2008/069; through the Proyectomission HEPHACOS under ESP-1473 and contract thesupport PITN-GA-2009-237920. European of Com- a B. FPI (MICINN) acknowledgesthe Zald´ıvar grant, use the with of reference financial the BES-2008-004688. IFT Also computation acknowledged cluster. is Acknowledgments We thank A. Ibarra forthe very useful MICINN, discussions. Spain, This underIngenio work has contracts PAU been CSD2007-00060 FPA-2007-60252 partially and and supported FPA-2007-60323; MULTIDARK by CSD2009-00064. Consolider- We thank as well the As a concluding remark, scanningrange the parameter is space necessary of the inscenario. order This to has make notthe been general procedure always exposed statements properly in about this incorporatedphysical paper predictions questions. in is of former very works. the easy to seesaw In implement this and sense, can be applied to other JHEP03(2011)034 , , 2 33 33 (see R ) for ,R (A.6) (A.4) (A.5) (A.2) η 32 A.1 ,R 2 23 c as: ,R 13 22 . R R ] ] (A.3) π to scan ciency factor, . . π 2 ] ] ffi 2 , . 32 π , π 2 [0 2 R 2 θ [0 . , , , ∈ 1 [0 ) ∈ [0 θ 2 2 23 1 , one scans s ∈ : ∈ ) or by checking eq. ( 2 1 φ 3 φ 2 23 c s φ 13 R 5.1 + φ , , . A convenient way to do it is by ). At this point we have used the 2 1 3 3 3 1 s , θ κ κ . Finally, we cross-check for each of s 2 , , 2 3 3 + (obviously, up to the sign) and exploit 1 1 κ c 2 2 is allowed to be much larger than 2 1 ), we see that 3 c M M 2 1 2 s 23 1 s 2 32 √ √ M = | R 3.2 | + R 2 : √ − 13 2 2 ≤ c 1 2 33 ( | s c | R 2 1 2 1 ≤ R | s c 23 | ≤ – 17 – 2 + | R c | 1 matrix is indeed consistent with the perturbativity | c 2 32 we derive ± | − R 1 2 to scan , R c c 13 ). Conversely, if the matrix elements 33 there are two of 23 φ , i , R R 3 e 2 , | 23 2 s A.1 φ 1 φ s i i 13 φ matrix. Then the rest is constructed automatically from or- e there are two values of i e | | R e = − 2 | | 1 c 23 1 c 3 | R c s = | R | = become too big, since they lead to a small e we derive the two possible values of 13 = 2 = c R 1 R 23 , and the orthogonality condition, to scan c (and those appearing below) is assumed to be a random number within 23 R . Then, instead of the previous scan in 33 2 R 13 φ ,R element (recall we prefer to impose perturbativity in the bottom-right part R 32 23 R R ). In this case, it pays to constrain from the beginning any of the entries in the 7 ). Thus we scan When leptogenesis constraints are included, most of the initially-allowed values for the Now, we have to scan the third complex angle, More in detail, using the parametrization ( R first-row entries of section first row, e.g. Once again, for eachthese value values that of the corresponding condition. One can dothe remaining that entries. directly by using equation ( For each value of (again, for each valuebounds of on using the of its interval. Now, forthe each perturbativity value of condition on but the perturbativity conditionup tells to us phases. that So, we can use the perturbativity upper bound on Here the phase thogonality, requiring a final cross-check of the perturbativity condition. corresponding entries in the same columning (row) cannot the satisfy perturbativity orthogonality without bound violat- satisfy ( their perturbativity constraints, thenumn normally would the do entries it ofbottom-right as the part well. first of row the Hence, and col- it makes sense to start imposing perturbativity in the JHEP03(2011)034 )is ) for (B.1) (B.2) (A.7) γ 3.2 e, matrices. -diagonal ff → − matrix is of R µ 2 L m the contribution scale is obtained ](implementedin ff − 3 M produces o , given by eq. ( . This constitutes the as ij ν 2 L ) 2 L e R . , Y m † ν . In our case, however, the m Y ) ν ]code.The ( 2 2 Y R at the 0 s ) is the contribution coming Y 2 1 24 L A ,switchingo ν c B.2 D Y = M + † ν 2 1 ij Y s ) all the soft terms are diagonal in the ) e 2 0 + X A 2 A ( 2 2 13 , we evaluate M s + down to 2 1 R 2 0 M c , X − m ( 1 + 2 2 0 M 2 1 (3 s , the RG running of ) for each point in the scan of the s 2 1 – 18 – 2 m X c 1 π . The value of = 8 3.1 2 L M from ij is given by m is normally much lower than its perturbativity bound − 2 L ± ) | 3 2 L 1 R m s 2 e s 13 m matrices (see appendix A). 13 R m | D − . R R R = =( = 2 L are the left- and right-handed slepton mass-squared matrices, ij 3 ) s m e 2 L ). A m the value of ( A.5 is diagonal. Below 13 and e R -diagonal entries of R Y ff 2 e is run down to low-energy (neutrino Yukawas do not play any role in this parametrization formula ( m − is the result of running 2 L , parametrization R 2 L 2 L -diagonal terms. At the seesaw scale, m − ff m in our case), but still is way larger than required by leptogenesis constraints. Now, m R 2 D scale to be the one where gauge couplings unify. We also extracted from SPheno the The rest of physical quantities (charged slepton mass matrices, gauge couplings, GUT We work with our own modified version of the SPheno [ We have adopted an mSUGRA framework, with universal soft terms at the GUT scale, As a final comment, recall that the parametrization of , 10 X X ∼ SPheno) to calculate the branching ratio at 1-loop level. RG-interval since right-handed neutrinos are decoupled). scale, charged lepton yukawas,M etc.) are takenparameters directly of from the SPheno, neutralinos which and charginos. imposes Finally, the we followed ref. [ from neutrino Yukawa couplings.from the The neutrino Yukawas, second evaluated atcontains term the the leading-log in approximation. ( o Thisfrom contribution the Finally, special interest since, amonglarger the o slepton mass matrices it is by far the one that develops Here where and the matrix of sleptonbasis trilinear in couplings. which At entries, due, essentially, tomain the source contribution of proportional flavor violation. to In this section we explaineach the point strategy in for the computing scan the branching of ratio the BR( M global changes of the sign of B Details of theB.1 numerical computations which replaces eq. ( completely general up toscan reflections changing is the completely sign of general det since all the relevant physical quantities are invariant under Note that this upper bound( on for each value of JHEP03(2011)034 ]. (1999) (2008) ] SPIRES (1996) 2442 D 59 ][ C 57 (2002) 206 D 53 (2002) 115013 on lepton flavour ) are obtained at the from the MEG 13 R θ Phys. Rev. γ V B 527 D 66 e hep-ph/0605151 , [ ]. Eur. Phys. J. → , Phys. Rev. hep-ph/0607263 , leptogenesis CP-violating ]. µ [ , (and γ as probes of neutrino mass Lepton flavor violating process i + γ ]. Impact of SPIRES M j e l Lepton flavour violation in the Phys. Rev. ][ Phys. Lett. ). On the other hand, the neutrino → , SPIRES , → i µ ][ 3.4 l (2006) 073006 (2006) 090 SPIRES Lepton-flavor violation via right-handed Waiting for ]. A new parametrization of the seesaw ][ and 11 γ D 74 µ → SPIRES JHEP – 19 – [ τ ), from eq. ( , hep-ph/0106245 parametrization, i [ − M matrix are indeed initial parameters, defined at the L ]. arXiv:0904.2080 Phys. Rev. V [ L , V ]. Large muon and electron number violations in supergravity arXiv:0910.2663 [ Charged lepton decays (1986) 961 ]. ]. ]. (2001) 269 Solar and atmospheric neutrino oscillations and lepton flavor SPIRES ). 57 ][ γ SPIRES and the (2009) 030 e, ][ Y B 520 matrix is obtained directly from SPheno, and can be used for the SPIRES SPIRES SPIRES 12 → (2010) 026 D ][ ][ ][ Flavour physics of leptons and dipole moments 2 L µ 05 m (a suitable average of JHEP , M Phys. Rev. Lett. ]. parametrization, we have used the SPheno code as well. However, the original hep-ph/9810479 JHEP Phys. Lett. , [ parametrization , , − L − arXiv:0801.1826 V L [ V hep-ph/0110283 hep-ph/0206110 hep-ph/9510309 SPIRES J.R. Ellis, J. Hisano, M. Raidal and Y. Shimizu, J. Girrbach, S. Mertens, U. Nierste and S. Wiesenfeldt, MSSM [ mechanism and applications in[ supersymmetric models [ experiment in degenerate and inverse-hierarchical neutrino models violation in supersymmetric models116005 with the right-handed neutrinos models parameters and Majorana phases neutrino Yukawa couplings in supersymmetric[ standard model theories 13 violating processes within SUSY seesaw J. Hisano, M. Nagai, P. Paradisi and Y. Shimizu, A. Kageyama, S. Kaneko, N. Shimoyama and M. Tanimoto, S. Lavignac, I. Masina and C.A. Savoy, S.T. Petcov and T. Shindou, F. Borzumati and A. Masiero, J. Hisano and D. Nomura, M. Raidal et al., S. Antusch, E. Arganda, M.J. Herrero and A.M. Teixeira, J. Hisano, T. Moroi, K. Tobe and M. Yamaguchi, Consequently, we modified the code, incorporating an iterative procedure to determine in a consistent way with all the boundary conditions (at low- and high-scale). In , from the beginning. But in the i i [8] [9] [6] [7] [4] [5] [1] [2] [3] [10] [11] M this way, the full computation of BR( References M seesaw scale, Yukawa eigenvalues, high-scale (and these are the ones in which we perform our scan of the parameter space). B.2 For the code is not prepared toIn introduce particular, the the original initial version parameters works according with given to values this of parametrization. right-handed neutrinos masses, JHEP03(2011)034 ]. , , ] ] , , (2011) SPIRES B 174 (2001) 171 B 739 New J. ][ , (2003) 275 192 153 B 618 hep-ph/0612289 [ Phys. Lett. hep-ph/9606211 [ , Nucl. Phys. , ]. (2008) 105 hep-ph/0301095 [ ]. (2006) 033004 Nucl. Phys. (2007) 064 (1997) 3 , 466 The see-saw mechanism, SPIRES γ ]. 04 Precision corrections in the ]. Astrophys. J. Suppl. ][ e, , D 73 SPIRES → B 491 ][ Hints on the high-energy seesaw (2003) 241 µ and leptogenesis JHEP SPIRES SPIRES , γ Constraints on SUSY seesaw parameters ]. ][ Comput. Phys. Commun. Phys. Rept. ][ + , The neutrino mass window for baryogenesis , ]. j l B 564 Phys. Rev. Parameterizing Majorana neutrino couplings in ]. , → Seven-year Wilkinson Microwave Anisotropy Observable seesaw and its collider signatures Quasi-degenerate neutrino mass spectrum, Nucl. Phys. i SPIRES hep-ph/0206174 Massive neutrinos and flavour violation l – 20 – [ , colliders ][ SPIRES ]. ][ − SPIRES [ e arXiv:0903.1408 [ Leptogenesis + Phys. Lett. e , ]. hep-ph/0302092 Baryogenesis without grand unification [ arXiv:0901.4596 SPIRES [ (2002) 229 ]. ][ Determining seesaw parameters from weak scale measurements? ]. ]. ]. Oscillating neutrinos and Leptogenesis and the violation of lepton number and CP at low SPIRES (2009) 003 hep-ph/0104076 [ ][ B 643 SPIRES hep-ph/0407325 05 (2003) 445 [ Flavour violation at the LHC: type-I versus type-II seesaw in minimal SPIRES SPIRES SPIRES (2009) 291 ]. ][ ][ ][ ][ arXiv:1007.3664 , JHEP Lepton flavour violation in supersymmetric models with seesaw mechanism B 665 hep-ph/0510404 , B 677 [ SPheno, a program for calculating supersymmetric spectra, SUSY particle decays SPIRES decay and leptogenesis (2001) 013 [ Nucl. Phys. collaboration, E. Komatsu et al., ]. ]. γ (2004) 202 , 09 + 6 e arXiv:1001.4538 [ → arXiv:0802.2962 hep-ph/0511062 hep-ph/0301101 hep-ph/0103065 SPIRES SPIRES A. Masiero, S.K. Vempati and O. Vives, W. Porod, D.M. Pierce, J.A. Bagger, K.T. Matchev and R.-j. Zhang, W. P. Buchmüller, Di Bari and M. Plümacher, F. Deppisch, H. Pas, A. Redelbach and R. Ruckl, J.N. Esteves et al., S. Davidson and A. Ibarra, J.A. Casas, A. Ibarra and F. Jimenez-Alburquerque, S.T. Petcov, W. Rodejohann, T. Shindou and Y. Takanishi, A. de W.-C. Gouvêa, Huang and S. Shalgar, E. Arganda, J.R. Ellis and M. Raidal, Probe (WMAP) observations: cosmological interpretation 18 (1986) 45 [ minimal supersymmetric standard model [ Phys. WMAP [ and SUSY particle production at [ µ Nucl. Phys. from leptogenesis and lepton flavor violation [ supergravity Phys. Lett. [ JHEP mechanism from the low-energy neutrino spectrum neutrino Yukawa couplings, LFV decays (2006) 208 the Higgs sector Ph.D. thesis in physics, Universidad de Autónoma Madrid, Madrid Spain (2008). energies M. Fukugita and T. Yanagida, S. Davidson, E. Nardi and Y. Nir, S. Pascoli, S.T. Petcov and C.E. Yaguna, N. Haba, S. Matsumoto and K. Yoshioka, J.A. Casas and A. Ibarra, [28] [29] [26] [27] [24] [25] [21] [22] [23] [19] [20] [17] [18] [14] [15] [16] [12] [13] JHEP03(2011)034 ] 315 ] ]. JHEP , Phys. Lett. , ]. SPIRES Ann. Phys. [ , hep-ph/0310123 [ Flavour issues in hep-ph/0605281 [ SPIRES ][ (1985) 36 Towards a complete theory of (2004) 89 ]. (2006) 010 B 155 09 B 685 On the anomalous electroweak baryon SPIRES ][ arXiv:0806.2832 CP violation in the SUSY seesaw: [ JHEP , Leptogenesis for pedestrians Phys. Lett. ]. , Nucl. Phys. The importance of flavor in leptogenesis , ]. – 21 – (2008) 053 ]. SPIRES ][ 09 hep-ph/0601083 CP violating decays in leptogenesis scenarios [ SPIRES ][ SPIRES JHEP ][ , (2006) 004 04 hep-ph/9605319 [ Flavour matters in leptogenesis hep-ph/0601084 [ JCAP hep-ph/0401240 , [ ]. ]. (1996) 169 (2006) 164 SPIRES SPIRES L. Covi, E. Roulet and F. Vissani, S. Davidson, J. Garayoa, F. Palorini and N. Rius, W. P. Buchmüller, Di Bari and M. Plümacher, A. Abada, S. Davidson, F.-X. Josse-Michaux, M. Losada and A.E. Riotto, Nardi, Y. Nir, E. Roulet and J. Racker, A. Abada et al., V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, G.F. Giudice, A. Notari, M. Raidal, A. Riotto and A. Strumia, B 384 [ leptogenesis and low energy (2005) 305 leptogenesis 01 number nonconservation in the early universe thermal leptogenesis in the[ SM and MSSM [37] [35] [36] [32] [33] [34] [30] [31] CHAPTER 5

COLLIDER BOUNDS ON DARK MATTER 64 Collider Bounds on Dark Matter JCAP10(2011)023 mbines the hysics P le heories beyond the as been observed in ic t ar phobic (WMAP Haze, dijet is work, we show that when sis, a light thermal fermionic 10.1088/1475-7516/2011/10/023 se of scalar dark matter, using (SM) particles and a light Dark doi: ne could ﬁt the data at the price g of dark matter to electron. d, strop A b [email protected] , osmology and and osmology and Bryan Zald´ıvar C a 10 GeV) that couples to electrons is mainly ruled out if one co !

dark matter theory, dark matter experiments, cosmology of t Recently, several astrophysical data or would-be signals h [email protected] rnal of rnal

ou An IOP and SISSA journal An IOP and 2011 IOP Publishing Ltd and SISSA E-mail: Laboratoire de Physique UniversitéParis-Su Théorique, F-91405 Orsay, France Instituto de Fisica Teorica,Nicolas IFT-UAM/CSIC, Cabrera 15, UAM Cantoblanco, 28049 Madrid, Spain b a c ! SM analysis with WMAP constraints. Wea also single-photon study events the simulation special to ca constrain the couplin Keywords: one takes into account theDark more Matter recent LEP ( and Tevatron analy different dark-matter oriented experiments.of In specific nature each of case, the o Matter coupling candidate: between the hadrophobic Standard (INTEGRAL, Model PAMELA)anomalies or lepto of CDF, FERMI Galactic Center observation). In th Abstract. Received July 18, 2011 Accepted October 2, 2011 Published October 18, 2011 Yann Mambrini light on the nature of dark matter When LEP and Tevatronwith combined WMAP and XENON100 shed J JCAP10(2011)023 3 6 7 8 9 1 2 3 7 10 ter 10 GeV he same ]. Many ! 1 ts for signals in atio spect WMAP upper bound ] signals even if contradicted an a simulation of events at 8 hobic dark matter candidates s and the excess of emission nic candidate in section IIIA, of couplings we have studied. rgy, and two jets [ In each case the nature of the hen implement the constraints coupling needed to reconciliate f light dark matter candidates rk matter with mass suppressed couplings to leptons es is fundamental in any kind of 1 tion of an excess of events which ays excess measured in PAMELA ], in which composite dark matter (dark lation rate to avoid the over-closure 13 ] and the CDF signal if it annihilate in [ 11 ] and CoGENT [ 7 –1– ] used the single-photon events at LEP to constraint 14 ] interpreted this excess by the introduction of a new gauge 3 ] in order to constraint the operator suppression scale, in t ]. On the other hand, some authors showed that a light dark mat 15 ] to explain the DAMA [ 10 6 , – ] which needed hadrophobic dark matter. 9 4 12 dark boson). Dark matter experiments had also given some hin ]). Some authors [ 2 A recent alternative explanation for these signals are given 4.1 Fitting4.2 with WMAP Tevatron constraints 4.3 XENON100 constraint 3.1 The3.2 fermionic case Scalar case leptophobic 1 5 Conclusion and prospect 1Introduction Very recently, the CDF collaborationinclude announced the a observa lepton (electron or muon), missing transverse ene 4 Complementarity with other experiments 2Themodels 3 Constraints from the thermal relic to the hadronic branching r Contents 1 Introduction We give our resultand in the consider case a ofDELPHI scalar contact experiment case operator [ in for a section fermio IIIB. For the later, we r with charged-leptonic couplings generates a tooof low the annihi Universe. In thisthe work, LEP we compute analysis the with rateconstraint. of a hadronic In thermal section dark II, matter hypothesis we will and re review the models and type atoms) is considered. fashion as is done in the literature for the fermionic DM. We t discoveries. Recently, the authorsthe of nature [ of the dark matter couplings, concluding that a da by the authors ofcould [ at theobserved same by the time Fermi explain Gammapredominantly into Ray these hadronic Space states. direct telescope Thereor detection [ was INTEGRAL also signal cosmic [ r couplings of the dark matter with the Standard Model particl studies have been done(see since e.g., then [ motivatingboson the with existence sizable o couplings to(a quarks, but with no or highly direct or indirect detectionwere modes. proposed in On [ one hand, some hadrop JCAP10(2011)023 (2.1) ) χ 5 γ µ g scale to be χγ ) )(¯ ] whereas scalar i χ q a real scalar DM µ 5 16 [ γ χγ ! µ µ ic final states in the γ )(¯ Z i i e that the dark matter q sis in section IV before q ive of UV completion: (¯ ) (2.2) ) µ i 2 i h s of interactions consistent γ i g Λ χχ χq q =0 (¯ in the expansion of the heavy )(¯ )(¯ i 2 i i ndard model fermions in terms i h χ ν ! strained by LEP analysis. This ation with a hard photon. The q g Λ i i ies the introduction of a second = y an universal generation/family q i q l hange of a l (¯ (¯ =0 g t 1) the neutrino couplings and 2) i h g )+ 2 2 i h i i g h , ! Λ ]. χ l g ] which was concerned by the leptonic Λ g Λ = 5 g √ i 17 γ 14 i µ,τ,ν )+ i µ = ≡ ! = ! χ i l µ h χγ g 1 . Note that we assumed lepton flavor to be τe,µ, ν , Λ , )(¯ )+ h τ e,µ, )+ χγ e = i i g i l l = g )(¯ i l g 5 i χl χχ g = l γ = –2– ; )(¯ µ µ )(¯ l i e l γ χ γ l g i g i i i Λ l ¯ l ¯ l ¯ l ¯ ( ( √ ( ( 2 2 i i 2 l i l l 2 i l g g g ≡ g Λ Λ Λ Λ u,d,c,s,b,t l . We will thus introduce hadronic and leptonic coupling = l 1 i h i i i i Λ ] made an analysis with relatively little model dependance, g ! ! ! ! 14 = = = = t A S V L L L L , such as l Axial : . Indeed there is no reason for the effective hadronic breakin g Vector : h channel : channel : − − and t s h , , is a Dirac fermion (except in section IIIB, where we consider g χ Electrophilic couplings (model A): Universal lepton couplings (model C) : Charged lepton couplings (model B) : We will consider 3 kinds of models which could be representat We will then consider the set of operators As we are interested in the ratio of the hadronic to the lepton Recently, the authors of [ Scalar Scalar • • • being the DM candidate. Throughout this paper, we will assum We begin with the casewith of a the fermionic requirement WIMP, of andenables Lorentz study us the invariance to 4 and describe type strongly theof con interaction an between effective WIMPs field and theory, sta propagator. in which However, contrarily we to keep the only description in the [ first term from the mono-jetconcluding event in of section Tevatron V. and XENON100 in the analy 2Themodels constants by pair production of pair of dark matter particles in associ interaction is motivated by Higgs-portal like models [ the possibility ofeffective hadronic scale, tree Λ level couplings. This impl constraints, we generalize the analysis taking into accoun χ candidate). A vectorial interaction is motivated by the exc the same than the leptonic one Λ particle DM annihilation, we willcoupling consider in without the loss hadronic of sector: generalit conserved in the dark matter interaction. JCAP10(2011)023 2 2 v the (3.2) (3.1) .We c ) and 2 ibility Λ 3.1 / e g = 2 e ng ratio in eq. ( Λ / 2 v 2 χ m 1, one needs to impose + . , taking into account the 100 0 l und to 1 4 χ 2 v ) 2 4 m ! Br hermal relic, asking for the J I,h nihilation cross section which 4 m / events in their data sets, but orial, scalar, axial or t-scalar) 2 s σ h t−Scalar − nic or universal leptonic), d model. They used the single- h ihilating WIMPs up to order 2 3 & χ 80 m ( 4) ). We then translated this limit on ]Ω (GeV) 2 χ / 1 + 2 " χ m 19 u,d,c,s,t,b v 2 4 by lue dashed-dotted), axial (green full-line) m 4 = 4 m 60 can be converted in a upper bound to − h − 2 2 h e s (1 ,m g − 3 / as a function of the dark matter mass for the cg 2 3 ! 2 √ m e / m + m g 40 2 v –3– √ / − J =4 I,l =Λ Λ (case A), into charged leptons pair (case B) or into s σ s e ≡ − ! e 20 2 s e | + 2 I τe,µ, ν , " e π = l Vector Scalar Axial 64 |M 2 l g 0 = 0 500 400 300 200 100 600 = I

1 pb to avoid an overclosed universe (but letting for the poss

Ω

e v Λ (GeV) d dσ J I & σ 1 − s 3 cm 26 − 10 Λ .DELPHIlowerlimiton × 3 " In the absence of resonances or coannihilation. $ to a limit on the ratio of hadronic to leptonic channel, Br 2 e σv for each type of couplings. We find where I=V, S, A,and t J=A, represents B, the C nature the of type the of coupling coupling (electronic, (vect charged lepto reproduce interpolated functions of this result in figure ( # Figure 1 with I=V, S, A, t. We then substitute of having another dark matteris candidate). given We for computed a the an final state with particles masses have found no discrepancy fromphoton the spectrum prediction from of the the DELPHI standar experiment to place upper bo different types of couplings:and t-channel vector scalar (red (magenta dashed), dotted). scalar (b LEP experiments have searched for anomalous single-photon expanding in powers of the relative velocity between two ann Λ the dark matter annihilation into 3Constraintsfromthethermalrelictothehadronicbranchi 3.1 The fermionicThe case LEP lower bound on the scale Λ relic density constraints. general leptons pairdensity (case to C). respect Moreover, the upper if bound dark given matter by WMAP is [ a t JCAP10(2011)023 ) is " 1) . l 3.5 0 (3.3) (3.4) (3.5) I,k tates. σ ! ). This /Br 2 1 h h χ 10 GeV has Br " 2 v $ ) : one thus can χ k 2 2 m h m )=0 )=0 χ 2 χ χ − m obtained by LEP (see m itude for the hadronic + 11 ( , in the electronic-type e m which gives 5 GeV in the case of an k χ ( GeV 9 ν m " m $ for the cross sections and 9 − = µ,τ,ν 2 χ B l nd on Ω − ronic channel) in other case v = χ A l 10 2 k m % 4 k ate, in the case of scalar and 10 r and t-scalar interactions, the m 0, and 0 otherwise) and m onic component as large as 80% upper bound limit (Ω 16 × m l,h × " /s. This can be summarize by: 5 − 2 3 − . 5 m v . +5 2 x> 2 χ χ 2 2 l 2 4 k . These behaviors can be understood 2 k cm ark matter of mass m m " m m $ 26 m ) (8 2 χ − l − 1 2 ) − 2 χ m 10 s k + 17 4 )=1if 3 2 k m /Br 2 k x m × ) with the value of Λ ) ,θ ,θ h − ( ) ) m ) ) χ 3 cm m l l e h + 4 χ H 2 χ 3.4 m − ). We combined the condition given in eq. ( Br 26 –4– χ m m m m " ( m V m − 2 χ J k 8 m gives a maximum value for the leptonic annihilation − − − − L θ ( m σv 22 10 e χ χ χ χ (1 + g 5 GeV. Neglecting × + )+ − × 4 e m m m m 2 χ 2 k v √ 2 ( ( ( ( Λ 3 4 χ $ ) / m 2 π m H H H H k I,k v m χ " ) σ . One can see that whatever is the nature of the coupling 8 + m $ 2 k m v 2 =Λ 2 χ = + v m + e )=Θ )=Θ )=Θ )=Θ V m v 2 k χ max h − χ χ χ χ σ σ J Ik m m 2 χ we expect for a dark matter mass m m m m σ ( ( ( ( + l 24(2 24 m A e ( 24( v ! ! 16. This corresponds to a 94% annihilation rate to hadronic s θ Λ . ! τ e,µ, g Λ Λ /Br computed through eq. ( 2 χ g g Λ $ max τe,µ, ν , = l h B l 480 GeV for for each type of couplings we considered (A, B and C, see figure g v σ = 4 θ /m 9 C l A,B,C u,d,c,s,t,b Λ V 2 k θ Br $ − = =4 = = 24 =4 = π σ m J h max l 10 v v v v θ − σ 192 1 max eV t,k × S,k V,k A,k √ σ σ σ 5 σ . = 2 ):Λ π 1 Λ 4 g The LEP constraint on Λ As an example, we can analytically evaluate the order of magn ) being the classical heaviside function (Θ 2 χ m max e H (Λ Θ given by: cross section coupling, one can simplify color factor and (electronic, charged-leptonic or universal leptonic), a d to the value of figure maximum value of the leptonic cross-section give a lower bou the results are shown in figure Of course, we ran the analysis with the complete formulation with (universal leptonic), for a vector interaction for example a very strong hadronicaxial interactions component (above in 90%). itsnature On of annihilation the the final coupling other plays st in hand, an one important for role, vecto case being (electronic an hadr coupling), or a 0% (i.e. no need of had calculate the hadronic contribution needed to satisfy WMAP corresponding to the thermal condition branching ratio electronic (case A) vector-like coupling ( JCAP10(2011)023 20 15 the velocity (GeV) χ m 10 per bound in the case of 20 a null hadronic branching Scalar Axial t and avoid the over-closure arged leptonic and universal the relic density constraints. mponents of the annihilation cause there are no possibility 5 needs a much larger hadronic c are the dark matter couplings undance for a DM mass above on corresponding to the kinetic the leptonic couplings. One can on or neutrino channels and the tions there is no such suppression, 15 (GeV) finteractions:vector(reddashed),scalar Vector t−Scalar χ Charged leptonic couplings 0 m el scalar (magenta dotted). Bounds coming 0

80 60 40 20

ccouplings(modelB,topright)anduniversal- 100

h (%) Br Universal leptonic couplings 10 20 –5– Scalar Axial 5 15 (GeV) Vector t−Scalar χ m 0 0

80 60 40 20

100

h (%) Br 10 ), where the scalar and axial interactions are suppressed by Scalar Axial 3.4 5 .MinimumhadronicbranchingrationeededtorespectWMAPup Vector t−Scalar Electronic couplings 0 0 We also observe that, paradoxically, the more electrophili

40 20 80 60

100

h Br (%) Figure 2 from expressions ( electronic couplings (model A, topleptonic left), couplings charged-leptoni (model C, bottom)(blue with dashed-dotted), 4 different axial types o (greenfrom full-line) LEP constraints and on t-chann leptonic couplings. so leading to possiblenotice large the contributions presence coming of onlyopening/closing from small of ”kinks” the different in annihilation the channels. branching fracti and the leptonic masses,contribution to respectively. ensure As a aof relic the consequence abundance Universe. one below However the for the WMAP vector limi and t-scalar interac this threshold. to fulfill the relichadronic abundance final states constraints become withleptonic thus charged models the lept dominant (B ones.ratio: and In C), this the corresponds there ch rate to exists are the a not mass threshold anymoreThe for mass necessary leptonic which (but with channels the can are hadronic be sufficient co to present) avoid to the fulfill relic overab (model A), the more hadrophilic it should also be. Indeed, be JCAP10(2011)023 . . . . n " 1 χ S P f σ χ m from (3.8) (3.6) (3.7) ) and m 45 . S P f 2 " σ ude that 815 GeV. = 10 GeV ≈ ! X P s -bottom. We S σ m 3 5. We realize that . 2 . ify thiserence,ff di we e obvious at the first 2 > 520GeV and 1 v the fermionic case. The 2 S γ was studied assuming the ntroduce a new scale Λ " nic dark matter ( η of 100 GeV per beam. We Λ 2 ower bound Λ e e π π g c candidate, we derive limits via the following scalar-type 1 /g gs in the case of scalar DM. A 32 32 γχχ S , shown in figure ar dark matter of 1 th the result shown in figure 2 S 2 e Λ + (which is the case in our analysis) , taking a dark matter candidate − → g Λ ]. We include as an example of a 8, which means a small difference c case, so in principle the bounds 2 χ P s " / / − 14 3 σ ∼ s, with negligible dependence on e m 12 / " + . P s 3 ≈ e 2 e 2 χ was taken directly from the simulation " σ = 10 GeV case, with a suppression scale m P m f s ¯ ee . cm ¯ of a scalar DM with a scalar interaction, ν we can deduce the lower limit on Λ σ χ 24 − v s γν χχ m − s S 2 1 S e σ g 10 ! √ and Λ → –6– / 520 GeV, which implies that in fact 2 S 2 ' /d.o.f. = 5 − = 2 e 2 Λ Λ e v ! g χ π S e s + S 4 S ≡ e π L σ s S ' 16 6 GeV, and a photon rapidity 4 2 v e ). If imposing g 1 Λ > 520 GeV, in order to be compatible with LEP bounds. s S,e coming from a signal due to a fermionic dark matter, and ] to simulate the distribution of number of events with photo from single-photon signals in the DELPHI experiment at LEP. ˜ σ ∼ , one could in principle try to deduce bounds on the amount γ -top, is a e ! 2 e e S 20 3 E g P f S /g σ ≡ S v , there is in principle no need for hadronic channel. We concl ) we obtained Λ 3 s S,e ) a cross-section 26 σ 3 − 10 ) one can show that, in the limit of × P s 3 σ ]. On the other hand, the signal process " . The background process 14 γ v s S E could be different if using those more rigorous cuts. To quant = 300 GeV, using the DELPHI luminosity of 650 pb σ With this bounds on Λ After computing the production cross section for the fermio e S /g S 20GeV (as in figure of our case with respect to the more correct result of [ Unfortunately, this single channel already gives, for 510 used MadGraph/MadEvent [ We assumed for simplicity that all data was taken at an energy 3.2 Scalar case We also checkedsight the that case we of can a apply the scalar same dark analysis. matter. In fact, It we could need not to b i In an analogous procedureon as the the suppression one scale Λ done above for a fermioni with mass of 10GeV, we deduce from the above expressions the l done by [ can extract from the above analysis that, for example, a scal a vector-like effective operator,The result, and shown compare in it figure directly wi scalar dark matter signal, the simulation of a energy these constraints are lesson restrictive Λ than in the fermioni needs a suppression scale Λ Λ following kinematical cuts: of hadronic channelexpression from for DM the annihilation, annihilation as cross-section we did above for into an electron-positron pair is: We thus observe that if we define Λ reproduced the bounds on Λ the lower limit on Λ(see figure However from figure ( Being bosonic one ( effective operator: We will consider a real scalar dark matter, which is produced LEP bounds are insufficient to constrain the nature of couplin we obtained JCAP10(2011)023 v s S nd σ (3.9) ], at a 90% 15 above which e /g   S v   s S,h eV. So in principle the LHC ˜ σ ntact coupling lagrangian into  =10GeV,andasuppressionscale   ctions of a scalar DM candidate.  χ  ering the total cross-section    dashed-green), to be compared with hows the signal+background coming    m  !  u,d,c,s,t,b =  h 2 h  cg    +  (fermionic dark matter) coming from a vector-like  ), with mass   v e –7–  g s 3.6 S,l  √  ˜ σ  /  ,herein(dotted-blue).(bottom)Distributionofphoton . It turns out that the scale Λ 1 (solid-red) coming from DELPHI experiment [ 4 e  τe,µ, ν , ! /g  = l S  2 l  ] for the different type of interactions (vectorial, scalar a g    21  = v   s S





        

         

 

  Λ .(top).Lowerlimitson =300GeV.Seebodytext. e /g S Figure 3 To run a more preciseCompHEP analysis, we and decided micromegas to implement [ the co would be able to constrain the nature of couplings and4Complementaritywithotherexperiments intera 4.1 Fitting with WMAP starts to be of the order of the thermal relic one, is around 5 T and the lower limits shown in figure effective operator, usingthe the correspondent same result cuts shown as in before, figure are shown ( similar conclusion holds for the Tevatron bounds, if consid C.L. For reference, the resulting limits on Λ energies in single-photonfrom events a at hypothetical DELPHI. scalar dark The matter, histogram as s in ( Λ JCAP10(2011)023 in er space ouplings, vents at LEP g, which is the 5 GeV for a vector ses, which are the ! . We see that WMAP 6 χ 100 m e analysis, we considered s not limit the lower bound eractions. After a look at and combined with WMAP and ]). We can easily understand vation of mono-jet excess at ble (limits of down or charm- 5 ) in the annihilation process. constraint on the relic density can be translated into a lower rder to respect WMAP upper WMAP we needed to increase 22 vatron analysis restricted even 1) dark matter with the scalar- 10) of σ contribution to respect WMAP . 80 0 ures ! (GeV) " e χ e 0175 and ran a scan on the parameter . m d-dotted) and axial (green full-line). /g , and the electronic coupling for scalar- 0 Axial h /g 60 g h ± g 1123 . 40 –8– as a function of the dark matter mass for the different =0 h 2 g h √ 11 GeV in the vectorial case (in total agreement with ). Whereas it excludes a broad region of the parameter 20 / e Λ g " ≡ χ ] made a similar analysis searching for mono-jet events for WMAP Vector Scalar ) keeping only the points respecting both astrophysical and h m χ 23 0 , 0 m ],Ω 400 300 200 100 500

19 h h Λ (GeV) which depend on the nature of the coupling and is represented ,Λ 10 GeV for a scalar interaction. For such values of hadronic c h l g are a factor 3 and 10 lower respectively [ ! √ h / χ is excluded because of the non-observation of mono-photon e . When combined with LEP analysis, a large part of the paramet Λ m 2 100 GeV. We have only plotted the limit on the up-type couplin e h ≡ /g " ]) it completely exclude leptophilic ( h h χ g 14 m WMAP Λ .CDFlowerlimiton , we decided to consider the most and the less conservative ca . Ω 2 4 for Contrarily to the LEP analysis, the center of mass energy doe " h 2 h accelerator constraints. Webounds, can the understand hadronic easilyfigure that contribution in depends o on the type of int space of the model (Λ Figure 4 figure Tevatron. To keepuniversal the leptonic logic couplings of (implying the a work smaller and hadronic keep a conservativ how the Tevatron constraintLEP imply analysis. some strong Indeed, tensionsthe to when hadronic reconciliate contribution LEPThis (and constraints then with thus, enters the in coupling conflict to with quarks the limit from the non-obser the Tevatron. These non-discoverybound of on any events Λ of this kind forbids a dark matter with hadrophilic couplings ( universal-leptonic coupling for vector-likelike interaction interaction, respectively. The results are shown in fig types of couplings: vector (red dashed), scalaraxial) in (blue the dashe fermionic DM case. We then applied the last 5 on Λ Last year, the authors of [ like interaction. Combiningfurther these the limits parameter space. with the recent4.2 Te Tevatron constraints from WMAP experiment [ figure6 of [ one we used throughtype the couplings paper on to Λ stay as conservative as possi interaction, and Ω space for a dark matter mass with small (which implies an upper bound on JCAP10(2011)023 e ]: ld 22 ) through e Mdm (GeV) Mdm (GeV) /g h is the process WMAP + LEP g γµ γµ f f f f χχg χ χ γµ χ χ γµ ndeed, the hadronic -jet events at Tevatron → rasfunctionofthedark (and thus qq ven more impressive for a h vents from LEP, Tevatron and g ratio needed to respect in ingent for dark matter mass o with XENON100 exclusion on that XENON100 is adding new analysis claiming for no detection dark matter in the early universe ed by the combined WMAP and Vectorial coupling Vectorial coupling WMAP + LEP + TEVATRON + XENON100 , XENON100 restrict even a larger s. 5 . Confirming the conclusions of [ 6 Λ avector-likeinteractionafterascanon ). 0 5 10 15 20 0 5 10 15 20 6 1 −2 1 −1 −3 −1 −2 −3 10 10 and 100 100 10 10 10 10 10 10 5 gh/ge gh/ge –9– 6 GeV (which would not have been the case if we took WMAP ! Mdm (GeV) Mdm (GeV) χ m γµ γµ f f f f . As we can see in figure χq χ χ γµ χ χ γµ ]. Their results are by far the more constraining one in the fie → WMAP + LEP + TEVATRON 24 χq at Tevatron, the nuclear recoil gives bound to Λ h Vectorial coupling Vectorial coupling .Hadronicratiocouplingfortheannihilationofdarkmatte .AfterapplyingtheconstraintofWMAP(topleft)mono-jete 0 5 10 15 20 0 5 10 15 20 h −2 −3 −1 −1 −2 −3 1 1 10 10 100 100 10 10 10 10 10 10 gh/ge gh/ge constraining Λ tensions when combinedbranching with fraction WMAP, required to LEPcould avoid and the enter overproduction Tevatron in of bounds. conflictlimits. not I only Indeed, with whereas Tevatron the results but s-channel als dark matter producti of direct detection experiments. One can easily understand the t-channel process Figure 5 part of the parameter space for 4.3 XENON100 constraint Recently, the XENON100 collaboration has releasedsignal several of dark matter [ for a vector-like couplingbelow 5GeV. the Tevatron Indeed, boundthe for meantime is such WMAP the low and mass, LEP moreand would the str would produce hadronic a have branchin clear beenscalar-like excess couplings, observed. in mono where The allLEP Tevatron analysis the constraints is parameter excluded are space by e allow Tevatron data (figure matter mass in theand case Λ of universal-leptonic couplings for XENON100 constraint (bottom right). See the text for detail upper bound). The results are shown in figures JCAP10(2011)023 .After h and Λ e ts from mono-jet events. h vector-like coupling to ata at the price of specific 12 GeV. rasfunctionofthedarkmatter cles and a light Dark Matter rameter space with light dark e of hadronic coupling needed ! Mdm (GeV) Mdm (GeV) χ studied the special case of scalar dTevatron(bottom)weobservethat as been observed in diff erent dark- m ] ee the text for details). 26 teraction after a scan on Λ ]). Whereas the scalar-like interaction is 20 GeV) should be largely hadrophobic for 10 GeV) is mainly excluded whatever is its 25 f f f f ! " χ χ χ χ – 10 – Allowed Allowed Scalar coupling WMAP + LEP Scalar coupling WMAP + TEVATRON 0 5 10 15 20 0 5 10 15 20 −1 −1 −2 −3 −2 −3 1 1 10 10 100 100 10 10 10 10 10 10 gh/ge gh/ge and another region definitively hadrophobic for .Hadronicratiocouplingfortheannihilationofdarkmatte 3 Which strangely coincides with the CoGENT excess signal [ 3 Figure 6 already excluded withoutthe the SM XENON100 still data, survivesmatter, in dark a matter narrow wit hadrophilic region of the pa mass in the case of electronic couplings for a scalar-like in candidate: hadrophilic orto leptophilic. respect WMAP We combined computedWe with showed the that the a rat LEP light and fermionic Tevatron dark constrain matter ( 5Conclusionandprospect Recently, several astrophysical data ormatter would-be oriented signals experiments. h nature of In the each coupling case, between the one Standard could Model fit (SM) the parti d type of interaction, whereas heavier candidates ( vectorial interaction, but excluded for scalar one. We also no point of the parameter space respects both constraints (s into account the previous XENON100 analysis [ applying the constraint of mono-jet events from LEP (top) an JCAP10(2011)023 ] ]; 511 ] On the SPIRES ][ , ]; . B. Zald´ıvar ]. , 28 100, arXiv:1008.1784 [ , SPIRES ][ arXiv:1105.3734 [ excess (2010) 123509 ) grant BES-2008-004688, o. However such unnatural A. Falkowski, G. Zaharijas and the spanish MICINNs uences is that models with jj ]; ]; ing is only to the bottom or wed that LEP and Tevatron arXiv:1003.2595 AlightscalarWIMPthrough ling. In this case, LEP limits ibility to escape such strong [ arf galaxies [ CSD2009-00064 W ments also to the LPT “Magic anks for JB De Vivie and M. M. McCullough, et al., , he coupling of dark matter to r constraints from synchrotron D82 eresting debates. The work was (2011) 083511 ]. Aconsistentdarkmatter TeV SPIRES SPIRES ][ ][ 96 Dark forces at the Tevatron . arXiv:1003.0014 (2011) 041301 [ D83 SPIRES ]; =1 . Phys. Rev. (2010) 043522 ][ s , D84 √ ,Ph.D.thesis,UniversitàdeglistudidiSiena, Implications of CoGeNT and DAMA for light production cross section and study of the dijet mass D82 SPIRES ANR-09-JCJC-0146 – 11 – Phys. Rev. ][ , Invariant mass distribution of jet pairs produced in (2010) 115005 WZ explanation for the CDF arXiv:1104.1375 arXiv:1104.0699 Metastable dark matter mechanisms for INTEGRAL ! [ [ + Phys. Rev. Z . , collisions at ¯ p D81 Phys. Rev. p arXiv:1104.3145 [ WW , ]; ]. Implications of CoGeNT’s new results for dark matter Baryonic arXiv:1006.3318 boson in [ (2011) 211803 (2011) 171801 Phys. Rev. SPIRES FERMILAB-THESIS-2010-51 SPIRES , +jetsfinalstateatCDF (2011) 256 [ W ][ 106 106 !ν FPA-2007-60252 The Kinetic dark-mixing in the light of CoGENT and XENON Measurement of B702 (2010) 022 ]; ]; 09 rays and DAMA/CoGeNT events collaboration, T. Aaltonen et al., γ SPIRES SPIRES arXiv:1007.1005 Phys. Lett. [ arXiv:1106.1066 Phys. Rev. Lett. M. Buckley, P. Fileviez Perez, D. Hooper and E. Neil, Y. Mambrini, JCAP J.M. Cline, A.R. FreykeV and F. Chen, [ interpretation for CoGeNT and[ DAMA/LIBRA A. Fitzpatrick, D. HooperWIMP and dark K.M. matter Zurek, S. Andreas, C. Arina,the T. Higgs Hambye, portal F.-S. and Ling CoGeNT and M.H. Tytgat, Phys. Rev. Lett. V. Cavaliere, spectrum in the Siena, Italy (2010), association with a CDF D. Hooper and C. Kelso, M.T. Frandsen, F. Kahlhoefer,DAMA J. March-Russell, and C. CoGeNT McCabe, Modulations [3] K. Cheung and J. Song, [2] D. Hooper, J. Collar, J. Hall, D. McKinsey and C. Kelso, [1] conclusion would be to supposedo that not DM apply. has Moreover, nocharm if electronic quark, at coup Tevatron XENON100 the bounds sameconstruction are time should not the be applicable hadronic excluded to coupl by FERMI last analysis of dw dark matter, usingelectron the with single-photon a complete eventsare simulation to not of constraint DELPHI able t eventslight to and electrophilic restrict sho couplings, theradiations couplings. explaining are INTEGRAL One excluded data of by o the Tevatron/LEP analysis. main conseq One poss References Consolider-Ingenio 2010 Programme underacknowledges grant as Multi- well Dark theunder the financial contract support of the FPI (MICINN The authors want toand thank F. particularly Bonnet T.Kado for Schwetz, for very E. enlightening useful us Dudas, withMonday discussions, the Journal adding LEP Club” analysis. special participantssupported Acknowledge for th enthusiastic by and the int french ANR TAPDMS Acknowledgments JCAP10(2011)023 , , e ] Riv. ]; , , in ]. ] ]. " − e Z + e SPIRES ][ warz mechanism SPIRES , SPIRES , ][ ][ , (2009) 1199 hep-ex/0406019 arXiv:0806.3581 [ and dark matter models , (2011) 131301 81 kinetic mixing in the " " (2011) 009 ]. Investigating electron " Dark matter search Z arXiv:1010.2752 Z (2009) 083528 (2007) 115001 [ 07 106 ZZ ]. , PoS(QFTHEP2010)027 D79 D75 ]. SPIRES (2005) 395 The arXiv:0903.1118 [ ]. JCAP ][ ]; ]; U(1), ectrons arXiv:0804.2741 , arXiv:0712.0562 ella, et al., (In)visible [ [ SPIRES chia, et al., ]. (2011) 412 ][ C38 MeV dark matter: Has it been detected? Rev. Mod. Phys. extension with kinetic mixing and SPIRES Searching for secluded dark matter via , SPIRES " ][ ]; ]; SPIRES SPIRES Phys. Rev. Z Phys. Rev. ][ Phys. Rev. Lett. , B697 (2010) 256 , ][ ][ SPIRES , [ (2008) 333 DAMA/LIBRA and leptonically interacting dark Leptophilic dark matter in direct detection Photon events with missing energy in Results from a search for light-mass dark matter LEP shines light on dark matter SPIRES SPIRES (2008) 023506 First results from DAMA/LIBRA and the combined – 12 – B827 arXiv:0907.3159 ][ [ Eur. Phys. J. ][ C56 gauge bosons , Electroweak and dark matter constraints on a Dark matter and a new gauge boson through kinetic " Phys. Lett. D77 Z , astro-ph/0309686 [ Light dark matter in leptophobic GeV The Stückelberg arXiv:1103.0240 Dark matter annihilation in the galactic center as seen by th [ arXiv:0907.0758 [ 209 arXiv:1011.3300 [ astro-ph/0307403 Nucl. Phys. arXiv:1011.1398 ]; ]; ]; ]. ]. ]; ]. ]. , , (2009) 083502 Eur. Phys. J. Leptophilic dark matter Phys. 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Andreas, C. Arina, T. Hambye, F.-S. Ling and M.H. Tytg CHAPTER 6

COMPLEMENTARITY AMONG DIFFERENT STRATEGIES OF DARK MATTER SEARCHES 80 Complementarity among different strategies of Dark Matter Searches JCAP11(2012)038 erived ocesses and hysics c,d P trophysical un- , mentary indirect le ar particle physics tive with the other th some reasonable ic set bounds on the cor- and Higgs portals. We t ! Z ar 10.1088/1475-7516/2012/11/038 s, [email protected] doi: , strop Gabrijela Zaharijas A b [email protected] , osmology and and osmology Michel H.G. Tytgat, C a e 1206.2352

dark matter theory, dark matter experiments, absorption and radiation pr In this work we confront dark matter models to constraints that may be d [email protected] rnal of rnal

ou An IOP and SISSA journal An IOP and 2012 IOP Publishing Ltd and Sissa Medialab srl F-91191 Gif sur Yvette, France Instituto de Fisica Teorica,Cantoblanco, IFT-UAM/CSIC, 28049 Nicolas Madrid, Cabrera Spain 15, UAM, E-mail: F-91405 Orsay, France Service de Physique UniversitéLibre de Théorique, CP225, Bruxelle Bld du Triomphe, 1050The Brussels, Abdus Belgium Salam InternationalStrada Centre Costiera for 11, Theoretical I-34014 Physics, Trieste,Institut Italy de Physique CEA/Saclay, Théorique, Laboratoire de Physique UniversitéParis-Sud, Théorique, [email protected] b c e d a c Bryan Zald´ıvar Abstract. from radio synchrotron radiation fromcertainties the and Galaxy, we taking compare intodark these account matter the to as searches. boundsmodels. set Specifically by First, we accelerator a generic apply andresponding effective our comple mass operator scale, analysis approach, and in to then, which three two case specific we popul UV completions, the Received June 26, 2012 Revised October 6, 2012 Accepted October 10, 2012 Published November 20, 2012 Yann Mambrini, Complementarity of Galactic radiocollider and data in constrainingdark WIMP matter models J ArXiv ePrint: show that for manysearches, candidates, and the could radio synchrotron evenassumptions limits give regarding are the the competi strongest astrophysical uncertainties. constraints (as of today)Keywords: wi " JCAP11(2012)038 7 8 – 5 ], 1 3 8 15 13 11 21 17 10 24 12 15 1 5 27 30 14 15 (see e.g. [ s gravitational ibility to actually in part due to the d by multi-purpose s and cosmology and ] and XENON100 [ 4 s, a Weakly Interacting and are actively pursued, comprehension of both funda- –1– , for different e ffective operators " σv ! Collider and complementary Indirect Detection bounds vs 5.1.1 Mono-events at5.1.2 colliders Synchrotron 5.1 E ffective operators 5.2 Higgs portal 5.3 Extra U(1) 3.5 Synchrotron signal for different magnetic field choices 3.1 Astrophysical uncertainties 3.2 Cross-check of3.3 semi-analytical calculation of Synchrotron synchrotron signal fluxes 3.4 for different choices of Synchrotron signal DM for density profile different choices of CR parameters 2.1 Semi-analytical approach 2.2 Numerical approach ]). While, as of today, we have no other evidence for DM than through it are dedicated to the search of DM in the Earth. These are supplemente Massive Particle (WIMP) has emergednatural as explanation one of of its the3 favourite cosmological candidates, abundance through thermal freeze-out 1 Introduction Dark Matter (DM) is oneunderstanding of its the nature most will important likelymental play issues an interactions in essential in and particle rôle our physic the structure of the Universe. Over the year 6 Conclusions and prospects A Annihilation cross-sections 4 Other indirect constraints 5 Constraints on Dark Matter Models 3 General astrophysical setup 2 Synchrotron radio emission in the Milky Way Contents 1 Introduction manifestations, the alleged weakobserve interaction DM of experimentally. WIMPs Several have strategiesto open have search been the for proposed, poss WIMPs. Direct detection experiments, such as CDMS [ JCAP11(2012)038 ]), on ]. SM 23 20 ], and ning for 19 est, and a ively. Such egions of the es, thus cross gy of charged otropies [ ales) and on two y dense region of g of DM induced parameters [ erse DM mass to een much studied s to those set by y constraining for ] which allows to quite stand on the ce, the dependence 22 ption of the gamma- hrough the remnants eutrinos, anti-matter of particles produced re DM is expected to irect detection experi- , y proved to give very ementary. Our analysis dark matter production To gauge the impact of 21 nt work we confront the es. We explore those by s. In particular, analysis on the annihilation cross e local CR fluxes, and used e the Galactic Centre (GC) ection experiments (see also LAT based on dSphs [ code [ ]. A radically di fferent, and a also to be most constraining for 1 10 – 6 Fermi GALPROP –2– and Higgs portals. We also show how colliders and radio ! Z ]), to our knowledge no analysis of specific particle physics 18 – 14 ] data and which are therefore suited to test electron propagation 26 ]). In general terms, the colliders data are comparatively more constrai ] some of us analysed bounds on effective couplings between dark matter and ] and radio [ 10 , 11 25 9 , , In [ The constraints from colliders and those from indirect searches do not 7 24 http://galprop.stanford.edu/webrun.php. 1 parameters sets using both those derivedtraditionally to to probe bracket uncertainty in this th type of uncertainty (MIN, MED, MAX setparameters of in the inner regions of our galaxy. explore the full setchecking of the CR semi-analytic propagation methodCR parameters and propagation and calibrating parameters up-to-date its on energy parameters. synchrotron loss signals, we sample a range of CR propagati of their annihilation (or decays)of in the astrophysical Milky environments, lik Way, nearbythe dwarf spheroidal Universe. galaxies The (dSphs) possible orin in remnants, cosmic general or rays an messengers, (CR),particles are and in high gamma-rays, the energy early or n universe. more generally an injection of ener complementary approach is to search for indirect detection of WIMPs, t particle physics experiments atbe colliders, produced, most and notablyof to the single-photon manifest LHC, or itself whe mono-jet aspertinent events missing constraints with energy, on in missing WIMP collision energy mass have and recentl interactions [ and CR propagation sets whichray were recently [ shown to give a good descri using both a semi-analyticof approach, the which radio allowsfull flux to numerical on control, for the calculation instan magnitude as of implemented the in magnetic the fields in our region of inter to constraints imposed on DM annihilations from the effectsame on footing. the In CMB particular, anis rather although light the DM radio candidates data —radio are as fluxes potentially we strongl suff ers show from also several in sources this of work — astrophysical the uncertainti modellin models implications have beenconstraints made from so far. synchrotroncolliders radiation Concretely data. in in the In the particular prese is Galaxy we based study at both to on which radio an extendspecific effective frequencie they DM operator are models, approach compl (so the we so-called put limits on energy sc low mass dark matterby candidates. dark This matter is annihilationthe essentially (gamma-rays, square. because etc.) the is flux section In proportional of particular, to DM interesting the basedMilky constraints inv Way. on While have the synchrotron been radiation measuredin constraints set synchrotron on the DM radiation literature have from already (e.g. b the in inner [ r particles from single-photon andstudies mono-jet are usually signals, compared at toe.g. LEP exclusion [ and limits set the by LHClow direct mass respect det dark matter candidates,ments, in while particular the below latter the are thresholdat more of constraining colliders d at is higher impeded. masses, where Interestingly, indirect searches tend synchrotron radiation limits compare to bounds set by JCAP11(2012)038 2 − ]. 27 close ]). In 2 at DM 28 endence we adopt ! ge region of synchrotron nihilation is lusions bear ]). However, 60 GHz. 5.1 is devoted to 16 > ∼ 5 (i.e. 10 MHz ith high angular ating or spinning we discuss the as- d (CMB) and its in DM simulations full numerical study. ve operators. There iffuse emission. The s on the modelling of ds. However thermal 3.1 osurveys,see[ ct searches and collid- future improvements of s. In part ised gas also contributes dels. Finally we give our but at higher frequencies, using the 45 MHz data. imately at ] and a 330 MHz observation of the higher frequency surveys. For example in ] we focus on a 45 MHz survey [ channel improve by a factor of G strength, as typical for the Galaxy, is devoted to study, numerically, the 38 we review the formalism of the semi- 18 µ , − µ 2 15 3.4 + – µ 12 100 GHz range (see for example [ –3– − we discuss our cross-check between semi-analytical we show the dependence of the synchrotron flux with we consider two specific UV completions, the Higgs 3.2 3.3 5.3 ] was used in [ 20 GHz is dominated by synchrotron emission of CR elec- 29 ∼ ) below), and following [ and . 6 ] that for light dark matter candidates it is beneficial to consider 5.2 10 GeV generate synchrotron emission which peaks at tens of MHz 2.16 18 , < ∼ 17 ] or eq. ( ]). Indeed the synchrotron flux is expected to be strongly enhanced is a semi-analytical (cross-checked by numerical) study of the dep 18 17 3.5 we discuss at lengths the astrophysical framework. In part ], authors show that the DM limits, for the 3 ]in[ 18 TeV when considering 408 MHz, with respect to the limits derived 30 portal, to which we apply the synchrotron constraints. While the conc < ∼ ! Additional motivation to consider the 45 MHz survey is that it covers a lar We present our results as a function of the systematic uncertaintie The paper is organised as follows. In section Synchrotron signal of electron by-products of dark matter WIMP self-an To constrain heavier DM candidates it is beneficial to consider 2 of the synchrotron flux on the magnetica field generic, normalisation. also knownwe Then, as confront section model-independent, the approach synchrotroners based constraints constraints. on we effecti derive In with parts other indire some resemblance, the constraintsconclusion are and quite prospects distinct for in the two mo (see figure 4 in [ figure 6 of [ electrons with energies resolution surveys have been used tofor constrain example a possible a DM 408 signal MHz ( SgrA* survey [ [ to the Galactic centre, as cusps in a DM density are generically found the sky (for a visual representationthe of past, the observations coverage of of the this Galactic and other Centre radi compact radio source (SgrA*) w it has been notedlower in frequencies. [ For instance, for a magnetic field of In the very high frequency radio (VHF) to microwave frequency range 2 Synchrotron radio emission in the Milky Way the discussion of synchrotron constraints on particle physics model and Z In section trophysical uncertainties, while in part and numerical approaches. Indiff erent part choices of Darkimpact Matter of profile, diff erent and choicessignal. Part of Part CR diffusion and propagation parameters to the the backgrounds due tothe standard limits astrophysics, based illustrating on potential radio data. analytical approach used in this work, and also present the setup for the mass of anisotropies represent thedust main becomes signal. relevant at Emission even higher by frequencies, small starting grains approx of vibr trons (accelerated in e.g.bremsstrahlung supernovae (free-free shocks) emission) on ofin Galactic electrons magnetic this on fiel range. the galactic ion At higher frequencies the Cosmic Microwave Backgroun 300 GHz), various astrophysicalradio processes sky contribute at to frequencies the below observed d generally expected to fall in the 100 MHz JCAP11(2012)038 . , δ a se v E 0 (2.1) (2.2) s of a K = e diffusion ic feedback xx ilation cross ed to have a le escape, i.e. K lemented in the ) is the electron , the mass in the analytical depen- ,briefly,fullynu- through adiabatic which could lie in ), we have checked x, E bed by the diffusion h )] ( e DM density is more he form 3.5 q L cal approach allows in sion in momentum space x, E ( n c V ns, so we refer collectively to these , − ) δ ) " ) ]. On top of that, the very presence ) c − 3 V x, E – 2 a ( ]. Having these uncertainties in mind, 1 ∇ v )(4 x, E n ( 2 ( 2 32 δ ∇ 3 p n p 4 2 − 1 xx − ] is a large scale survey and we test its data p ) K (4 [ 27 ∂ δ ∂p –4– 3 x, E ∇· pp ( , which is usually expressed using Alfven speed = K pp ˙ pn 2 )+ xx p ! K in the galactic medium is governed by a transport equa- K around 20 kpc and half-thickness ∂ ∂ 3 ∂p ∂p x, E pp ] ( h q + − K R 33 = ) ) = 0. x, E ∂t ( ,p ]. h L ∂n 22 ± , r, ] or cored through scattering of dark matter particles by stars in the den 21 ( . In the following we will assume, as customary in literature, that th n ) is the number density of electron per unit energy, and 31 xx K )= code [ x, E ( n ,z,p Cosmic rays propagate in the diff usive halo which is usually approximat In order to compute the synchrotron signal, the propagation and energy losse The propagation of electrons We dedicate the next two sections to describe a semi-analytical and h DM annihilation produce the same amount of electrons and positro is the convection velocity and re-acceleration is described as diffu 3 R ( c particles as ’electrons’. stellar cusp observed in the vicinity of the SgrA* [ V defined as and is determined by the coefficient where in central regions ofsection may DM be halos, placed andinner by therefore regions considering strong of radio constraints emission theprocesses from on might Galaxy erase SgrA*. DM is the However annih baryon putative dark dominated matter and cusp it [ is possible that baryon source term. Thecoe transportcient ffi through magnetic turbulence can be descri coefficient is spatially independent and has an energy dependence of t the robustness of that approachGALPROP against a full numerical computation, as imp tion, which can be written as [ contraction [ we take advantage offor the a fact DM that contribution farther 45robustly away MHz determined. from [ the Galactic Centre, where th of a black hole might disrupt the DM profile, both by making it steeper, cylindrical shape with radius Galactic electron population need to be modelled. the range of 1n to 15 kpc. The spatial boundary conditions assume free partic some cases fordence more of physical synchrotron insight flux (for on instance, the we strength use of it the to magnetic find field, the see merical approach used to derive synchrotron flux. While semi-analyti JCAP11(2012)038 ← (see with (2.8) (2.7) (2.5) (2.3) 1 x − x, E s 4kpcis ( · G ly through ∼ ] we neglect h ) vanishes at GeV ) which at the 35 L to reach , /τ 2 x, E s ( ess 34 x, E ). The solution of E )] ( n function E . b g[ 2 x, E )= x, E 1 GeV. ( safe assumption for our /λ ) ( , 2 q s E ) n δ ( ed in terms of the param- semi-analytical approach, ˜ t ng during its propagation. s ) 2 ), since for the synchrotron b z ) − ∆ ,E 1 x 2 0 n s ≈ K ) (2.4) 1) x E (∆ 4 ) (2.6) x, E cal calculation, see the next section. ( ) − s ( − q ˜ t with energy b ) e , [ +( s x, E s )= 2 s ( x, E / x nL x ∂ q ( 3 ,E ∂E b (2 s . Additional bremsstrahlung losses of $ ← 1 x, E − x ˜ t − ( ˜ − t )=˜ q s ∆ )] · ← 0 exp x, 1 ( n x, E G ( πK x, E , such that in practice the radial boundary GeV s 1) 2 4 ( n x, E h ) and ˜ τ ( n − E 16 # ∆˜ R ( –5– dE − ∇ . The unique argument of the Green function is 0 xG 2 x, E ∞ ) ∞ 3 ) ( −∞ K , because the energy dependence enters only in this E s )= )= = d ˜ ]. The Green function for which t n % = s s " n x ) can be re-expressed as 2 ˜ − t n ) as ∆ ) , 35 , x, E DZ 0 ) 3 E − ,E s ( ) " E ) 2.3 1 s 2.1 x ( K x K x b ] × [ ˜ t 1 πλ x, E )= ∂ ← x, E ( 34 ← ( =4 √ =( ˜ t ˜ n ( q )= λ 2 ∂ x, ) x, E ( · −∇ = ( x )). If at the level of propagation one considers that energy losses x, E )= n G ( ) x, E ( )˜ G n x, E λ,L ( x, E ( 2.4 ∂t ( ˜ /b and (∆ G δ and rewrite eq. ( s ∂n ˜ t E 4 ( , and has a general solution of the form − s ˜ t dE ) encodes the energy loss rate. Cosmic ray electrons loose energy main ! = may be expressed as [ )), then in ( ˜ t x, E h ( E ∼ m 8 10 4 10 3 10 6 10 6 10 2 10 4 10 –9– 05 10 [cm ...... ], derived using ! . L m 4 3 2 6 6 1 3 1 0 24 [1 + L and )) has a spatial profile, which affects the synchrotron K " exp B m ∝ we allow for the magnetic field to have a higher effective L 2.20 ≡ ) ) in the inner Galaxy, on synchrotron fluxes in section 0 K 4 4 4 4 1 [kpc] B ρ, z 15 10 10 ( ] in taking h should in principle depend on the diffusion model assumed (or locally B L 18 (1 + magnetic field as customary in literature: m (see eq. ( " ) in the inner Galaxy. We arbitrary set L B ], consistent with [ K rad total 25 1a 2a 1b 2b U PD ]. However the value of the field in the inner Galaxy is considerably le and MIN MED MAX Model 51 (1 + m – unchanged ) is the unit step function as a function of R " r 48 " B ]. − B 55 IG ), since the propagation in the Galactic medium is intimately related w R . Upper three raws: parameter sets derived using a semi-analytical Several 3D models of the large scale magnetic field are implemented in Contributions to the energy losses for electrons are assumed to come onl is more or less well constrained, and we take the value of 6 following [ 8 ]). However, as our ROI is away from the Galactic disk and the small-scal " chrotron and IC processes in ourthe semi-analytical radiation approach, density as commented ab as the field extends into our ROI (i.e. B between a constant magnetic field and the exponential form defined above that work that an actual extent of vice versa component is deduced mainly based on the synchrotron emission. 53 component is expectedparametrisation to for dominate a the total value of the magnetic field, w Table 1 MIN/MED/MAX anti-proton fluxes at Earth from an exotic Galactic component [ while leaving The parameters measurements [ known and we rewrite the normalisation of ( rows: parameters from [ magnetic field. We follow [ where Θ( normalization cover our ROI. We thereforeof fix overall the normalisation spatial dependence of the B field an Faraday rotation measures and polarized synchrotron radiation, see e.g. [ shown to reproduce theconsistent gamma-ray with diffuse the radio data data well. at Last 22 row: MHz — plain 94 di GHz frequencies, [ order micro-Gauss. The best available constraints in determining t JCAP11(2012)038 ] 18 gnetic to be ytical ,which 3 2 higher rad − U Hz. In [ ∼ ds, for this 9 in our ROI. to constrain the measured region of low ich is generally emission comes s smaller as the n the numerical tor of tudy performed in ile for the region of )), for the otherwise ] using their particular ere we take in our ROI. approach we have used ecomes less suppressed, 3.6 ifference between taking 18 er synchrotron fluxes (by we take 8 eV cm eoretical modelling. imises a synchrotron signal GALPROP code rad he U GALPROP , and for this latitude, the semi- ◦ quite different from a simple CR propa- -like structures appear sub-dominant with 10 r ROI and electron energies of interest, we ] and WMAP haze at microwave frequencies, bbles like structures Centerd at the Galactic ∼ ture of magnetic fields or the CR propagation 57 , 56 – 10 – constant works very well. rad U ], witness of possibly more complicated configuration of the ma , which corresponds to its value at intermediate latitudes 10 degrees) the agreement between the two predictions is 59 rad (NG) approaches. We see that in our region of interest at ∼ U estimations are typically larger for large angles and vice versa, GALPROP ]andPlanck[ GALPROP 58 we show a comparison of synchrotron spectra obtained with the semi-anal . 1 1 However, as commented in section D, the data of synchrotron flux we used Finally, we have checked a good agreement with the results of [ Concerning the frequency of observation we focus on the data taken at 45 M Note however that the conditions in the inner Galaxy might be 9 ISRF energy densities, inspired byThat the in values turn implemented implies inabout higher t overall the energy same losses, factor, and if therefore IC low losses are the dominant ones, see eq. ( dark matter models is the one taken at a latitude of analytical and numerical approacheslatitude are the approximation in of very taking good agreement. In other wor in the maps useddistance in from the the galactic NGexplaining centre approach. why increases, the so Typically the this synchrotron radiation flux density b get flux estimations. However, in the semi-analytical estimations used h see figure gation setup assumed here.centre and In extending particular, to observations 50 of deg the in bu latitudes in gamma rays [ constant, which turns outthis to be work, as a is good justified approximation in for the the next particular section. s For the value of conditions, cannot be reliably modeled. observed by WMAP [ fields and CR propagationrespect parameters in to that the region. standardcaution While that components bubble before of their the origin diffuse is emission understood, in the ou actual struc setup. The biggest di fference between our works is that we assume a fac 3.2 Cross-check of semi-analytical calculation ofIn synchrotron figure fluxes (SA) and numerical/ it was shown that in asynchrotron flux wide is range very of small, frequencies whileof (22 going low to to mass 1420 lower frequencies MHz) WIMPs. max the Thesubdominant change RMS in with temperature respect noise to of the this survey systematics is errors 3500 involved K, in wh th a constant radiation density generally good (within a factorfrom of low-energy 1.5). electrons This whichaverage is diffuse properties not surprising, over (as as large inapproach) the distances, 45-MHz the is the SA not d method)latitudes, expected SA and to calculation proper gives be largerlarge spatial large. fluxes latitudes, profiles than it (as the In is NG the i addition, counterpart, opposite. wh we That observe too that is expected, in as the in the SA the intermediate latitudes (of is approximately the value read offthe ISRF maps used by same parameter set. JCAP11(2012)038 = = s ibu- ρ " would ρ s r =5kpc s =0, i.e. no approaches, r K ations: ]. Therefore, 30 60 and [ ) and isothermal ]). As this profile ar, this profile is 3 11 − ape, and shown in 3.1 y higher than those 45 ( [ e found in DM-only 3 − 25 range nclude baryonic feedback 3 S-A, bottom N-G, bottom − S-A, all S-A,leptons all N-G, all N-G,leptons all 53 GeV cm . 20 ended core. Smaller values of this value can vary in the 0.2–0.8GeV =1 e. 041 GeV cm s G (which corresponds to . ρ = 10 GeV. A frequency of 45 MHz was µ =0 096) GeV cm . DM s 0 15 ρ = 10 m ± GC Latitude [deg] B 113 . – 11 – . 0 24 kpc, . 10 2 " ± ρ 43 . = 35 s r 5 11, . ]. However, in this section we also show the prediction for the =0 18 [ α 0 10

= 21 kpc, for an NFW proﬁle, and

1e+06 1e+09 1e+08 1e+07 1e+10

ƒ /MHz) ( [K Temperature Synchrotron

] s 2.5 r , . One should also keep in mind that the overall normalisation of DM distr ), using the following values of parameters, consistent with observ 3 3 − − 3.3 . Comparison between semi-analytical (S-A.) and numerical/Galprop (N-G) is uncertain, being in the (0 , synchrotron signals scale with 2 " range. In the parameter sets we chose, the local value of DM density is set to ρ 3 Note that we make a conservative choice by choosing a rather ext However, depending on the analysis, the uncertainty window on − 10 11 31 GeV cm 43 GeV cm . . in addition to thefigure di fferences in the signal caused by the DM profile sh for Isothermal profile, 0 3.3 Synchrotron signal for different choices ofIn DM the remainder density of profile the text we will focus on two DM profiles, NFW, eq. enhancement of the magnetic field in our ROI). Figure 1 result in fluxes more similar to those obtained with the NFW profil used, assuming a magnetic field normalised to cm synchrotron signal inmodified the to case have of asimulations a shallower and modified inner it Einasto slope(parameters describes profile. we than better use the results are In usual of particul Einasto simulations profil which i (ISO) eq. ( for diff erent annihilation channels, and DM mass of 0 tion of NFW in that region. is ’bulkier’ at distances 1 kpc from the GC, the DM signals are generall JCAP11(2012)038 ]. rs 26 "= 0, – a 80 24 s of CR v ved using t, with a g to muons In a parallel y radio data . code [ 1 4 kpc, 1a', EinB 4 kpc, 1a', NFW 4 kpc, 1a', ISO ' ' ' to confront CR oaches, we also 'z 'z 'z 60 . le s regions in the ti-proton fluxes, 3 The main reason ementary to CR e 10 GeV ' " Χ is case and produces clei data, it is found m deg ! GALPROP refore impact of these electrons diff use out of gamma-ray predictions b ]), but also large values r also a plain diff usion GALPROP 40 ], where whole sky radio 26 fied Einasto, NFW and ISO 61 45 MHz, bb, 20 are obtained in a fit to B/C data. 0 K 0 9 8 8 7 7 6 6 based parameter sets in general have lower 10 10 10 10 10 10 10

! ! ! ! ! ! !

1 5 1 5 1 5 1

K MHz T Υ " ! % $ # 2.5 LAT gamma-ray measurement. We will use the – 12 – GALPROP 80 Fermi .The is assumed for this plot (see text for the remaining parameters) 3 3 4 kpc, 1a', NFW 4 kpc, 1a', ISO 4 kpc, 1a', EinB ' ' ' − 'z 'z 'z 60 10 GeV ' Χ 1 kpc are essentially excluded (see also [ " m , % deg ! ∼ Μ $ b ], and few models within a 2 sigma scatter of the Bayesian analysis, and consequently higher values Μ h 40 43 GeV cm . code is used to derive a set of CR parameters which provide a good δ 25 L ] has shown, however that some models with re-acceleration ( 45 MHz, =0 26 ! ρ 20 GALPROP . We show a comparison of synchrotron signal calculated with these choice 1 ], a full Bayesian analysis of a ﬁt of CR models is made using . Comparison between synchrotron signals for a DM mass of 10 GeV annihilatin 25 ]the 15 kpc show some tension with radio data, arguing that MIN/MAX sets of paramete 0 Astudyin[ In parallel to the above mentioned analysis based on semi-analytical appr 7 6 6 8 8 7 24 10 10 10 10 10 10 > ∼ ! ! ! ! ! !

1 5 1 5 1 5

h MHz T K Υ $ # " ! % 2.5 propagation parameters in figure As discussed above, MED/MIN/MAX setsa of CR semi-analytical propagation description parametersrequirement were of deri to CR produce propagation, minimal,at medium, a and Solar and a position. maximal fit DM to These generated models B/C an were measuremen recently reanalysed in [ 3.4 Synchrotron signal for different choices of CR parameters Figure 2 (left) and bthermal quarks profile. (right figure) for three DM density profiles: modi too strong radio emission at low frequencies. We therefore conside similar to the case 1afor above) this are is in that tension with production the of large CR set secondary of radio electrons data. is enhanced in th best fit model from [ fit to the CRwork, data, [ and at the same time reproduce the gamma-ray data well. data and to derive theof best such fit model model are and consistent its with scatter. the It issee shown table that In [ parameters on electronGalaxy. population, This and work its concludestowards that synchrotron the both emission galactic MIN in anti-centre. andthat variou MAX small Demanding models halo consistency are sizes with disfavoredL b B/C nu data together with B/C measurements where considered, testing the and propagation of electrons is done using a CR propagation setup as shown in tab That in turn meansour that ROI, diff usion and is contribute higher instead in more this at set higher of latitudes, models, as i.e. seen in figur values of parameters nuclei data. present somewhat extreme choices when compared to observations compl use results of three analysis based on a numerical calculation with the JCAP11(2012)038 3 ]) ]. fic − 17 26 (3.6) affects 80 GALPROP IC. Note the NFW g to muons g form: . NFW DM 1 3GeVcm see e.g. [ . r analysis, the dN/dE 4 kpc, model 1a 10 kpc, model 2a & & MAX MED MIN z z PD 60 =0 ence between the ! )), for a given value 10 GeV ρ ptions of the overall & " Χ m deg value for the magnetic uency. However in this ! which can perfectly be G, the electron energies 2.16 .Awaytoseethisis b e are going to work with h the radio data in [ µ spectrum , the energy dependence 40 B tailed in table 2 6 E 45 MHz, bb, ! , ,(seeeq.( 20 E B 1 √ " 2 0 B 2 9 8 7 6

10 10 10 10

K MHz T Υ " ! % $ # B

+ could be disfavored (or not available) by the 2.5 α ) one observes two extremal cases: one in which E ! – 13 – 3.6 ∝ 80 ) ν,B ( and the electron energy as an moderately optimistic scenario, and the ’1a’, ), the magnetic field influences the flux through the energy F 4 kpc, model 1a 4 kpc, model 1b 10 kpc, model 2a 10 kpc, model 2b 3 ν & & & & MIN z z z z PD MAX MED 60 − 2.18 10 GeV & Χ " m , $ deg , the correspondent ! Μ that matters are encoded in the energy losses. Indeed, given a speci # b Μ B 40 B 43 GeV cm . 45 MHz, =0 represents here the rest of energy-losses, here assumed to be only 20 ! α , and a given annihilation channel, it can be expressed in the followin ρ ν . Comparison between synchrotron signals for a DM mass of 10 GeV annihilatin In the remainder of the text we will adopt the MED model together with where As can be seen in eq. ( 0 7 6 9 8

10 10 10 10

MHz T K Υ $ # " ! % 2.5 frequency Figure 3 (left) and b quarks (right figure) for di fferent CR propagation setups, de producing those desired frequenciesproduced by are our always DM smaller candidatesonly in than dependence the 1 on range GeV, [1-200] GeV. So in practice, for ou electron spectrum produced by thesynchrotron data DM. with However, frequency as of in 45 this MHz, study and magnetic w fields cancels in this particular analysis. From ( that since both synchrotron and IC losses scale with energy as losses, and through the electron energy. In principle, the electron of the magnetic field that for a fixedfield electron which injection maximises the spectrum synchrotronsection there flux we at exists want a an to given optimum synchrotron understand freq this fact in more detail. profile and profile is assumed here. model (PD) often used in literature and shown to be consistent wit based CR propagation model, together with the Isothermal profile and In this section wenormalisation study how of DM the limits magnetic change depending field on in the our assum ROI. It has already been noticed ( as a moderately conservative setup. 3.5 Synchrotron signal for different magnetic field choices the maximisation of the flux with respect to the magnetic field the following: inemitted synchrotron the frequency assumption of considering a one-to-one correspond JCAP11(2012)038 ld G. µ LAT 26 have an eld at the % Fermi d lack active GC e magnetic field B ), assuming the , thus decreasing s actually the one 100 3.6 B field configurations. 80 √ ing radio data to i) al, i.e. / 1 n this work. 60 ∼ es, for which the flux scales ite galaxies in a single joint 40 F he NFW profile and the MED = GeV 10 X = GeV 100 X m m 20 factors (line of sight integrals of DM for which the flux is maximal scales J G. B µ G)  ( 10 26 GC 8 B % ). Lines represent the results in the semi-analytical 6 GC increases, and the other, in which synchrotron is – 14 – max GC B B 4 5 B . , 0 3 ] and ii) constraints based on the measurement of the G value in our region of interest is assumed, and the sec- ≈ µ 19 2 ROI 10 B % = 8 eV/cm LAT collaboration from the non-observation of dwarf spheroidal ]. These limits are among the strongest to date and uncertainties GC 20 B rad

U 1e+08 1e+07 1e+06 1e+09

100000

ƒ /MHz) ( [K temperature Synchrotron

] Fermi

, thus increasing as 2.5 2 / 3 . For B rad shows the shape of synchrotron flux as a function of the value of magnetic fie ∼ U 4 correspondent to this case. The value of √ F α . Flux predicted at 10 deg offthe GC, as a function of the value of magnetic fi ∝ as increases. In other words, there will be an intermediate value of the B The main strength of considering dSphs is that they are DM-dominated an In the following we will show DM limits using two cases for magnetic Figure max GC B B Figure 4 4 Other indirect constraints In the followingthose we derived by will the compareGalaxies (dSphs) the in limits gamma-rays we [ derive from consider CMB anisotropy spectra [ involved in their calculation are complementary to the onesastrophysical derived i production of gamma-rays. We will refer to an analysis of th idea about the maximumvalues of of the flux already by direct differentiation of ( at GC. Taking into account only IC (apart from synchrotron of course), one can GC (roughly, at our ROI, we have likelihood fit and includes the effects of uncertainties in data which, for the first time, combines multiple Milky Way satell as as for which synchrotron becomesmaximising the the dominant flux. energy loss, and this value i The first one, a standard approach. We assume adiff usion DM model. annihilating directly to electrons, using t synchrotron loss is negligible withwith respect to the rest of energy loss ond case will be the normalisation which maximises the synchrotron sign actually the dominant energy loss, after which the flux scales as JCAP11(2012)038 is − 5to e ]. In √ + 62 e and the 12 ]. The phys- 66 3 branching to ], based on the signal could be – / quark states, as ic medium that ationally bound ong these states, lation modes for sdirectly he near future, as 20 b a. As constraints 63 ]. The constraints e a factor of , rotons have a lesser ed in this work. We 68 o leptons and hadrons 20 T sensitivity increases ovements of the limits nation epoch is that it s on , when a pair of electron- onic states, in which case ain parameters related to ived in [ uare-root of the integration ower redshifts and increases 1000 [ he amplitude of some of the h as Pan-STARRS < ∼ z 15 to 15. As we focus mainly on < ∼ √ LAT mission was estimated in [ channel worsened by a 1 τ Fermi channels, renormalised by a branching factor of – 15 – e and 15 to illustrate potential improvement in our plots. are significantly weaker (this likely holds true also for can be characterised by missing energy and the emission √ µ µ χ )weusei)thelimitsto ]. These data can then be used to constrain theoretical models. We 7 70 ) could provide a factor of 3 increase in the number of detected dSphs , ] and Atacama Cosmology Telescope 2008 data [ 13 and 69 6 67 ]. 100 GeV, we will use 20 < ∼ channel, however these limits were not published) and ii) limit In order to translate these limits in terms of democratic couplings t CMB constraints arise from redshifts in the range 100 e http://www.darkenergysurvey.org/. http://pan-starrs.ifa.hawaii.edu/public/. 3. We also show the projected sensitivity of the PLANCK telescope in t 12 13 / WMAP (7-year) [ density squared, which definederived the in DM such annihilation an rates). analysis over The 10 impr years of the gamma-ray spectra are very similar for all quark channels. are somewhat sensitive towhich a the portion dominant of DMimpact the on energy annihilation the is channel: CMB; carried on away annihi by the neutrinos contrary or the stored annihilation in mode p which produce results in an increased amountthe of free width electrons, of which survive lastoscillation to scattering l peaks surface, in consequentlycome the suppressing temperature t from and highstructure, polarisation redshifts this CMB well set power of beforestructure spectr constraints the formation. does formation not Detailed depend of constraints on any have highly sizeable been uncert gravit recently der ical effect of energy injection of (exotic) particles around the recombi 5 increase in sensitivity.Dark Additionally, Energy new Survey optical surveyscorresponding (suc to an overallDM increased masses constraining power (shown in figures the low-energy regime, thetime. sensitivity increases However, as in the roughly high-energymore the (limited sq linearly background with regime) the integration LA time. Thus, 10 years of data could provid we combine the published limits to the most effective one.therefore show The CMB limits limits on only for hadronic the channels democratic were coupling not to deriv lept this channel; dSphs constraints on 1 of a single photon [ calculated in [ the produced and measured in collider experiments.positron collides At LEP, at for example highthe energy production a of plethora a of DM final candidate states are produced. Am 5.1 Effective operators 5.1.1 Mono-events at colliders The very sameultimately interactions can responsible produce for synchrotron DM fluxes, annihilation are in the the ones galact by which a DM 5 Constraints on Dark Matter Models JCAP11(2012)038 th (5.1) .This e . ]. Introducing 100 vant energies, the 71 5.3 ors while black lines ) plus background ith heavy mediator the Standard Model. on data, while dashed ¯ χχ nts [ and γ ) operators respectively. f A → 5.2 (EFT) to capture features ). “CMS(had)” means CMS O − es a common language which e + ) are used as source of processes e ], and by the CMS Collaboration 5.1 , 73 ) f i ) or axial ( Γ f ¯ V from some collider studies, as well as from f O )( l,q [GeV] χ X i stands for a single jet. A mono-jet final state the limits we obtained on the leptonic and m Γ j 5 χ (¯ – 16 – ] using results from the CDF collaboration with 2 f 7 1 G G G G Λ     , ), vector ( 6 f S and thus on (in this case) hadronic annihilation cross- = ]. 8 O q f i scalar, LEP scalar, , obtained by LEP and CMS as function of the DM mass ] from ATLAS study [ 10 q O 9 ] and put lower bounds on the eff ective coupling Λ 8 vector, CMS(had) for instance), discussed in sections " and Λ scalar, synch.(lep).26 scalar, scalar, synch.(lep).10 scalar, Z vector, synch.(had).10 vector, synch.(had).26 l are SM quarks and q ) for scalar ( one can define effective couplings between (e.g. fermionic) DM to SM µ γ i 5 100

1000 γ

10000

) processes, one can compare the theoretical background predictions wi q

[GeV] ; min µ ¯ νν ), where γ γ ], and more recently [ ¯ χχ by → j 72 =(1; f . Lower bounds on the effective scales Λ − ]. The analogous types of operators defined in eq. ( i → e + 10 e As an illustration, we show in figure A similar study has been made [ qq Tevatron [ new physics scales Λ fermions first choose to use thefrom a powerful broad machinery class of of WIMP effectiveonly models. field available degrees theory In of particular we freedomProvided assume are that this the at is the dark true rele matter forallows particle all one itself energies to and of compare interest, the the constraints EFT from provid different types of experime Figure 5 synchrotron data for abounds certain to choice quark oflines parameters operators. are (see bounds text Solid fromcorrespond for LHC lines to details (CMS) are or leptonic bounds LEP. ones. Red from lines synchrotron correspond to radiati quark operat where Γ hadronic effective scales, Λ itself [ like (¯ (e.g. the real data produced at LEP [ effective approach can be(Higgs understood boson as or a extra limit U(1) of microscopic models w Performing a Monte Carlo simulation studying the signal ( is then used to put constraints on Λ section, in the very same way as in [ JCAP11(2012)038 → ,in ,by signal χχ obs F no bckgd e notice that ) in our ROI, , a HASLAM el vation of dwarf , and using the pper bounds on omment on this DM he present data, rgy. It becomes F onic coupling. We DM limits without of the data. obtained from syn- es. Indeed, whereas for details). ematic uncertainties ving lower frequency t seems reasonable to l suited to gauge the i A o that region, ible -given the energies old is of course higher in chrotron emission. These y on the value of the mag- s originating in annihilations ling the astrophysical emission ) is around 300 GeV (600 GeV) 30% residuals when compared to q ( from the inverse processes (¯ l ∼ ) (see appendix ee/qq " 5.1 σv ! magnetic field in our ROI, with the hypothesis into limits on annihilation cross sections. We is the uncertainty considered in each case (see is the rms temperature noise, taken to be 3500 K. i – 17 – σ µG σ ,1 for di fferent values of the magnetic fields, and natures of we have included the bounds on Λ 7 become weaker as DM mass increases because of the phase ,where1 5 σ l,q no bckgd and = f +2 6 Collider and complementary Indirect Detection bounds obs F vs ≤ 30%. However, in our analysis, we do not attempt to model specific ] can be used as a template. It is a full sky radio map with the best LAT and the measurement of CMB anisotropies by the WMAP and ACT. ∼ DM 29 F ). In the case of 9 Fermi ] it has been shown that by extrapolating the HASLAM data down to 45 MHz, to and 18 8 ), generated by the operators defined in eq. ( Our procedure is to compute the synchrotron flux coming from DM ( We also translated these limits on Λ For illustration, in figure In [ in the case of scalar/vectorial interaction with universal leptonic/hadr χ ¯ and then constrain the result with the available data corresponding t requiring that figures model astrophysical synchrotron emission, one is left with on the astrophysical model, expressed as a background uncertainty in % the figures and also show how this limits change as a function of the syst and therefore no DM detection couldassume be that claimed. the current Based uncertainty on (i.e.is this systematics) analysis at in i model aastrophysical level signals. of Instead, weassuming decide any to contribution apply from a conservative astrophysical 95% background, CL and which we lab the actual 45 MHz data. Those residuals do not have the proper morphology of a DM maps sensitive to test a possibleof contribution light from dark softer electron matter candidates. angular resolution and sensitivity.contribution Indeed, from a the 408 harder MHz population frequency of is galactic wel electrons, while lea m present the result in figures for low masses, inbehaviour the below. hadronic channel the bounds becomes weaker. We c that the dark matterNFW+MED signal set-up. lies within Wecan clearly 5% already see of give uncertainty that stronger of limit thenetic on the synchrotron field, the background radiation, and eff ective scales, is withthere independentl complementary t are to no the ”threshold bounds effect” from for accelerator synchrotron search radiation at large DM mass, on galaxies by the DM couplings, in comparison with limits coming from LEP, CMS, obser 5.1.2 Synchrotron chrotron data at 45 MHz with a 26 and 10 notice that the bounds on Λ ee/qq space reduction, up toof some the point in experiment- which toLHC it produce is experiments the kinematically than pair imposs at of LEP). DM particles The (the lower thresh bounds on Λ In order to408 model MHz the map synchrotron [ signal of a standard astrophysical origin and relatively independentthen of interesting the to DM comparebounds this mass obtained limit as by to the one thethe LEP measures one leptonic/hadronic and derived annihilation CMS missing from can cross-section ene syn also be converted directly into u JCAP11(2012)038 ic 50 50 .Purple 3 = 5 GeV = 5 GeV 40 40 X X = GeV10 = GeV20 = GeV40 = GeV60 = GeV80 = GeV10 = GeV20 = GeV40 = GeV60 = GeV80 LHC, axial LHC, LHC, axial LHC, X X X X X X X X X X = 100 GeV = 100 = 100 GeV = 100 LHC, vector LHC, LHC, vector LHC, m m X X m m m m m m m m m m m m of products of a pure leptonic o data. At the rons will radiate G G 30 30   l gives the weaker ely sensitive to the 43 GeV/cm . based on the Galactic =0 dline). kinematic closure of the 20 20 ! ρ both cases, comparison with d normalisation has been set to Background Uncertainties [% of data] Uncertainties Background Background Uncertainties [% of data] Uncertainties Background Hadronic Channel, B = 26 Channel, Hadronic Hadronic Channel, B = 10 Channel, Hadronic 10 10 ts based on the measurement of CMB dashed line) limits coming from Dwarf 0 0

1e-25 1e-26 1e-27 1e-28 1e-24 1e-25 1e-26 1e-27 1e-28

« « /s] [cm v) ( /s] [cm v) (

3 max 3 max – 18 – 50 50 ) and that synchrotron bounds largely compete with , assuming DM couples democratically to all charged-leptons µG " = 5 GeV = 5 GeV 40 40 X X = GeV10 = GeV20 = GeV40 = GeV60 = GeV80 = GeV10 = GeV20 = GeV40 = GeV60 = GeV80 X X X X X X X X X X = GeV 100 = GeV 100 LEP, scalar LEP, scalar LEP, vector m m X X m m m m m m m m m m m m σv ! G. ), with the same astrophysical setup, but now the magnetic ﬁeld µ 6 G G 30 30   20 20 Background Uncertainties [% of data] Uncertainties Background Background Uncertainties [% of data] Uncertainties Background , especially at low masses. For low masses, the annihilation into hadron Leptonic Channel, B = 26 Channel, Leptonic Leptonic Channel, B = 10 Channel, Leptonic 10 10 " σv ! . Same as in ﬁgure ( . Synchrotron bounds on 0 0

1e-28 1e-29 1e-25 1e-26 1e-27 1e-24 1e-26 1e-27 1e-28 1e-24 1e-25

« « /s] [cm v) ( [cm v) ( /s]

3 max max 3 G. Astrophysical setup: MED diffusion model, NFW profile with µ channel, and radiates much less synchrotron emission than in the case of b ¯ (left panel) orbounds to coming all from kinematically colliders10 are available shown quarks explicitly. (right The panel). magnetic fiel In Figure 7 Figure 6 Comparing the 2 figures,value of we the first magnetic notice field that (10 or the 26 results are not extrem normalisation has been set to 26 bounds on bounds from other type of experiments. Secondly, the hadronic channe lines represent present (solid line)galaxies. and Brown 10-years line projection in (dot- anisotropies the left (solid) panel and represents near current DM future limi reach based on the Planck data (dashe states gives an electronb spectrum much softer, especially after the synchrotron emission, dwarf Galaxies or CMB) get weaker because the flux channel: a lotat of frequencies soft lower electron thanhigh produced 45 mass by end, MHz later the and decays bounds will of coming not from final be indirect state bounded detections had by searches ( current radi JCAP11(2012)038 very otron, ent of y small of about n the case In the case DM der to render our knowledge m erenceiff in the no background, case of leptonic suming a vector onsidered here), an exclude those effective operator e case of leptonic or channel the exclu- a bound assuming case of allowing at er bounds on axial r (for light masses) espect to the scalar 6 rtainty bounds could predictions for differ- case, is due to quark o background, (orange " y up to the TeV range raction is vectorial, the to develop a term inde- 10 GeV. Furthermore, to σv " ger than tens of GeV, and ! e (in general) than collider # σv ! value to decrease. Since there no σv channel opens, it gives an important b ¯ b ), one realizes by direct inspection of those A . We should also emphasise that even the 2 χ , with the “optimum” choice of magnetic field, – 19 – 7 /m increases, as in the the case of leptonic channels (for , and thus not suppressed. The consequence of this is 2 χ DM m m 2 GeV). the bounds on hadronic channels, now compared to collider ∼ 6 and proportional to result, where we supposed that all the data are generated by DM synchr G, studied in the previous section. We see in this case an improvem 2 µ v = 26 In the comparison with accelerator constraints, one realises that in the We also show in figure Concerning the accelerator constraints, from the study of the We run a similar analysis in figure GC or similar (for heavierwhere masses) LHC than starts synchrotron loosing constraints, sensitivity.is obviousl The however LEP bound one coming order from of a scalar magnitude stronger for light masses of channel, the synchrotron bounds inband) the conservative is case in of general assuming as n effective competitive interaction. as It the starts bound tobeyond coming be 100 from more GeV, competitive LEP space for studies, ofof masses as parameter lar CMS space bound beyond from the a reach vector of interaction, the it accelerator. is I all the way slightly bette have an idea ofsynchrotron how more well competitive we than would LEP need or CMS, to we understand include the in background figure in or DM self-annihilation is proportional to 1 pendent on that in general, vector interaction produce weaker bounds. B uncertainties the synchrotron searches startbounds, to independently be more of competitiv the DM mass (in the casebounds of on vectorial LEP). and axialsynchrotron interactions. bounds Again, for start the conservative to case of be more competitive than collider bounds f that background is known within 5%, at 95% CL. We observe that for these alread expressions that normally theand vector axial operator case. gives This larger is values, because with r its Lorentz structure allows for synchrotron can exclude thechannels, collider even bounds in for thesion conservative all background is the assumption. already mass effective For range,of hadronic for for the masses th background, beyondinteraction. synchrotron 10 GeV. could As be before, able depending to on exclude current collid synchrotron bounds with respect to collider bounds. If the DM inte conservative gives already limits competitive withgive LEP the bounds, best whereas limit the 5% obtained unce by an indirect detection experiment. contribution to the flux, causing the maximum allowed of axial coupling, collidermost bounds 5% of are uncertainties always infrom stronger. the colliders background, In already the the bounds atbehaviour optimistic from 10 for GeV synchrotron masses c in lightermasses the than thresholds. case 5 GeV, of In with vectorial the respect interaction. hadronic to case, The the when d leptonic the tens of GeV, in the case of vectorial interaction, for ATLAS as well as for CMS. more threshold after thethe bottom flux mass decreases smoothly (the when top is heavier than the ranges c ent kind of e ffective interactions (cf. appendix which all thresholds are below JCAP11(2012)038 ]). 0.5 0.5 26 ion of , ses, ad- 24 he future where we = 5 GeV = 5 GeV X X = GeV10 = GeV20 = GeV40 = GeV60 = GeV80 = GeV10 = GeV20 = GeV40 = GeV60 = GeV80 LHC, axial LHC, LHC, axial LHC, 0.4 0.4 9 X X X X X X X X X X = 100 GeV = 100 = 100 GeV = 100 LHC, vector LHC, LHC, vector LHC, m m X X m m m m m m m m m m m m ches, and that ng complemen- and G G   8 0.3 0.3 s are plotted. Dots teractions. The magnetic ent status see, [ next years with the new 0.2 0.2 ies is well motivated. To ion; or LHC-ATLAS (right- e case of a democratic coupling Background Uncertainties [% of data] Uncertainties Background Background Uncertainties [% of data] Uncertainties Background Hadronic Channel, B = 10 Channel, Hadronic B = 26 Channel, Hadronic 0.1 0.1 and from low frequency arrays like 0 0 14

1e-26 1e-27 1e-28 1e-24 1e-25 1e-26 1e-27 1e-28 1e-25

« « /s] [cm v) ( /s] [cm v) (

3 max 3 max – 20 – 0.5 0.5 G. Astrophysical setup: MED diffusion model, NFW profile = 5 GeV = 5 GeV µ X X = GeV20 = GeV40 = GeV60 = GeV80 = GeV10 = GeV10 = GeV20 = GeV40 = GeV60 = GeV80 0.4 0.4 X X X X X X X X X X = GeV 100 = GeV 100 LEP, scalar LEP, scalar LEP, vector m m X X m m m m m m m m m m m m , with the same astrophysical setup, but with the magnetic field 8 G G coming from DM synchrotron fluxes as function of background un- 0.3 0.3   " σv ! G. . 3 µ 0.2 0.2 Background Uncertainties [% of data] Uncertainties Background Background Uncertainties [% of data] Uncertainties Background It is also worth noticing that the analysis of the Galactic diffuse emiss Leptonic Channel, B = 26 Channel, Leptonic Leptonic Channel, B = 10 Channel, Leptonic 0.1 0.1 16 43 GeV/cm . which will survey low frequencies 10-240 MHz and is the pathfinder for t LAT will bring further insights on CR production and propagation proces . Bounds on . The same as in figure 15 0 0 =0

1e-27 1e-28 1e-29 1e-24 1e-25 1e-26 1e-27 1e-28 1e-24 1e-25 1e-26

« « [cm v) ( /s] v) ( /s]

[cm The impact of the control of the background is illustrated in ﬁgures To summarise, our analysis shows that synchrotron bounds, beside bei

ρ max 3 max 3 Fermi http://www.skatelescope.org/. http://www.rssd.esa.int/index.php?project=Planck. http://www.lofar.org/ 14 15 16 extensive studies of thethis astrophysical purpose background at consistent these progresshigh frequenc is quality expected data to coming be from achieved the in PLANCK the mission tary, could be potentially stronger than those obtained from collider sear Figure 9 Figure 8 certainties assumptions [in %represent of the data] bounds atwith coming leptons 95%CL. from (as Different LEP the choicespanel), (left-panel), synchrotron for for bounds) specifically DM democratic and th mas field assuming coupling normalisation a with scalar has quarks, interact been for set both to vector 10 and axial in ditionally constraining the Galactic synchrotron emission (for a curr We leave such dedicated studies for future works. LOFAR, SKA facility. the normalisation set to 26 with JCAP11(2012)038 dark – right), the e LEP data e of WIMPs effective DM otron bounds tabilised by a 8 resent collider ion is axial-like at the energies ger, and on the always stronger, V and assuming vel. o cases for which ould be a particle hich is to take our ounds are stronger ft-hand-side of the o synchrotron data, y possible underlying r or weaker than the on. The conclusion is nsions of the SM with opic theory in general as its limitations. This ude/probe the e ffective he case of a 20 GeV DM edictions, that are hidden lings to which it reduces at e bounds from colliders are s on specific microscopic models. – left), the synchrotron bounds – right) is of course even better, 9 9 , or kinetic mixing, portal. In particular we G (figure G, (figure ! µ µ Z – 21 – = 26 = 26 GC For all these reasons, it is of interest to consider more GC B B 17 ). For 8 G. The case of µ 60 GeV, even with background uncertainties of 40%, the synchrotron can = 10 ! G (see figure µ χ GC ] for two very recent examples of mono-jet/single-photon studie – left). For vector-like interactions, the synchrotron bounds were m B 74 8 A mediator. Thismatter component candidate can naturally be present by(the an construction. Higgs, existing or For particle instance a or it new c a scalar) companion or of a new the gauge boson (Z’). A candidate. It(discrete should or be continuous, possibly a gauged) massive symmetry. and weakly interacting particle, s = 10 In this and the following section, we study the two simplest exte Generally speaking, any fundamental theory that addresses the abundanc Looking at hadronic channels, we see that for uncertainties of 5% (figure See [ • • 17 GC requires two fundamental ingredients: DM particles, the so-called Higgs-portal and dots correspond toright-hand-sides, situations to for stronger which boundscandidate the from with colliders. synchrotron scalar bounds For leptonicprovided instance, are interactions uncertainties t stron is on more(figure the constrained background by radi are smaller than 10%, than by th show the level ofmodels precision for that would diff erent bebounds DM required for masses. to the bethe corresponding The able synchrotron mass. dots to limit excl on The isrespective top lines (for collider of with the bounds. the whole no iso-mass range dots So, of lines correspond for uncertainties) rep t each stronge iso-mass, the regions on the le fundamental models of darkanalysis matter. of Hence constraints the from purpose synchrotron of radiation this to section, a more w microscopic le synchrotron bounds are stronger thanand those for coming from LHC,and if for the interact B are always better thanthat, collider for uncertainties bounds, on in thethan the background the of case LEP 5%, of bounds, orcoupling vector independently less, to interacti the of SM synchrotron the b leptons. DM mass and of the nature of the no matter what the uncertainties considered, for DM masses above 5 Ge considered (be in thepredicts some early specific universe relations or betweenlow the energies. at various colliders). Finally, eff ective a coup in specific Also, theory an a may effective microsc have operator implications approach. or pr 5.2 Higgs portal The effective operator approach is quitemicroscopic powerful, theories as it with encompasses a man is minimum set in of particular free parameters, the but case it when h other degrees of freedom become relevant rule out the colliderweaker, bounds. as we If discussed theare above, always interaction and stronger. is for vector-like, masses th of 10 GeV and beyond, synchr JCAP11(2012)038 ] ], H 75 77 (5.4) (5.3) (5.2) ]. Some ]. Other G and 26 esses [ s logically 76 77 µ DM density tudied in [ servative case d that all the ]. Other studies real singlet scalar is simply ] or the restriction diator is the Higgs 79 ound in [ unchanged, thereby M Higgs doublet S onservative case A), e to modelling of an elds (10 78 in the paper: a NFW H WMAP constraint, we plest case of one singlet ications of an augmented ,with H S † H 2 →− S , S 2 4 HS v W λ ray or positrons, but there is not yet and 3 g M 4 HS S − − 2 λ 3 γ 4 3 HS κ S λ + S − ] of 2 S − 4 λ and the SM Higgs boson (and the existence of µ – 22 – 82 = – − S HS = † the SU(2) gauge coupling. 2 80 2 S S ). We also represented the limits we obtain based on g H HSS m σ 2 S 1 C 2 2 µ κ − particle must be stable to be a dark matter candidate. This symmetry on the model, S L⊃− (left) the synchrotron flux in our ROI expected for the Higgs- 2 Z terms. We also require that the true vacuum of the theory satisfies 10 3 κ and is 1 κ S is the W-boson mass and W = 246 GeV is the vev of the Higgs field, and the H-SS coupling is M v We show in figure Different aspects of scalar singlet extension of the SM has already been s The perhaps simplest extension of the SM consists in the addition of a = 0, thereby precluding mixing of % G) as a function of the dark matter mass, for a Higgs of 125 GeV and candidates that S dark matter profile, with MED diff usion model and a local normalisation of a respect the WMAP constraintthe (at 45 5 MHzsignal radio is data due in to synchrotronand the radiation when two (labelled we cases ”no allowed backgr”,astrophysical the discussed background extremely signal of above: c to standard lie origin whenB). (labelled within we We ”5% the do background, suppose 5% less this of con for uncertainties two du typical astrophysical setups, as studied above µ and the singlet where is achieved by imposing a determine their radio synchrotronanalyse emission, the and parameter space combined that with is the experimentally allowed. analysis of constraints from synchrotron radiation. with probed the model by indirect searches [ portal model for two different representative values of the magnetic fi whereas a preliminary analysisauthors of considered its dark thewhereas matter possibility several consequences of authors can looked explain beauthors at looked f the at the the DAMA consequences consequencesof on of and/or the the earlier COGENT invisible Higgs XENON100 parameter exc width data at space [ LHC due [ to an hypothetical 125 GeV Higgs signal [ For the case of interest, the eliminating the possible to generalise thisalready scenario provides to a very more useful thanscalar framework one to sector. singlet, analyse The the the generic most sim impl general renormalisable potential involving the S field, that couples toboson the SM itself, through and the Higgs the field. model In is this case, usually the called me the “Higgs portal”. Although it i $ cosmologically problematic domain walls). In this case, the mass of the JCAP11(2012)038 . ± 1 or G. − µ s W HSS γ 3 C ,even lusion e, dark ity (see ± . cm 3 = 26 tron data − 29 W constraints − B 10 prospect ( ! erve that mainly pairs, but with a 1 ient to counterbal- b ¯ − 3 GeVcm 100 GeV, except in . b ter space, explaining , as the amplitude is ilation into LAT data on galactic LAT gamma-ray and 1 erhaps emphasize that very low value of =0 ). However, nowadays, − into s 200kms ± ! 3 # ρ W v ), and just above the # cm M Fermi Fermi H σv 26 " ! m − s H east for our energies of interest-, can not ! 10 m m pair. However, the synchrotron ra- , as its kinetic energy has dropped s × ± H S m 3 Γ ]. Also, with the same assumption on ere. Instead, we use the full numerical analysis and from there we derive the new bounds. i W 82 + #∼ 2 H HSS essentially at rest, whereas its kinetic energy σv C m " S − – 23 – 10 GeV [ 2 CM ! is also due for kinetic reasons. Indeed, as soon as the E s m 10 25. Consequently, for annihilation in the Galaxy, the points M∝ ], which is one of the important results of our analysis. Indeed, final state is by far the dominant one due to the large gauge m/ . 81 ± : the annihilation cross section is , 10 ! W 20 GeV are already close to be excluded for small values of the G), and even DM masses below 40 GeV are excluded for 80 , as the process is dominated by the gauge interaction and not the T HSS µ , and on the other hand, an Isothermal profile, with diff usion model C ! 3 final state is the dominant one for − HSS b ¯ , again giving a very low synchrotron emission. These two narrow regions C ’s to the Higgs. Thus, for singlet candidate with a mass slightly below b = 10 ]) and gives even better result than the projected AMS-02 sensitiv 80 GeV). Indeed, in a region around the pole, the enhancement of the cross W B 81 ]. HSS ! C 80 ]) if the DM signal lies in the 10% uncertainty of the background. The exc S being the CM energy of the annihilating 81 m 43 GeVcm . (see table I) and a local normalisation of a DM density of CM =0 , its kinetic energy at the freeze-out temperature allows its annih E In case B, allowing the DM signal to stay within 5% uncertainty, we obs For the (NFW + MED) setup, we observe that, in the more conservative cas The second dip seen in figure channel is open, the 18 ± ! Those models, where velocity Alfvén can not be neglected — at l ρ ± W 18 gamma-rays [ by synchrotron emission is also similar to the one sets by the two narrow regions, namely near the Higgs pole (2 all the parameter space of the model is excluded for masses lighter than of proportional to to estimate suppression factors in every annihilation channel, be studied directly by the Semi-Analytical approach followed h is required to match the WMAP constraint, antimatter) and is not specific to synchrotron. matter masses below magnetic fields ( This result is better that the one can obtain based on the at freeze-out was about W with figure 3 of [ dwarf galaxies study excludethe only CR propagationalready model reach (MED), a one higher(see can figure level 2 see of of that precision [ than the those current based synchrotron on the PAMELA posi “1a” ance the small value of respecting WMAP are away from the pole and the enhancement is not suffic threshold ( M coupling of the are thus very di fficult tothe observe arguments with this we type gave of above signal. are We should valid p for any kind of, indirect detection section due to the Breit-Wigner form of the amplitude implies that a for small values of small value of PAMELA positron data [ diation produced in the Galaxy comes from The synchrotron flux is thusthe largely first reduced dip in this seen region in of figure the parame Yukawa ones (the below the threshold. Consequently, its main decay in the Galaxy is such slightly lighter singlet cannot annihilate into JCAP11(2012)038 e " wn the χ atter m due to 12 GeV. very low o of 40%. HS ! ds (even in λ χ rand Unified ely for m rotron bounds tup considered XENON. ger groups than rough the Higgs wing us to claim nds coming from n would be even if 5-10% (at 95% swithinthisnew eady excluded for tion, as can be the ion. ed above (where an d uncertainties are of sult is then essentially inner galaxy. Indepen- his case the synchrotron 5% of the measurement is s excluded. In the region of ! to be embedded into bigger GUT ! are among the best motivated exten- ! Z the exclusion of the coupling (1) symmetry in addition to the Standard ! 11 U ), given some reasonable knowledge about the W – 24 – M G for the magnetic field in our ROI, and compare µ " . GC B (right) the precision on the background that is required and an associated G one needs to suppose that µ Y 10 ]. Indeed, one can consider, to first approximation, a massive 85 3 times greater than the one we obtained for the simplest singlet = 10 ! ) masses, except around the region of the Higgs-pole, and also for GC W B M ! G. For µ ], assuming the Higgs can decay into DM particles with a branching rati 79 8GeV, where XENON100 starts loosing sensitivity. In this sense, synch ] that in general the scalar singlet model is disfavoured by XENON100 data in = 26 Finally, some words regarding the comparison with XENON100 bounds. It was sho One should notice that our analysis can also be applied to a vectorial dark m We also show in figure Of course, these conclusions are weaker for the second astrophysical se , synchrotron constraints are stronger than those of XENON100, assuming reasonabl ! 83 W χ GC in [ 1% (at 95% CL)case or of less, the for effective approach,Einasto DM an profile masses equally (not below valid shown 40 optionstronger GeV. here), for conclusions Again, can the about as DM enhance the distribu commented the exclusion above signal power for by of the a the factor synchrotron 3, radiat allo here, where electrons diffusedently more of and the the magnetic DM field is used, less the cusped model in is the excluded if backgroun flux should be a factor In this case, everythingrelatively which low is masses to those the bounds left are of comparable the to brown the curvethrough ones i the coming Higgs from portalvector as [ a scalar particle with three internal degrees of freedom. In t region of low ( B to be ableportal. to We exclude/discover observe aunreasonable that, synchrotron except precision in signal would the beCL) for two of needed a narrow the to pole singlet observed measure regions flux DM any is discuss th type due of to synchrotron fluxes), radiation, the model is alr synchrotron data, assuming a value of 26 due to the annihilationindependent of of the the precise singlet value to of exclude the model. This re m LHC [ For completeness we also included in the analysis a comparison with bou Neutral gauge sectors with an additional dark scalar extension.stronger One than thus the deduces one we that presented the in constraints the singlet we scalar would5.3 scenario. obtai Extra U(1) it directly with the exclusion coming from XENON100. We see that effectiv backgrounds uncertainties of 20% atthe 95%CL. conservative case Also, of as no-background) expected, do better synchrotron than boun XENON100, for masses M Model (SM) hypercharge U(1) background. For illustration, we show in figure are complementary to XENON100 bounds, asmass, they as are well able as to the constrain region the of region large of mass ( SU(5) or SO(10) allow the SM gauge group and U(1) sions of the SM,gauge and sector give of theTheories the possibility theory. (GUTs) that a and Extra dark appear gauge matter symmetries systematically candidate are lie in predicted string in constructions. most G Lar JCAP11(2012)038 , DM 200 m G G 180 µ µ constraint very B=26 B=10 160 5% background 5% ed) Excluded by 140 ng no background; G for the magnetic . Right) Maximum astrophysical setup, 200 µ 3 G (red, orange) and 120 3 − µ − [GeV] e BR of the Higgs. e 26 X 100 m usion by XENON100, while ncertainties of 5% (see text Excluded by uncertainties of 80 space of the model, with the 3GeVcm 100 . 60 43 GeV cm . 80 =0 40 ! =0 ρ 60 ! 20 ρ

50 40 30 20 10 Background uncertainties [% of data] of [% uncertainties Background 40 [GeV] X m for the Higgs-portal model, as a function of DM – 25 – 200 HSS λ 20 = 40% G G 180   inv BR B=26 B=10 5% uncert. 5% uncert. 1% No backgr. No 15 160 XENON100 (2012) XENON100 excl. excl. by backgr. No excl. excl. by <= uncert. 1% 140 excl. excl. by uncert. <= 40% excl. by uncert. <= 20% 10 120 [GeV] 1 X 100 0.1 m

0.01

0.001

hss 80 ⁄ is scanned while requiring that the relic abundance respects WMAP 60 HSS 40 λ . Constraints on the coupling . Left) Synchrotron ﬂux at 45 MHz produced from the Higgs-portal model. For e 20 . Diﬀerent colours show the exclusion by synchrotron data, assuming: R χ 1e+07 1e+06 1e+09 1e+08

100000

m ƒ Temperature [K ( [K Temperature ] /MHz) . We presented the result for a magnetic field profile normalised to 26 2.5 σ G (green, light green). Dashed line shows bound coming from data, assumi µ Figure 11 Figure 10 the coupling field at GC, as wellthe as black NFW+MED dashed set-up. line Black is solid the line LHC shows limit the excl assuming that at most a 40% invisibl mass at 5 No-background choice, Blue) Excluded20%, by uncertainties and of Green) 40%, Excluded Yellow) by uncertainties of less than 1%. Here we assum 10 background uncertainties allowed tosame exclude conventions as a above. point in the parameter whereas dotted line showsfor the details). bound assuming Red a and full green background with bands u are for NFW+MED+ while light-green and orange correspond to ISO+1a model+ JCAP11(2012)038 , 89 (5.5) (5.6) gauge ! ich can (1) and Z erated at D gamma-ray der the SM U ariant kinetic ctors through omenology of the (1) but neutral with D stified in recent string U µν ) candidate is the lightest ˜ X 0 ψ , µν ˜ j B could be the lightest (and thus (1)’s because there is no reason ! Ψ δ 2 # 0 µ the gauge field of U 2 2 j j ψ − D (1) quantum numbers. These states (1) (hence the “dark” denomination m µ µ M D µν ˜ γ D X " j ˜ U X ¯ Ψ ]. In such framework, the extra the particles from the hybrid sector, µν log j ˜ and U 86 X j j D being the SU(2) generators, and ! q 1 4 i a extension of the SM can be decomposed into j Y ]. q T − + 97 i D – 26 – ), j µν ψ a µ ! µ ˜ B 2 W D D µν a ˜ g π µ ˜ γ B Y ]. The DM candidate i 16 ˜ g gT 4 1 ¯ ψ 88 + = − i µ δ ! ˜ i (1), mixing with the SM hypercharge through both mass and X SM ! D + L propagator. After diagonalising of the current eigenstates that U ˜ g (1) but not charged under is made of particles which are charged under the SM gauge group contains states with SM = D µ Y ]. One of the first models of dark matter from the hidden sector q (1)s are special compared to GUT ˜ ! U X L 87 + [ , U × ], but has also been studied within a model independent approach [ µ µ ˜ ! ˜ B µν 93 B being hybrid mass states [ – F Y ˜ j g µν 90 Y SU(2) Y M q F × ( ]. i 2) Visiblesector yrdsector Hybrid 96 − and δ/ (1)s generated in such scenarios see e.g. [ ! µ j ∂ ]. For typical smoking gun signals in such models, like a monochromatic U particle of the dark sector. The for this gauge group). the dark sector is composed by the particles charged under are fundamental because they actthe as kinetic a mixing portal they between induce the two at previous loop se order. respect to the SM gauge symmetries. The dark matter ( SU(3) The m Several papers considered that the key of the portal could be the gauge inv Notice that the sum is on all the hybrid states, as they are the only ones wh The matter content of any dark U(1) From these considerations, it is easy to build the effective Lagrangian gen being the gauge field for the hypercharge, the particles from the hidden sector, Ψ 95 = • • • , µ µ i ˜ mixing ( boson would act asgauge a group) portal and between the the “visible” “dark sector. world” (particles not charged un for the SM particlesextra to be charged under them; for a review of the phen stable) particle ofconstructions this [ secluded sector. Such a mixing has been ju groups. Brane-world contribute to the with a massive additional B kinetic mixings, can be found in [ 94 with ψ three families of particles: one loop: D line, see [ JCAP11(2012)038 ! ]. # ly Z 98 ach ded (5.8) (5.9) (5.7) nthe which DM ! so quite m 4 Z /M and obtained 1 erve a similar d factor in the δ fermions state, ∝ 0 in 2 ψ bosons for ) after diagonalization .Indeed,thetwo ! µ ! 20 W Z θ W |M| Z generates an effective ,X θ 2 o be able to distinguish the secondary products M µ boson which induces an 2 δ . µ µ B W ) sin θ Z 0 sin 2 ψ and φX φX δ 2 µ δ sin γ Z δ sin 0 SM SM +4 2 ¯ M ψ +4 2 − f the model in the appendix of ref. [ µ 2 α → ) ) + cos ) µ ! µ ! to the SM – right). W φZ B Z B θ in windows to avoid juxtapositions of 0 = 2 W . W ! 12 ψ θ → θ δ 2 and atomic parity violation constraint). Z φ µ W sin couplings to first order 0 +sin θ B − M 2 sin sin ψ 0 2 sin g δ Z W channel exchange of the 0 ψ 0 − θ − ¯ µ − sin ψ ψ 3 − 3 µ µ 0 γ 2 W s 2 Z – 27 – 0 ) diagonal and canonical, we can write after the δ θ ψ ) before the diagonalization, ( W W ¯ ψ µ 2 ! µ + cos − W ) for points respecting the WMAP and electroweak /M ˜ W 5.5 X ! θ δ θ 3 , 2 µ sin Z 2 and Z ; µ 0 2 ! parameter φZ ˜ W B ψ δ M α Z /M ρ 200 GeV the value of the flux begins to be quite weak for (cos (cos ! W . The kinetic mixing parameter m − φ 2 φ θ Z , and a coupling of Z +4 # SM (cos 120 GeV (see figure ! = ! (1 2 ψ M D M Z ! Z ) ˜ g α " − channel process SM D > to M =sin =sin = cos ! the synchrotron flux emitted at 45 MHz for different values of DM ( ± q ψ − 19 DM ! s µ µ µ m = = =1 SM 12 < Z Z A m , ψ δ ! L φ α Z M cos breaking portal too, except that there are diff erent fine-tuned pole regions for e = 3 trough our analysis, keeping in mind that our results stay complete ! . The fluxes in both cases lie in a region of parameter space between 5% to Y ! Z D Z ˜ g ) after the electroweak breaking. D U(1) µ q ! × ,Z µ L 2. We have restricted the fluxes for each We show in figure One observes that the conclusions we have obtained for the Higgs portal are al Z / ! One can find a detailed analysis of the spectrum and couplings o Our notation for the gauge fields are ( Z 20 19 with, to first order in We do thisbehaviour exercise to for that the in astrophysical the NFW+MED Higgs setup. portal around We first two poles, obs channels giving a good relic density are the M general by a simple rescaling of the kinetic mixing makes the gauge kinetic terms of eq. ( we computed the effective We took precision tests (including Z width, valid for the SU(2) masses after a scan on ( regions of other parameteramplitude space square far of from the the pole region, due to the reduce and ( mass of the and + (-) sign if interaction on nucleons. Developing the covariant derivative on SM and 6 Conclusions and prospects In this work, we derived theof constraints dark on synchrotron matter emission annihilation from in our Galaxy, based on the radio 45 MHz data. We expan 100% of the measured flux and could thus be probed in a near future. is almost a factorsynchrotron 1/6 fluxes. compared In to this a casea 125 one dark GeV needs matter Higgs a signal precision exchange, better if factor than that 5% we t recover i the fluxes. However, when coupling of SM states JCAP11(2012)038 200 180 istent. In ]. We have y to gauge 160 ]. t bounds on -portal model 5% background 5% 26 ! action of this 18 – d found that it Z – 140 everypertinent axy in the range 24 ical uncertainties ders and indirect 16 , providing therefore 120 N, MED and MAX ing a full background s allowed to be able to gh the scope of our 14 [GeV] ], and a full numerical ions and treat targets X 100 mi-analytical approach m 35 t dark matter candidates, ysics models in question, , 80 , where the DM profiles are 34 ]. Both of these observations 60 20 40 20

50 40 30 20 10 Background uncertainties [% of data] of [% uncertainties Background LAT observations of nearby dwarf spheroidal is scanned while requiring good relic abundance, δ – 28 – 200 ], benefiting from the strengths of both in particular Fermi G (light colours). Dashed line shows bound coming from 22 µ 180 , 21 160 140 , the coupling code [ DM 120 . A frequency of 45 MHz was used, assuming a magnetic field profile m ]), we have considered several other sets of CR parameters, which ! [GeV] X Z 23 100 m m GALPROP 80 G G G G G G       G. 60 G (dark colours) and 10 µ µ masses, 5% uncert. 5% No backgr. No ). For every ! 40 Z =50GeV, =50GeV, B=10 =50GeV, B=26 ], as well as the limits set by CMB anisotropies [ =200GeV, =200GeV, B=10 =200GeV, B=26 =100GeV, B=10 =100GeV, B=26 Z’ Z’ 5.5 Z’ Z’ Z’ Z’ m m . Left) Scatter plot of predictions for synchrotron flux coming from a m m m m 20 19 10 GeV, for which direct detection limits are weaker, or altogether inex

∼ 1e+07 1e+06 1e+09 1e+08

100000

ƒ Temperature [K ( [K Temperature ] /MHz) One of the purpose of our work is to put in perspective various indirec The discussion of the astrophysical propagation setup occupies a large fr 2.5 DM for different normalised to 26 Figure 12 defined in ( data, assuming nowith background; uncertainties of whereas 5% dottedexclude at 95%CL. a line Right) point shows Maximum in background the the uncertaintie bound parameter space assum of the model. upon the current literatureto in model several the aspects: particle propagation a) in by the using intergalactic both medium a [ se independent probes of DM self-annihilation signatures. give stringent indirectdiff erent limits than on the dark one matter we annihilation focus cross on to sect derive the radio DM constraints, between 1–100 work. Together withuncertainties a of usual the setmodels astrophysical of of conditions CR propagation on [ propagationare the parameters shown radio to used be signals, previousl also consistent (MI explored with the CR, impact gamma ofis ray the generally magnitude and/or mild, of synchrotron ranging the data a magnetic [ factor field of in 3–4 our for ROI magnetic an fields in the inner Gal analysis based on the aspects of the work, b)more considering intermediate robustly Galactic constrained, latitudes andon c) the taking magnetic into fields.bounds account on We the DM have large annihilation shown astrophys cross that section, synchrotron confirming emissions previous can results giv [ m dark matter models,results complemented is with broader, collidersearches this are constraints. was supposed in to Althou part put very motivated relevant by limits the on relatively fact ligh that both colli this spirit, weincluding have forecasts) the briefly constraints reviewed from (and applied to particle ph galaxies [ JCAP11(2012)038 anni- uction r space − W + usion of the rovides very ng challenge ion produced onable values W tarity between with a generic he near future, , provided some nd signal could certainties in the the charge cosmic ative synchrotron idates. Moreover, l, and assuming a ere it is dominated s for DM could be ic approach to DM raints can even be hose based on LEP icular, in the case of t at some price. Pro- present in our Galaxy. information. We have y on the astrophysical hold for M couples to hadrons, sonance effects become effects are inexistent in rotron radiation on two servative setups, where the astrophysical setup ep in mind that missing , but some control of the of the effective operator t (and therefore indirectly . For eff ective couplings to t searches, with a particular portal, because of substantial anni- ! Z and measurement of di ffuse emission in 21 portals. This is motivated by the fact that ! Z – 29 – The effective approach is powerful, but reaches its limits when re This work illustrates again, if necessary, that a multi-signal approach p We have applied our results to various DM particle models, starting http://ams.cern.ch/. 21 phenomenology, we have considered the constraintsspecific set DM by radio models: synch the so-called Higgs and relevant. To assess such effects, and for the own sake of a more microscop vided the uncertainty oncuspy the profile background (NFW), could and beis a assessed excluded, large at except the magnetic near 5% field, resonance leve (the most Higgs of pole), the or Higgs close portal the paramete thres these models are amongeach the of simplest these extensions modelsapproach, of and the provide thus SM fully provide with self-consistentshown complementary, DM that UV albeit cand radio completions model data dependent, may put severe constraints on these models, bu quarks, the synchrotronprediction have of a this strong signal, potentialconstraints one can too, cannot improve but robustly overdwarf due claim the constraint to that collider generally the huge perform bounds. most un better. conserv As expected, if D we supposed thatby all DM the annihilation, radio themeasurements data limits (photon+missing are obtained energy). saturated arebetter by already than In the competitive the synchrotron with this onesmodelling. t radiat derived case from too, CMB, Concretely, depending radiogive allowing on the const that the best uncertaint DM limit obtained contribute by to current 5% indirect detection the signals backgrou approach based onof eff ective parameters, operators. and forcomparable There, a to we conservatively those have chosen ofeff ective shown ROI, colliders, couplings synchrotron and that, to sometimes searche for leptons, even reas we do have better. shown In that, part even for very con synchrotron bounds and theparameter last space XENON100 of bounds the when considering model. the excl For the case of the hilation, which is impeded inan the effective Galaxy operator (by approach). construction both We have these also shown a very good complemen hilation into leptons, themay be constraints more are conservative (smaller stronger,background magnetic is or field, also less alternatively, required cuspy profile) to exclude (most of) the parameter space. for both futurePLANCK observations will and be able theoretical to studyby works. Galactic the emission To in dust the pave emission, frequencygas) the range content mapping of wh road, with our Galaxy. unprecedented in Togetherray precision with t spectra improvements dus in we measurement of will soon be getting from AMS–02 complementary information onat DM colliders phenomenology. is likelyenergy If to may not give on be the directly the strongestOur related results to constraints, long present the one (part actual run, of) should DMemphasis the that DM ke state on is prod of supposedly synchrotron art radio in data,control confronting indirec a may very be promising signal gained for on dark the matter mundane, astrophysical background, a fascinati JCAP11(2012)038 , p- and rsay (A.2) (A.3) (A.1) ]. V Rept. 8 , O s , S 24 [ . O =0 2 ye, and Roberto the French ANR v , period. Finally, the erators trophysique de Paris # for lighter candidates. 2 obe ide further leverage on v 4 f ; of this project. The work is the relative velocity of m # v 2 v +5 2 f 4 f 2 f m m m 2 χ − ]. 2 χ ]. m , + 17 4 2 f 2 m LAT, models of propagation and energy 2 f v − m ) m SPIRE 4 χ 2 f 2 χ SPIRE − IN m m IN 2 χ m ][ 8 Fermi ][ − m 22 , for different effective operators 2 χ " – 30 – )+ − the DM mass, while m 2 f 4 χ ( 2 f 2 f 2 χ 2 χ σv χ m 2 f 2 χ ! m m m m m Particle dark matter: Evidence, candidates and constraints 8 m m m + − − 2 χ + − 1 1 m 2 f 1 ! ! m hep-ph/0404175 [ ! 4 4 hep-ph/0002126 [ 4 24(2 24 Λ Λ 1 1 Λ π π 1 " " π 8 48 48 × × = = = v v v (2005) 279 S (2000) 793 A V Nonbaryonic dark matter: Observational evidence and detection method σ σ σ 405 63 is the fermion mass and f m Phys. Rept. Prog. Phys. are given by: A [2] G. Bertone, D. Hooper and J. Silk, [1] L. Bergstrom, O The expressions for the annihilation cross-sections coming from the op for hospitality during completion of this work. A Annihilation cross-sections gramme under grantport Multi- of Dark the CSD2009-00064. FPIPITN-GA-2009-237920 (MICINN) (UNILHC) grant B.Z. of BES-2008-004688,UniversitéParis the acknowledges and Sud, European the the for Commission, financial contracts theof FPA2010-17747 as hospitality su and M.T. during well is the as supported development group LPT-Orsay by for of the support IISN and and hospitality. an G.Z. ULB-ARC is grant. grateful He to the also Institut thanks d’As the LPT-O Acknowledgments The authors want toLineros thank for Illias very Cholis, usefulTAPDMS Alessandro discussions ANR-09-JCJC-0146 Cuoco, and Timur and opinions. Delaha the Y.M. 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CONCLUSIONS

Next I mention brieﬂy the main conclusions of the works presented in this thesis.

Regarding the Seesaw mechanism and Lepton Flavour Violation:

The extended believe that in supersymmetric Seesaw models BR(µ eγ) depends • → strongly on the θ13 value, is not correct, as a careful analytical and numerical study shows. In this analysis we have scanned in a complete way the Seesaw parameter space using two alternative parametrisations, obtaining consistent results.

The potential dependence of BR(µ eγ) en θ comes mainly from the Y †Y matrix, • → 13 ν ν where Yν is the neutrino Yukawa matrix. In one of the parametrisations we used (the so-called VL-parametrisation) this magnitude is trivially insensitive to θ13 or any other observable parameter. This fact causes BR(µ eγ) to be also insensitive. In the other → † parametrisation (the so-called R-parametrisation), the quantity Yν Yν depends a priori on θ13, but this dependence disappears when a complete scan of the parameter space is performed. Hence, the results of the two parametrisations are consistent, as expected.

In this work we have obtained very simple rules to scan in a complete way the R-matrix • of the second parametrisation, while respecting the perturbativity of the Yukawa couplings. These rules are valid for the supersymmetric as well as for the non-supersymmetric Seesaw.

An important result is that the values of BR(µ eγ) in the supersymmetric Seesaw • → are typically larger than those reported in the literature (and above the experimental bounds). Essentially this is due to the fact that a complete scan of the R-matrix allows access to regions where the matrix entries are typically larger than those considered so far. 122 Conclusions

When the condition of successful leptogenesis is imposed, the qualitative conclusions • do not change; i.e. the requirement of leptogenesis does not introduce extra dependences of BR(µ eγ) on θ . However it does have an effect on the surviving values → 13 of BR, which tend to be lower.

Summarising, the conclusion of the ﬁrst part of the thesis is that a complete scan of the parameter space of the R matrix (for which we provide precise rules) is necessary, in order − to arrive to general conclusions about predictions of the Seesaw scenario. This has not been taken into account in part of the works in previous literature, and thus we consider it an interesting line of research, which can be exploited in future projects 123

Regarding Dark Matter Phenomenology:

We have computed the percentage of hadronic coupling (with respect to leptonic one) • of Dark Matter with Standard Model particles, which is compatible with data coming from LEP, Tevatron, WMAP and XENON experiments. The bounds coming from LEP (Tevatron) are obtained from the study of ﬁnal-state topologies consisting of a single photon (jet) plus missing energy.

We have demonstrated that for thermally produced fermionic Dark Matter, whose in- • teractions with the Standard Model can be described by an effective theory, the bounds are such that light DM candidates ( 10 GeV) are basically excluded whatever the type of interactions they have with the SM fermions. Besides, heavier candidates ( 20GeV) are excluded if they have scalar-like interactions. If they have vector-like interactions, their couplings should be highly hadrophobic.

An important conclusion is that models with light candidates, which are mainly cou- • pled to electrons (which are favoured by experiments as INTEGRAL), are excluded by the bounds coming from LEP and Tevatron.

More exotic models, as those containing DM candidates which do not couple to elec- • tron or lightest quarks, can escape the conclusions mentioned above, because then the bounds coming from LEP and Tevatron can not be applied.

We have also computed the predictions of several Dark Matter Models for the syn- • chrotron radiation fluxes reaching the Earth at radio frequencies, which has been used to put bounds on those models from existent experimental constraints. We have done this taking into account all sources of uncertainties, as the galactic magnetic field, the cosmic ray diffusion model and the dark matter density profile. The calculation has been performed semi-analytically and verified by full numerical analyses.

We have compared the bounds coming from synchrotron radiation, with those coming • from various colliders: LEP, Tevatron and LHC, as well as with other bounds coming from indirect detection, as White Dwarfs and CMB measurements. Also, for one particular model, we have made the comparison with bounds coming from XENON experiment of direct detection.

The conclusion is that the bounds coming from synchrotron radiation are competitive • with those from the rest of experiments. This conclusion depends of course on the assumed uncertainties for the background contributions, coming from standard astrophysical sources. However, for reasonable uncertainties, of the order of 5–10%, the data coming from synchrotron are able to exclude regions of the parameter space of these models for which other experiments are less sensitive, although there are some regions for which synchrotron bounds are less constraining. 124 Conclusions CHAPTER 8

CONCLUSIONES

A continuacion´ menciono brevemente las conclusiones principales de los trabajos de esta tesis.

En relacion´ al mecanismo de Seesaw y Lepton Flavour Violation:

La creencia (extendida entre la comunidad) de que en modelos Seesaw supersimetricos´ • BR(µ eγ) depende fuertemente del valor de θ , no es correcta, como demuestra → 13 un cuidadoso estudio tanto anal´ıtico como numerico,´ en el cual se ha escaneado el espacio de parametros´ del Seesaw de forma completa usando dos parametrizaciones alternativas.

La potencial dependencia de BR(µ eγ) en θ13 se da principalmente a traves´ de la • † → matriz Yν Yν , donde Yν es la matriz Yukawa de los neutrinos. En una de las parametrizaciones utilizadas (la llamada “parametrizacion´ V ”) esta magnitud es trivialmente in- − L sensible a θ , o a cualquier otro parametro´ observable; por lo que BR(µ eγ) lo es 13 → tambien.´ En la otra parametrizacion´ (“parametrizacion´ R”), la cantidad Y †Y depende − ν ν en principio de θ13, pero esa dependencia desaparece al hacer un scan completo del espacio de parametros.´ Por tanto, el resultado es consistente en las dos parametrizaciones, como era de esperar.

En el trabajo obtenemos unas reglas muy simples para escanear de forma completa la • matriz R de la segunda parametrizacion,´ respetando la condicion´ de que los acoplos de Yukawa sean perturbativos (estas reglas son validas´ tanto para el Seesaw supersimetrico´ o el no supersimetrico)´ .

Un resultado importante es que los valores de BR(µ eγ) en el Seesaw supersimetrico´ • → son t´ıpicamente mas´ grandes que muchos de los reportados en la literatura (y por 126 Conclusiones

encima de las cotas experimentales). Esto se debe basicamente´ a que cuando se realiza el scan completo, se accede a regiones de valores la matriz R que no hab´ıan sido consideradas antes.

Cuando se incluye el requisito de una leptogenesis´ exitosa en el analisis´ el resultado no • cambia en esencia; es decir, la condicion´ de leptogenesis no introduce dependencias extras de BR(µ eγ) en θ . Sin embargo s´ı afecta a sus valores permitidos, pues → 13 tiende a seleccionar valores bajos del BR.

En resumen, la conclusion´ de la primera parte de la tesis es que un scan completo del espacio de parametros´ de la matriz R (para el cual damos reglas precisas) es necesario en orden a sacar conclusiones generales sobre las predicciones del escenario del Seesaw. Esto no ha sido tomado en cuenta en una parte de los trabajos en la literatura previa, y por tanto lo consideramos como una l´ınea de investigacion´ interesante para ser explorada en proyectos futuros. 127

En relacion´ con la fenomenolog´ıa de Materia Oscura:

Hemos calculado el porcentaje de acoplo hadronico´ necesario (con respecto al leptonico)´ • de la Materia Oscura con las part´ıculas del Modelo Estandar´ para que esta´ sea consistente con los datos de WMAP, XENON, LEP y Tevatron. Para los dos ultimos´ experimentos los datos relevantes provienen de considerar estados ﬁnales de un foton´ (en el caso de LEP) o un jet (en el caso de Tevatron) mas´ energ´ıa perdida.

Hemos mostrado que si la Materia Oscura esta´ hecha de fermiones, esta´ generada • de forma termica´ y su interaccion´ con las part´ıculas observables puede ser descrita con una teor´ıa efectiva, entonces los candidatos ligeros ( 10GeV) estan´ basicamente´ excluidos cualquiera sea el tipo de interaccion´ que tengan con el resto de fermiones del Modelo Estandar.´ Ademas,´ los candidatos mas´ pesados ( 20GeV) estan´ tambien´ excluidos si esas interacciones son de tipo escalar y estan´ forzados a tener acoplos hadronicos´ muy suprimidos si son de tipo vectorial.

En este mismo marco general, una conclusion´ importante es que los modelos con can- • didatos ligeros principalmente acoplados a electrones, que son favorecidos por experimentos como INTEGRAL, resultan estar excluidos por las cotas de LEP y Tevatron.

Modelos mas´ exoticos,´ los cuales por ejemplo contengan candidatos a Materia Oscura • que no se acople al electron´ o a los quarks mas´ ligeros, pueden escapar las conclusiones anteriores, ya que entonces las cotas de LEP y Tevatron no pueden aplicarse.

Se ha calculado las predicciones de varios modelos de Materia Oscura sobre los flu- • jos de radiacion´ de sincrotron´ a frecuencias de radio, y por tanto, hemos sido capaces de poner cotas a los modelos considerados provenientes de datos experimentales ex- istentes. Esto se ha hecho tomando en cuenta todas las fuentes de error presentes en el calculo´ estandar´ del flujo, como son el campo magnetico´ de la galaxia, el modelo de difusion´ de rayos cosmicos´ o el perfil de densidad de energ´ıa de la Materia Oscura. El calculo´ se ha hecho de forma semi-anal´ıtica, y ha sido verificado por herramientas numericas.´

Hemos comparado la potencia de las cotas provenientes de los ﬂujos de radiacion´ sin- • crotron,´ con la de las que provienen de los colisionadores de part´ıculas, como LEP, Tevatron y LHC; as´ı como con la de otras cotas de deteccion´ indirecta, como ﬂujos desde Enanas Blancas y mediciones del CMB. Ademas,´ para uno de los modelos com- paramos tambien´ con las cotas provenientes del experimento XENON de deteccion´ directa.

La conclusion´ es que las cotas provenientes de la radiacion´ de sincrotron´ son competi- • tivas con las del resto de los experimentos. Esta conclusion´ depende de las incertidumbres que se asuman para las contribuciones de “background”, proveniente de procesos astrof´ısicos estandar.´ Pero para incertidumbres razonables, del orden del 5–10%, los 128 Conclusiones

datos de sincrotron´ son capaces de excluir regiones del espacio de parametros´ de los escenarios anteriores, insensibles a los otros experimentos; si bien en otras regiones los datos de sincrotron´ son menos sensibles. Appendices

APPENDIX A

CORRECTED BOLTZMANN EQUATION

In this appendix I continue to analyse the Boltzmann equation, without assuming a Maxwell- Boltzmann distribution function for the bath particles which, as we have seen, should not be -a priori- valid for the mbath T regime. The collision operator CB of the Boltzmann equation -i.e. the RHS of (3.33)- should describe how the number density of a species changes in time, assuming the system is a FLRW gas of particles with deﬁned distribution functions. First of all, I revert back the simpliﬁed Boltzmann equation (3.38), which can be rewritten as

3 d(nχ a) 2 2 a− = σv bb¯ χχ¯ nb σv χχ¯ bb¯ nχ CY , (A.1) dt → − → ≡ where I made explicit that in principle the annihilation cross-section of the bath differs from that of DM. nb is the number density of the bath particles. Next I will show that this equation corresponds to a thermal bath which (as the DM) follows a Maxwell-Boltzmann distribution function. Indeed we have

2 CY σv bb¯ χχ¯ nb (A.2) ⊃ → n2 4 d3 p = b i (2π)4δ (4)(p p ) M 2 f eq f eq , eq 2 ∏ (2 3)2E in − out | | b b¯ (nb ) i=1 π i where the equilibrium distributions f eq have zero chemical potentials. So the only way to recover the original -exact- collision operator

4 3 d pi 4 (4) 2 C (2π) δ (p p ) M f f¯ (A.3) B ⊃ ∏ ( 3) E in − out | | b b i=1 2π 2 i

is if the fb can be expressed as a product of functions of µb and Eb only, which is eq eq realised in the Maxwell-Boltzmann form, where f (µ ,E )/ f (E ) eµb/T = n /n . b b b b b ≈ b b 132 Corrected Boltzmann equation

Search for a more complete expression

The original equation (3.33), after doing the reasonable simpliﬁcation of considering the DM (but not the bath particles) to follow a Maxwell-Boltzmann distribution can be rewritten as

4 3 3 d(nχ a) d pi 4 (4) 2 a− (2π) δ (p p ) M dt ≈ ∏ ( )3 E in − out | | i=1 2π 2 i [ f f¯ f f ¯ (1 f )(1 f¯ )] P A , (A.4) × b b − χ χ ± b ± b ≡ − where P and A are DM production and annihilation terms, respectively. Taking into account

3 3 d pb d pb¯ 1 = dE dE¯ ds , (A.5) ( )3 E ( )3 E 4 b b 2π 2 b 2π 2 b¯ 64π 3 3 3 3 d pχ d pχ¯ d pχ d pχ¯ (2π)4δ (4)(p p )= ( )3 E ( )3 E in − out ( )3 E ( )3 E 2π 2 χ 2π 2 χ¯ 2π 2 χ 2π 2 χ¯ 3 (3) (2π)δ(√s E E ¯ )(2π) δ (p +p ¯ ) × − χ − χ χ χ 1 2 2 4 = (s 2m ) 4m dΩ ¯ , 32π2s − χ − χ χχ we can express P as

1 1 2 2 4 2 P = dE f dE¯ f¯ ds (s 2m ) 4m dΩ ¯ M . (A.6) 64π4 b b b b 32π2s − χ − χ χχ | | On the other hand, the annihilation term A contains a term proportional to fχ fχ¯ fb fb¯ , which allows us to factorise the integral, so having

1 (E µ )/T (E ¯ µ ¯ )/T A dE e− χ − χ dE¯ e− χ − χ (A.7) ⊃ 4 χ χ 64π 3 3 d pb d pb¯ 4 (4) 2 ds (2π) δ (p p ) M f f¯ × ( )3 E ( )3 E in − out | | b b 2π 2 b 2π 2 b¯ 1 (Eχ µχ )/T (Eχ¯ µχ¯ )/T = 4 dEχ e− − dEχ¯ e− − 64π 1 2 2 4 2 ds (s 2m ) 4m ( fb f¯ ) √ dΩ ¯ M . × 32π2s − b − b b |Eb,b¯ = s/2 bb| | in the absence of the f f¯ factor, the above expression would be equal to n n ¯ σv ¯ . b b χ χ χχ¯ bb So the complete expression for A is: →

2 1 E /T A = n n ¯ σv ¯ + dE e− χ (A.8) χ χ χχ¯ bb 64π4 χ → 1 1 2 2 4 2 dsdΩbb¯ (s 2m ) 4m M [ fb¯ fb fb fb¯ ] √ . × 32π2 s − b − b| | ± ± Eb,b¯ = s/2 133

It is straightforward to show from (A.6) and (A.8) that in the limit where fb,b¯ follow a Maxwell-Boltzmann law, (A.4) reduces to the familiar equation (A.1).

Let us take a closer look at (A.8). There we see that the bath distribution functions fb,b¯ are evaluated at E ¯ = √s/2, with √s 2m . This means that even if in the large majority b,b ≥ χ of cases E T, for the reaction to occur one must have quite energetic bath particles: b ∼ E m . Thus energy conservation forces the energy of the bath particles to be close to the b ∼ χ DM mass, for the process to take place. When this happens, the bath distribution function has a value very similar to the one given by its Maxwell-Boltzmann approximation, because E m >> T. b ∼ χ A similar situation occurs for the production term (A.6). The integrals over Eb,b¯ are sensitive to mχ , because if mb << s:

2 s = 2m + 2E E¯ 2 p p¯ cosθ ¯ 2E E¯ (1 cosθ ¯ ) (A.9) b b b − | b|| b| bb ≈ b b − bb So the integration limits can be expressed as

∞ ∞ Eb Eb¯ . (A.10) mb s/4Eb

So for very large masses mχ , s -and thus- Eb are very large, making the distribution function fb very suppressed, and close in value to its Maxwell-Boltzmann approximation.

To summarise: the energy conservation Eb +Eb¯ = Eχ +Eχ¯ forces the bath distributions functions to behave very similarly to their Maxwell-Boltzmann approximations. 134 Corrected Boltzmann equation LIST OF FIGURES

2.1 Feynman diagrams which give rise to li l jγ. ν˜X and l˜X represent the mass eigen- → 0 states of sneutrinos and charges sleptons, and χ˜ −, χ˜ the charginos and neutralinos. (Taken from “Oscillating neutrinos and µ e,γ”. J.A. Casas and A. Ibarra. → Nucl.Phys. B618 (2001) 171-204)...... 12 2.2 Dominant diagram in the process l l γ, in the mass-insertion approximation. L˜ i → j i are the slepton doublets in the basis where the gauge interactions and the charged- lepton Yukawa couplings are ﬂavour-diagonal. (Taken from “Oscillating neutrinos and µ e,γ”. J.A. Casas and A. Ibarra. Nucl.Phys. B618 (2001) 171-204). ... 12 → 3.1 Evolution of the scale factor a(t) for different geometries of the universe. (Taken from “Course on Astrophysics”. Balsa Terzic. PHYS 652 -2008)...... 22 3.2 Evolution of the comoving number density of DM with temperature. (Taken from “TASI 2008 Lectures on Dark Matter”. Dan Hooper, FERMILAB-CONF-09-025A.) ...... 32 3.3 Different bounds to DM-nucleon elastic scattering cross-section, coming from direct detection searches. (Taken from “Direct Search for Dark Matter”. Josef Jochum. Talk given at “IFT 2012 Xmas Workshop”)...... 37 136 LIST OF FIGURES