Minimalistic Dark Matter Extensions of the Standard

Dissertation zur Erlangung des Doktorgrades der Fakult¨ at f¨ ur Mathematik und Physik der Albert-Ludwigs-Universit¨ at Freiburg im Breisgau

Minimalistic Dark Matter Extensions of the Standard Model

by Oliver Fischer

A thesis submitted in partial fulfillment for the degree doctor rerum naturalium

June, 2013

Dekan: Prof. Dr. Michael R˚ u˘zi˘cka Referent: Prof. Dr. Jochum van der Bij Koreferent: Prof. Dr. Stefan Antusch M¨ undliche Pr¨ ufung: 17.06.2013

i To my Family: past, present, and future.

Abstract In this thesis, we consider simple extensions of the Standard Model (SM), that do not change its fundamental chiral structure. The extra fields typically are singlets or triplets under the electroweak group. We show that such extensions are able to explain a number of problems, like dark matter, neutrino masses or possible anomalies in the Higgs decays at the LHC. Concerning the DM problem, the main topic of this thesis, we consider the two simplest DM models, namely the real scalar singlet, which we generalize to N identical singlet fields, and the real scalar triplet. Further we combine a singlet and a triplet into two different multicomponent DM models. All of these models can account for the DM abundance and evade the direct detection experiments. The mass of the respective DM candidate tends to be on the TeV scale, but can become as light as 35 GeV in one of the multicomponent models. To explain the neutrino masses, we consider the leptino model, a simple extension of the SM gauge group related to the baryon-minus-lepton number (B-L ). The model adds three righthanded neutrinos, and pairs of fermion singlets with a fractional lepton number, which we call leptinos. One leptino of each pair belongs to the odd representation of a Z2 symmetry. This renormalizable model leads naturally to the inverse seesaw mechanism, for the SM-like neutrino masses. The leptino masses are on the TeV scale. The lightest Z2 -odd leptino is a natural DM candidate, which has to be annihilated resonantly with the Z ′ boson, or the heavy Higgs boson associated with B-L. The combination of astrophysical and collider constraints limit the Z ′ mass from above by 5 TeV, and the B-L -Higgs boson by 3 TeV. The observed baryon asymmetry can be generated in the Resonant Leptogenesis scenario, without fine tuning. The apparent enhancement of the diphoton signal at the LHC compared to the SM case suggests the existence of unaccounted charged particles. Therefore we consider an extended Hill model. This model features a scalar singlet, which mixes with the SM-Higgs doublet, and a fermion triplet, apart from the SM field content. We fit the two scalar bosons to the Higgs-like events at LEP-II at 98 GeV and to the scalar boson at the LHC at 125 GeV. The triplet field, together with the two scalar bosons, can explain the observed enhancement of the diphoton rate for masses below 90 GeV. The model also predicts a reduction of the tree-level decays of the scalar boson at 125 GeV, compared to the SM case. For completeness, also quadruplets and quintuplets are considered, and the odd N -plets can be part of a multicomponent DM scenario.

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Contents Contents

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List of Figures

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List of Tables

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Declaration of Authorship

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Acknowledgements

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1 Introduction 1.1 Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Dark Matter Formalism 2.1 Big Bang Cosmology . . . . . . . . . . . . . . . 2.2 Thermal Relic Abundance . . . . . . . . . . . . 2.2.1 The Global Z2 Symmetry . . . . . . . . 2.2.2 Boltzmann Equation . . . . . . . . . . . 2.2.3 Evaluation of the Relic Abundance . . . 2.3 Direct Dark Matter Detection . . . . . . . . . . 2.3.1 Spin-Independent Cross Section . . . . . 2.3.2 Spin-Dependent Cross Section . . . . . 2.4 Indirect Detection . . . . . . . . . . . . . . . . 2.5 Observational and Experimental Constraints . 2.5.1 Cosmological Parameters from the CMB 2.5.2 Direct Detection Experiments . . . . . . 2.5.3 MicrOMEGAs . . . . . . . . . . . . . .

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3 Scalar Dark Matter 3.1 The O(N ) Singlet Model . . . . . . 3.1.1 The Model . . . . . . . . . 3.1.2 O(N ) Relic Abundance . . 3.1.3 WIMP-Nucleon Scattering . 3.1.4 Astrophysical Limits . . . . 3.1.5 Summary . . . . . . . . . . 3.2 The Triplet Model . . . . . . . . . 3.2.1 The Model . . . . . . . . . 3.2.2 Mass Splitting . . . . . . .

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CONTENTS

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3.2.3 Triplet Relic Abundance . . . 3.2.4 WIMP-Nucleon Scattering . . Multicomponent Scalar Dark Matter: 3.3.1 The Model . . . . . . . . . . 3.3.2 Z2 × Z2 Relic Abundance . . 3.3.3 WIMP-Nucleon Scattering . . Multicomponent Scalar Dark Matter: 3.4.1 The Model . . . . . . . . . . 3.4.2 Mass Mixing . . . . . . . . . 3.4.3 Perturbativity . . . . . . . . 3.4.4 Dark Matter Assessment . . . 3.4.5 Singlet Scenario . . . . . . . 3.4.6 Triplet Scenario . . . . . . . . 3.4.7 Mixed Scenarios . . . . . . . Summary of the Results . . . . . . .

. . . . . . . . . . . . . . . . . . Z2 × Z2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . Z2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Leptonic Dark Matter 4.1 The Leptino Model . . . . . . . . . . . . . . . . . . . 4.1.1 Gauge Sector . . . . . . . . . . . . . . . . . . 4.1.2 Anomaly Cancellations . . . . . . . . . . . . 4.1.3 Scalar Sector . . . . . . . . . . . . . . . . . . 4.1.4 Yukawa Sector . . . . . . . . . . . . . . . . . 4.2 Dark Matter Study . . . . . . . . . . . . . . . . . . . 4.2.1 Z ′ Boson Resonant Annihilation . . . . . . . 4.2.2 Heavy Higgs Boson Resonant Annihilation . . 4.2.3 Direct Detection . . . . . . . . . . . . . . . . 4.2.4 Extension to Nℓ Families . . . . . . . . . . . 4.3 Dark Radiation . . . . . . . . . . . . . . . . . . . . . 4.3.1 Effective Relativistic Degrees of Freedom . . 4.3.2 Decoupling Temperature . . . . . . . . . . . . 4.4 Resonant Leptogenesis . . . . . . . . . . . . . . . . . 4.4.1 Baryon Number Violation . . . . . . . . . . . 4.4.2 Lepton flavour Asymmetry . . . . . . . . . . 4.4.3 Asymmetry Generation in the Leptino Model 4.4.4 Departure from Thermal Equilibrium . . . . 4.4.5 Numerical Evaluation . . . . . . . . . . . . . 4.5 Summary of the Results . . . . . . . . . . . . . . . . 5 Higgs Phenomenology 5.1 Mathematical Formalism . . . . . . . . . . . . 5.1.1 Partial Higgs Decays at Tree-Level . . 5.1.2 Partial Higgs Decay to Diphoton . . . 5.1.3 Recent LHC Results on Higgs Decays 5.2 Particle Constraints on Dark Matter Models . 5.2.1 The O(N ) Model . . . . . . . . . . . . 5.2.2 The Triplet Model . . . . . . . . . . . 5.2.3 The Z2 × Z2 Model . . . . . . . . . .

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CONTENTS

5.3

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5.2.4 The Z2 Model . . . . . . . . . 5.2.5 The Leptino Model . . . . . . . The Hill Model with Triplet Fermions 5.3.1 The Model . . . . . . . . . . . 5.3.2 Experimental Input . . . . . . 5.3.3 The Diphoton Signal . . . . . . 5.3.4 Results . . . . . . . . . . . . . Summary of the Results . . . . . . . .

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6 Discussion 113 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7 Appendix 7.1 Glossary . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Constants and Conventions . . . . . . . . . . . . . 7.3 The Scalar One-Loop Integrals . . . . . . . . . . . 7.3.1 One-Point Function . . . . . . . . . . . . . 7.3.2 Two-point Function . . . . . . . . . . . . . 7.4 Feynman Rules of the Scalar Dark Matter Models 7.4.1 The O(N ) Model . . . . . . . . . . . . . . . 7.4.2 The Triplet Model . . . . . . . . . . . . . . 7.4.3 Z2 × Z2 Model . . . . . . . . . . . . . . . . 7.4.4 Z2 Model . . . . . . . . . . . . . . . . . . . 7.5 Yukawa Matrices For Resonant Leptogenesis . . . . References

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List of Figures 2.1

3.1

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3.5 3.6 3.7

Top: Spin-independent elastic WIMP-nucleon cross-section as function of WIMP mass. The thick blue line represents the XENON 100 limit. It’s confidence level is 90%, taking into account relevant systematic uncertainties. The yellow/green band denote the expected sensitivity of this run. (For further information on this figure see Ref.[38] and references therein). Bottom: 90% CL upper limits on spin-dependent WIMP-nucleon cross section, for hard and soft annihilation Chan- Nels over a range of WIMP masses. Systematic uncertainties are included. See Ref. [39] for further details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . √ Parameter space for O(N) singlet dark matter, namely the coupling ωΦ ( N ) over the mass mΦ . The different colored lines represent N = 1, 5, 10 consistent with the abundance constraint. The uncertainty of ΩDM is too small to show on a logarithmic scale. The coupling ωΦ is largely independent of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-independent Φ-nucleon cross section (σ SI ) in the O(N ) model. The blue line represents exclusion limits from the XENON experiment. The upper limit of a band of a given color represents the values for σ SI as returned by a computation with the program micrOMEGAs. The lower limit of a band of a given color represents the strangeness-corrected value of σ SI defined in eq. (2.92). . . . . . . . . . . . . . . . . . . . . . . . . . . Feynman diagrams contributing the O(N ) singlet self interaction process. The Φr denote the singlet fields of flavour r while h denotes the physical Higgs boson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameter space N over the mass of the real scalar N −plet. N is formally taken as a continuous parameter. The abundance constraint from WMAP is used for the correlation between singlet-Higgs coupling and mass of the Φi fields. The red area represents the parameter space excluded by direct search experiments, while the green area shows the limits of perturbativity. The blue shaded region indicates the exclusion limits given by the elliptic cluster observations. . . . . . . . . . . . . . . . . . . . . . . . . . . Self energy induced by three-point interaction. . . . . . . . . . . . . . . . Self energy induced by four-point interaction. . . . . . . . . . . . . . . . . Dark matter abundance of the triplet model as a function of the mass mΨ . The width of the colored area represents the variation of the triplet-Higgs coupling ωΨ , between zero and one. The horizontal lines represent value of the abundance as observed by WMAP, and its uncertainty. . . . . . . .

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LIST OF FIGURES 3.8

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Leading order one-loop diagrams, representing the dominating effective triplet-nucleon interaction operators. The labels n and n′ denote the nucleus, and a corresponding W –boson induced state, respectively, for instance proton and neutron. . . . . . . . . . . . . . . . . . . . . . . . . Direct search constraints on the triplet-Higgs coupling constant ωΨ . Values for the coupling above the red line result in a spin independent Ψ nucleon scattering cross section, which is excluded by the XENON 100 experiment. The black line in the figure represents the perturbative limit on the coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Left: The triplet fraction ΩΨ of the DM abundance for the two values of the singlet-triplet mixing coupling η = 0, 1 and mΦ = 100 GeV. Right: The singlet fraction ΩΦ of the DM abundance, using the parameters η = ωΦ = ωΨ = 1 and two different triplet masses. The blue line represents the O(N ) model for the same set of parameters and N = 1. . . . . . . . The red curve denotes the value of mΨ so that the enhancement factor f = 1/ωΦ maximizes the singlet-Higgs coupling. The width of the band reflects the variation of the singlet-triplet coupling parameter η between zero and one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass parameter space of the Z2 × Z2 model respecting the abundance and perturbative constraint. Here the direct dependence of maximum mass values on the other parameter has been neglected. This leads to an uncertainty of O(20 %). . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass parameter space for the Z2 × Z2 model. The green area represents the combination of abundance and perturbative constraint on the masses, with the coupling constant η = 1. The gray area represents LEP limits on the branching ratios of the Z boson. The red area represents the XENON 100 exclusion limits, where the strangeness corrected value of the spin independent WIMP nucleon scattering cross section, as given by eq. (2.92), has been used. . . . . . . . . . . . . . . . . . . . . . . . . . . Left: The perturbative upper limit on the mass-splitting parameter c for three different values of δ. Right: The perturbative upper limit on the mixing angle δ for three different values of the mass-splitting parameter c. In both panels the lines are upper exclusion contour lines. . . . . . . Left: Abundance of the singlet scenario as a function of the mass m1 . The red area represents values which are allowed by the constraints on the parameter space. For the green curve, sδ = 0.3 has been used. For m1 = 457 GeV the two lines diverge due to κ > 1, see eq. (3.79). Right: Direct detection constraints on the S1 mass. The blue area represents the XENON 100 exclusion limits, the red line represents the abundance constraint on the coupling of the S1 to the Higgs boson. The strangeness corrected value for σ SI , as defined in eq. (2.92), has been used. . . . . . Abundance of the Z2 model in the triplet scenario. The red area represents values for the abundance, allowed for a maximum variation of The Higgs couplings ω1 , ω2 , the mixing angle sδ and the mass-splitting parameter c. The black line represents the WMAP7 reported abundance constraint, its width reflects the uncertainty. . . . . . . . . . . . . . . . .

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LIST OF FIGURES 3.17 Left: Mass-splitting parameter between the S1 and the S ± fields. The red area represents the WMAP exclusion from the abundance constraint, the gray area represents the perturbative constraint on the Higgs couplings. Right: Upper limits on the absolute value of the three different mass splittings between the S1 , S ± and the S2 fields. . . . . . . . . . . . . . . . 3.18 The relic density as a function of mass m1 for three different scenarios. The width of the blue and green area represents the variation of ω1 , ω2 between 0 and 1. For the singlet scenario c = 1.4 and sδ = 0 have been used, the mixed scenario uses c = 1.2 and 0.35 ≤ sδ ≤ 0.75, excluding the range given in eq. (3.73). The triplet scenario uses sδ = 0.9 and c = 1.1 . 3.19 Allowed range for the mass parameter m1 as a function of the mass splitting parameter c. The red area represents parameter sets being excluded by WMAP, the dark gray area represents the partial Z boson decay width into S ± violating LEP-I limits and the light gray area separates the singlet from the triplet scenario. The two black lines separate the singlet from the triplet scenario. The abundance constraint demands sδ < 0.3 for parameter sets above the upper black line, while below the lower black line sδ > 0.75 is allowed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Parameter space in the relaxed singlet model. The red area represents a DM relic abundance being too small compared to the value reported by WMAP. The blue area represents the abundance and direct search constraints on the model parameters being mutually exclusive. The gray area denotes the decays of the Z boson into S ± violating LEP-I exclusion limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 Parameter space of the relaxed triplet scenario. The red area represents the WMAP exclusion limits, the blue area denotes the combination of the abundance constraint and the direct search exclusion limits on the spin-independent S1 -nucleon scattering cross section (σ SI ). The green area denotes the upper bound from perturbativity and the gray area the suppression of the abundance by the coannihilations of the S ± . The strangeness corrected value for σ SI , as defined in eq. (2.92), has been used. 4.1 4.2

4.3

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The triangle graph giving rise to the anomaly. . . . . . . . . . . . . . . . . DM relic abundance as a function of the candidate mass for MH2 = 800 GeV and MZ ′ = 2 TeV, for two different choices of the scalar mixing angle. For the other parameters, see the text. . . . . . . . . . . . . . . . . Relic density as a function of the DM candidate mass around the Z ′ peak. The lower curve, for gBL = 0.5, is for illustrative purposes only, being already excluded by LEP. . . . . . . . . . . . . . . . . . . . . . . . . Variation of the relic density (at the Z ′ resonance) with g2 for a choice of gBL (see the text for further details), for (left) MZ ′ = 2 TeV and (right) MZ ′ = 3 TeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . √ Z ′ exclusions from CMS data, at s = 7 TeV for the combination of 4.7 fb−1 in the electron channel and 4.9 fb−1 in the muon channel. The dotted black lines refer to the two main benchmark models of this analysis.

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LIST OF FIGURES 4.6

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Existence of a suitable DM candidate: the allowed region is the one above the dashed curves, in the gBL −MZ ′ plane, Left: for g2 = 0. Right: for g2 = g2min (gBL ), which minimizes the Z ′ width, hence allowing for a smaller gBL (and therefore a smaller cross section) per fixed MZ ′ . The red shading combinations are forbidden by LEP (eqs. (4.50)–(4.56), respectively), the black (solid) lines are the LHC exclusion, as in table 4.2. . . . . . . . . . Left: DM relic abundance as a function of the DM mass for MH2 = 800 GeV, for some values of the vev v ′ and sin α = 0.1. The blue shading represents values of the vev for which an allowed DM mass exists, for this MH2 , as taken from the right panel. Right: Allowed region for v ′ for which a DM mass yields the correct relic density, as a function of the heavy Higgs boson mass, for the three values of sin α = 0.05, 0.1, 0.3. The LEP exclusion is as in eqs. (4.50)–(4.56). . . . . . . . . . . . . . . . Spin-independent direct searches for maximum Yukawa couplings yS2 ∼ MS2 /v ′ . Only allowed masses are plotted, from figure 4.7(right), for g2 = g2min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Left: Total Z ′ width for selected gBL values (solid lines refer to Nℓ = 1, dashed lines to Nℓ = 3). Right: percentage variation between the 2 models in the gBL − g2 plane, for MZ ′ = 2.0 TeV. For the heavy neutrino masses, see the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Left: n ≡ Nℓ = 3 is not shown as always disallowed when Higgs resonant annihilation is considered. Here, sin α = 0.1. Right: Allowed parameter range for resonant Z ′ annihilation for n ≡ Nℓ = 2, 3 families of leptinos. The curves are for g2 = g2min (gBL ). . . . . . . . . . . . . . . . . . . . . . Abundance of S2 DM particles for Nℓ = 1 depending on the Z ′ boson mass. In this graph the following conditions are used: MS2 = MZ′ /2 (resonant annihilation), g2 = 4.6gBL (minimization of decoupling temperature of the right-handed neutrinos) and gBL = MZ ′ /6 TeV (LEP limit). The horizontal line denotes the WMAP9 reported value for the DM abundance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic representation of the energy dependence of the gauge configurations as a function of the Chern-Simons number. Sphalerons correspond to the maxima of the curve. . . . . . . . . . . . . . . . . . . . . . . . . . Diagrams generating the CP asymmetry ǫ in the decays of νh . A summation over the internal flavour index β is implied. The X represents a Majorana mass insertion. The loop diagrams are lepton-flavour and lepton-number violating. . . . . . . . . . . . . . . . . . . . . . . . . . . . Feynman diagrams corresponding to processes relevant for the Boltzmann equations in leptino model Leptogenesis. The particles are: heavy neutrino νh νh′ ), left-handed lepton ℓ of arbitrary flavour, physical Higgs H, generic SM (anti) fermion (f¯) f and the B − L–gauge boson Z ′ . . . . . . Magnitudes for the decay parameter and the Z ′ scattering parameter as defined in the text, for the range of temperature z relevant for Leptogenesis in the leptino model. . . . . . . . . . . . . . . . . . . . . . . . . . . The black line represents the baryon asymmetry, generated by the lightest pair of heavy neutrinos in the leptino model, within the Resonant Leptogenesis scenario. The red line denotes the value for the asymmetry as inferred from WMAP and big bang Nucleosynthesis. . . . . . . . . . . .

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LIST OF FIGURES 5.1

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5.6

5.7

One-loop diagram contributing to the partial decay width with of the Higgs boson into two photons. The particle in the loop carries electric charge and couples to the Higgs boson. . . . . . . . . . . . . . . . . . . . Left: Partial decay width of the SM Higgs boson into a pair of real scalar singlet DM fields Φ. The abundance constraint has been used to correlate mass and coupling. The horizontal line represents the exclusion constraints from the total Higgs width. Right: Maximum value for N so that the partial Higgs decay width is at most 1 MeV. . . . . . . . . . . . Left: Parameter space of the scalar triplet field. The gray area is excluded due to ΓΨ ≫ O(1 MeV). The red area denotes an enhancement of the diphoton signal at the LHC compatible with the preliminary results: Rγ = (1.7 ± 0.3)RγSM . The triplet-Higgs coupling ωΨ is taken to be negative. Right: The red area represents the enhancement of the diphoton rate, respecting the abundance constraint and with −1 < ωΨ < 0. . . . . . . . The red line denotes the allowed abundance for the triplet fraction in the Z2 ×Z2 model as a function of the singlet mass. The gray area is excluded due to the constraint from the total Higgs width. . . . . . . . . . . . . . The background confidence 1 − CLb as a function of the test mass mH . Full curve: observation; dashed curve: expected background confidence; dash-dotted line: the position of the minimum of the median expectation of 1−CLb for the signal plus background hypothesis, when the signal mass indicated on the abscissa is tested. The horizontal solid lines indicate the levels for 2σ and 3σ deviations from the background hypothesis. For details, see Ref. [109]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Allowed 1-σ range in the (mass–coupling) plane for one triplet to match the enhancement in the Higgs-to-diphoton signal. The red (solid) contour lines are for s2α = 0.1, while the black (dashed) line represents the variation of s2α with ±0.05, as from LEP [109]. The allowed region is between the lines. The shadings correspond to excluded regions. . . . . . . . . . . . . Allowed 1-σ range in the (mass–coupling) plane for (left) one quadruplet and for (right) one quintuplet, to match the enhancement in the Higgs to diphoton signal, for s2α = 0.1. The black (dashed) lines represent a variation of s2α of ±0.05, as from LEP [109]. . . . . . . . . . . . . . . . .

xiii

. 98

. 100

. 101

. 103

. 107

. 109

. 111

List of Tables 4.1 4.2 4.3

Quantum number and Z2 parity assignments for chiral fermion and scalar fields. For a definition of the respective particle, see text. . . . . . . . . . 65 95% C.L. exclusions for the benchmark models of interest. Couplings smaller than those in the table are allowed. . . . . . . . . . . . . . . . . . 71 List of particles in thermal equilibrium, their B-L -number and the according relativistic degrees of freedom for the temperature T in GeV. . . . 82

5.1

Signal strength Ri of the Higgs boson observed at the LHC with respect to the SM expectation, and their average (without correlations), at Mh = 125 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.1 7.2 7.3

Feynman rules of the O(N ) model. . . . . . . . . . . Feynman rules of the triplet model. . . . . . . . . . . Feynman rules of singlet-triplet mixing term in the Z2 × Z2 model. . . . . . . . . . . . . . . . . . . . . . Feynman rules of the Z2 model, in unitary gauge. . .

7.4

xv

. . . . . . . . . . . . . . Lagrangian . . . . . . . . . . . . . .

. . . . . . . . of the . . . . . . . .

. 121 . 122 . 122 . 124

Declaration of Authorship Ich, Oliver Fischer, erkläre hiermit, dass ich die vorliegende Arbeit mit dem Titel Minimalistic Dark Matter Extensions of the Standard Model ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Die aus anderen Quellen direkt oder indirekt u ¨bernommenen Daten und Konzepte sind unter Angabe der Quelle gekennzeichnet. Insbesondere habe ich hierf¨ ur nicht die entgeltliche Hilfe von Vermittlungs- beziehungsweise Beratungsdiensten (Promotionsberater/-beraterinnen oder anderer Personen) in Anspruch genommen. Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ähnlicher Form einer Pr¨ ufungsbeh¨ orde vorgelegt.

xvii

Acknowledgements My first Thank You is directed at Prof. Dr. J. J. van der Bij, for providing me with the opportunity to work in his group, and for offering to me the very interesting subjects of study, which led to the writing of this thesis. I want to direct a special Thank You to Dr. L. Basso, for his invaluable collaboration, his careful and repetitive reading of countless manuscripts, and his helpful advice. Also a big Thank You goes to V. Bertone, A. Banfi und S. Moser, respectively, for valuable feedback on different stages of the manuscript. It is said, behind each successful man stands a woman, which is true in my case, provided the creation of this thesis is considered a success. For never-ending support, patience, and love, i thank you, Lezanne. To all others who supported me in all kinds of ways, and whom have not been mentioned above, i thank you as well. The work, which gave rise to this thesis, was supported by the Deutsche Forschungsgemeinschaft through the Research Training Group grant GRK 1102 Physics at Hadron Accelerators and by the Bundesministerium f¨ ur Bildung und Forschung within the F¨ orderschwerpunkt Elementary Particle Physics.

xix

Chapter 1

Introduction One of the most astounding revelations of the twentieth century for our understanding of the Universe was, that ordinary baryonic matter is not the dominant form of matter. Today we know that the up to now unobservable dark matter (DM) is about five times more abundant and has yet to be detected in the laboratory. First hints about the existence of non luminous matter came from the angular velocities of stars in the Milky Way. Those velocities should have been sufficient for the stars to escape the galaxy, but they did not [1]. Later F. Zwicky found that the luminous mass of the Coma cluster was insufficient to gravitationally bind the residual galaxies [2]. Vera Rubin and collaborators found that rotation curves of galaxies flatten with larger distances to the center, which is inconsistent with Newtonian gravity and the luminous mass profile [3]. A more advanced probe for the distribution of DM is gravitational lensing, which uses the light-bending properties of large mass densities. The gravitational pull of a large foreground mass bends the light rays from a more distant background object toward the observer. This can result in an Einstein Ring image of the lensed object, where the radius of the ring allows to infer the amount of lensing mass. In general, the lensing masses appear to be much larger compared to the luminous mass of the associated objects. Another astrophysical probe on DM is the bullet cluster, which is the result of a collision between a sub-cluster and the large cluster 1E 0657-56. It shows a clear separation of the baryonic mass fraction (residing in the hot gas between the galaxies) and the gravitational mass. The latter appears completely undisturbed by the strong lag of the large amount of gas, which yields limits on the DM interaction properties and its amount compared to that of the gas [4]. The most important probe of the properties of the early universe is the cosmic microwave background (CMB). It originates from the time when neutral atoms started to form from the primordial plasma. This moment, when the universe became transparent to photons, is called the last scattering. The photons, from this point on propagating freely in all directions, make up the CMB which has been discovered by Penzias and Wilson in 1964 [5]. Its perfect blackbody spectrum was found in 1989 by the Cosmic Background Explorer (COBE) experiment [6]. The anisotropies in the spectrum are 1

2

1. INTRODUCTION

due to the energy loss of photons escaping from dense, primordial regions, and acoustic oscillations of the primordial plasma. Therefore indications of the initial density perturbations that allowed for the formation of early gravitational wells as well as dynamics of the photon-baryon fluid can be inferred from the CMB. The Wilkinson Microwave Anisotropy Probe (WMAP) was launched in 2001 with the mission to more precisely measure the anisotropies in the CMB. Recently the nineyear results have been released [7]. An essential observation from these results is, that total and baryonic mass densities are different, concluding that baryonic matter is not the only form of matter in the universe. A discrepancy existed between the prediction of cosmological N-body simulations [8], and astronomical observations, regarding the number of satellite galaxies. The fact, that astronomers observed an order of magnitude less satellite galaxies compared to the numerical predictions, was called the ”missing satellite problem”. However, a study from a survey of the Hubble Space Telescope of the Perseus Cluster, which is located about 250 million light years from Earth, showed that small, dwarf spheroidal galaxies are stable, while larger galaxies are being torn apart by the tidal forces of the cluster potential [9]. Interpreting this situation, as the dwarf galaxies being stabilized by considerable cores of dark matter, leads to the conclusion that massive dwarf galaxies need not necessarily be very luminous, which alleviates the missing satellite problem. For a review of the DM situation, see for instance Ref. [10]. There are several interpretations of the DM situation. One is to assume it to be formed by massive compact halo objects (MACHOs), for instance brown or white dwarfs, planets and dust. Since these objects are made up of baryonic matter, they violate WMAP observed values on the total baryonic energy density of the universe [7]. Also recent micro lensing analyses of the Large Magellanic Cloud have ruled out MACHO dominated halos [11]. Another way to interpret the DM problem, is to assume a modification of Newtonian dynamics (MOND) [12]. This leads to a modification of General Relativity (GR), and aims in particular at explaining the rotation curves. Some discrepancy between prediction and observation remains, however [13], and the effect of a modification of such a fundamental theory like GR on the other branches of physics is not well under control. The interpretation of DM as representing a type of elementary particle, is the most studied and most successful. The DM particle has to meet certain conditions: first, it must not carry an electric charge, as it would not be dark otherwise. Second, it has to be massive, in order to interact gravitationally with the visible baryonic matter. Third, it must be stable on cosmological time scales, which implies either a (possibly new) symmetry, or a lifetime which is larger than the age of the universe. If it is a particle, which is not accounted for by the Standard Model of particle physics (SM), a fourth condition is given by collider constraints, namely its non-observation, especially by the Large Electron Collider (LEP) and the Large Hadron Collider (LHC).

3 To address the question, whether the SM contains a particle which meets the above conditions, we consider its structure: The SM describes the known fundamental forces except gravity, and contains sixteen confirmed particles, seven of which were predicted prior to their experimental discovery. Recently a new particle has been discovered, which appears to be a scalar boson with a mass of 125 GeV. It is compatible with the Higgs boson, which is responsible for the SM particle masses. The field content of the SM consists of six quarks, six leptons and four force carriers and the mass-generating Higgs boson. Quarks and leptons are fermions, coming in three generations, the force carriers are the gauge bosons. The model has been thoroughly probed up to energies on the TeV scale, and has led to spectacular results such as the precision measurement of the anomalous magnetic moment of the electron. The SM is confirmed at the LHC, through the absence of unexpected phenomena, by the four experiments ATLAS, CMS, Alice and LHCb [14, 15, 16, 17]. A small discrepancy remains with the Higgs to diphoton decay rate [18]. The only SM candidates, meeting the DM conditions from above, are the neutrinos. Although massless in the mathematical formulation of the SM, the observation of neutrino flavor oscillations indicates, that at least two of them are massive [19]. This makes the neutrinos a candidate for a weakly interacting, massive and stable DM. However, primordial neutrinos were relativistic at the time of structure formation, which leads to a conflict of predicted and observed large-scale structure [20]. Furthermore, the upper bound on the neutrino masses, renders their contribution to the energy density of the universe negligibly small [21]. Besides the DM riddle, the universe has more mysteries to offer. In the primordial plasma, all symmetries are exact, meaning the asymmetry parameters are exactly zero. In this case, the Big Bang would have produced equal numbers of baryons and anti baryons, which would have annihilated completely into radiation. Our very existence implies, that the baryon asymmetry is non-zero, which requires a violation of the baryon symmetry in the early universe. The baryon asymmetry can be measured via the energy density of the baryons and that of the photons, which are given to high precision by WMAP [7]. Big Bang Nucleosynthesis (BBN), which is a period from a few seconds to a few minutes after the Big Bang, is another, independent measure on the baryon asymmetry. With nuclear physics and known reaction rates, the elemental abundances of the light elements of the early Universe can be calculated [22]. The thus inferred value of the baryon asymmetry is in agreement with the baryon abundance from WMAP analyses. In 1967, Sakharov formulated three famous conditions, which are necessary, to generate a baryon asymmetry: a baryon number violating process must exist, it must be violating the charge parity symmetry (CP ), and the associated process has to be out of chemical equilibrium with the primordial plasma [23]. In the SM, baryon number violation is given by the triangle anomaly, and Sphaleron processes can change baryon- and lepton-number [24, 25]. The weak interactions violate CP , and at the electroweak phase

4

1. INTRODUCTION

transition a departure from thermal equilibrium is present [26, 27]. However, though formally the Sakharov conditions can be met within the SM, electroweak baryogenesis requires the Higgs mass to be less than 80 GeV [26], which is excluded by the recent LHC results [14, 15]. Yet another puzzle is given with the relativistic energy content of the early universe. At the time of the last scattering, which is visible today via the CMB, the relativistic degrees of freedom (RDF) can be inferred, for instance via WMAP measurements. Particles with masses below ∼ 1 eV were relativistic at this time, which leads to a contribution of the SM photons and neutrinos.

Recent fits of Cosmological models to CMB yield an amount of RDF, that is larger compared to the value obtained from the theoretical prediction [28], based on the SM particle content [7, 29]. The unaccounted relativistic energy is as much as the contribution of one extra light neutrino, which can be interpreted as the existence BSM physics. The absence of a viable DM candidate in the SM, the RDF discrepancy and the inability to generate the observed baryon asymmetry, BSM physics is necessary. Many theories have been created over the last decades, of which the most popular DM models shall be listed below. For a review on the respective model, and other models, see Ref. [30] and references therein. • Most investigated among those theories is Supersymmetry (SUSY), which presumes a symmetry between bosons and fermions, creating a superpartner for each

SM particle. The non-observation of SUSY particles up to now leads to the conclusion that the superpartners live on a different mass scale than the SM particles. SUSY often also invokes R-parity, which is a symmetry that prevents proton decay and stabilizes the lightest supersymmetric particle as a DM candidate. • Compactified extra dimensions lead to a quantization of momenta in Fourier modes

∼ 1/R2 , where R is the compactification radius. The set of Fourier modes, called

Kaluza-Klein states, appears for each particle species. If translational invari-

ance along the fifth dimension is postulated, then a new discrete symmetry called Kaluza-Klein parity exists and the Lightest Kaluza-Klein particle (LKP) is stable and act as DM. In most models the LKP is the first excitation of the photon. • The axion is a pseudo-Goldstone boson corresponding to a broken global symmetry. This broken symmetry was introduced By Roberto Peccei and Helen Quinn in order to solve the strong CP problem. It is a pseudo Goldstone boson because the corresponding symmetry is broken through the chiral anomaly. Realized in the singlet sector of the theory, one can arrange for very small masses of the axion of the order of µeV. Axions can act as dark matter, but one needs more complicated processes than ordinary thermal freeze-out from equilibrium.

5 The above models all start with a motivation for the respective theory, which is not directly related to the dark matter problem, but nevertheless leads to a natural inclusion of dark matter candidates. Also the models, presented and discussed in this thesis have external motivations. The first example is given by the model with scalar singlets. The motivation is based on simplicity; it is the simplest possible extension of the SM, and it should be considered a benchmark model. This model is subsequently extended with a scalar triplet to make a two-component form of dark matter. The combination of a singlet and a triplet is not arbitrary. It arises quite naturally in a model of unification, coming from the group SU (5). The second model to be considered is the so-called leptino model. The main purpose of this model is to give a natural explanation for the smallness of the neutrino masses. This is done within a modified form of the seesaw mechanism, known in the literature as the inverse seesaw. The advantage of this modification is, that the Dirac Yukawa couplings of the neutrinos are comparable to the quark and lepton Yukawa couplings. The seesaw does not come from the Planck scale, but from a more accessible scale. The model that is presented here is the first example, where the inverse seesaw appears automatically due to the choice of representations, whereby there is a relation to the existence of an extra Z ′ -boson. The last model is also motivated from simplicity and a possible connection with SU (5). However, its main motivation is the explanation of some deviations in the Higgs signal at the LHC from the expectation in the simplest standard model. Altogether the models show, that with simple and rather minimal extensions of the SM, a large amount of phenomenology can be explained. It is therefore not necessary, to impose large extensions like Supersymmetry or Kaluza-Klein models.

1.1

Structure of this Thesis

The work presented in this thesis has been partially published in Refs. [31, 32, 33]. In chapter 2 the theoretical framework for a quantitative evaluation of DM models is defined. Section 2.1 is a very brief review on Big Bang Cosmology. In section 2.2 the computation of the relic abundance, of a weakly interacting massive particle (WIMP), is described. Direct DM searches are discussed in section 2.3, and the spin-dependent and -independent WIMP-nucleon scattering cross sections, respectively, are presented in two separate paragraphs. The indirect search is described in section 2.4, which is done for completeness, since the models investigated in this thesis do not contribute to the cosmic fluxes in an observable way. Section 2.5 presents recent experimental constraints on the DM abundance, the direct search exclusion limits on the WIMP-nucleon scattering cross section, and the numerical tool micrOMEGAs, which was used for the numerical studies.

6

1. INTRODUCTION

Chapter 3 discusses four different scalar DM models with different phenomenology. In the first section, the O(N ) model is discussed, which adds a Z2 symmetry and N real scalar singlet fields to the matter content of the SM. The constraint on the model parameters from the WMAP observation, and direct search exclusion limits, respectively, is investigated In section 3.2 the triplet model is discussed, which adds a Z2 symmetry and a real SU (2)L triplet field to the SM. The neutral triplet component is 168 MeV lighter than the charged ones due to radiative corrections, which makes it literally a WIMP. The abundance constraint on the model parameters is investigated, along with the direct search constraints on the triplet-Higgs coupling parameter. The Z2 × Z2 model is discussed in section 3.3. This multicomponent DM model

adds two different Z2 symmetries to the SM, and a real scalar singlet plus a real scalar SU (2)L triplet field to its matter content. The singlet and triplet fields reside in the odd representation of different Z2 symmetries. The abundance and direct search constraints on the model parameters are investigated, along with LEP-I constraints on the charged triplet components. Section 3.4 presents the Z2 model, which adds one global Z2 symmetry, and the real scalar singlet and triplet fields from sections 3.1 and 3.2 to the SM. The singlet and triplet fields both reside in the odd representation of the Z2 symmetry. The mass mixing due to spontaneous symmetry breaking is discussed, and the constraints on the model parameters from the abundance constraint and direct searches are investigated, and LEP-I limits on the Z boson are taken into account. Chapter 4 discusses the leptino model. The model is described in section 4.1 in four paragraphs. The first paragraph treats the gauge sector, the second sector evaluates the gauge anomalies. The third paragraph presents the scalar sector, and the Yukawa sector is discussed in the fourth paragraph. The model features a global Z2 symmetry and the following additional fields: a Z ′ boson and a heavy Higgs boson, both associated to the baryon-lepton number, three right-handed neutrinos, and a number of fermions with fractional baryon-lepton number, called leptinos. Due one of the leptinos being in the odd representation of the Z2 symmetry, it becomes a cold DM candidate. In section 4.2 a quantitative study on the leptino DM abundance and the leptino-nucleon scattering cross section is performed, and the abundance and direct search constraints on the model parameters are investigated. The right-handed neutrinos (νR ) naturally contribute to the relativistic energy content of the early Universe. The assumption that the νR explain the observed discrepancy, results in constraints which are discussed in section 4.3. The leptino model employs the inverse seesaw mechanism to generate the light neutrino masses, which also results in two quasi mass-degenerate heavy neutrinos, with masses on the TeV scale. Such particles are natural candidates to provide for Resonant

7 Leptogenesis, which is discussed within the leptino model in section 4.4. In the first section of chapter 5 we review partial Higgs decay widths, at tree level into scalars, fermions and bosons, and at one-loop level into diphotons. Recent LHC results on the partial Higgs decay widths are presented in the last paragraph of this section The constraints of the partial Higgs decay widths are used in section 5.2 to constrain the DM models discussed in chapter 3. In section 5.3 the Hill model, as first presented in Ref. [34], which features two Higgslike bosons instead of a single Higgs boson, is extended with a fermionic isospin N -plet. The experimental results from LHC and LEP-II yield the masses and the scalar mixing angle of the two Higgs-like bosons. With this input, the contribution of the N -plet to the observed Higgs to diphoton rate is discussed. The results, obtained in chapters 3, 4 and 5, are summarized and discussed in the first section of chapter 6, the conclusions are presented in the second and last section.

Chapter 2

Dark Matter Formalism In this chapter, a brief review of the theoretical foundation, and the mathematical formalism of selected properties, of WIMP-like cold dark matter is presented. In section 2.1, the basic principles of Big Bang Cosmology [35] are sketched, the Cosmological Principle is defined and some basic parameters are introduced. The mathematical derivation of the DM abundance is reviewed in section 2.2 [36]. The fundamental assumptions are, that the DM is a thermal relic of the early Universe and that the DM particle is massive and weakly interacting. Section 2.3 presents the mathematical framework on DM-nucleon scattering processes [37]. In two separate paragraphs, the phenomenologically different spin-dependent and spin-independent interaction cross sections are presented, respectively. The indirect detection of dark matter in cosmic fluxes is outlined in section 2.4 [36]. In section 2.5 recent experimental constraints and observations concerning the cosmological parameters are presented in two subsections. The first subsection displays the recent results from the nine-year analysis of WMAP data [7], which are relevant for this thesis. The second subsection summarises the most stringent direct dark matter detection constraints, coming from the XENON 100 [38] and Ice Cube [39] experiment, respectively. In the last subsection, the numerical tool micrOMEGAs [40, 41] is described, which is used for the numerical evaluation of the DM relic abundance throughout this thesis.

2.1

Big Bang Cosmology

Isotropy in cosmology is defined in the following way: the value of a cosmological observable has to be independent of the direction of observation. Homogeneity is defined through the absence of a preferred reference frame. The cosmological principle states that the Universe is homogeneous and isotropic. From this statement follows, that the Universe can be described by the adiabatic expansion from an initial, hot and dense state. Roughly speaking, this implies that every comoving observer in the cosmic fluid has the same history. The metric of space-time on large distance scales can be written 9

10

2. DARK MATTER FORMALISM

in the Friedmann-Robertson-Walker (FRW) form: 2

2

ds = dt − a(t)

2

dr2 2 2 2 2 + r dθ + sin θdφ , 1 − kr2

(2.1)

where the constant k determines the topology of the spatial sections. For a flat universe, k = 0, which we shall assume in the following. In this case, we can without loss of generality take the scale factor a(t0 ) = 1 at the present time t0 . The coordinates r, θ, φ are comoving spherical coordinates. For a homogeneous and isotropic Universe with the cosmological constant equal to zero, the Einstein equations reduce to the FRW equations: 8πG ρ, 3 4πG = s (ρ + p) , 3

H2 = a ¨ a

(2.2) (2.3)

where H = a/a ˙ is the Hubble constant, p and ρ denote the pressure and energy density, respectively, and G is Newton’s constant. These equations can be combined to yield the continuity equation ρ˙ = −3H(p + ρ) .

(2.4)

The equation of state of a cosmic fluid i is described by a number wi defined by pi = w i ρ i ,

(2.5)

ρ i = ωi p i ,

(2.6)

with the equation-of-state parameter ωi being a constant, and i indicating the type of cosmic fluid under consideration. Cosmic fluids which are usually considered are radiation (p = 0), non-relativistic matter (p = 3), and the cosmological constant (p = −1). Further, since the expansion of the Universe is adiabatic, we have d (sa3 ) = 0 , dt

(2.7)

with s being the entropy density, which is a conserved quantity. The Friedmann equations relate the expansion rate of the Universe to its matter and energy content and enforce energy and entropy conservation per comoving volume. The expansion rate of the universe as a function of time can be determined by specifying the matter or energy content. According to eq. (2.6), this relation between the cosmic temperature and time is given by a(t) ∝

t t0

2

3(1 + ω) ,

(2.8)

where t0 denotes the present time, such that a(t0 ) = 1. Non-relativistic matter, indicated with the index m, has negligible pressure, ωm = 0 and thus am ∝ t2/3 . For radiation and

11 relativistic matter, indicated by the index r, the equation-of-state parameter is ωr = 1/3, which yields ar ∝ t1/2 .

The scale factor of the Universe follows the dominant contribution. In the early

Universe, radiation was dominating the energy content, which implies that the scale factor evolved proportional to t1/2 . The energy content was given by a relativistic gas or plasma of primordially produced particles. The number density, energy density, and pressure of a species labeled i, are given by Z gi fi (p)d3 p , (2π)3 Z gi E(p)fi (p)d3 p , (2π)3 Z gi |p|2 fi (p)d3 p , (2π)3 3E(p)

ni = ρi = pi =

(2.9) (2.10) (2.11)

where gi is the number of spin states of the particles. In thermal equilibrium at a temperature T the particles are either in Fermi-Dirac or Bose-Einstein distributions, f (p) =

1 eE(p)/T

±1

,

(2.12)

where the plus sign denotes half integer spin and the minus sign integer spin, respectively. This definition of the particle distributions, yields the important relation between the temperature of the particle species i and its number density: ζ(3) gi T 3 , π2 3 r n , = 4 B mi T 3/2 −mi /T e , = gi 2π

nrB = nrF nnr M

(2.13)

where the indices B, F, M of the number density n refer to bosonic, fermionic and general matter distributions, respectively, and the superscripts r, nr denotes relativistic, and non-relativistic matter, respectively. The above correlation of fermionic and bosonic number densities leads to the definition of the effective number of relativistic degrees of freedom gef f =

X

bosons

gi

Ti T

4

4 Ti 7 X + . gi 8 T

(2.14)

fermions

The temperature T is defined by the temperature of the primordial plasma. If only particles in thermal equilibrium with the plasma are considered, all the Ti are equal to T.

12

2. DARK MATTER FORMALISM The total energy density of the Universe, which resides in relativistic species, is

obtained by adding the contributions of each species ρ=

π2 gef f T 4 . 30

(2.15)

With the definitions from above, the entropy density of relativistic species can be expressed as s=

2π ′ g T3 , 45 ef f

(2.16)

′ 4 3 where gef f is defined by substituting (Ti /T ) with (Ti /T ) in eq. (2.14). The entropy

density is a very important quantity for cosmological considerations. It is observable today via the cosmic microwave background, and it is directly related to the energy contribution of relativistic neutrinos to the early Universe, and the evaluation of thermal relic abundances.

2.2

Thermal Relic Abundance

In this section, the mathematical foundation for the computation of the abundance of a thermal relic is presented. In the first subsection, the Z2 symmetry is motivated, which is used by all the models in thesis with a DM candidate to provide stability on cosmological time scales. The second subsection motivates the Boltzmann equation of the number density of the thermal relic, while the third subsection evaluates the relic abundance. For an excellent review on the dark matter formalism, see Ref. [42].

2.2.1

The Global Z2 Symmetry

The possibly most important condition, that any DM model has to meet, is the absence of decay channels of the DM candidate into SM fields. Therefore the DM models which are studied in this thesis have a global Z2 symmetry. This symmetry has two inequivalent irreducible representations, which we shall label as “even” and “odd”. The SM fields are assigned to the even representation, so that the SM Lagrangian is not affected by this additional global symmetry. Additional fields, which are supposed to be DM candidates, are assigned to the odd representation. Invariance of the model Lagrangian with respect to the global Z2 symmetry implies, that the DM fields can only appear in even powers. This automatically prevents terms which would allow the lightest Z2 -odd field to decay into SM fields.

2.2.2

Boltzmann Equation

A particle species, denoted by the index i is in chemical (or thermal) equilibrium with a number of species, denoted by the index j, if the processes of the form fi f¯i ↔

X j

fj f¯j .

(2.17)

13 are identical in both directions. In this case, the number of particles of species i in a comoving volume Ni = a3 ni is given by its equilibrium value neq i (T ). Once species i decouples from thermal equilibrium, its number density ni , as given by eq. 2.13, is depleted with a rate proportional to the total annihilation cross section and to the product of the number density of i and ¯i: X j

σij |v|n2i ,

(2.18)

where σij = σi¯i→j¯j is the annihilation cross section of species i into species j, and the Møller velocity is given by p (pi · p¯i )2 − mi m¯i . |v| = Ei E¯i

(2.19)

Notice, that eq. 2.18 includes the assumption, ni = n¯i . The inverse process creates particle species i (and ¯i) with a rate proportional to (neq )2 . The departure from equilibrium i

of the species i can be expressed by the Boltzmann equation dni 2 + 3Hni = − hσA |v|i n2i − (neq i ) dt

(2.20)

where the term proportional to H expresses the dilution coming from the expansion of the universe. hσA |v|i is the thermally averaged total annihilation cross section hσA |v|i =

1 2 (neq i )

Z

gi

dp¯3i eq eq dp3i g f f σA |v| ¯ i (2π)3 (2π)3 i ¯i

(2.21)

with g being the internal degrees of freedom, p the three-momentum, f eq the distribution function as defined in eq. 2.12, mi (Ei ) the mass (energy) of particle species i and σA the total annihilation cross section into all particles that are kinematically accessible.

2.2.3

Evaluation of the Relic Abundance

For eq. (2.20), two asymptotic regimes can be identified: for very high temperatures, T ≫ mi , the number density ni is given by the equilibrium density, and therefore

suppressed exponentially due to the Hubble expansion: neq i

= gi

mi T 2π

3/2

e−mi /T .

(2.22)

For small temperatures, T ≪ mi , the equilibrium density can be neglected, thus the

terms 3Hni and hσA |v|i n2i deplete the number density. For sufficiently small ni , the an-

nihilation process becomes negligible compared with the expansion rate. The comoving number density for species i is fixed, which is often referred to as freeze out. We define the quantity Y =

n , s

(2.23)

14


where s is the entropy density, as in eq. (2.16). Omitting the trivial expansion term, the Boltzmann equations from eq. (2.20) can be expressed as Y˙ = − hσA |v|i s Y 2 − (Y eq )2 .

(2.24)

From this the following evolution equation can be derived dY =− dx

r

πgef f m hσA |v|i s Y 2 − (Y eq )2 , 2 45G x

(2.25)

where gef f are the effective degrees of freedom of the thermal plasma, as defined in eq. 2.14. Similarly as in the case of eq. (2.20), two asymptotic regimes are present: g 3/2 e−x hef f x g hef f

Y eq ∝

Y eq ∝

x ≫ 1,

(2.26)

x ≪ 1.

(2.27)

Due to its exponential suppression, Y eq can be neglected for x ≫ 1. Thus the equation following equation is obtained:

1 1 = + Y∞ Y (Xf )

r

π m 45G

Z

∞ Xf

dx

hσA |v|i q gef f (x) , x2

(2.28)

where Xf is the freeze out temperature. Neglecting 1/Y (Xf ) and assuming that hσA |v|i is constant, eq. (2.28) yields

Y∞ ≈

s

Solving iteratively d ln (Y eq ) = dT

Xf 45G 1 . πgef f (Xf ) m hσA |v|i

r

πgef f hσA |v|i Y eq δ(δ + 2) , 45G

(2.29)

(2.30)

with δ being a numerical, small, and constant number, the freeze out temperature can be obtained. Typical values for Xf , obtained for the models discussed in chapters 3 and 4 are 20 to 40. The energy density of species i can now be expressed as ρi = mi ni = mi Y∞ s0 ,

(2.31)

where s0 is the entropy density of the CMB photons, given by their temperature T0 as s0 = 3.915

2π 3 T . 45 0

(2.32)

Altogether, the energy density of a thermally decoupled massive particle is given by the relic density: Ωi h2 =

Xf 3 × 10−38 cm2 s0 Y∞ mi h2 p ≃ . ρcrit hσA |v|i gef f (Xf )

(2.33)

15

2.3

Direct Dark Matter Detection

If the Milky Way DM halo is composed of WIMPs, the WIMP flux on the Earth can be estimated to be of the order 105 (100 GeV/mDM ) cm−2 s−1 . Direct detection experiments aim to measure the rate R and energies ER of the nuclear recoils. The differential event rate, usually expressed in terms of counts/kg/day/keV for a WIMP with mass mDM and a nucleus with mass mN is given by ρ0 dR = dER mN mDM

Z

∞

vf (v)

vmin

dσI (v, ER )dv , dER

where ρ0 = 0.389 GeV/cm3 is the local WIMP density,

dσI dER (v, ER )

(2.34) is the differential

cross section for the WIMP-nucleus elastic scattering and f (v) is the WIMP velocity distribution in the detector frame. In the galactic rest frame the velocity distribution is given by fgal (~v ) =

(

N e−v

2 /¯ v2

0

v < vesc v > vesc

,

(2.35)

with the average velocity of the Solar system |¯ v | = 220 km/s and the escape velocity of

the halo being |vesc | = 550 km/s. In the Earth rest frame the velocity distribution is then

f⊕ (~v , t) = fgal (~v + ~v⊙ + ~v⊕ (t))

(2.36)

with the velocity of the sun ~v⊙ = (0, 220, 0)+(10, 13, 7) km/s, the periodic Earth velocity

|~v⊕ | = 30 km/s.

Assuming that the mass of the WIMP particle is of the order of the nucleus mass

and the interaction is non-relativistic, the recoil energy of the nucleon is given by ER =

2µ2 v 2 cos2 θlab ∼ 10 keV , mN

(2.37)

with the reduced mass µ = mDM mN /(mDM + mN ) and the scattering angle in the lab frame θlab . This formula yields a minimum velocity given by vmin =

s

mN ER , 2µ2

(2.38)

while the maximum velocity of the DM flux is given by the local escape velocity vesc . The total event rate (per kilogram per day) is obtained by integrating the differential event rate over all the possible recoil energies: R=

Z

∞ ET

ρ0 dER mN mDM

Z

∞ vmin

vf (v)

dσI (v, ER )f v, dER

(2.39)

where ET is the threshold energy of the detector, the smallest detectable recoil energy.

16

2. DARK MATTER FORMALISM In general the cross section can be separated into a spin-independent (scalar) and a

spin-dependent contribution: dσI = dER

dσI dER

+

SI

dσI dER

(2.40)

SD

The full differential cross section can be expressed as dσI mN 2 2 (ER ) + σ0SD FSD (ER ) , = 2 2 σ0SI FSI dER 2µ v SI/SD

where σ0

(2.41)

are spin-independent/-dependent cross sections at zero momentum trans-

fer and the F (ER ) are momentum dependence encoding form factors. The contributions to the spin-dependent cross section arise from couplings of the WIMP field to the quark axial current q¯γµ γ5 q.

2.3.1

Spin-Independent Cross Section

Consider an effective scalar four-point operator of the form ¯N , ON = f N χχ ¯ N

(2.42)

where χ denotes the DM particle with mass mDM , N = n, p denotes the nucleon with n, p being a neutron and proton, respectively, and the effective scalar coupling between χ and N given by f N . The differential cross section, as obtained from the above operator, is given as

dσI dER

SI

=

mN σF 2 (ER ) , 2µ2 v 2

(2.43)

with the nuclear form factor F 2 (ER ) and the WIMP-nucleus cross section σ. The form factor can be qualitatively understood as a Fourier transform of the nucleon density and is usually parametrized in terms of the momentum transfer as 2

F (q) =

3j1 (q R1 ) q R1

2

exp −q 2 s2 ,

(2.44)

with j1 being a spherical Bessel function, s ≃ 1 fm a measure of the nuclear skin thickness √ and R1 = R2 − 5s2 with R ≃ 1.2 A1/2 fm. The form factor is normalized to unity at

zero momentum transfer, so that F (0) = 1. The χ–nucleus cross section is given by σ=

4µ2 [Zf p + (A − Z)f n ]2 , π

(2.45)

where Z, A are the nuclear charge and mass respectively. The coefficients f n/p are again the effective couplings between χ and nucleon as in (2.42).

17 In reality WIMPs do not interact with the nucleon but with its quark currents. Analogous to (2.42), an effective quark-WIMP operator can be defined: Oq = gq q¯q χχ ¯ ,

(2.46)

with gq being a (dimensionful) coupling constant. The quark content of the nucleon can be expressed in terms of the matrix elements fqN mN = hN |mq q¯q|N i ,

(2.47)

with mN (mq ) being the nucleon (quark) mass. Roughly speaking, the coefficient fqN is giving the projection of the quark current in the nucleon and it accounts for the contribution of quark q to the nucleon mass. This allows us to relate the operator in eq. (2.46) with the nucleon by projecting the quarks into the nucleon spinors hN |

X q

¯N Oq |N i = χχ ¯ N

X q

mN gq fqN mq

!

.

(2.48)

The effective coupling gq of quark- to DM-current is model dependent. Note that due to the Yukawa coupling of the Higgs boson to the quark currents, the direct dependence on the quark masses vanishes. The quark masses are relevant only for determining the coefficients fqN . The coefficients fqN of the light quarks q can be obtained by using experimental input: the value of the light quark masses, the ratio of quantities Bq = hN |¯ q q|N i for

u, d and s quarks and the value of the pion-nucleon sigma term (which has the largest uncertainty). The pion-nucleon sigma term is defined as σπN = and the quantity σ0 =

mu + md (Bu + Bd ) , 2

mu + md (Bu + Bd − 2Bs ) 2

(2.49)

(2.50)

is estimated from chiral perturbation theory or from the baryon mass differences. The strangeness of the nucleon can be expressed as y = 0.02

and

z=

Bu − B s ≈ 1.49 , Bd − Bs

(2.51)

where the value for y can be found in Ref. [43]. With the definitions α=

Bu 2z − (z − 1)y = , Bd 2 + (z − 1)y

(2.52)

18


the light quark coefficients in the proton can be expressed as fdp =

2σπN , mp (1 + α)

mu md

1+ mu p αf , md d σπN ms . md u 1+ m m p md

fup = fsp =

(2.53) (2.54) (2.55)

Similarly, expressions for the light quark coefficients of the neutron are obtained: fdn =

2σπN α , (1 + α) u 1+ m m n md

mu fnp , md α σπN y ms . md u 1+ m m n md

fun = fsn =

(2.56)

(2.57) (2.58)

In the literature the following experimental values are used: σπN = 55 MeV

and

σ0 = 35 MeV ,

(2.59)

both with relative uncertainties above 0.1. The quark mass ratios are given by mu = 0.553 ± 0.043 md

and

ms = 18.9 ± 0.8 . md

(2.60)

N Thus the numerical values for the light quark coefficients fu,d,s are found to be

fdp = 0.034, fup = 0.021, fsp = 0.016 ,

(2.61)

fdn = 0.011, fun = 0.062, fsn = 0.016 .

(2.62)

The nucleon mass can be expressed in the form of matrix elements of the trace of the energy momentum tensor Tµν at vanishing momentum transfer [44] ¯ N = hN |Tµµ |N i , mN N

(2.63)

with the naive estimate of Tµµ =

X q

mq q¯q +

β(αs ) a a G G , 4αs µν µν

(2.64)

where the sum includes all the quarks q with mass mq , Gaµν is the gluonic field strength tensor and

2 αs 2 + O(αs3 ) , β(αs ) = − 9 − nh 3 2π

(2.65)

19 is the Gell-Mann-Low function, with nh the number of heavy quarks. The last term in eq. (2.64) prevents the nucleon from becoming massless in the chiral limit. The heavy quarks enter via virtual states. The contribution of such states can be expressed using the heavy quark expansion, in leading order given by the triangle graph: X h

¯ → − 2 αs nh Ga Ga + O(α2 ) . mh hh s µν µν 3 8π

(2.66)

An evaluation of the heavy quark contributions, yields the nucleon matrix elements hN |Tµµ |N i = hN |

X ˜ s) β(α mq q¯q |N i , Gaµν Gaµν + 4αs

(2.67)

q=u,d,s

˜ s ) = −9α2 /(2π). In neglecting the light quark contribution the following idenwith β(α s tity is obtained:

˜ ¯ N = hN | β(αs ) Gaµν Gaµν |N i . mN N 4αs

(2.68)

Therefore the heavy quark content of the nucleon is obtained by combining eq. (2.66) and eq. (2.68): hN |

X h

¯ |N i ≈ − 2 nh hN | αs Ga Ga |N i , mh hh 3 8π µν µν =

2 ¯N . nh mN N 27

(2.69) (2.70)

Thus the the total spin independent scattering cross section of the dark matter particle on a nucleon N is proportional to SI σN ∝

m 2 X N (fiN )2 , v

(2.71)

i,n

where the index i indicates all the light and heavy quarks.

2.3.2

Spin-Dependent Cross Section

The effective Lagrangian for spin dependent interactions of the fermion χ with the fermionic nucleus ΨN at zero momentum transfer is given by ¯ N γ5 γ µ ΨN , L SD = ξN χγ ¯ 5 γµ χ Ψ

(2.72)

with the effective coupling ξN between DM and nucleon currents. This leads to the squared amplitude

SD 2 AN = 192 (ξN Mχ MN )2 .

(2.73)

For interactions at rest, the γ0 component of the pseudo-vector current, eq. (2.72), van¯ 5 γi Ψ leads to a three-dimensional vector current. This ishes. The resulting interaction Ψγ

20


vector current is proportional to the angular momentum J, which yields the definition ~ A A JA , J~N = SN |JA |

(2.74)

A are the expectation values of the spin content of the nucleon N in a nucleus where SN

with A nucleons. A over spin states and taking into account Performing the non-trivial summation of J~N

the J dependence the WIMP-nucleus squared amplitude can be expressed as |ASD |2 = 256

2 JA + 1 ξp SpA + ξn SnA Mχ2 MA2 . JA

(2.75)

This reduces to eq. (2.73) in the special case of the nucleon and leads to the cross section at rest for a point-like nucleus σ0SD =

2 16 µ2χ JA + 1 ξp SpA + ξn Sna . π JA

(2.76)

A are obtained from nuclear calculations or from simple nuclear models. The quantities SN

They are estimated to be ∼ 0.5 for nuclei with an odd number of protons or neutrons and

∼ 0 for an even number. Thus no strong enhancement is expected for SD interactions. When q 6= 0 the vectors J~p/n are not collinear, so that three form factors need to be

introduced. They correspond to the three coefficients of the quadratic function of ξp2 , ξn2 and ξp ξ n in the squared amplitude. With the isoscalar and isovector combinations a 0 = ξp + ξn ,

(2.77)

a 1 = ξp − ξn ,

(2.78)

the SD recoil energy distribution for a fixed WIMP velocity reads SD 16 µ2χ dσA Θ ((Emax (v) − E) = (S00 (q)a20 + S01 (q)a0 a1 + S11 (q)a21 ) . dE 2JA + 1 Emax (v)

(2.79)

The coefficients Si i(q) are the nuclear structure functions which take into account both the magnitude of the spin in the nucleon and its spatial distribution. They are normalized such that S00 (0) = C(JA )(Sp + Sn )2 ,

(2.80)

S11 (0) = C(JA )(Sp − Sn )2 ,

(2.81)

S01 (0) = 2 C(JA )(Sp + SN )(Sp − Sn ) , (2 JA + 1)(JA + 1) . where C(JA ) = 4 π JA With this normalization one recovers the cross section at rest as in eq. (2.76).

(2.82) (2.83)

21 After taking into account the velocity distribution, the distribution for the number of events over the nuclei recoil energy for spin dependent interaction reads 8 Mdet t ρ0 dN SD = (S0 0(q)a20 + S01 (q)a0 a1 + S11 (q)a21 I(E) . dE 2 J A + 1 Mχ

(2.84)

The form factors are calculated from detailed nuclear models including the momentum dependence. For nuclei for which the SD form factor has not been computed precisely it can be described with a Gauss distribution: Sij (q) = Sij (0) exp (−q 2 R2 /4) .

(2.85)

The form factors have a similar q dependence S00 (q) S01 (q) S11 (q) ≈ ≈ . S11 (0) S00 (0) S01 (0)

(2.86)

Therefore the same effective radius R can be used for all form factors. When Z exchange is dominant, which implies S11 being the dominant form factor, the A−dependence of the effective radius is given by 1/3

RA = 1.7 A

1/3

− 0.28 − 0.78 A

q 1/3 2 − 3.8 + (A − 3.8) + 0.2 fm .

(2.87)

The number of events for SD interactions can be calculated via the assumption of a the Gauss distribution, to retrieve RA for a given detector material.

2.4

Indirect Detection

Indirect detection experiments search for the products of dark matter annihilation. In matter dense regions of the universe, two dark matter particles can annihilate to produce Standard Model particle-antiparticle pairs. These annihilation products can in principle be detected indirectly through an excess of the respective flux over the astrophysical background. The decay products typically searched for are gamma rays, antiprotons or positrons. The detection of such a signal does not yield a conclusive evidence for dark matter, as the astrophysical background sources are not yet fully understood. For reviews on indirect dark matter detection, see for instance Ref. [30]. The observed flux of dark matter annihilation products i¯i can be written as 1 dNi Φi (ψ, E) =< σv > dE 4πm2DM

Z

ds ρ2 (r(s, ψ)) ,

(2.88)

l

with < σv >, ρ being the thermally averaged annihilation cross section and density of the annihilating dark matter particles respectively, l is the line of sight and dNi /dE is the associated spectrum. Further the coordinate s runs along the line of sight, forming an angle ψ with the galactic center.

22

2. DARK MATTER FORMALISM In introducing the quantity J(ψ), the factors depending on the halo profile can be

separated from those, depending on particle physics: 1 J (ψ) = 8.5 kpc

1 0.3 GeV/cm3

2 Z

ds ρ2 (r(s, ψ)) .

(2.89)

l

With the definition of J(∆Ω), as the average of J(ψ) over a spherical region of solid angle, ∆Ω, centered on ψ = 0, the flux coming from a solid angle, ∆Ω, can be expressed as Φi (∆Ω, E) ≃ 5.6 × 10

−12

dNi dE

< σv > pb

mDM −2 J (∆Ω) ∆Ω cm−2 s−1 . TeV

(2.90)

Since the thermally averaged annihilation cross section < σv > is specified by the abundance constraint, the model-specific signatures are found in the spectra of the annihilation products dNi /dE. Decay products which are, in principle, observable on Earth, are photons, neutrinos, positrons and anti-protons. Due to their charge, the propagation of the latter particles is affected by galactic fields and has to be approximated by a diffusion equation. This inserts another uncertainty and thus a loss of predictivity. A neutrino flux Φν could in principle be detected, for instance with the Ice Cube detector [45], in particular when the neutrinos are very energetic, to be distinguished from astrophysical backgrounds. Up to now, however, no signal has been observed [39, 46] Due to the fact that the dark matter candidates, to be investigated in the chapters 3 and 4, do not directly annihilate into photons, their annihilation in the universe will not lead to observable signatures in the γ flux Φγ . The neutral scalar DM candidates decay via a Higgs boson into the heaviest SM particle pair that is kinematically available. This means, depending on their mass, b¯b, W W, tt¯ for the O(N ) singlets and leptinos, and W W for the neutral triplet. Those decay products carry a well defined amount of energy. However, only the fraction of those particles that decays into γ, e+ or p¯, can be observed, with a smeared out energy distribution. This stands against the uncertainty from the astrophysical input, J, which varies with more than a factor of three for different choice of the halo profile [47], and the uncertainty of diffusive propagation of charged anti-particles. Thus, because of the absence of a clean signal in the photon flux, we will not consider the prospects of indirect detection of the dark matter models in this thesis. We remark that the Z2 model, due to the potentially large mass splitting between neutral and charged components, might yield a viable flux of charged antimatter, compatible with different observations [48, 49, 50], and possibly with the recent news from the AMS collaboration [51]. Due to time constraints, however, this feature of the model has not been investigated here.

23

2.5

Observational and Experimental Constraints

In this section, experimental constraints on dark matter aspects are presented. In subsection 2.5.1, the most recent nine year results from the WMAP experiment are displayed, which are of relevance for this thesis. Subsection 2.5.2 outlines the XENON and Ice Cube experiments, which yield the most stringent exclusion limits on spinindependent and spin-dependent WIMP-nucleon scattering cross sections, respectively. In subsection 2.5.3 the numerical tools micrOMEGAs is introduced, which was used for the numerical studies of the DM models of the following chapters.

2.5.1

Cosmological Parameters from the CMB

The combination of WMAP data and improved astrophysical data rigorously tests the standard cosmological model and places new constraints on its basic parameters and extensions [52]. The analysis of WMAP9 data in Ref. [7] determined the parameters of the simplest 6-parameter cold dark matter model with a cosmological constant (ΛCDM). The most relevant parameters for this thesis are: Nef f

= 3.84 ± 0.40 ,

H0 = (70.0 ± 2.2) km/s/Mpc , Ωb = 0.0463 ± 0.0024 ,

ΩDM h2 = 0.1138 ± 0.0045 .

(2.91)

H0 is the Hubble constant, which yields the relation between distance and relative velocity of a cosmic object, such as a galaxy or a cluster of galaxies. Nef f is the effective number of neutrinos in the early Universe. This observation stands against the SM predicted value of ∼ 3. Ωb denotes the baryonic matter content of the Universe, which

is directly related to the baryon asymmetry ηB , and ΩDM is given by the difference between observed and baryonic matter content of the Universe. It is commonly ascribed to the thermal abundance of a relic WIMP. The last line of eq. (2.91) is often referenced in the following chapters with “WMAP observation”. This yields an important quantitative condition for any dark matter model with a cold dark matter candidate.

2.5.2

Direct Detection Experiments

For a review on direct dark matter detection, see for instance Ref. [37]. WIMPs can be detected via their elastic collisions with terrestrial nuclei, with predicted event rates ranging from 10−6 to 10 events per kilogram detector material and day, which makes low background and high mass detectors essential. The recoil energy of the scattered nucleus is transformed into a measurable signal, such as charge, light or phonon’s. The motion of the Earth through the halo induces a seasonal variation of the total event

24


rate and a forward-backward asymmetry in a directional signal, which serve as a unique signature for a halo-borne WIMP. Cryogenic experiments operated at sub-Kelvin temperatures (CDMS [53], CREST [54], EDELWEISS [55]) and detectors based on liquid noble elements such as Xe (XENON 100 [38], ZEPPELIN-II [56]) and Ar (WARP [57]) are currently most sensitive to WIMPnucleon cross sections (for spin-independent couplings). The elastic scattering of a WIMP on liquid noble elements produces a low-energy nuclear recoil, which loses its energy through ionization and scintillation. Both signals allow to distinguish between electron and nuclear recoils. Natural xenon does not contain any long-lived radioactive isotopes (apart from the double beta emitter 136 Xe which has a half-life larger than 0.5×1021 years). The high mass of the Xe nucleus is favorable for WIMP scalar interactions and the presence of two isotopes with unpaired neutrons (129 Xe, spin 1/2, 26.4% and

131 Xe,

spin 3/2, 21.2%) offers excellent sensitivity to the

WIMP spin-dependent interaction. The high density (3 g/cm3 ) and high atomic number (Z=54, A=131.3), allow to build self-shielding, compact dark matter detectors. The XENON 10 collaboration has operated a 15 kg (active mass) dual phase detector time projection chamber in the Gran Sass Underground Laboratory (LNGS), in WIMP search mode from August 24, 2006 to February 14, 2007. From a total of 1800 events in the 58.6 live-days of blind WIMP search data, 10 events were observed in the WIMP search region, with 7.0+1.4 −1.0 events expected based on statistical (Gaussian) leakage alone.

Conservative limits with no background subtraction were calculated for spin-independent

WIMP cross sections. The 90% C.L. upper limit at a WIMP mass of 100 GeV/c2 is 8.8×10−44 cm2 [58]. The XENON 100 experiment used a 161 kg (62 kg WIMP target) detector at LNGS and acquired data over 100.9 live days in 2010 [38]. The principle is similar to XENON10, with a few important differences: an active, liquid Xe veto shield surrounds the inner WIMP target, the signal and high-voltage feed-through, as well as the cracklier are moved outside the Pb/PE shield, the double-walled stainless steel cryostat is made of low-radioactivity steel with

mZ . 2

(3.43)

The singlet-triplet coupling η on eq. (3.41) adds the the annihilation of triplet into singlet fields (and vice versa) to the total annihilation cross section (σA ) if kinematically allowed. Therefore, σA is expected to grow with increasing η, which reduces the fraction of either singlet or triplet component, depending on which mass is smaller, implying ΩZ2 ×Z2 (η = 0) > ΩZ2 ×Z2 (η 6= 0) ,

(3.44)

which increases the upper bound on the respective mass range. The left panel of Fig. 3.10 shows this in detail for the triplet fraction fΨ . For η = 1 and with mΦ < mΨ , the upper bound on the triplet component mass is given with mcrit Ψ (η = 1) = 2.8 TeV .

(3.45)

We note, that for mΦ ≃ mΨ it is mcrit Ψ (1) = 2.6 TeV. However, the constraint from

eq. (3.41) demands mΦ to be small, so that the upper bound from eq. (3.45) is valid.

42

3. SCALAR DARK MATTER

mφ [GeV]

10000

1000

100

κ=1 κ=0 500

1000 1500 2000 2500 3000 mΨ [GeV]

Figure 3.12: Mass parameter space of the Z2 × Z2 model respecting the abundance and perturbative constraint. Here the direct dependence of maximum mass values on the other parameter has been neglected. This leads to an uncertainty of O(20 %). For η > 0 and mΨ < mΦ , the perturbative constraint on ωΦ is relaxed. The right panel of Fig. 3.10 shows the effect of η = 1 quantitatively for mΨ = 0.2, 1.0 TeV, compared to the case of η = 0. The relaxation of the abundance constraint on ωΦ shifts the upper bound on the singlet mass to mcrit Φ (η = 1) = 4.3 TeV .

(3.46)

In general the mcrit Φ(Ψ) are dependent on η, mΨ , mΦ . However, this dependence is small as can be seen in Fig. 3.12, where two lines represent the combination of abundance and perturbative constraint for the two values of η = 0, 1.

3.3.3

WIMP-Nucleon Scattering

The direct search constraints affect the singlet part of the Z2 × Z2 model only, since

the parameter ωΨ remains unconstrained from WMAP observations as in section 3.2.3. The presence of the coupling parameter η can relax the abundance constraint on ωΦ for

mΨ < mΦ . In other words, a tuning of η allows to match the WMAP observation even for ωΦ ≈ 0. Thus the abundance constraint on ωΦ is valid only for mΨ > mΦ .

Because the spin-independent singlet-nucleon interaction cross section is proportional

to the square of the singlet-Higgs coupling, it follows with eq. (3.40): SI σΦ ∝

2 ωΦ . 1 − fΨ

(3.47)

SI are inversely proAccording to eq. (2.39), the XENON 100 exclusion limits on σΦ

portional to the cosmological density of the singlet fraction, ρΦ . With eq. (3.39), the

43

10000 WMAP

mφ [GeV]

1000 LEP-I 100

XENON 100 10 10

100

1000

10000

mΨ [GeV] Figure 3.13: Mass parameter space for the Z2 × Z2 model. The green area represents the combination of abundance and perturbative constraint on the masses, with the coupling constant η = 1. The gray area represents LEP limits on the branching ratios of the Z boson. The red area represents the XENON 100 exclusion limits, where the strangeness corrected value of the spin independent WIMP nucleon scattering cross section, as given by eq. (2.92), has been used.

SI is increases as sensitivity of the XENON limits is gets reduced in the same way as σΦ

a function of the triplet fraction fΨ . Therefore, the direct search constraints limit mΦ from below with 41 GeV as in the O(N ) model with N = 1, and are insensitive to fΨ . In Fig. 3.13 all the constraints on the Z2 × Z2 model are summarized. Since mΨ >

mZ /2 > 41 GeV, the XENON limits, denoted by the blue area, are constant over the whole mass range of mΨ . The red area represents the combination of the abundance constraint and perturbativity, for a value of the coupling constant η = 1.

3.4

Multicomponent Scalar Dark Matter: Z2 Model

This section introduces the Z2 model, which adds one global Z2 symmetry, and the real scalar singlet and triplet fields from sections 3.1 and 3.2 to the matter content of the Standard Model. The singlet and triplet fields both reside in the odd representation of the Z2 symmetry. Including all the renormalizable terms in the Lagrangian leads to mass mixing of the singlet and triplet fields after spontaneous symmetry breaking.

44

3. SCALAR DARK MATTER The four Z2 -odd mass eigenstates of the model are linear combinations of the singlet

and triplet. Due to the mass mixing, the constraints on the model parameters from the abundance constraint and direct searches deviate strongly from the individual models and from the multicomponent scenario from the previous section. In particular, it is possible that all the fields have masses on the weak scale, without violating the abundance constraint from WMAP, the direct search constraint from XENON 100 and the perturbativity of the couplings. The electroweak precision tests are not affecting this model more severely than the triplet model, because the gauge bosons couple to the interaction eigenstates. If the dark matter fields are light, detection at the LHC is a possibility. The production cross section, via vector-boson fusion, can be as large as ∼ fb [69]. The larger mass splittings of the neutral to the charged components, allow decays of the components being accompanied by more than a low-energetic pion.

3.4.1

The Model

The matter content of the SM is extended with one real singlet field Φ and one real SU (2)L triplet field as defined in sections 3.1 and 3.2 to the scalar sector. Contrary to the Z2 × Z2 Model, one Z2 symmetry is added. The Φ and Ψ fields reside in the odd representation of the Z2 symmetry.

The Lagrangian of the Z2 Model reads: LZ2 = LSM + LΦ1 + LΨ + Lmix ,

(3.48)

where the term LSM contains the SM fields while the terms LΦ and LΨ are as defined in eq. (3.1) for N = 1 and (3.14) respectively. The singlet-triplet mixing term is given by Lmix = κH † τ i HΨi Φ ,

(3.49)

with H being the SM Higgs doublet, and κ is a dimensionless coupling constant. The singlet-triplet mixing term in eq. (3.49) is the only renormalizable term, which is Z2 invariant, beyond the ones inherited from the O(N ) and the triplet model, apart from |Ψ|2 Φ2 , which does only contribute to the DM self interactions, which is not very re-

strictive on the model parameters.

As the the self couplings λΦ , λΨ do not affect the abundance, we shall neglect them in the following. The remaining parameters of the Z2 model are the two masses mΦ , mΨ , the singlet- and triplet-Higgs coupling, ωΦ and ωΨ , respectively. The Feynman rules of the Z2 model in unitary gauge, are summarised in Table 7.4 in the Appendix.

3.4.2

Mass Mixing

The spontaneous breaking of the electroweak symmetry leads to a mixing of the neutral interaction eigenstates Φ, Ψ0 through the singlet-triplet mixing term (3.48). The mixed

45 fields S1 and S2 are linear combinations of the interaction eigenstates, which can be expressed as Φ Ψ0

!

=

cδ

sδ

−sδ

cδ

!

S1 S2

!

,

(3.50)

where sδ (cδ ) are the sine (cosine) of the mixing angle δ. For conformity, the charged triplet components are relabeled to S ± ≡ Ψ± and the mass mc ≡ mΨ + ∆m, where ∆m is the loop-induced mass splitting due to the gauge couplings of the triplet component.

The physical masses of the S−fields are given by the eigenvalues of the mass matrix M=

1 2 2 κv m2Ψ

m2Φ 1 2 2 κv

!

.

(3.51)

The eigenstate Si to the eigenvalue m2i , i = 1, 2 is then the physical field as defined in eq. (3.50). The m2i , obtained via the characteristic polynomial, are given by q 1 2 2 2 2 4 = mΦ + mΨ − (∆) + κ v 2 q 1 m22 = m2Φ + m2Ψ + (∆)2 + κ2 v 4 , 2 m21

(3.52) (3.53)

with ∆ = m2Φ − m2Ψ . From the above definitions for the physical masses, the following identities arise:

m21 + m22 = m2Φ + m2Ψ

(3.54)

(m22 − m21 )2 = ∆2 + κ2 v 4 .

(3.55)

In transforming the mass matrix (3.51) into the S−field basis, using the rotation matrix in eq. (3.50), and identifying it with diag(m1 , m2 ), a set of conditions, connecting the mass parameters and the mixing angle, is given: cδ sδ

−sδ cδ

!

m2Φ 1 2 2 κv

1 2 2 κv m2Ψ

!

cδ

sδ

−sδ

cδ

!

=

m21

0

0

m22

!

.

(3.56)

The first condition comes from the off-diagonal matrix elements: 1 sδ cδ ∆ + (c2δ − s2δ ) κ v 2 = 0 . 2

(3.57)

This leads to the quadratic equation for s2δ (or equivalently c2δ ): 1 ∆2 (s2δ − s4δ ) − (1 − 4 s2δ + 4 s4δ ) κ2 v 4 = 0 , 4 which is solved for s2δ

∆ 1 1− √ . = 2 ∆2 + κ2 v 4

(3.58)

(3.59)

46


The identity in eq. (3.55) allows to express eq. (3.59) as a function of the physical masses s # " 2 v4 κ 1 . s2δ = 1− 1− 2 (m21 − m22 )2

(3.60)

In solving eq. (3.57) for κ an auxiliary equation is obtained: κ v 2 = −t2δ m21 + m22 − 2 m2c ,

(3.61)

where eq. (3.54) and the trigonometric identities were used, and with the definitions t2δ = tan 2δ. Eq. (3.56) yields the following equation for the mass m2 : m22 = s2δ m2Φ + c2δ m2Ψ + sδ cδ κ v 2 .

(3.62)

Together with relation (3.54), this can be expressed as m22 = t2δ m21 + (1 − t2δ ) m2c + tδ κ v 2 .

(3.63)

The mass splitting between m1 and mc can be defined via a mass-splitting parameter c, as

mc . m1

c≡

(3.64)

Altogether, the eqs. (3.61) and (3.63) lead to expressions for the coupling constant κ and the mass m2 , as functions of the model parameters m1 , c, δ:

c2 − 1 1+ = c2δ m2 κ = 2 tδ c2 − 1 21 . v

m22

m21

(3.65) (3.66)

Thus in the limit of δ going to zero, the mass m2 converges to mc and κ to zero. It is convenient to define the mass splitting between the S ± and the S2 , analogous to the definition in eq. (3.64), as c2 ≡

3.4.3

m2 = m1

q c2 − s2δ cδ

.

(3.67)

Perturbativity

Due to the Feynman rules, derived from the Lagrangian in eq. (3.48) and displayed in Table 7.4, the Higgs couplings of the fields S1 and S2 are given by − ω1 ≡ 4 sδ cδ κ − c2δ ωΦ − s2δ ωΨ ,

−ω12 ≡ −2 (s2δ − c2δ )κ − sδ cδ (ωΦ − ωΨ ) , −ω2 ≡ −4sδ cδ κ − s2δ ωΦ − c2δ ωΨ .

(3.68) (3.69) (3.70)

47 While ωΦ , ωΨ are free parameters, κ is a function of the model parameters m1 , c, δ. In defining the Lagrange parameters ωΦ , ωΨ as functions of the effective Higgs couplings ω1 , ω2 , the latter become the free parameters of the model. This choice of parameter set gives − ω12 =

c2δ

s δ c δ ω1 2κ − 2 − s δ c δ ω2 , 2 − sδ cδ − s2δ

4sδ cδ κ − c2δ ω2 − s2δ ω1 . s2δ − c2δ

−ωΨ =

(3.71) (3.72)

These couplings diverge for tδ → 1, and they are linearly dependent on κ. With κ diverging for m1 > v or δ → π/2 according to its definition in eq. (3.66), the perturbativity of the theory is violated with |ω12 |, |ωΨ | > 1 in this regime, as well as for tδ ≃ 1.

In order to define a perturbative limit on ω12 , ωΨ , we define the following limits on

the model parameters: |κ| ≤ 1

and

sδ ∈ / [0.65, 0.75] .

(3.73)

The limit on κ defines upper bounds for the model parameters c, δ, which represent the perturbative constraint on ω12 and ωΨ v2 + 1, tδ m21 v2 = arctan . m21 (c2 − 1)

cmax = δmax

s

(3.74) (3.75)

The left panel of Fig. 3.14 shows the resulting upper bound for the mass-splitting parameter c as a function of the mass, for two different, allowed values of the mixing angle. For sδ → 1(0), the upper bound becomes tighter (relaxed). The right panel of Fig. 3.14 shows the resulting upper bound on sδ as a function of the mass, for two given values of the mass-splitting parameter c. An increased (reduced) value of c implies a tighter (relaxed) upper bound on sδ for a given mass. Note, that the black lines represent the separation of the singlet and the triplet scenario, which are defined below. In the right panel of the figure, the excluded range of the mixing angle is within the black lines.

3.4.4

Dark Matter Assessment

Among the two neutral fields S1 and S2 , the former is the DM candidate due to its smaller mass. The S1 couples to the Higgs boson via ω1 , as defined in eq. (3.68). It also interacts with the gauge fields proportional to sδ . This implies that, for a very small value of δ, the Higgs coupling is dominant, and the abundance constraint translates into a constraint on ω1 that is comparable to the constraint on ωΦ in the O(N ) model for N = 1. For δ → π/2, the gauge couplings become dominant, and the dependence of the

abundance on the Higgs couplings becomes suppressed.

48


100

1

sδ = 0.3 sδ = 0.7

10

triplet-scenario singlet-scenario

0.1

1

sδ

c-1

singlet scenario

0.1

triplet scenario 0.01

0.01 0.001 10

100 m1 [GeV]

c = 1.17 c = 1.4

0.001

1000

0

250

500 m1 [GeV]

750

1000

Figure 3.14: Left: The perturbative upper limit on the mass-splitting parameter c for three different values of δ. Right: The perturbative upper limit on the mixing angle δ for three different values of the mass-splitting parameter c. In both panels the lines are upper exclusion contour lines. Consider now the contribution of other fields to the S1 total annihilation cross section. Due to their earlier decoupling with respect to the S1 freeze-out temperature, the coannihilations of the Si , Sj are suppressed by a Boltzmann factor ln B = −Xf

mi + mj − 2 m1 m1

,

(3.76)

with the masses mi , mj ∈ (m2 , mc ) and Xf the freeze-out temperature of the S1 . MicrOMEGAs considers these processes up to ln B = −13.8. For the coannihilations of the

S ± , with mi = mj = c m1 , this yields the function for the mass-splitting parameter c: fc (Xf ) = 1 +

6.9 . Xf

(3.77)

This function yields a benchmark, whether S ± coannihilations are being included in the computation of the abundance. Since typical WIMP freeze-out temperatures can vary between 20 and 40, for values of the mass-splitting parameter given by c ≥ fc (20) = 1.345, coannihilations are not considered by micrOMEGAs. On the other hand, for

c ≤ fc (40) = 1.1725 the coannihilations of the other particles are excluded.

3.4.5

Singlet Scenario

We define the singlet scenario, so that the S1 abundance is dependent on the mass m1 and the Higgs coupling ω1 , comparable to the O(N ) model for N = 1, by constraining the mixing angle and mass-splitting in the following way: sδ ≤ 0.3

and

c ≥ 1.4 .

(3.78)

Because of the small value of sδ , the annihilation cross section is dominated by the Higgs channel, the strength of which is controlled by ω1 . The coannihilations of the S ± fields

49 are negligible, according to the lower bound on the mass-splitting c and eq. (3.77). Notice that the choice for the limiting values for sδ and c is somewhat arbitrary. However, we will see in the following that it is justified. In this scenario, the abundance constraint can be matched by tuning the parameters m1 , ω1 , and, to a lesser extent, δ. Via eq. (3.66), perturbativity limits the mass range for m1 for a given δ from above in the following way:

246 GeV , m1 < p tδ (c2 − 1)

(3.79)

which for sδ = 0 and c ≥ 1.4 recovers the limits on the mass range from the O(N )

model with N = 1, which is due to the perturbativity constraint |ω1 | ≤ 1. This is

illustrated by the red area in the left panel of Fig. 3.15, which represents allowed values for the abundance. The mass is limited from above by m1 ≤ 3.4 TeV, which is the same

as for the O(N ) model. The green curve in the figure represents the case of sδ = 0.3 and c = 1.4. It diverges from the red area at m1 = 457 GeV, in agreement with the formal upper bound on m1 from eq. (3.79). This divergence is due to ω12 becoming larger than one, growing proportional to m21 , for this value of m1 . This increases σA through contributions from coannihilations of the form S1 S2 → h → f f¯, suppressing

the abundance. An increase of c does increase the Boltzmann suppression, but at the same time it increases the coupling ω12 . Considering the mass spectrum in the singlet scenario, the parameter c ≥ 1.4 deter-

mines the mass splitting between the S1 and the S ± fields. The relative mass splitting between the S ± and the S2 is given by eq. (3.67). For sδ = 0 it is c2 = c, and the two heavier fields become mass degenerate, apart from the radiative corrections to the S ± . = 1.05 for large c and sδ = 0.3. The maximum value for c2 is given with cmax 2 Subsequently, the S2 , S ± form a set of fields that is similar to the fields as defined

in the triplet model. We want to point out two differences. First, the S2 tends to be heavier than the S ± . Second, the relative mass difference between S2 and S ± does not converge to zero for large masses. In the singlet scenario, the coannihilations and the perturbative constraints on the parameter c are negligible for m1 ∼ mH /2. Therefore, the direct detection constraints

affect the mass range in the same manner as in the O(N ) model for N = 1. This places the lower bound of 41 GeV on m1 , which is not affected by a variation of sδ between zero and 0.3, so that the mass range for the S1 is given by 41 GeV ≤ m1 ≤ 3.4 TeV. Note that the strangeness corrected value for σ SI , as defined in eq. (2.92), has been used.

3.4.6

Triplet Scenario

We define the triplet scenario, so that the abundance is qualitatively close to that of the triplet model, by the following constraints: sδ ≥ 0.75

and

c < 1.17 .

(3.80)

50


1 1e-06

1e-08

σ

SI

Ωh2

[pb]

0.01

0.0001

1e-10

sδ = 0.0 sδ = 0.3

1e-06 10

100

1000

10000

1e-12

XENON 100 δ=0 sδ=0.3 25

m1 [GeV]

50 m1 [GeV]

75

100

Figure 3.15: Left: Abundance of the singlet scenario as a function of the mass m1 . The red area represents values which are allowed by the constraints on the parameter space. For the green curve, sδ = 0.3 has been used. For m1 = 457 GeV the two lines diverge due to κ > 1, see eq. (3.79). Right: Direct detection constraints on the S1 mass. The blue area represents the XENON 100 exclusion limits, the red line represents the abundance constraint on the coupling of the S1 to the Higgs boson. The strangeness corrected value for σ SI , as defined in eq. (2.92), has been used. Because of the small value for c, the coannihilations of the S ± fields are contributing to σA , resulting in an increased sensitivity of the abundance on the couplings ω2 , ω12 , ωΨ , compared to the singlet scenario. The large value of sδ yields strong gauge couplings of the S1 field, which results in a reduced sensitivity to ω1 , compared to the singlet scenario. The masss-splitting parameter c is very powerful in controlling the contributions from the S ± , S2 to the abundance, and their coannihilation strength. For c close to one, the Boltzmann suppression of the S ± , S2 abundances becomes negligible, which increases the S1 abundance. For a suitable choice of c, sδ , the coannihilations of the S2 still contribute to σA , while its abundance is still suppressed. In this extreme case, Ω can take smaller values compared to the abundance in the triplet model for the same mass. On the other hand, if c ≃ 1, the abundance can be much larger than in the triplet

model due to the contributions of the S ± and the S2 to the DM abundance.

Altogether, the abundance is mainly sensitive to the variation of m1 and the parameter c, and only secondarily to a tuning of the Higgs couplings. Fig. 3.16 shows how the variation of the parameters ω1 , ω2 , sδ , c affect the abundance. The upper boundary of the red area is given by sδ = 0.75 and c, c2 ≃ 1. The lower boundary is given by

sδ = 1 − ε, with ε → 0 for numerical stability, and c ∼ cmax . The variation of the model

parameters allows for the S1 mass in the range of

1.33 TeV ≤ m1 ≤ 6.65 TeV ,

(3.81)

to match the abundance constraint. The left panel in Fig 3.17 shows the maximum allowed values for the mass-splitting parameter, for m1 in the range given by eq. (3.81). It is apparent from the right panel in Fig 3.17, that the upper bound on the absolute value of the three mass splittings, given by (m2 − m1 ), (m2 − mc ), (mc − m1 ), become

51

0.1

Ωh2

0.01 0.001 0.0001 WMAP Triplet Scenario

1e-05 10

100 1000 m1 [GeV]

10000

Figure 3.16: Abundance of the Z2 model in the triplet scenario. The red area represents values for the abundance, allowed for a maximum variation of The Higgs couplings ω1 , ω2 , the mixing angle sδ and the mass-splitting parameter c. The black line represents the WMAP7 reported abundance constraint, its width reflects the uncertainty. rather large for m1 → 5 TeV. Accordingly, the mass splittings are confined to the range 0 ≤ m2 − m1 ≤ 19.4 GeV ,

(3.82)

−0.168 GeV ≤ m2 − mc ≤ 16.6 GeV ,

(3.83)

0.168 GeV ≤ mc − m1 ≤ 2.8 GeV ,

(3.84)

which may have consequences for collider searches, as it opens up more decay channels. 20 Mass splitting [GeV]

0.001

WMAP

c-1

Perturbativity

0.0005

16

mc - m1 m2-m1 m2-mc

12 8 4 0

2

3

4 5 m1 [TeV]

6

7

2

3 4 m1 [TeV]

5

Figure 3.17: Left: Mass-splitting parameter between the S1 and the S ± fields. The red area represents the WMAP exclusion from the abundance constraint, the gray area represents the perturbative constraint on the Higgs couplings. Right: Upper limits on the absolute value of the three different mass splittings between the S1 , S ± and the S2 fields.

52


3.4.7

Mixed Scenarios

1

Ωh2

0.01

0.0001

1e-06

mix triplet singlet WMAP 25

50 m1 [GeV]

75

100

Figure 3.18: The relic density as a function of mass m1 for three different scenarios. The width of the blue and green area represents the variation of ω1 , ω2 between 0 and 1. For the singlet scenario c = 1.4 and sδ = 0 have been used, the mixed scenario uses c = 1.2 and 0.35 ≤ sδ ≤ 0.75, excluding the range given in eq. (3.73). The triplet scenario uses sδ = 0.9 and c = 1.1 We define the mixed scenario by the range that remains uncovered by the singlet and triplet scenario: 0.3 < sδ < 0.75

and

1.17 < c < 1.4 .

(3.85)

This scenario is represented in the two panels of Fig. 3.14 as the roughly rhomboid areas, defined by the intersections of the four lines in each panel. The mixed scenario is displayed in Fig. 3.18, together with the singlet and the triplet scenario. We choose the mass range between 10 and 100 GeV to illustrate the effects of coannihilations and the gauge interactions of the S1 . The blue and green areas represent the mixed and triplet scenario, respectively. The width of each area is given by the variation of the Higgs couplings ω1 , ω2 , between 0 and 1. The red line represents the lower contour line for the singlet scenario. The black line denotes the WMAP observed dark matter abundance, which implies that the triplet scenario is excluded in this low mass range, while the mixed scenario is able to match the constraint for a tightly constrained mass range. The gauge couplings of the mixed and triplet scenario lead to the Z and W ± boson resonances, which are absent in the lower contour line of the singlet scenario. This is due to the Boltzmann suppression of the coannihilations with growing c, and to the interaction strength being proportional to sδ , which makes the magnitude of the resonance proportional to s2δ . The resonance of the weak gauge bosons is shifted to smaller

53

10000 2

Ω hWMAP > 0.117

m1 [GeV]

1000 0.7

2

Ω h < 0.106

LEP-I 10 0.001

0.01

0.1

1

c-1

Figure 3.19: Allowed range for the mass parameter m1 as a function of the mass splitting parameter c. The red area represents parameter sets being excluded by WMAP, the dark gray area represents the partial Z boson decay width into S ± violating LEP-I limits and the light gray area separates the singlet from the triplet scenario. The two black lines separate the singlet from the triplet scenario. The abundance constraint demands sδ < 0.3 for parameter sets above the upper black line, while below the lower black line sδ > 0.75 is allowed. values of m1 when comparing the mixed to the triplet scenario. This is because of the associated process, S1 S ± → W ± → f 0 f ± , with f 0 , f ± being neutral and charged Stan-

dard Model fields, respectively. The center-of-mass energy for dark matter annihilation processes is given by the sum of the dark matter masses, which implies for the above process: Ecm = m1 (1 + c) .

(3.86)

We define the relaxed singlet scenario by sδ < 0.3

and

c < cmax .

(3.87)

so that the contributions of the coannihilations of the S ± can reduce the abundance. Thus a reduction of c allows for smaller values of ω1 to match the abundance constraint. We say that the abundance constraint on the Higgs coupling is relaxed. With sδ = 0 and ω1 ∼ 0, the abundance is given as a function of the mass splitting.

For each mass there exists a value c > 1, so that the coannihilations alone satisfy the abundance constraint. Fig. 3.19 illustrates this critical value for c. For parameter sets in the red area, the resulting value for Ω is excluded by WMAP observations. For the S1 mass in the range 1.33 TeV ≤ m1 ≤ 1.68 TeV, the coannihilations have the appropriate annihilation efficiency to match the abundance constraint, so that c = 1 is allowed. The

two black lines in the figure represent the perturbative constraint. For a given c, a mass

54


1.5 1.4 XENON 100

c

1.3 1.2 LEP-I

1.1

WMAP

1 30

40 m1 [GeV]

50

Figure 3.20: Parameter space in the relaxed singlet model. The red area represents a DM relic abundance being too small compared to the value reported by WMAP. The blue area represents the abundance and direct search constraints on the model parameters being mutually exclusive. The gray area denotes the decays of the Z boson into S ± violating LEP-I exclusion limits. m1 above the upper black line requires sδ < 0.3, while masses below the lower black line allow for sδ > 0.75. Together with the gray area, denoting 1.17 ≤ c ≤ 1.4, the domains of the three scenarios can be identified.

For c close to border of the red area in Fig. 3.19, the abundance constraint on ω1 is relaxed such, that it can become arbitrarily small. This allows for a sufficiently small spin-independent S1 nucleon scattering cross section, not violating the XENON 100 exclusion limits for m1 ≤ 41 GeV.

The narrow area between the exclusion limits from XENON 100, WMAP and LEP-

I in Fig. 3.20, represents the fraction of parameter space in agreement with all the constraints. The LEP-I limits come from upper bounds on the Z boson decay width as in eq. (3.43). The blue area represents the value of σ SI , as given by the abundance constraint on ω1 , being excluded by the XENON 100 exclusion limits. The red area denotes where coannihilations suppress the abundance below the WMAP observed value. Thus, the lower bound on the mass m1 is given by 35 GeV in the relaxed singlet scenario. We define the relaxed triplet scenario with sδ > 0.75

and

1.4 < c < cmax .

(3.88)

This definition allows to suppress the effect of coannihilations, which in turn increases the sensitivity of the abundance to ω1 . We notice, that the condition on the mass splitting limits m1 from above, with 238 GeV.

55

10

S1S1 -> W+W-

XENON 100

c-1

κ>1

1

WMAP

0.1 20

40

60 m1 [GeV]

80

Figure 3.21: Parameter space of the relaxed triplet scenario. The red area represents the WMAP exclusion limits, the blue area denotes the combination of the abundance constraint and the direct search exclusion limits on the spin-independent S1 -nucleon scattering cross section (σ SI ). The green area denotes the upper bound from perturbativity and the gray area the suppression of the abundance by the coannihilations of the S ± . The strangeness corrected value for σ SI , as defined in eq. (2.92), has been used. Fig. 3.21 shows the areas of parameter space for which the WMAP constraint on the abundance can be met by a variation of ω1 . For values of c and m1 as defined by the red area, the coannihilations are too efficient. The green area denotes the perturbative constraint. For masses m1 ∼ MW , the W boson production threshold is reached. Due

to virtual W production, this threshold is given with 75 GeV. For m1 at the threshold and beyond, the four-point interaction between the S1 and the W bosons dominates the total annihilation cross section efficiency, which is too large to match the abundance constraint for allowed masses. Thus the upper limit of m1 in this scenario is given by 75 GeV.

The WMAP observation constrains ω1 for m1 , c in the allowed range, which leads to a prediction for σ SI . The blue area in Fig. 3.21 shows the resulting the XENON 100 exclusion limits on the parameter space. Altogether, the relaxed triplet scenario confines the mass m1 to the range 35 GeV ≤ m1 ≤ 75 GeV.

The mass splitting in this scenario can become very large. In particular for c = 5.7

is possible, and with m2 = c/cδ , c2 is constrained by perturbativity. This leads to a decay spectrum with large mass splittings for m1 ≃ 41 GeV and sδ = 0.75 49.2 GeV ≤ m2 − m1 ≤ 310 GeV ,

(3.89)

49.2 GeV ≤ mc − m1 ≤ 193 GeV ,

(3.90)

0 GeV ≤ m2 − mc ≤ 117 GeV .

(3.91)

56


3.5

Summary of the Results

In sections 3.1, 3.2 we introduced two simple models with phenomenologically different cold dark matter candidates. By employing the numerical tool micrOMEGAs, we performed a quantitative, numerical study on the respective model parameters, so that those could be constrained by the WMAP observation on the the dark matter abundance (referred to as abundance constraint) and by the exclusion limits from the direct dark matter search experiment XENON 100. The O(N ) model, discussed in section 3.2 contains N real scalar singlet fields, which reside in the odd representation of a global Z2 symmetry. The abundance constraint from WMAP, together with a perturbative constraint on the singlet-Higgs coupling, yields an upper limit on the number of fields N . The singlet DM (ΦDM) mass also receives a lower bound of 4.5 GeV and an upper bound of 3.4 TeV in this way. The combination of abundance and direct search constraint on the the spin-independent ΦDM-nucleon scattering cross section (σ SI ) yields the lower bound of 41 GeV for the ΦDM mass, and limits the model parameter N from above with O(100) for masses being about 200 GeV, apart from the Higgs-resonant regime, where N is unbounded. The predicted ΦDM-nucleon interaction cross section is close to the XENON 100 exclusion limits for ΦDM masses below 100 GeV. This implies that future direct detection experiments can in principle detect the O(N ) model when the masses are light. If the ΦDM mass is on the TeV scale, however, direct detection is not feasible. The astrophysical limit on the dark matter self-interaction cross section, taken from elliptic cluster research, turned out to be negligible compared to the others. The triplet model, described in section 3.2, contains a real scalar SU (2)L triplet field, which resides in the odd representation of a global Z2 symmetry. Due to radiative corrections to the masses of its components, the neutral triplet component becomes a viable DM candidate. The total annihilation cross section (σA ) of the triplet DM (ΨDM) is dominated by gauge interactions, its couplings to the Higgs boson affect the abundance only quantitatively. The abundance constraint, together with the perturbative constraint on the triplet-Higgs coupling, allow for a ΨDM mass between 1.8 TeV and 2.6 TeV. Due to exact cancellations, σ SD is suppressed by α4 . The XENON 100 exclusion limits affect the couplings to the Higgs boson for ΨDM masses below 100 GeV. The constraint on the coupling has is strongest for masses around 30 GeV, where the upper bound is given with 0.3. For the ΨDM masses as required by the abundance constraint, XENON 100 exclusion limits are not restrictive. The direct LHC search for a triplet field as defined above is not very promising. For masses on the TeV scale, as required by the abundance constraint, the pair production cross section of charged or neutral triplet components via vector-boson fusion, is below attobarn (ab) [69], which is too small to be observable at the LHC.

57 The multicomponent dark matter model introduced in section 3.3 combines a real scalar singlet with a real scalar triplet from the two previous sections. The singlet and triplet fields transform odd with respect to two different Z2 symmetries, which results in two separate relic abundance fractions. Compared to the O(N ) model, the upper bound on the mass of the singlet component, which is due to the combination of abundance and perturbativity constraint, is relaxed to 4.3 TeV. The triplet mass can take arbitrary values between MZ /2 and 2.8 TeV. The XENON 100 exclusion limits on σ SI affect only the singlet fraction with the lower limit of 41 GeV as in the O(N ) model for N = 1. The most relevant aspect of this model is, that the triplet mass can be small without violating the abundance constraint. At the LHC, several hundred monojet plus track events with 100 f b−1 can be expected, as long as mΨ < ∼ 2MW [69]. The decays of triplet components into singlet components plus a SM field, are forbidden due to the two different Z2 symmetries and

thus do not affect the detection prospects. The Z2 model, as introduced in section 3.4, combines a singlet and an SU (2)L triplet field, which transform odd with respect to a global Z2 symmetry. Spontaneous symmetry breaking yields the fields S1 , S2 , S ± as mass eigenstates. The perturbativity of the Higgs-couplings yields constraints on the mixing angle and the mass splitting between the lightest S1 and the charged S ± fields. The model can be separated in three distinct scenarios. The singlet scenario is comparable to the O(N ) model for N = 1, with suppressed coannihilations and gauge couplings. The dominant parameters are the S1 -Higgs coupling and its mass m1 . The perturbativity constraint together with the combination of abundance and direct search constraints leads to the allowed mass range of 41 GeV ≤ m1 ≤ 3.4 TeV. The S2 and S ± are nearly mass degenerate in this scenario, which results in detector signatures as

difficult to detect as in the case of the triplet model. The mass splitting between S ± and S1 , however, can be of order 100 GeV for m1 > ∼ 1 TeV, which allows for decays of the form S ± → S1 f ± , with f ± being any Standard Model field with an even charge.

The triplet scenario, which is defined so that coannihilations are included in the total

annihilation cross section of the S1 , and with dominating gauge couplings, is qualitatively very close to the triplet model as described above. Due to the variation of the model parameters, and the presence of the S2 field, the allowed range for the S1 mass is given by 1.33 TeV ≤ m1 ≤ 6.65 TeV in this scenario. The combination of abundance and

perturbative constraint leads to upper bounds on the mass splitting between the S1 and the S ± of 2.8 GeV. The bounds on the splitting between S2 , S ± and S2 , S1 are given by 16.6 GeV and 19.4 GeV, respectively.

The mixed scenario allows a variation of the mixing angle and the mass splitting between the S1 and the S ± , constrained only by perturbativity. Two interesting sub scenarios emerge. The relaxed singlet scenario demands a small mixing angle, but allows

58


for the inclusion of coannihilations. This relaxes the abundance constraint on the S1 Higgs coupling, which in turn allows for a reduction of the lower bound on m1 to 35 GeV. The relaxed triplet scenario is defined with the sine of the mixing angle close to one, and the coannihilations being suppressed. Due to the strong gauge couplings of the S1 for large mixing angles, the abundance constraint limits m1 from above with 75 GeV. The combination of abundance and direct search constraint limits the mass m1 from below to 35 GeV. In the relaxed scenarios, some of the mass-splittings between the S1 , S ± , S2 can become larger than MW . This has consequences for collider searches, because a mass splitting above the MW threshold allows for the decays S2 → S ± W ∓ and S ± → S1 W ± .

Those decays lead to more detectable signatures at the LHC compared to the rather soft SM particles which are possible in the triplet scenario.

Chapter 4

Leptonic Dark Matter In this chapter, a detailed study of the leptino model is presented. The model is based on the class of minimal Z ′ models, see for instance Refs. [73, 74, 75, 76, 77], and it has been published in Ref. [32]. The leptino model is natural in the sense, that only renormalizable terms are included in the Lagrangian. It offers a viable candidate for cold dark matter, it has additional light fields which can add to the energy content of the early universe, and it can generate the observed baryon asymmetry in the so-called Leptogenesis scenario. Furthermore, it is testable in collider experiments. In the first subsection of section 4.1, the model is described. It extends the gauge sector of the SM with a U (1)B−L gauge factor, with B-L being the baryon-minus-lepton number. In the next subsection the anomalies are evaluated, and the scalar sector is dealt with in the third subsection. In the last subsection, which describes the Yukawa sector, pairs of singlet fermions with fractional lepton number are added, which are called leptinos. The light neutrino masses are generated through mass-mixing of the left-handed and righthanded neutrinos with one of the leptinos, which is referred to as the inverse seesaw mechanism in the literature [78, 79, 80, 81]. To prevent the leptinos from spoiling the inverse seesaw, a global Z2 symmetry is added, with respect to which one leptino per pair transforms odd. In section 4.2 the properties of the leptino abundance of a Z2 -odd leptino being the DM candidate, are discussed, for the case of one pair of leptinos. The abundance constraint from WMAP [7] as well as the collider constraints from LEP [82] and the LHC [83, 84] are employed to constrain the model parameters. The direct search constraint from XENON 100 [38], is taken into account as well. In the last subsection, the effect of enlarging the number of leptinos is investigated. The contribution of the right-handed neutrinos (νR ) to the relativistic degrees of freedom (RDF) in the early Universe is studied in section 4.3. It is discussed, whether the νR can explain the discrepancy between the SM predicted RDF, and the observed value [7, 29]. Section 4.4 presents the concept of the Leptogenesis scenario. In the leptino model, pairs of heavy, quasi mass-degenerate neutrinos are present, due to the inverse seesaw 59

60

4. LEPTONIC DARK MATTER

mechanism, with masses O(1) TeV. This leads to the Resonant Leptogenesis scenario, which allows for the generation of the observed baryon asymmetry [7, 22]. The results obtained in the sections of this chapter, are summarised in section 4.5.

4.1

The Leptino Model

In this section, the leptino model [32] is presented in four subsections. Subsection 4.1.1 presents the gauge sector of the model, which extends the SM with a U (1)B-L factor, where B-L is the baryon-minus-lepton number, and by the associated Z ′ boson. In subsection 4.1.2, the anomalies, and resulting conditions on the fermion sector, are considered. The scalar sector is presented in subsection 4.1.3, where the gauge and scalar bosons are considered, which obtain masses after breaking of the SU (2)L × U (1)Y × U (1)B-L symmetry. In subsection 4.1.4, the Yukawa sector is presented, and further

fields are added to the matter content of the model in order to employ the inverse seesaw mechanism.

4.1.1

Gauge Sector

The classical gauge invariant Lagrangian, obeying the SU (3)C × SU (2)L × U (1)Y × U (1)B−L gauge symmetry, can be decomposed as:

L = LY M + Ls + Lf + LY ,

(4.1)

where the indices Y M, s, f, Y stand for Yang-Mills, Scalar, Fermion and Yukawa sector respectively. The Yang-Mills sector contains the kinetic terms of the gauge sector, which is given by the field strength tensors 1 a aµν 1 1 1µν 1 2 2µν κ 1 2µν 1 W − Fµν F − Fµν F − Fµν F , LY M = − Gαµν Gαµν − Wµν 4 4 4 4 2

(4.2)

with G, W, F 1,2 being the gluonic-, weak- and U (1)−field strength tensors respectively. The real parameter κ < 1 allows for a mixing of the two U (1) fields. The general field strength tensor is defined as Aiµν = ∂µ Aiν − ∂ν Aiµ + gf ijk Ajµ Akν ,

(4.3)

with the gauge field Aiµ , the coupling constant g and the group structure constant f ijk . Note that the last term in the Lagrangian (4.1) can be rotated away by a suitable choice of basis of the U (1) fields. In the gauge field basis in which the kinetic terms in LY M are diagonal the covariant derivative reads Dµ ≡ ∂µ + igS T α Gµα + igW τ a Wµ a + ig1 Y Bµ + i(g2 Y + gBL YB-L )Bµ′ ,

(4.4)

61

Vρc

Vµa

Vνb

Figure 4.1: The triangle graph giving rise to the anomaly. with gS , gW , g1 the strong, weak, hypercharge coupling constants, T α , τ a , Y the respective generators and Gαµ , Wµa , Bµ the gluon, isospin, and hypercharge fields respectively. Bµ′ is the gauge field related the B-L gauge factor. It couples to hypercharge and baryonlepton number via the coupling constants g2 and gBL respectively. In this case, g2 is a mixing gauge coupling.

4.1.2

Anomaly Cancellations

For an introduction to anomalies, see for instance [85]. If a gauge symmetry is anomalous, violation of renormalizability and unitarity is implied. A criterion for the absence of anomalies as the triangular fermion loop with external gauge bosons is described in Ref [86], as illustrated in Fig. 4.1. In general its contribution to the anomaly is proportional to Aabc = Tr

hn

o i T a, T b T c ,

(4.5)

where the T i are the generators of the respective gauge groups. Aabc = 0 for every a, b, c guarantees the absence of gauge anomalies. The [SU (2)L ]3 and SU (2)L [U (1)Y ]2 triangle graphs vanish automatically due to the traceless Pauli matrices, while [U (1)Y ]3 and [SU (2)L ]2 U (1)Y anomalies are proportional to the trace over all doublets Trd , of the electric charge Q, which is given by Trd Q = Nf [Nc (Qu + Qd ) + Qℓ ] = 0 ,

(4.6)

with Nf , Nc being the number of fermion generations and colors respectively, Qx , x = u, d, ℓ is the electric charge of the up, down, and electron respectively. Conventionally, the contribution of doublets is written before the ones of singlets, quarks before leptons and all fermions are left-handed. The contributions of SU (3)C to the SM anomaly, [SU (3)C ]3 , SU (2)L [SU (3)C ]2 , SU (3)C [SU (2)L ]2 and SU (3)C [U (1)Y ]2 also vanish due to the generators, namely the Pauli and Gell-Mann matrices, being traceless. The remaining anomaly, [SU (3)C ]2 U (1)Y ,

62


is proportional to the trace of the electric charge over the quarks, Trq Q, which is Trq Q = Nf Nc [(Qu − Qd ) − Qu + Qd ] = 0 .

(4.7)

Therefore the SM is anomaly free and extending its matter content does not create anomalous triangle graphs as long as those particles are singlets with respect to the SM gauge group. Including the U (1)B−L gauge factor and reevaluating the expressions (4.5) yields vanishing contributions from the anomalies which involve a single SU (3)C or SU (2)L factor, and the following non-trivial contributions for each generation: 1 1 1 =0 [SU (3)C ] U (1)B−L : Trq (YB−L ) = 3 2 · − − 3 3 3 1 2 [SU (2)L ] U (1)B−L : Trd (YB−L ) = 2 3 · − 1 = 0 3 8 2 9 1 − − − −1=0 [U (1)Y ]2 U (1)B−L : Tr(YB−L Y 2 ) = 18 18 18 18 1 2 1 2 [U (1)B−L ]2 U (1)Y : Tr((Y YB−L )= − + −1+1=0 9 9 9 2 1 1 3 [U (1)B−L ]3 : Tr(YB−L ) = − − − 2 + 1 = −1 9 9 9

2

(4.8) (4.9) (4.10) (4.11) (4.12)

Considering also the U (1)-gravitational anomalies, the associated expressions are U (1)Y − grav. : U (1)B−L − grav. : Tr(YB−L ) =

Tr(Q) = 0 2 9

−

1 9

−

1 9

− 2 + 1 = −1 .

(4.13) (4.14)

For the anomalies (4.12) and (4.14) to be zero, one right-handed lepton has to be added per generation. Not to spoil the other anomalies this right-handed lepton needs to be a singlet with respect to the SM gauge groups. We identify this particle with the righthanded neutrino which is added to the matter content of the theory to keep it free of anomalies. Adding more fermions, which are a singlets with respect to the SM gauge groups but have nonzero YB−L charge requires for example the inclusion of partner fermions with the opposite YB−L charge in order not to spoil (4.12) and (4.14).

4.1.3

Scalar Sector

The complex scalar field χ is added to the scalar sector of the model. This field is a singlet with respect to the SM gauge group and carries B-L number, for the spontaneous symmetry breaking of the U (1)B-L . Together with the SM Higgs doublet, the Scalar Lagrangian is given by [87] Ls = (Dµ H)† Dµ H + (Dµ χ)† Dµ χ − V (H, χ) ,

(4.15)

63 with the potential V (H, χ) = m2 H † H + µ2 | χ |2 +λ1 (H † H)2 + λ2 | χ |4 +λ3 H † H | χ |2 ,

(4.16)

where H and χ are the complex scalar Higgs doublet and singlet fields, respectively. If the parameters of the potential satisfy λ1 λ2 −

λ23 > 0 λ1 , λ2 > 0 , 4

(4.17)

V (H, χ) is bounded from below for hHi, hχi = 6 0. Then the two fields develop vacuum

expectation values (vevs). In unitary gauge, the scalars can be parametrized as 0

|H| ≡

h+v √ 2

!

, |χ| ≡

h′√ +v ′ 2

,

(4.18)

where h, h′ are real fields and the vevs v, v ′ are determined by the evaluation of the minimum of the potential (4.16): −4λ2 m2 + 2λ3 µ2 , 4λ1 λ2 − λ23 −4λ1 µ2 + 2λ3 m2 . 4λ1 λ2 − λ23

v2 = v ′2 =

(4.19) (4.20) (4.21)

From the scalar potential the mass matrix of the h, h′ fields reads M02

2λ1 v 2

=

λ3 v v ′

λ3 v v ′ 2λ2 v ′2

!

.

(4.22)

Diagonalizing M02 leads to a light and a heavy Higgs field, which are linear combinations of the interaction fields h, h′ : h1 h2

!

=

cos α − sin α sin α

cos α

!

h h′

!

,

(4.23)

with the masses m2h1

2

= λ1 v + λ2 v −

m2h2

= λ1 v 2 + λ2 v ′2 +

The mixing angle − π2 ≤ α ≤

π 2

′2

q

q

(λ1 v 2 − λ2 v ′2 )2 + (λ3 v v ′ )2 ,

(4.24)

(λ1 v 2 − λ2 v ′2 )2 + (λ3 v v ′ )2 .

(4.25)

is a free parameter of the model. It can be expressed as:

tan 2α =

λ3 vv ′ . λ1 v 2 − λ2 (v ′ )2

(4.26)

64

4. LEPTONIC DARK MATTER The gauge boson mass spectrum is derived from the kinetic terms of the scalar

bosons. In the unitary gauge parametrization the kinetic terms are 1 (h + v)2 g 2 |W1 − W2 |2 + (gW3 − gY B − g2 B ′ )2 , 8 1 ′ (h + v ′ )2 (gBL YB-L (χ)B ′ )2 , 2

(Dµ H)† Dµ H = (Dµ χ)† Dµ χ =

(4.27) (4.28)

where the terms |∂ µ h, h′ |2 have been omitted as well as the vector indices of the gauge

fields. YB-L (χ) is the baryon-lepton charge of the χ field. From eq. (4.27), the SM relation for the W boson mass, mW = g v/2, is recovered. The W3 , B, B ′ fields mix with each other 



Bµ



cw −sw cθ′

    W3µ  =  sw Bµ′ 0

cw cθ ′ sθ ′

sw sθ ′



Aµ



  −cw sθ′   Zµ  Zµ′ cθ ′

(4.29)

with the mass eigenstates Aµ , Zµ , Zµ′ being the photon, Z- and Z ′ -boson, sw (cw ) the sine (cosine) of the weak mixing angle θw and sθ′ (cθ′ ) the sine (cosine) of the mixing angle − π4 ≤ θ′ ≤ π4 . This angle is related to the coupling constants in the following way p 2 g2 Σg , tan 2θ = ΣB-L − Σg ′

with ΣB-L = g22 + 4

v ′2 2 2 Y (χ) gBL v 2 B-L

and

(4.30)

Σg = g 2 + gY2 .

(4.31)

The masses of the physical eigenstates are given by MA = 0 , MZ

=

v 2

MZ ′

=

v 2

(4.32)

1 q 2 1 2 (ΣB-L + Σg ) − (ΣB-L − Σg ) + (2g2 )2 , 2 1 q 2 1 2 2 . (ΣB-L + Σg ) + (ΣB-L − Σg ) + (2g2 ) 2

(4.33) (4.34)

In the limit of the mixing coupling g2 → 0, the Z ′ and Z boson masses are given by the

following equations:

v MZ = 2

q g 2 + gY2

and

2 MZ ′ = YB-L (χ) gBL v ′ .

(4.35)

Note that the SM value for the Z−boson is recovered. The mixing of the Z− and the Z ′ −bosons is not arbitrary. LEP-I measurements at the Z-boson peak constrain the

mixing angle to be less than O(10−3 ) [88]. Further, from a combination of LEP-I and LEP-II data the mass-over-coupling ratio of the Z ′ boson is bounded to be bigger than several TeV. In Ref. [76] the authors reanalyzed the LEP data for a model with the same gauge sector as the one presented in this chapter, while a specific bound for g2 = 0 can

65 Field SU (2)L Y Y B-L Z2

ℓL 2 1 − 2 −1 +

eR 1

νR 1

S1 1

S2 1

−1

0

0

0

−1 +

1 3

− 13

−1 +

+

−

H 2 1 2 0 +

χ 1 0 2 3

+

Table 4.1: Quantum number and Z2 parity assignments for chiral fermion and scalar fields. For a definition of the respective particle, see text. also be found in Refs. [73, 89].

4.1.4

Yukawa Sector

Three right-handed neutrinos νR and Nℓ pairs of right-handed singlet fermion fields S1i , S2i are introduced. Anomaly cancellation demands YB-L (S1i ) = −YB-L (S2i ). Further,

a global Z2 symmetry is added, to avoid S1 –S2 field mixing.

Considering renormalizable operators only, the neutrino terms of the Yukawa sector Lagrangian read s2 s1 D e R − y N (νR )c S1 χ − yij ℓLk Hν (S1i )c S1j χ† − yij (S2i )c S2j χ + h.c. , (4.36) LYν = −ykl ki i k l

where the indices k, l ∈{1, 2, 3} run over the SM generations and i, j ∈{1, ..., Nℓ }.

H, χ are the SU (2)L , U (1)B-L –symmetry breaking Higgs fields, ℓL and νR are the left-

handed lepton doublet and right-handed neutrino singlet, respectively, and S1 , S2 are right-handed fermions which are singlets with respect to the SM gauge group. The gauge invariance of LYν enforces the conditions YB-L (S1 ) + YB-L (χ) = −YB-L (νR )

and

2YB-L (S1 ) = ±YB-L (χ) .

(4.37)

With YB-L (νR ) = −1 due to demanding the cancellation of the B-L -anomaly, two sets

of solutions exist. One set is described in Ref. [90]. The set investigated in the following assigns the charges YB-L (S1 ) =

1 3

and

YB-L (χ) =

2 , 3

(4.38)

which fixes YB-L (S2 ) = −1/3. Because their YB-L charge is a fraction of the SM leptons and they do not carry a color charge, the fermionic S1,2 fields are labeled “leptinos”.

A summary of the particle content and the respective quantum numbers is found in Table 4.1. After spontaneous breaking of the SU (2)L and U (1)B-L symmetries the two Higgs fields acquire vacuum expectation values. The first three terms of eq (4.36) lead to a

66


non-diagonal mass matrix for the interacting neutrinos, in the (νlc , νR , S1 ) basis: 

0

 † M I =  MD 0

where

MD

 MN  , MS1

0 MN

yN MN = √ v ′ , 2

yD MD = √ v , 2



0

MS1 =

(4.39)

√

2 y s1 v ′ .

(4.40)

While it is not restrictive to consider the y N as real 3×3 matrices for Nℓ = 3, the Yukawa couplings y ν and y s1 are in general 3 × 3 complex matrices. In the limit of MS1 ≪ MN eq. (4.39) can be diagonalized, which yields the 3 × 3 neutrino mass matrices Mν l Mν2h = Mν2h′

−1 T −1 ∼ MD MN MS1 (MN ) (MD )† ,

(4.41)

2 2 ∼ MD + MN ,

(4.42)

where the index νl , νh , νh′ refers to the mass eigenstates, given by one light and two almost mass degenerate neutrino fields. Eq. (4.39) is referred to as the inverse seesaw mechanism in traditional literature [78, 79, 80, 81]. For simplicity, the case Nℓ = 1 is considered in the following, and the index i is omitted unless otherwise specified. By convention, the inverse seesaw mechanism is implemented in the third generation of leptons only. y N is now a 3–component vector and a suitable basis can be chosen such that y N and the parameter y s1 are real. The above formulas simplify so that M I is a 3 × 3 matrix and eq. (4.42) gives the masses of the 3 neutrino eigenstates, where νh and νh′ combine into a quasi-Dirac fermion.

The left- and right-handed neutrinos of the first two generations obtain a Dirac mass term, for which O(10−12 ) Yukawa parameters are required to fit the light neutrino masses. The third generation neutrino mass matrix is given by eq. (4.39). It can be diagonalized to yield the physical mass eigenstates of the neutrinos as linear combinations of the favor eigenstates



νl





νL



     νh  = U  νR  , νh′ S1

(4.43)

with the 3 × 3 unitary matrix U being parametrized as 

1

 U = 0

0 C23

0 −S23

0

 

C13

0 S13

 

C12

S12

0



     S23   0 1 0   −S12 C12 0  , C23 −S13 0 C13 0 0 1

(4.44)

where Sij (Cij ) = sin αij (cos αij ). In good approximation, the angle defining the mixing between νR and S1 is found to be very close to maximal, i.e., α23 ∼ π/4, while the angle

2 ≪ which parametrizes the mixing between νL and S1 is very small: S13 ∼ MD MS1 /MN

67 1. Altogether, U is simplified to 

C12

S12

0

S √12 2

C √12 2 − C√122

√1 2 √1 2

 S12 √ U ≃  − 2



 , 

(4.45)

yD v MD = N ′ controls the mixing between νL and νR . Neglecting interMN y v generational mixing, eq. (4.42) can be rewritten as

where S12 ∼

Mν l ∼

−1 T −1 ∗ T MD MN MS1 (MN ) (MD )

=

√

2y

s1

yD yN

2

v2 . v′

(4.46)

To ensure compatibility with LEP exclusion limits, we consider v ′ = O(10) TeV. Limits on the SM-like neutrino require Mνl < 1 eV. In this regime eq. (4.46) reduces to y

s1

yD yN

2

< 10−10 .

(4.47)

If the heavy neutrino masses are assumed to be in the range of O(100) GeV, Yukawa couplings are y N ∼ O(10−2 ). Then y D ∼ O(10−5 ) implies y s1 < O(10−3 ). In the non-interacting neutrino sector the S2 mass is given by MS2 =

√

2 y s2 v ′ ,

(4.48)

which is unconstrained by neutrino physics and is thus a free parameter.

4.2

Dark Matter Study

For the leptino field S2 , as introduced in the previous section, there are no decay processes since the field is odd with respect to the Z2 symmetry. Without a decay channel, the S2 leptino is stable and thus a candidate for WIMP-like cold dark matter. To investigate the leptino relic abundance, the free parameters of the model have to be considered. The scalar sector entails the two masses Mh1 , Mh2 , the two vevs v, v ′ and the mixing angle α. Due to the recent LHC [91] discovery, the mass Mh1 = 125 GeV and v = 246 GeV are fixed. Also, though not yet strongly constrained, it is implied that sin(α) ≤ 0.2. The scale of the second vev is assumed to be v ′ = O(10 TeV), which defines the B-L breaking scale, while the mass Mh2 remains unconstrained.

The Yukawa couplings of the extra neutrino fields are largely unconstrained and D = 10−4 , can be considered as free parameters as well. For concreteness, the values yν3 M = 0.06 and y S1 = 10−5 are used, in order to obtain masses m yν3 νl = 0.1 eV and 3

Mνh = Mνh′ = 600 GeV for v ′ = 10 TeV.

68


sin(α)=0.1 sin(α)=0.01

Ωh2

10000

100

1

200

400

600 800 1000 1200 1400 MS [GeV] 2

Figure 4.2: DM relic abundance as a function of the candidate mass for MH2 = 800 GeV and MZ ′ = 2 TeV, for two different choices of the scalar mixing angle. For the other parameters, see the text.

69 The neutrino Yukawa couplings, the mass Mh1 and the vevs v and v ′ as well as α will stay fixed to these values unless otherwise specified. The free parameters are therefore MZ ′ , MH2 , gBL , g2 and the dark matter mass MS2 . For a first assessment, the values sin(α) = 0.1, gBL = 0.1, g2 = 0, MH2 = 0.8 TeV and MZ ′ = 2.0 TeV are used. The coupling constant gBL and the Higgs mixing angle α are expected to have a strong effect on the computation of the relic abundance, as they control the interaction strength of the Z ′ boson and of the Higgs bosons, respectively, with the DM particle and with the remaining particles of the model. For MS2 = 500 GeV, the computation of the relic abundance yields [40, 41] Ωh2 = 662 .

(4.49)

Varying the above fixed Yukawa couplings by two orders of magnitude does not affect this numerical result, leading to the conclusion that the masses of the extra neutrinos do not affect the computation of the relic density in a significant way. We remark, that this statement is relaxed later, due to the Z ′ resonance and its relevance for the evaluation of the abundance constraint. Figure 4.2 shows the relic density as a function of the DM mass MS2 for two different values of sin(α). It is apparent, that for MS ∼ 6 MZ ′ /2 and MS ∼ 6 MH /2, Ω is = = 2

2

2

always several orders of magnitude larger than the WMAP7 observation. To achieve

compatibility between computation and observation, the leptino relic density has to be reduced, which can be done in two different ways: one is to reduce the vev v ′ while keeping MS2 fixed (increase of the Yukawa coupling), which will make the heavy Higgs resonant annihilation more efficient. The other possibility is to increase the coupling gBL to enhance the Z ′ annihilation efficiency. However, an arbitrarily small relic density is forbidden, due to the upper limit on gBL due to RGE analysis [92] and to the LEP-I lower limit on v ′ : v ′ > 9 TeV

for

g2 = 0 .

(4.50)

This first assessment leads to the conclusion, that for the S2 particle to be the dark matter, it must annihilate efficiently via a resonant heavy Higgs or via the Z ′ boson. Also the precise value of α is not relevant in the resonant regions as long as it is small.

4.2.1

Z ′ Boson Resonant Annihilation

Figure 4.3 shows in more detail that the annihilation is only sufficient to match the abundance constraint around the Z ′ resonance. Due to LHC direct searches the Z ′ boson mass has to be larger than O(2 TeV), implying for the DM mass scale to be O(1 TeV). As a starting point for the investigation the mixing gauge coupling is set to g2 = 0. The abundance thus is inversely proportional to the square of the gauge coupling: Ωh2 ∼ (gBL )−2 .

70


10000

gBL = 0.1 gBL = 0.21 gBL = 0.5

1000

Ωh

2

100 10 1 0.1 0.01 0.6

0.7

0.8

0.9

1 1.1 MS [TeV]

1.2

1.3

1.4

1.5

2

Figure 4.3: Relic density as a function of the DM candidate mass around the Z ′ peak. The lower curve, for gBL = 0.5, is for illustrative purposes only, being already excluded by LEP.

MZ’ = 2 TeV, MS = 0.99 TeV

MZ’ = 3 TeV, MS = 1.47 TeV

2

2

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

Ωh

2

0.3

0.05

0.05 gBL = 0.21 gBL = 0.20

0 -1

-0.5

0 g2

0.5

gBL = 0.37 gBL = 0.32

0 1

-1

-0.5

0 g2

0.5

1

Figure 4.4: Variation of the relic density (at the Z ′ resonance) with g2 for a choice of gBL (see the text for further details), for (left) MZ ′ = 2 TeV and (right) MZ ′ = 3 TeV.

71

p,p → Z' → e e +µ µ gBL

+ -

0.8 g2min

U(1)B-L

0.6

+ MZ' = 2.5 TeV MZ' = 2.2 TeV MZ' = 2.0 TeV MZ' = 1.8 TeV MZ' = 1.5 TeV MZ' = 1.0 TeV

0.4 0.2 0 -1

-0.5

0

0.5 g2

√ Figure 4.5: Z ′ exclusions from CMS data, at s = 7 TeV for the combination of 4.7 fb−1 in the electron channel and 4.9 fb−1 in the muon channel. The dotted black lines refer to the two main benchmark models of this analysis.

MZ ′ (TeV) 2.5 2.2 2.0 1.8 1.5 1.0 0.5

gBL for g2min > 0.8 0.81 0.58 0.39 0.24 0.13 0.05

gBL for g2 = 0 0.75 0.45 0.31 0.22 0.13 0.08 0.03

Table 4.2: 95% C.L. exclusions for the benchmark models of interest. Couplings smaller than those in the table are allowed.

72


1.2 1

0.6

gBL

0.8 0.4

0.6 0.4

0.2 LEP exclusion LHC exclusion g2 = 0

0 1000

2000 3000 MZ’ [GeV]

0.2 g2 = g2min(gBL)

0

4000

5000

1000

2000 3000 MZ’ [GeV]

4000

5000

Figure 4.6: Existence of a suitable DM candidate: the allowed region is the one above the dashed curves, in the gBL − MZ ′ plane, Left: for g2 = 0. Right: for g2 = g2min (gBL ), which minimizes the Z ′ width, hence allowing for a smaller gBL (and therefore a smaller cross section) per fixed MZ ′ . The red shading combinations are forbidden by LEP (eqs. (4.50)–(4.56), respectively), the black (solid) lines are the LHC exclusion, as in table 4.2. Through eq. (4.34) the lower limit on the vev v ′ translates into an upper limit on the Z ′ parameters: 3 MZ ′ v′ = 2 gBL

s

1−

4MZ2 ′

g22 v 2 . − v 2 (g 2 + gY2 )

(4.51)

In Figure 4.3 the Z ′ mass is 2 TeV, which yields the upper bound for the B-L−coupling gBL ≤ 1/3 once the limit from (4.50) is used. Demanding Ωh2 ≃ 0.1 imposes the lower

limit of gBL > 0.21, as in Fig. 4.3, thus constraining the gauge coupling to a narrow range for a specific MZ ′ . To investigate the effect of the gauge coupling g2 on Ω, we recall that Ω ∼ 1/σ. Since

the process under consideration is resonant, it is

P a2S2 i a2i Pi σ= . m2Z ′ Γ2Z ′

(4.52)

with the phase space factor Pi and ai containing spin-related and combinatorial factors and the coupling of particle i to the Z ′ −boson (Yi g2 + YB-L i gBL )2 . Performing the sum P over all particles coupling to the Z ′ −boson it is i a2i Pi → ΓZ ′ . Thus the relic density

Ω becomes proportional to the Z ′ width ΓZ ′ , which means that reducing the width also

leads to a reduction of the relic density. The total Z ′ width in the approximation of massless fermions and neutrinos, with Nℓ generations of leptinos, and with massive and degenerate S2 fields is given by: ΓZ ′ =

MZ ′ 2 A1 gB-L + A2 g22 + A3 g2 gBL , 12π

(4.53)

73 with the quantities A1 A2 A3

s 2MS2 2 1 Nℓ 1− + = 8− 2 18 MZ ′ Nℓ = 5− 8 13 3 − Nℓ + . = 2 2

(4.54)

Interpreting eq. (4.53) as a function of g2 and keeping all the other parameters fixed, the value for g2 that minimizes ΓZ ′ can be derived as a function of the parameters gBL and Nℓ : g2min (gBL ) = −2

16 − Nℓ gBL . 40 − Nℓ

(4.55)

For g2 6= 0, the LEP constraints on the vev v ′ given in (4.50) have to be adjusted. For Nℓ = 1, the LEP-I constraint is

v ′ > 6.7 TeV

with

g2 = g2min ,

(4.56)

which in turn relaxes the upper limit on gBL . For instance for the same setting as in Figure 4.3 the upper limit is now given by 0.448 instead of 0.333. To quantify the effect of g2 on Ω two sets of values are chosen for MZ ′ and gBL such that for g2 = 0 the abundance constraint is just met, then g2 is varied between -1 and +1 for both sets. Figure 4.4 shows that in general Ω increases by a factor ≃ 3 for

|g2 | → 1, but it is reduced when g2 assumes small and negative values and minimized

for g2 = g2min . Therefore, fixing g2 = g2min (gB-L ) according to eq. (4.55), minimizes the relic density which in turn allows for a smaller value of gBL still matching the abundance constraint. This lowers the limit on gBL by roughly 5% compared to the one given for g2 = 0. LHC direct searches for a Z ′ −boson allow to constrain the coupling parameters gBL

and g2 . Figure 4.5 presents the 95% C.L. exclusion limits based on the CMS data for √ the combination of 4.7(4.9) fb−1 in the electron (muon) channels at s = 7 TeV [83]. In table 4.2 the inferred upper limits on the coupling parameter gBL for a given Z ′ boson mass are shown for the cases g2min and g2 = 0. The ATLAS analysis for ∼ 5 fb−1 [84] is

less constraining than the CMS results, and shall therefore be neglected.

Next, the dependence of the leptino abundance on the parameters gBL and MZ ′ is discussed. For parameter sets (gBL , MZ ′ ), the abundance is computed as a function of the leptino mass MS2 . If a value for the mass exists, that produces an abundance compatible with the abundance constraint the point in parameter space under consideration is accepted, otherwise it is rejected. In this manner, the parameter space is scanned in the range 0.5 TeV ≤ MZ ′ ≤ 5 TeV and for the two cases g2 = 0 and g2 = g2min (gBL ). We note, that the resonance condition needs to be fulfilled throughout the scan.

74

4. LEPTONIC DARK MATTER Figure 4.6 shows the results of the above described investigation, the left (right) panel

represents the case of g2 = 0 (g2min ). The respective boundaries are representing LEP, LHC and abundance constraint respectively. Parameter sets for mass and coupling of the Z ′ boson on the right of the line denoting the LHC constraint, below the one denoting the LEP-I constraint and above the line denoting the abundance constraint, are allowed. The Z ′ boson parameters are constrained to MZ ′ ≤ 3.5 (5.0) TeV and gBL ≤

0.58 (1.12) for the g2 = 0 (g2min ) case by collider searches and cosmological implications. The resonant regime translates the upper limit on MZ ′ to on a upper limit for the S2 mass: MS2 ≤ 2.5 TeV.

4.2.2

Heavy Higgs Boson Resonant Annihilation

10000

v’/TeV = 30 9 < v’/TeV < 11.7

1000

12

Ωh

2

v’ [TeV]

100 10 1 0.1 0.01 200

10 0.2

0.1

0.05

8

LEP excl. for g2=0 LEP excl. for gmin 2

6 300

400 MS [GeV] 2

500

600

1000

2000

3000 4000 MH [GeV]

5000

6000

2

Figure 4.7: Left: DM relic abundance as a function of the DM mass for MH2 = 800 GeV, for some values of the vev v ′ and sin α = 0.1. The blue shading represents values of the vev for which an allowed DM mass exists, for this MH2 , as taken from the right panel. Right: Allowed region for v ′ for which a DM mass yields the correct relic density, as a function of the heavy Higgs boson mass, for the three values of sin α = 0.05, 0.1, 0.3. The LEP exclusion is as in eqs. (4.50)–(4.56). When the main annihilation channel of the S2 is via the heavy Higgs boson, the relic 2 ∼ v ′2 , using density is proportional to the B-L-breaking vev since Ωh2 ∼ 1/σ ∼ 1/ys2

eq. (4.48).

Thus the lower limits on v ′ from LEP translate into a minimal abundance of leptino DM, which is shown in the left panel of fig. 4.7. Demanding the resonant S2 annihilation to proceed via the Higgs channel strongly constrains the vev v ′ from above as a function of the heavy Higgs mass MH2 , v ′ ≤ 11.7 TeV, as can be seen in the right panel of figure

4.7. It is interesting to note that this upper bound on v ′ is independent of the scalar mixing angle α. At the same time, the heavy Higgs mass is constrained from above depending on the value of α. For instance for sin α = 0.1, MH2 ≤ 1.8 (3.0) TeV for

g2 = 0 (g2min ). Considering g2min instead of g2 = 0 leads to a reduction of the lower limit on v ′ , which is the only effect of the parameter g2 in the resonant Higgs regime.

WIMP Nucleon cross section [cm2]

75

1e-43 XENON 100 Exclusion 1e-44 1e-45 1e-46 1e-47 1e-48 200

sin(α) = 0.1 sin(α) = 0.3 300

400

500 600 MS [GeV]

700

800

900

2

Figure 4.8: Spin-independent direct searches for maximum Yukawa couplings yS2 ∼ MS2 /v ′ . Only allowed masses are plotted, from figure 4.7(right), for g2 = g2min .

4.2.3

Direct Detection

The dominant contribution to the spin independent leptino-nucleon cross section σ SI comes from the light scalar boson h1 , which can be interpreted as the SM-like Higgs boson. According to eq. (2.45), σ SI is sensitive to both the masses of the light Higgs boson and of the leptino. The coupling of h1 to the nucleon is modified with a factor cos(α) compared to that of the SM Higgs. The coupling of h1 to leptinos is proportional to sin(α), so that σ SI is proportional to (sin (α) cos (α))2 . Figure 4.8 shows that, even for the rather large value of sin α = 0.3, σ SI is several orders of magnitude below the exclusion limits from XENON100. At most it is O(10−44 )cm2 for a DM candidate mass of around 50 GeV. Note that the curve representing sin(α) = 0.3 extends only up to MS2 = 500 GeV, due to the constraints from subsection 4.2.2. Considering the spin-dependent leptino-nucleon cross section σ SD , we notice, that the Z − Z ′ −mixing leads to a non-trivial dependence of the cross section on the parameters MZ , MZ ′ , gBL and g2 , while the parameters of the scalar sector, namely the masses Mh2 , Mh1 and the mixing angle sin(α), are negligible. For MZ ′ /gBL > MZ /g, σ SI becomes even smaller compared to the purely Z boson mediated one. Therefore only for excluded Z ′ −masses below 90 GeV the cross section becomes bigger than 10−10 pb. For MS2 , MZ ′ = 1 GeV a maximum of σ SI → 10−2 pb is reached.

76

4. LEPTONIC DARK MATTER However, for MZ ′ > 100 GeV the cross section σ SI drops to O(10−40 ) pb, which is

far below the current experimental limits of O(10−5 ) pb [30, 39].

4.2.4

Extension to Nℓ Families

Extending the number of leptinos from Nℓ = 1 to Nℓ = 3 leads to an application of the inverse seesaw mechanism to more generations, which in turn leads to an enriched fermion spectrum, with more heavy neutrino particles. This affects Ω in two ways: through the total Z ′ width ΓZ ′ and through the number of effective degrees of freedom gef f (eq. (2.33)). To quantify the resulting effect on Ω, the change of effective degrees of freedom is considered first. In the SM, gef f = 106.75 for temperatures bigger than the top-quark mass. In the leptino model with Nℓ = 3, there are three νR neutrinos and six S1 and S2 leptinos in addition to the matter content of the SM. With the spin-statistical factor of 7/8 and the two degrees of freedom per Majorana field this yields 3 ∆gef f =

7 · 2 · (3 + 6) = 15.75 . 8

(4.57)

In the case of Nℓ = 1 there are again three νR and additionally two leptinos, which yields 1 ∆gef f = 7/8 · 2 · 5 = 8.75 .

(4.58)

i SM i Together with the SM value, the ratio of gef f = gef f + ∆gef f , i = 1, 3 is 1 gef f 3 gef f

= 0.94 .

(4.59)

The effect from changes of the Z ′ width, due to eq. (4.53), yields a value of ΓZ ′ at the minimum which is different for the cases Nℓ = 1, 3. Direct computation yields Γ1Z ′ = 1.09 . Γ3Z ′

(4.60)

However, for MZ ′ fixed, the masses of the fermions depend on the parameter gBL through eq. (4.51) provided that their Yukawa couplings remain fixed as well. Threshold effects also can reduce the branching ratios Z ′ → νh,h′ , which affects ΓZ ′ according to the number of heavy neutrinos, implying that Γ3Z ′ is affected differently than Γ1Z ′ .

In Figure 4.9, this is shown in detail for y N = 0.042. The larger partial decay width into light neutrinos for Nℓ = 1 is almost exactly compensated. In the chosen setup for the Yukawa couplings, the fractional variation of the total Z ′ width is below 1% in most of the parameter space, getting above 5% only in a limited region around small values of the gauge couplings.

77 By making use of eq. (4.60), the variation of Ω with Nℓ can be quantified by the conservative, combined ratio:

Ω1 . 1.03 . Ω3

(4.61)

Notice, that this ratio is of the same order as the uncertainties of the computation by MicrOMEGAs, and the one from the WMAP analysis.

10

MZ' = 2.0 TeV

0.8

gBL

ΓZ' (GeV)

MZ' = 2.0 TeV

(1G-3G)/1G δΓZ' = 15 % δΓZ' = 5 % δΓZ' = 1 % δΓZ' = 0 δΓZ' = -1 %

0.6

2 gBL=0.4 gBL=0.2 gBL=0.1

0.4

10

0.2 0 -1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1 g2

g2

Figure 4.9: Left: Total Z ′ width for selected gBL values (solid lines refer to Nℓ = 1, dashed lines to Nℓ = 3). Right: percentage variation between the 2 models in the gBL − g2 plane, for MZ ′ = 2.0 TeV. For the heavy neutrino masses, see the text. An effect to be taken into account is the presence of Nℓ Z2 −odd fields S2i . In general

these fields will freeze-out at different times, with the heavier particles decoupling earlier, if they have a mass hierarchy. In this case, the respective abundance is Boltzmannsuppressed, and the contribution of the heavier fields to the DM abundance via decays is further suppressed by the mass ratio. Also the contribution to the total annihilation cross section via coannihilations of the heavier leptinos is Boltzmann suppressed. In the limit of the relative mass differences of the leptinos going to zero, the Boltzmann suppression is inefficient, and each species contributes with Ωi to the total abundance, which is given by ΩNℓ =

X i

Ωi = Nℓ · Ω1 ,

(4.62)

where the last equality is true when the masses are exactly degenerate, and Ω1 is the abundance for the case Nℓ = 1. Due to eq. (4.62), the lower bound on the relevant √ coupling (gBL or yS2 ), given by the abundance constraint, increases by a factor Nℓ . For resonant heavy Higgs annihilation the relevant Yukawa coupling is ys2 = MS2 /v ′ , so for a √ fixed mass MS2 the vev v ′ has to be reduced by Nℓ . The left panel of Figure 4.10 shows that for the cases Nℓ = 1, 2, the constraint on the vev and the abundance constraint are not mutually exclusive for an area of parameter space. However in the case of Nℓ = 3 these two constraints become mutually exclusive. The right panel of Figure 4.10 shows that also the resonant Z ′ annihilation can satisfy the abundance constraint even for Nℓ = 3 and mass degenerate S2 fields. This represents the most constraining scenario, so that all other possible combinations, i.e. when just

78


two generations are degenerate or when a possibly tight mass hierarchy is present, will be between the case of 1 generation and the case of 3 generations, degenerate in mass.

12

1.2

LEP exclusion (gmin 2 ) n=1 n=2

1

LEP excl.

v’ [TeV]

0.8 gBL

10

0.6 0.4

8

n=3 n=2 n=1 LHC exclusion

0.2 0 1000 MH

2

2000 [GeV]

3000

1000

2000 3000 MZ’ [GeV]

4000

5000

Figure 4.10: Left: n ≡ Nℓ = 3 is not shown as always disallowed when Higgs resonant annihilation is considered. Here, sin α = 0.1. Right: Allowed parameter range for resonant Z ′ annihilation for n ≡ Nℓ = 2, 3 families of leptinos. The curves are for g2 = g2min (gBL ).

4.3

Dark Radiation

The total relativistic energy of the universe at the time of last scattering is defined as ρrad = ργ + ρν + δρ ,

(4.63)

where the indices γ, ν refer to the cosmic microwave background (CMB) and the cosmic background of SM neutrinos (CνB) respectively, and δρ denotes a contribution unaccounted for by neutrinos and photons. Eq. (4.63) contains one of the major predictions of standard Cosmology, namely the contribution of SM neutrinos to the relativistic energy content. The non-CMB energy density can be parametrized as ρrad

# " 4 π2 4 7 4 3 (Nν + δN ) , = T 1+ 15 γ 8 11

(4.64)

with Tγ =2.728 K being the present day temperature of the CMB. The parameter Nν is the contribution of the massless SM neutrinos. Including flavour oscillations and the non-instantaneous decoupling of the neutrinos, Nν is computed to be 3.046[28]. In the SM, the parameter δN is zero. In general this parameter is nonzero if other particle species are present which are relativistic at the time of the last scattering. The 9 year results of WMAP yield a value of δN = 3.84 ± 0.40 [7], which seems to indicate

the presence of additional light particle species.

79 The leptino model, as discussed in this chapter, includes three right-handed neutrinos νR , and Nℓ Z2 -even leptinos. Due to the Z ′ boson, coupled to the B-L number, all the fields are in kinetic equilibrium with the thermal bath for high temperatures. If any of these particles is relativistic at the time of the last scattering, it contributes to the relativistic energy content of the early Universe. The leptinos have masses O(1) TeV, and the inverse seesaw mechanism mixes one leptino with one left- and one righthanded neutrino, yielding a light, SM-like neutrino and two heavy fermions. Therefore, the number of relativistic neutrinos in the leptino model is given by three SM-like light neutrinos, plus 3 − Nℓ right-handed neutrinos.

4.3.1

Effective Relativistic Degrees of Freedom

A generic relativistic, right-handed neutrino νR contributes to the energy content of the universe as

7 gR π 2 4 T , 8 30 R

ρR =

(4.65)

where gR = 2 are the internal degrees of freedom of the neutrino and TR is the neutrino temperature. With this definition, the parameter δN as introduced in eq. (4.64) can be expressed as δN =

11 4

4 3

TR Tγ

4

,

(4.66)

which is completely determined by the temperature of the right-handed neutrinos. The latter can be related to the temperature of the thermal bath by considering the thermodynamical properties of the early universe. The entropy density of the thermal bath is defined as s = gef f

T3 , a3

(4.67)

with the number of relativistic degrees of freedom of the thermal bath as defined in eq. (2.14). In general the decoupling of species i at temperature Ti reduces the effective degrees of freedom of the primordial plasma by removing its contribution to gef f . Conservation of the entropy density over the period of decoupling demands i−1 3 i 3 gef f Ti = gef f Ti−1 ,

(4.68)

where the index i (i − 1) refers to the thermal bath before (after) decoupling of species i.

An entropy transfer occurs, which affects the temperature of the remaining thermalized particles: Ti = ci(i−1) Ti−1 ,

with

ci(i−1) =

i−1 gef f i gef f

!1

3

.

(4.69)

i−1 i Since gef f > gef f it follows that Ti < Ti−1 . This is interpreted as a reheating of the

thermal bath through the decoupling of a particle species.

80

4. LEPTONIC DARK MATTER When the next lighter species labeled (i − 1) decouples, it is Ti−1 = c(i−1) (i−2) Ti−2 ,

which ultimately allows to express the temperature of species i in terms of the CMB temperature Ti = ci (i−1) c(i−1) (i−2) ... ce γ Tγ .

(4.70)

The above can be summarized in the simple expression Ti = c i γ T γ ,

(4.71)

quantifying the suppression of the temperature of species i with respect to the photon background temperature. By substituting eq. (4.71) into eq. (4.66) and summing over the 3 − Nℓ fermionic νR ,

the contribution of the right-handed neutrinos is obtained: X R

δNR = (3 − Nℓ )

11 R 2 gef f

!4

3

,

(4.72)

R is defined by the decoupling temperature of the right-handed neutrinos T . where gef R f

If the decoupling of the νR occurs before the QCD phase transition, the hadronic R ∼ 100, which yields δN ∼ (3 − N )0.021. This phase degrees of freedom render gef R ℓ f

transition is assumed to take place between 150 and 500 MeV. If the νR are freezing out between the muons and the electrons it is δNR ∼ 0.4 (3 − Nℓ ).

4.3.2

Decoupling Temperature

√ T2 , with gef f as in eq. (2.14), In the early universe, the expansion rate H = 1.66 gef f m P leads to a cooling of the primordial plasma, which in turn suppresses number densities n(T ) and interaction rates Γ(T ) of its constituents. The decoupling of a species i occurs, when its interaction rate Γi becomes smaller than the expansion rate. Subsequently, the decoupling temperature Ti is defined by Γi (Ti ) = 1, H(Ti )

(4.73)

where the interaction rate and number density of species i are defined as Γi =

X f

ni σif |v|

and

ni =

3 ζ(3) gi T 3 , 4 π2

(4.74)

and with the interaction cross section σif , the Riemann zeta function ζ(3) = 1.20206, the degrees of freedom gi and the thermally averaged Møller velocity |v| = 1.

Considering the leptino model, the interaction between the right-handed neutrinos

νR and the SM particles is mediated by the Z ′ boson. Thus the low energy interaction ′ f ′ , with f (f ′ ) being an SM fermion, is cross section for processes of the type νR f → νR

81 given by σRf =

s a2R a2f 12 π MZ4 ′

,

(4.75)

with the center of mass energy s = T 2 , the Z ′ mass MZ ′ and the couplings of species νR , f to the Z ′ aR = |gBL YB-L (νR )|

and

af = |gBL YBL (f ) + g2 Y (f )| .

(4.76)

Notice, that the index R (f ) in eq. (4.75) indicates the right-handed neutrinos (SM fermions). From eq. (4.73), an expression for the decoupling temperature of the right-handed neutrinos is obtained: #1 p 1.66 gef f (TR )π 3 48 MZ4 ′ 3 P . TR = 3 ζ(3) gi mP a2R f a2f "

(4.77)

This relation yields the decoupling temperature for the two different values of g2 = 0, g2min and the according limits on the vev from LEP (see eqs. (4.50), (4.56)). Using Table 4.3 this leads to TR >

(

0.39 ± 0.02 GeV, g2 = 0

g2 = g2min

0.70 ± 0.04 GeV

,

(4.78)

where the two limits represent the uncertainty whether or not to include the quarks into the interaction rate, due to the resulting temperature overlapping with the QCD phase transition. However, in both cases the effective degrees of freedom, gef f , are O(100), which suppresses the contribution of the νR to the energy density according to eq. (4.72). In defining the scale of the decoupling temperature with TR = y · 0.1 GeV, eq. (4.77)

can be rearranged to



MZ ′  = |gBL |

P h f

2 Y (f ) YB-L (f ) + 2 ggBL p gef f (y)

i2

1/4 y 3 gR  2.05 TeV . 

(4.79)

By correlating the B-L breaking vev with the decoupling temperature, via eq. (4.51), an upper limit as a function of the decoupling temperature emerges:  X ′ YB-L (f ) + v