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Phase transitions : applications to physics beyond the Standard Model Blinov, Nikita 2015

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Phase Transitions: Applications to Physics Beyond theStandard ModelbyNikita BlinovMathematical Physics, BSc, University of Alberta, Edmonton, 2010a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Physics)The University Of British Columbia(Vancouver)August 2015© Nikita Blinov, 2015AbstractDespite their phenomenological successes, the Standard Models (SMs) of particle physics andcosmology remain incomplete. Several theoretical and observational problems cannot be ex-plained within this framework, including the hierarchy problem, dark matter (DM), and thebaryon asymmetry of the Universe. The objective of this thesis is to investigate phenomeno-logical and theoretical aspects of the solutions to these issues. We consider two kinds of phasetransitions that can occur in the early or late Universe in extensions of the SM, that can beeither responsible for dark matter and/or baryon asymmetry production or may be used toconstrain possible models of new physics.In the first part we analyze string theory-inspired models where the Universe transitionsfrom matter- to radiation-dominated evolution just before Big Bang Nucleosynthesis throughout-of-equilibrium decays of a scalar modulus field. We employ these decays to produce DMand for baryogenesis. We study the phenomenology of these scenarios and its implications forhigh-scale physics.The second part of this thesis is dedicated to thermodynamic and quantum phase transitionsin the early and late Universe, respectively. In the former case, we investigate the dynamics ofthe electroweak phase transition when the electroweak symmetry is broken down to electromag-netism in the Inert Doublet Model, a simple extension of the SM that can account for DM. Suchtransitions can generate the baryon asymmetry in a process called electroweak baryogenesis.Some extensions of the SM also predict similar transitions through quantum tunnelling thatbreak the colour and electromagnetic symmetries, indicating that our ground state is unsta-ble. We use these arguments to put new constraints on the Minimal Supersymmetric StandardModel.iiPrefaceParts II and III of this thesis are based on published and unpublished work in collaborationwith other authors. Parts I and IV (as well as certain sections in Parts II, III) are originalexpository materials.A version of Chapter 3 has appeared in N. Blinov, J. Kozaczuk, A. Menon, and D. E. Morris-sey, Confronting the moduli-induced lightest-superpartner problem, Phys.Rev. D91 (2015), no. 3035026, [arXiv:1409.1222]. This thesis contains an expanded introduction in Sections 3.1, 3.1.1and 3.1.2. I participated in the construction of models in Sections 3.3, 3.4. I was also respon-sible for the majority of analytic and numerical calculations in Sections 3.2, 3.3, 3.4 and 3.5,as well as their composition. David Morrissey was the supervisory author on this work andcontributed analytic results to these and other sections. Jonathan Kozaczuk provided variousindirect detection constraints in Sections 3.3 and 3.4. Arjun Menon cross-checked the resultsof Sec. 3.2. All authors contributed to the composition of the manuscript.A longer version of Chapter 4 appeared in N. Blinov, D. E. Morrissey, K. Sigurdson, andS. Tulin, Dark Matter Antibaryons from a Supersymmetric Hidden Sector, Phys.Rev. D86(2012) 095021, [arXiv:1206.3304]. I performed the calculations in Section 4.3, and contributedto Section 4.5. Sean Tulin was responsible for Section 4.5.1, while David Morrissey wroteSections 4.4 and 4.5.3. David Morrissey and Kris Sigurdson were the supervisory authors inthis work. All authors contributed to the composition and editing of the manuscript; theappendix that appears in the published work has been excluded from this thesis, because I didnot write it.Chapter 6 is based on N. Blinov and D. E. Morrissey, Vacuum Stability and the MSSM HiggsMass, JHEP 1403 (2014) 106, [arXiv:1310.4174]. I contributed the majority of the numericalresults in Section 6.3.3 and Appendix B as well as the proof in Appendix A. David Morrisseywas the supervising author and contributed Section 6.5. We jointly composed and edited themanuscript.Chapter 7 is based on work in N. Blinov, S. Profumo, and T. Stefaniak, The ElectroweakPhase Transition in the Inert Doublet Model, JCAP 2015 (2015), no. 07 028, [arXiv:1504.05949].Here I contributed most of the analytic and numerical results in Sections 7.3 and 7.5. Tim Stefa-niak was responsible for checking collider constraints for models of Section 7.5. Stefano Profumowas the supervising author and wrote Section 7.4. I composed the manuscript; all authors wereinvolved in editing it.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiI Standard Models and Their Extensions . . . . . . . . . . . . . . . . . . 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 The Standard Model of Particle Physics . . . . . . . . . . . . . . . . . . . . . . . 31.1.1 Symmetries and Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Gauge Fixing and Ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 Spontaneous Symmetry Breaking and the Physical Spectrum . . . . . . . 81.2 The Standard Model of Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.1 The Expanding Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.3 The First 400 000 Years . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3 Outstanding Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.3.1 Hierarchy Problem and Naturalness . . . . . . . . . . . . . . . . . . . . . 191.3.2 Baryon Asymmetry of the Universe . . . . . . . . . . . . . . . . . . . . . . 211.3.3 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Aspects of Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26iv2.2 Superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 The Minimal Supersymmetric Standard Model . . . . . . . . . . . . . . . . . . . 322.4 Supersymmetry Breaking and Supergravity . . . . . . . . . . . . . . . . . . . . . 352.4.1 Global Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . 352.4.2 Super-Higgs Mechanism and Supergravity . . . . . . . . . . . . . . . . . . 382.4.3 Models of Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . 402.5 Status of Experimental Searches for Supersymmetry . . . . . . . . . . . . . . . . 45II Moduli Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Moduli Induced Lightest Superpartner Problem . . . . . . . . . . . . . . . . 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.1.1 Moduli from Compactification . . . . . . . . . . . . . . . . . . . . . . . . 523.1.2 Moduli Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.1.3 Moduli Reheating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.1.4 Non-Thermal Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1.5 Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.6 Gravitino Production and Decay . . . . . . . . . . . . . . . . . . . . . . . 583.2 Moduli Reheating and the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3 Variation #1: Hidden U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.1 Setup and Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.2 Decays to and from the Hidden Sector . . . . . . . . . . . . . . . . . . . . 633.3.3 Hidden Dark Matter from Moduli . . . . . . . . . . . . . . . . . . . . . . 643.3.4 Constraints from Indirect Detection . . . . . . . . . . . . . . . . . . . . . 663.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.4 Variation #2: Asymmetric Hidden U(1) . . . . . . . . . . . . . . . . . . . . . . . 693.4.1 Mass Spectrum and Decays . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4.2 Moduli Reheating and Asymmetric Dark Matter . . . . . . . . . . . . . . 713.4.3 Relic Densities and Constraints . . . . . . . . . . . . . . . . . . . . . . . . 723.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.5 Variation #3: Hidden SU(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.5.1 SU(N)x Mass Spectrum and Confinement . . . . . . . . . . . . . . . . . . 753.5.2 Connectors to the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.5.3 Moduli Reheating and Hidden Dark Matter . . . . . . . . . . . . . . . . . 773.5.4 Hidden Gluino Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 Dark Matter Antibaryons from a Supersymmetric Hidden Sector . . . . . . 84v4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2 Supersymmetric Hylogenesis Model . . . . . . . . . . . . . . . . . . . . . . . . . 864.2.1 Hidden Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.2 Baryon Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3 Hylogenesis Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3.1 CP-violating Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.2 Decays and Annihilations of SUSY States . . . . . . . . . . . . . . . . . . 934.3.3 Boltzmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4 Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.4.1 Minimal Transmission of Supersymmetry Breaking . . . . . . . . . . . . . 1004.4.2 Mediation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.5 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.5.1 Induced Nucleon Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.5.2 Precision Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.5.3 High-Energy Colliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109III Tunnelling and First Order Phase Transitions . . . . . . . . . . . . . 1115 Quantum Tunnelling in Field Theory . . . . . . . . . . . . . . . . . . . . . . . 1125.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.2 Tunnelling at Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.3 Tunnelling at Finite Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186 Charge and Colour Breaking in the MSSM . . . . . . . . . . . . . . . . . . . 1216.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2 Parameters and Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.2.1 Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.2.2 Parameter Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.3 Limits from Vacuum Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.3.1 Existence of a CCB Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . 1266.3.2 Computing the Tunnelling Rate . . . . . . . . . . . . . . . . . . . . . . . . 1266.3.3 Results and Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.4 Implications for the MSSM Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . 1286.5 Comparison to Other Stop Constraints . . . . . . . . . . . . . . . . . . . . . . . . 1326.5.1 Precision Electroweak and Flavour . . . . . . . . . . . . . . . . . . . . . . 1326.5.2 Direct Stop Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.5.3 Stop Bound States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134vi6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347 The Electroweak Phase Transition in the Inert Doublet Model . . . . . . . 1357.1 Electroweak Baryogenesis in the Standard Model and Beyond . . . . . . . . . . . 1357.2 The Inert Higgs Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.3 Phase Transitions in the Inert Doublet Model . . . . . . . . . . . . . . . . . . . . 1407.3.1 IDM at Tree-Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.3.2 Finite-Temperature Corrections . . . . . . . . . . . . . . . . . . . . . . . . 1417.3.3 Electroweak Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . 1447.4 Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.5 Benchmark Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151IV Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1538 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160A Minimality of the Action Under Path Deformations . . . . . . . . . . . . . . 189B An Approximate Empirical Bound for Charge and Colour-Breaking Vacuain the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192C Renormalization Group Equations in the Inert Doublet Model . . . . . . . 195viiList of TablesTable 1.1 Field content of the Standard Model in the gauge eigenbasis. Only one gener-ation of fermions is shown; the other two families have identical gauge charges. 6Table 2.1 Particle content in the MSSM before and after electroweak symmetry break-ing, along with colour, electromagnetic charge and R-parity assignments. . . 35Table 4.1 New superfields in the hidden sector, with quantum numbers under U(1)′, B,and R-parity. Chiral supermultiplets X1,2, Y1,2,H also include vector partnersXc1,2, Y c1,2,Hc with opposite charge assignments (not listed). . . . . . . . . . . 87Table 6.1 MSSM scalar potential parameter scan ranges. The values of other parametersto be considered are described in the text. . . . . . . . . . . . . . . . . . . . . 125Table 7.1 Input parameters for the three benchmark scenarios discussed in the textalong with critical and nucleation temperatures, the transition strength andthe signal strength for h → γγ. The masses, given in GeV, are pole massesand the couplings λi are specified at Q = MZ . Temperatures are also given inGeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146viiiList of FiguresFigure 1.1 Number of energy and entropy relativistic degrees of freedom, g∗ and g∗S ,respectively, for the Standard Model as a function of temperature. The largedrop in the number of degrees of freedom at T ∼ 200MeV corresponds to theQCD phase transition when quarks and gluons became confined in hadrons. 14Figure 1.2 Comoving number density N = nχa3 (solid line) of a self-conjugate particle χas a function of mχ/T . The equilibrium number density is shown as a dashedline. Freeze-out occurs at mχ/T ∼ 20. . . . . . . . . . . . . . . . . . . . . . . 18Figure 1.3 Possible contributions to the Higgs mass parameter from fermions (left graph)and scalars (middle and right graphs). The momentum integrals associatedwith these graphs are ultraviolet (UV) divergent and give rise to the fine-tuning problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 2.1 Renormalization group evolution of the gauge couplings in the SM (dashedlines) and the MSSM (solid lines). In the MSSM gauge couplings appear tounify at around µ ≈ 2× 1016 GeV. . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.2 Example contributions to soft SUSY-breaking parameters from heavy mes-senger loops (bold lines) to MSSM gauginos (left) and sfermions (right) ingauge mediation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 2.3 Typical Large Hadron Collider (LHC) production mechanisms for gluinosand squarks (left) and stops (right). . . . . . . . . . . . . . . . . . . . . . . . 46Figure 2.4 Representative contributions to Bs → µ+µ− in the SM (left) and the MSSM(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 2.5 Representative contributions to electron EDM in the MSSM at one (left) andtwo (right) loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.1 (Left) Solution of the modulus equation of motion for an initial displacementof MPl and modulus mass mϕ = 100TeV. (Right) Evolution of the energydensity in the modulus oscillations (solid) and radiation (dashed). Becausethe oscillation energy density dilutes slower with the expansion of the Uni-verse, it will eventually dominate the energy density in radiation. . . . . . . 56ixFigure 3.2 Relic density and constraints from indirect detection (ID) for a mixed Higgsino-wino LSP produced by moduli reheating as a function of µ/M2 and m3/2.The modulus parameters are taken to be mϕ/m3/2 = 100, c = 1, and Nχ = 1.Contours of the LSP mass in GeV are given by the dashed grey lines. Thesolid black contours show where Ωχ01 = Ωcdm. The solid red line shows whereTRH = Tfo: to the left of it we have TRH > Tfo; to the right TRH < Tfo and theproduction is non-thermal. The remaining lines correspond to bounds fromID for different galactic DM distributions, and the area below and to theright of these lines is excluded. . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 3.3 Contours of the hidden neutralino χx1 mass in GeV (dashed grey) and moduli-generated relic abundance Ωχh2 (solid red) as a function of of µ′/Mx andm3/2. The moduli parameters are taken to be mϕ = m3/2, c = 1, and Nχ = 1,with the hidden-sector parameters as described in the text. . . . . . . . . . . 65Figure 3.4 Constraints from indirect detection on hidden U(1)x neutralino DM producedby moduli decays for mx = mχ/2, ξ = 0.1, as well as (c=1, mϕ=m3/2) (left),and (c = 10, mϕ = 2m3/2) (right). The green shaded region is excluded byFermi GC observations and the blue shaded region is excluded by COMP-TEL. Both exclusions assume an Einasto galactic DM profile. The thick solidand thin dotted contours correspond to the exclusions assuming the NFWand cored profiles, respectively. The green and blue dashed lines show theboundaries of the stronger exclusion obtained assuming a contracted profilewith γ = 1.4. Above and to the right of the solid red line, the hidden LSPdensity is larger than the observed DM density. The solid and dash-dottedorange lines shows the exclusion from deviations in the CMB for f = 0.2 andf = 1, respectively, with the excluded areas above and to the right of thelines. Note that the entire c = 1 parameter space is excluded by the CMBconstraint for f = 1. The gray shaded region at the bottom has a hiddenvector mass mx < 20 MeV that is excluded by fixed-target experiments. . . . 68Figure 3.5 Abundance of Ψ and Φ in the κ − m3/2 plane. The right y axis shows theΨ mass mΨ = µY . Solid red contours show the fraction of the measuredabundance made up by Ψ and Φ and their anti-particles. The dashed greylines show the fractional asymmetry between DM and anti-DM. The blueregion is excluded by the CMB bound and the green by direct detection. . . 72Figure 3.6 Abundance of Ψ and Φ in the gx −m3/2 plane. Solid red contours show thefraction of the measured abundance made up by Ψ and Φ and their anti-particles. The dashed grey lines show the Ψ mass in GeV. The green regionis excluded by direct detection. . . . . . . . . . . . . . . . . . . . . . . . . . . 73xFigure 3.7 Relic abundance of the hidden gluino ˜Gx (solid black) after moduli reheatingas a function of the hidden gauge coupling gx for N = 2, mϕ = m3/2 =100 TeV, c = 1, Nx = 1, and cx/cv = 1/9. The lifetime of the lightest MSSMsuperpartner, assumed to be a Higgsino-like neutralino, is shown in lightblue for µ = 150 GeV, NF = 3, and λu = 0.75. The vertical solid grey linecorresponds to T xRH ≈ Tfo, while the dashed horizontal line shows τχ01 = 1 s. . 79Figure 4.1 The three steps of hylogenesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 4.2 Representative diagrams contributing to X1 → qiq˜Rj q˜Rk decays which areresponsible for the generation of the baryon asymmetry. . . . . . . . . . . . . 92Figure 4.3 Allowed masses for the scalar Φ and fermion Ψ components of dark matter.For a fixed value of nΨ/nΦ, the shaded region shows the entire mass range ofΨ (blue) and Φ (red) that reproduces ΩDM/Ωb ≈ 5 and satisfies the stabilityrequirement |mΨ−mΦ| < me +mp. Shifting ΩDM/Ωb by +(−)6% moves theallowed region right (left), as indicated by the dashed contours. . . . . . . . 93Figure 4.4 Solutions to the reheating Eqs. (4.26a, 4.26b, 4.26c) and DM production,described by Eq. (4.29). Here Nϕ = ρϕa3/mϕ and Ni = nia3 for i =B, Ψ, Φ, Ψ¯, Φ∗. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 4.5 Solution to the Boltzmann equations for the yields Yi = ni/s as a function ofthe scale factor a. The plot on the left shows the evolution for the case whenthe transfer reaction ΦΦ ↔ ΨΨ is turned off, while the plot on the rightshows the outcome when it is active. The transfer drives the dark matterpopulation into lighter state, Ψ in this case. The DM (anti-DM) abundanceis indicated by solid (dashed) lines, with dark (light) lines referring to thefermion (scalar) component. The parameters used are described in the text. 98Figure 4.6 The ratio nΨ/nΦ for the allowed range of mass splittings ∆m = mΦ −mΨand relevant values of the hidden gaugino mass mχ. At each point in theplane the DM abundance is fixed to be ΩDM/Ωb = 5.0. Contours of constantmΨ (in GeV) are also shown. The gray contour shows the CMB constraintfor DM annihilations from Ref. [5]. Points to the right of this line are excluded. 99Figure 4.7 IND processes at leading order in chiral effective theory (left, center). Graydot shows effective B transfer operator, generated by ˜P , X exchange in ourmodel (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Figure 4.8 Total IND cross section (σv)IND = (σv)NΨ→KΦ∗IND + (σv)NΦ→KΨ¯IND over allowedrange for mΨ, with mΦ = (ΩDM/Ωb)mp − mΨ ≈ 5mp − mΨ. The effectivebaryon transfer mass scale is ΛIND = 1 TeV. Cases I, II, III correspond todifferent baryon transfer models considered in Eqs. (4.12,4.52). . . . . . . . . 106Figure 4.9 Proton and neutron lifetimes for different baryon transfer models (cases I,II, III) considered in Eqs. (4.12) and (4.52). Black line/gray regions showlifetime range for any r, while blue curves correspond to particular r values. . 107xiFigure 5.1 (Left) A typical potential with a false vacuum at φ+. (Right) The particlemotion interpretation of the bounce equation of motion. . . . . . . . . . . . 116Figure 5.2 The bounce solution for a two dimensional potential evaluated using twodifferent methods. The left plot shows the field profiles as a function of thecoordinate ρ, while the right plot shows the tunnelling path in the φ1 − φ2plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 6.1 The two loop SM-like MSSM Higgs mass as a function of Xt/mQ3 computedusing FeynHiggs as described in Sec. 6.4. The shaded teal region correspondsto 123 GeV < mh < 127 GeV, a range that encompasses the approximatetheoretical uncertainty in mh around the measured value mh ≈ 125GeV [6].Here we have taken mA = 1000GeV, mQ3 = mU3 = 750GeV and µ =250GeV. These parameters are discussed in Sec. 6.2. . . . . . . . . . . . . . 123Figure 6.2 Limits from metastability and the existence of a local SM-like (SML) vacuumalone for tanβ = 10, µ = 300 GeV, mA = 1000 GeV, and m2U3 = m2Q3 . Allpoints shown have a global CCB minimum and a local SML minimum. Thered points are dangerously unstable, while the blue points are consistent withmetastability. The green dashed line is the analytic bound of Eq. (6.3) andthe black dotted line corresponds to Eq. (6.4), the empirical bound fromRef. [7]. The values of the other MSSM parameters used here are describedin the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Figure 6.3 Metastability bounds relative to the MSSM Higgs mass. The coloured bandscontain models for which 123 GeV < mh < 127 GeV. Pink models havean absolutely stable SML vacuum, blue points have a global CCB minimum,while red points are unstable on cosmological time scales. The green dashedline is the analytic bound of Eq. (6.3) and the black dotted line is Eq. (6.4).The orange dashed line is an approximate empirical bound discussed in Ap-pendix B. The grey dot-dashed contours are lines of constant lightest stopmass (in GeV). MSSM parameters used here are described in the text. . . . 130Figure 6.4 Metastability with the correct Higgs mass, 123 < mh < 127 GeV. The la-belling is the same as in Fig. 6.3, and the relevant MSSM parameter param-eters are varied one at a time as summarized in Table 6.1. . . . . . . . . . . 131Figure 6.5 Points in the Xt−mQ3 plane with 123 GeV < mh < 127 GeV as well as ex-clusions from metastability (red points) from precision electroweak ∆ρ (greenpoints) and flavour BR(B → Xsγ) (orange points). The MSSM parametersused are the same as in Fig. 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . 133Figure 7.1 Schematic representation of electroweak baryogenesis. . . . . . . . . . . . . . 137Figure 7.2 Evolution of the temperature-dependent effective potential (free energy) inthe Standard Model around the critical temperature Tc. . . . . . . . . . . . . 138xiiFigure 7.3 Phase transition strength as a function of the heavier IDM scalar masses,taking mA = mH± . The remaining parameters are chosen as in the bench-mark models of Table 7.1, which are shown by black dots. The lines for BM1and BM3 terminate where the inert doublet develops a non-zero vev, φ 6= 0,as described in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Figure 8.1 Constraints on the spin-independent scattering cross section as a functionof DM mass (for the model of Sec. 3.4) from LUX [8], XENON10 S2 onlyanalysis [9], CDMSLite [10] and CRESST-Si [11]. The dashed lines show theexpected cross section for various combinations of gx (see Eq. (3.53)). . . . . 157Figure B.1 The deterioration of the empirical bound of Eq. (B.6) for m2U3/m2Q3  1or m2U3/m2Q3  1. For these parameter ranges the assumption of SU(3)CD-flatness that motivated Eq. (B.6) breaks down and it cannot be used toreliably model the boundary between the metastable and unstable parameterregions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194xiiiGlossaryADM asymmetric dark matterAMSB Anomaly Mediated Supersymmetry BreakingBBN Big Bang NucleosynthesisCC charged currentCKM Cabibbo-Kobayashi-MaskawaCMB Cosmic Microwave BackgroundDM dark matterEDM electric dipole momentEOM equation of motionEW electroweakEWBG electroweak baryogenesisEWPT electroweak phase transitionEWSB electroweak symmetry breakingFRW Friedmann-Robertson-WalkerGR General RelativityGUT Grand UnificationHS hidden sectorIDM Inert Higgs Doublet ModelIND induced nucleon decayLHC Large Hadron ColliderxivLSP lightest supersymmetric particleMSSM Minimal Supersymmetric Standard ModelNGB Nambu-Goldstone bosonOR O’RaifeartaighQCD Quantum ChromodynamicsSM Standard ModelSSB spontaneous symmetry breakingSUGRA supergravityUV ultravioletvev vacuum expectation valueWIMP weakly interacting massive particleWZ Wess-ZuminoxvAcknowledgementsThis thesis is the culmination of five years of work which would not have been possible withoutthe help and support of many people. First I would like to thank the faculty, post-docs, studentsand staff at TRIUMF, UBC and UC Santa Cruz. I am especially grateful to Peter Winslowand Jonathan Kozaczuk with whom I had many illuminating discussions about field theory, theUniverse and everything.All research presented in this thesis is the product of a collective effort. I am indebted tomy collaborators Sean Tulin, Kris Sigurdson, Jonathan Kozaczuk, Carlos Tamarit, Stefano Pro-fumo, Tim Stefaniak and Arjun Menon for their guidance and patience for my many questions.I am grateful to my doctoral committee, Jeremy Heyl, Alison Lister, David Morrissey andKris Sigurdson, for carefully reading this manuscript and for providing feedback throughoutthe program.Most importantly, I would like to thank David Morrissey for being the best advisor a studentcan imagine. His mentorship, support and encouragement have been invaluable. Thank youfor sharing your passion for physics and for always keeping your door open.I am fortunate to able to thank many friends in Vancouver, Edmonton and abroad. Thankyou for regularly pulling me away from theory back into the real world.Finally, I could not have completed this work without the incredible support from my familyin Canada and Russia.Thank you, all of you.xviПосвящается маме и папе, за бесконечную помощь и любовьи Даше Зинченко, потому что она очень просилаxviiPart IStandard Models and TheirExtensions1Chapter 1IntroductionThe Standard Model (SM) of particle physics has proven to be an extremely successful theoryof strong and electroweak interactions. The Higgs boson, responsible for the breaking of theelectroweak (EW) symmetry, has finally been observed at the Large Hadron Collider (LHC) [12,13], confirming a 50 year old theoretical prediction by Higgs [14], Englert and Brout [15] andGuralnik, Hagen and Kibble [16].1 All SM parameters have been measured. On the cosmologicalfrontier, the Planck experiment has observed the Cosmic Microwave Background (CMB) withunprecedented precision, with the resulting data in near-perfect agreement with the StandardModel of cosmology [17], confirming our understanding of the physical processes that occurredover 13 billion years ago.These are just some of the latest successes of the Standard Models in a long history ofcorrect predictions that included new fundamental particles like the electroweak gauge bosonsand the top quark, and baryon acoustic oscillations and the CMB power spectrum. There are,however, several strong indications of the existence of physical phenomena unexplained by theSMs. Before describing these shortcomings and thereby motivating the following chapters, wediscuss the structure of the SM in Sec. 1.1. The essential aspects of the Standard Model ofcosmology are described in Sec. 1.2. In Sec. 1.3 we outline some of the central issues that themodels of new physics we consider attempt to address. These include the hierarchy problem,the nature of dark matter and the creation of the baryon asymmetry of the Universe.In this thesis we use the following conventions. For the spacetime metric we use the “mostlyminus” version with ηµν = diag(1,−1,−1,−1) in flat space. Unless stated otherwise, formulaeare given in natural units with ~ = c = 1, so all dimensionful quantities are measured in unitsof energy (typically GeV).1For reasons of brevity we will refer to the Higgs-Englert-Brout-Guralnik-Hagen-Kibble boson as the Higgsboson as is commonly done.21.1 The Standard Model of Particle Physics1.1.1 Symmetries and InteractionsThe fundamental interactions among known particles are well described by gauge field theories.The SM is a gauge field theoretic description of the strong, weak and electromagnetic forces. Allinteractions are completely determined by symmetries and particle content of the theory. Thegauge symmetry of the SM is SU(3)C × SU(2)L × U(1)Y . The first factor, SU(3)C describesQuantum Chromodynamics (QCD) - interactions of quarks and gluons, which at low energiesmanifest themselves in the existence of colour-neutral hadrons, such as protons and neutronsand their binding into nuclei. The remaining factor SU(2)L×U(1)Y is the electroweak theory: aunification of the weak and electromagnetic interactions, notably responsible for β radioactivityof certain nuclei and the existence of atoms.Gauge symmetry is an invariance of a theory under local (i.e., spacetime dependent) trans-formations. The implementation of such symmetries requires the existence of a vector gaugefield in the adjoint representation of the corresponding group. In QCD this is the gluon Aaµ,where a is the adjoint SU(3)C index running from 1 to 8. The weak triplet W aµ (a = 1, 2, 3) andthe hypercharge Bµ are the gauge bosons of SU(2)L and U(1)Y , respectively. After electroweaksymmetry breaking (EWSB) W aµ and Bµ mix to give the mass eigenstates: W± and Z bosons,and the photon γ, as discussed in Sec. 1.1.3.The fermion content of the SM consists of three generations of quarks (particles with SU(3)Ccharge) and leptons (those with only SU(2)L × U(1)Y charges). All stable matter is made ofthe first generation fermions (written in SU(2)L space)Q =(ud)L: (3,2,+1/6)L =(νe)L: (1,2,−1/2)ucR : (3,1,−2/3)dcR : (3,1,+1/3)ecR : (1,1,+1),where the SU(3)C × SU(2)L × U(1)Y charges are given in parentheses on the right hand side.The two heavier generations have identical structure. The fermions above are two-componentWeyl fields; two component spinors are reviewed in Ref. [18]. The physical propagating statesare four-component Dirac fermions.2 For example the electron field is (now written in Dirac2Whether the neutrinos are Dirac or Majorana is still unknown.3space)e =(eLeR),where eR = (ecR)†. Thus, the SM is a chiral gauge theory, with left- and right-handed fermionstransforming in different representations of the gauge group. The physical consequence ofthis structure is violation of parity P in physical processes, such as β decay [19]. Generalchiral theories suffer from anomalies – violations of the gauge symmetries arising from quantumcorrections. However, the charges of the SM fermions are such that every possible gaugeanomaly cancels [20].The final ingredient of the SM is the Higgs boson. The chiral nature of the SM pro-hibits gauge-invariant mass terms for the gauge bosons and fermions. Gauge invariance canbe preserved if the masses are generated dynamically through spontaneous symmetry break-ing (SSB), when a new scalar field, charged under SU(2)L × U(1)Y acquires a vacuum expec-tation value (vev). A priori, boson and fermion mass generation are independent. The SM iseconomical in that it uses the same Higgs boson H with charges (1,2, 1/2) to give masses toall SM states simultaneously. Note that there is no right-handed ν in the SM, so νL remainmassless. This is in conflict with neutrino flavour oscillation observations. We do not includeR-type neutrinos since the origin of the ν mass is still unknown. The field content of the SMis summarized in Tab. 1.1.The interactions of the SM is are given by the most general renormalizable Lagrangianconsistent with symmetries and field content of the model. Poincaré invariance allows theLagrangian to be written as space integral of a density L which is constructed from Lorentzinvariant objects evaluated at the same spacetime point (as required by causality and locality);the action is then the full spacetime integral of L . Lorentz invariance is implemented byensuring there are no uncontracted vector or spinor indices. General gauge transformations ofa field ψ in the fundamental representation take the formψ → eitaαa(x)ψ, (1.1)where αa(x) is a spacetime-dependent gauge parameter and ta are the Lie group generators.Spacetime derivatives of ψ do not transform in a simple way when αa is a function of x. It istherefore useful to define the covariant derivativeDµψ =(∂µ + igstaCAaµ + igtaLW aµ + ig′Y Bµ)ψ, (1.2)which has a simple transformation ruleDµψ → eitaαa(x)Dµψ, (1.3)4when the gauge fields transform as discussed below. Above, gs, g and g′ are the strong, weakand hypercharge gauge couplings, respectively. It is now easy to write down gauge invariantkinetic terms for the SM fermions and the Higgs field:Lkin ⊃∑iiψ¯iσ¯µDµψi + (DµH)† (DµH) . (1.4)Note that gauge invariance completely fixes the interactions of the matter (and Higgs) fieldswith the gauge bosons. Gauge invariance also allows for the following renormalizable Higgsself-interactions:LHiggs = −VHiggs = −µ2|H|2 − λ|H|4. (1.5)To ensure Eq. (1.3) holds, gauge fields must transform under an infinitesimal gauge trans-formation asAaµ → Aaµ + fabcαb(x)Acµ −1g∂µαa(x), (1.6)where g is the gauge coupling and fabc are the group structure constants satisfying[ta, tb] = ifabctc. (1.7)The structure constants vanish for an Abelian group. One can construct a gauge invariant fieldstrength tensor F aµν from the vector fields:F aµν = ∂µAaν − ∂νAaµ + ifabcAbµAcν , (1.8)where Aaµ stands for any of the three SM gauge fields. The non-vanishing Lorentz invariantcombinations are thenLkin ⊃−14GaµνGa µν −14W aµνW a µν −14BµνBµν (1.9)−θQCD32pi2Gaµν ˜Ga µν −θEW32pi2W aµν˜W a µν −θB32pi2Bµν ˜Bµν , (1.10)where Gaµν , W aµν and Bµν are the field strengths associated with SU(3)C , SU(2)L and U(1)Y ,respectively. In the second line ˜F aµν = µνσρF a σρ/2 is the dual field strength. One can showthat F ˜F terms are in fact total derivatives. These contribute a surface term (at spacetimeinfinity) that vanishes, unless the vacuum gauge field configuration has a non-trivial winding.Such configurations can exist for non-Abelian gauge theories. In particular, the SU(3)C termgives rise to the strong CP problem [21]. The electroweak vacuum angle θEW can be removedfrom the Lagrangian using appropriate global B (baryon number) and L (lepton number)transformations [22].5Gauge Eigenstates SU(3)C SU(2)L U(1)YAaµ 8 1 0W aµ 1 3 0Baµ 1 1 0Q 3 2 +1/6L 1 2 −1/2ucR 3 1 −2/3dcR 3 1 +1/3ecR 1 1 +1H 1 2 +1/2Table 1.1: Field content of the Standard Model in the gauge eigenbasis. Only one gener-ation of fermions is shown; the other two families have identical gauge charges.Next, we have the Yukawa interactionsLYukawa = HQYuucR −H†QYddcR −H†LYeecR + h.c., (1.11)where  is the two component Levi-Civita symbol with 12 = +1 and Yi are general complex3 × 3 matrices in family space. The complete SM Lagrangian at the classical level then takesthe formLSM = Lkin +LHiggs +LYukawa. (1.12)1.1.2 Gauge Fixing and GhostsEvaluating physical quantities in the quantum theory requires the computation of time orderedproducts of fields using path integrals like [23, 24]〈0|Tf(x1, x2, . . . )|0〉 =∫[DA]f(x1, x2, . . . ) exp(iS[A])∫[DA] exp(iS[A]) , (1.13)where f(x1, x2, . . . ) is a gauge invariant product of fields inserted at positions xi. In writingthe above we have suppressed dependence of the path integrals on fields other than the gaugefield A. Gauge symmetry of the theory is a redundancy in the number of degrees of freedomspecified by A. Thus, the unrestricted integration over A counts contributions from physicallyequivalent gauge configurations multiple times. This overcounting factorizes in the numeratorand denominator of Eq. (1.13) and therefore cancels in the ratio.A second related issue arises when we try to evaluate these path integrals in perturbationtheory where we need to compute the propagator of A, which is the inverse of the quadratic part6of the equation of motion operator. For example, for an Abelian gauge field with a canonicalkinetic term −FµνFµν/4, the momentum space propagator Dνρ is the Green’s function definedby(−k2gµν + kµkν)Dνρ = iδρµ. (1.14)The operator on the left hand side is not invertible, since it annihilates any tensor proportional tokνkρ, which corresponds to the unphysical longitudinal polarizations of A. These zero modes areresponsible for the infinite factors in the path integrals mentioned above. The inversion can beperformed and the propagator constructed if the integration is restricted to field configurationsthat are not related to each other by gauge transformations, i.e., if we explicitly fix the gauge.A particular gauge is specified by a functional F that satisfiesF [Agµ] = 0. (1.15)The superscript g emphasizes that this relation is satisfied only for a unique choice of gaugetransformation. This condition can be enforced in the path integral by employing the Faddeev-Popov trick [23]. After manipulation, the Lagrangian density is modified by∆L = − 12ξ(F [A])2 + η¯(x)M(x, y)η(y), (1.16)whereM(x, y)η(y) =∫d4yδF [Aθµ(x)]δθ(y)∣∣∣∣∣θ=0η(y), (1.17)and θ is an infinitesimal gauge transformation parameter. The auxiliary fields η¯ and η areGrassmanian (anti-commuting) scalar fields, called ghosts, that are integrated over in the pathintegral. They are required for the cancellation of unphysical gauge field degrees of freedomin perturbative computations at one- and higher loop orders. The parameter ξ is real andparametrizes the family of gauges specified by F . The first term in the gauge fixing Lagrangianof Eq. (1.16) contains pieces quadratic in A that modify the operator in Eq. (1.14), allowing itto be inverted.In unbroken gauge theories such as QCD it is often useful to work in a Lorentz covariantgauge, with a popular choice beingF [A] = ∂µAµ. (1.18)With this gauge fixing Eq. (1.14) becomes(−k2gµν + (1−1ξ)kµkν)Dνρ = iδρµ, (1.19)7which can be solved for Dνρ, because the operator on the left hand side is now invertible.Different forms of F lead to different forms of the vector propagator.There exists a more convenient choice of the gauge fixing functional for spontaneously brokentheories. With SSB, the Higgs kinetic terms in Eq. (1.4) give rise to a mixing between the gaugebosons and Goldstone bosons via bilinear terms likeLkin ⊃ igAaµ(∂µH†ta〈H〉 − 〈H†〉ta∂µH), (1.20)where 〈H〉 is the Higgs field vev. To avoid mass matrix diagonalization, the mixing can beremoved by choosingF a[A] = ∂µAaµ − iξg(H†ta〈H〉 − 〈H†〉taH). (1.21)This family of gauges is known as the Rξ gauges. The full SM Lagrangian which contains gaugefixing for the entire SU(3)C ×SU(2)L×U(1)Y and the corresponding ghost terms is presentedin, e.g., [23, 25].1.1.3 Spontaneous Symmetry Breaking and the Physical SpectrumThe SU(2)L × U(1)Y symmetry forbids mass terms for EW gauge bosons and SM fermions.To match observations the SM generates masses dynamically through spontaneous symmetrybreaking. The order parameter of the symmetry is the expectation value of the Higgs field; ageneral vev can always be chosen (using a global SU(2)L × U(1)Y rotation) to be〈H〉 =(0v/√2)(1.22)with v real and positive. At tree-level this pattern of symmetry breaking arises when we takeµ2 < 0 in VHiggs (see Eq. (1.5)). From the Higgs kinetic terms in Eq. (1.4) we obtain mass termsfor the SU(2)L × U(1)Y gauge bosons. The mass matrix in the (W 1,W 2,W 3, B) basis is [26]M2gb =14v2g2 0 0 00 g2 0 00 0 g2 −gg′0 0 −gg′ g′ 2. (1.23)The upper left 2×2 block has two degenerate eigenvalues that correspond to mass of the chargedW bosonsW± = 1√2(W 1 ∓ iW 2) (1.24)8with the valuem2W =14g2v2. (1.25)The (canonically normalized) combinationZ = cWW 3 − sWB (1.26)receives the massm2Z =14(g2 + g′ 2)v2, (1.27)where cW (sW ) are the cosine (sine) of the Weinberg angle, defined byc2W = cos2 θW =g2g2 + g′ 2. (1.28)The orthogonal combinationA = sWW 3 + cWB (1.29)remains massless and is identified with the photon. Thus the full SM gauge symmetry SU(3)C×SU(2)L × U(1)Y is broken down to SU(3)C × U(1)em. From the kinetic terms of Eq. (1.4) itfollows that A couples to matter with electromagnetic chargeQ = t3L + Y (1.30)in units of e = gsW = g′cW . The measured masses of theW and Z bosons are 80.385±0.015GeVand 91.1876± 0.0021GeV [27]. Together with the Higgs vev, v ≈ 246GeV, these numbers setthe fundamental mass scale of the SM. In the following chapters we will refer to masses andenergies of order O(100GeV) as electroweak scales.As emphasized above, fermion mass generation is a priori unrelated to that of the gaugebosons. The SM is economical in that it uses the same Higgs field for both tasks. The fermionmass matrices can be read off from Eq. (1.11) by substituting the Higgs vev, Eq. (1.22), for H:LYukawa = −v√2uLYuucR −v√2dLYddcR −v√2eLYeecR + h.c. (1.31)Note the absence of a mass term for the neutrinos due to lack of a right handed ν. Thecomplex Yukawa matrices Yi can be diagonalized using two unitary transformations, U and V ,corresponding to separate rotations of the L and R type fermions in family space, respectively,f ′L = UffL, f ′R = VffR, (1.32)9such thatMf =v√2UTf YfVf (1.33)is diagonal. All gauge interactions, except the quark charged current (CC) terms, are invariantunder these rotations; dropping the primes, the CC interactions can be written asLcc = −g√2W+µ (u¯Lσ¯µVCKMdL + ν¯Lσ¯µeL) + h.c., (1.34)where VCKM = U †uUd is the unitary Cabibbo-Kobayashi-Maskawa (CKM) matrix. Note thatthere is no equivalent matrix in the lepton sector because the neutrinos are massless in the SM.There are 4 physical parameters in the CKM matrix, including 1 phase [26]. The existence ofthis complex phase implies that Lcc is not invariant under CP.The SM and relevant field-theoretic details are described in, e.g., Refs. [20, 23, 25, 26, 28–32]. The historical development of the electroweak theory and chromodynamics is discussed inRefs. [33–35].1.2 The Standard Model of Cosmology1.2.1 The Expanding UniverseOn cosmological scales the Universe appears approximately isotropic and homogeneous. Themost convincing evidence for this is the CMB, which is isotropic to a few parts in 10−5 [36]. Anygeneral-relativistic description of our Universe should therefore possess these features. The mostgeneral metric satisfying isotropy and homogeneity is the Friedmann-Robertson-Walker (FRW)metric, with [36, 37]ds2 = dt2 − a(t)2(dr21− kr2+ r2dΩ2), (1.35)where k = −1, 0,+1 corresponds to constant negative, zero or positive spacial curvature; dΩ2 =dθ2+sin2 φdφ2 is the usual metric on a sphere. Various observations, including the latest resultsfrom Planck [38, 39] suggest that the Universe is, to a very good approximation, flat with k = 0.Then the physical (proper) distance to an object at radial coordinate r is given by a(t)r. Thefunction a(t) is therefore called the scale factor. The present-day scale factor, a(t0), is usuallytaken to be 1. When discussing astrophysical observables it is common to use the redshift zinstead of a:z(t) = a(t0)a(t)− 1, (1.36)10where t0 is the current time. This is a useful quantity since a photon emitted at time t withwavelength λ is observed at time t0 to have wavelength (1 + z(t))λ.The Einstein equations determine the evolution of the metric in the presence of an energydensity distribution [37]Rµν = 8piG(Tµν −12gµνT), (1.37)where Rµν is the Ricci tensor (computed from the metric), Tµν is the energy-momentum tensor(the source for the gravitational field), T = Tµµ , and G is Newton’s constant. We will modelenergy densities as perfect fluids, that is, continuous distributions that are isotropic in theirrest frame. The state of such fluid is specified by energy density ρ and isotropic pressure p.The energy momentum tensor for a perfect fluid can be written as [37]Tµν = (ρ+ p)UµUν + pgµν , (1.38)where Uµ is the fluid 4−velocity. In its rest frame, we have Uµ = (1, 0, 0, 0).For an FRW metric Eq. (1.37) reduces to the two Friedmann equations [37]a¨a= −4piG3ρ− kρ+ 3p, (1.39)andH2 ≡(a˙a)2= 8piG3ρ− ka2, (1.40)where the Hubble parameter H determines the rate of expansion of the Universe. The currentvalue of the expansion rate, H0, is often written in terms of a dimensionless number h asH0 = 100h km sec−1 Mpc−1, with h = 0.673 ± 0.0012 [27]. Equation (1.40) is frequentlywritten in dimensionless form as1 = Ω + Ωk, (1.41)withΩ = ρρc, Ωk = −ka2H2, (1.42)where the critical density ρc isρc =3H28piG. (1.43)If the sum of the fluid densities in the Universe is equal to the critical density, the spatialcurvature k must vanish. The density parameters Ω are often used to specify the energy11density composition of the Universe.Another useful relationship is the covariant continuity (or energy-momentum conservation)equation∂tρ+ 3H(ρ+ p) = 0, (1.44)which follows from the diffeomorphism/coordinate invariance of the matter actionSM =∫d4x√−gLM , (1.45)where g is the determinant of the metric and LM is the matter Lagrangian density. For theSM, LM is given in Eq. (1.12). Note that the factor√−g gives the minimal coupling of matterto gravity. We will make use of this coupling when discussing moduli in Chapters 3 and 4.The fluids relevant to cosmology obey equations of state of the formp = wρ, (1.46)with w = 0, 1/3,−1 for matter/dust (non-relativistic particles), radiation (relativistic species)and vacuum energy, respectively. In each case the continuity equation, Eq. (1.44), can be solvedto yieldρ ∝ a−3(1+w). (1.47)For example dust (w = 0) energy density dilutes as a−3, corresponding to the growth of thecomoving volume with a3. If the total energy density ρ is dominated by one fluid, the Friedmannequation, Eq. (1.40), can be solved for the time dependence of the scale factor. For a spatiallyflat Universe (k = 0) we finda ∝t23(1+w) w 6= −1exp(Kt) w = −1,(1.48)where K is a constant and we only wrote down the expanding solutions. Thus, the change inthe rate of expansion depends on the composition of the energy density of the Universe.Using the above results we can can decompose the total density parameter Ω, defined inEq. (1.42), into componentsΩ = Ωr + Ωm + ΩΛ, (1.49)where the three contributions correspond to radiation (w = 1/3), matter (w = 0) and vacuumenergy (w = −1). All present measurements are consistent with Ω ≈ 1 and Ωk = 0, sothroughout this work we will assume that the Universe is spatially flat with k = 0 [27].121.2.2 ThermodynamicsWe will often be interested in the present day abundances of particles that interacted with thehot plasma of the early Universe. The thermodynamic properties of an ensemble of particlesare contained in the phase space distribution functions fi(p, T ) where i labels particle species, pis the momentum and T is the temperature. From these distributions we can compute numberand energy densities, and pressure [40, 41]:ni = gi∫d3p(2pi)3fi(p, T ), (1.50)ρi = gi∫d3p(2pi)3fi(p, T )Ei(p), (1.51)pi = gi∫d3p(2pi)3fi(p, T )p23Ei(1.52)where E2i = p2 +m2i and gi is the number of internal degrees of freedom. In kinetic equilibriumand at zero chemical potential the distributions have the Bose-Einstein or Fermi-Dirac form:fi(p, T ) =1exp(Ei(p)/T )± 1, (1.53)where + is for fermions and − is for bosons. At a given temperature T it is useful to partitionthe plasma into relativistic (T  mi) and non-relativistic (T  mi) components. Assumingkinetic equilibrium, Eqs. (1.50), (1.51), (1.52) can be evaluated in these limits analytically. Inthe relativistic limitni = giζ(3)pi2T 31 bosons34 fermions, (1.54)ρi = gipi230T 41 bosons78 fermions, (1.55)pi = ρi/3. (1.56)In the non-relativistic limit both distributions reduce toni = gi(miT2pi)3/2exp(−mi/T ), (1.57)ρi = mini, (1.58)pi = niT. (1.59)Note that the energy density and pressure of relativistic species are parametrically larger thanthose of non-relativistic states. This means that the total energy density and total pressure ofthe heat bath are well approximated by the relativistic contributions alone. Summing over all1311010010−5 10−4 10−3 10−2 10−1 100 101 102 103NumberofRelativisticD.o.FsT [GeV]g∗(T )g∗S(T )Figure 1.1: Number of energy and entropy relativistic degrees of freedom, g∗ and g∗S ,respectively, for the Standard Model as a function of temperature. The large dropin the number of degrees of freedom at T ∼ 200MeV corresponds to the QCD phasetransition when quarks and gluons became confined in hadrons.relativistic species we find the total radiation density and pressure:ρR =pi230g∗(T )T 4 (1.60)pR ≈ ρR/3, (1.61)where g∗ counts the number of relativistic degrees of freedom at temperature Tg∗(T ) =15pi4∑igi(TiT)4 ∫ ∞xi(u2 − x2i )1/2u2exp(u)± 1 du (1.62)≈∑i=bosonsgi(TiT)4+ 78∑i=fermionsgi(TiT)4,where xi = mi/Ti, Ti is the temperature of species i, and T is the photon temperature; the lastequality follows for xi  1 and the observation that non-relativistic species do not contributesignificantly to ρR.3 The integral in Eq. (1.62) smoothly removes non-relativistic species fromthe summation. This function for the SM fields is shown in Fig. 1.1 as the solid blue curve.The large decrease in g∗ at T ∼ 200MeV corresponds to the QCD phase transition when freequarks and gluons condensed into hadrons.We can use the above results to derive the evolution of radiation and entropy density in the3The function g∗ for pressure is numerically different from g∗ defined above due to different integrands inEq. (1.51) and (1.52). pR = ρR/3 exactly only when xi = 0.14expanding Universe. In the absence of energy injection, the First Law of thermodynamicsdU + PdV = 0, (1.63)with total energy U and pressure P , as applied to the radiation bath givesa3dρR + (ρR + pR)da3 = 0. (1.64)Using Eq. (1.61) we can write this asρ˙R + 4HρR = 0. (1.65)In equilibrium, entropy S is conserved:0 = dS = d(sV ) = d(ρV ) + pdVT, (1.66)where we used the fundamental thermodynamic relation, TdS = dU + pdV , in the second stepand defined the entropy density s as [36]s =∑iρi + piTi. (1.67)As for the energy, the entropy density is dominated by relativistic species. Using the definitionsof Eq. (1.51) and (1.52) we can rewrite this ass = 2pi245g∗S(T )T 3, (1.68)whereg∗S(T ) =454pi4∑igi(TiT)3 ∫ ∞xi(u2 − x2i )1/2u2exp(u)± 1 du (1.69)≈∑i=bosonsgi(TiT)3+ 78∑i=fermionsgi(TiT)3,When all species have the same temperature g∗S ≈ g∗; however, when a relativistic speciesdecouples from the bath and Ti 6= T , g∗S 6= g∗. This can be seen in Fig. 1.1 at low temperatures,where g∗S is shown as the solid red curve. The difference is due to the decoupling of neutrinosat T ∼ 1MeV and the subsequent e+e− annihilation for T . 2me (when they become non-relativistic). The annihilations deposit energy into the photon bath, but not into the neutrinos(since they are decoupled), leading to T/Tν = (11/4)1/3 at late times [36, 40].When a massive species decouples from the plasma, its comoving number N = na3 is(approximately) conserved until present day, unless it decays. Such relics will play key roles15in Chapters 3 and 4 where we consider dark matter and baryon production. At high enoughtemperatures, a particle χ is kept in chemical equilibrium with the thermal bath throughinteractions like χχ¯ ↔ XX¯, where X is part of the plasma. For example for T & 1GeV,χ could be a quark and X a gluon. If equilibrium persists sufficiently long, and χ is stable,it becomes non-relativistic as the Universe cools and since nχ ∝ exp(−m/T ), its density willbecome negligible. However, if the interactions that keep χ in equilibrium are not fast enoughto keep up with the expansion of the Universe, the number density will become frozen-in. Thishappens approximately when the interaction rateΓ = 〈σv〉nχ (1.70)falls below the Hubble expansion rate H, where 〈σv〉 is the thermally averaged interactionrate (defined below). This freeze-out mechanism is quantitatively described by the Boltzmannequation. In its most fundamental form the Boltzmann equation describes the time evolutionof the particle phase space density f(p)L[f ] = C[f ], (1.71)where L = d/dt is the Liouville operator, given by [40–42]L = ∂∂t−H |p|2E∂∂E, (1.72)for a spatially homogeneous distribution in the FRW spacetime and C[f ] is the collision termthat encodes particle interactions. By integrating over phase space, one obtains a time evolutionequation for number densities,n˙i + 3Hni = gi∫C[fj ]d3p(2pi)3Ei(1.73)The general collision term C is a complicated expression that depends on matrix elements forthe forward and reverse reactions, as well as the phase space densities of all of the interactingparticles. We will be primarily interested in decays and 2 → 2 scattering reactions. Severalassumptions can be used to simplify C[f ]:1. Kinetic and thermal equilibrium is attained immediately by the annihilation products Xand X¯, implying that their interactions with the plasma must be relatively strong. Thiswill be the case when X is a SM state.2. Maxwell-Boltzmann statistics is valid, which is a good assumption for non-degenerategases with T . 3m [42].16Under these assumptions, the collision term for a reaction of the type 12 → 34 reduces tog1∫C[fj ]d3p(2pi)3E1= −〈σv〉(n1n2 − neq1 neq2 ), (1.74)where〈σv〉12→34 =∫σvMøle−E1/T e−E2/Td3p1d3p2∫e−E1/T e−E2/Td3p1d3p2, (1.75)vMøl =√(p1 · p2)2 −m21m22E1E2, (1.76)and neq is given by Eq. (1.57). The cross section σ for 12 → 34 is computed using the usualFeynman rules. For many cases of interest decoupling of a particle from the plasma occurs whenit is already non-relativistic, in which case Eq. (1.75) admits a simple expansion in m/T 1 [42].We can now write down the simplest Boltzmann equation for a single self-conjugate speciesχ annihilating into SM particles:n˙χ + 3Hnχ = −〈σv〉(n2χ − (neqχ )2). (1.77)The solution of this equation for mχ = 100GeV and σ ∼ 10−11 GeV−2 is shown in Fig. 1.2.Note that at early times the comoving number density N = nχa3 tracks the equilibrium, untilthe χχ annihilations freeze-out. This occurs at xf = mχ/Tfo ∼ 20, so χ decouples when it isalready non-relativistic. In this simple scenario it is possible to analytically estimate both thefreeze-out point xf and the final abundance of Ωχ. The freeze-out temperature can be foundusing the decoupling condition, Eq. (1.70) [40, 43]:xf = ln(√454pi5gχ√g∗mχMPl〈σv〉)− 12lnxf , (1.78)where MPl = 2.435× 1018 GeV is the reduced Planck mass. In particular, note that xf is onlylogarithmically sensitive to mχ and 〈σv〉, so xf ∼ 20 is generic for weak-scale masses and cross-sections. The approximate abundance can be obtained by matching the early-time solution(which tracks neqχ ) with the late-time solution (where neqχ can be neglected) [40], with the resultΩχ =mχnχρc= h−2(2× 10−10 GeV−2〈σv〉)(xf20)(g∗(Tfo)85)−1/2. (1.79)Note that the abundance is inversely proportional to the annihilation cross-section: the moreeffective χχ annihilations are, the smaller the final abundance.The above results hold for a single particle with a cross section constant with T . In realistic1710−1410−1210−1010−810−610−410−2100100 101 102 103Nmχ/TNχNeqFigure 1.2: Comoving number density N = nχa3 (solid line) of a self-conjugate particleχ as a function of mχ/T . The equilibrium number density is shown as a dashedline. Freeze-out occurs at mχ/T ∼ 20.models, there can be other states present in the plasma that interact with the long-lived particleof interest. The annihilation rate can have important features as a function of T , such asresonances when intermediate states go on shell. These and other effects can significantlymodify the final abundance [43].1.2.3 The First 400 000 YearsHaving discussed thermodynamics in an expanding Universe, we can begin to reconstruct thethermal history of the Universe. Here we highlight some of the key events. We start at t ∼ 10−43s, corresponding to temperatures T ∼ 1018 GeV, where a theory of quantum gravity is requiredeven for a qualitative description. Shortly after this, inflation or another mechanism operates tosolve the horizon and flatness problems [44]. The evolution down to T ∼ 103 GeV has not beenprobed experimentally, but many models of baryogenesis and neutrino mass generation operateabove this energy [45, 46]. As we cool further and cross the TeV threshold at t ∼ 10−13 s, weenter the realm of energies that have been probed experimentally. At T ∼ 100GeV (t ∼ 10−11s), the Universe undergoes the electroweak phase transition (EWPT) where SU(2)L × U(1)Yis broken down to U(1)em. The QCD chiral phase transition at T ∼ 200MeV (t ∼ 10−5 s)occurs when quarks and gluons condensed into hadrons. Neutrinos decouple at T ∼ 1MeV(t ∼ 1 s), fixing the neutron to proton ratio. Big Bang Nucleosynthesis begins shortly after atT ∼ 0.1MeV (t ∼ 102 s) when the primordial plasma is cool enough to not disassociate themajority of newly formed nuclei. The matter density becomes comparable to radiation densityat T = 1 eV (t = 104 years), initiating structure formation. Finally, baryons recombine with18electrons at T ∼ 0.1 eV (t ∼ 4× 105 years), making the Universe transparent to photons. Thefree-streaming photons from that era make up the CMB.The above history is the canonical account of early Universe cosmology. In particular,it assumes that the Universe remains radiation-dominated until matter-radiation equality atT = 1 eV. As we will discuss in Chapters 4 and 3, this idea is experimentally grounded onlyfor T . 5MeV. There we will consider modifications to this radiation-dominated scenario.1.3 Outstanding ProblemsDespite their innumerable successes, the Standards Models are incomplete. Below we list someof the major motivations for new physics. In the following, chapters we consider models thatattempt to explain one or more of the following: the hierarchy problem, the baryon asymmetryof the Universe and dark matter. These are discussed in detail in Sections 1.3.1, 1.3.2 and 1.3.3,respectively. This is far from an exhaustive list. Other important issues in particle physicsinclude the origins of neutrino masses and the flavour structure of the SM, the strong CP andthe cosmological constant problems [47]. The latter, in particular, indicates that somethingfundamental is missing in our understanding of gravity and particle physics.1.3.1 Hierarchy Problem and NaturalnessThe Higgs boson mass was measured to be ∼ 125 GeV [6]. In order to relate this observationto other parameters in the theory, we must compute quantum corrections shown in Fig. 1.3,which contain ultraviolet (UV) divergent integrals over spacetime, that must be regulated. Forexample, a fermion with mass Mf , Nc colours and Yukawa coupling yhf¯f/√2 shifts the massbym2h = m20 −y2Nc8pi2Λ2 + . . . , (1.80)where the loop integral corresponding to the left diagram in Fig. 1.3 has been regulated usingthe momentum cutoff Λ; the ellipsis stands for terms that are at most logarithmically divergentas Λ → ∞. The SM is a valid effective field theory for energies below Λ. If Λ is much biggerthan the electroweak scale ∼ 100 GeV, then to get mh = 125 GeV we must tune the tree-levelparameter m20 very precisely to cancel the large contribution from new physics at scale Λ. Weexpect new degrees of freedom to exist at several possible scales associated with, e.g., neutrinomass generation, unification, or, at the very least, quantum gravity. Taking Λ ∼ MPl, wefind that m20 has to be chosen with a precision of one part in 1032. This is called the fine-tuning or naturalness problem. Equivalently, it is a statement about the enormous hierarchyin energy scales between gravitational and electroweak physics. In absence of symmetry, onewould naturally expect mh ∼MPl and similar strengths of weak and gravitational interactions.The disturbing precision of the required cancellation has been a strong motivation for thedevelopment of models where the quadratic divergence in Λ is absent and mh is natural. The19Figure 1.3: Possible contributions to the Higgs mass parameter from fermions (left graph)and scalars (middle and right graphs). The momentum integrals associated withthese graphs are UV divergent and give rise to the fine-tuning problem.solutions can be divided into several categories. First, we may introduce a symmetry thatregulates the size of these large quantum corrections. For example, suppose that in addition tothe new fermion, there exist a pair of complex coloured scalars φL,R of mass Ms that couple tothe Higgs boson viaL ⊃ −λ2(v + h)2(|φL|2 + |φR|2). (1.81)The corresponding quantum corrections are computed from the middle and right graphs ofFig. 1.3; only the right diagram is quadratically divergent with the resultm2h = m20 +λNc8pi2Λ2 + . . . (1.82)In particular, if λ = y2 the Λ2 contributions to the Higgs self-energy cancel exactly with thatcoming from the fermion, and no fine tuning would be required to have mh at the electroweakscale. The cancellation of quadratic divergences does not depend on the mass difference Mf −Ms; the logarithmic divergences (not written) cancel if Mf = Ms. This is exactly what happensin supersymmetry, which we discuss in Ch. 2. Shift symmetries h→ h+α can also forbid largequantum corrections, resulting in technically natural mh. Such symmetries are associated withNambu-Golsdstone bosons (NGBs) of spontaneously broken global symmetries. Since NGBsare massless and the Higgs is not, the symmetry must be broken explicitly in analogy to pionmass generation and the breaking of the chiral SU(2) by non-degenerate quark masses. Theclass of models where the Higgs boson is a pseudo-NGB are known as Little Higgs theories [48].In both cases, naturalness demands the existence of new states that couple strongly to theHiggs with masses close to the electroweak scale. Other explanations include compositeness ofthe Higgs [49] and extra dimensions [50, 51]It is also possible that the Higgs mass is not natural and fine-tuning is required for life toexist. However, such anthropic arguments are less convincing for the case of Higgs mass (asopposed to the cosmological constant), since one can get a habitable Universe even when mh ∼MPl [52]. Anthropic reasoning often emerges in the context of the string theory landscape [53].201.3.2 Baryon Asymmetry of the UniverseRelativistic field theory requires the existence of antimatter. Each anti-particle has oppositegauge charges but otherwise identical properties to the corresponding particle [45]. It is thenpuzzling why the local Universe seems to be composed of entirely of matter and not anti-matter. It is conceivable that the Universe is actually baryon-symmetric – domains of matterare spatially separated from domains of antimatter. However, pair annihilation at interfacesof such regions would produce unacceptably high gamma ray fluxes, unless the regions areseparated by distances greater than 10 Mpc (roughly the size of the Virgo cluster) [40]. Still,annihilation is inevitable, unless the domains are separated by enormous voids. These voidswould appear as inhomogeneities in the CMB [45, 54].Ignoring the above difficulties, suppose the Universe is baryon-symmetric. At high tempera-tures baryons are kept in equilibrium with the plasma by pair annihilations with 〈σv〉 ∼ m−2pi =5 × 10−5 GeV2. These interactions freeze-out via the mechanism discussed in Sec. 1.2.2 whenTfo ≈ 20MeV (from Eq. (1.70)), resulting in baryon abundance that is 9 orders of magnitudetoo small [40]. This means that either some mechanism separated baryons from antibaryonsbefore the annihilations depleted the abundance, or, there was a slight excess of matter overantimatter, such that only the asymmetry remained at late times. In the former case, themass of baryons contained within the horizon at that time was a fraction of a solar mass, sothe mechanism that separates cluster-sized masses of baryons and antibaryons would have toviolate causality [40]. Thus, in this work we consider only the latter possibility.The observed baryon to photon ratio, η = nB/nγ = 6.19 × 10−10 suggests that at temper-atures above 1 GeV the excess of quarks over antiquarks was tiny: about one more quark perevery thirty million antiquarks [40]. This asymmetry can be an initial condition or it can begenerated dynamically. The initial condition hypothesis is impossible to test, so we considertheories of dynamical baryogenesis instead. Sakharov outlined the general conditions necessaryto generate this asymmetry in the early Universe [55]:1. Baryon number must be violated in order to generate B 6= 0 from a symmetric initialstate.2. C (charge conjugation invariance) and CP (charge-parity conjugation invariance) mustbe violated to produce more baryons than antibaryons. Given a process i({si}, {pi}) →f({sf}, {pf}) that produces baryon number, its charge conjugate i¯ → f¯ yields oppositebaryon number.4 If the rates are equal, there is no net asymmetry produced. Since localLorentz invariant theories conserve CPT, CP invariance is equivalent to time-reversalinvariance T. If the rate for the T-conjugate process f({−sf}, {−pf}) → i({−si}, {−pf})is equal to i({si}, {pi}) → f({sf}, {pf}), no net asymmetry is produced.3. There must be a departure from thermal equilibrium, since the rates for i→ f and f → iare equal in equilibrium.4{si,f} and {pi,f} represent the sets of spins and momenta in the initial or final state.21The SM has all of the above ingredients: non-perturbative SU(2)L processes can provideB 6= 0, while the single phase in the CKM matrix violates C and CP. The departure from thermalequilibrium is achieved if the electroweak phase transition is first order: bubbles of phase withnon-vanishing Higgs vev nucleate within the plasma, expand and collide until SU(2)L×U(1)Yis broken everywhere. This process is described in more detail in Sec. 7.1. In the SM, the EWphase transition proceeds in that way only if the Higgs boson is light, i.e., mh ≤ 70 GeV [56],in conflict with the observed value of ∼ 125 GeV. Thus this mechanism does not work in theSM alone and new physics is required.In this work we focus on the third requirement, the departure from equilibrium. We considertwo different ways to achieve this. In Ch. 4, we investigate a scenario with an out-of-equilibriumdecay of a new heavy particle. In Ch. 7, we augment the SM with extra matter to ensure thatthe EW phase transition is first order and therefore suitable for baryogenesis.1.3.3 Dark MatterObservations of radial velocities of galaxies in the Coma cluster provided the first hint forthe existence of dark matter [57–59]. Later, measurements of orbital speeds of visible objectsinside galaxies (i.e., galactic rotation curves) indicated that the galactic mass distribution mustextend beyond luminous matter [60].5 Consider a visible object on a stable orbit around agalaxy. Assuming Newtonian gravity is valid, the orbital speed for a body at distance r fromthe galactic center isv =√GM(r)r, (1.83)where M(r) is the mass enclosed by the trajectory. In particular, if only luminous mattercontributes to M , beyond the visible part of the galaxy M is constant, so the orbital speed mustdecrease as v ∝ r−1/2. However observations of many galaxies indicate that the speed does notdecrease at all, but flattens out, implying that M ∝ r in this region. This suggests the existenceof a spherical dark matter halo that extends beyond luminous matter, with density falling asρ ∝ r−2. This gives rise to the idea that galaxies are embedded in halos of non-luminous, darkmatter (DM) [61]. Since the initial measurements on cluster scales, the DM hypothesis hasreceived support from an array of observations over a wide range of distance scales, from sub-galactic to cosmological. For example, gravitational lensing of galaxies and clusters is sensitiveto the total amount of matter in the lens [62]. These measurements unequivocally show thatthe amount of matter in these objects is much larger than what is visible. The possibility thatthis missing matter is in the form of massive compact halo objects (MACHOs) is excluded bymicrolensing surveys of nearby satellite galaxies [62].An even more stringent limit on the baryonic contribution to total matter abundance in5These measurements are usually done using the 21 cm line from Hydrogen hyperfine splitting and opticalsurface photometry [61].22the Universe comes from Big Bang Nucleosynthesis (BBN). Nucleosynthesis is the earliestepoch in the history of the Universe that is probed observationally, corresponding to timeson the order of 1 second after the Big Bang. Standard BBN has been very successful inpredicting the present-day abundances of the light elements such as helium, deuterium andlithium [63].6 Nucleosynthesis predictions are very sensitive to the nucleon to photon ratio ηwith the equilibrium abundance of element with atomic number A proportional to ηA−1 [40].As a result, the observed light element abundances constrain the baryonic contribution to totalmatter density to be [27]Ωb ∼ mbnγη/ρc . 0.053. (1.84)This upper bound on baryonic matter in nucleons combined with total matter density measure-ments from dynamical (i.e., rotation curves) means and lensing imply that most of the mattercontent of the Universe is in the form of DM. Note that, if there exist exotic quark “nugget”bound states with a sufficiently strong binding, they will not participate in in BBN and DMcan be of baryonic origin – see Ref. [66] for a review of this possibility.The most precise determination of DM abundance in the Universe comes from fits to cos-mological data from the expansion of Universe, galaxy clustering, CMB power spectra andother measurements. The CMB power spectrum, in particular, is sensitive both to the totalamount of matter and its baryonic fraction. For temperatures above about 1 eV, the primordialplasma is ionized and the baryon fluid is coupled to the photon fluid via electromagnetic inter-actions (for a review of CMB physics, see, e.g., Refs. [27, 67]).7 This coupled fluid oscillatesin the gravitational potential established by DM over/under-densities, leading to temperatureanisotropies on top of a nearly perfect black body spectrum. The CMB power spectrum ofthese anisotropies has been subject of intense experimental study, culminating with the Planckmission, which provided the most precise measurements of the temperature power spectrumto date.8 Fitting the standard cosmological model to the latest Planck data yields a baryonabundance [17] ofΩb = 0.049± 0.002 (1.85)and total matter abundance ofΩm = 0.315± 0.013. (1.86)This incredibly precise measurement of the matter density on cosmological scales again requires6At present, the prediction for 7Li is inconsistent with the CMB measurement of η. This is known as theLithium problem [64]. Possible solutions can include astro- and nuclear physics effects, as well physics beyondthe Standard Model [65].7In the astrophysics community “baryon” refers to all SM particles, including baryons and leptons [68].8Planck is able to detect angular variations of ∼ 0.1◦ on the sky. Ground based telescopes can attain evenhigher resolution.23a large component of non-baryonic DM.Finally, DM is necessary for the formation of large scale structure consistent with whatis observed today in galactic surveys [69]. The growth of the initial density perturbations inthe primordial plasma occurs at different rates in the matter and radiation-dominated phases.Gravitational collapse is more efficient during matter domination, which does not occur earlyenough with only baryonic matter to reproduce the observed matter power spectrum [62]. Ifmatter has a significant non-baryonic component that is not strongly coupled to photons, thenthe matter dominated epoch can begin earlier. The DM hypothesis thus provides an excellentfit to the matter power spectrum from cosmological scales down to ∼ 0.1 Mpc (see, e.g., [70]).The validity of the cold (i.e., non-relativistic), non-interacting DM fluid on small scales (. 10kpc, corresponding to small galaxies and inner regions of larger ones) has been less certain,as several apparent conflicts between astrophysical observations and many-body simulationsemerged. These difficulties are reviewed in Ref. [71, 72] and are called “core vs cusp”, “missingsatellites” and “too big to fail” problems. The first of these is the statement that DM-onlymany-body simulations predict a DM density profile with a cusp near the galactic center,which provides a poor fit to the inner regions of real galactic rotation curves. Moreover, thesesimulations produce many more small satellite galaxies (dwarves) than observed in the vicinityof the Milky Way. This is the missing satellites problem. Some of these missing DM halosare so massive that it is puzzling that they have failed to form stars. These issues are facetsof the same problem: DM-only simulations predict more substructure than what is observed.This has motivated modifications of the standard cold DM paradigm that included warm (semi-relativistic) and self-interacting DM. The former allows DM to stream out of gravitational wells,washing out structure of sizes smaller than the free-streaming length; self-interacting DM resultsin a hotter halo core that prevents the formation of cusps [68]. However, as emphasized above,the simulations that lead to these small scale structure anomalies have not included baryonicphysics. These effects include gas outflows, energy feedback from stars and supernovae andcan significantly alter galactic evolution and the form of the resulting DM distribution throughgravitational interactions [68].Having discussed the successes and difficulties of the dark matter framework, it is importantto note that it is not the only logical possibility to explain the above purely gravitationalobservations. In principle, this can also be done by modifying general relativity. In practice,this is difficult because of the vast range of length scales spanned by these observations (see [73]for an example of problems faced by modified gravity). Hence in this thesis we assume thatDM is a new particle or a set of particles. With this in mind, DM must be long-lived, massive,non-relativistic during structure formation and interact only weakly with photons. The SM ofparticle physics lacks such a particle and we must look beyond the SM for suitable candidates.Even with the above requirements, the nature of DM is difficult to determine from cosmo-logical data alone, with DM masses in viable models spanning ∼ 31 orders of magnitude from10−15 GeV for axion DM to 1016 GeV for “WIMPzillas”, with interaction strengths varying24across an even wider range [74]. We will focus on a well-motivated class of DM candidateswith masses in the range 1 − 1000GeV and weak-scale couplings to ordinary matter. This isthe domain of the weakly interacting massive particle (WIMP). Stable particles in this massrange frequently arise in extensions of the SM that address other issues, such as the hierarchyproblem. Perhaps the best studied examples of WIMPs are the neutralinos of supersymmetrictheories [75], where DM production occurs through the thermal freeze-out process described inSec. 1.2.2.Much of the experimental effort in the search for DM is focused on WIMPs. The experimentsfall into three broad categories. Direct detection searches for collisions of DM particles in thehalo with nuclei in ultra-sensitive detectors. The present status and future prospects for theseexperiments are reviewed in Ref. [76]. There have been many observed event excesses in thelast few years, but they have not been corroborated by competing groups. The second class ofexperiments searches for energetic DM annihilation products from astrophysical sources, suchas the Galactic Center and dwarf galaxies. The products can be cosmic rays, diffuse or line-likegamma rays or neutrinos. Both indirect and direct detection are subject to many astrophysicaluncertainties, such as DM substructure and poorly understood backgrounds [77, 78]. Finally,there are accelerator based methods that seek to produce DM and associated particles in thelaboratory, as opposed to detecting relic DM. Since DM is stable and is expected to be weaklyinteracting, it escapes the detector giving a “missing energy” signal. Searches for such signalsgive a complementary probe of the “dark sector”, since astrophysical uncertainties are absent.For a brief review of recent work in this direction see Ref. [79].25Chapter 2Aspects of Supersymmetry2.1 IntroductionThe search for solutions to problems in particle physics and cosmology described in the previouschapter has lead to the development of a wide variety models for physics beyond the standardmodels. Supersymmetry (SUSY) has emerged as one of the most fruitful of these ideas, pro-viding a powerful framework for tackling problems of naturalness, dark matter, baryogenesis,and gauge coupling unification [80]. Aside from these phenomenological benefits, SUSY en-ables effective study of field theories in non-perturbative regimes through its dualities andnon-renormalization theorems (for a detailed introduction, see, e.g., Ref. [81]). For example,it is possible to prove confinement in some theories with extended SUSY [82]. Finally, SUSYprovides a connection to most consistent formulations of string theory, which require super-symmetry, at least at string scales.All possible symmetries of the scattering (S) matrix have been analyzed in the “no-go”theorem of Coleman and Mandula [83]. The Coleman-Mandula theorem states that the fullsymmetry group of the S-matrix is necessarily the direct product of the Poincaré spacetimesymmetry and any internal symmetries. This means that it is not possible to augment spacetimesymmetries with, e.g., gauge symmetries. That is, the generators of the new symmetries mustbe Poincaré scalars, carrying no spacetime indices. Otherwise, the S-matrix would be trivialand no interactions would occur. The theorem assumes bosonic generators (those that obeycommutation relations) for the symmetry groups, leaving the possibility of anti-commuting(fermionic) generators.1 In this sense, SUSY is the unique extension of the usual Poincarésymmetry that incorporates fermionic generators into the group algebra [84]. Qualitatively, theaction of the SUSY generators Q on states of definite statistics is [80]Q|boson〉 = |fermion〉, Q|fermion〉 = |boson〉. (2.1)1Another assumption is the existence of a mass gap between the vacuum and first excited state, which does notoccur in conformal field theories (CFTs). However, CFTs do not contain particle-like excitations and, therefore,are not appropriate for the description of particle physics near electroweak scales.26In order for the above to make sense, Q must carry a fermionic (i.e., a spacetime) index.Supersymmetry adds to the standard Poincaré algebra of translations, boosts and rotations,the following relations2 [81, 85]{QAα , QBβ } = {Q†α˙ A, Q†β˙ B} = 0, (2.2)[Pµ, QAα ] = [Pµ, Q†α˙ A] = 0, (2.3){QAα , Q†α˙ B} = −2σµαα˙PµδAB. (2.4)Here α and α˙ are two-component spinor indices; A, B = 1, . . . ,N label different generators;Pµ is the energy momentum operator (generator of spacetime translations); σµ = (1, σi) whereσi are the Pauli matrices. Theories with N > 1 are called extended supersymmetry. In thesetheories fermions arise in vector pairs – one can always write down gauge invariant mass terms.This is unlike the SM, which is a chiral theory. Thus N = 1 SUSY is considered to be moreapplicable for extending the SM [81]. In the following, we work only with N = 1, so we candrop the labels A and B.Chapters 3, 4 and 6 make extensive use of supersymmetric model building. Thereforethis chapter is devoted to the introduction of some of the relevant techniques. The outline ofthis chapter is as follows. In Sec. 2.2 we describe the superspace formalism for writing downsupersymmetric theories. In Sec. 2.3 we introduce the minimal supersymmetric extension ofthe SM and discuss some benefits of SUSY. We discuss supersymmetry breaking in Sec. 2.4 andreview current status of experimental searches in Sec. 2.5.2.2 SuperspaceIn the following chapters we will frequently need to construct supersymmetric models. Inthis section we outline the formalism of superspace, which efficiently encodes interactions ofsupersymmetric theories.Poincaré invariance is guaranteed if we construct the action S from the unrestricted space-time integral over a Lorentz scalar Lagrangian density L :S =∫d4xL . (2.5)Similarly, it is useful to implement supersymmetric invariance using an integral over a “su-perspace”, where SUSY transformations are represented simply as translations. As with theLorentz group, it is useful to work with irreducible representations of the symmetry algebra.Superspace facilitates this as well, by allowing us to write a supermultiplet (which containsbosonic and fermionic degrees of freedom) as a single superfield. To achieve this, superspaceis parametrized by the spacetime coordinate xµ (mass dimension -1) and Grassman number(anti-commuting) spinors θα and θ¯α˙ (mass dimension -1/2). A general function on superspace2Here we show only a subset of the entire supersymmetry algebra. The complete set can be found in Ref. [85].27can be expanded in a Taylor series that terminates due to the anti-commuting nature of θ andθ¯. For example, a general Lorentz scalar superfield is given by [86]:T = a+ θξ + θ2b+ θ¯χ¯+ θ¯σ¯µθvµ + θ2θ¯ζ¯ + θ¯2c+ θ¯2θη + θ4d, (2.6)where Greek (Roman) letters denote fermions (bosons). Note that there are 8 complex bosonicand 8 complex fermionic degrees of freedom. The superspace translations by an infinitesimalspinor αθα → θα + α,θ¯α˙ → θ¯α˙ + ¯α˙, (2.7)xµ → xµ + ∆µ,with ∆µ = iσµθ¯ + i¯σ¯µθ, generate the required supersymmetric transformations that mixbosons with fermions. The corresponding generators areQα = i∂∂θα− (σµθ¯)α∂µ, Qα˙ = i ∂∂θ¯α˙− (σ¯µθ)α˙∂µ. (2.8)An infinitesimal transformation is then3δT = T (x+ ∆, θ + , θ¯ + ¯)− T (x, θ, θ¯) = −i(Q+ Q)T, (2.9)from which one can derive explicit transformation laws for all components of T . For example,the lowest and highest components of the superfield T transform as [86]a→ a+ ξ+ χ¯¯,d→ d+ i2∂µ(ζ¯σ¯µ+ ησµ¯). (2.10)Note that the highest component of any superfield transforms by a total spacetime derivative.Therefore any action of the formS =∫d4xT |θ4 + h.c. =∫d4x∫d4θT + h.c. (2.11)is invariant under supersymmetry transformations. In the last step we wrote the highest com-ponent of T as an integral over the Grassmannian coordinates. Differentiation and integrationwith Grassmannian numbers are reviewed in Refs. [80, 86].The general superfield of Eq. (2.6) is a reducible representation of SUSY, meaning thatthere are objects with fewer degrees of freedom that have simple supersymmetric transformationproperties [81, 86]. Irreducible representations are obtained by imposing restrictions on T . The3Different authors use various normalizations for the SUSY transformations; see, e.g., Ref. [80].28first relevant case is the chiral superfield Φ, defined byD¯Φ = 0, (2.12)where we introduced the SUSY covariant derivativesDα =∂∂θα− i(σµθ¯)α∂µ, D¯α˙ =∂∂θ¯α˙− i(σ¯µθ)α˙∂µ. (2.13)These operators obey{Dα, Dβ} = 0, {Dα, D¯β˙} = 2iσµαβ˙∂µ. (2.14)The most general solution of Eq. (2.12) is conveniently written in terms of a new coordinateyµ = xµ + iθ¯σ¯µθ (in which D¯α˙ = ∂/∂θ¯α˙):Φ = φ(y) +√2θψ(y) + θ2F (y). (2.15)The above expression can be related to the components of the general superfield, Eq. (2.6), byexpanding y. The chiral supermultiplet contains a complex scalar φ, a Weyl fermion ψ and acomplex scalar auxiliary field F . If φ has canonical mass dimension 1 (in 4D), then ψ and F havedimensions 3/2 and 2, respectively. The only dependence on θ¯ enters through yµ. To constructrealistic theories (i.e., those with interactions), we take products of chiral superfields Φi likeΦiΦj and ΦiΦjΦk; these superfields are also chiral, because the SUSY covariant derivativesdefined in Eq. (2.13) obey the Leibniz (product) rule. Moreover, by writing the superchargesQ and Q from Eq. (2.8) in terms of y, one can easily show thatδφ =√2ψ,δψα = −i√2(σµ†)α∂µφ+√2αF,δF = −i√2¯σ¯µ∂µψ. (2.16)Note that the highest component of a chiral superfield transforms by a total derivative, so thesupersymmetry-invariant action can also include terms of the formS ⊃∫d4x∫d2θW (Φ) + h.c., (2.17)where W is a holomorphic function of chiral superfields called the superpotential. The super-potential must have mass dimension 3, so the most general renormalizable expression for WisW = 12m2ijΦiΦj +13λijkΦiΦjΦk. (2.18)29Interactions of the component fields can be derived from W by noting that [85]ΦiΦj |θθ = φiFj + φjFi − ψiψj (2.19)ΦiΦjΦk|θθ = Fiφjφk − ψiψjφk + permutations, (2.20)where the fields on the right hand side are functions of y only.Finally, the canonically normalized kinetic terms are contained inS ⊃∫d4θΦ†Φ = ∂µφ∂µφ+ iψ¯σ¯µ∂µψ + F †F. (2.21)Note that F does not have a kinetic term. In fact, its equation of motion is an algebraic (asopposed to a differential) equation that can be used to eliminate F from the Lagrangian:F †i = −δWδΦi∣∣∣∣Φi=φi. (2.22)Thus F is said to be an auxiliary field.To implement gauge symmetries we require invariance of the action under local phase rota-tions of the (matter) chiral superfieldsΦ → eiqΩΦ, (2.23)where q is the charge of Φ under an Abelian gauge symmetry. In order for the transformed fieldto remain a chiral superfield, the transformation parameter, Ω, must also be a chiral superfield,which follows from the definition in Eq. (2.12). Since Ω is a complex quantity, we absorbthe factor i into its definition. The kinetic terms of Eq. (2.21) are not invariant under thistransformation sinceΦ†Φ → Φ†eq(Ω†+Ω)Φ. (2.24)As in non-supersymmetric field theory, we must introduce a compensating gauge field V andinclude it into the kinetic terms, which gives rise to gauge interactions. The appropriate formfor the new gauge invariant kinetic energy is∫d4θΦ†e2qV Φ, (2.25)where V is realV = V †, (2.26)30and transforms asV → V − 12(Ω + Ω†) (2.27)under a gauge transformation. The above transformation is much more general than a standardgauge transformation. This freedom can be exploited to eliminate components of V . Whenwriting interactions in terms of component fields it is extremely helpful to use the Wess-Zumino(WZ) gauge [85], in whichV = Aµθ¯σ¯µθ + θ¯θ¯θλ+ θθθ¯λ¯+12θθθ¯θ¯D (2.28)V 2 = 12AµAµθθθ¯θ¯ (2.29)V 3 = 0. (2.30)Thus, the physical degrees of freedom of a vector superfield consist of a vector Aµ, Weyl fermionλ and an auxiliary scalar fieldD. Even after this simplifying gauge choice, there remains freedomto perform the standard gauge transformations, under whichAµ → Aµ + ∂µα(x), (2.31)while λ and D remain unchanged; here α corresponds to the imaginary part of the scalarcomponent of Ω, which follows from Eq. (2.27) [86]. In the WZ gauge, the gauge invariantkinetic term of Eq. (2.25) gives∫d4θΦ†e2qV Φ = (Dµφ)†Dµφ+ iψ¯σ¯µDµψ + F †F −√2q(φ†ψλ+ h.c.) + qφ†φD, (2.32)where Dµ = ∂µ − iqAµ is the covariant derivative. Finally, the gauge kinetic terms are con-structed fromWα(y, θ) = −14D¯2DαV = λα +Dθα +i2Fµν(σµσ¯νθ)α + i(σµ∂µλ)αθθ, (2.33)which is a chiral superfield. Gauge invariance ofWα is easily proven using commutation relationsof Dα and D¯β˙ in Eq. (2.14). Canonically normalized kinetic terms for the vector A and gauginoλ come from∫d2θ(14− iθCP32pi2)WαWα + h.c. = −14FµνFµν + iλ¯σ¯µ∂µλ+θCP32pi2Fµν ˜Fµν +12D2. (2.34)Note that D is an auxiliary field with no proper kinetic terms. Its equations of motion can besolved algebraically:D = −φ†iqiφi (2.35)31The θCP term for an Abelian theory is a total derivative and can be dropped. The abovediscussion can be generalized to non-Abelian gauge symmetries [80, 85, 86].To summarize, the Lagrangian density of a renormalizable supersymmetric Abelian gaugetheory can be written asL =∫d4θΦ†ie2qiV Φi +∫d2θ(14WαWα +W (Φ) + h.c.). (2.36)The interactions of the component fields can be read off from Eqs. (2.20), (2.32), and (2.34).However, this form slightly obfuscates the scalar potential V of the model. A more usefulrepresentation of the scalar potential is obtained by fixing the auxiliary fields Fi and D usingtheir equations of motion (Eqs. (2.22) and (2.35)), which givesV = F †i Fi + 12D2 =∣∣∣∣δWδΦi∣∣∣∣2+ 12g2∣∣∣∣∣∑iφ†iqiφi∣∣∣∣∣2(2.37)where in δW/δΦi is taken to be a function of the scalar components of chiral superfields Φjonly. For a non-Abelian gauge theory, the charge qi is replaced by the group generator T aappropriate for the representation of Φi.2.3 The Minimal Supersymmetric Standard ModelIt is now easy to specify the simplest realistic supersymmetric theory: the Minimal Supersym-metric Standard Model (MSSM). All SM matter fields are promoted to chiral superfields, whilegauge bosons become real gauge superfields. The superpotential of the model isW = ucRYuQHu − dcRYdQHd − ecRYeLHd + µHuHd, (2.38)and the superfields have the same gauge charges as their analogues in the SM (see Sec. 1.1.1).One immediate novel feature is that we need two Higgs doublets Hd (hypercharge Y = −1/2)and Hu (Y = +1/2) to write this superpotential. 4 In the SM, we were able to couple bothup and down type fermions to the same doublet by using the conjugate field H†. Since W is aholomorphic function, complex conjugates do not appear – another Higgs doublet with oppositehypercharge is required.5Unlike the Yukawa interactions of Eq. (1.11), this superpotential is not the most generalrenormalizable functional of the chiral superfields consistent with gauge symmetries. For ex-ample, one can also write down the gauge invariant operators [80]W∆B=1 =12λ′′ijkucRidcRjdcRk, W∆L=1 =12λijkLiLjecRk + λ′ijkLiQjdcRk + µ′iLiHu, (2.39)4The doublets are sometimes called H1 and H2 instead of Hd and Hu, respectively.5Another doublet is also needed for anomaly cancellation since they introduce chiral fermions [80].320.40.50.60.70.80.911.11.21.30 2 4 6 8 10 12 14g ilogµ/MZg1g2g3Figure 2.1: Renormalization group evolution of the gauge couplings in the SM (dashedlines) and the MSSM (solid lines). In the MSSM gauge couplings appear to unifyat around µ ≈ 2× 1016 GeV.which violate baryon and lepton numbers by one unit. For non-zero values of, e.g., λ′′ and λ,these interactions lead to a large proton decay rate. Such dangerous terms can be forbiddenby R-parity, a new type of discrete Z2 symmetry with SM fields having charge +1 and theirsuperpartners having charge −1. In the superspace language, each superfield has the charge ofits lowest component. So, for example, the quark and lepton superfields are odd under R-parity,while the Higgs and gauge multiplets are even. Thus terms in Eq. (2.39) are forbidden.The physical content of R-parity is that superpartners are always produced in pairs andthat the lightest supersymmetric particle (LSP) is stable. If it is electrically neutral, it canbe a good dark matter candidate [75]. As was alluded to in the previous sections, the MSSMhas several other enticing features. First, in the supersymmetric limit, quadratically-divergentquantum corrections to the physical Higgs boson mass cancel exactly among superpartnersvia the mechanism described in Sec. 1.3.1. Second, the matter content in the MSSM is moreconsistent with gauge coupling unification at the scale of ∼ 2× 1016 GeV, the so-called GrandUnification (GUT) scale. The running of gauge coupling in the SM and MSSM is shown inFig. 2.1 by dashed and solid lines, respectively. The common value of the couplings at GUTscale suggests that the SM gauge group descends from a simple gauge group such as SU(5)with a single gauge coupling [27].The spectrum of the MSSM consists of SM particles and their superpartners, along with theextended Higgs sector. Electroweak symmetry breaking occurs when the two Higgs doubletsHd and Hu acquire expectation values vd and vu; these are usually specified using the SM vevv = v2d + v2u = 246.22GeV and tanβ = vu/vd. Symmetry breaking mixes the components of33gauge-eigenstate superfields; the resulting particle content of the MSSM is shown in Tab. 2.1.In addition to the SM-like Higgs, there are CP odd neutral and charged scalars. Each SMfermion (4 degrees of freedom on-shell) has a pair of complex scalar superpartner “sfermions”.The gluon has a strongly-interacting Majorana fermion partner - the gluino. Electroweak gaugeboson superpartners (gauginos) mix with Higgs superpartners (the Higgsinos) to create spin 1/2neutralinos and charginos. In this table and below, we use the convention for R-type sfermionsthat f˜ cR = f˜ †R. The details of electroweak symmetry breaking and MSSM mass spectrumcomputation can be found in, e.g., Refs. [80, 81, 87] and (partially) in the following chapters.Despite its attractive theoretical features, the exactly-supersymmetric MSSM is alreadyruled out since it requires near degeneracy between SM states and their superpartners. Thiscan be seen through the “supertrace” [80, 88]Str(m2) =∑s(−1)2s(2s+ 1)Tr(m2s) = 0, (2.40)where the sum is over particles of spin s. This relation is valid for renormalizable theories, evenif supersymmetry is broken spontaneously at tree-level. For example, the contribution of firstgeneration quark superfields decouples, yielding [86]Str(m2) = m2u˜R +m2u˜L +m2d˜R +m2d˜L− 2(m2u +m2d) = 0, (2.41)which implies that the first generation squarks are at the MeV scale! Since the supertracerelationship holds for renormalizable interactions, SUSY cannot be broken spontaneously inthe MSSM sector. We conclude that if supersymmetry has anything to do with nature, it mustbe broken in a “hidden” sector, such that physics looks explicitly non-supersymmetric in theMSSM, or “visible” sector. Supersymmetry-breaking effects must be communicated from thehidden to the visible sector via quantum (loop) effects and non-renormalizable interactions.Before discussing the possible mechanisms of supersymmetry breaking and its communica-tion to the visible sector, we note that in the MSSM, renormalizable SUSY breaking effects canbe parametrized by [80]Lsoft = −(M3g˜ag˜a +M2˜W a˜W a +M1 ˜B ˜B + h.c.)−(u˜†RAu ˜QHu − d˜†RAd ˜QHd − e˜†RAe˜LHd + h.c.)− ˜Q†m2Q ˜Q− ˜L†m2L˜L− u˜†Rm2u˜u˜R − d˜†Rm2d˜d˜R − e˜†Rm2e˜ e˜R−m2Hu |Hu|2 −m2Hd |Hd|2 − (bHuHd + h.c.) , (2.42)which is the most general SUSY-breaking Lagrangian with only positive mass dimension (soft)operators. Taking operators with positive dimension ensures that we do not introduce any newquadratic divergences (which SUSY was designed to cure in the first place). Using dimensionalanalysis, it is easy to see that any 2 point function corrections arising from these couplings will34Name Gauge Eigenstates Mass Eigenstates SU(3)C U(1)em RNeutral Higgs H0u, H0d h, H, A 1 0 +1Charged Higgs H±u , H±d H± 1 ±1 +1up squarks u˜L,R, c˜L,R, t˜L,R u˜1,2, c˜1,2, t˜1,2 3 +2/3 -1down squarks d˜L,R, s˜L,R, b˜L,R d˜1,2, s˜1,2, b˜1,2 3 -1/3 -1sleptons e˜L,R, µ˜L,R, τ˜L,R e˜1,2, µ˜1,2, τ˜1,2 1 -1 -1sneutrinos ν˜e, ν˜µ, ν˜τ same 1 0 -1gluino g˜ same 8 0 -1neutralinos ˜H0u, ˜H0d , ˜W 3, ˜B ˜Ni, i = 1, 2, 3, 4 1 0 -1charginos ˜H±u , ˜H±d , ˜W± ˜Ci, i = 1, 2 1 ±1 -1Table 2.1: Particle content in the MSSM before and after electroweak symmetry breaking,along with colour, electromagnetic charge and R-parity assignments.have magnitude ∼ m2soft, where msoft is the generic scale of couplings appearing in Eq. (2.42).With these terms included the SM particles and their superpartners are no longer degenerate.As a result the cancellation of the leading quadratic divergence is not exact; fortunately, theremainder is proportional to m2soft, instead of Λ2 as in Sec. 1.3.1. Thus, the hierarchy betweenthe EW and gravitational scales is reduced, assuming msoft MPl.In Eq. (2.42) the couplings m2i , Ai are real and complex matrices in family space, respec-tively. In total, there are more than 100 new parameters beyond those found in the SM [80, 89].Therefore, studying the full parameter of the MSSM with generic soft breaking terms is in-tractable. Moreover, arbitrary choices for these parameters generally suffer from phenomeno-logical problems, such as flavour mixing and CP violation, which are severely constrained by,e.g., bounds on rare decays such as µ → eγ and K − K oscillation measurements [80]. Thisis known as the SUSY flavour problem. This motivates the search for a mechanism of super-symmetry breaking, that can generate Eq. (2.42) and predict the various couplings in terms of(hopefully) a small number of fundamental parameters. The proliferation of parameters canalso be seen as being advantageous, since there are many new possible CP-violating sources,which is important for baryogenesis, as discussed in Sec. 1.3.2.2.4 Supersymmetry Breaking and Supergravity2.4.1 Global Supersymmetry BreakingIn the previous section we determined that if supersymmetry is realized in nature, it cannotbe an exact symmetry, since we do not observe any superpartners mass-degenerate with knownparticles. In this section we discuss how to break supersymmetry spontaneously and how theseeffects are communicated to the visible sector in realistic theories.If supersymmetry is broken spontaneously, the vacuum |0〉 is not invariant under SUSY35transformations, that isQα|0〉 6= 0, Qα˙|0〉 6= 0. (2.43)Our first goal is to determine whether a given model breaks SUSY. Using the supersymmetryalgebra, Eq. (2.4), we can express the Hamiltonian asH = 14(Q1Q†1 +Q†1Q1 +Q2Q†2 +Q†2Q2), (2.44)so Eq. (2.43) implies〈0|H|0〉 > 0 (2.45)when supersymmetry is spontaneously broken. Since kinetic terms cannot have an expectationvalue in a homogeneous vacuum, we conclude that the potential alone must have an expectationvalue. The SUSY potential (see Eq. (2.37)) is positive definite, so 〈V 〉 > 0 implies eitherF i 6= 0 (F -term breaking), (2.46)for some i orDa 6= 0 (D-term breaking). (2.47)There is an alternative way to see that these are the conditions required for spontaneous SUSYbreaking, that will be useful when discussing supergravity. If the vacuum state is not invariant,that is Eq. (2.43) is satisfied, we have〈0|δT |0〉 6= 0, (2.48)for some superfield T (either chiral or vector), where δT is defined in Eq. (2.9). Let us specializeto the case where T is a chiral superfield. Looking at the transformation of the component fields,Eq. (2.16), we see that only δψ can have a non-zero expectation value without breaking Lorentzinvariance6 when√2αF 6= 0. (2.49)Thus we recover the condition for F -term SUSY breaking. The same analysis can be repeatedwith a vector superfield to re-derive D-term breaking.A simple renormalizable example of F -term breaking is the O’Raifeartaigh (OR) model [80,6It is also possible to do this using a non-zero expectation value of a fermion bilinear [90].3681, 91] with the superpotentialWOR = −k2Φ1 +mΦ2Φ3 +y2Φ1Φ23. (2.50)The corresponding F -terms areF1 = −k2 +y2(φ∗3)2 (2.51)F2 = mφ∗3 (2.52)F3 = mφ∗2 + yφ∗1φ∗2. (2.53)In particular, we cannot simultaneously have F1 = 0 and F2 = 0, so 〈V 〉 > 0 and SUSY isspontaneously broken. For example, in the vacuum where all scalar VEVs vanish, V = |F1|2 =k4 > 0, as required by (2.45). To confirm that SUSY is broken, we can evaluate the spectrumof the model, which consists of 6 real scalars with squared masses (assuming real m and k2)0, 0, m2, m2, m2 ± yk2 (2.54)and 3 Weyl fermions with mass squared0, m2, m2, (2.55)which demonstrates the mass splitting among spin 0 and spin 1/2 partners. Note that thisspectrum satisfies the supertrace relationship, Eq. (2.40). While the vanishing scalar massesare lifted by quantum corrections [81], the massless fermion remains massless at all orders inperturbation theory. The appearance of a massless fermion, the Goldstino, is a general featureof spontaneous supersymmetry breaking, and is completely analogous to the appearance ofa Nambu-Goldstone boson in a theory with spontaneously broken continuous symmetry. Thedifference in statistics is due to the fermionic nature of the SUSY generators. The existence of aGoldstino in a general theory can also be seen from the fact that the linear combination [80, 86]η = F iψi +1√2Daλa (2.56)is an eigenvector of the fermion mass matrix with eigenvalue 0.The OR model relies on the existence of an explicit dimensionful parameter in the superpo-tential, which some find unsatisfying, since the scale of SUSY breaking √F1 = k is effectivelyput in by hand. This motivated the search for dynamical, non-perturbative mechanisms thatgenerate this scale from dimensionless parameters. For a recent review of such mechanisms see,e.g., Refs. [92–94]. Quantitative discussion of non-perturbative processes is only made possi-ble by the existence of dualities (weakly coupled descriptions of strongly coupled physics) andholomorphy of the superpotential (which severely restricts the couplings that can appear in the37action) [95].2.4.2 Super-Higgs Mechanism and SupergravityPhenomenological difficulties associated with a massless fermion aside, we will usually thinkof SUSY descending from a string theoretic framework, where SUSY is realized as a localsymmetry [96]. In local supersymmetry, or supergravity (SUGRA), the Goldstino is not partof the physical spectrum, but rather is absorbed as an extra polarization state of a “gaugefermion” associated with SUGRA, the spin 3/2 gravitino (superpartner of the spin 2 graviton).In the process, the gravitino becomes massive. This super-Higgs mechanism is analogous tospontaneous breaking of (bosonic) SM gauge group SU(2)L × U(1)Y when W and Z bosonsacquire masses. Note that the gravitino mass, m3/2, must vanish in the supersymmetric limit,as well as in the limit where gravity is completely decoupled (MPl →∞). For F -term breaking,this suggests that the gravitino mass must scale as [80]m3/2 ∼FMPl. (2.57)The above relation can be made more precise. While the construction of the full SUGRA-invariant action [90] is beyond the scope of this work, we will display some useful results. First,because gravity is non-renormalizable, we cannot restrict ourselves to renormalizable “kinetic”and superpotential terms. In the global SUSY limit, this means we must consider Lagrangiandensities of the formL =∫d4θK(Φ†ie2qV ,Φj) +∫d2θ(14fab(Φ)W aW b +W (Φ) + h.c.), (2.58)where K, fab are arbitrary functions called the Kähler potential and gauge kinetic function [97];note that fab must be a chiral superfield for the above to be SUSY invariant; here a and b areadjoint group indices. The SUGRA potential can be expressed entirely in terms of a Kählerfunction GG = 1M2PlK + ln(|W |2M6Pl)(2.59)and the gauge kinetic function fab as [90]V = M4PleG(Gi¯GiG¯ − 3)+ g22M4Pl(Ref−1ab )GiT aijφjGkT bklφl, (2.60)whereGi =∂G∂φi, Gı¯ =∂G∂φ∗i, Gi¯ =∂G∂φi∂φ∗j, (2.61)38and Gi¯ is the matrix inverse of Gi¯. Equation (2.60) is the generalization of Eq. (2.37) tolocal supersymmetry; the first and second terms correspond to F and D contributions in globalSUSY. Note that the potential of Eq. (2.60) is no longer positive-indefinite, so the global SUSY-breaking criterion, Eq. (2.45), is no longer appropriate. We can still employ the alternativecondition, Eq. (2.48), from which it follows that either [90, 98]Gi¯G¯ 6= 0, (2.62)orRef−1ab GiT bijφj 6= 0. (2.63)These conditions generalize F - and D-term breaking, respectively, to supergravity. Both casesrequire that there exists an i such that Gi 6= 0. For example, for the canonical choice of theKähler potentialK = Φ†iΦi, (2.64)we have G¯ = φj/M2Pl + (W †)−1(∂jW )†, Gi¯ = δijM2Pl, so Eq. (2.62) simplifies toM2PlδWδΦi∣∣∣∣Φi=φj+φ∗iW 6= 0. (2.65)Note that in the limit MPl → ∞ we recover the global SUSY F -term breaking condition,Eq. (2.46) (via the equation of motion, Eq. (2.22)).We can now describe how the gravitino obtains its mass. After SUSY breaking, the SUGRALagrangian density contains a mixing term between the gravitino Ψµ and the Goldstino Ψ =(η, η†)T 7i2MPleG/2ΨLµγµΨL + h.c., (2.66)where G is the expectation of Eq. (2.59) in the SUSY breaking vacuum. Just like in the caseof bosonic local symmetries, there exists a “unitary” gauge (here a local SUSY transformation)that removes this mixing, leaving behind only the physical propagating state – the massivegravitino [90]. The gravitino mass parameter can be read off from the remaining mass termi2MPleG/2ΨµσµνΨν , (2.67)7Here we have momentarily switched to 4 component notation because the Rarita-Schwinger equation, theequation of motion for a spin 3/2 field, is most easily specified using Dirac spinors [99].39som3/2 = eG/2MPl = eK/2M2Pl|W |M2Pl. (2.68)This result can be simplified, if, in addition to canonical kinetic terms, we also assume avanishing cosmological constant [86]m3/2 ≈Ftot√3MPl, (2.69)where|Ftot|2 =∣∣∣∣∣δWδΦi∣∣∣∣Φi=φj+φ∗iWM2Pl∣∣∣∣∣2(2.70)parametrizes the total amount of supersymmetry breaking from non-vanishing F -terms. Thegravitino mass is often used as a proxy for the scale of supersymmetry breaking.2.4.3 Models of Supersymmetry BreakingIn Sec. 2.3 we determined that phenomenologically viable models of SUSY must be softlybroken. We parametrized this breaking by a generic Lagrangian with many new parameters.We noted that arbitrary choices of these parameters lead to phenomenological problems. Wenow seek to relate these parameters to a fundamental theory of spontaneous SUSY breaking.Suppose that SUSY is broken in a hidden sector (HS) by a collection of non-MSSM fieldsthat couple weakly to the MSSM via some messenger fields. There are three well-studied classesof SUSY breaking mechanisms, which differ in the nature of the messengers.Even in the absence of any coupling between the SUSY-breaking and visible sectors, gravitycan mediate these effects to the MSSM. We will see that this scenario yieldsmsoft ∼ m3/2 (gravity mediation). (2.71)Note that this implies (via Eq. (2.57)) that the scale of SUSY breaking must beF ∼ msoftMPl ∼ (1011 GeV)2, (2.72)where we assumed that msoft ∼ 1TeV.We demonstrate the appearance of soft SUSY breaking terms in the visible sector using asimple example. Suppose the SUSY-breaking sector has superpotential WH(Zr), a function ofHS fields Zr. Furthermore, assume there is no explicit coupling between Zr and visible sectorfields Φi, so the full superpotential takes the formW = WV (Φi) +WH(Zr), (2.73)40where WV is the superpotential in the visible sector. Taking the minimal Kähler function andusing Eq. (2.60) yields the potentialV = eK/M2Pl(∣∣∣∣φ∗iWM2Pl+ ∂WV∂φi∣∣∣∣2+∣∣∣∣z∗rWM2Pl+ ∂WH∂zr∣∣∣∣2− 3 |W |2M2Pl). (2.74)Note that even though we started with no explicit coupling between the visible and hiddensectors, the supergravity potential automatically generates mixings between the two. Whenthe HS fields acquire SUSY-breaking expectation values parametrized as [97]WH = µM2Pl, zr = arMPl,∂WH∂zr= crµMPl, (2.75)and one finds the gravitino mass (using Eq. (2.68))m3/2 = e|ar|2/2µ. (2.76)Here µ has dimensions of energy, while ar and cr are dimensionless. These substitutions allowone to expand Eq. (2.74) in m3/2/MPl  1:V ≈∣∣∣∣∣∂̂WV∂φi∣∣∣∣∣2+m23/2|φi|2 +(m3/2[|ar|2 + cra∗r − 3]̂WV +m3/2φi∂̂WV∂φi+ h.c.), (2.77)where ̂WV = exp(|ar|2/2)WV . The first term matches the global SUSY F -term contribution(see Eq. (2.37)). The following terms are soft SUSY-breaking parameters, including a universalmass for all visible scalars, bi- and tri-linear interactions, all set by the scale m3/2. Theseoperators match onto the general soft SUSY-breaking Lagrangian of Eq. (2.42). At this pointwe are only missing the gaugino mass terms, which arise from the term14MPleG/2∂f∗ab∂z∗¯G¯kGkλaλb, (2.78)in the SUGRA Lagrangian [90]. Note that this requires a non-minimal gauge kinetic functionfab; assuming these effects are generated by SUGRA interactions, this term yields a gauginomass m1/2 ∼ m3/2. The scenario just described is called minimal supergravity (mSUGRA) orconstrained MSSM (cMSSM) and has the virtue of having very few parameters. Moreover, theuniversal scalar mass suppresses dangerous flavour and CP-violating effects that plague genericSUSY-breaking parameters. For these reasons, much experimental and phenomenological efforthas been spent on constraining and excluding mSUGRA [80], despite the fact that the extremelystrong assumptions (namely the form of the Kähler potential and the superpotential) are notwell-motivated theoretically. For more general choices of K and W , scalar masses are no longeruniversal, and once again flavour observables become a problem.One way to avoid flavour problems is to ensure that SUSY-breaking is communicated to41the MSSM in a way that does not mix different generations. This can be achieved usingmessengers (new chiral superfields) that are charged under SU(3)C × SU(2)L × U(1)Y . Sincegauge interactions do not violate flavour at tree-level, we expect the resulting soft terms to bephenomenologically viable. This mechanism is known as gauge mediation. Consider a pair ofchiral superfields Φ and Φ¯ that are charged under SU(3)C × SU(2)L × U(1)Y and that coupledirectly to a field X whose vev breaks SUSY:W ⊃ XΦ¯Φ, (2.79)where X = M + θ2F . The SUSY breaking vev F splits the masses of the messenger scalarsand fermions, but these effects are communicated to the MSSM only indirectly via loops ofmessenger and gauge fields. For example, the gaugino mass arises at one loop, while soft scalarmass squared parameters appear at two loops from diagrams like those shown in Fig. 2.2. Thus,we expect the leading contributions to gaugino and scalar masses to be [100]Mg˜ ∼g216pi2FM, m2f˜ ∼(g216pi2FM)2. (2.80)In either case, the messengers couple to MSSM through gauge interactions and the resultingsoft parameters are flavour diagonal, alleviating the SUSY flavour problem. Interestingly, inorder to obtain the precise leading order (in F/M2) predictions for the soft parameters, onedoes not need to evaluate diagrams like those in Fig. 2.2 explicitly. Instead, these results arecompletely determined by the beta functions and anomalous dimensions of the theory [101].Due to the non-renormalization of the superpotential [81], X dependence may only enter inthe Kähler potential and gauge kinetic function, i.e., via renormalizations of matter and gaugekinetic terms, respectively. The former is related to anomalous dimensions, while the latter tobeta functions of gauge couplings. One can then use the reality of the Kähler potential andthe holomorphy of the gauge kinetic function to determine their dependence on X and therebyextract the soft parameters which arise from terms like∫d4θ(X†XM2)Q†Q, (2.81)and∫d2θ(XM)W aW a + h.c. (2.82)in the Lagrangian. For example, for the gaugino masses this procedure givesMa =Ng2a16pi2FM, (2.83)where N is the number of messenger fields and where a labels the gauge group. The scalar soft42masses and cubic A-terms are found to depend only on beta function coefficients and grouptheoretic invariants [80].From Eqs. (2.57) and (2.80) it is clear thatmsoft ∼g216pi2(MPlM)m3/2 (gauge mediation), (2.84)so msoft  m3/2 unless M ∼ MPl. In the latter limit, however, gravitational effects becomeimportant and the flavour problem is reintroduced. Thus, in gauge mediation, the gravitino isusually the LSP and it is much lighter than the rest of superpartners. This has several interestingcollider (e.g., displaced vertices) and cosmological implications due to its weak (gravitationalstrength) interactions [80, 100].Our last example of theories of SUSY breaking is motivated by extra-dimensional construc-tions. When discussing gravity mediation we mentioned that, in general, because gravitationalinteractions are flavour blind, there is a severe flavour problem due to inter-generational mixingof SM superpartners. This is because the Kähler potential that gives rise canonical kineticterms, Eq. (2.64), is not radiatively stable; 8 in the presence of gravitational interactions athigh scales one expects [81]K =(δij +cijX†XM2Pl)Φ†iΦj , (2.85)where X is the chiral superfield with the SUSY-breaking vev. The coefficients cij are notdiagonal and O(1); such terms arise from integrating out heavy string states that do not nec-essarily respect flavour symmetries [102]. These undesirable terms can be suppressed if theSUSY-breaking sector is physically sequestered from the visible sector in an extra dimension.The above scenario is implemented as follows [102]. The MSSM and SUSY-breaking sectorsare embedded into different 3-branes, a distance R apart in an extra dimension. It is thennatural to have a “separable” superpotential of the form given in Eq. (2.73). Moreover, ifsupergravity (and heavy string) states are the only degrees of freedom allowed to propagatein the bulk of the extra dimension, the effects of SUSY-breaking on the MSSM brane will besuppressed bye−M5R, (2.86)where M5 is the higher-dimensional Planck scale. Thus, if the size of the extra dimension issuch that M5R 1, the undesirable flavour violating effects will be suppressed.In the above limit, the leading order effects arise due to anomalous global symmetries of theclassical SUGRA action [103]. In the derivation of the low-energy effective theory for SUGRAwith canonically normalized gravitational kinetic terms, one is forced to make a certain set field8The superpotential is not renormalized in perturbation theory, so the special form of Eq. (2.73) is radiativelystable.43redefinitions. While the classical action is invariant under these, the path integral measure isnot, and as a result, new terms in the effective action are generated at one loop, similar to themore familiar chiral anomalies of quantum field theory. The resulting terms in the action dependon auxiliary fields of the supergravity multiplet. Since the supergravity multiplet couples to theSUSY-breaking sector, these auxiliary fields can inherit a SUSY-breaking vev and give rise tosoft terms. This scenario is called Anomaly Mediated Supersymmetry Breaking (AMSB) andthe resulting soft terms have magnitudemsoft ∼g216pi2FMPl∼ g216pi2m3/2 (anomaly mediation). (2.87)An efficient way to derive these effects is to use the super-conformal compensator formal-ism [100, 102]. The soft parameters are completely determined in terms of gauge beta functionsand anomalous dimensions. For example, gaugino masses are given by [81]Ma =bag2a16pi2FMPl, (2.88)where a labels the gauge group and ba is the corresponding β function coefficient. The sfermionmasses depend on the anomalous dimension matrix γij [80](m2f˜ )ij =12µdγijdµF 2M2Pl. (2.89)Note that while Eq. (2.89) contains non-flavour diagonal contributions, they are proportionalto MSSM Yukawa couplings and therefore do not introduce significant flavour violation beyondwhat is already present in the SM. While AMSB is very predictive (all masses are determinedin terms of a single parameter F/M2Pl), it is not a complete model, since Eq. (2.89) can giverise to tachyonic sfermion masses. For example, for the the right-handed selectron we have [80]m2e˜R = −99g413200pi2F 2M2Pl. (2.90)This is a general problem for matter that is charged under a gauge group that is not asymp-totically free. Thus, in general AMSB requires additional dynamics to be a viable theory ofsupersymmetry breaking and the resulting spectrum of superpartners depends on the nature ofthese modifications.There is additional motivation to consider AMSB contributions to soft masses. In theabsence of extra-dimensional sequestering between the MSSM and the SUSY-breaking sectors,normal supergravity contributions to the soft scalar masses are important, since there is nosymmetry that can forbid terms like Eq. (2.81), since X†X is invariant under all symmetries.In contrast, gaugino mass terms of the form of Eq. (2.82) require the SUSY-breaking field Xto be a gauge singlet, which can be difficult to realize, especially in theories with dynamical44Figure 2.2: Example contributions to soft SUSY-breaking parameters from heavy mes-senger loops (bold lines) to MSSM gauginos (left) and sfermions (right) in gaugemediation.supersymmetry breaking [102]. Thus, it is natural for these terms to be forbidden, and as aresult, the leading contributions to gaugino masses come from anomaly mediation. This givesrise to split spectra, with light gauginos (since Eq. (2.88) is loop suppressed relative to F/MPl)and heavy scalars with mf˜ ∼ F/MPl. This class of models will be the subject of Chapter 3.The above discussion of SUSY breaking scenarios is far from an exhaustive list. Each casedescribed above suffers from drawbacks: the flavour problem in gravity mediation, potentiallydangerous gravitino cosmology in gauge mediation and tachyonic slepton masses in anomaly me-diation. Thus, fully satisfactory mechanisms of SUSY breaking often require further dynamicsbeyond the basic scenarios.2.5 Status of Experimental Searches for SupersymmetryThe most important states in the MSSM spectrum for maintaining naturalness (i.e., smallfine-tuning of the Higgs mass parameter) are the stop and the gluino. The stop cancels thequadratic divergence in the Higgs self-energy associated with the top, while the gluino hasindirect effects through large radiative corrections to the squark masses [80]. As a result, thesestates are two of the primary targets for direct searches for SUSY at colliders. With no signalsreported, the most stringent constraints (as of March 2015) come from the ATLAS and CMSexperiments at the LHC based on 20 fb−1 of data.9 Typical production mechanisms of squarksand gluinos at the LHC are shown in Fig. 2.3. Note that in R-parity conserving SUSY, thesuperpartners are produced in pairs. Once produced, squarks undergo a cascade of decays thateventually terminates with the LSP, here taken to be the lightest neutralino ˜N1. The neutralinoescapes the detector, resulting in “missing” momentum transverse to the beam direction. Theprecise limits on stop masses depend on the spectrum of a given model (especially the massof the LSP); for generic values of m˜N1 , ATLAS and CMS exclude mt˜ . 700GeV (see, e.g.,Refs. [104, 105]). 10 The production cross section for gluinos and first generation squarks isgreater than for stops, resulting in bounds that reach mg˜ ∼ 1400GeV and mq˜ ∼ 1500GeV[105–107]. Dominant production of sleptons and “electroweakinos” (charginos and neutralinos)occurs through electroweak interactions and has correspondingly smaller rates, compared tothe QCD production of gluinos and squarks [80]. The resulting bounds probe LSP masses of9The LHC is the culmination of the energy frontier searches for SUSY. For a brief summary of earlier resultsfrom LEP and Tevatron see Ref. [80].10Near certain degenerate regimes, e.g., mt˜1 = mt +m ˜N1 , the bounds can be significantly weaker.45pp g˜g˜ q˜′q′q˜qpp t˜t˜ N˜1ttN˜1Figure 2.3: Typical LHC production mechanisms for gluinos and squarks (left) and stops(right).∼ 350GeV, lightest chargino masses of ∼ 700GeV and slepton masses of ∼ 300GeV [108, 109].Supersymmetric particles can also have indirect effects on measurements of processes thatare already present in the SM. For example, various Higgs production and decay rates aremodified in the MSSM. Neither ATLAS nor CMS observe statistically significant deviationsfrom SM predictions [27]. This can be interpreted as a lower limit on the mass of additionalMSSM states, such that their effects on Higgs rates are suppressed. Alternatively, this result canbe seen as an “alignment” limit of the theory, where MSSM Higgs boson interactions effectivelyreduce to those of the SM, without the need for heavy MSSM states [110]. However, even if thetree-level couplings of the lightest MSSM Higgs are the same as in the SM, the rates for loopinduced processes, namely h ↔ gg, Zγ, γγ can still be modified by O(10%) in the presenceof light stops, staus (. 350GeV) and charginos (. 250GeV) (see, e.g., Refs. [111, 112]). With3000 fb−1 of data, the LHC will be able to probe Higgs couplings to gluons, photons and Zboson with a precision of about . 5% [113]. The complementarity between direct searchesfor superpartners and precise measurements of Higgs boson properties is discussed in detail inRef. [114].Flavour physics experiments can also be extremely sensitive indirect probes of supersym-metric states [115, 116]. For example, the process Bs → µ+µ− has been recently seen at LHCb,with the observed rate in agreement with the SM prediction [117]. However, the SM rate is sosmall (due to loop, CKM and helicity suppressions – see the left graph of Fig. 2.4) that generic,TeV scale, new physics contributions would give a larger result. Agreement of the measuredrate with SM therefore places strong constraints on the scale of new physics. In particular, inthe MSSM additional contributions to this amplitude arise from the exchange of new scalarsas shown in the right graph of Fig. 2.4. However, within the MSSM with a particular flavourstructure for the soft terms, these constraints are important only in the large tanβ regime,rendering them very model dependent [116].A variety of other low energy precision probes of SUSY are available [118]. Some of the mostimportant tests are measurements of electric dipole moments (EDMs), which are sensitive toCP violation (crucial for baryogenesis) and flavour structure in the MSSM. One of the strongest46µµbstWWZ bs µµC˜t˜t˜H, AFigure 2.4: Representative contributions to Bs → µ+µ− in the SM (left) and the MSSM(right).e e e eγ γγf˜N˜ eh, H, Af˜Figure 2.5: Representative contributions to electron EDM in the MSSM at one (left) andtwo (right) loops.limits to date has been recently obtained by ACME [119] for the electron EDM. The exact limiton superpartner masses depends on which loop order the EDM contributions occur at; as shownin Fig. 2.5 there are both one and two loop diagrams contributing to fermion EDMs. In theabsence of cancellations among different phases (i.e., tuning), the ACME result translates intoa lower bound on msoft of 1 to 3 TeV [119].Finally, models with DM candidates can be probed on the cosmological frontier with directand indirect detection. These were described in Sec. 1.3.3 and will be discussed in more detailin Chapters 3 and 4. Complementarity between direct and indirect detection and LHC searchesis discussed in Ref. [120].47Part IIModuli Decays48Chapter 3Moduli Induced LightestSuperpartner Problem3.1 IntroductionString theory is the only known consistent framework that can incorporate both the StandardModel of particle physics and General Relativity (GR) in a theory that is free of ultravioletdivergences. Instead of point particles, the fundamental degrees of freedom are strings, whosefinite size regulates UV behaviour. Consistency suggests the existence of extra dimensions (10or 11 in total), while low energy observations indicate that we live in a 3 + 1 dimensionalworld. The extra dimensions must then be small and nearly unobservable. Compactificationleads to a landscape of ∼ 10500 vacua [53], each differing by the shape of the extra dimensions.Each vacuum corresponds to vastly different physics at low energy, with distinct particle andsymmetry content. The lack of a unique vacuum, or at least an easily identifiable class ofphysical vacua, has been a major criticism of string theory. At the same time, this landscapemay provide an anthropic explanation for problems such as the smallness of the cosmologicalconstant.While difficult to test at every day energies or even at the LHC, viable string vacua havegeneric features that are important at energies far below the Planck scale. First, many stringmodels enjoy spacetime supersymmetry.1 Nature at low energies is not supersymmetric. How-ever, if the scale of supersymmetry breaking is not too high, superpartners can be observeddirectly at colliders or as cosmological relics from the early Universe.The second feature is the presence of moduli – light (compared to MPl) scalar fields with onlyhigher-dimensional couplings to other light species. In string theory, each vacuum is labelledby a set of continuous parameters which determine the size and shape of the compactifieddimensions [121–124]. These parameters are expectation values of the moduli. While thecouplings of moduli to matter are model dependent, they are always non-renormalizable, i.e.,1Exceptions exist but these often have tachyons in the spectrum [53].49they are suppressed by a heavy scale Λ. For moduli originating from compactification, Λ ∼MPl = 2.435×1018 GeV. Such a coupling is too weak to be probed directly at present, but it canhave significant implications for the cosmological history of the Universe, the mass spectrum ofthe supersymmetric partners of the SM, and the density of dark matter today [123, 125, 126].In Sections 3.1.1 and 3.1.2 we justify two key moduli properties that we will repeatedlymake use of. These properties are1. Interactions of moduli with matter occur through higher-dimensional operators suppressedby a heavy scale,2. The mass of the lightest modulus is typically on the order of m3/2, the supersymmetrybreaking scale.The latter point is closely connected to supergravity and supersymmetry breaking which isdiscussed in detail in Sec. 2.4.A modulus field can alter the standard cosmology if it is significantly displaced from theminimum of its potential in the early Universe, as can occur following primordial inflation [127].The modulus will be trapped by Hubble damping until H ∼ mϕ, at which point it will beginto oscillate. The energy density of these oscillations dilutes in the same way as non-relativisticmatter, and can easily come to dominate the expansion of the Universe.2 This will continueuntil the modulus decays at time t ∼ Γ−1ϕ , transferring the remaining oscillation energy intoradiation. At this point, called reheating, the radiation temperature is approximately [133]TRH ∼ (5MeV)(MPlΛ)( mϕ100 TeV)3/2, (3.1)where Λ is the heavy mass scale characterizing the coupling of the modulus to light matter. Toavoid disrupting primordial nucleosynthesis, the reheating temperature should be greater thanabout TRH & 5MeV [134], and this places a lower bound on the modulus mass.Acceptable reheating from string moduli therefore suggests mϕ ∼ m3/2 & 100 TeV. Thishas important implications for the masses of the SM superpartner fields. Surveying the mostpopular mechanisms of supersymmetry breaking mediation described in Sec. 2.4.3, the typicalsize of the superpartner masses ismsoft ∼m3/2 gravity mediation(LMPlM)m3/2 gauge mediationLm3/2 anomaly mediation(3.2)where L ∼ g2/(16pi2) is a typical loop factor and M  MPl/L is the mass of the gaugemessengers. Of these mechanisms, only anomaly mediation (AMSB) allows for superpartnersthat are light enough to be directly observable at the LHC [102, 135]. Contributions to the2Such an early matter-dominated phase might also leave an observable signal in gravitational waves at multiplefrequencies [128] or modify cosmological observables [129–132].50soft terms of similar size can also be generated by the moduli themselves [136, 137], or othersources [138–141]. However, for these AMSB and AMSB-like contributions to be dominant, thegravity-mediated contributions must be suppressed [102], which is non-trivial for the scalar softmasses [142–146]. An interesting intermediate scenario is mini-split supersymmetry where thedominant scalar soft masses come from direct gravity mediation with msoft ∼ m3/2, while thegaugino soft masses are AMSB-like [147–152].Moduli reheating can also modify dark matter production [133, 153–156]. A standardweakly-interacting massive particle (WIMP) χ will undergo thermal freeze-out at temperaturesnear Tfo ∼ mχ/20 as shown in Sec. 1.2.2. If this is larger than the reheating temperature,the WIMP density will be strongly diluted by the entropy generated from moduli decays. Onthe other hand, DM can be created non-thermally as moduli decay products. A compellingpicture of non-thermal dark matter arises very naturally for string-like moduli and an AMSB-like superpartner mass spectrum [133]. The lightest (viable) superpartner (LSP) in this casetends to be a wino-like neutralino. These annihilate too efficiently to give the observed relicdensity through thermal freeze-out [157–159]. However, with moduli domination and reheating,the wino LSP can be created non-thermally in moduli decays, and the correct DM density isobtained for M2 ∼ 200 GeV and mϕ ∼ 3000 TeV.This scenario works precisely because the wino annihilation cross section is larger than whatis needed for thermal freeze-out. Unfortunately, such enhanced annihilation rates are stronglyconstrained by gamma-ray observations of the galactic centre by Fermi and HESS, and thenon-thermal wino is ruled out even for very conservative assumptions about the DM profile inthe inner galaxy (e.g., cored isothermal) [158, 159]. A wino-like LSP can be consistent withthese bounds if it is only a subleading component of the total DM density. Using the AMSBrelation for M2 in terms of m3/2, this forces mϕ/m3/2 & 100, significantly greater than thegeneric expectation [159]. The problem is even worse for other neutralino LSP species, sincethese annihilate less efficiently and an even larger value of mϕ  m3/2 is needed to obtainan acceptable relic density. Furthermore, mϕ > 2m3/2 also allows the modulus field to decayto pairs of gravitinos. The width for this decay is typically similar to the total width to SMsuperpartners [160–163]. For mϕ  m3/2 > 30 TeV, the gravitinos produced this way willdecay to particle-superpartner pairs before nucleosynthesis but after the modulus decays, andrecreate the same LSP density problem that forced mϕ  m3/2 in the first place.These results suggest a degree of tension between reheating by string-motivated moduli(with mϕ ∼ m3/2 and Λ ∼ MPl) and the existence of a stable TeV-scale LSP in the minimalsupersymmetric standard model (MSSM). This tension can be resolved if all relevant modulihave properties that are slightly different from the naïve expectation; for example mϕ  m3/2and BR(ϕ→ ψ3/2ψ3/2)  1 [163, 164], an enhanced modulus decay rate with mϕ ∼ m3/2 andΛ < MPl [165], or a suppressed modulus branching fraction into superpartners [166]. Given thechallenges and uncertainties associated with moduli stabilization in string theory, we focus onwhat seem to be more generic moduli and we investigate a second approach: extensions of the51MSSM that contain new LSP candidates with smaller relic densities or that are more difficultto detect than their MSSM counterparts.In this chapter we investigate extensions of the MSSM containing additional hidden gaugesectors as a way to avoid the moduli-induced LSP problem of the MSSM. Such gauge extensionsarise frequently in grand-unified theories [167] and string compactifications [168, 169]. Weassume that the dominant mediation of supersymmetry breaking to gauginos is proportionalto the corresponding gauge coupling, as in anomaly or gauge mediation, allowing the hiddensector gauginos to be lighter than those of the MSSM if the former have a smaller couplingconstant [170–172]. We also focus on the case of a single light modulus field with mϕ ∼ m3/2and Λ ∼MPl, although similar results are expected to hold for multiple moduli or for reheatingby gravitino decays.This chapter is organized as follows. In Sec. 3.1.1 we review the origin of moduli fromcompactification and highlight their properties. We discuss moduli masses in Sec. 3.1.2. InSec. 3.1.3 we review moduli cosmology and the resulting non-thermal production of LSPs. Next,in Sec. 3.2 we examine in more detail the tension between moduli reheating and a stable MSSMLSP. In the subsequent three sections we present three extensions of the MSSM containing newLSP candidates and examine their abundances and signals following moduli reheating. Thefirst extension, discussed in Sec. 3.3, comprises a minimal supersymmetric U(1)x hidden sector.We find that this setup allows for a hidden sector LSP with a relic density lower than that ofthe wino and which is small enough to evade the current bounds from indirect detection. InSec. 3.4 we extend the U(1)x hidden sector to include an asymmetric dark matter candidate andfind that it is able to saturate the entire observed DM relic density while avoiding constraintsfrom indirect detection. In Sec. 3.5 we investigate a pure non-Abelian hidden sector, andshow that the corresponding gaugino LSP can provide an acceptable relic density and avoidconstraints from indirect detection, although it is also very strongly constrained by its effect onstructure formation and the cosmic microwave background. Finally, Sec. 3.6 is reserved for ourconclusions. This chapter is based on work published in Ref. [1] in collaboration with JonathanKozaczuk, Arjun Menon and David Morrissey.3.1.1 Moduli from CompactificationAs a toy example of a modulus field associated with compactification, consider (non-supersymmetric)5D gravity - the Kaluza-Klein model, which attempted to unify GR with electromagnetism [173,174]. The Einstein-Hilbert action in 5D isS5 = −12M35∫S1×R1,3d5x√−GR5[G], (3.3)52where we parametrize the 5D metric as [51, 96]GMN =(ϕ−1/3(gµν − ϕAµAν) −ϕ2/3Aν−ϕ2/3Aµ −ϕ2/3). (3.4)Here M5 is the 5D Planck mass, G is the determinant of the metric, R5 is the 5D Ricci scalarand M,N = 1, . . . 5, and µ, ν = 1, . . . , 4. The fields in Eq. (3.4) are functions of xµ only, andare constant over the fifth dimension x5. This means we are considering only the zero modes inthe Kaluza-Klein expansion; higher modes receive masses on the order of the compactificationscale and can be integrated out of the low-energy dynamics. Using this metric parametrizationwe can rewrite S5 as an effective 4D actionS5 = −12(2piRM35 )∫d4x√−g(R4[g] +14ϕFµνFµν −16ϕ−2∂µϕ∂µϕ,)(3.5)where R is the radius of the compactified extra dimension, R4 is the 4D Ricci scalar (a functionalof gµν only), Fµν = ∂µAν − ∂νAµ. We can now identify 2piRM35 = M2Pl. Finally, using the fieldredefinitions ϕ → exp(−√6ϕ/MPl) and F → MPlF/√2, we get canonical kinetic terms for ϕand FS5 = −12M2Pl∫d4x√−gR4[g] +∫d4x√−g(−14FµνFµνe−√6ϕ/MPl + 12∂µϕ∂µϕ). (3.6)The scalar ϕ is the desired modulus. Note that its interactions with the vector field F arenon-renormalizable and are suppressed by the heavy scale MPl.The compactified action of Eq. (3.6) illustrates the second important property of moduli:there is no tree-level potential for ϕ. Without additional dynamics these fields remain massless.Because different values of the fields correspond to physically distinct vacua, in a quantumtheory one wants to stabilize moduli at particular values. Moreover, at low energies thesenew massless degrees of freedom can be inconsistent with precise tests of gravitational inter-actions (see Ref. [58] and references therein). Moduli stabilization is therefore necessary. Itis usually achieved through the inclusion of background fluxes (gauge fields) and/or gauginocondensation [175, 176]. In the next section, we argue that the lightest modulus mass is oftenon the order of supersymmetry breaking scale, independently of the stabilization mechanism.The above example is a simple illustration of a much more complicated process. In realisticstring theory compactifications, there are six (or seven) compact dimensions curled up into a(possibly) complex manifold. As a result, there can be O(100) moduli, all of which must bestabilized [175].3.1.2 Moduli MassesModuli stabilization can be divided into two categories. In supersymmetric stabilization, themodulus ϕ acquires a mass without relying on SUSY breaking, such that mϕ  m3/2 is possible.53However, once the potential is tuned to reproduce the small cosmological constant, one findsthat the mass of the lightest modulus is parametrically related to m3/2 [177], as we arguebelow. Even if this is not true and mϕ  m3/2, moduli decay to gravitinos which generallycause similar cosmological problems. This point is discussed in more detail in Sec. 3.1.6. Thesecond case corresponds to the modulus acquiring a mass mϕ ∼ m3/2 from SUSY breakingeffects.Moduli masses can be easily related to m3/2 for the simple (i.e., somewhat unrealistic) caseof the canonical Kähler potential of Eq. (2.64). Assuming the moduli superfields Φi have nosuperpotential, the SUGRA potential, Eq. (2.60), for these fields reads [154]V ⊃M2PleG|ϕi|2 = m23/2|ϕi|2, (3.7)where G the expectation value of the Kähler function defined in Eq. (2.59); we have usedEq. (2.68) in the last step to rewrite this term after SUSY breaking. This shows that the mod-ulus mass is on the order of the SUSY-breaking scale. Note that the precise relationship dependson the form of the Kähler potential and the possible existence of non-perturbative contributionsto the superpotential (see, e.g., Ref. [178] for examples). Alternative, more rigorous derivationsof the mϕ ∼ m3/2 relationship for more general models can be found in Refs. [123, 124].The above result relied on the absence of a modulus potential at the supersymmetric leveland a canonical Kähler function. Non-perturbative effects from instantons and fermion con-densation can give rise to a modulus superpotential of the form [179]W = W0 − Λ3e−bΦ/MPl , (3.8)where the parameters W0, Λ and b can be determined in string theory [177]. It is usually as-sumed that |W0|  Λ3 ∼M3Pl, while b ≤ 2pi and depends on the nature of the non-perturbativeeffects. Moreover, moduli generally have logarithmic Kähler functions likeK = −3 ln(Φ + Φ†), (3.9)with the functional form related to the volume of the compact dimensions parametrized byΦ [177]. The resulting scalar potential gives rise to a supersymmetric anti-de Sitter (AdS)vacuum with a negative cosmological constant and the modulus stabilized with an expectationvalue 〈ϕ〉 ∼ b−1MPl ln(Λ3/W0). The energy density of this vacuum is VAdS ∼ −W 20 /M2Pl,while the mass of the fluctuations is mϕ ∼ exp(G/2)〈ϕ〉. Our vacuum is observed to bede Sitter (dS), with a small positive cosmological constant, so a realistic theory must upliftthe AdS minimum to yield a positive vacuum energy density. This can be achieved throughsupersymmetry breaking, which gives a positive contribution to the energy density, as was shownexplicitly for the O’Raifeartaigh model in Sec. 2.4; the contribution of the SUSY breakingsector to the vacuum energy was found to be VOR ∼ κ4, where κ2 determines the scale of54the F -term vev. The requirement of a tiny positive cosmological constant then forces κ ∼(W0/MPl)1/2. This relates the scales of SUSY breaking and moduli stabilization, such thatthe modulus mass is now given by mϕ ∼ m3/2 ln(Λ3/W0). Note that the modulus mass isenhanced only logarithmically relative to m3/2. More complicated superpotentials can give riseto a larger hierarchy between mϕ and m3/2, but this typically requires additional tuning ofparameters [177].Once the modulus develops a non-trivial potential, it can significantly modify the cosmo-logical evolution of the Universe, as we discuss in the following sections.3.1.3 Moduli ReheatingA modulus field ϕ is very likely to develop a large initial displacement from the minimum of itspotential before or during the course of primordial inflation [127, 180]. Hubble damping willtrap the modulus until H ∼ mϕ, at which point it will start to oscillate coherently. This is shownin the left plot of Fig. 3.1, where we take an initial displacement of O(MPl) and mϕ = 100TeV.For even moderate initial displacements, the energy density in these oscillations will eventuallydominate over radiation, as shown in the right plot of Fig. 3.1. The time evolution of theaverage modulus oscillation energy density for H < mϕ is given by [40]ρ˙ϕ + 3Hρϕ + Γϕρϕ = 0, (3.10)where Γϕ is the modulus decay rate. For a modulus field with MPl-suppressed couplingsΓϕ =c4pim3ϕM2Pl, (3.11)where c is a model-dependent number with a typical range of 10−3 < c < 100 [159].3 As themodulus oscillates, it decays to radiation with the radiation density becoming dominant oncemore when H ∼ Γϕ.Thermodynamics in the early Universe was described in Sec. 1.2.2. Moduli decays injectenergy into the plasma, which can be accounted for by straightforward modifications of theequations in that section. For example, the evolution of the radiation density ρR follows fromthe First Law of thermodynamics as in Eq. (1.65):dρRdt+ 3H(ρR + pR) = Γϕρϕ, (3.12)where pR is the radiation pressure. The right hand side is the rate of energy injection intothe bath, of which moduli decays are assumed to be the dominant source. Contributions fromDM annihilation can also be included, but these do not make much difference when the DM is3 Values of c much larger than this can be interpreted as corresponding to a suppression scale Λ < MPl.55-0.4-0.200.20.40.60.810.1 1 10 100ϕ/MPlmϕt10−410−310−210−11001010.1 1 10 100ρ/(m2 ϕM2 Pl)mϕtoscillationsradiationFigure 3.1: (Left) Solution of the modulus equation of motion for an initial displacementof MPl and modulus mass mϕ = 100TeV. (Right) Evolution of the energy densityin the modulus oscillations (solid) and radiation (dashed). Because the oscillationenergy density dilutes slower with the expansion of the Universe, it will eventuallydominate the energy density in radiation.lighter than the modulus field. The radiation density is used to define the temperature throughρR =pi230g∗(T )T 4, (3.13)where g∗(T ) is the effective number of relativistic degrees of freedom [181]. Reheating is said tooccur when radiation becomes the dominant energy component of the Universe, correspondingto H(TRH) ' Γϕ. Following Ref. [133], we define the reheating temperature TRH to be:TRH =(90pi2g∗(TRH))1/4√ΓϕMPl (3.14)' (5.6 MeV) c1/2(10.75g∗)1/4( mϕ100 TeV)3/2.Here MPl ' 2.4×1018 GeV is the reduced Planck mass. The reheating temperature TRH shouldexceed 5MeV to preserve the predictions of primordial nucleosynthesis [134].4 For c = 1 thisimplies that mϕ & 100 TeV.4 We have adjusted for our slightly different definition of TRH relative to Ref. [134] in the quoted bound.563.1.4 Non-Thermal Dark MatterModuli decays can also produce stable massive particles, such as a self-conjugate dark mattercandidate χ [133]. This is described bydnχdt+ 3Hnχ =NχΓϕmϕρϕ − 〈σv〉(n2χ − n2eq), (3.15)where 〈σv〉 is the thermally averaged annihilation cross-section, neq = gTm2χK2(mχ/T )/2pi2 isthe equilibrium number density, with g being the number of internal degrees of freedom and Nχis the average number of χ particles produced per modulus decay.5 Values of Nχ ∼ 1 are usuallyexpected when χ is the LSP [178, 183]. Together, Eqs. (3.10, 3.12, 3.15) and the Friedmannequation form a closed set of equations for the system.The general solution of these equations interpolates between three distinct limits [133, 153,155]. For reheating temperatures above the thermal freeze out temperature Tfo of χ, the finalχ density approaches the thermal value. When TRH < Tfo, annihilation may or may not besignificant depending on 〈σv〉 and Nχ. Smaller values imply negligible χ annihilation afterreheating and a final relic density of about [155]Ωχh2 '34Nχ(mχmϕ)TRH(s0ρc/h2)(3.16)' (1100)Nχ( mχ100 GeV)(TRH5 MeV)(100TeVmϕ),where s0 is the entropy density today and ρc/h2 is the critical density. Larger values of Nχ or〈σv〉 lead to significant annihilation among the χ during the reheating process, giving a relicdensity of [154, 155]Ωχh2 'mχΓϕ〈σv〉sRH(s0ρc/h2)(3.17)' (0.2)(mχ/20TRH)(3× 10−26cm3/s〈σv〉)(10.75g∗)1/2' (200) c−1/2( mχ100 GeV)(3× 10−26cm3/s〈σv〉)(100 TeVmϕ)3/2(10.75g∗)1/4.We emphasize that the expressions of Eqs. (3.16,3.17) are only approximations valid to withina factor of order unity. In what follows we solve this system numerically using the methods ofRefs. [184, 185]. For TRH < Tfo and Nχ not too small, the reannihilation scenario is usually therelevant one [155].5This includes χ produced in direct decays, as well as rescattering [182] and decay cascades.573.1.5 Scaling RelationsIt is instructive to look at how the relation of Eq. (3.17) scales with the relevant couplings andmasses [170]. Motivated by the MSSM wino in anomaly mediation, we will assume that thedark matter mass scales with a coupling gχ according tomχ = rχg2χ(4pi)2m3/2, (3.18)for some parameter rχ. We will assume further that the dark matter annihilation cross sectionscales with the coupling as well,〈σv〉 = kχ4pig4χm2χ, (3.19)for some parameter kχ. For an AMSB-like wino, the r and k parameters are [133]r2 ' 1, (3.20)k2 ' 2[1− (mW /M2)2]3/2[2− (mW /M2)2]2→ 1/2, (3.21)with gχ = g2 ' 0.65, and the last expression neglects coannihilation with charginos, which canbe suppressed at low reheating temperatures [155].With these assumptions, the thermal χ abundance isΩthχ h2 ' (5.5× 10−3)r2χkχ(mχ/Tfo20)( m3/2100 TeV)2(106.75g∗)1/2(3.22)independent of the specific mass or coupling. This is no longer true of non-thermal DM producedby moduli decays, where the mass dependence is different. Rewriting Eq. (3.17) subject to theassumptions of Eqs. (3.18,3.19), we obtainΩχh2 ' 15 c−1/2(r3χ/kχr32/k2)(gχg2)2(m3/2mϕ)3( mϕ100 TeV)3/2(10.75g∗)1/4. (3.23)This result shows that reducing the coupling or the modulus mass suppresses the non-thermalrelic density. It also makes clear that mϕ > m3/2 is needed to obtain an acceptable winoabundance within the reannihilation regime.3.1.6 Gravitino Production and DecayOur previous discussion of moduli reheating did not take gravitinos into account. Modulican also decay to gravitinos if mϕ > 2m3/2, and the corresponding branching ratio BR3/2 isexpected to be on the order of unity unless some additional structure is present [160–163]. Formϕ ∼ 2m3/2, the gravitinos will decay at about the same time as the moduli and our previous58results for the moduli-only case are expected to apply here as well. On the other hand, ifmϕ  m3/2 and BR3/2 is not too small, the gravitinos produced by decaying moduli are likelyto come to dominate the energy density of the Universe before they themselves decay. Weexamine this possibility here, and show that our results for moduli decay can be applied to thisscenario as well after a simple reinterpretation of parameters.If the gravitino is not the LSP, it will decay to lighter particle-superpartner pairs withΓ3/2 =d4pim33/2M2Pl, (3.24)where d = 193/96 if all MSSM final states are open and d = (1+3+8)/8 = 3/2 if only gauginomodes are available [186]. These decays will not appreciably disrupt BBN for m3/2 & 30 TeV,but they can produce a significant amount of LSP dark matter.For mϕ  m3/2, the modulus will decay much earlier than the gravitino (unless c  d).The gravitinos produced by moduli decays at time ti ' Γ−1ϕ will be initially relativistic withp/m3/2 = mϕ/2m3/2. Their momentum will redshift with the expansion of the Universe, andthey will become non-relativistic at timetnr 'd4c(m3/2mϕ)Γ−13/2, (3.25)where we have assumed that the Universe is radiation-dominated after moduli reheating. Thus,the gravitinos produced in moduli decays become non-relativistic long before they decay form3/2/mϕ  1 (and c ∼ d). While tnr < t < Γ−13/2, the gravitinos will behave like matter. Thequantity m3/2n3/2 begins to exceed the (non-gravitino) radiation density at timet ' dc(1− BR3/2BR3/2)2(m3/2mϕ)3Γ−13/2 (3.26)Again, this is much earlier than the gravitino decay time for m3/2/mϕ  1 unless BR3/2 or c/dis suppressed.6The scenario that emerges for m3/2  mϕ, c ∼ d, and BR3/2 ∼ 1 is very similar to a secondstage of moduli reheating: the gravitinos produced in moduli decays become non-relativisticand come to dominate the energy density of the Universe until they decay at time t ' Γ−13/2, atwhich point they reheat the Universe again. Dark matter will also be created by the gravitinodecays, with at least one LSP produced per decay (assuming R-parity conservation). The largegravitino density from moduli decays can interfere with nucleosynthesis or produce too muchdark matter, and is sometimes called the moduli-induced gravitino problem [160–163].We will not discuss gravitinos much for the remainder of this chapter. Instead, we will focusmainly on the case of mϕ ∼ m3/2, where the presence of gravitinos does not appreciably changeour results [155]. However, our findings can also be applied to scenarios with mϕ  m3/2, c ∼ d,6We have assumed radiation domination here, but a similar result holds for matter domination.59and BR3/2 ∼ 1 with the moduli decays reinterpreted as gravitino decays (i.e., mϕ → m3/2,c→ d, Nχ → 1).3.2 Moduli Reheating and the MSSMThe discussion of Sec. 3.1.3 shows that the LSP relic density is enhanced in the moduli-decayscenario relative to thermal freeze out unless the fraction of decays producing LSPs Nχ isvery small. In Ref. [159], this observation was used to put a very strong constraint on wino-like LSPs produced by moduli decays. In this section we apply these results to more generalMSSM neutralino LSPs, and we argue that the MSSM has a moduli-induced LSP problem formϕ ∼ m3/2, c ∼ 1, and Nχ not too small. See also Refs. [187, 188] for related analyses.Consider first a wino-like LSP with an AMSB-like mass. Direct searches at the LHC implythat the mass must lie above mχ01 & 270 GeV if it is nearly pure wino [189], although smallermasses down to the LEP limit mχ±1 & 104 GeV are possible if it has moderate mixing with aHiggsino [190]. Examining Eq. (3.23), the moduli-induced wino relic density (in the reannihila-tion regime) tends to be larger than the observed DM density, and indirect detection places aneven stronger bound of Ωχh2 . 0.05 [159]. Fixing mχ = 270 GeV, a relic density of this size canbe obtained with the very optimistic combination of parameter values c = 100, mϕ = 2m3/2,and rχ/r2 . 0.3. Such a reduction in rχ/r2 can arise from supersymmetry-breaking thresholdcorrections [191, 192] or moduli-induced effects [136, 137], but requires a significant accidentalcancellation relative to the already-small AMSB value of r2 [159].A small effective value of rχ < r2 could also arise from |µ|  |M2| and a correspondingHiggsino-like LSP. The reduction in the relic density in this case is countered by a smallerannihilation cross section: for µ  mW , heavy scalars, and neglecting coannihilation we havegχ ' g2, rχ = (µ/M2), and kχ ' (3+2t2W + t4W )/128 ' 0.03 [157] (where tW ≡ tan θW , with θWthe Weinberg angle). To investigate this possibility in more detail, we set mϕ/m3/2 = 1, 10, 100and c = 1, and compute the moduli-induced LSP relic density for various values of µ/M2 andm3/2. In doing so, we fix M2 to its AMSB value with c = 1 and mϕ = m3/2, and we computethe annihilation cross section in DarkSUSY [193, 194]. For the other MSSM parameters, weset tanβ = 10, mA = 1000 GeV, m˜ = 2000 GeV for all scalars, and we fix At such thatmh = 126± 1 GeV.Fixing mϕ/m3/2 = 1, 10 we find no Higgsino-like points with Ωχh2 ≤ 0.12, i.e., for valueswhich would appear to be generically expected from string theory.7 Smaller relic densities arefound for mϕ/m3/2 = 100, and the results of our scan for this ratio are shown in Fig. 3.2. TheLSP relic density is smaller than the total DM density to the left and below the solid black line,while the grey dashed contours show the LSP mass. To the right of the red line, the reheatingtemperature lies above the freeze-out temperature and the resulting density is thermal. Thecoloured dashed contours correspond to bounds from indirect detection for different DM density7 We also fail to find any such points for c = 100 and mϕ = 2m3/2.60m3/2[100TeV]µ/M2mϕ/m3/2 = 100NFWEinastoCored RC = 0.1 kpcContr. γ = 1.2123456789100.1 1-1-0.500.511.52log 10ΩN˜1/Ωcdm1000700400100LEP Excl.TRH < Tf.o.Figure 3.2: Relic density and constraints from indirect detection (ID) for a mixedHiggsino-wino LSP produced by moduli reheating as a function of µ/M2 and m3/2.The modulus parameters are taken to be mϕ/m3/2 = 100, c = 1, and Nχ = 1.Contours of the LSP mass in GeV are given by the dashed grey lines. The solidblack contours show where Ωχ01 = Ωcdm. The solid red line shows where TRH = Tfo:to the left of it we have TRH > Tfo; to the right TRH < Tfo and the productionis non-thermal. The remaining lines correspond to bounds from ID for differentgalactic DM distributions, and the area below and to the right of these lines isexcluded.profiles, excluding everything below and to the right of them.8 This figure also shows a funnelregion with very low relic density along the mχ01 = 500GeV contour corresponding to an s-channel A0 pseudoscalar resonance.In general, for mϕ ∼ m3/2, we find that a Higgsino-like LSP also tends to produce too muchdark matter when it is created in moduli reheating. As for the wino, this can be avoided forlarger values of mϕ/m3/2, as demonstrated by Fig. 3.2, although one must still ensure that thevery heavy modulus does not decay significantly to gravitinos.These results can be extended to an arbitrary MSSM neutralino LSP. In general, mixingwith a bino will further suppress the annihilation cross section, leading to an overproduction ofdark matter for mϕ ∼ m3/2. The only loophole we can see is a very strong enhancement of theannihilation from a resonance or coannihilation [43]. This requires a very close mass degeneracyeither between 2mχ and the mass of the resonant state, or between mχ and the coannihilatingstate, with coannihilation further suppressed at low reheating temperatures. The only otherviable LSP candidates in the MSSM are the sneutrinos. These annihilate about as efficiently asa Higgsino-like LSP [195], and therefore also tend to be overproduced. The MSSM sneutrinosalso have a very large scattering cross section with nuclei, and bounds from direct detection8The details of our indirect detection analysis will be presented in the next section.61permit them to be only a small fraction of the total DM density [196].Having expanded slightly on the findings of Ref. [159], we conclude that a neutral MSSM LSPis typically overproduced in moduli reheating unless mϕ  m3/2 (with tiny BR3/2), Nχ  1,or the decay coefficient c  100 is very large. None of these features appears to be generic instring compactifications. We call this the moduli-induced MSSM LSP problem. For this reason,we turn next to extensions of the MSSM with more general LSP candidates that can potentiallyavoid this problem.3.3 Variation #1: Hidden U(1)The first extension of the MSSM that we consider consists of a hidden U(1)x vector multipletX and a pair of hidden chiral multiplets H and H ′ with charges xH,H′ = ±1. Motivated bythe scaling relation of Eq. (3.23), we take the characteristic gauge coupling and mass scale ofthe hidden sector to be significantly less than electroweak, along the lines of Refs. [197–199].The LSP of the extended theory will therefore be the lightest hidden neutralino. We alsoassume that the only low-energy interaction between the hidden and visible sectors is gaugekinetic mixing. Among other things, this allows the lightest MSSM superpartner to decay tothe hidden sector. In this section we the investigate the contribution of the hidden LSP tothe dark matter density following moduli reheating as well as the corresponding bounds fromindirect and direct detection.3.3.1 Setup and SpectrumThe hidden superpotential is taken to beWHS = −µ′HH ′, (3.27)and the soft supersymmetry breaking terms are−Lsoft ⊃ m2H |H|2 +m2H′ |H ′|2 +(−b′HH ′ + 12Mx ˜X ˜X + h.c.). (3.28)The only interaction with the MSSM comes from supersymmetric gauge kinetic mixing in theformL ⊃∫d2θ 2XαBα, (3.29)where X and B are the U(1)x and U(1)Y field strength superfields, respectively.We assume that the gaugino mass is given by its AMSB value [198]Mx = bxg2x(4pi)2m3/2, (3.30)62where bx = 2 and gx is the U(1)x gauge coupling. Since pure AMSB does not provide a viablescalar spectrum in the MSSM, we do not impose AMSB values on the scalar soft terms in thehidden sector. However, we do assume that they (and µ′) are of similar magnitude to theirAMSB values, on the order of (g2x/16pi2)m3/2. This could arise if the dynamics that leads to aviable MSSM spectrum also operates in the hidden sector and that its effects are proportionalto the corresponding gauge coupling.For a range of values of µ′ and the soft terms, the scalar components of H and H ′ willdevelop vacuum expectation values,〈H〉 = η sin ζ, 〈H ′〉 = η cos ζ. (3.31)Correspondingly, the hidden vector boson Xµ receives a massmx =√2gxη. (3.32)The scalar mass eigenstates after U(1)x breaking consist of two CP-even states hx1,2 (with hx1the lighter of the two) and the CP-odd state Ax. The fermionic mass eigenstates are mixturesof the hidden Higgsinos and the U(1)x gaugino, and we label them in order of increasing massas χx1,2,3. Full mass matrices for all these states can be found in Refs. [200, 201].3.3.2 Decays to and from the Hidden SectorKinetic mixing allows the lightest MSSM neutralino to decay to the hidden sector. It can alsoinduce some of the hidden states to decay back to the SM. We discuss the relevant decay modeshere.The MSSM neutralinos connect to the hidden sector through the bino. For AMSB gauginomasses, the bino soft mass is significantly heavier than that of the wino, and the lightestneutralino χ01 tends to be nearly pure wino. Even so, it will have a small bino admixture givenby the mass mixing matrix element N11. In the wino limit, it can be approximated by [157]|N11| =cW sWm2Z(M2 + sin 2βµ)(M1 −M2)(µ2 −M22 ). (3.33)With this mixing, the lightest MSSM neutralino will decay to the hidden sector through thechannels χ01 → χxk + Sx, where χxk are the hidden neutralinos and Sx = hx1,2, Ax, Xµ are thehidden bosons, with total width [200]Γχ01 =2g2x|N11|24pimχ01 (3.34)= (1.3× 10−16 sec)−1|N11|2( 10−4)2 ( gx0.1)2( mχ01100 GeV).The corresponding χ01 lifetime should be less than about τ . 0.1 s to avoid disrupting nucle-63osynthesis. This occurs readily for MSSM gaugino masses below the TeV scale and  not toosmall.In the hidden sector, the χx1 neutralino will be stable while the other states will ultimatelydecay to it or to the SM. To ensure that χx1 is able to annihilate efficiently, it should also beheavier than the vector Xµ. This implies that the hidden vector will decay to the SM throughkinetic mixing, or via X → hx1Ax. For mx > 2me, the vector decay width to the SM isΓ(X → SM + SM) = R′α2mx3, (3.35)where R′ is a constant on the order of unity that depends on the number of available finalstates. This decay is much faster than τ = 0.1 s for  & 4× 10−10 and mx & 2mµ.Of the remaining hidden states, the longest-lived is typically the lightest CP-even scalar hx1 .The structure of the hidden sector mirrors that of the MSSM, and this scalar is always lighterthan the vector at tree level. Loop corrections are not expected to change this at weak coupling.As a result, the hx1 decays exclusively to the SM through mixing with the MSSM Higgs scalars(via a Higgs portal coupling induced by gauge kinetic mixing) or through a vector loop [200].This decay is typically faster than τ = 0.1 s for  & 2× 10−4 and mhx1 & 2mµ [201].Light hidden sectors of this variety are strongly constrained by fixed-target and precisionexperiments [202, 203]. For dominant vector decays to the SM, the strongest limits for mx >2mµ come from the recent BaBar dark photon search [204], and limit  . 5 × 10−4. As thevector mass approaches mx = 20 MeV, fixed-target searches become relevant and constrainthe mixing  to extremely small values [202, 203]. In this analysis, we will typically choosemx > 20 MeV and  ∼ 10−4 so that the hidden sector is consistent with existing searches.3.3.3 Hidden Dark Matter from ModuliModuli decays are expected to produce both visible and hidden particles and reheat bothsectors. The superpartners created by moduli decays will all eventually cascade down to thehidden neutralino LSP. Kinetic mixing can allow the hidden LSP to thermalize by scatteringelastically with the SM background through the exchange of X vector bosons. The rate ofkinetic equilibration depends on the typical energy at which the LSP is created, the reheatingtemperature, and the mass and couplings in the hidden sector [155]. For optimistic parametervalues we find that it is faster than the Hubble rate for TRH & 5 MeV, and we will assume herethat such thermalization occurs.If the net rate of superpartner production in moduli decays is unsuppressed and the χx1annihilation cross section is moderate, the χx1 LSPs will undergo additional annihilation toproduce a final relic density as described in Eq. (3.17). The relevant annihilation modes of theLSP are χx1χx1 → hx1hx1 , Xhx1 , XX. Computing the corresponding annihilation rates using themethod of Ref. [42] near T ∼ TRH, we find that the XX final state typically dominates providedit is open, as we will assume here. Using these rates, we compute the relic abundance of χx1 by640.511.522.533.50.1 1m3/2[100TeV]µ′/MxΩχh2 and mχx1 for gx = 0.130105110310.30.120.06Figure 3.3: Contours of the hidden neutralino χx1 mass in GeV (dashed grey) and moduli-generated relic abundance Ωχh2 (solid red) as a function of of µ′/Mx and m3/2.The moduli parameters are taken to be mϕ = m3/2, c = 1, and Nχ = 1, with thehidden-sector parameters as described in the text.numerically solving the system of equations presented in Sec. 3.1.3. In doing so, the decays ofthe MSSM LSP and all hidden states are treated as being prompt.Before presenting our numerical results, it is instructive to examine the parametric depen-dence of the approximate solution of Eq. (3.17). Writingµ′ = ξ Mx, (3.36)and focusing on a hidden Higgsino-like LSP with ξ ≤ 1, we obtain gχ = gx and rχ = 2ξin Eq. (3.23). Thus, smaller values of ξ and gx are expected to produce decreased χx1 relicabundances.The results of a full numerical analysis are illustrated in Fig. 3.3, where we show the contoursof the final χx1 abundance (solid red) and DM mass (dashed grey) in the ξ−m3/2 plane for mϕ =m3/2, c = 1, and Nχ = 1. The range of m3/2 considered corresponds to M2 ∈ [100, 1000] GeV,and the hidden sector parameters are taken to be gx = 0.1, tan ζ = 10, and mx = 0.2 GeV,mAx = 10 GeV. The shape of the abundance contours in Fig. 3.3 is in agreement with thescaling predicted by Eq. (3.23). We also see that ξ = µ′/Mx < 1 is typically needed to avoidcreating too much dark matter, and this implies some degree of fine tuning for hidden-sectorsymmetry breaking. Larger values of ξ are allowed when the moduli decay parameter c is greaterthan unity, since this leads to a higher reheating temperature and more efficient reannihilation.653.3.4 Constraints from Indirect DetectionWhile this extension of the MSSM can yield an acceptable hidden neutralino relic densityfrom moduli reheating, it is also constrained by indirect detection (ID) searches for DM.9 Thepair annihilation of hidden neutralinos can produce continuum photons at tree level from cas-cades induced by χx1χx1 → XX with X → ff¯ , as well as photon lines at loop level throughkinetic mixing with the photon and the Z. These signals have been searched for by a num-ber of gamma-ray telescopes, and limits have been placed on the corresponding gamma-rayfluxes. We examine here the constraints on the χx1 state from observations of the galactic cen-tre (GC) gamma ray continuum by the Fermi Large Area Telescope (Fermi-LAT) [205], as wellas from observations of the diffuse photon flux by the INTEGRAL [206], COMPTEL [207],EGRET [208, 209], and Fermi [210] experiments. For the GeV-scale dark matter masses weare considering, these observations are expected to give the strongest constraints [211, 212].10We also study bounds from the effects of DM annihilation during recombination on the cosmicmicrowave background (CMB) [213, 214].The continuum photon flux from χx1 pair annihilation into hidden vectors is given bydΦγdEγ=〈σv〉χχ→XX8pim2χdN totγdEγ×∫dl ρ2(l), (3.37)where 〈σv〉χχ→XX is the thermally averaged annihilation rate at present, ρ(l) is the dark mat-ter density along the line of sight l, and dN totγ /dEγ is the total differential photon yield perannihilation, defined asdN totγdEγ≡∑fBRfdNfγdEγ(3.38)where BRf is the branching fraction of the XX state into the final state f .In our calculations, we use the results of Refs. [215, 216] to estimate the partial yieldsdNfγ /dEγ by interpolating between the results for the values of mχ and mχ/mx listed in thesestudies. For the dark matter density profile, we consider four distributions that span therange of reasonable possibilities: Navarro-Frenk-White (NFW) [217], Einasto [218, 219], con-9 Constraints from direct detection are not relevant; the χx1 LSP is a Majorana fermion, and scatters off nucleimainly through a suppressed Higgs mixing coupling [201].10We have also examined constraints from monochromatic photon line searches and found the continuumconstraints significantly more stringent for the small values of  allowed by fixed target experiments.66tracted [211], and cored NFW [211]. These take the formsρ(r) ∝[rRs(1 + rRs)2]−1(NFW)e−2/α[(rRs)α−1](Einasto)[(rRs)γ (1 + rRs)3−γ]−1(contracted)[rc+(r−rc) Θ(r−rc)Rs(1 + rc+(r−rc) Θ(r−rc)Rs)2]−1(cored). (3.39)Here, r is the radial distance from the GC and Θ is a step function. Following Refs. [159, 211], wefix the scale radius to be Rs = 20 kpc and the Einasto parameter α = 0.17. For the contractedprofile we set γ = 1.4 and for the cored profile we set the core radius to be rc = 1 kpc, asin Ref. [211]. In all four cases, we fix the overall normalization such that ρ(r = 8.5 kpc) =0.3 GeV/cm3.Using these halo profiles, we are able to compute the gamma-ray fluxes from hidden darkmatter created in moduli decays and compare them to limits derived from observations ofthe GC and the diffuse gamma-ray background. For the GC signal, we use the limits on〈σv〉/m2χ∫ EmaxEmin dEγ dNtotγ /dEγ computed in Ref. [211] in several energy bins [Emin,i, Emax,i] andeach of the four DM profiles described above. For the diffuse gamma ray background, we usethe flux limits compiled and computed in Ref. [212].In addition to measurements of cosmic gamma rays, observations of the CMB also provide asignificant limit on DM annihilation [213, 214]. The energy released by dark matter annihilationaround the time of recombination will distort the last scattering surface, and hence affect theCMB anisotropies. The limit derived from this effect is [220–222]fΩ2χΩ2cdm〈σv〉CMB ≤ (2.42× 10−27 cm3/s)( mχGeV), (3.40)where 〈σv〉CMB is the thermally averaged cross section during recombination and f is a constantefficiency factor parametrizing the fraction of energy transferred to the photon-baryon fluid,which can typically range from f ≈ 0.2−1.0 [222]. We will vary f across this range to illustrateits effect on the resulting constraint.These observations put very strong constraints on hidden dark matter when it is producedin moduli decays. The corresponding ID and CMB bounds are shown in Fig. 3.4 in the m3/2−gxplane. We fix the moduli parameters to mϕ = m3/2 and c = 1 in the left panel and mϕ = 2m3/2and c = 10 in the right. The relevant hidden-sector parameters are taken to be ξ = 0.1 andmx = mχ/2. The solid red line shows where Ωχ = Ωcdm, with the region above and to the rightof the line producing too much dark matter. The green shaded regions show the exclusion fromFermi observations of the GC assuming the Einasto DM profile of Eq. (3.39) rescaled by theexpected dark matter fraction (Ωχ/Ωcdm)2, while the blue shaded regions show the exclusion67g xm3/2 [100 TeV]ID and CMB Constraints, mϕ = m3/2, c = 1, mχ = 2mx0.010.111 2 3 4 5 6 7 8 9 10Excluded by Fermi GCExcluded by COMPTELCMBmx < 20 MeV0.1110100Ωχ>ΩCDMg xm3/2 [100 TeV]ID and CMB Constraints, mϕ = 2m3/2, c = 10, mχ = 2mx0.010.111 2 3 4 5 6 7 8 9 10Excluded by Fermi GCExcluded by COMPTELCMBmx < 20 MeV0.1110100Ωχ > ΩCDMAllowedFigure 3.4: Constraints from indirect detection on hidden U(1)x neutralino DM producedby moduli decays for mx = mχ/2, ξ = 0.1, as well as (c=1, mϕ=m3/2) (left), and(c = 10, mϕ = 2m3/2) (right). The green shaded region is excluded by Fermi GCobservations and the blue shaded region is excluded by COMPTEL. Both exclusionsassume an Einasto galactic DM profile. The thick solid and thin dotted contourscorrespond to the exclusions assuming the NFW and cored profiles, respectively.The green and blue dashed lines show the boundaries of the stronger exclusionobtained assuming a contracted profile with γ = 1.4. Above and to the right of thesolid red line, the hidden LSP density is larger than the observed DM density. Thesolid and dash-dotted orange lines shows the exclusion from deviations in the CMBfor f = 0.2 and f = 1, respectively, with the excluded areas above and to the rightof the lines. Note that the entire c = 1 parameter space is excluded by the CMBconstraint for f = 1. The gray shaded region at the bottom has a hidden vectormass mx < 20 MeV that is excluded by fixed-target experiments.from COMPTEL under the same conditions. Exclusions for other profiles are also shown by theparallel contours.11 We have also considered the corresponding constraints from INTEGRAL,EGRET, and Fermi diffuse gamma ray observations, but these do not exclude any additionalparameter space and so are not included in Fig. 3.4 for clarity. Limits from CMB distortionsare shown by the solid and dash-dotted orange lines, for f = 0.2 and 1 respectively, with theexcluded region above and to the right of the contours. The dashed black lines are contoursof the hidden LSP χ mass in GeV, with the region where mx = mχ/2 < 20 MeV excluded byfixed target experiments [202].For generic moduli parameters, c = 1 and mϕ = m3/2, we find that constraints from indirectdetection and CMB observations nearly completely rule out this scenario even with optimisticchoices for the DM halo properties and CMB energy injection efficiency. However, for c = 1011The thick green and blue dashed lines show the boundaries of the regions excluded for a more aggressivecontracted profile with γ = 1.4. For clarity, we do not shade the interior of these. The thick solid and thin dottedcontours correspond to the NFW and cored profiles, respectively.68and mϕ = 2m3/2, the hidden neutralino relic density can become sufficiently small to evade thestrong limits from ID and the CMB, despite the relatively large χx1 annihilation cross section.In this case, a second more abundant contribution to the total dark matter abundance would beneeded. Note as well that the remaining allowed region corresponds to sub-GeV hidden sectormasses that could potentially be probed in current and planned precision searches [203].3.3.5 SummaryWith optimistic but reasonable choices for the moduli parameters, a light hidden sector neu-tralino LSP produced in moduli reheating can be consistent with current DM searches. Evenso, the scenario is tightly constrained by indirect detection and CMB measurements. The chal-lenge here is precisely the same as in the MSSM: to avoid overproducing the neutralino LSPduring moduli reheating, the annihilation rate must be large relative to the standard thermalvalue 〈σv〉 ∼ 3 × 10−26 cm3/s, and such an enhanced rate is strongly constrained by indirectDM searches. To avoid these bounds while not creating too much dark matter, the annihilationrate must be large enough that the LSP relic abundance is only a small fraction of the totalDM density.The U(1)x hidden sector does slightly better than the MSSM in this regard for two reasons.First, the hidden gauge coupling can be taken small (as can ξ = µ′/Mx), which helps to reducethe LSP relic abundance as suggested by Eq. (3.23). And second, the hidden LSP can bemuch lighter than an MSSM wino or Higgsino, leading to smaller photon yields below theprimary sensitivity of Fermi-LAT. The strongest constraints for such light masses come fromCOMPTEL, which are less stringent than those from Fermi. Since the large late-time hiddenneutralino annihilation rate is the primary hindrance to realizing this set-up, one might consideranalogous scenarios in which the CMB and indirect detection signatures are suppressed; weaddress this possibility in the following section.Before moving on, let us also comment on the spectrum in the hidden sector. To avoida large fine tuning, the hidden scalar soft terms must be relatively small, on the same orderor less than the hidden gaugino mass. Given the large values of m3/2 considered, the scalarsoft masses must be sequestered from supersymmetry breaking. They must also receive newcontributions beyond minimal AMSB, and the b′ bilinear soft term must not be too muchlarger than (µ′)2. All three features require non-trivial additional structure in the underlyingmechanisms of supersymmetry breaking or mediation [223, 224].3.4 Variation #2: Asymmetric Hidden U(1)As a second extension of the MSSM, we investigate a theory of hidden asymmetric dark matter(ADM) [225–228]. In the ADM framework, the DM particle has a distinct antiparticle, and itsabundance is set mainly by a particle-antiparticle asymmetry in analogy to baryons, and thistends to suppress indirect detection signals from late-time annihilation if very little anti-DMis present [5, 229–231]. The ADM theory we consider is nearly identical to the hidden U(1)x69theory studied in Sec. 3.3, but with an additional pair of vector-like hidden chiral superfields Yand Y c with U(1)x charges xY = ±1. We assume that a small asymmetry in the Y density isgenerated during moduli reheating, in addition to the much larger symmetric density, and wecompute the resulting relic densities and experimental signals.3.4.1 Mass Spectrum and DecaysThe superpotential in the hidden sector is the same as that considered in Sec. 3.3 up to a newmass term for the Y and Y c multiplets,W ⊃ −µY Y Y c. (3.41)We also include the new soft supersymmetry breaking terms−Lsoft ⊃ m2˜Y |˜Y |2 +m2˜Y c |˜Y c|2 − (bY ˜Y ˜Y c + h.c.). (3.42)As in Sec. 3.3, we fix the hidden gaugino mass to its AMSB value with bx = 2(1+1), accountingfor the new superfields. We also do not impose minimal AMSB values for the scalar softterms, but take them (as well as µ′ and µY ) to be of similar size to the gaugino soft mass.Finally, we arrange parameters so that the hidden Higgs scalars develop expectation values andspontaneously break the U(1)x.The mass spectrum of the hidden sector follows the minimal model considered in Sec. 3.3,but now a new Dirac fermion Ψ of mass mΨ = µY and two complex scalars Φ1,2. The scalarmass matrix in the (˜Y , ˜Y c∗) basis isM2˜Y =(|µY |2 +m2˜Y− δ˜D b∗YbY |µY |2 +m2˜Y c+ δ˜D,), (3.43)where δ˜D = g2xη2 cos 2ζ + xY gxg′v2 cos 2β/2. Taking m2˜Y= m2˜Y cfor convenience, the masseigenvalues arem21,2 = |µY |2 +m2˜Y ∓√δ˜2D + |bY |2. (3.44)In what follows we will refer to the lighter scalar Φ1 as Φ.This theory preserves both the usual R-parity as well as a non-anomalous global U(1) flavoursymmetry among the Y and Y c multiplets, and can support multiple stable states. The numberof stable particles depends on the mass spectrum. To allow for dominantly asymmetric darkmatter, we will focus on spectra with mχx1 > mΨ +mΦ such that the decay χx1 → Ψ + Φ∗ ispossible, and the only stable hidden states are Ψ and Φ. If this channel is not kinematicallyallowed, the χx1 neutralino will also be stable and can induce overly large gamma ray signalsas in the previous section. We also choose soft masses such that mx < mΦ, mΨ to allow bothstates to annihilate efficiently into hidden vectors. With this mass ordering, the lightest hidden70states will be the vector Xµ and the hidden Higgs hx1 . Both will decay to the SM in the sameway as in the minimal model of Sec. 3.3. The lightest MSSM neutralino will also continue todecay to the hidden sector through gauge kinetic mixing, now with additional decay modesχ01 → ΨΦ1,2. As before, the net χ01 lifetime is expected to be short relative to the cosmologicaltimescales of interest.3.4.2 Moduli Reheating and Asymmetric Dark MatterThe Ψ and Φ states will both act as ADM if they are created in the moduli reheating pro-cess slightly more often than their antiparticles. The production of the asymmetry can beaccommodated within a set of Boltzmann equations similar to Eq. (3.15) as follows:dnΨdt+ 3HnΨ = (1 + κ/2)NΨΓϕmϕρϕ − 〈σv〉Ψ(nΨnΨ − (neqΨ )2) (3.45)− 〈σv〉trans(n2Ψ − ν2n2Φ)dnΦdt+ 3HnΦ = (1 + κ/2)NΦΓϕmϕρϕ − 〈σv〉Φ(nΦnΦ∗ − (neqΦ )2) (3.46)− 〈σv〉trans(ν2n2Φ − n2Ψ),with a similar set of equations for the anti-DM Ψ and Φ∗, but with κ → −κ. Here, NΨ andNΦ are the mean number of Ψ and Φ produced per modulus decay. This includes particlescreated directly in moduli decays, rescattering, and from the cascade decays of other states.The thermally-averaged cross sections 〈σv〉Ψ,Φ describe the ΨΨ and ΦΦ∗ annihilation, while〈σv〉trans in each equation corresponds to the transfer reaction ΨΨ ↔ ΦΦ mediated by U(1)xgaugino exchange with ν = 2 (mΨ/mΦ)2K2(mΨ/T )/K2(mΦ/T ).Asymmetry generation in this scenario is parametrized by the constant κ. It could arisedirectly from moduli decays or from the interactions of intermediate moduli decay productsalong the lines of one of the mechanisms of Refs. [232–238]. Indeed, this theory can be viewedas a simplified realization of the supersymmetric hylogenesis model studied in Ref. [2]. Relativeto that work, we undertake a more detailed investigation of the relic density resulting fromdifferent choices for the moduli parameters, and we do not attempt to link the DM asymmetryto the baryon asymmetry.The annihilation cross section 〈σv〉Ψ is dominated by the ΨΨ → XX channel to hiddenvector bosons and is given by〈σv〉Ψ =116pig4xm2Ψ(1− m2xm2Ψ)3/2(1− m2x2m2Ψ)−2(3.47)' (1.5× 10−24 cm2/s)( gx0.05)4(1GeVmΨ)2. (3.48)71m3/2[100TeV]mΨ[GeV]κΩadm/Ωcdm for gx = 0.1 and  = 10−4CMBExcl.Ωadm/ΩcdmDM Frac.1234567891010−5 10−4 10−3 10−2 10−1123451010.10.950.110−5 10−10DDExcl.Figure 3.5: Abundance of Ψ and Φ in the κ−m3/2 plane. The right y axis shows the Ψmass mΨ = µY . Solid red contours show the fraction of the measured abundancemade up by Ψ and Φ and their anti-particles. The dashed grey lines show thefractional asymmetry between DM and anti-DM. The blue region is excluded bythe CMB bound and the green by direct detection.The scalar annihilation rate is similar. For the transfer reaction, we have〈σv〉trans ≈g4x8pi√1−m2Φm2Ψ∣∣∣∣∣3∑k=1(A∗2k −B2k)mχxkm2χxk +m2Ψ −m2Φ∣∣∣∣∣2, (3.49)where Ak = Z∗11Pk3 and Bk = Z12Pk3 with Pk3 the HS gaugino content of χxk and Zij is aunitary matrix that diagonalizes the scalar mass matrix of Eq. (3.43). Note that the transferreaction can be suppressed relative to annihilation for mχx1 > mΨ +mΦ.3.4.3 Relic Densities and ConstraintsTo investigate the relic densities of Ψ and Φ in this theory following moduli reheating and thecorresponding constraints upon them, we set all the dimensionful hidden parameters to be fixedratios of the U(1)x gaugino soft mass Mx = 4g2xm3/2/(4pi)2:mAx = 10µ′ = 50µY = 100mx = 250b1/2Y = 250m˜Y = Mx. (3.50)With these choices, the mass spectrum for gx = 0.1 and m3/2 = 200 TeV ismΨ = 1GeV, mΦ = 0.97GeV, mχx1 = 5.1GeV, mx = 0.51GeV, mhx1 = 0.5GeV.This mass ordering coincides with the spectrum described in in Sec. 3.4.1.72m3/2[100TeV]gxΩadm/Ωcdm for κ = 5× 10−3 and  = 10−4Ωadm/ΩcdmmΨ123456789100.05 0.1 0.15 0.2 0.25 0.31010.11051DD ExcludedFigure 3.6: Abundance of Ψ and Φ in the gx −m3/2 plane. Solid red contours show thefraction of the measured abundance made up by Ψ and Φ and their anti-particles.The dashed grey lines show the Ψ mass in GeV. The green region is excluded bydirect detection.In Fig. 3.5 we show the dark matter abundance Ωadm = ρadm/ρc of Ψ and Φ (and theirantiparticles) relative to the observed abundance Ωcdm in the κ−m3/2 plane for gx = 0.1, = 10−4, mϕ = m3/2, and c = 1. Contours of Ωadm/Ωcdm = 0.1, 1, 10 are given by solid redlines. The grey dashed lines in this figure correspond to the net residual anti-DM abundanceRΨ + RΦ, where RΨ = ΩΨ/ΩΨ and similarly for Φ. Not surprisingly, larger values of theproduction asymmetry parameter κ lead to smaller residual anti-DM abundances. In thisfigure we also show in blue the region of parameters that is excluded by CMB observations, aswell as the region excluded by direct detection in green. These constraints will be discussed inmore detail below.The ADM abundance in the gx−m3/2 plane is shown in Fig. 3.6 for κ = 5× 10−3,  = 10−4,mϕ = m3/2, and c = 1. Again, contours of Ωadm/Ωcdm = 0.1, 1, 10 are given by solid red lines.We also plot contours of the Ψ mass with dashed grey lines. As before, the shaded green regionis excluded by direct detection searches.The region excluded by CMB observations in Fig. 3.5 (shaded blue) coincides with largervalues of the residual anti-DM abundances RΨ + RΦ. These residual abundances provide anannihilation mode that injects energy into the cosmological plasma during the CMB era [5], asdiscussed in Sec. 3.3.4. Accounting for exclusively asymmetric annihilation and the multipleDM species, the result of Eq. (3.40) translates into2f∑i=Ψ,Φ(Ωi + Ωi¯Ωcdm)2 Ri(1 +Ri)2〈σv〉imi< 2.42× 10−27 cm3/sGeV . (3.51)73The CMB exclusion shown in Fig. 3.5 uses f = 1, but other values in the range f = 0.2−1.0 yieldsimilar results. The boundary of the excluded region is also nearly vertical and independent ofm3/2. This can be understood in terms of an approximate cancellation of factors of m3/2 = mϕin the combination Ω2adm〈σv〉/m, while Ri is determined primarily by κ. In addition to thelimits from the CMB, we have also computed the bounds from indirect detection as describedin Sec. 3.3. These searches yield exclusions very similar to that from the CMB and are omittedfrom Fig. 3.5.Direct detection searches also place a significant constraint on this ADM scenario. Kineticmixing of the hidden U(1)x with hypercharge links the hidden vector to charged matter withan effective coupling proportional to −e cW . In the present case, the dark matter consists ofDirac fermions and complex scalars charged under U(1)x, and this allows a vectorial couplingof these states to the X gauge boson. Together, these two features induce a vector-vectoreffective operator (for mx & 20 MeV) connecting the DM states to the proton that gives riseto spin-independent (SI) scattering on nuclei. The Ψ-proton scattering cross section isσp =2c2W e2g2xµ2npim4x. (3.52)A similar expression applies to the scalar Φ. This gives rise to an effective SI cross section pernucleon (in terms of which experimental limits are typically quoted) ofσ˜n = (Z2/A2)σp (3.53)' 2× 10−38 cm2(2ZA)2( 10−3)2 ( gx0.1)2 ( µn1 GeV)2(1 GeVmx)4.Comparing this result to the exclusions of low-mass DM from LUX [8], XENON10 S2 onlyanalysis [9], CDMSLite [10] and CRESST-Si [11], we obtain the green exclusion regions shownin Figures 3.5 and 3.6.3.4.4 SummaryThis hidden U(1)x extension of the MSSM can account for the entire relic dark matter abun-dance in the aftermath of moduli reheating while being consistent with existing constraints fromdirect and indirect detection. Even though the DM annihilation cross section is much largerthan the standard thermal value, a strong DM-anti-DM asymmetry allows for a significant totaldensity while suppressing DM annihilation signals at late times. Limits from direct detectionsearches can also be evaded for light DM masses below the sensitivity of current experiments.To achieve a strong DM asymmetry, a relatively large asymmetry parameter κ & 10−3is needed. We have not specified the dynamics that gives rise to the asymmetry in modulireheating, but more complete theories of asymmetry generation suggest that values this largecan be challenging to obtain [2, 233–235]. Furthermore, as in the symmetric hidden sectortheory considered previously, the spectrum required for this mechanism to work requires scalar74sequestering and scalar soft masses of the right size.3.5 Variation #3: Hidden SU(N)The third extension of the MSSM that we consider consists of a pure supersymmetric SU(N)xgauge theory together with heavy connector matter multiplets charged under both SU(N)x andthe MSSM gauge groups.12 In contrast to the two previous extensions, we do not have to makeany strong assumptions about the scalar soft mass parameters for the theory to produce anacceptable LSP relic density. In particular, this extension can work in the context of a mini-splitspectrum where the scalar superpartners are much heavier than the gauginos [147–152].3.5.1 SU(N)x Mass Spectrum and ConfinementThe hidden states below the TeV scale consist of the SU(N)x gluon and gluino. The hiddengluino soft mass isMx = rxg2x(4pi)2m3/2, (3.54)where rx = 3N if it is generated mainly by AMSB effects. In the discussion to follow, we willconsider additional heavy matter charged under SU(N)x with large supersymmetric mass µF .For µF  m3/2, the coefficient rx will be unchanged [135]. However, when µF . m3/2, thevalue of rx can be modified by an amount of order unity that depends on the soft masses ofthese states [139, 192]. We consider deviations in rx away from the AMSB value but still of thesame general size.Below the hidden gluino mass, the hidden sector is a pure SU(N)x gauge theory. It istherefore guaranteed to be asymptotically free, and the low-energy theory of hidden gluonsshould undergo a confining transition at some energy scale Λx to a theory of massive glueball(and glueballino) bound states. The one-loop estimate of the confinement scale givesΛx = Mx exp(− 3rx22Nm3/2Mx). (3.55)Demanding that the SU(N)x gluino be lighter than the lightest MSSM neutralino typicallyforces Λx to be very small. For example, setting Mx < 1000 GeV, rx = 3N , and requiringthat Mx < M2 (with its value as in AMSB, M2 ' m3/2/360), one obtains Λx < 10−61 GeV.Thus, we will neglect SU(N)x confinement in our analysis and treat the hidden gauge theoryas weakly interacting.12See also Refs. [239, 240] for previous studies of this scenario in a slightly different context.753.5.2 Connectors to the MSSMThe lightest MSSM superpartner must be able to decay to the hidden sector for this extension tosolve the MSSM moduli relic problem. Such decays can be induced by heavy matter multipletscharged under both the MSSM gauge groups and SU(N)x. Following Ref. [239], we examinetwo type of connectors.The first set of connectors consists of NF pairs of chiral superfields F and F c with charges(1, 2,∓1/2;N) under SU(3)C×SU(2)L×U(1)Y×SU(N)x with a supersymmetric mass term [239]W ⊃ µFFF c. (3.56)For µF & m3/2, the heavy multiplets can be integrated out supersymmetrically to give [239]−∆L ⊃∫d4θ g2xg22(4pi)22NFµ4FW †x α˙W† α˙WαxWα (3.57)⊃ αxα22NFµ4F[˜G†x(σ¯ ·∂)˜W Gµνx Wµν + (Gµνx Wµν)2]. (3.58)Similar operators involving the U(1)Y vector multiplet will also be generated, and additionaloperators will also arise with the inclusion of supersymmetry breaking. The wino operator ofEq. 3.58 allows the decay ˜W 0 →W 0Gx ˜Gx, whose rate we estimate to beΓ ∼4(N2−1)N2F8pi(4pi)2α2xα22 |N12|2m9χ01µ8F(3.59)' (7× 105 s)−1(N2−1)N2F |N12|2( αx10−3)2( mχ01270 GeV)9(100 TeVµF)8,where mχ01 is the mass of the lightest MSSM neutralino, |N12| is its wino content, and thefiducial value of mχ01 corresponds to the AMSB value of M2 for m3/2 ' 100 TeV. Note thatthese sample parameter values lead to decays after the onset of primordial nucleosynthesis.The second set of connectors that we consider consists of the same NF heavy multipletsF and F c together with P and P c multiplets with charges (1, 1, 0, N¯) [239]. This allows thecouplingsW ⊃ λuHuFP + λdHdF cP c + µFFF c + µPPP c. (3.60)Neglecting supersymmetry breaking, integrating out the heavy F and P multiplets at one-looporder generates operators such as [239]−∆L ⊃∫d2θ g2xλ2u(4pi)22NFµ2FW xαWx α˙ Hu ·Hd (3.61)⊃ αx(λ2u4pi)2NFµ2F[˜Gxσµσ¯ν ˜HdHuGµνx +Gµνx Gxµν Hu ·Hd]. (3.62)76where we have set µP = µF and λd = λu for simplicity. Additional related operators arise whensupersymmetry breaking is included. The first term in Eq. 3.62 induces the decay χ01 → GxG˜x,whose rate we estimate to beΓ ∼4(N2−1)N2F8piα2x(λ2u4pi)2|N13|2v2um3χ01µ4F(3.63)' (1× 10−6 s)−1(N2− 1)N2F |N13|2( αx10−3)2(λu0.75)4( mχ01200 GeV)3(100 TeVµF)4,where |N13| describes the ˜Hd content of the MSSM LSP. This decay can occur before primordialnucleosynthesis, even for very large values of µF & 100 TeV.Finally, let us mention that the exotic doublets F and F c will disrupt standard gaugeunification. This can be restored by embedding these multiplets in 5 and ￿5 representations ofSU(5) and limiting the amount of new matter to maintain perturbativity up to the unificationscale [239]. The latter requirement corresponds to N ×NF ≤ 5 for µF ∼ 100 TeV.3.5.3 Moduli Reheating and Hidden Dark MatterThe treatment of dark matter production by moduli reheating in this scenario is slightly differ-ent from the situations studied previously. The key change is that the visible and hidden sectorsare unlikely to reach kinetic equilibrium with one another after reheating for µF,P & m3/2. Asa result, it is necessary to keep track of the effective visible and hidden temperatures indepen-dently.To estimate kinetic equilibration, let us focus on the wino operator of Eq. 3.58. This givesrise to Gxγ → Gxγ scattering with a net rate of Γ ∼ T 9/µ8F . Comparing to the Hubblerate, kinetic equilibration requires Teq & (µ8F /MPl)1/7. On the other hand, the reheatingtemperature after moduli decay is on the order TRH ∼ (m33/2/MPl)1/2. Thus, we see that TRHis parametrically smaller than Teq for µF & m3/2. A similar argument applies to the Higgsinteraction in the second term in Eq. 3.62.The total modulus decay rate is the sum of partial rates into the visible and hidden sectors,Γϕ =c4pim3ϕM2Pl= Γv + Γx =cx + cv4pim3ϕM2Pl, (3.64)where cx and cv describe the relative hidden and visible decay fractions. Moduli decays willreheat both sectors independently, and self-interactions within each sector will lead to self-thermalization. The total radiation density is the sum of the two sectors, ρR = ρv + ρx. Wewill also define effective temperatures within each sector byρv =pi230g∗T 4, (3.65)ρx =pi230g∗xT 4x , (3.66)77where g∗ and T refer to the visible sector, and g∗x and Tx to the hidden. Since the hiddenand visible sectors do not equilibrate with each other after reheating, entropy will be conservedindependently in both sectors.Just after reheating, we also haveρv =(cvc)ρR, ρx =(cxc)ρR. (3.67)Given the first equality, we now define the reheating temperature to beTRH =(cvc)1/4[90pi2g∗(TRH)]1/4√ΓϕMPl, (3.68)corresponding approximately to the visible radiation temperature when H = Γϕ. In the sameway, we also define the reheating temperature in the hidden sector to beT xRH = (cx/cv)1/4(g∗/g∗x)1/4TRH. (3.69)The number density of SU(N)x gaugino dark matter evolves according to Eq. (3.15) butwith two important modifications. First, the quantity Nχ now corresponds to the mean numberof hidden gauginos produced per modulus decay. This includes production from direct decays,decay cascades (including decays of the lightest MSSM neutralino), and re-scattering. Thesecond key change is that the thermal average in 〈σv〉 is now taken over the hidden-sectordistribution with effective temperature Tx ' T xRH.The thermally-averaged SU(N)x gaugino cross section can receive a non-perturbative Som-merfeld enhancement from multiple hidden gluon exchange if the hidden confinement scale isvery low, as we expect here [241, 242]. This enhancement can be written as a rescaling of theperturbative cross section,〈σv〉 = Sx〈σv〉pert. (3.70)The perturbative cross section can be obtained by modifying the SU(3)C gluino result [243] bythe appropriate colour factor:〈σv〉pert =3N216(N2 − 1)14pi(g4xM2x). (3.71)The Sommerfeld enhancement factor is [241–243]Sx = A/(1− e−A), (3.72)with A = piαx/v, for v =√1− 4M2x/s. In the perturbative cross section, the characteristicmomentum transfer is √s ' 2Mx, and αx should be evaluated at this scale. However, thetypical momentum transfer leading to the non-perturbative enhancement is √s ∼ 2vMx [243].7810−110010110210310−4 10−3 10−2 10−110−610−410−2100102104106108ΩG˜x/ΩcdmLifetime[sec]gxN = 2, c = 1, cx/cv = 1/9BBNτχ01T xRH > TfoFigure 3.7: Relic abundance of the hidden gluino ˜Gx (solid black) after moduli reheatingas a function of the hidden gauge coupling gx for N = 2, mϕ = m3/2 = 100 TeV,c = 1, Nx = 1, and cx/cv = 1/9. The lifetime of the lightest MSSM superpartner,assumed to be a Higgsino-like neutralino, is shown in light blue for µ = 150 GeV,NF = 3, and λu = 0.75. The vertical solid grey line corresponds to T xRH ≈ Tfo,while the dashed horizontal line shows τχ01 = 1 s.In our calculation, we estimate v '√3T xRH/2Mx and take A to beA ' pi2vαx[1 + 11N6piαx ln(v)]−1, (3.73)where αx in this expression is evaluated at 2Mx.In Fig. 3.7 we show the relic density of hidden gluinos produced by moduli reheating as afunction of gx for mϕ = m3/2 = 100 TeV, c = 1, Nx ∼ 1, and cx/cv = 1/9. We also showin this figure the lifetime of the lightest MSSM superpartner in seconds, which we take to bea Higgsino-like neutralino with µ = 150 GeV, along with N = 2, µF = m3/2, NF = 3, andλu = 0.75. As expected from the estimate of Eq. (3.23), smaller values of the gauge couplinggx  g2 are needed to obtain an acceptable relic density.For very small gx, the hidden gluino mass becomes small enough that the reheating tem-perature exceeds the freeze-out temperature, and the final density is given by the thermalvalue. This corresponds to the plateau, where the abundance is only weakly dependent on thegauge coupling. At intermediate gx, freeze-out happens in the matter dominated phase, whereΩ˜Gx ∝M−3x ∝ g−6x [185], resulting in the turn-over. The abundance continues to decrease untilnon-thermal production takes over, corresponding to the straight section for gx & 4 × 10−3.Note as well that very small values of gx also increase the lifetime of the lightest MSSM stateto τ > 1 s. This can be problematic for nucleosynthesis, and will be discussed in more detailbelow.793.5.4 Hidden Gluino BoundsWe found previously that for Mx < M2 and AMSB-like masses, the SU(N)x confinement scaleis negligibly small relative to the Hubble scale today. This implies that the hidden gluon willbe a new relativistic degree of freedom in the early Universe. A nearly massless hidden gluonwill also interact significantly with the relic hidden gluinos, which has significant implicationsfor dark matter clustering and its imprint on the CMB.New relativistic particles are constrained by primordial nucleosynthesis and the CMB. Thenumber of corresponding degrees of freedom is often written in terms of an effective number ofadditional neutrino species, ∆Neff. If the hidden gluon is the only new light state below thereheating temperature and 5 MeV < TRH < mµ, we have [199]∆Neff '(47)(N2 − 1)(cxcv), (3.74)where cx and cv correspond to the hidden and visible branching fractions of the moduli. Thecurrent upper bound (95% c.l.) on ∆Neff from primordial nucleosynthesis is [244, 245]∆Neff . 1.0 at T ∼ TBBN. (3.75)This bound can be satisfied for smaller N provided (cx/cv) < 1. If we reinterpret our moduliresults in terms of heavy gravitino decay, the corresponding ratio is cx/cv = (N2−1)/12 if onlygaugino modes are open and cx/cv = 12(N2 − 1)/193 if all MSSM channels are available [186].A similar limit on Neff can be derived from the CMB [246]. However, the net effect of thehidden gluon and gluino on the CMB is more complicated than just a change in ∆Neff, as wewill discuss below.A more significant challenge to this scenario comes from the relatively unsuppressed inter-actions among the hidden gluons and gluinos. Self-interactions among dark matter particles arestrongly constrained by observations of elliptical galaxies and the Bullet Cluster [247–249].13Furthermore, we find that the relic hidden gluinos remain kinetically coupled to the hiddengluon bath until very late times. This generates a pressure in the dark gluino fluid that inter-feres with its gravitational collapse into bound structures. A study of this effect lies beyondthe scope of this chapter, and we only attempt to describe some of the general features here.In this scenario, moduli reheating generates a bath of thermal gluons with temperatureTx ∼ (cx/cv)1/4T . Arising from a non-Abelian gauge group, the gluons will interact withthemselves at the rateΓ ∼ α2xTx ∼ (10−12eV)(cxcv)1/4( αx10−4)2(T2.7K). (3.76)13 Dark matter interactions close to these upper bounds can help to resolve some of the puzzles of large-scalestructure [240, 250–253].80This is easily larger than the Hubble rate today, H ∼ 10−33 eV, and we expect the hidden gluonto remain in self-equilibrium at the present time. One of the key features of such non-Abelianplasmas at temperatures well above the confinement scale is that the gluon field is screened byits self-interactions [254, 255]. Correspondingly, the electric and magnetic components of thegluon develop Debye masses on the order of [256],mE ∼√αx Tx (3.77)mB ∼ αxTx. (3.78)Relic hidden gluinos will interact with the hidden gluon bath through Compton-like scat-tering. This can proceed through a t-channel gluon with no suppression by the hidden gluinomass. Modifying the calculation of Refs. [257], we find that the corresponding rate of momen-tum transfer between a relic gluino and the gluon bath is much larger than the Hubble rateeven at the present time. We also estimate that for moderate αx and mϕ ∼ 100 TeV the rateof formation of gluino-gluino bound states, which are expected to be hidden-colour singlets inthe ground state [258–260], is much smaller than the Hubble rate at temperatures below thebinding energy.Together, these two results imply that the relic gluinos remain kinetically coupled to thegluon bath. The pressure induced by the gluons will drive gluinos out of overdense regionsand interfere with structure formation, analogous to the photon pressure felt by baryons beforerecombination. This is very different from the behaviour of standard collisionless cold darkmatter, and implies the hidden gluinos can only be a small fraction of the total dark matterdensity. This fraction, can be constrained using observations of the CMB and by galaxy surveys.A study along these lines was performed in Ref. [261], and their results suggest that the fractionfx = Ω˜Gx/Ωcdm must be less than a few percent, depending on the temperature ratio Tx/T '(cx/cv)1/4.14 Hidden gluino interactions may also modify the distribution of dark matter ongalactic scales [264].3.5.5 SummaryThis supersymmetric hidden SU(N)x extension can produce a much smaller non-thermal LSPrelic density than the MSSM, and has only invisible annihilation modes that are not constrainedby indirect detection. However, the hidden gluino LSP remains in thermal contact with a bathof hidden gluons, and thus can only make up at most a few percent of the total dark matterdensity. Obtaining such small relic densities is non-trivial and leads to new challenges, as wewill discuss here.From Fig. 3.7 we see that reducing the gauge coupling gx lowers the non-thermal hiddengluino density until TRH ∼ Tfo, at which point the relic abundance becomes approximately14 A relic population of millicharged particles will have a similar effect. This was considered in Refs. [262, 263],and a limit of fx . 1% was obtained.81constant in gx. At the same time, Eq. (3.63) shows that smaller values of gx also suppressthe decay rate of the lightest MSSM superpartner. If such decays happen after the onset onprimordial nucleosynthesis, they can disrupt the abundances of light elements [265, 266]. Thedirect two-body decays χ01 → ˜GxGx are invisible. However, the operator of Eq. 3.62 also givesrise to the semi-visible three-body mode χ01 → h0 ˜GxGx if it is kinematically allowed. Thedecay products of the Higgs boson will be significantly hadronic, and can modify light-elementabundances. The branching fraction of this three-body mode depends on the available phasespace. Taking it to be Bh ∼ 10−3 and estimating the Higgsino yield as in Sec. 3.2, we find thatHiggsino lifetimes below τχ01 . 1−100 s are allowed [265]. This can occur for larger values ofN , NF , or λu, or smaller values of µ or µF . Note that reducing µF below mϕ/2 is dangerousbecause it would lead to the production of stable massive F and P states which would tend tooverclose the Universe.An acceptable hidden gluino relic density with a sufficiently rapid MSSM decay can beobtained in this scenario, but only in a very restricted and optimistic region of parameters.For example, with rx = 3N/5, gx = 0.01, N = 2, NF = 3, λu = 0.75, cx/cv = 1/9, andmϕ = 2m3/2 = 2µF = 100 TeV, we obtain Ω ˜Gx/Ωcdm = 0.023 and τχ01 = 0.01 s. Compared tothe parameters used in Fig. 3.7, the greatest effect comes from the small value of rx relative tothe minimal AMSB value (rx = 3N). Such a reduction could arise from threshold correctionsdue to the heavy multiplets [192].3.6 ConclusionsIn this work we have investigated the production of LSP dark matter in the wake of modulioscillation and reheating. For seemingly generic string-motivated moduli parameters mϕ =m3/2, c = 1, Nχ ∼ 1, we have argued that the MSSM LSP is typically created with anabundance that is larger than the observed dark matter density. The exception to this is awino-like LSP, which has been shown to be inconsistent with current bounds from indirectdetection. We call this the MSSM moduli-induced LSP problem.To address this problem, we have studied three gauge extensions of the MSSM. In the first,the MSSM is expanded to include a lighter hidden U(1)x vector multiplet with kinetic mixingwith hypercharge that is spontaneously broken by a pair of chiral hidden Higgs multiplets. Thekinetic mixing interaction allows the lightest MSSM superpartner to decay to the lighter hiddensector LSP. If this LSP consists primarily of the hidden Higgsinos and is sufficiently light, it willannihilate very efficiently. The resulting hidden LSP relic abundance after moduli reheating canbe small enough to be consistent with current bounds from indirect detection and the CMB. Inthis case, a second more abundant component of the DM density is needed. The spectrum ofscalar soft terms required in this theory can also be challenging to obtain for the large valuesof m3/2 & 100 TeV considered.The second extension of the MSSM that we studied has an asymmetric dark matter candi-date. The underlying theory in this case was again a kinetically-mixed U(1)x vector multiplet82spontaneously broken by a pair of chiral hidden Higgs, but now with an additional pair of chiralmultiplets Y and Y c. For a range of parameters, the two stable states in this theory are theDirac fermion Ψ and the lighter complex scalar Φ derived from Y and Y c. If Ψ or Φ obtaina significant particle anti-particle asymmetry in the course of moduli reheating, they can ac-count for the entire DM density. A large production asymmetry leads to a very small residualanti-DM component, which allows the asymmetric abundances of Ψ and Φ to be consistentwith limits from indirect (and direct) detection. However, the production asymmetry requiredfor this to work is relatively large, and may be difficult to obtain in a more complete theoryof asymmetry generation. This theory also faces the same scalar soft term requirement as thesymmetric hidden U(1)x extension.The third extension of the MSSM consists of a pure non-Abelian SU(N)x vector multipletat low energies. This sector can connect to the MSSM through additional heavy multipletscharged under both the visible and hidden gauge groups, allowing for decays of the lightestMSSM superpartner to the SU(N)x gluino. Acceptable hidden gluino relic densities can beobtained for smaller values of the SU(N)x gauge coupling. This implies a potential tensionwith primordial nucleosynthesis from late MSSM decays, and leads to a negligibly small hiddenconfinement scale. In contrast to the two previous extensions, light scalar superpartners are notrequired and this mechanism can work in the context of mini-split supersymmetry [147–152].While this theory is not constrained by standard indirect detection searches, the coupling ofthe hidden gluino to a bath of hidden gluons leads to non-standard DM dynamics that requirethe hidden gluino density to be only a few percent of the total DM density. It is very difficultto obtain relic densities this small in this scenario.Our main conclusion is that it is challenging to avoid producing too much LSP dark matterin the course of string-motivated moduli reheating. For seemingly generic modulus parameters,the relic density in the MSSM is either too large or at odds with limits from indirect detection.This may be a hint that the properties of moduli (in our vacuum at least) differ from the generalexpectations discussed above [124, 178]. Alternatively, this could be an indication of new lightphysics beyond the MSSM. We have considered three examples of the latter possibility in thischapter and have shown that they can produce a stable LSP abundance that is consistentwith current observations and limits. Even so, these three extensions all lead to a significantcomplication of the MSSM and require a somewhat fortuitous conspiracy of parameters forthem to succeed. A more direct solution might be the absence of a stable LSP through R-parity violation, or simply the absence of light superpartners and very large mϕ ∼ m3/2.83Chapter 4Dark Matter Antibaryons from aSupersymmetric Hidden Sector4.1 IntroductionThe apparent coincidence between the densities of dark and baryonic matter, given by ΩDM/Ωb ≈5, may be a clue that both originated through a unified mechanism. A wide variety of modelshave been proposed along these lines within the framework of asymmetric DM [225, 232, 267–277]; see Ref. [278] for a review. In these scenarios, DM carries a conserved global charge, andits relic abundance is determined by its initial chemical potential. Moreover, if the DM chargeis related to baryon number (B), then the cosmic matter coincidence is naturally explained forO(5 GeV) DM mass.In this chapter, we explore model-building, cosmological, and phenomenological aspects ofhylogenesis (“matter-genesis”), a unified mechanism for generating dark and baryonic mattersimultaneously [232, 279]. This model is an extension of the scenario that was presented inSec. 3.4. Hylogenesis requires new hidden sector states that are neutral under SM gauge in-teractions but carry non-zero B. CP-violating1 out-of-equilibrium decays in the early Universegenerate a net B asymmetry among the SM quarks and an equal-and-opposite B asymmetryamong the new hidden states. The Universe has zero total baryon number, but for appropri-ate interaction strengths and particle masses, the respective B charges in the two sectors willnever equilibrate, providing an explanation for the observed asymmetry of (visible) baryons.The stable exotic particles carrying the compensating hidden antibaryon number produce thecorrect abundance of dark matter. Put another way, DM consists of the missing antibaryons.The minimal hylogenesis scenario, described in Refs. [232, 279], has the following threeingredients:1. DM consists of two states, a complex scalar Φ and Dirac fermion Ψ, each carrying B =−1/2.1C is charge conjugation and P is parity.842. A Dirac fermion X, carrying B = 1, that transfers B between quarks and DM throughthe gauge invariant operators [280]X ucRi dcRj dcRk , XΨΦ (4.1)where i, j, k label generation (colour indices and spinor contractions are suppressed).3. An additional U(1)′ gauge symmetry that is kinetically mixed with hypercharge andspontaneously broken near the GeV scale, producing a massive Z ′.2With these ingredients, hylogenesis proceeds in three stages, which we illustrate schematicallyin Fig. 4.1:1. Equal (CP-symmetric) densities of X and X¯ are created non-thermally, e.g., at the endof a moduli-dominated epoch when the Universe is reheated through moduli decay to atemperature TRH in the range of 5 MeV . TRH . 100 GeV  mX [133].2. The interactions of Eq. (4.1) allow X to decay to uRi dRj dRk or Ψ¯Φ∗, and similarly for X¯.With at least two flavours of X, these decays can violate CP leading to slightly differentpartial widths for X relative to X¯, and equal-and-opposite asymmetries for visible andhidden baryons.3. Assuming Φ and Ψ are charged under U(1)′, the symmetric densities of hidden particlesannihilate away almost completely, with ΨΨ¯ → Z ′Z ′ and ΦΦ∗ → Z ′Z ′ occurring very effi-ciently in the hidden sector, followed by Z ′ decaying to SM states via kinetic mixing. Theresidual antibaryonic asymmetry of Φ and Ψ is asymmetric DM. Likewise, the symmetricdensity of visible baryons and antibaryons annihilates efficiently into SM radiation.Both Ψ and Φ are stable provided |mΨ −mΦ| < mp +me, and they account for the observedDM density for mΨ +mΦ ≈ 5mp, implying an allowed mass range 1.7 . mΨ,Φ . 2.9 GeV.On the phenomenological side, hylogenesis models possess a unique experimental signature:induced nucleon decay (IND), where antibaryonic DM particles scatter inelastically on visiblebaryons, destroying them and producing energetic mesons. If X couples through the “neutronportal” uRdRdR, IND produces pi and η final states, while if X couples through the “hyperonportal” uRdRsR, IND produces K final states. These signatures mimic regular nucleon decay,with effective nucleon lifetimes comparable to or shorter than existing limits; however, presentnucleon decay constraints do not apply in general due to the different final state kinematicsof IND. Searching for IND in nucleon decay searches, such as the Super-Kamiokande experi-ment [281] and future experiments [282–284], therefore offers a novel and unexplored means fordiscovering DM.2We use different notation for hidden sector states and parameters in this chapter compared to Ch. 3, in orderto more closely match with the original literature of Refs. [232, 279].85XX¯3q3q¯Ψ¯,Φ∗Ψ,ΦHidden SectorVisible Sector3qΨ,ΦFigure 4.1: The three steps of hylogenesis.Although the minimal hylogenesis model described above successfully generates the cosmo-logical baryon and DM densities, two puzzles remain. Is there a natural framework to considerDM as a quasi-degenerate scalar/fermion pair? Is there a mechanism to ensure the quantumstability of the GeV-scale masses for hidden sector scalars? Supersymmetry (SUSY) can provideanswers to both questions; the DM pair (Φ,Ψ) forms a supermultiplet with B = −1/2, andthe stability of the GeV-scale hidden sector and the (Φ,Ψ) mass splitting is ensured naturally,provided SUSY breaking is suppressed in the hidden sector compared to the visible sector.The goal of this chapter is to embed hylogenesis in a supersymmetric framework of naturalelectroweak and hidden symmetry breaking, and to study in detail the cosmological and phe-nomenological consequences. In Sec. 4.2, we present a minimal supersymmetric extension ofthe hylogenesis theory described above. We also address the origin of the nonrenormalizablenucleon portal operator X ucRi dcRj dcRk. In Sec. 4.3, we investigate the cosmological dynamicsof supersymmetric hylogenesis, showing explicitly the range of masses and parameters that canexplain the correct matter densities. Section 4.4 contains a discussion of how such parametervalues can arise in a natural way from various mechanisms for supersymmetry breaking. InSec. 4.5 we investigate the phenomenology of our model, including IND signatures, colliderprobes, and DM direct detection. Our results are summarized in Sec. 4.6. This chapter isbased on Ref. [2], completed in collaboration with David Morrissey, Kris Sigurdson and SeanTulin; an alternative supersymmetric model based on Higgs portal mixing is also presented inthe Appendix of that work.4.2 Supersymmetric Hylogenesis ModelIn this section, we present an extension of the Minimal Supersymmetric Standard Model (MSSM)that can account for the dark matter and baryon densities through a unified mechanism of hy-logenesis. In order to organize our discussion, it is useful to divide our model into three sectors,given by the superpotential termsW = WMSSM +WHS +Wtrans . (4.2)86superfield U(1)′ B Rhidden baryons X1,2 0 +1 −1Y1 0 −1/2 iY2 +1 −1/2 ihidden U(1)′ H +1 0 +1Z ′ 0 0 0Table 4.1: New superfields in the hidden sector, with quantum numbers under U(1)′,B, and R-parity. Chiral supermultiplets X1,2, Y1,2,H also include vector partnersXc1,2, Y c1,2,Hc with opposite charge assignments (not listed).First, WMSSM corresponds to the superpotential of the usual MSSM with weak-scale super-partners; this is the visible sector. Second, we introduce a hidden sector comprised of newstates which carry B, but are uncharged under the SM gauge group, and whose interactionsare described by the superpotential WHS. The third term Wtrans corresponds to operatorsresponsible for B transfer between the visible and hidden sectors. Baryon transfer operatorsgenerate equal-and-opposite B asymmetries within the two sectors, and lead to IND signaturesin nucleon decay searches.4.2.1 Hidden SectorThe hidden sector of our model consists of (i) four vector-like chiral superfields carrying nonzeroB, denoted X1,2 and Y1,2, with charge-conjugate partners Xc1,2 and Y c1,2,3 and (ii) a U(1)′gauge sector, with gauge boson Z ′ and gauge coupling e′, spontaneously broken by a vectorpair of hidden Higgs supermultiplets H,Hc. Table 4.1 summarizes these exotic fields. Thesuperpotential is given byWHS =∑a=1,2ζaXaY 21 + ζ¯aXca(Y c1 )2 + γY1Y c2H + γ¯Y c1 Y2Hc+ µXaXaXca + µYaYaY ca + µHHHc , (4.3)which includes Yukawa-type interactions with couplings ζ1,2, ζ¯1,2, γ, γ¯, and vector masses µX1,2 ,µY1,2 , µH . We also assume a canonical Kähler potential for these multiplets. Note as well thatwe have extended R-parity to a ZR4 for the Y (c)i multiplets. Aside from allowing the couplingslisted above, this extension does not lead to any novel features in the present case, beyond thoseimposed by the standard R-parity.After symmetry breaking in the hidden sector, the superfields Y1,2 and Y c1,2 mix to formtwo Dirac fermions Ψa (a = 1, 2) and four complex scalars Φb (b = 1, 2, 3, 4) with B = −1/2.Among these, the lightest states Ψ1 and Φ1 are stable DM. The fermionic mass terms for Y1,23Two species X1,2 are required for CP-violating decays (see Sec. 4.3), while two species Y1,2 are needed tocouple them to the gauge-singlet X fields.87are (in two-component notation)Lferm = −(Y c1 , Y c2)MY(Y1Y2)+ h.c. , MY ≡(µY1 γ¯ηcγη µY2). (4.4)where η ≡ 〈H〉, ηc ≡ 〈Hc〉 are the hidden Higgs vacuum expectation values (vevs). This massmatrix can be diagonalized by a biunitary transformation V TMY U † = diag(mΨ1 ,mΨ2). Thescalar mass terms in the basis ˜Y ≡ (˜Y1, ˜Y2, ˜Y c∗1 , ˜Y c∗2 )T areLscalar = −˜Y †M2˜Y˜Y . (4.5)The 4× 4 mass matrix M2˜Yreceives contributions from F terms from Eq. (4.3), D terms, andsoft SUSY-breaking terms−Lsoft ⊃ m2Y1 |˜Y1|2 +m2Y2 |˜Y2|2 +m2Y c1 |˜Y c1 |2 +m2Y c2 |˜Y c2 |2 (4.6)+(b1 ˜Y1 ˜Y c1 + b2 ˜Y2 ˜Y c2 + γAγ ˜Y1 ˜Y c2H + γ¯Aγ¯ ˜Y c1 ˜Y2Hc + h.c.), (4.7)We haveM2˜Y ≡(M†Y MY − δ + m2˜Y∆†∆ MY M†Y + δ + m2˜Y c), ∆ ≡(b1 γAγηγ¯Aγ¯ηc b2), (4.8)also defining m2˜Y≡ diag(m2Y1 ,m2Y2), m2˜Y c ≡ diag(m2Y c1,m2Y c2 ), δ ≡ e′2(η2c − η2)× diag(0, 1). Thescalar mass matrix can be diagonalized by a unitary transformation ZM2˜YZ† = diag(m2Φ1 ,m2Φ2 ,m2Φ3 ,m2Φ4).Similar mass matrices arise for the X supermultiplets; for simplicity we assume that the fermionstates X1,2 and scalar states ˜X1,2, ˜Xc1,2 are all mass eigenstates.The U(1)′ gauge sector consists of the Z ′ gauge boson, with mass m2Z′ = 2e′2(η2 + η2c ), the˜Z ′ gaugino, and the hidden Higgsinos ˜H, ˜Hc. The three neutralinos have the mass matrixM =M ′ −√2e′η√2e′ηc−√2e′η 0 µH√2e′ηc µH 0, (4.9)which can be brought into a diagonal form using a unitary transformation P , such thatP †MP = diag(mχ1 ,mχ2 ,mχ3). The U(1)′ gauge superfield mixes kinetically with the MSSMhypercharge,4−L ⊃ κ2∫d2θBαZ ′α + h.c. , (4.10)where Z ′α and Bα are the U(1)′ and U(1)Y supersymmetric gauge field strengths, respectively,4The kinetic mixing parameter in Ch. 3 was called .88with the mixing parameter κ 1.The full particle content of the hidden sector after the spontaneous breaking of U(1)′ consistsof the following mass eigenstates: three neutralinos χi; three hidden Higgs scalars h, H, A; twoDirac fermions Ψi; four complex scalars Φi; and a massive gauge boson Z ′. The lightest Diracfermion Ψ1 and complex scalar Φ1 are stable due to their masses and B charge assignments —they make up the dark matter. All other states either annihilate or decay into Standard Modelparticles as described in Sec. 4.3.Now that we have presented the ingredients for the hidden sector states, we make someremarks:• We assume that the mass scales of the hidden sector parameters lie at the GeV scale(with the exception of the X states). For the soft terms, this can be accomplished byassuming that SUSY-breaking is suppressed in the hidden sector (see Sec. 4.4). However,the SUSY-preserving vector mass terms present a hidden µ-problem; we ignore this issue,but in principle this can be solved by introducing an additional hidden singlet analogousto the NMSSM.• Since X1,2 mediates baryon transfer between the visible and hidden sectors (described be-low), IND signatures are more favourable if the DM states (Φ1,Ψ1) are mostly aligned withthe Y1 supermultiplet. However, nonzero mixing with Y2 is induced by SUSY-breakingand hidden Higgs vevs resulting in a DM-Z ′ coupling that is essential for annihilation ofthe symmetric DM density.• We have imposed B as a global symmetry. Since gravitational effects are expected to vio-late global symmetries, B violation could arise through Planck-suppressed operators, po-tentially leading to DM particle-antiparticle oscillations that can erase the hidden baryonasymmetry [285–288]. In our SUSY framework, these effects are forbidden by the ZR4 ex-tension of R-parity. For example, the B = −1 operators W ∼MY 21 , Y1Y2Hc are allowedby U(1)′ and can lead to DM oscillations, but they are not invariant under R-parity. If ZR4descends from an anomaly-free gauge symmetry, such as U(1)B−L spontaneously brokenby two units, it cannot be violated by gravity [289, 290] and these operators are forbidden.Thus there exists a consistent embedding of ZR4 in a gauge symmetry that excludes theMajorana mass terms for Y1,2 that could erase the hidden asymmetry by oscillations.4.2.2 Baryon TransferBaryon number is transferred between the hidden and visible sectors through superpotentialterms Wtrans. The hidden baryon states X1,2 are coupled to the operator U ciDcjDck, whereU ci , Dcj are the usual SU(2)L-singlet quark superfields (i, j, k label generation). We focus onthe case involving light quarks (U c ≡ U c1 , Dc ≡ Dc1, Sc ≡ Dc2), corresponding to the “hyperon89portal” [280]:Wtrans =∑a=1,2λaMαβγXaU cαDcβScγ , (4.11)with SU(3)C indices α, β, γ and nonrenormalizable couplings λ1,2/M . Although hylogenesisis viable for any generational structure, Eq. (4.11) is the most interesting case for IND signa-tures. In contrast to non-SUSY hylogenesis models, the “neutron portal” coupling X1,2U cDcDcvanishes by antisymmetry. SUSY hylogenesis therefore favours IND involving K final states,rather than pi, η final states allowed in generic non-SUSY models.The simplest possibility to generate the nonrenormalizable coupling in Eq. (4.11) is tointroduce a vector-like colour triplet supermultiplet P with global charges B = −2/3 andR = 1. There are three cases to consider:5Wtrans =λ′1,2X1,2PαU cα + λ′′ αβγP cαScβDcγ + µPPαP cα (case I)λ′1,2X1,2PαDcα + λ′′ αβγP cαU cβScγ + µPPαP cα (case II)λ′1,2X1,2PαScα + λ′′ αβγP cαDcβU cγ + µPPαP cα (case III)(4.12)The SU(3)C × SU(2)L × U(1)Y quantum numbers for P are (3, 1, 2/3) for case I (up-type),and (3, 1,−1/3) for cases II and III (down-type). In all cases P c carries the opposite charges.The choice between the cases in Eq. (4.12) makes little difference for hylogenesis cosmology.However, the different cases affect the IND signals, manifested in the ratio of the rates ofp→ K+ to n→ K0 channels, discussed in Sec. 4.5.Integrating out P and P c at the supersymmetric level generates the superpotential operatorof Eq. (4.11) with λa/M ≡ λ′aλ′′/µP together with the (higher-order) Kähler potential term(for case I, with similar operators for cases II and III)K ⊃ |λ′′|2|µP |2[(Dc†Dc)(Sc†Sc)− (Sc†Dc)(Dc†Sc)]. (4.13)Including supersymmetry breaking leads to additional operators. In particular, the holomorphicsoft scalar coupling bP P˜ P˜ c (or a squark-gaugino loop with a gaugino mass insertion) gives riseto the four-fermion operator XucRdcRscR that plays a central role in IND.4.3 Hylogenesis CosmologyWe turn next to a study of the early Universe dynamics of our supersymmetric model ofhylogenesis. To summarize the main ingredients:• We assume that the Universe is dominated at early times by a long-lived non-relativisticstate ϕ (e.g., an oscillating modulus field), which decays and reheats the Universe before5We consider the interactions of cases II and III separately, although in general both may arise simultaneously.The simultaneous presence of both sets of couplings leads to strangeness-violating interactions that may beconstrained by flavour violation constraints that we do not consider here.90the onset of Big Bang nucleosynthesis (BBN) [133].• Nonthermal CP-symmetric densities of X1 and ˜X1 states are populated through ϕ decays.Depending on their specific origin, the scalar or the fermion can be created preferen-tially [127]. CP-violating decays of X1 and ˜X1 generate equal-and-opposite asymmetriesin quarks and hidden sector baryons (Ψ,Φ), while the total baryon number is conserved.6• A hidden U(1)′ gauge sector allows for cascade decays of heavier B = −1/2 states (Φ2,Ψ2,etc.) into the lightest states Φ ≡ Φ1 and Ψ ≡ Ψ1 that are DM. Both states can be stableprovided the condition |mΨ −mΦ| < mp +me is met. Also, the symmetric DM densitiesannihilate efficiently through the light Z ′, which decays to SM states via kinetic mixingwith hypercharge.Below, we first compute the CP asymmetries for X1 and ˜X1 decays. Second, we ensure thatthe successful predictions of BBN are not modified by hidden sector decays into SM particles.Third, we solve the system of Boltzmann equations for hylogenesis, incorporating all of theaforementioned ingredients, to compute the baryon asymmetries. There are significant differ-ences compared to nonsupersymmetric hylogenesis [232]; in particular, the DM masses mΦ, mΨand the ratio of Φ to Ψ states can be different, with implications for IND phenomenology.4.3.1 CP-violating AsymmetriesVisible and hidden B asymmetries are produced by CP violation in the partial decay widths ofX1X1 → uRd˜Rs˜R + dRs˜Ru˜R + sRu˜Rd˜R , X1 → Ψ¯iΦ∗j , (4.14)due to interference between tree-level and one-loop amplitudes, shown in Fig. 4.2. The corre-sponding CP asymmetry isX ≡1ΓX1[Γ(X1 → uRd˜Rs˜R)− Γ(X¯1 → u¯Rd˜∗Rs˜∗R) + perms.](4.15a)=3[Im(λ∗1ζ1ζ∗2λ2)mX1 + Im(λ∗1ζ¯∗1 ζ¯2λ2)mX2]m3X164pi3M2(m2X2 −m2X1)(|ζ1|2 + |ζ¯1|2). (4.15b)We assume that ΓX1 is dominated by the two-body decay to Ψ¯iΦ∗j final states.7 For X > 0, apositive net B asymmetry is generated in the visible sector. By CPT invariance, the decay ratesfor X1 and X¯1 are equal, and so an equal-and-opposite (negative) B asymmetry is generatedin the hidden sector. Additional contributions to X from X1 → uRdRsR, arising throughSUSY-breaking or at one-loop, may be subleading provided squark decays are kinematicallyavailable.6We neglect CP-violating decays of X2 and ˜X2, which in principle can also contribute to B asymmetries.7In what follows, flavour indices i, j for hidden sector states are implicitly summed over in final states.91X1ΨciΦjud˜Rs˜RX2X1ud˜Rs˜RFigure 4.2: Representative diagrams contributing to X1 → qiq˜Rj q˜Rk decays which areresponsible for the generation of the baryon asymmetry.The baryon asymmetry can also be generated through the decays of the scalar componentof the X1 superfield, ˜X1, via interference of supersymmetrizations of the diagrams in Fig. 4.2.In the supersymmetric limit, the CP-asymmetry due to ˜X1 is equal to Eq. (4.15). However, ˜X1decay can populate preferentially Ψ or Φ, due to the different hidden sector decay ratesΓ( ˜X1 → Φ∗iΦ∗j ) =|ζ¯1|216pimX1 , Γ( ˜X1 → Ψ¯iΨ¯j) =|ζ1|216pimX1 . (4.16)For X1 decays, the primordial ratior ≡ nΨ/nΦ (4.17)of charge densities nΨ,Φ is equal to unity. However, ˜X1 decays can deviate from r = 1 for|ζ1| 6= |ζ¯1|. As we discuss below, IND signals can be significantly enhanced if the heavier stateis overpopulated compared to the lighter state (e.g., r  1 for mΨ > mΦ).The dark matter abundance is given byΩDMΩb= 2(mΨr +mΦ)mp(1 + r), (4.18)where we have neglected the contributions from the DM anti-particles. This is appropriate inthe limit of completely asymmetric DM populations. The allowed DM mass window, includingthe uncertainty in ΩDM/Ωb ≈ 5, is then1.3 GeV ≤ mΨ, mΦ ≤ 3.4 GeV. (4.19)More specifically, Fig. 4.3 shows the allowed mass range for mΨ (blue) and mΦ (red), for agiven value of r. In the r → 0 (∞) limit, only Φ (Ψ) is populated and its mass is required to beapproximately 5mp/2 to explain the DM density; the underpopulated Ψ (Φ) state is constrainedwithin the range of Eq. (4.19) by the stability condition |mΨ −mΦ| < mp +me.92n Ψ/nΦDark Matter Mass [GeV]10−210−11001011021.5 2.0 2.5 3.0 3.5mΦmΨFigure 4.3: Allowed masses for the scalar Φ and fermion Ψ components of dark matter.For a fixed value of nΨ/nΦ, the shaded region shows the entire mass range of Ψ(blue) and Φ (red) that reproduces ΩDM/Ωb ≈ 5 and satisfies the stability require-ment |mΨ − mΦ| < me + mp. Shifting ΩDM/Ωb by +(−)6% moves the allowedregion right (left), as indicated by the dashed contours.4.3.2 Decays and Annihilations of SUSY StatesAside from the stable DM states Φ1 and Ψ1, the hidden sector contains numerous states thatdecay, producing additional SM radiation. These decays, listed below, must occur with alifetime shorter than about one second to avoid conflicts with BBN predictions.• The Z ′ gauge boson decays to SM states via kinetic mixing with the photon, requiringκ & 10−11 (mZ′/GeV)−1 [291], while κ . 10−3 is consistent with existing limits formZ′ ∼ GeV [202, 292].8• For the hidden Higgs states, the heavy CP-even state decays H → Z ′Z ′, while the CP-oddstate decays A→ Z ′h, through U(1)′ gauge interactions. Since the lighter CP-even stateh is necessarily lighter than the Z ′, it must decay to Standard Model fermions either vialoop-suppressed processes [293] or via D-term mixing with the MSSM Higgs [200]. For hmasses above the two-muon threshold, the mixing process dominates requiring κ & 10−5.• The heavy dark states Φi (i > 1) and Ψ2 cascade down to Φ1 and Ψ1 by emitting Z ′ andh bosons.8We assume a stronger condition κ & 10−8g∗(mZ′/GeV)−1(TRH/GeV)3/2 such that the hidden and visiblesectors are in kinetic equilibrium at T < TRH. [291]93• The hidden neutralinos can decay χi → Φ1Ψ¯1,Φ∗1Ψ1 provided this channel is open (as-sumed below). If this channel is closed, then the lightest state χ1 is stable, providingan additional DM component, and must annihilate efficiently via the t-channel processχ1χ1 → Z ′Z ′.In addition, the lightest supersymmetric particle within the MSSM decays to hidden statesthrough mixing of the hidden and MSSM neutralinos induced by κ. This mixing generally hasa negligible effect on the mass eigenvalues [200].The symmetric DM densities of Ψ1Ψ¯1 and Φ1Φ∗1 annihilate to Z ′ gauge bosons. In the casewhere DM is nearly aligned with the Y2 multiplet, the cross sections are given by [291]〈σv〉ΨΨ¯→Z′Z′ =e′416pim2Ψ√1−m2Z′/m2Ψ, 〈σv〉ΦΦ∗→Z′Z′ =e′416pim2Φ√1−m2Z′/m2Φ . (4.20)To have 〈σv〉 & 3× 10−26 cm3/s for efficient annihilation, we require e′ & 0.03 [229].The presence of light hidden neutralinos allows for the chemical equilibration of baryonnumber between Φ and Ψ. The most important process is Φ1Φ1 ↔ Ψ1Ψ1 which transfers theB asymmetry from the heavier DM state to the lighter state. This effect is phenomenologicallyimportant for IND, potentially quenching the more energetic down-scattering IND processes.The transfer arises from the supersymmetrization of the hidden gauge and Yukawa interactions,which, in the mass basis, takes the formL ⊃ΦiΨj[(−√2e′Z∗i4Vj2P1k − γZ∗i1Vj2P2k − γ¯Z∗i2Vj1P3k)PL+(√2e′Z∗i2Uj2P ∗1k − γ∗Z∗i4Uj1P ∗2k − γ¯∗Z∗i3U∗j2P ∗3k)PR]χk + h.c. (4.21)In the limit where the dark matter is mostly aligned with the Y2 supermultiplet, the interactionsimplifies toL ⊃√2e′Φ1Ψ1 (aPL + bPR)χ1 + h.c., (4.22)where a = −√2e′Z∗14, b =√2e′Z∗12, and χ1 is the U(1)′ gaugino. Note that even if the mixingdue to U(1)′ breaking can be neglected, the scalars ˜Y2 and ˜Y c∗2 will still mix via the soft b-term.For mΦ ≥ mΨ the s-wave contribution to the thermalized Φ1Φ1 → Ψ1Ψ1 cross section for thisinteraction takes the form〈σv〉ΦΦ→ΨΨ =18pi(M2 +m2Φ −m2Ψ)2√1−m2Ψm2Φ××(2m2Φ[(|a|4 + |b|4)m2χ + (|a|2 + |b|2)(ab∗ + a∗b)mχmΨ + 2|a|2|b|2m2Ψ ]−m2Ψ|(a2 + b2)mχ + 2abmΨ|2). (4.23)94In our numerical calculations we use a = b = e′ for simplicity. In this case the transfer crosssection reduces to〈σv〉ΦΦ→ΨΨ = e′4 (mχ +mΨ)22pi(m2χ +m2Φ −m2Ψ)2(1−m2Ψm2Φ)3/2. (4.24)The cross section for the reverse process Ψ1Ψ1 → Φ1Φ1 is related to Φ1Φ1 → Ψ1Ψ1 by thedetailed balance condition〈σv〉ΦΦ→ΨΨ =(neqΨ /neqΦ)2 〈σv〉ΨΨ→ΦΦ, (4.25)where the equilibrium distributions neqi are given in Eq. (4.34). We discuss depletion of theheavier DM state in more detail below. Before doing so, however, let us mention that thetransfer process is not generic, and may be absent in other constructions. In particular, this istrue of the alternate Higgs portal model presented in the Appendix of Ref. [2].4.3.3 Boltzmann EquationsThe generation of the visible and hidden B asymmetries during reheating is described by asystem of Boltzmann equations:ρ˙ϕ =− 3Hρϕ − Γϕρϕ (4.26a)s˙ =− 3Hs+ ΓϕTρϕ (4.26b)n˙B =− 3HnB + (XNX + ˜XN ˜X)Γϕρϕmϕ. (4.26c)Here, ρϕ is the energy density of the modulus field ϕ, s is the entropy density, and nB is thevisible B charge density (the hidden B asymmetry is −nB). Also, NX, ˜X is the average numberof X1 or its superpartner produced per modulus decay, while X, ˜X is the CP-asymmetry fromX1, ˜X1 decay. In the supersymmetric limit X = ˜X . The modulus decay rate Γϕ determinesthe reheat temperatureTRH =[454pi3g∗(TRH)]1/4√ΓϕMPl. (4.27)The total modulus decay rate is given by [133, 294, 295]Γϕ =m3ϕ4piΛ2, (4.28)where we take Λ = 2.43 × 1018 GeV to be the reduced Planck constant. Along with theFriedmann equation H2 = (8piG/3)(ρϕ + ρr), where ρr = (pi2/30)g∗T 4, Eqs. (4.26) form aclosed set and can be solved using the method of Refs. [184, 185]. Here g∗ is the energy density9510−3010−2910−2810−2710−2610−2510−2410−2310−2210−2110−20101 102 103 104 105NimϕaNϕρra4NΨ +NΦ −NΨ¯ −NΦ∗NBFigure 4.4: Solutions to the reheating Eqs. (4.26a, 4.26b, 4.26c) and DM production, de-scribed by Eq. (4.29). Here Nϕ = ρϕa3/mϕ and Ni = nia3 for i = B, Ψ, Φ, Ψ¯, Φ∗.number of relativistic degrees of freedom. Instead of entropy, one can also solve for radiationdensity. We take mϕ = 2000 TeV (corresponding to TRH ≈ 270 MeV), NX = N˜X = 1 andX = ˜X = 3.68×10−4. This decay asymmetry can be generated for example by taking |λa| ≈ 1,|ζa| ≈ 0.1,9 maximal CP-violating phase and M ≈ mX1 ≈ mX2/3 ≈ 1 TeV. These parametersreproduce the observed baryon asymmetry ηB = nB/s ≈ 8.9 × 10−11. Numerical solutions tothe reheating equations for these parameters are shown in Fig. 4.4. The modulus field ϕ decaysinto radiation and the heavy states X, which immediately decay asymmetrically in the visibleand hidden sectors, generating the baryon asymmetry and the dark matter abundance.The production of dark matter and its dynamics are described by a system of four Boltzmannequations for Ψ1, Φ1 and their antiparticles10 which take the formn˙i = −3Hni + Ci + (Ni,XNX +Ni, ˜XN ˜X)Γϕρϕmϕ, (4.29)for i = Ψ1, Ψ¯1, Φ1,Φ∗1. Here Ni,X is the average number of species i produced per X decay.For i = Ψ (we drop the subscript 1 from hereon) we haveNΨ ≡ NΨ,X +NΨ, ˜X =Γ(X¯ → ΨΦ) + 2Γ( ˜X∗ → ΨΨ)ΓX. (4.30)9The magnitude of the coupling constants is chosen to be consistent with hidden sector SUSY-breaking asdiscussed in Sec. 4.4.10We assume that the heavier dark states Ψ2 and Φi, i > 1, decay to Ψ1 and Φ1 sufficiently fast.96Similar definitions hold for Ψ¯, Φ and Φ∗, so thatNΨ +NΦ −NΨ¯ −NΦ∗ = 2X + 2 ˜X . (4.31)The last term on the right hand side of Eq. (4.29) describes the production of the species ithrough modulus decay into X which promptly decays into i. The quadratic collision termsCi describe the particle-antiparticle annihilations as well as the transfer reaction ΨΨ ↔ ΦΦ,required by supersymmetry. The collision terms for i = Ψ, Φ areCΨ =− 〈σv〉ΨΨ¯→Z′Z′[nΨnΨ¯ − (neqΨ )2]− 〈σv〉ΨΨ→ΦΦ[n2Ψ −(neqΨ /neqΦ)2 n2Φ](4.32)CΦ =− 〈σv〉ΦΦ∗→Z′Z′[nΦnΦ∗ − (neqΦ )2]− 〈σv〉ΦΦ→ΨΨ[n2Φ −(neqΦ /neqΨ)2 n2Ψ], (4.33)whereneqi =gi2pi2Tm2iK2(mi/T ) (4.34)is the Maxwell-Boltzmann equilibrium number density for a particle of mass mi with gi internaldegrees of freedom. The collision terms for the antiparticles are identical, with the replacementsΨ → Ψ¯ and Φ → Φ∗.The solutions to the Boltzmann equations for the yields Yi = ni/s are shown in Fig. 4.5 formΨ = 1.9 GeV, mΦ = 2.2 GeV, mχ = 5 GeV, e′ = 0.05. We consider two cases. In the ploton the left, we show the limit where 〈σv〉ΦΦ↔ΨΨ = 0; this can occur when the rate is mixing-suppressed, for a heavy gaugino, or within models with alternative symmetric annihilationmechanisms (see Appendix of Ref. [2]). With the transfer turned off, the scalar and fermionDM sectors are decoupled. The resulting DM abundances are determined by the X and ˜Xdecay asymmetries. In this limit, the dark sector reduces to two independent copies of thestandard asymmetric DM scenario. We show the case where Ψ and Φ are populated equally bythe ˜X decays, but, in general, the asymmetries can be different for the scalar and fermion DM,as discussed in Sec. 4.3.1.In the plot on the right we show the result when the transfer is efficient, driving the darkmatter population into the lighter state Ψ. Since the asymmetry is also transferred into thelighter state, the ΨΨ¯ annihilation rate is enhanced, resulting in a highly asymmetric finalabundance. The heavier state, on the other hand, freezes out with nearly equal abundancesof particle and anti-particle, which are about an order of magnitude smaller than that of Ψ.The transfer reaction does not affect the production of the net hidden sector baryon numbernΨ + nΦ − nΨ¯ − nΦ∗ . Its evolution is shown in Fig. 4.4. Note thatnΨ + nΦ − nΨ¯ − nΦ∗ = 2nB (4.35)as required by B conservation in hylogenesis.The ratio of the abundances of Ψ to Φ is important for IND. We study the effect of varying9710−1210−1010−810−610−410−2101 102 103 104 105Y imϕaΨΨ 6↔ ΦΦYΨYΨ¯YΦYΦ∗10−1210−1010−810−610−410−2101 102 103 104 105Y imϕaΨΨ ↔ ΦΦYΨYΨ¯YΦYΦ∗Figure 4.5: Solution to the Boltzmann equations for the yields Yi = ni/s as a functionof the scale factor a. The plot on the left shows the evolution for the case whenthe transfer reaction ΦΦ ↔ ΨΨ is turned off, while the plot on the right showsthe outcome when it is active. The transfer drives the dark matter population intolighter state, Ψ in this case. The DM (anti-DM) abundance is indicated by solid(dashed) lines, with dark (light) lines referring to the fermion (scalar) component.The parameters used are described in the text.the mass splitting ∆m = mΦ − mΨ and the gaugino mass on nΨ/nΦ in Fig. 4.6. For eachpoint in the parameter space, we solve the reheating and DM production equations and plotthe final value of nΨ/nΦ. The reheat temperature, asymmetry and gauge coupling strengthare the same as for Fig. 4.5. Setting the DM abundance to the observed value fixes mΨ.Light gauginos and large DM mass splittings make the transfer more efficient, increasing theabundance of the lighter state relative to the heavier one. For small ∆m or heavy mχ thetransfer rate is suppressed.If the symmetric density does not annihilate efficiently, residual annihilations during theCMB era can inject enough energy to alter the power spectrum. The WMAP7 constraint onthe annihilation rate for Dirac fermions or complex scalars is given by [5]2ΩiΩi¯Ω2DMf〈σv〉mi< 2.42× 10−27 cm3/sGeV , (4.36)where i = Ψ, Φ and Ωi/ΩDM is the fraction of total DM abundance in species i. This constraintis shown in Fig. 4.6 by the gray line (parameter space to the right is excluded). For theparameters we have chosen, symmetric annihilation is only marginally efficient, and transfer98∆m[GeV]mχ [GeV]TRH = 267 MeV, ΩDM/Ωb = 5.0mΨ00.10.20.30.40.50.60.70.80.911 10 1000102030405060n Ψ/nΦ2.22.12.01.91.81.61.7Figure 4.6: The ratio nΨ/nΦ for the allowed range of mass splittings ∆m = mΦ − mΨand relevant values of the hidden gaugino mass mχ. At each point in the plane theDM abundance is fixed to be ΩDM/Ωb = 5.0. Contours of constant mΨ (in GeV)are also shown. The gray contour shows the CMB constraint for DM annihilationsfrom Ref. [5]. Points to the right of this line are excluded.processes through the light gaugino help achieve efficient annihilation (hence, the large mχregion is excluded). For larger gauge coupling e′ = 0.1, symmetric annihilation is much moreefficient, and CMB constraints are evaded in the entire parameter region in Fig. 4.6.Finally, let us mention that we have not included any baryon washout processes in ourBoltzmann equations. For TRH  mX1 , mq˜, the only such processes that are allowed kine-matically are ΨΦ ↔ 3q¯ and the corresponding crossed diagrams. These transitions require theexchanges of massive intermediate P and X states to occur. These processes are therefore welldescribed by effective operators of the kindL ∼ 1Λ3INDudsΨΦ, (4.37)where the scale ΛIND is defined in Section 4.5 (see Eq. (4.52)) and the order of fermion con-tractions depends on which UV completion is used in Eq. (4.12). We find that the corre-sponding cross sections are safely smaller than the stringent limits found in Ref. [296] providedmX , mP & 300 GeV. For example, we can estimate the cross section for ΨΦ ↔ 3q¯ as〈σv〉 ∼(14pi)3 m4Ψ1Λ6IND=(4× 10−21 GeV−2)(1 TeVΛIND)6( mΨ13 GeV)4. (4.38)99The authors of Ref. [296] found that washout is negligible formDM〈σv〉 . 10−18 GeV−1, (4.39)which is easily satisfied by Eq. (4.38). Even if these processes were important, the X1 decayasymmetry can be adjusted to compensate, as long as the couplings satisfy the conditionsimposed by SUSY breaking discussed in the next section. Thus, our omission of baryon washouteffects is justified.4.4 Supersymmetry BreakingOur model for asymmetric antibaryonic dark matter typically requires light hidden scalars withmasses of a few GeV to obtain an acceptable dark matter abundance. For such masses to betechnically natural, the size of soft supersymmetry breaking felt by the light states should alsobe near the GeV scale. This is much smaller than the minimal scale of supersymmetry breakingfelt in the MSSM sector, which must be close to or above the TeV scale to be consistent withcurrent experimental bounds.Such a hierarchy between visible and hidden sector soft terms can arise if the hidden sectorfeels supersymmetry breaking more weakly than the visible. We examine here the necessaryconditions for this to be the case based on the interactions required for hylogenesis. We alsodiscuss a few specific mechanisms of supersymmetry breaking that can give rise to the requiredspectrum. Motivated by the desire for large moduli masses, which are frequently on the sameorder as the gravitino mass, we pay particular attention to anomaly mediation.4.4.1 Minimal Transmission of Supersymmetry BreakingThe interactions we have put forward in Section 4.2 will transmit supersymmetry breakingfrom the MSSM to the hidden sector. Thus, a minimal requirement for small hidden-sector softterms is that these interactions do not themselves create overly large hidden soft masses. Webegin by studying these effects.The states that we wish to remain light derive from the Y (c)1,2 and H(c) chiral multiplets.These multiplets do not couple directly to the MSSM, but they are connected indirectly bytheir interactions with the X(c) states and U(1)′−U(1)Y gauge kinetic mixing. Thus, the X(c)multiplets and the gauge kinetic mixing will act as mediators to the hidden states.Beginning with the X multiplets, they will feel supersymmetry breaking from their directcouplings to the quarks and the triplet P (c) given in Eq. (4.12). The transmission of super-symmetry breaking can be seen in the renormalization group (RG) equations of the soft scalar100squared masses of X(c)a and P (c) (assuming interactions as in case I):(4pi)2dm2Xadt= 6|λ′a|2(m2Xa +m2Uc +m2P + |Aλ′a |2) + 4|ζa|2(m2Xa + 2m2Y + |Aζa |2)(4pi)2dm2Xcadt= 4|ζ¯a|2(m2Xca + 2m2Y c + |Aζ¯a |2) (4.40)(4pi)2dm2Pdt=∑a2|λ′a|2(m2Xa +m2Uc +m2P )−323g23|M3|2 + . . .(4pi)2dm2P cdt= 4|λ′′|2(m2Sc +m2Dc +m2P c)−323g23|M3|2 + . . .where the Ai are trilinear soft terms corresponding to superpotential operators, and we havedropped subleading hypercharge contributions for P (c). We see that P and P c will typicallyobtain full-strength (TeV) soft masses from their direct coupling to the gluon multiplet, andthese will be passed on to X in the course of RG evolution. Recall as well that λ′1,2 and λ′′must both be reasonably large for hylogenesis to work.Turning next to the Y multiplets, we find for the Y1(4pi)2dm2Y1dt=∑a8|ζa|2(m2Xa +m2Y1 + |Aζa |2)+ 2|γ|2(m2H +m2Y1 +m2Y c2 + |Aγ |2)(4.41)(4pi)2dm2Y c1dt=∑a8|ζ¯a|2(m2Xca +m2Y c1 + |Aζ¯a |2)+ 2|γ¯|2(m2Hc +m2Y c1 +m2Y2 + |Aγ¯ |2).The Y2 multiplets are also charged under the U(1)′ hidden gauge symmetry, which mixeskinetically with hypercharge with strength κ. This leads to additional contributions to therunning [197, 297, 298]. At leading non-trivial order in κ, we have(4pi)2dm2Y2dt= 2|γ¯|2(m2Hc +m2Y c1 +m2Y2 + |Aγ¯ |2)−8e′2(|M ′|2 + κ2|M1|2) + 2e′2SZ′ − 2κ√35g1e′SY(4.42)(4pi)2dm2Y c2dt= 2|γ|2(m2H +m2Y1 +m2Y c2 + |Aγ |2)−8e′2(|M ′|2 + κ2|M1|2)− 2e′2SZ′ + 2κ√35g1e′SY ,where SZ′ = tr(Q′m2), SY = tr(Y m2), g1 =√5/3 gY , and M1 is the hypercharge gaugino(Bino) mass. The RG equations for the soft mass of H (Hc) has the same form as for Y c2 (Y2)but with signs of the last two “S” terms reversed.In addition to these RG contributions, there is D-term mixing between hypercharge andU(1)′. After electroweak symmetry breaking in the MSSM sector, this generates an effective101Fayet-Iliopoulos [197, 297] term in the hidden sector of the formV ⊃ e′22(|H|2 + |Y2|2 − |Hc|2 − |Y c2 |2 − ξFI)2 , (4.43)with ξFI = −κ(gY /2e′)v2 cos(2β), where v ≈ 174 GeV and tanβ is the ratio of MSSM Higgsvevs. This term can be absorbed by shifting the hidden-sector soft masses by m2i → (m2i −Q′iξFI).The RG equations we have presented here are valid down to the scale msoft where the MSSM(and X and P (c) if their supersymmetric masses are near msoft) states should be integrated out.This will generate additional threshold corrections to the hidden-sector soft masses. However,these lack the logarithmic enhancement of the RG contributions and are typically subleading.Thus, putting these pieces together we can make estimates for the minimal natural values ofthe soft terms in the hidden sector. In terms of msoft ∼ M3 ∼ TeV and ∆t = ln(Λ∗/msoft)(where Λ∗ is the scale of supersymmetry-breaking mediation), we findmP (c) & msoft (4.44)mX(c) & |λ′|msoft(√∆t6)(4.45)mY (c)1& |ζ λ′|msoft(√∆t6)2(4.46)mY (c)2,mH(c) & max|γ ζλ′|msoft(√∆t6)3/2, κM1(√∆t6),√κgY2e′v. (4.47)Note that√∆t ≈ 6 for Λ∗ = MPl. We see that the soft masses of the Y2 and H multipletscan be naturally suppressed relative to the MSSM for relatively small couplings. For example,choosing γ = e′ = 0.05, λ′1 = 1, κ ∼ 10−4 and ζ = 0.1 yields soft masses for the Y (c)2 and H(c)multiplets below a few GeV. Therefore the direct coupling of the MSSM to the hidden sectorneed not induce overly large supersymmetry breaking in the hidden sector.4.4.2 Mediation MechanismsWe consider next a few specific mechanisms to mediate supersymmetry breaking to the MSSMand the hidden sector that will produce a mass hierarchy between the two sectors. Motivatedby our desire for large moduli masses, which in supergravity constructions are frequently re-lated closely to the gravitino mass [177, 179], the mechanism we will focus on primarily isanomaly mediation. However, we will also describe a second example using gauge mediationwith mediators charged only under the SM gauge groups.With anomaly mediated supersymmetry breaking (AMSB) [102, 135], the leading-order102gaugino mass in the hidden sector isM ′ = b′e′2(4pi)2m3/2 , (4.48)where b′ = −4 is the one-loop U(1)′ beta function coefficient. A similar expression applies tothe MSSM gaugino soft masses, but with e′2b′ replaced by the corresponding factor. Based onthis comparison, we see that a much lighter hidden gaugino will arise for small values of thehidden gauge coupling [198]. For example, with MSSM gaugino masses in the range of a fewhundred GeV, the hidden gaugino mass will be a few GeV for e′/gSM ∼ 0.1, corresponding toe′ ∼ 0.05.The hidden-sector scalar soft masses will also be parametrically smaller than those of theMSSM if the corresponding Yukawa couplings are smaller as well. The explicit AMSB expres-sions for Y1 and Y c2 arem2Y1 =m23/2(4pi)4γ2(3γ2 − 4e′2) + 6γ2∑aζ2a + 6∑aζ2aλ′2a + 4∑a,bζ2aζ2b (2 + δab) ,(4.49)m2Y c2 =m23/2(4pi)4[γ2(3γ2 − 4e′2) + 2γ2∑aζ2a − 16e′2].We also have m2H = m2Y c2 , while m2Hc = m2Y2 are given by the same expressions with γ → γ¯ andζa → ζ¯a. The latter point also applies to m2Y c1 relative to m2Y1 but with λ′a → 0 as well. Thus,we find GeV-scale soft masses for Y (c)2 and H(c) (and TeV-scale MSSM soft masses) for thefiducial values ζa ∼ 0.1, γ ∼ e′ ∼ 0.05, and m3/2 ∼ 100 TeV. Note that in these expressions wehave neglected kinetic mixing effects which are negligible for κ < 10−3  e′/g1, as we assumehere.The result of Eq. (4.49) shows that the AMSB scalar squared masses can be positive ornegative, depending on the relative sizes of the gauge and Yukawa couplings. This featurecreates a severe problem in the MSSM where minimal AMSB produces tachyonic sleptons.We assume that one of the many proposed solutions to this problem corrects the MSSM softmasses without significantly altering the soft masses in the hidden sector [223, 299]. In contrastto the MSSM, negative scalar soft squared masses need not be a problem in the hidden sectordue to the presence of supersymmetric mass terms for all the multiplets. In particular, thesupersymmetric mass terms we have included in Eqs. (4.3, 4.12) for the vector-like hiddenmultiplets can generally be chosen so that only the H and Hc multiplets develop vevs.Let us mention, however, that supersymmetric mass terms are problematic in AMSB. Inparticular, a fundamental supersymmetric mass term Mi will give rise to a corresponding holo-morphic bilinear soft “bi” term of size bi ∼ Mim3/2. If bi  m2soft, |Mi|2, such a term willdestabilize the scalar potential. To avoid this, we must assume that the supersymmetric mass103terms we have written in Eqs. (4.3, 4.12) are generated in some other way, such as from the vevof a singlet field.11 A full construction of such a remedy lies beyond the scope of the presentchapter, but we expect that it can be achieved in analogy to the many similar constructionsaddressing the corresponding µ−Bµ problem within the MSSM [223, 299] or beyond [198].A second option for the mediation of supersymmetry breaking that preserves the MSSM-hidden mass hierarchy is gauge mediation by messengers charged only under the SM gaugegroups [197, 297]. The soft masses generated in the hidden sector in this case can be deducedfrom the RG equations, up to boundary terms at the messenger scale on the order of κmsoft,which are safely small. Unfortunately, the U(1)′ gaugino mass generated in this scenario onlyappears at very high loop order, and tends to be unacceptably small [197]. This can be resolvedif there are additional gravity-mediated contributions to all the soft masses on the order of afew GeV. The gravitino mass in this case will be on the same order as the hidden states. If itis slightly lighter, it may permit the decay Ψ1 → ψ3/2 + Φ1 (for mΨ1 > m3/2 +mΦ1).4.5 Phenomenology4.5.1 Induced Nucleon DecayDark matter provides a hidden reservoir of antibaryons. Although baryon transfer interactionsare weak enough that visible baryons and hidden antibaryons are kept out of chemical equilib-rium today, they are strong enough to give experimentally detectable signatures of DM-inducednucleon destruction. In these events, a DM particle scatters inelastically on a nucleon N = p, n,producing a DM antiparticle and mesons. For SUSY models, the simplest IND events are thoseinvolving a single kaon,ΨN → Φ∗K , ΦN → Ψ¯K . (4.50)We consider only the lightest DM states Ψ ≡ Ψ1 and Φ ≡ Φ1; the heavier states are notkinematically accessible provided their mass gap is larger than (mN −mK) ≈ 400 MeV. Bothdown-scattering and up-scattering can occur (defined as whether the heavier DM state is in theinitial or final state, respectively), but up-scattering is kinematically forbidden for |mΨ−mΦ| <mN −mK .Assuming the hidden states are not observed, IND events mimic standard nucleon decayevents N → Kν, with an unobserved neutrino ν (or antineutrino ν¯). Nucleon decay searchesby the Super-Kamiokande experiment have placed strong limits on the N lifetime τ for thesemodes [281]:τ(p→ K+ν) > 2.3× 1033 years , τ(n→ K0Sν) > 2.6× 1032 years. (4.51)11 Note that we could have |µP |, |µX |  m3/2 without any problems. In this case, the threshold corrections tothe light soft masses from integrating out the heavy multiplets precisely cancel their leading contributions fromRG, leading to a zero net one-loop AMSB contribution.104Ψ,Φ Φ†, Ψ¯p, n K+,K0Σ0,Λ0Ψ,Φ Φ†, Ψ¯p, n K+,K0P˜XΨ ΦdusFigure 4.7: IND processes at leading order in chiral effective theory (left, center). Graydot shows effective B transfer operator, generated by ˜P , X exchange in our model(right).However, these bounds do not in general apply to IND, due to the different kinematics. ForN → Kν, the K has momentum pK ≈ 340 MeV. IND events are typically more energetic:680 . pK . 1400 MeV for down-scattering, and pK . 680 MeV for up-scattering (if allowed).12The Super-Kamiokande analysis assumes: for p→ K+, K+ is emitted below C̆erenkov thresholdin water, corresponding to pK . 550 MeV; for n→ K0S , the K0S is emitted with 200 < pK < 500MeV. Therefore, IND is largely unconstrained by standard nucleon decay searches. The limitsin Eq. (4.51) only constrain up-scattering IND in a subset of parameter space, whereas down-scattering provides typically the dominant contribution to the total IND rate [279].Next, we compute the IND rates within our supersymmetric model, starting from the baryontransfer superpotential in Eq. (4.12). The vector-like squarks ˜P , ˜P c mix through the SUSY-breaking term Lsoft ⊃ bP ˜P ˜P c to generate the mass eigenstates ˜P1,2, with masses m˜P1,2 . Theleading contribution to IND arises from tree-level ˜P1,2 exchange, shown in Fig. 4.7, giving (intwo-component notation)Leff =1Λ3IND×αβγ(dαRsβR)(uγRΨR)Φ (case I)αβγ(sαRuβR)(dγRΨR)Φ (case II)αβγ(uαRdβR)(sγRΨR)Φ (case III), 1Λ3IND≡∑a=1,22ζ¯∗aZ31V ∗11bPλ′aλ′m2˜P1m2˜P2mXa.(4.52)Here, we have neglected higher derivative terms, and ΛIND characterizes the IND mass scale.The different cases, corresponding to different baryon transfer interactions in Eq. (4.12), leadto different fermion contractions.The effective IND rate for nucleon N = p, n isΓ(N → K) = nΨ(σv)NΨ→KΦ†IND + nΦ(σv)NΦ→KΨ¯IND (4.53)where nΨ,Φ are the local DM number densities and (σv)IND is the IND cross section. The IND12For fixed DM masses, IND is either bichromatic or monochromatic, depending on whether up-scattering isallowed or not; the range in pK corresponds to the allowed mass range 1.4 . mΦ,Ψ . 3.3 GeV. If other hiddenstates Ψa≥2 and Φb≥2 are kinematically accessible, the IND spectrum can have additional spectral lines.105p ® K+HILp ® K+HIILp ® K+HIIILn ® K0HIILn ® K0HILn ® K0HIIILdown-scat.up- & down-scat.down-scat.2.0 2.2 2.4 2.6 2.80.050.100.501.005.00mY@GeVDHΣvLIND@10-39cm3sDFigure 4.8: Total IND cross section (σv)IND = (σv)NΨ→KΦ∗IND + (σv)NΦ→KΨ¯IND over allowedrange for mΨ, with mΦ = (ΩDM/Ωb)mp −mΨ ≈ 5mp −mΨ. The effective baryontransfer mass scale is ΛIND = 1 TeV. Cases I, II, III correspond to different baryontransfer models considered in Eqs. (4.12,4.52).lifetime can be expressed asτ(N → K) = 1Γ(N → K)= (1 + r)(ΩDM/Ωb)mp2ρDM[r(σv)NΨ→KΦ†IND + (σv)NΦ→KΨ¯IND](4.54)with local DM mass density ρDM = mΨnΨ +mΦnΦ, and assuming the local ratio r ≡ nΨ/nΦ isthe same as over cosmological scales. The IND cross section is estimated as(σv)IND ≈m4QCD16piΛ6IND≈ 10−39 cm3/s ×(ΛIND1TeV)−6, (4.55)with QCD scale mQCD ≈ 1 GeV.13 For r ∼ O(1), the IND lifetime isτ(N → K) ≈ 1032 yrs ×((σv)IND10−39 cm/s)−1( ρDM0.3GeV/cm3)−1, (4.56)which is exactly in the potential discovery range of existing nucleon decay searches, providedthe baryon transfer scale ΛIND is set by the weak scale.More quantitatively, we compute (σv)IND using chiral perturbation theory, which providesan effective theory of baryons and mesons (and DM) from the underlying quark-level interactionin Eq. (4.52), following the same methods applied to standard nucleon decay [300] and withadditional input from lattice calculations of hadronic matrix elements [301]. We refer thereader to Ref. [279] for further details. Fig. 4.8 shows numerical results for the total crosssection (σv)IND, over the allowed mass range mΨ for r = 1, for the three types of interactions13The cross section (σv)IND also depends on the DM masses mΨ,Φ, which for the purposes of dimensionalanalysis are comparable to mQCD.106caseIIIcase IIcaseIr=1r=5r=15103210331034103210331034ΤHp®K+L ‰ HTeVLINDL6@yrsDΤHn®K0L‰HTeVLINDL6@yrsDFigure 4.9: Proton and neutron lifetimes for different baryon transfer models (cases I,II, III) considered in Eqs. (4.12) and (4.52). Black line/gray regions show lifetimerange for any r, while blue curves correspond to particular r values.in Eq. (4.52).14 This calculation agrees well with our previous estimate in Eq. (4.55). However,since the typical IND momentum is comparable to the chiral symmetry breaking scale ≈ 1GeV(i.e., where the effective theory breaks down), we regard these results as approximate at best.The different rates for different cases (for fixed ΛIND) satisfy(σv)p→K+IND,III = (σv)n→K0IND,III , (σv)p→K+IND,I = (σv)n→K0IND,II , (σv)p→K+IND,II = (σv)n→K0IND,I (4.57)as a consequence of strong isospin symmetry [279]. The kinks correspond to up-scatting kine-matic thresholds; to the left and right, only down-scatting is allowed, while in the center bothup- and down-scattering occur.More generally, we show in Fig. 4.9 the allowed range for p → K+ and n → K0 INDlifetimes. We consider masses mΨ,Φ consistent with ΩDM/Ωb ≈ 5, for arbitrary r in the range0 < r < ∞, and we take ΛIND = 1 TeV. For case I (II), the allowed region is shown in gray,with a smaller (larger) lifetime for n→ K0 than p→ K+. For case III, shown by the black line,the p and n IND lifetimes are equal, modulo SUSY radiative corrections and isospin-breakingthat we neglect. Specific values for r are shown by blue curves. The solid blue curves show theIND lifetimes for r = 1, corresponding to the calculation in Fig. 4.8. For r 6= 1, the IND ratecan be enhanced if the heavier state is overpopulated (e.g., r > 1 for mΨ > mΦ or r < 1 formΨ < mΦ); on the other hand, the IND rate can be highly suppressed if the heavier state is14The total rate n → K0 includes both K0S and K0L final states, and the individual channels n → K0S andn → K0L are (approximately) half the total rate.107depleted and up-scattering is kinematically forbidden. The dashed (dotted) blue curves showthe IND lifetimes for r = 5 (r = 1/5).We make a number of comments:• The IND lifetimes scale as τ(N → K) ∝ Λ6IND. Taking ΛIND in the range 500GeV−5TeVcorresponds to lifetimes of 1030 − 1036 years. Lifetimes that can be probed in nucleondecay searches correspond to energy scales accessible in colliders (see below).• Both channels p→ K+ and n→ K0 provide complementary information, and which onedominates depends on the underlying heavy states mediating baryon transfer.• The largest IND rates in Fig. 4.9 correspond to Φ-dominated DM (r  1) with mΦ > mΨ,and IND is dominated by ΦN → Ψ¯K down-scattering.Lastly, we note that while the observation of IND would be a smoking gun signal for hylogenesis,a nonobservation does not rule out hylogenesis as a baryogenesis mechanism. The IND rate canbe suppressed if (i) the effective scale ΛIND lies beyond the TeV-scale (due to small couplingsor large mass parameters), (ii) baryon transfer in the early Universe involves heavier quarkflavours, and/or (iii) the heavier DM state is depleted while up-scattering IND is kinematicallyblocked.4.5.2 Precision ProbesAt energies well below the weak scale, the light states in the hidden sector interact with theSM primarily through the gauge kinetic mixing interaction. The most important effect of thismixing is an induced coupling of the Z ′ vector boson to the SM fermions f given by−L ⊃ −κcWQemf f¯γµZ ′µf , (4.58)where Qemf is the electric charge of the fermion and cW is the cosine of the weak mixing angle.Direct searches for a light Z ′ limit κcW . 10−3 for mZ′ . 1GeV [202, 292, 302] with significantimprovements expected in the coming years [303–305].The dark matter states in our scenario consist of a Dirac fermion or a complex scalar with adirect coupling to the Z ′ vector. With the mixing interaction connecting the Z ′ to SM fermions,this state will efficiently mediate spin-independent elastic scattering of the DM states off nuclei.We estimate the cross section per target nucleon to be [232]σSI0 = (5× 10−39 cm2)(2ZA)2( µnGeV)2(e′0.05)2( κ10−4)2(0.3 GeVmZ′)4, (4.59)where µn is the DM-nucleon reduced mass. While this cross section is quite large, the massesof the DM particles in this scenario lie below the region of sensitivity of most current di-rect detection DM searches, including the specific low-threshold analyses by COGENT [306],CDMS [307], XENON10 [9], and XENON100 [308]. For a DM mass below 3 GeV, this cross108section lies slightly below the current best limit from CRESST [11]. Proposed low-thresholdsearches for DM scattering with nuclei or electrons are expected to improve these limits [309].4.5.3 High-Energy CollidersThe new heavy states required for hylogenesis couple directly to the SM and can potentially beprobed in high-energy colliders such as the Tevatron and the LHC. In particular, the effectiveinteractions induced by the vector-like quark multiplets (P, P c) can generate monojets andmodify the kinematic distributions of dijets. We discuss here the approximate limits thatexisting collider data places on the masses of these multiplets, although we defer a detailedanalysis to the future.Monojet signals arising from the effective four-fermion interaction (XucRdcRscR)/M2 presentin the minimal hylogenesis model were considered previously in Ref. [279]. More recent searchesfor monojets by the ATLAS [310] and CMS [311] collaborations limit the corresponding massscale M to lie above 0.5 − 3 TeV. Note, however, that in our supersymmetric formulation thecorresponding four-fermion operator is only generated once supersymmetry-breaking effects areincluded. This weakens the correlation between the monojet signal and the operator responsiblefor hylogenesis, although the limit does typically force the P (c) multiplets to be at least asheavy as a few hundred GeV. On the other hand, this operator is directly related to the INDinteraction. An alternative signal that can arise directly from the superpotential interactionis a “monosquark” q˜∗ ˜X final state, with the squark decaying to a jet and missing energy. Inboth cases, collider limits may be weakened through cascade decays in the hidden sector, whichcould produce additional hidden photons or Higgs bosons that decay to SM states.A second way to probe our supersymmetric UV completion of hylogenesis is through thekinematic distributions of dijets, which can be modified by the direct production of the tripletP scalars (which are R-even). On-shell production of scalar P˜ states via the interactions ofEq. (4.12) can produce a dijet resonance. For heavier masses, the primary effect is describedby the non-minimal Kähler potential operator of Eq. (4.13), which gives rise to a four-quarkcontact operator. Studies of dijet distributions by ATLAS [312] and CMS [313, 314] put limitson the masses of the P (c) scalars of 1-10 TeV, although the specific limits depend on the flavourstructure of the quark coupling in Eq. (4.12) present in the underlying theory.4.6 ConclusionsThrough the mechanism of hylogenesis, the cosmological densities of visible and dark mattermay share a unified origin. Out-of-equilibrium decays during a low-temperature reheating epochgenerate the visible baryon asymmetry, and an equal antibaryon asymmetry among GeV-scalehidden sector states. The hidden antibaryons are weakly coupled to the SM and are the darkmatter in the Universe.We have embedded hylogenesis in a supersymmetric framework. By virtue of its weakcouplings to the SM, SUSY-breaking is sequestered from the hidden sector, thereby stabilizing109its GeV mass scale. The DM consists of two states, a quasi-degenerate scalar-fermion pair ofsuperpartners. We studied in detail one particular realization of supersymmetric hylogenesis,considering several aspects:• We constructed a minimal supersymmetric model for hylogenesis. Hidden sector baryonsare chiral superfields X and Y , with B = 1 and −1/2, respectively. The lightest Y statesare DM, while X decays in the early Universe generate the B asymmetries.• In addition, we introduced a vector-like SU(3)C triplet to mediate B transfer betweenvisible and hidden sectors, and a hidden Z ′ gauge boson (with kinetic mixing) to depleteefficiently the symmetric DM densities.• We showed that hylogenesis can successfully generate the observed B asymmetry dur-ing reheating. We computed the CP asymmetry from X decay and solved the coupledBoltzmann equations describing the cosmological dynamics of hylogenesis.• We studied how SUSY breaking is communicated between the visible and hidden sectorsthrough RG effects. We also examined predictions within an AMSB framework. Whileanomaly mediation explains the late-time reheating epoch from moduli decay, we havenot explicitly addressed the issues of tachyonic slepton masses in the visible sector andthe origin of SUSY mass terms in the hidden sector.• Antibaryonic DM annihilates visible nucleons, causing induced nucleon decay to kaonfinal states, with effective nucleon lifetime in the range 1030 − 1036 years. DM can bediscovered in current nucleon decay searches, and this signal remains unexplored.• Collider searches for monojets and dijet resonances provide the strongest direct constraintson our model, and these signals are correlated with IND. Lifetimes of 1030 − 1036 yearscorrespond to energy scales ΛIND ∼ 0.5− 5 TeV that can be probed at the LHC.• DM direct detection experiments and precision searches for hidden photons constrain theZ ′ kinetic mixing, although our model remains consistent with current bounds.We emphasize that our specific model was constructed to illustrate general features of hy-logenesis, and certainly there are many other model-building possibilities along these lines.Nevertheless, it is clear that supersymmetric hylogenesis provides a technically natural and vi-able scenario for the genesis of matter, explaining the cosmic coincidence between the dark andvisible matter densities and predicting new experimental signatures to be explored in collidersand nucleon decay searches.110Part IIITunnelling and First Order PhaseTransitions111Chapter 5Quantum Tunnelling in Field Theory5.1 IntroductionThe present part of the thesis explores two different applications of first-order phase transitions,that is, transitions where an order parameter changes discontinuously. This order parameter isthe set of expectation values of some scalar fields. At zero temperature these transitions proceedvia quantum tunnelling through a potential barrier. At finite temperature, thermal fluctuationsof fields can overcome a free energy barrier in process called “thermal tunnelling”. As we willsee, these two types of tunnelling are conveniently described using the same formalism.We will be interested in evaluating the rate of tunnelling events either at zero or at fi-nite temperature. This computation is the field theory analogue of Wentzel-Kramers-Brillouin(WKB) method for evaluating barrier penetration probabilities in ordinary quantum mechan-ics. In Sec. 5.2 we evaluate the tunnelling rate at zero temperature. Section 5.3 describes theequivalent computation for a thermodynamic system at finite temperature, which is relevantto phase transitions in the early Universe. In Sec. 5.4 we outline the numerical methods usefulfor the study of such transitions and show examples. These methods are used in the followingchapters.5.2 Tunnelling at Zero TemperatureQuantum tunnelling occurs when a system can minimize its energy by transitioning to a lowerenergy state through a classically forbidden region in configuration space. The problem oftunnelling in field theory was originally addressed in the seminal papers of Coleman andCallan [315, 316]. In general, the energy of the ground state field configuration |φ〉 can bedetermined from the matrix element〈φ| exp(−Hτ/~)|φ〉 =∑nexp(−Enτ/~)〈φ|n〉〈n|φ〉, (5.1)112where τ = it is the Euclidean time and {|n〉} is a complete set of eigenstates of the HamiltonianH. We make ~ explicit in this section to make use of the semi-classical expansion of the pathintegral. The eigenvalues of physical Hamiltonians are bounded from below, that is there existsa lowest energy E0. By considering the large Euclidean time limit τ →∞ we can solve for E0:E0 = −~ limτ→∞1τln〈φ| exp(−Hτ/~)|φ〉. (5.2)For a stable state, E0 is real. Now consider the case shown in the left plot of Fig. 5.1: thevacuum at φ+ is not the lowest energy configuration and therefore cannot be the ground state.If the system is found in this configuration, it must somehow evolve into the true vacuum atφ−. This means that |φ+〉 is not an eigenstate of the Hamiltonian, but its energy E+ can bedefined through analytic continuation [317], such that Eq. (5.2) is still valid. E+ then acquiresa non-zero imaginary part, related to the decay rate of the state |φ+〉 throughΓ = −2~ImE+. (5.3)Our goal is then to compute the leading contribution to this decay rate. The imaginary part ofthe energy arises due to quantum tunnelling from the false vacuum into the true ground state.Perturbation theory can capture only small deviations from a vacuum state, so this computationis done in the semi-classical limit (~→ 0) in the path integral formalism.1 In ordinary quantummechanics this reduces to the WKB approximation for tunnelling amplitudes.First, let us consider the case of a single scalar field φ with the Euclidean actionSE =∫d4x[12(∂φ)2 + U(φ)], (5.4)where U is a potential function with a false vacuum, such as the one shown in the left plot ofFig. 5.1. The required matrix element can be computed using〈φ+| exp(−Hτ/~)|φ+〉 =∫[Dφ] exp(−SE [φ]/~), (5.5)where the path integral on the right hand side is over field configurations such that φ(±τ/2) =φ+. In the semi-classical limit ~ → 0, the dominant contributions to the path integral comefrom the vicinity of stationary points of the action (via the method of steepest descent - see,e.g., Ref. [318]). In our case, there is a minimum of the action that corresponds to the trivialfield configuration that just sits at φ+. It is easy to see that such “motion” does not contributeto the decay rate, since the matrix element is real. Thus, we should look for the leadingimaginary contribution to the path integral. There is another solution to the classical equationof motion (EOM), called the bounce, that is a saddle point of the of the action, which, as we will1The more correct statement of the semi-classical regime is the limit of the dimensionless quantity S/~ →∞,where S is the action.113show, gives the desired imaginary part. The idea is to compute the path integral in Eq. (5.5)by expanding the action around this saddle point solution to quadratic order and performingthe Gaussian integral. The fact that φ¯ is a saddle configuration means that the operatorδ2SEδφ(x)δφ(y)= δ4(x− y)(−∂2 + δ2Uδφ2(φ¯)). (5.6)has a negative eigenvalue, i.e., it is not positive definite. As we will emphasize, this is the keyfeature that generates an imaginary part in E+.The bounce φ¯ satisfies the equation of motionδSEδφ= 0 ⇒ ∂2φ = U ′(φ). (5.7)Note the “wrong” sign of the potential gradient, which is due to the fact that we work inEuclidean space. The boundary conditions arelimτ→±∞φ(τ,x) = φ+. (5.8)Moreover, for the action to remain finite, the spacial derivatives of φ must also vanish at infinity,so, by continuity, we must havelim|x|→∞φ(τ,x) = φ+. (5.9)A theorem due to Coleman, Glaser and Martin [317, 319] states that solutions that are O(4)-symmetric have smaller actions than non-symmetric solutions. Here O(4) is just the Euclideanversion of the Lorentz group, O(1, 3). This means that we can look for solutions that are only afunction of the O(4) invariant variable ρ = (xµxµ)1/2 = (τ2 + x2)1/2. Using the hypersphericalcoordinate expression for the four-dimensional Laplacian (and neglecting the angular part), theEOM reduces tod2φdρ2+ 3ρdφdρ= U ′(φ). (5.10)This has the form of the equation of motion for a particle moving in the inverted potential−U , subject to a friction force with a time-dependent coefficient 3/ρ. This observation is usefulfor understanding the qualitative behaviour of the solutions. Using the definition of ρ, theboundary conditions for an O(4)-symmetric solution collapse into a single requirementlimρ→∞φ(ρ) = φ+. (5.11)Using the particle picture we can discuss the possible solutions. First, we have the trivialsolution where the particle just sits at φ+. As mentioned above, this does not contribute to thedecay rate. We also have the bounce solution, where the particle starts at φ+ at early timesτ → −∞ (corresponding to ρ→∞) rolls off the maximum at φ+, bounces off the potential wallat some finite time, say ρ = 0 and position φ∗ and then returns and comes to rest back at φ+ at114τ → +∞ (also corresponding to ρ → ∞).2 This motion and the inverted potential are shownschematically in the right plot of Fig. 5.1. Therefore to solve for the bounce, we need to findφ(ρ = 0) = φ∗ such that limρ→∞ φ(ρ) = φ+. This makes the solution of Eq. (5.10) non-trivial,since one of the boundary conditions is unknown. Numerical approaches to this problem arediscussed in Sec. 5.4.With the classical solution in hand, we can expand the action in the path integral of Eq. (5.5)around φ¯:SE [φ] = SE [φ¯] +12∫d4xd4yφ(x) δ2SEδφ(x)δφ(y)(φ¯)φ(y) + . . . , (5.12)where φ represents quantum fluctuations around φ¯ and the ellipsis stands for higher orderterms in the expansion of SE . Using this expansion and Eqs. (5.2, 5.3, 5.5) we can see that theresulting decay rate per unit volume has the formΓ/V = A exp(−B/~), (5.13)where B = SE [φ¯] is the Euclidean action evaluated on the classical bounce solution φ¯ andthe factor of volume arises from integrating over possible bounce locations [317]. The pre-exponential factor is obtained from the functional Gaussian integral using the second term inEq. (5.12). It is very difficult to compute because it involves determinants of the differentialoperator of Eq. (5.6) [320, 321]. Therefore, it is usually estimated on dimensional grounds [322].For example, for electroweak scale parameters, we will take[A] = M4 ⇒ A = (100 GeV)4 . (5.14)In any case, the controlling exponential factor depends on B only. Note that the determinantsthat appear in A are precisely where the aforementioned imaginary part of the matrix elemententers the calculation. Here an analogy with a matrix integral is useful:∫ +∞−∞exp(−12xiSijxj)dnx = (2pi)n/2√detS. (5.15)If the quadratic form Sij is not positive definite, it has a negative eigenvalue, which meansthat the right hand side can be imaginary. In this case the integral must be defined throughanalytic continuation. Translating this to field theory, if the operator of Eq. (5.6) has a negativeeigenmode, the Gaussian integral will have an imaginary part.3Above we have outlined the computation of the false vacuum decay rate at zero temperature.2The time at the turn-around is a free parameter - this just means that we can translate the solution in spaceand time to obtain another solution with the same action. This has important consequences for the computationof the path integral.3This depends on how many negative modes there are. It is assumed that there is only one. This has beenshown to be the case in a wide range of physical theories [323].115Uφφ+ φ−−Uφφ+ φ−φ∗Figure 5.1: (Left) A typical potential with a false vacuum at φ+. (Right) The particlemotion interpretation of the bounce equation of motion.We have ignored many technical details, namely the treatment of the zero mode associatedwith the translation of the center of the bounce (which we took to be at ρ = 0) as well as thecomputation of the pre-exponential factor. More careful considerations can be found in theoriginal literature (e.g., Refs. [315, 316]), as well as in pedagogical reviews, Refs. [317, 324–326].Computation of the determinantal prefactors is discussed in Ref. [320, 321]. The O(4)/Lorentzinvariance of bubbles is considered in [327].5.3 Tunnelling at Finite TemperatureThe early Universe at finite temperature can be described as a closed system in approximatelocal thermodynamic equilibrium [40]. Thermodynamic properties of this system are thendescribed by the canonical ensemble with the partition function ZZ = Tre−βH =∑a〈φa|e−βH |φa〉, (5.16)where β = 1/T , H is the Hamiltonian and the trace is over a complete set of states {|φa〉} [328].This is reminiscent of the amplitude in Eq. (5.5). In fact, the partition function can be writtenas a path integral:Z =∫Dφ exp(−∫ β0dτ∫d3xLE), (5.17)whereLE is the Euclidean (i.e., imaginary time) Lagrangian corresponding toH. The boundaryconditions on the fields φ follow from Eq. (5.16): we must have φ(0,x) = φ(β,x), i.e., the116path integral is over field configurations periodic in β.4 This is completely analogous to theT = 0 case, but with a compact Euclidean time interval τ ∈ [0, β] and periodic (anti-periodic)boundary conditions for the bosonic (fermionic) fields in the theory.The various phases of the theory at a given temperature T are determined by the minimaof the free energy FF = −T lnZ = E − TS, (5.18)where E and S are the average energy and entropy of the system, respectively. If a phase oflower free energy exists, the system will undergo a transition to that state. In the presenceof a barrier between the two phases, this will occur through thermal tunnelling, which wasfirst treated by Affleck and Linde [329, 330]. As before, the false vacuum decay rate can beextracted from the the imaginary part of F via Eq. (5.3) (with E+ → F ).5 As in the T = 0case, the leading contribution to ImF can be obtained using the semi-classical approximation.It is typically assumed that finite-T bounce solution does not depend on τ and is therefore onlyO(3) symmetric, that is φ¯ = φ¯(r = |x|) [330, 332]. See Ref. [330] for exceptions to this latterassumption. With these caveats in mind, the finite temperature bounce action can be writtenasSE [φ] = S3/T =1T∫d3x(12(∂φ)2 + U(φ)), (5.19)from which the bounce EOM is found to bed2φdr2+ 2rdφdr= U ′(φ), (5.20)with boundary conditions given by Eq. (5.11) (with the replacement ρ → r). Formally theonly difference between Eq. (5.20) and the T = 0 result, Eq. (5.10), is the coefficient of theφ˙ “friction” term. The resulting tunnelling probability per unit volume again has exponentialform [332]Γ/V = A(T ) exp(−S3[φ¯]/T ), (5.21)where, as beforeA(T ), is usually estimated using dimensional analysis to be of order O(T 4) [332].4The corresponding integral for fermionic fields is anti-periodic in β.5There are important subtleties related to the convexity of F and the equivalent quantity at T = 0, theeffective action. See Ref. [331] for a pedagogical description of these issues.1175.4 Numerical MethodsIn Sections 5.2 and 5.3 we saw that the evaluation of tunnelling rates requires the solution ofthe system of ordinary differential equationsd2φdρ2+ α− 1ρdφdρ= ∇U(φ), (5.22)subject to the boundary conditionslimρ→∞φ(ρ) = φ+,dφdρ(ρ = 0) = 0 (5.23)where α is the dimension (α = 3, 4 for finite and zero temperature tunnelling, respectively).In writing Eqs. (5.22) and (5.23) we have extended the single scalar case to multiple fielddirections, where φ is a vector of all relevant scalar fields and φ+ is the location of the falsevacuum. The second boundary condition (BC) ensures that the solution is finite when ρ→ 0.This BC is very weak, since it does not fully determine the behaviour at ρ = 0. In particular,the bounce point φ(ρ = 0) ≡ φ∗ must be found. When there is only one field, this can be doneusing the shooting method used in one-dimensional eigenvalue problems: one simply solves fora φ∗ such that the first of Eq. (5.23) is satisfied to desired numerical precision, using standardone dimensional root finding algorithms (such as bisection). This becomes intractable withmore than one field, since guessing just the right φ∗ in many dimensions is virtually impossible.Several methods of solving the multi-field bounce equation, Eq. (5.22), have been proposed,including Refs. [333–335]. Below we briefly describe two different classes of algorithms.The algorithm of Ref. [334] (and Ref. [333] on which it is based) exploits the fact theEq. (5.22) simplifies when α = 1. In this limit, there is no damping term in the equation andit takes the form of a Newtonian EOM for a particle moving in an inverted potential −U (seeSec. 5.2). The resulting motion can be obtained by minimizing6 the discretized actionSα[φ(ρ)] = ΩαN−1∑n=0dρ(ndρ)α−1[12(φn+1 − φndρ)2+ U(φn)](5.24)subject to the constraintsU(φ0) = U(φ+), U(φn) ≥ U(φ+) ∀ n < N, (5.25)andφN−1 = φ+. (5.26)In the above equations Ωα = 2piα/2/Γ(α/2), N is the number of lattice sites and dρ is the6Note that this would not work for the true (α > 1) bounce motion since it is a saddle point, not a minimum,of the action.118lattice spacing. The optimization is computationally costly with multiple field directions sinceone often needs N ∼ O(100) per field direction.The above procedure yields the α = 1 solution that serves as an estimate φe of the truebounce point φ∗. In the second part of the algorithm, deformation from α = 1 to a different αis performed in small steps. If the solution does not change much from α to α + dα, then theright hand side of Eq. (5.22) can be linearized and the resulting equation discretized and solvedusing standard numerical linear algebra methods.The alternative method used in Ref. [335] does not rely on discretization. Instead, thestrategy is to make a guess for the tunnelling path φ(s) parametrized by a arbitrary “time”s. For fixed φ(s) this reduces the problem to a single differential equation for s(ρ), which canbe solved using the shooting method. The initial path φ(s) is iteratively refined to find thetrue bounce φ(s(ρ)); the optimal choice of φ(s) minimizes the action (see Appendix A). Morequantitatively, we can rewrite Eq. (5.22) asd2φds2(dsdρ)2+[d2sdρ2+ α− 1ρdsdρ]dφds= ∇U(φ), (5.27)by applying the chain rule. Since the path parametrization is arbitrary, let us demand that ithas unit speed:∣∣∣∣dφds∣∣∣∣= 1. (5.28)We can then use this to show thatd2φds2· dφds= 0. (5.29)Taking the inner products of the EOM with dφ/ds we getd2sdρ2+ α− 1ρdsdρ= dUds. (5.30)This is the equation of motion along the parametric path. It has the same form as the equationof a bounce for a single field variable and can be easily solved using the shooting method.There is another equation contained in the EOM associated with deformations of φ(s), whichis obtained by subtracting Eq. (5.30) from Eq. (5.27):d2φds2(dsdρ)2=(∇− dφds∂∂s)U. (5.31)This equation is used to deform the path φ(s) toward the true bounce. This algorithm hasbeen implemented in the public code CosmoTransitions (CT) [335].In Ch. 6 we make extensive use of CT, so we validated its results by implementing in C++1190.00.20.40.60.81.01.20.0 2.0 4.0 6.0 8.0 10.0 12.0φ¯(ρ)ρφ1 C++φ2 C++φ1 CTφ2 CT-0.50.00.51.01.52.0-0.5 0.0 0.5 1.0 1.5φ 2φ1φ¯ C++φ¯ CT-1.5-1-0.500.511.5U(φ)Figure 5.2: The bounce solution for a two dimensional potential evaluated using two dif-ferent methods. The left plot shows the field profiles as a function of the coordinateρ, while the right plot shows the tunnelling path in the φ1 − φ2 plane.the algorithm of Ref. [334] described above. In Fig. 5.2 we show a sample bounce configuration(computed using these different methods) for a toy two dimensional potentialU(φ) =(φ21 + φ22) (1.8(φ1 − 1)2 + 0.2(φ2 − 1)2 − δ), (5.32)with δ = 0.4. This potential is used as a test case in Ref. [335]. With this value of δ the trueand false minima are at φ = (1.04637, 1.66349) and (0, 0), respectively. The left plot shows theevolution of each field component with the parameter ρ. The right plot shows the trajectoryin field space, which interpolates between the two vacua. We found that the resulting bounceactions differ by less than 10% between the two methods.120Chapter 6Charge and Colour Breaking in theMSSM6.1 IntroductionSupersymmetry predicts a scalar superpartner for every fermion in the Standard Model (SM) [80].While these scalar fields help to protect the scale of electroweak symmetry breaking from largequantum corrections (see Sections 1.3.1 and 2.3), they can also come into conflict with ex-isting experimental bounds. This tension is greatest for the scalar top quarks (stops). Onthe one hand, the stops must be heavy enough to have avoided detection in collider searches.On the other hand, smaller stop masses maximize the quantum protection of the electroweakscale [336, 337].In the minimal supersymmetric extension of the Standard Model (MSSM), there is anadditional constraint on the stops implied by the discovery of a Higgs boson with mass nearmh = 125 GeV [12, 13]. Specifically, the stops must be heavy enough to push the (SM-like)Higgs mass up to the observed value [338, 339]. After electroweak symmetry breaking, the twogauge-eigenstate stops t˜L and t˜R mix to form two mass eigenstates, t˜1 and t˜2 (mt˜1 ≤ mt˜2). Thecorresponding mass-squared matrix in the (t˜L t˜R)T basis is [80]M2t˜ =(m2Q3 +m2t +DL mtXtmtX∗t m2U3 +m2t +DR), (6.1)where Xt = (A∗t−µ cotβ) is the stop mixing parameter, m2Q3,U3 and At are soft supersymmetry-breaking parameters, µ is the Higgsino mass parameter, tanβ = vu/vd is the ratio of the twoHiggs expectation values, and DL,R = (t3−Qs2W )m2Z cos 2β are the D-term contributions. Thestops generate the most important quantum corrections to the mass of the SM-like Higgs stateh0 in the MSSM. Decoupling the heavier Higgs bosons (mA  mZ), the h0 mass at one-loop121order is [340–342]m2h ' m2Z cos2 2β +34pi2m4tv2[ln(M2Sm2t)+ X2tM2S(1− X2t12M2S)], (6.2)where MS = (mQ3mU3)1/2. The first term is the tree-level contribution and is bounded aboveby m2Z . The second term in Eq. (6.2) is the sum of one-loop top and stop contributions. Thiscorrection is essential to raising the mass of the SM-like MSSM Higgs mass to the observedvalue.The contribution of the stops to the h0 mass depends on both the mass eigenvalues andthe mixing angle. Without left-right stop mixing, at least one of the stops must be very heavy,mt˜ & 5 TeV, to obtain mh ' 125 GeV [343]. This leads to a significant tension with thenaturalness of the weak scale [336, 337]. This tension can be reduced by stop mixing, with thelargest effect seen in the vicinity of the maximal mixing scenario of Xt ' ±√6MS [344]. Thisis shown in Fig. 6.1, where the maximal mixing scenario corresponds to the maxima of mh as afunction of Xt/MS . However, such large values of Xt/MS require a large value of At (small µis needed for naturalness [345]) which can induce new vacua in the scalar field space where thestops develop vacuum expectation values (vevs). The lifetime for tunnelling to these charge-and colour-breaking (CCB) vacua must be longer than the age of the Universe to be consistentwith our existence.The existence of CCB stop vacua in the MSSM has been studied extensively [7, 346–354]. Under the assumption of SU(3)C ×SU(2)L×U(1)Y D-flatness, an approximate analyticcondition for the non-existence of a CCB stop vacuum is [348, 349]A2t < 3(m2Q3 +m2U3 +m22) , (6.3)where m22 = m2Hu + |µ|2 and m2Hu is the Hu soft mass squared parameter. Generalizations toless restrictive field configurations [349, 352–354] and studies of the thermal evolution of suchvacua [355–357] have been performed as well. Relaxing the requirement of absolute stabilityof our electroweak vacuum and demanding only that the tunnelling rate to the CCB vacua issufficiently slow provides a weaker bound. The tunnelling rate was computed in Ref. [7], wherethe net requirement for metastability was expressed in terms of the empirical relationA2t + 3µ2 . 7.5(m2Q3 +m2U3) . (6.4)In this chapter we attempt to update and clarify the stability and metastability bounds on theparameters in the stop sector of the MSSM. We expand upon the previous body of work byinvestigating the detailed dependence of the limits on the underlying set of stop parameters.Furthermore, we relate our revised limits to recent Higgs and stop search results at the LHC.The outline of this chapter is as follows. In Sec. 6.2 we specify the ranges of MSSM param-eters and field configurations to be considered. Next, in Sec. 6.3 we investigate the necessary12290.0100.0110.0120.0130.0140.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0mh[GeV]Xt/mQ3123 GeV < mh < 127 GeVtanβ = 10tanβ = 50Figure 6.1: The two loop SM-like MSSM Higgs mass as a function of Xt/mQ3 computedusing FeynHiggs as described in Sec. 6.4. The shaded teal region corresponds to123 GeV < mh < 127 GeV, a range that encompasses the approximate theoreticaluncertainty inmh around the measured valuemh ≈ 125GeV [6]. Here we have takenmA = 1000GeV, mQ3 = mU3 = 750GeV and µ = 250GeV. These parameters arediscussed in Sec. 6.2.conditions on the underlying stop and Higgs parameters for the scalar potential to be stableor safely metastable. We then compare the constraints from metastability to existing limits onthe MSSM stop parameters from the Higgs mass in Sec. 6.4, as well as to direct and indirectstop searches in Sec. 6.5. Finally, we conclude in Sec. 6.6. Some technical details related toour tunnelling calculation are expanded upon in Appendices A and B. This chapter and theaforementioned Appendices are based on work published in Ref. [3] in collaboration with DavidMorrissey.6.2 Parameters and PotentialsIn our study, we consider only variations in the scalar fields derived from the superfields Q3 →(t˜L, b˜L)T , U c3 → t˜∗R, Hu → (H+u ,H0u), and Hd → (H0d , H−d ). To make this multi-dimensionalspace more tractable, we further restrict ourselves to configurations where b˜L = H+u = H−d = 0and the remaining fields (and MSSM parameters) are real-valued. Previous studies of CCBvacua in the stop direction suggest that this condition is not overly restrictive [349].1236.2.1 Scalar PotentialUnder these assumptions, the tree-level scalar potential becomesVtree = V2 + V3 + V4 (6.5)whereV2 = (m2Hu + |µ|2)(H0u)2 + (m2Hd + |µ|2)(H0d)2 − 2bH0uH0d +m2Q3 t˜2L +m2U3 t˜2R (6.6)V3 = 2yt(AtH0u − µH0d) t˜Lt˜R (6.7)V4 = y2t[t˜2Lt˜2R + t˜2L(H0u)2 + t˜2R(H0u)2]+ VD , (6.8)withVD =g′28[(H0u)2 − (H0d)2 +13t˜2L −43t˜2R]2+ g28[−(H0u)2 + (H0d)2 + t˜2L]2 + g236(t˜2L − t˜2R)2 .(6.9)In writing this form, we have implicitly assumed that the stops are aligned (or anti-aligned) inSU(3)C space, so that t˜L and t˜R may be regarded as the magnitudes of these colour vectors(up to a possible sign). It is not hard to show that such an alignment maximizes the likelihoodof forming a CCB minimum.In our analysis of metastability, we use the tree-level potential of Eq. (6.5) with the pa-rameters in it taken to be their DR running values defined at the scale MS . However, we alsocompare our metastability results to a full two-loop calculation of the Higgs boson mass. Whilethis is a mismatch of orders, we do not expect that including higher order corrections willdrastically change our metastability results for two reasons. First and most importantly, theformation of CCB vacua is driven by the trilinear stop coupling At, which is already present inthe tree-level potential. Second, when a CCB vacuum exists, the large stop Yukawa couplingyt ∼ 1 implies that it typically occurs at field values on the order of MS [349]. Thus we do notexpect large logarithmic corrections from higher orders.Including higher-order corrections in the tunnelling analysis is also challenging for a num-ber of technical reasons. Turning on multiple scalar fields, the mass matrices entering theColeman-Weinberg corrections to the effective potential become very complicated and multi-dimensional [356]. These corrections can be absorbed into running couplings by an appropriatefield-dependent choice of the renormalization scale [347]. In doing so, however, the otherwisefield-independent corrections to the vacuum energy (which are not included in the Coleman-Weinberg potential) develop a field dependence. These vacuum energy corrections must beincluded to ensure the net scale independence of the effective potential [358, 359]. Beyond theeffective potential, kinetic corrections (i.e., derivative terms in the effective action) will also berelevant for the non-static tunnelling configurations to be studied. Furthermore, the effectivepotential and the kinetic corrections are both gauge dependent [360, 361]. The gauge depen-124Parameter Values|mQ3 | [300, 3000] GeVm2Q3/m2U3 0.3 , 1, 3Xt [−10, 10]× |mQ3 |µ 150, 300, 500 GeVmA 1000 GeVtanβ 5, 10, 30Table 6.1: MSSM scalar potential parameter scan ranges. The values of other parametersto be considered are described in the text.dence of the effective potential can be shown to cancel on its own for static points [362, 363].However, to ensure the gauge invariance of the non-static tunnelling configuration and thus thedecay rate, kinetic corrections must be included as well [364, 365]. For these various reasons,we defer an investigation of higher-order corrections to metastability to a future work.6.2.2 Parameter RangesWithout loss of generality, we may redefine H0u and H0d such that b and H0u are both positive.This ensures that the unique SM-like vacuum (with t˜L = t˜R = 0) has tanβ = 〈H0u〉/〈H0d〉 > 0,and thus 〈H0d〉 > 0 as well. By demanding that a local SM-like vacuum exists, b, m2Hu , and m2Hdcan be exchanged in favour of v =√〈H0u〉2 + 〈H0u〉2, tanβ, and the pseudoscalar mass mA:b = 12m2A sin(2β) (6.10)m2Hu = −µ2 +m2A cos2β +12m2Z cos(2β) (6.11)m2Hd = −µ2 +m2A sin2β −12m2Z cos(2β) . (6.12)Moving out in the stop directions, we may also redefine t˜L and t˜R such that t˜L is positive.The parameter ranges we investigate are motivated by existing bounds on the MSSM andnaturalness. We typically scan over (m2Q3 , Xt) while holding other potential parameters fixed.We also consider discrete variations in m2U3/m2Q3 , tanβ, µ, and mA. The corresponding rangesare specified in Table 6.1. For the remaining supersymmetry breaking parameters, we choosemf˜ = 2 TeV and Af = 0 for all sfermions other than the stops, as well as M1 = 300 GeV,M2 = 600 GeV, and M3 = 2 TeV. To interface with the Higgs mass calculation, we take theseto be running DR values defined at the input scale MS = (mQ3mU3)1/2. We also use runningDR values of yt, g′, g, and g3 at scale MS when evaluating the potential.6.3 Limits from Vacuum StabilityA necessary condition on the viability of any realization of the MSSM is that the lifetime of theSM-like electroweak vacuum at zero temperature be longer than the age of the Universe. This125will certainly be the case if the electroweak vacuum is a global minimum, and it can also be truein the presence of a deeper CCB minimum provided the tunnelling rate is sufficiently small.More stringent conditions can be derived for specific cosmological histories [356]. While colour-broken phases in the early Universe can have interesting cosmological implications, such as forbaryogenesis [355–357], we focus exclusively on the history-independent T = 0 metastabilitycondition.6.3.1 Existence of a CCB VacuumThe first step in a metastability analysis is to determine whether a CCB minimum exists.Such minima are induced by a competition between the trilinear A and quartic couplings λin the potential, and one generally expects 〈φ〉CCB ∼ A/λ [349]. We use this expectationas a starting point for a numerical minimization of the potential, Eq. (6.5), employing theminimization routine Minuit2 [366]. For every MSSM model, we choose the starting point tobe 〈φi〉CCB = ξiAt, where ξi ∈ [−1, 1] is chosen randomly. The global CCB vacua we find aregenerally unique, up to our restrictions of H0u, t˜L ≥ 0. If no global CCB minimum is found, theminimization is repeated several times with new ξi values. If the global minimum turns out tobe the EW vacuum, the model is considered to be Standard Model-like (SML).6.3.2 Computing the Tunnelling RateWhen a deeper CCB vacuum is found, the decay rate of the SML vacuum is computed using theCallan-Coleman formalism [315, 316], where the path integral is evaluated in the semi-classicalapproximation as described in Ch. 5. The decay rate per unit volume is given byΓ/V = C exp(−B/~) , (6.13)where B = SE [φ¯] is the Euclidean action evaluated on the bounce solution φ¯. The bounceis O(4)-symmetric, depending only on ρ =√t2 + x2, and satisfies the classical equations ofmotion subject to the boundary conditions ∂ρφ¯|ρ=0 = 0 and limρ→∞ φ¯ = φ+, where φ+ is thefalse-vacuum field configuration. The pre-exponential factor C is obtained from fluctuationsaround the classical bounce solution. It is notoriously difficult to compute [320, 321], and istherefore usually estimated on dimensional grounds [322]. We use[C] = M4 ⇒ C = (100 GeV)4 . (6.14)The metastability of the SM-like vacuum then requiresΓ−1 & t0 ⇒ B/~ & 400, (6.15)where t0 = 13.8 Gyr is the age of the Universe. Our choice of scale for C corresponds tothe SM-like vacuum, and provides a reasonable lower bound on C. Larger values of C would126increase the decay rate, implying that the limits we derive are conservative.Finding the bounce φ¯ is straightforward in one field dimension, since the equation of motioncan be solved by the shooting method. This method reduces the problem to a root-finding taskfor the correct boundary conditions and relies on the unique topology of the one-dimensionalfield space. Unfortunately, this strategy becomes intractable with more than one field di-mension. Several methods of solving the multi-field bounce equation of motion have beenproposed [333, 334, 356, 367]. Some of these are described in Sec. 5.4. In the present analysiswe use the public code CosmoTransitions [335].1CosmoTransitions (CT) implements a path deformation method similar to the that sug-gested in Ref. [356]. Once a pair of local minima are specified, CT fixes a one-dimensional pathbetween them in the field space. Along this path, the one-dimensional bounce solution canbe computed using the shooting method. In Appendix A, we show that the action computedfrom the bounce solution for any such fixed path is necessarily greater than or equal to theunconstrained bounce action. The fixed path in field space is then deformed by minimizing aset of perpendicular gradient terms to be closer to the true bounce path through the field space.This procedure is iterated until convergence is reached. We exclude any points where CT failsto converge.This path deformation approach has several advantages over other methods. Here, thebounce equation of motion is solved directly, while many other approaches involve minimizationof a discretized action as part of the procedure. This is numerically costly, since one needs botha fine lattice spacing to evaluate derivatives accurately, and a large ρ domain to accommodatethe boundary condition at infinity. Path deformation involves no discretization or large-scaleminimization. As a result CosmoTransitions is quite fast for our four-field tunnelling problem.We also cross check the CT results in two ways. First, we have compared CT to thediscretized action methods of Refs. [333, 334] for a set of special cases, and we generally findagreement between these approaches as shown in Sec. 5.4. Second, we also compute the bounceaction independently along the optimal path determined by CT, allowing us to estimate thenumerical uncertainty on the bounce. Finally, let us emphasize once more that even if the pathdetermined by CT is not the true tunnelling trajectory, our result in Appendix A implies thatit still provides an upper bound on the bounce action, and thus a lower bound on the tunnellingrate.We note that recently a new program, Vevacious [369], has been released that can alsobe used to study metastability in field theories with many scalar fields. While we do not usethis code, we share some similarities with their approach in that we both employ Minuit forpotential minimization and CT for tunnelling rates. Moreover, as mentioned above, we alsocarried out extensive independent checks of the tunnelling calculation.1We modify the code slightly, replacing an instance of scipy.optimize.fmin by scipy.optimize.fminboundin the class pathDeformation.fullTunneling. This allows CosmoTransitions to better deal with very shallowvacua. The same modification has been used in Ref. [368] (see Footnote 1).1276.3.3 Results and ComparisonWe begin by presenting our limits from metastability alone, without imposing any other con-straints such as the Higgs mass requirement. This allows for a direct comparison with the resultsof Ref. [7]. In Fig. 6.2 we show a scan over Xt and m2Q3 while keeping fixed mA = 1000 GeV,tanβ = 10, µ = 300 GeV, and m2U3/m2Q3 = 1. Every point shown is a model with a global CCBvacuum. The red points have a tunnelling action B/~ < 400, and are therefore unstable oncosmological time scales. The blue points have a metastable SM-like vacuum with B/~ > 400.Also shown in the figure is the analytic bound (green dashed line) of Eq. (6.3), and the empiricalresult (black dotted line) from Ref. [7] given in Eq. (6.4).The shape of the regions shown in Fig. 6.2 can be understood simply. As expected, theexistence of a CCB vacuum requires a large value of At/MS . The cutoff at the upper-leftdiagonal edge corresponds to the absence of a CCB vacuum. Above and to the left of thisboundary, the SML minimum is a global one and the EW vacuum can be absolutely stable.There is also a lack of points below a lower-right diagonal edge. Here, one of the physical stopsbecomes tachyonic, and the SML vacuum disappears altogether. At low values of A2t , we seethat the CCB region is squeezed between the SML region (on the left) and the tachyonic stopregion (on the right), giving rise to the cutoff seen in the lower left corner.It is apparent from Fig. 6.2 that we find much more restrictive metastability bounds on theMSSM than the empirical relation of Eq. (6.4) from Ref. [7]. We also see that the analytic boundof Eq. (6.3) tends to underestimate the existence of CCB vacua, and that it accidentally linesup fairly well with the lower boundary of metastability. It is not clear why our results shouldbe so much more restrictive than those found in Ref. [7], but we are confident that the pathdeformation method of CT (and our several cross-checks) gives a robust upper bound on thebounce action. We find qualitatively similar results for the other parameters ranges describedin Table 6.1. The quantitative results for these ranges will be presented in more detail belowin the context of the Higgs mass.6.4 Implications for the MSSM Higgs BosonAs discussed in the Introduction, there is a significant tension in the MSSM between obtainingthe observed Higgs boson mass and keeping the stops relatively light. This tension is reducedwhen the stops are strongly mixed. To obtain such mixing, large values of Xt are needed. Wehave just seen that large values of Xt can lead to dangerous CCB minima. In this section wecompare the relative conditions imposed by each of these requirements.To calculate the physical h0 Higgs boson mass, we use FeynHiggs 2.9.5 [370]. We also usethis program together with SuSpect 2.43 [371] to compute the mass spectrum of the MSSMsuperpartners. As inputs, we take mpolet = 173.1 GeV and αs(mZ) = 0.118 [58]. Our results areexhibited in terms of variations on the fiducial MSSM parameters tanβ = 10, µ = 300 GeV,mA = 1000 GeV, and m2U3 = m2Q3 . The other MSSM parameters are taken as in Section 6.2.2.In Fig. 6.3 we show points in the Xt-MS plane (where MS = (mQ3mU3)1/2) that produce128105106107108105 106 107 108 1093(m2 Q3+m2 U3)[GeV2]A2t + 3µ2 [GeV2]UnstableMetastableAnalytic BoundEmpirical BoundFigure 6.2: Limits from metastability and the existence of a local SM-like (SML) vacuumalone for tanβ = 10, µ = 300 GeV, mA = 1000 GeV, and m2U3 = m2Q3 . All pointsshown have a global CCB minimum and a local SML minimum. The red pointsare dangerously unstable, while the blue points are consistent with metastability.The green dashed line is the analytic bound of Eq. (6.3) and the black dotted linecorresponds to Eq. (6.4), the empirical bound from Ref. [7]. The values of the otherMSSM parameters used here are described in the text.a Higgs mass in the range 123 GeV < mh < 127 GeV. All other parameters are set totheir fiducial values described above. The pink (blue) region are models with a global SML(CCB) vacuum. The red points are excluded by metastability. The dashed lines show theapproximate CCB condition of Eq. (6.3), the empirical limit of Eq. (6.4), and our own attemptat an empirical limit on metastability to be discussed below. The requirement of metastabilitycuts off a significant portion of the allowed range at very large |Xt|. Also shown are contoursof constant mt˜1 , the lightest stop mass (grey dot-dashed lines).In Fig. 6.4 we show the additional dependence of the Higgs mass and the metastabilitybounds on other relevant MSSM parameters. All parameters are set to their fiducial valuesexcept for those we vary one at a time. In the top row we show results for tanβ = 5 (30) onthe left (right). Reducing tanβ decreases the tree-level contribution to the MSSM Higgs mass,and so larger values of MS are needed to raise mh to the observed range. These larger valuesalso lead to shallower CCB minima and lower tunnelling rates. Larger values of tanβ do notappear to differ much from tanβ = 10.In the middle row of Fig. 6.4 we show results for µ = 150 (500) , GeV on the left (right).We do not see a large amount of variation in the exclusions from metastability, which is notsurprising given that generally have Xt ' At  µ. Setting µ = −300 GeV also produces verysimilar results.1290.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]50010001500SMLCCBS < 400Analytic BoundOur Empirical BoundEmpirical BoundFigure 6.3: Metastability bounds relative to the MSSM Higgs mass. The coloured bandscontain models for which 123 GeV < mh < 127 GeV. Pink models have anabsolutely stable SML vacuum, blue points have a global CCB minimum, whilered points are unstable on cosmological time scales. The green dashed line is theanalytic bound of Eq. (6.3) and the black dotted line is Eq. (6.4). The orange dashedline is an approximate empirical bound discussed in Appendix B. The grey dot-dashed contours are lines of constant lightest stop mass (in GeV). MSSM parametersused here are described in the text.In the bottom row of Fig. 6.4 we show the same metastability limits for m2U3/m2Q3 = 0.3 (3.0)on the left (right). For these unequal values, there is a tension between minimizing the quadraticterms in the potential and reducing the quartic terms through SU(3)C D-flatness. Unequalsquark vevs also tend to reduce the effective trilinear term. Together, these effects reduce themetastability constraint somewhat, but do not eliminate it.In summary, the constraint imposed by CCB metastability rules out a significant portion ofthe MSSM stop parameter space that can produce a Higgs mass near the observed value. Thelimits are strongest on the outer branches at large |Xt|. Varying other MSSM parameters withinthe restricted ranges we have considered does not drastically alter this result. By comparison,the empirical bound from Ref. [7] does not rule out any of the stop parameter space consistentwith the Higgs mass.As a synthesis of these results, we have attempted to obtain an improved empirical boundon stop-induced metastability. We find the approximate limitA2t .(3.4 + 0.5 |1− r|1 + r)m2T + 60m22 , (6.16)where m2T = (m2Q3 +m2U3), m22 = (m2Hu + µ2), and r = m2U3/m2Q3 . Let us emphasize that this1300.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]tanβ = 5SMLCCBS < 400Analytic BoundOur Empirical BoundEmpirical Bound0.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]tanβ = 30SMLCCBS < 400Analytic BoundOur Empirical BoundEmpirical Bound0.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]µ = 150 GeVSMLCCBS < 400Analytic BoundOur Empirical BoundEmpirical Bound0.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]µ = 500 GeVSMLCCBS < 400Analytic BoundOur Empirical BoundEmpirical Bound0.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]m2U3/m2Q3 = 0.3SMLCCBS < 400Analytic BoundOur Empirical BoundEmpirical Bound0.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]m2U3/m2Q3 = 3.0SMLCCBS < 400Analytic BoundOur Empirical BoundEmpirical BoundFigure 6.4: Metastability with the correct Higgs mass, 123 < mh < 127 GeV. The la-belling is the same as in Fig. 6.3, and the relevant MSSM parameter parametersare varied one at a time as summarized in Table 6.1.limit is very approximate and only applies to smaller values of µ, larger values of mA, moderatetanβ, and r not too different from unity. Details on the derivation of this bound are given in131Appendix B.6.5 Comparison to Other Stop ConstraintsThe metastability conditions we find exclude parameter regions with large stop mixing. Thismixing can produce one relatively light stop mass eigenstate as well as a significant mass splittingbetween the members of the ˜Q3 sfermion doublet. These features are constrained indirectly byelectroweak and flavour measurements, as well as by direct searches for a light stop. In thissection we compare these additional limits to the bounds from metastability.6.5.1 Precision Electroweak and FlavourThe most important electroweak constraint on light stops comes from ∆ρ, corresponding tothe shift in the W mass relative to the Z. In the context of highly mixed stops motivated bythe Higgs mass, this effect has been studied in Refs. [372, 373]. We have computed the shift∆ρ due to stops and sbottoms using SuSpect 2.43 [371], which applies the one-loop resultscontained in Refs. [374, 375]. With a Higgs mass of mh ' 125 GeV, the preferred range is∆ρ = (4.2± 2.7)× 10−4 [372].Supersymmetry can also contribute to flavour-mixing. Assuming only super-CKM squarkmixing (or even minimal flavour violation [376]), the most constraining flavour observable isfrequently the branching ratio BR(B → Xsγ). It receives contributions in the MSSM fromstop-chargino and top-H+ loops. These contributions tend to cancel each other such that thecancellation would be exact in the supersymmetric limit [377]. With supersymmetry breaking,the result depends on the stop masses and mixings, tanβ, µ, and the pseudoscalar mass mA.Constraints on light stops from BR(B → Xsγ) were considered recently in Refs. [373, 378]. TheSM prediction is BR(B → Xsγ) = (3.15 ± 0.23) × 10−4 [379], while a recent Heavy FlavourAveraging Group compilation of experimental results finds BR(B → Xsγ) = (3.55 ± 0.24 ±0.09) × 10−4 [380]. We have investigated the limit from BR(B → Xsγ) and other flavourobservables using SuperIso 3.3 [381] assuming only super-CKM flavour mixing.In Fig. 6.5, we show the exclusions from flavour and electroweak bounds for model pointswith 123 GeV < mh < 127 GeV for tanβ = 10, and mA = 1000 GeV, µ = 300 GeV, andm2Q3 = m2U3 in the Xt−mQ3 plane. We impose the generous 2σ constraints ∆ρ ∈ [−1.2, 9.4]×10−4 and BR(B → Xsγ) ∈ [2.86, 4.24] × 10−4 and show them together with the metastabilityconstraint from the previous Section. The green points show the regions excluded by ∆ρ whilethe orange points show those excluded by BR(B → Xsγ).The exclusion due to ∆ρ can be understood in terms of the large stop mixing inducedby Xt, which generates a significant splitting between the mass eigenstates derived from the˜Q3 = (t˜L, b˜L)T SU(2)L doublet. This constraint depends primarily on the stop parameters, andis mostly insensitive to variations in µ, mA, and tanβ. While this bound overlaps significantlywith the limit from metastability, there are regions where only one of the two constraints applies.The limits from ∆ρ are also weaker for m2Q3 > m2U3 .1320.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]50010001500SMLCCBS < 400∆ρ excludedBR(B → Xsγ) excludedFigure 6.5: Points in the Xt−mQ3 plane with 123 GeV < mh < 127 GeV as well asexclusions from metastability (red points) from precision electroweak ∆ρ (greenpoints) and flavour BR(B → Xsγ) (orange points). The MSSM parameters usedare the same as in Fig. 6.3.Limits from BR(B → Xsγ) are less significant for this set of fiducial parameters with amoderate value of tanβ. However, this branching fraction depends significantly on µ, mA, andtanβ, and the limit can be much stronger or much weaker depending on the specific valuesof these parameters. We do not attempt to delineate the acceptable parameter regions, butwe do note that the constraint from metastability can rule out an independent region of theparameter space.6.5.2 Direct Stop SearchesStops have been searched for at the LHC in a diverse range of final states, and these studies ruleout stop masses up to 200-600 GeV, depending on how the stop decays (see, e.g., Refs. [382–385]). While the large stop mixing that occurs in the region excluded by metastability consid-erations can produce lighter stops, the stop masses in this dangerously metastable region arenot necessarily light, as can be seen in Fig. 6.3. Thus, metastability excludes parameter rangesbeyond existing direct searches.Note as well that metastability does not place a lower bound on the mass of the lighteststop. For example, a very light state can be obtained for m2U3  m2Q3 and Xt = 0. This scenariois not constrained by metastability, and can generate a SM-like Higgs boson mass consistentwith observations for sufficiently large values of m2Q3 [386].22 A lower limit on the light stop mass in this scenario can be obtained from its effect on Higgs production133Our results also have implications for future stop searches and measurements. Should apair of stops be discovered, a variety of methods can be used to determine the underlyingparameters in the stop mass matrix through precision measurements at the LHC [388–391] or afuture e+e− collider [391–393]. If these stop parameters turn out to lie within the dangerouslyunstable region, corresponding to larger values of |Xt|, we can conclude that new physics beyondthe MSSM must be present.6.5.3 Stop Bound StatesAn additional phenomenon that can potentially occur in the MSSM when At is very large is theformation of a t˜Lt˜∗R bound state through the exchange of light Higgs bosons [394, 395]. Sucha state could have the quantum numbers of a Higgs field and mix with the MSSM Higgs fieldsto participate in electroweak symmetry breaking [394–397]. If this occurs, our results on themetastability of the MSSM may no longer apply. Calculating the critical value of At for when abound state arises is very challenging, but under a set of reasonable approximations Ref. [396]finds that it requires At/MS & 15. While this lies beyond the region considered in the presentchapter, it is conceivable that a full numerical analysis would yield a lower critical value forthis ratio.6.6 ConclusionsIn this chapter we have investigated the limits on the stop parameter space imposed by vacuumstability considerations. A SM-like Higgs boson with a mass of ∼ 125 GeV in the MSSM pointsto a particular region of the parameter space if naturalness of the EW scale is desired. In thisregime, the two stop gauge eigenstates are highly mixed, and this can induce the appearanceof charge- and colour-breaking minima in the scalar potential. Quantum tunnelling to thesevacua can destabilize the electroweak ground-state.We have studied the conditions under which stop mixing can induce CCB vacua and wehave computed the corresponding tunnelling rates. We find that metastability provides animportant constraint on highly-mixed stops. We have also considered constraints from flavourand precision electroweak observables and direct stop searches, which are sensitive to a similarregion of the MSSM parameter space. Metastability provides new and complimentary limits,with a different dependence on the underlying parameter values.The metastability limits we have derived provide a necessary condition on the MSSM. Theyapply for both standard and non-standard cosmological histories. Let us emphasize, however,that the MSSM parameter points that we have found to be consistent with stop-induced CCBlimits may still be ruled out by more general stability considerations, such as configurationswith more non-zero scalar fields. Fortunately, our own SML vacuum appears to be at leastsafely metastable.and decay rates [387].134Chapter 7The Electroweak Phase Transition inthe Inert Doublet Model7.1 Electroweak Baryogenesis in the Standard Model andBeyondThe Universe appears to be made entirely of matter, with all observed antimatter consistentwith secondary production from, e.g., cosmic ray collisions. It was argued in Sec. 1.3.2 thata dynamical explanation of this baryon asymmetry of the Universe (BAU) necessarily invokesnew physics. We also mentioned that SM alone can, in principle, generate a small asymme-try. This occurs through the mechanism of electroweak baryogenesis (EWBG) which operatesduring the electroweak phase transition (EWPT). This transition occurs when the Higgs fieldspontaneously acquires an expectation value, thereby breaking the electroweak symmetry downto electromagnetism. This mechanism is illustrated in Fig. 7.1, which conveniently captureshow each of Sakharov’s requirements from Sec. 1.3.2 is satisfied. If the EWPT is first order, itproceeds via the nucleation and subsequent expansion of bubbles of the broken v 6= 0 phase.This constitutes a departure from thermal equilibrium. In the symmetric phase baryon (B)and lepton numbers (L) are violated by non-perturbative SU(2)L processes [326, 398]9qL ↔ 3¯`L (7.1)called sphaleron transitions. Note that in these processes ∆B = ∆L, so (B − L) number isconserved, while (B + L) is violated. The resulting rate of baryon density production can beestimated as [399]dnBdt= −3Γsph(T )T∆F, (7.2)where Γsph is the sphaleron rate and ∆F is the difference in free energy between stateswith ∆B = +1. The sphaleron rate Γsph can be found using semi-classical methods sim-135ilar to those presented in Ch. 5 and depends exponentially on the energy scale Esph(T ) ∼(8pimW /g2)(v(T )/v0) as Γsph ∝ exp(−Esph/T ), where v(T ) is the Higgs vev at temperature Tand v0 = v(0) = 246GeV [56]. Note that in the symmetric phase v = 0 and Esph ≈ 0 andthere is no exponential suppression for the sphaleron rate. In this phase the sphaleron ratecan be estimated from scaling arguments to be Γsph/V ∼ α5WT 4 [400] and the coefficient ofproportionality can be extracted from lattice simulations [401].Ordinarily, increasing the B charge costs energy so ∆F > 0, meaning that the sphalerontransitions tend to wash out any existing baryon number. However, these processes can be biasedto produce slightly more baryons than anti-baryons if there is another asymmetry generatedby C and CP violating interactions of the plasma particles with the Higgs field [56, 402]. Notethat these charge asymmetries can be only created in the vicinity of the bubble wall, where thesphaleron rate is suppressed as discussed below. Thus this asymmetry must diffuse farther intothe symmetric phase before it can be “re-processed” by the sphalerons. This scattering anddiffusion process is described by a complicated network of quantum transport equations [56].In the last step, the baryon asymmetry is captured by the expanding bubble.Electroweak baryogenesis in the SM fails for two reasons. The first problem is that theamount of CP violation available in the SM is insufficient. The second issue is related to thedynamics of the phase transition itself. If the sphaleron transitions are active near the boundaryand inside the bubble of true phase, any baryon number that is captured by the bubble willbe washed out. The amount of wash out depends on the sphaleron rate in a given phase;integrating Eq. 7.2 over the timescales of the EWPT, ∆tEW ∼ H−1, one finds the wash outfactor [361]nB(∆tEW)nB(0)∼ exp (−Esph(Tc)/Tc) , (7.3)where Esph is defined above and we assumed that the transition occurs at around a temperatureTc (defined below). Requiring that the argument of the exponential is not much bigger than 1(such that wash out is not so severe) requires [56, 361]vcTc& 1, (7.4)where vc is the Higgs expectation value at T = Tc. The quantity vc/Tc is often referred to as the“strength” of the transition. The phase transition occurs after the symmetric and EW-breakingphases become degenerate, so Tc is defined as the temperature at which this happens. In Fig. 7.2we show the evolution of the effective potential with temperature for the SM. Above Tc thesymmetric phase is energetically favourable; the two phases become degenerate at T = Tc.As the Universe cools further, it transitions into the symmetry breaking phase via thermaltunnelling described in Sec. 5.3. The computation of the effective potential is described indetail in Sec. 7.3. For the SM, we find that in this (perturbative) approximation vc/Tc ∼ 0.17,so the wash out rate is too great – the transition is not strong enough. This inadequacy of the136v 6= 0 v = 0ψRψL ✟✟✟CPBnBSU(2)L sphaleronsbubble wallFigure 7.1: Schematic representation of electroweak baryogenesis.EWPT in the SM is confirmed by non-perturbative lattice simulations, which suggest that thetransition is not even first-order, but rather a smooth cross-over for mh & 75GeV [56]. In thefollowing sections we will focus on improving the strength of the transition by considering aparticular extension of the SM.Among the many proposed mechanisms of baryogenesis, EWBG is particularly attractive.Because EWBG is driven by the Higgs field, any new particles that seek to resolve the afore-mentioned problems in the SM implementation of EWBG must couple strongly to the Higgs;moreover, they must also be abundant in the primordial plasma at the time of the EW phasetransition, so they cannot be too heavy.1 The most salient feature of EWBG is therefore arequirement of new physics close to the EW scale. This aspect of EWBG makes it very predic-tive and falsifiable. Many simple models of EWBG have already been excluded either by lackof direct discovery of the new light states, or by limits from electric dipole moment searches,which are extremely sensitive to CP violation. However, many well-motivated possibilities stillremain. Below we consider an extension of the SM known as the inert doublet model, which inaddition to improving the phase transition strength, can also account for the dark matter relicdensity.7.2 The Inert Higgs DoubletThe simplest extension of the Standard Model (SM) that includes two SU(2)L Higgs doubletsis known as the Inert Higgs Doublet Model (IDM). In the IDM, the extra doublet has nocoupling to SM fermions and is odd under a postulated new Z2 discrete symmetry, whereas allSM fields are Z2-even. Such symmetry makes the lightest Z2-odd particle (LOP) from the extradoublet stable and, thus, a potential weakly interacting massive particle (WIMP) dark mattercandidate. The symmetry also eliminates numerous terms in the interaction Lagrangian of themodel containing an odd number of extra “inert” scalars.The IDM was introduced originally as one possible, generic scenario for electroweak symme-1This requirement can be relaxed - see the discussion at the end of Sec. 7.5.137-1000-5000500100015002000250030003500400045000 5 10 15 20 25 30 35V(h,T)[GeV4]h [GeV]T > TcT = TcT < TcFigure 7.2: Evolution of the temperature-dependent effective potential (free energy) inthe Standard Model around the critical temperature Tc.try breaking (EWSB) [403]. Only subsequently was it realized that the IDM naturally featuresa WIMP DM candidate [404, 405], possibly providing a thermal relic density compatible withthe inferred universal DM abundance. Numerous studies have subsequently investigated theDM and collider phenomenology of the model (see, e.g., Ref. [406–412]).An additional early motivation to consider the IDM as an appealing augmentation of the SMscalar structure was to allow for a relatively heavy SM-like Higgs while remaining compatiblewith constraints from electroweak precision observables, and without severe fine tuning [405,412]. Although this motivation has somewhat faded after the discovery of a SM-like Higgsboson at the LHC with a mass of ∼ 125 GeV [12, 13], this important discovery decreases thenumber of free parameters in the theory by one, and places interesting and stringent constraintson the IDM phenomenology [413].In the present study we are concerned with the nature of the electroweak phase transi-tion (EWPT) in the IDM, and, specifically, with determining which physical parameters drivethe strength of the phase transition, making it more or less strongly first-order, or second-order. This question is intimately related with the possibility to produce the observed baryon-antibaryon asymmetry in the Universe at the electroweak phase transition as discussed in theprevious section. A strongly first-order phase transition (in a quantitative sense we shall makeclear below) is a necessary ingredient to (i) achieve the necessary out-of-equilibrium conditions,occurring on the boundary of broken and unbroken electroweak phase, and to (ii) shield abaryon asymmetry captured in the broken electroweak phase region from sphaleron wash-out.While necessary, a strongly first-order phase transition is not a sufficient condition. The CPviolating sources of the SM are known to be insufficient to generate the necessary asymmetry in138the number density of baryons compared to antibaryons during the electroweak phase transition.The unbroken Z2 symmetry in the IDM precludes any new source of CP violation, and thusthis model per se cannot accommodate successful electroweak baryogenesis (EWBG). However,the IDM might be in effect a good approximation at low energy of a broader constructionthat includes such additional CP violating sources at higher energies. Several suggestions ofplausible effective higher-dimensional operators have been made in the literature [414, 415]. Wewill not discuss this aspect any more, as it falls outside the scope of this study.The nature and strength of the electroweak phase transition in the IDM has been subjectof several studies, with increasingly refined treatment of the effective potential [416–420]. Forexample, Ref. [416] utilized only the high-temperature form of the effective potential with-out including the zero-temperature Coleman-Weinberg terms. These were then shown to bequantitatively important for the phase transition strength in Ref. [418], where the full one-loopeffective potential was used. Alternative SU(2)L representations of the inert scalar were consid-ered in Ref. [419], where it was argued that in general, higher representations are less successfulin satisfying experimental and theoretical constraints, thereby further motivating the study ofthe doublet case.With the exception of Ref. [420], the primary focus has been on the Higgs funnel regime(described in more detail in Sec. 7.4). Indeed, we will confirm the findings of Refs. [417, 418]that this is the only region of parameter space that can successfully saturate the DM abundanceand provide a strong-enough first-order EWPT. In Ref. [420] it was emphasized that the IDMcan be useful for the EWPT even if the LOP provides only a subleading component of DM.In this work we go beyond previous studies by utilizing a state-of-the-art treatment of thefinite-temperature and zero-temperature effective potential including renormalization group,daisy resummation improvements and one-loop model parameter determination. As we discussin great detail, strongly first-order EWPT in the IDM requires sizable quartic couplings thatenhance quantum corrections to masses. This is important in the context of DM phenomenologysince DM particle production in the early Universe often relies on resonance and thresholdeffects [413]. In addition, we also ensure that the phase transition completes by evaluatingbubble nucleation rates.Unlike previous studies which primarily utilized large numerical scans of the parameterspace, here we take an orthogonal approach: we restrict our attention to a few, key benchmarkmodels, motivated by the requirement of having a viable dark matter particle candidate andrepresenting different features in the DM phenomenology. Based on these benchmarks, wethen discuss how the EWPT depends on the physical model parameters. We identify the keyphysical inputs that drive the phase transition to the interesting regime where it is stronglyenough first-order to accommodate successful electroweak baryogenesis. We will see that in allbut one case the demand for a strongly first-order EWPT is in tension with either the relicabundance requirement or with experimental probes.Our central finding is that the main driver of the strength of the phase transition is the139mass difference between the lightest inert scalar and the heavier scalars. Thus, we extend theresults of Refs. [416, 418] to other regions of IDM parameter space. For large enough masssplittings, but for light enough heavy scalars, we find a phase transition strength (as measuredby the ratio vc/Tc, as we discuss in detail below) which increases with the mass splitting.The remainder of this chapter is organized as follows. In Sec. 7.3 we give a brief introductionto the IDM, thereby clarifying our conventions, discuss quantum and finite-temperature correc-tions to the effective potential, and outline the computation of the phase transition strength.The essential features of DM phenomenology are reviewed in Sec. 7.4. In Sec. 7.5 we studythe electroweak phase transition in several benchmark models motivated by the various DMscenarios available in the IDM. We conclude in Sec. 7.6. This chapter is based on unpublishedwork in collaboration with Stefano Profumo and Tim Stefaniak [4].7.3 Phase Transitions in the Inert Doublet Model7.3.1 IDM at Tree-LevelThe IDM is a particular realization of the general type I Two Higgs Doublet Model (2HDM)(see, e.g., Ref. [421] for a review) which features an additional Z2 symmetry. The SM doubletH is even under Z2, while the new (inert) doublet Φ is odd. If we take Φ to have hypercharge+1/2, the most general renormalizable potential consistent with these symmetries is then givenby [413]:V0 = µ21|H|2 + µ22|Φ|2 + λ1|H|4 + λ2|Φ|4 + λ3|H|2|Φ|2 + λ4|H†Φ|2 +λ52[(H†Φ)2 + h.c.]. (7.5)Conventionally, within CP-conserving Higgs sectors, the physical states are decomposedinto CP-even and CP-odd scalars. One should keep in mind, however, that in the IDM thereis no observable that can actually distinguish between the CP-even or CP-odd character of theinert Higgs bosons. In the absence of a vacuum expectation value (vev) for Φ, the doubletsdecompose asH =(G+1√2(v + h+ iG0)), Φ =(H+1√2 (H + iA)). (7.6)Below we will consider the thermal evolution of the effective scalar potential in the early Uni-verse. In general, spontaneous breaking of Z2 can occur, in which case we must also include avev for the neutral component of Φ which we will indicate with φ.The lightest Z2-odd particle is stable, and potentially provides a viable particle dark mattercandidate. The Z2 symmetry also forbids Yukawa couplings of Φ to SM fermions (assumed tobe even under Z2), which eliminates tree-level flavor-changing neutral currents. Either H orA can be the LOP, and since gauge interactions with SM states do not distinguish betweenthe two, they are effectively equivalent from the standpoint of phenomenology. Below we will140indicate the LOP as H, but all statements made with respect to DM phenomenology and theelectroweak phase transition remain true after the replacements H → A and λL → λS (thelatter determines the coupling of the LOP to the SM Higgs) [417].In the electroweak vacuum, the tree-level masses of the new states are given bym2h = µ21 + 3λ1v2,m2H = µ22 + λLv2,m2A = µ22 + λSv2,m2H± = µ22 +12λ3v2, (7.7)where λL = (λ3 + λ4 + λ5)/2 and λS = (λ3 + λ4 − λ5)/2. In what follows we employ theone-loop effective potential defined in the next section to fix µ21 and λ1 from the physical valuesv = 246.22 GeV and mh ≈ 125 GeV. The remaining parameters of the model are specifiedusing the three physical masses mH , mA and mH± , along with λL and λ2. The masses arerelated to potential parameters using the one-loop relations from Ref. [413], while λL and λ2are given at scale MZ .7.3.2 Finite-Temperature CorrectionsThe effective potential at finite temperature T can be written asVeff = V0 + V1 + VT , (7.8)where V0, V1 and VT are tree-level, one-loop temperature-independent and -dependent pieces,respectively. The tree-level potential V0 has been given in Eq. (7.5). The temperature-independent one-loop correction has the Coleman-Weinberg form [317, 331, 332]:V1 =∑ini64pi2m4i (v, φ)(ln m2i (v, φ)Q2− Ci). (7.9)The sum is over all particle species coupling to the doublets; ni is the number of degreesof freedom (positive for bosons and negative for fermions), Ci are renormalization-scheme-dependent constants (Ci = 1/2 for transverse gauge bosons and 3/2 for everything else in theMS scheme); m2i (v, φ) is the field-dependent squared mass for each species. In writing theabove, we have implicitly absorbed the counterterms into V1; the temperature-dependent partis ultraviolet finite.The field dependent masses in the IDM for the SM vector bosons and fermions are, respec-tively,m2W =14g2(v2 + φ2), m2Z =14(g2 + g′2)(v2 + φ2), m2γ = 0 (7.10)141andm2f =12y2fv2, (7.11)with the corresponding bosonic degrees of freedom ni = 6, 3, 2 for i = W, Z, A, and fermionicdegrees of freedom ni = −12, −12, −4 for i = t, b, τ .The field-dependent neutral CP-even, CP-odd and charged scalar mass eigenstates are ob-tained by diagonalizingM2h =(µ21 + 3λ1v2 + λLφ2 2λLφv2λLφv µ22 + 3λ2φ2 + λLv2)(7.12)M2A =(µ21 + λ1v2 + λSφ2 λ5φvλ5φv µ22 + λ2φ2 + λSv2)(7.13)M2± =(µ21 + λ1v2 + 12λ3φ2 12(λ5 + λ4)φv12(λ5 + λ4)φv µ22 + λ2φ2 + 12λ3v2). (7.14)Notice that for φ = 0 the (22) components reduce to the expressions in Eq. (7.7).The leading order quantum corrections give rise to a renormalization scale-dependent po-tential. One can choose the renormalization scale Q to minimize the size of higher order k-loopcorrections which scale with (lnm2/Q2)k. The scale choice can be important when a parameterin the potential is very different from the electroweak vev ∼ 246 GeV. We thus choose to usethe renormalization group (RG) improved effective potential to minimize the scale dependence.The potential parameters are replaced by their running values, evaluated at the scale Q. Therelevant one-loop β functions are given in Appendix C.The leading order temperature-dependent corrections to the effective potential take theform [332]VT =T 42pi2(∑i=bosonsniJB[m2i (v, φ)/T 2]+∑i=fermionsniJF[m2i (v, φ)/T 2]), (7.15)where the J functions are defined asJB(x) =∫ ∞0dt t2 ln[1− exp(−√t2 + x)], (7.16)JF (x) =∫ ∞0dt t2 ln[1 + exp(−√t2 + x)]. (7.17)These functions admit useful high-temperature expansions which allow us to study the phase142structure as a function of T analytically (as long as the expansion is justified):T 4JB[m2/T 2]= −pi4T 445+ pi212T 2m2 − pi6T(m2)3/2 − 132m4 ln m2abT 2+O(m2/T 2)(7.18)T 4JF[m2/T 2]= 7pi4T 4360− pi224T 2m2 − 132m4 ln m2afT 2+O(m2/T 2), (7.19)where ab = 16af = 16pi2 exp(3/2 − 2γE). The T 2 terms in the expressions above illustratesymmetry restoration at high temperatures. The non-analytic m3 term in Eq. (7.18) can beresponsible for the barrier between the high T phase (at the field origin) and low T phase thatbreaks SU(2)L × U(1)Y .Note that symmetry restoration signals the breakdown of perturbation theory — higherorder diagrams become important. This can be accounted for by performing a resummation ofdaisy diagrams [422–424]. The resummation is performed by adding finite-temperature correc-tions to the boson masses in Eq. (7.16):m2 → m2 + cT 2, (7.20)where c is computed from the infrared limit of the corresponding two-point function. For theSM Higgs doublet we findc1 =18g2 + 116(g2 + g′2) + 12λ1 +112λL +112λS +112λ3 +14y2t +14y2b +112y2τ . (7.21)The various components of the inert doublet receive similar contributions (but without contri-butions from the fermions):c2 =18g2 + 116(g2 + g′2) + 12λ2 +112λL +112λS +112λ3. (7.22)These expressions are in agreement with those in Refs. [418, 425, 426]. We implement thesecorrections by replacing µ2i → µ2i + ciT 2 in the scalar mass matrices, Eqs. (7.12, 7.13, 7.14).2The thermal masses of the gauge bosons are more complicated. Only the longitudinalcomponents receive corrections. The expressions for these in the SM can be found in Ref. [425],but it is easy to modify them to include the contribution of an extra Higgs doublet. For thelongitudinally polarized W boson, the result ism2WL = m2W + 2g2T 2. (7.23)This includes contributions from gauge boson self-interactions, two Higgs doublets and all threefermion families. The masses of the longitudinal Z and A are determined by diagonalizing the2There are subleading thermal corrections to off-diagonal self-energies suppressed by additional powers ofcoupling constants and vevs which are usually neglected.143matrix14(v2 + φ2)(g2 −gg′−gg′ g′2)+(2g2T 2 00 2g′2T 2). (7.24)The eigenvalues can be written asm2ZL,AL =12m2Z + (g2 + g′2)T 2 ±∆, (7.25)where∆2 =(12m2Z + (g2 + g′2)T 2)2− g2g′2T 2(v2 + φ2 + 4T 2). (7.26)7.3.3 Electroweak Phase TransitionArmed with the finite-temperature effective potential, we can now study the structure of theEWPT. The key property we intend to investigate is the transition strength, which sets thebaryon number wash-out rate inside a bubble of broken phase (for a recent review of electroweakbaryogenesis, see, e.g., Ref. [56]). In order to suppress sphaleron wash-out in the regions ofbroken electroweak phase, the relevant condition is typically quantified by requiring that [398]vcTc& 1, (7.27)where vc is the Higgs vev at the critical temperature Tc, defined as the temperature at whichthe origin is degenerate with the electroweak-breaking vacuum.Note that it has been shown that this baryon number preservation condition (BNPC) is aquantity which is manifestly not gauge-invariant [361]. A gauge invariant BNPC can be howeverderived from the high-T expansion of the dimensionally reduced effective action and the criticaltemperature Tc must be obtained using the gauge invariant prescription of Ref. [361], whichemploys expansions in powers of ~ of the potential and vev. Near the critical temperature,O(~) contributions to the potential are as important as the tree-level terms, so the ~ expansionfails. This is also why an all-orders ring diagram resummation discussed in Sec. 7.3.2 is needed.A consistent gauge-invariant method for implementing the ring resummation for the effectivepotential evaluated at the minimum was also demonstrated in Ref. [361]. We will be interested instudying tunnelling and nucleation temperatures, which require the evaluation of the potentialaway from the minima. For this reason, below we employ the standard BNPC of Eq. (7.27) anduse the full one-loop effective potential to study IDM phases. We will argue that our resultsdo not depend strongly on the issues of gauge invariance. We leave the full gauge-invarianttreatment of the IDM to future work.Finally let us note that the physical phase transition does not begin at Tc, but rather ata lower nucleation temperature Tn, at which the bubble formation rate exceeds the Hubble144expansion rate. Equivalently, the probability of nucleating a bubble of broken phase within oneHubble volume is close to 1 [332]. The nucleation rate per unit volume is evaluated in Sec. 5.3.If this rate is too slow, the false vacuum is metastable and the transition does not complete. Theabove conditions translate into a requirement on the bounce action S3(Tn)/Tn . 140, whichdefines Tn [332]. We evaluate the nucleation temperature for a given model with a first-orderphase transition using the CosmoTransitions package [335].7.4 Dark MatterThe requirement of a thermal relic abundance for the LOP matching the observed DM densityin the Universe of Ωcdmh2 = 0.1199± 0.0022 [17], or at least of not over-producing such densityvia thermal production (“subdominant IDM”, see, e.g., Ref. [420]) naturally selects four distinctsectors of the model’s parameter space:1. a low mass regime, with a LOP mass, mH , well below half the observed SM-like Higgsmass, mH . mh/2;2. a resonant or funnel region, mH ∼ mh/2, i.e., a mass range where LOP annihilationproceeds predominantly through quasi on-shell Higgs s-channel exchange;3. an intermediate mass regime, with a LOP mass of mh/2  mH . 500GeV;4. a heavy mass regime, with a LOP mass between 500GeV a few TeV.In the first case, the low mass regime, the DM pair-annihilation predominantly proceedsvia the pair production of the heaviest kinematically accessible fermion (τ leptons, b quarks)through h exchange. The lower the LOP mass, the larger the λL,S couplings need to be in orderto produce a large enough pair-annihilation cross section. The allowed mass values range downto values close to the classical Lee-Weinberg lower mass limit for WIMPs [427], for this class ofmodels somewhere in between 3 and 4 GeV. Direct detection limits from XENON10 [9] probesuch combinations of masses and couplings quite tightly, such that only a small mass windowbelow 5− 7GeV remains.3As the mass of the LOP approaches the resonant condition mH ∼ mh/2, the resonant Higgsexchange allows for much smaller values of the λL,S couplings, and direct detection constraintscan be readily evaded. The relevant mass window left unconstrained by XENON100 [429] andLUX [8] has a width of approximately 10− 15GeV centered around mh/2 [430].The mass regions above and below the resonance mH = mh/2 are actually slightly differentfrom each other: Above the resonance, the pair production of WW ∗ in the final state of DM pairannihilation processes becomes increasingly important, even if λL,S = 0, because the four-pointinteraction through gauge couplings, independent of λL,S , starts contributing significantly. As3The exact limit depends on the different possible choices of nucleon matrix elements, especially those con-nected with the strange quark content of nucleons [428].145MH MA MH± λL λ2 Tc Tn vc/Tc µγγBM1 66 300 300 1.07× 10−2 0.01 113.3 110.3 1.5 0.90BM2 200 400 400 0.01 0.01 116.1 113.7 1.5 0.93BM3 5 265 265 -6× 10−3 0.01 118.2 116.3 1.3 0.90Table 7.1: Input parameters for the three benchmark scenarios discussed in the text alongwith critical and nucleation temperatures, the transition strength and the signalstrength for h → γγ. The masses, given in GeV, are pole masses and the couplingsλi are specified at Q = MZ . Temperatures are also given in GeV.a result the values of λL,S giving the “correct” relic density are pushed to increasingly (withLOP mass) large, negative values.For larger and larger LOP masses, the cross section for LOP pair annihilation to gaugebosons becomes very large, such that the thermal relic density is systematically below theuniversal dark matter density for any combination of model parameters. Barring non-thermalproduction mechanisms, in this intermediate mass region the LOP cannot be the dominantdark matter constituent [413, 420].Finally, at about mH ' 500 GeV, for λL,S ' 0 cancellations between scalar t- and u-channelexchange diagrams and the four-point interaction diagram alluded to above allow, again, for asufficiently large thermal relic density. Such cancellations are suppressed by driving λL,S awayfrom zero. Thus, tuning λL,S for increasing values of mH generally allows one to achieve thecorrect relic density for mass values from mH & 500GeV up into the multi-TeV range.7.5 Benchmark ModelsThe benchmark models specified in Ref. [413] demonstrated various aspects of DM phenomenol-ogy and the possibility for the IDM to influence the h → γγ rate. Unfortunately, noneof the suggested scenarios exhibits a strongly first-order EW phase transition. In this Sec-tion, we identify alternate benchmark models which can potentially yield a strongly first-orderEW phase transition, while having disparate properties for the lightest Z2-odd particle. Allour benchmark models are compatible with constraints from Higgs collider bounds and ratemeasurements, which has been explicitly checked using the tools HiggsBounds [431–433] andHiggsSignals [434], where the model predictions have been calculated using a SARAH-generatedSPheno version [435–438]. In the following discussion we mostly focus on the interplay betweenthe dark matter phenomenology and the strengths of the EWPT.Our key finding is that the requirement of a strongly first-order phase transition generallyleads to a large mass splitting between the LOP and the other scalars in the IDM. Our bench-mark models are summarized in Tab. 7.1, along with the corresponding critical and nucleationtemperatures, as well as phase transition strengths, as parametrized by the ratio vc/Tc. In eachcase the masses of the A and H± are chosen to ensure a strongly first-order phase transition.In Fig. 7.3 we show the dependence of the transition strength on these parameters. The lines146corresponding to BM1 and BM3 terminate where the potential develops a non-inert (φ 6= 0)vacuum first during thermal evolution. This vacuum can then either continuously evolve intothe SM/inert (φ = 0) vacuum at T = 0 or it can persist to low temperatures. In the latter case,the EWPT can occur in two steps. Such models are viable if the inert vacuum is deeper thanthe new one at T = 0 and both transitions complete (i.e., nucleation rate(s) are large enough).Two step electroweak phase transitions have been investigated in detail in Refs. [439, 440]. Inthis work we consider only simple one step transitions, hence the truncation. Notice that inRef. [418] the strength of the EW phase transition in models with multiple phase transitionsteps was always weaker, see Fig. 3 and 4 in Ref. [418].First, let us consider model BM1, where LOP production in the early Universe is predomi-nantly set by near-resonant s-channel Higgs exchange. This scenario has been recently examinedin the context of phase transitions in Ref. [418]. Even more recently, it has also been suggestedas a possible explanation [441] of the Fermi-LAT gamma-ray excess (see Ref. [442] and referencestherein). As discussed above, the on-resonance requirement forces mH ∼ mh/2, but allows λLto be small enough to be consistent with direct detection constraints. Here DM production doesnot rely on interactions with A or H±, so their masses can be essentially chosen freely, as longas the resulting quartic couplings λi (through Eq. (7.7)) satisfy perturbativity and constraintsfrom electroweak precision observables (EWPO), which we check with 2HDMC [443]. In orderto satisfy the BNPC of Eq. (7.27), one needs to increase the coupling of the new scalars to h,which, in turn increases the splitting of A and H± relative to H. For this and the followingmodels we choose mA = mH± to minimize the impact of splitting these states from H on thePeskin-Takeuchi T function [405] and to reduce the number of parameters. This assumptioncan be somewhat relaxed, but the results are qualitatively similar. This benchmark representsthe only class of scenarios where the thermal LOP relic density (which we calculated with themicrOMEGAs code [444]) matches the observed DM universal density, and where the EW phasetransition is strong enough.When mA, mH± & 340 GeV, the corresponding loop corrections to mH are large and requireµ22 < 0. This causes a second minimum to appear in the potential at T = 0. As mA = mH± isincreased further, this minimum quickly becomes deeper than the SM one, corresponding to thetermination of the blue curve in Fig. 7.3 at mA ∼ 350GeV. This behaviour was also observedin Ref. [418].In this scenario, the LOP mass mH has been chosen slightly above the kinematic thresholdof the decay h → HH in order to evade constraints from direct searches for invisible Higgsdecays and Higgs rate measurements. These however become important for our benchmarkscenario BM3 (see below).The second benchmark BM2 in Tab. 7.1 represents the intermediate mass regime. Hereannihilation into gauge bosons is efficient and DM is generally underabundant, unless there isa cancellation among different amplitudes [413]. The cancellation depends, as indicated above,on how close λL,S → 0, i.e., on how degenerate the IDM Higgs sector is. In our benchmark,14700.511.522.5200 250 300 350 400 450v c/TcmA = mH± [GeV]mH = 66 GeVmH = 200 GeVmH = 5 GeVBM1BM2BM3Figure 7.3: Phase transition strength as a function of the heavier IDM scalar masses,taking mA = mH± . The remaining parameters are chosen as in the benchmarkmodels of Table 7.1, which are shown by black dots. The lines for BM1 and BM3terminate where the inert doublet develops a non-zero vev, φ 6= 0, as described inthe text.such a cancellation requires mH ≈ mA ≈ mH± with a maximum splitting of ∼ 10 GeV. Thesesmall splittings lead to small couplings of the new states to h and therefore an insufficientlystrong phase transition. Thus the phase transition requirement forces thermal relic DM to beunderabundant. The observed DM density can be explained here, however, by invoking non-thermal production mechanisms (e.g., the decay of a heavy particle) or with the existence ofadditional DM particles (e.g., axions). The multitude of “non-standard” production mechanismshas been recently reviewed in Ref. [74].The final benchmark model, BM3, belongs to the light-mass regime, and is another examplethat requires further ingredients to be fully consistent with the phenomenology of the DM sector.For mH < mh/2, decays of the SM Higgs to invisible final states become possible, with a decayrate [405]Γ(h→ HH) =v2λ2L8pimh(1−4m2Hm2h)1/2. (7.28)Requiring consistency with the observed 95% C.L. upper limit on the branching fraction,BR(h → HH) ≤ 17% [445], provides a strong constraint on the coupling λL of |λL| . 0.007,while a large value |λL| & 0.4 is required to sufficiently deplete the DM abundance [413]. Theseproblems can be remedied by softly breaking the Z2, which would allow H to decay [446, 447].148As in the previous example, another explanation for DM is then needed. An alternative possi-bility is to provide the LOP with new annihilation modes, e.g., to new light vector bosons [448],or a mechanism to dilute the thermal relic density, such as an episode of late entropy injec-tion [449, 450].DM phenomenology aside, it is again easy to get a strongly first-order phase transition witha large mass splitting between H and A, H±. We note that this scenario requires a significanttuning of parameters, because a small LOP mass requires near cancellation of tree-level andloop contributions. For λL > 0, this leads to negative values of µ22 which can result in theappearance of a new φ 6= 0 minimum.In all three cases, the first-order transition is driven by the non-analytic (m2)3/2 terms (seeEq. 7.18) due to A and H±, while the gauge boson contributions are not as important. Thisexplains the common feature of large splittings between H and A, H± among the benchmarkscenarios. These lead to large couplings between h and the new states, enhancing the size ofthermal corrections. This appears to be a generic requirement for increasing the strength ofthe phase transition in the IDM. It is also important to emphasize that thermal corrections tothe crucial (m2)3/2 terms from A and H± are not subject to gauge invariance issues that affectthe gauge sector contributions. As a result, we expect these arguments to remain valid in thecontext of a fully gauge invariant treatment. This can be further tested in a toy model withall gauge coupling constants set to 0, thereby completely eliminating gauge dependence fromthe effective potential.4 We checked that such a simplified analysis gives quantitatively similarresults for critical temperatures and transition strengths when the scalar couplings are large.The high T expansion of the effective potential also provides a simple explanation for theshape of the curves in Fig. 7.3. In this limit the transition strength is proportional to thecoefficient of the v3 term [332]. For the IDM scalars such terms arise from the non-analyticcontributions proportional to (µ22+λSv2c )3/2 (assuming mA = mH± , as above, and ignoring daisycontributions for simplicity), which behaves as v3 only when λSv2c  µ22. Thus when λSv2c  µ22,the transition strength is independent of IDM parameters, corresponding to the plateau of thegreen curve in Fig. 7.3. In the opposite limit, the IDM gives an additional contribution tothe cubic coefficient, so the transition strength scales as λ3/2S ∼ m3A, as illustrated by themonotonically increasing sections of the curves in Fig. 7.3.5For heavy masses m2/T 2  1 (with T ∼ 100 GeV), the IDM states thermally decouple, butthis does not mean that they have no impact on the phase transition. When µ22  |µ1|2, theheavy doublet can be integrated out to yield a SM effective theory with the potentialV0 = µ2|H|2 + λ|H|4 + κ|H|6 + . . . (7.29)where the dots stand for higher mass dimension operators. The parameters µ2, λ and κ can be4We thank Michael Ramsey-Musolf for pointing this out to us.5The precise scaling is modified by Daisy corrections, O(µ22/λSv2c ) terms, finite-T and renormalization groupeffects.149related to those in the fundamental IDM by equating the effective potentials for the two modelsat a matching scale Q ∼ µ2. For example, one-loop matching yieldsκ = 124pi2µ22(λ3L + λ3S + λ33/4), (7.30)while µ2 and λ are determined below the matching scale by fixing the vev and the Higgs mass.With the presence of a dimension-six term in the potential, the barrier required for a stronglyfirst-order transition can be generated if λ . 0 and µ2 + cT 2 > 0 for T ∼ Tc, where c encodesthermal corrections from SM states only [425]. Scenarios of this type have been considered, e.g.,in Refs. [451–454]. One immediate difficulty is that in the IDM κ is generated only at one-loop,so in order for this operator to be significant for field values of around the electroweak vev, onemust overcome the loop suppression, suggesting that the combination λ3L+λ3S +λ33/4 cannot betoo small. This again forces a large splitting between the IDM states, meaning that the heavyDM scenario described in Section 7.4 cannot be realized together with a strongly first-orderphase transition. Such large couplings can run into perturbativity problems and invalidate theexpansion used to generate the effective field theory.We briefly comment on the discovery prospects of the new IDM states at the LHC. Due tothe Z2 symmetry, the IDM states can only be produced pairwise at colliders. Successive decaysof the heavier IDM states A and H± into the LOP and a Z or W boson, respectively, can giverise to multilepton signatures [412, 455, 456]. In a recent analysis [456] of LHC searches forsupersymmetric particles with two leptons plus missing transverse energy in the final state inthe context of the IDM, mass limits of up to mA . 140 GeV for LOP masses mH . 55 GeV andcharged Higgs masses around 85−150 GeV have been derived. While these limits partly exceedprevious limits from the LEP collider, they are not yet sensitive to the parameter regions thatyield a strongly first-order phase transition as required for successful electroweak baryogenesis,see Fig. 7.3.The new IDM states can also have an indirect effect on precision Higgs measurements. Inparticular, the new charged state H± provides an additional contribution to the loop-inducedh → γγ and γZ rates. These effects have been recently studied in Refs. [457, 458] in thecontext of the 125 GeV Higgs boson. Modifications of these branching fractions by O(10%)are a generic feature of our benchmark scenarios, as we show below. The h→ γγ rate has theform [421, 457–461]Γ(h→ γγ) =α2GFm3h128√2pi3∣∣∣∣ASM +λ3v22m2H±A0(m2h4m2H±)∣∣∣∣2, (7.31)where the leading contributions to the SM amplitude ASM ≈ −6.56 + 0.08i come from Wbosons and top quarks. The second term is the new contribution from H±, where A0 is a loopfunction with the property limx→0A0(x) = 1/3 [461]. For our benchmarks we have λ3 > 0, so150one expects a suppression of h→ γγ relative to the SM.6 Note that for fixed µ22, the amplitudefor the H± contribution tends to a constant value 1/3 as λ3 is increased. This means that inthe limit of a large mass splitting between H and H±, which is required for a strongly firstorder phase transition, the branching fraction is reduced by ∼ 10%. This effect was also noticedin Refs. [417, 420] for models similar to our BM1 and BM2, respectively. The deviations toBR(h→ γγ) induced by H± are shown in Tab. 7.1 in terms of the SM normalized signal strengthµγγ = (σ BR)/(σ BR)SM. While they are still consistent with the present measurements fromATLAS [462] and CMS [463], the LHC should reach a precision of 4–8% for µγγ [113, 445],thereby definitively testing the benchmark scenarios in Tab. 7.1.While our benchmarks feature sizable deviations of BR(h→ γγ) from the SM expectation,we note that it is possible to avoid this by taking H± to be nearly degenerate with H, andusing A alone to drive the phase transition to be strongly first order. However, in this case,efficient coannihilation of H with H± during freeze-out generally results in a very small relicabundance [464]. The near degeneracy is also required by constraints on the oblique T parameterwhen mA  mH± [405]. For example, taking mH± = 70GeV, mA = 370GeV and otherparameters as in BM1 results in a strongly first order phase transition, an order of magnitudesmaller relic abundance and only a ∼ 3% depression of µγγ relative to the SM.7.6 Discussion and ConclusionsWe studied the structure of the electroweak phase transition in the inert Higgs doublet model,utilizing a set of three benchmark scenarios that feature a potentially viable dark matter par-ticle. Our choices for the three benchmark models essentially exhaust all possible prototypicalsetups for particle dark matter in the inert doublet model. While only one of the benchmarkshas a dark matter particle with a thermal relic density matching the observed dark matterdensity, the other two (under- and over-abundant) can be made viable by invoking additionalproduction mechanisms or a scenario where the thermal relic density is diluted away, respec-tively.The key finding of our study is that in all cases where the model possesses a reasonableparticle dark matter candidate, the inert scalar spectrum can be arranged in such a way so as toproduce a strongly first-order electroweak phase transition. Central to achieving such a phasetransition is to postulate a large enough splitting between the dark matter candidate and theheavier inert scalars. The physics driving this result is simple: Large mass splittings genericallycorrespond to large couplings between the inert scalars and the Standard Model Higgs; These,in turn, increase the magnitude of non-analytic ∼ (m2)3/2 terms in the temperature-dependenteffective potential and thus the potential barrier between the field origin and the SU(2)L ×U(1)Y -breaking phase.6Various limits on LOP-Higgs coupling discussed above force |λL| to be small, such that the H mass isprimarily determined by µ22 (at tree level, see Eq. (7.7)). If the charged Higgs H± is heavier than H then thisforces λ3 > 0.151The mass splitting under consideration cannot be arbitrarily large. For large enough values,for example, the phase structure of the model becomes more complicated, with possible non-zerovacuum expectation values for the inert doublet and multiple-step phase transitions. While,based upon the results of Ref. [418] the latter possibility is not expected to yield strongerelectroweak phase transitions than in the single-step case, this is an interesting possibilitywhich we leave for future studies.The question of how to embed large-enough CP violating sources in detail was also left unan-swered here. It will be interesting to study whether such a source (for example an additionalgauge-singlet complex scalar, see Ref. [465]) significantly impacts the electroweak phase tran-sition and dark matter phenomenology, and, with this, the conclusions reached in the presentstudy.152Part IVConclusions153Chapter 8Conclusion and OutlookIn this thesis we have investigated extensions of the Standard Model of particle physics thatseek to address one or more of the problems outlined in Ch. 1. These included theoreticalissues, such as the hierarchy problem, as well as observational questions of dark matter and thebaryon asymmetry of the Universe. We classified our discussion in terms of phase transitions– events in the history of the Universe, when an order parameter, such as an energy densityof a cosmological species (Part II) or the expectation value of a scalar field (Part III), changesabruptly, modifying subsequent evolution. To conclude, below we summarize the main resultsand outline prospects for experimental tests of these ideas.In Chapters 3 and 4 we considered string theory motivated cosmological scenarios with late-time reheating. In many realistic string compactifications long lived scalar fields called modulidominate the energy density of the Universe until their decay just before the onset of primordialnucleosynthesis. This relatively late phase transition from matter to radiation domination canbe responsible for dark matter and baryon asymmetry genesis.Chapter 3 was dedicated to the production of massive supersymmetric particles during theera of moduli reheating. Such states can account for the observed dark matter relic abundance.However, due to the low reheating temperature, TRH & 5MeV, DM generation is non-thermaland the lightest supersymmetric particle (LSP) must rely on large annihilation cross sectionsto deplete its number density to acceptable levels. In the Minimal Supersymmetric StandardModel (MSSM) with a sub-TeV wino LSP, this generally leads to a conflict with gamma rayobservations of the Galactic Center. In an attempt to preserve the string-motivated modulicosmology, we considered three extensions of the MSSM gauge structure. These share thecommon feature that the additional gauge sector contains the true LSP of the theory, allowingthe wino to decay. The first two models discussed were based on an additional U(1)x, kineticallymixed with the MSSM hypercharge. We explored the possibility of (symmetric) self-conjugateand (asymmetric) U(1)x-charged DM, finding that the former case still suffers from a largeindirect detection rate, generally inconsistent with observations (except for extreme choicesof parameters). The asymmetric DM model is compatible with experiment. However, bothcases require light scalar particles, which is puzzling in the context of the MSSM itself, where154LHC bounds suggest that the scalar superpartners are heavy. This issue was addressed inthe third example, which relied on a pure SU(N)x gauge theory and therefore did not requireany unnaturally light scalars. While this case provided a solution to the moduli-induced LSPproblem in the MSSM, we found that this model can only account for a small fraction of DMin order to be consistent with structure formation in our Universe. Thus, none of the threemodels discussed provided a fully satisfactory (theoretically and/or experimentally) solutionto the moduli problem. These conclusions hold for generic moduli parameters, relatively lowscale of SUSY breaking and R-parity conserving supersymmetry. The results of our study cantherefore be interpreted in two ways. First, they may be taken as hint that moduli properties,such as the relation of the modulus mass to the SUSY breaking scale and its coupling toordinary matter, are not generic in our string vacuum. The second interpretation suggeststhat R-parity is violated, such that supersymmetry does not give rise to a stable dark mattercandidate. Alternatively, the scale of SUSY breaking (and therefore the modulus mass) can bemuch larger than expected, leading to a higher reheating temperature.The late-time moduli decays discussed above occur far out of equilibrium and thereforeprovide a viable setting for generating the baryon asymmetry of the Universe. This idea wasdeveloped in Ch. 4, where we considered a unified origin for the matter-antimatter asymmetryand dark matter, through the mechanism of hylogenesis. In this scenario the moduli decayproducts give rise to an excess of visible baryons over antibaryons, and an equal asymmetry inhidden antibaryons (such that total baryon number is conserved) that populate a “dark sector”.The hidden antibaryons are the dark matter. The structure of the dark sector is very similarto the asymmetric U(1)x extension used in Ch. 3. The operator that mediates baryon transferbetween the visible and hidden sectors also gives rise to a novel direct detection signature –induced nucleon decay (IND). This arises when a DM particle (which carries antibaryon number)scatters inelastically off a nucleon, destroying it and producing another hidden state and ameson. In a nucleon decay detector, this looks like a standard nucleon decay event (spontaneousemission of a meson, together with missing momentum), but with different kinematics for theoutgoing meson. The resulting effective nucleon lifetime is accessible by present and futurenucleon decay searches. The strongest limits at present come from the Super-Kamiokandeexperiment [466]. However, these limits are not directly applicable due different kinematics ofstandard and induced nucleon decay. Because the experiments optimize their event selectionfor standard nucleon decay, the resulting efficiency for IND events can be as small as 5% [467]– the effective constraints on IND rates and the underlying parameters in the model are weak.These limits can be improved by relaxing event selection (so that more IND events pass thecuts), tailoring analyses specifically for IND and by the next generation of experiments, suchas Hyper-Kamiokande [468].The operators needed for hylogenesis can also be probed at colliders. For example, theeffective baryon transfer operator (XucRdcRdcR)/M2 contributes to the LHC production of jetsand missing energy (via the decay X → ΨΦ∗). The resulting limits from the 8 TeV run constrain155M & 2.6−3.5TeV, depending on the flavour structure [469]. An ultraviolet-complete model thatgenerates the above operator also gives rise to dijet resonances, which modify QCD predictionsof dijet mass and angular distributions. The latest results from ATLAS [470, 471] and CMS [472,473] are consistent with SM predictions and set lower limits on the scale suppressing contactoperators (roughly corresponding to the mass of the P scalars discussed in Sec. 4.2.2) that insome cases reach ∼ 12TeV.1 These limits depend on the flavour structure of the completetheory. However, note that the effective scale enters the CP asymmetry parameter defined inSec. 4.3.1 and for large enough values the baryon asymmetry produced during hylogenesis willbe insufficient to explain the observed value. As this thesis is being completed, the LHC ispreparing to begin Run 2 at near design energy of 13 TeV. Stronger limits from this new energyfrontier will be instrumental in testing hylogenesis.The baryon transfer operator is only one of two portals between the visible and dark sectors.The implementation of hylogenesis presented in Ch. 4, as well as two models in Ch. 3 also featurea kinetic mixing interaction that can be probed from multiple directions. First, irrespective ofwhether or not the hidden sector contains a DM candidate that saturates the relic abundance,the hidden vector can be produced in fixed target/beam dump experiments and at the flavourfactories (see Ref. [203] for a review). This search strategy has the advantage of being relativelymodel independent. Indeed, if the vector decays back to SM particles, the signal is a functiononly of the vector mass and the kinetic mixing parameter. As discussed in Sections 3.3.2 and4.3.2, there exist certain minimal values of this mixing parameter in the range 10−5−10−4 fromthe requirement of sufficiently fast decays of certain hidden sector states. While this parameterrange is not probed by current measurements for hidden photons (for GeV-scale vector masses),the next generation of experiments such as HPS and Belle II will be sensitive to the interestingparameter region [203, 474]. If no signal is found, our implementation of hylogenesis and therelated models of Ch. 3 will be in tension with bounds on energy injection during primordialnucleosynthesis.A complementary set of probes of the U(1)x hidden sector paradigm is available whenthe hidden sector contains a viable dark matter candidate. For example, the kinetic mixinginteraction gives a direct detection rate that is very sensitive to the hidden photon mass –see Eqs. (3.53) and (4.59). These rates also depend on the kinetic mixing parameter andthe hidden gauge coupling. However, these quantities are bounded from below as discussedabove, suggesting that there is a also a minimum rate for scattering on nucleons. This roughlycorresponds to the lower dashed line in Fig. 8.1, where we also demonstrate the best constraintsin the low mass regime from various experiments. Models with scattering rates far belowthe lower dashed line will generally overclose the Universe or have problematic late-decayingparticles in the hidden sector. We note, however, that predictions of direct detection signals aresubject to astrophysical uncertainties such as the local DM density and velocity distributions.Another disadvantage is that these searches are very model dependent. For example, when the1The numbers cited in Sec. 4.5.3 hold for older LHC results with a partial Run 1 data set.15610−4210−4110−4010−3910−3810−3710−361 2 3 4 5 6 7 8 9 10σSIn[cm2]mΨ,Φ [GeV]gx = 10−4gx = 10−5CDMSLite (2013)XENON10 S2 (2011)LUX (2013)CRESST Si (2002)Figure 8.1: Constraints on the spin-independent scattering cross section as a function ofDM mass (for the model of Sec. 3.4) from LUX [8], XENON10 S2 only analysis [9],CDMSLite [10] and CRESST-Si [11]. The dashed lines show the expected crosssection for various combinations of gx (see Eq. (3.53)).DM candidate is a Majorana fermion, as in Sec. 3.3, the resulting interactions only allow forspin-dependent scattering for which experimental limits are much weaker than for the spin-dependent case. With these caveats in mind, the next 5–10 years should bring improvements inexperimental sensitivity of an order of magnitude or better in the light mass regime [76]. FromFig. 8.1 it is clear that this will probe a significant portion of viable parameter space.A second important probe of cosmological dark matter, indirect detection, was discussed indetail in Ch. 3. It was shown, despite significant astrophysical uncertainties, that continuumgamma ray observations provide a sensitive test of models with non-thermal dark matter. Thisis because the large annihilation rates needed to produce the correct relic density also lead tosignificant annihilation at late times. These limits, in particular those coming from Fermi-LATwill continue to improve, as more data is collected. With an expected mission lifetime of 10years, it is plausible that the Fermi-LAT sensitivity will improve by an order of magnitude,translating to more stringent constraints of models of light non-thermal DM, even if it is asubdominant component [475].In the above discussion we emphasized the fact that the models considered in Part II aretestable at readily available energies, despite the fact that some of the core dynamics originatesfrom speculative high scale physics. In contrast, Part III studies phase transitions associatedwith electroweak scales. In Ch. 6 we studied the implications of naturalness of the Higgs mass,mh, and its observed value on the vacuum structure of the MSSM. Supersymmetry introducesmany new field directions in the scalar potential. Minima of the potential in these directions157generally break charge and colour symmetries. If such a minimum is deeper than the one we livein, our vacuum can be destabilized by quantum tunnelling, corresponding to a phase transitionfrom colour preserving to a colour broken phase. Interestingly, the appearance of these minimais correlated with mh through quantum corrections due to stops, superpartners of the top quark.We computed the tunnelling rates to various charge and colour-breaking minima and placed newconstraints on the MSSM parameter space. These limits are complementary to experimentalresults described in Sec. 2.5, since they reach a different region of the stop parameter spaceand are independent of the nature of the LSP. The experimental limits on stops, on the otherhand, are quite sensitive to the mass of the LSP. If stops are discovered, our results will usefulin determining whether the electroweak vacuum is stable, as discussed in Sec. 6.5.2.The transitions studied above occur late in the evolution of the Universe, when the temper-ature is effectively zero. At finite temperature, however, a similar phase transition occurs evenin the Standard Model, when the electroweak symmetry is broken down to electromagnetismwhen the Higgs field acquires an expectation value. If this transition proceeds through bubblenucleation and is sufficiently strong, it can potentially explain the production of the observedbaryon asymmetry in the Universe. In the SM, this process produces too little baryon numberfor reasons explained in Sec. 7.1. This can be remedied by including additional matter thatcouples strongly to the Higgs field. In Ch. 7 we investigated this possibility within one of thesimplest possible extensions of the SM: the inert doublet model (IDM). Despite its simplicity,the IDM is able to produce an electroweak phase transition appropriate for baryogenesis, andeven account for DM for certain choices of parameters. Motivated by different classes of DMphenomenology, we presented three benchmark scenarios that generated a strongly first orderphase transition. In all but one case, we found that the requirement of a good DM candidateand a strong electroweak transition are mutually exclusive. This is because appropriate levelsof DM annihilation often rely on precise relationships among the IDM particle masses or valuesof parameters that are inconsistent with precision SM Higgs measurements. In contrast, a suf-ficiently strong electroweak transition requires large couplings of the Higgs to the new states,which in turn translates into large mass splittings in the IDM sector. This has important im-plications for collider limits on IDM, which are typically obtained by recasting SUSY searchesfor neutralinos and charginos for the IDM H, A, and H± states (see, e.g., Ref. [456] for recentresults). For example, in dilepton searches, larger splittings between the IDM states give riseto highly boosted lepton pairs. Interestingly, this can actually reduce the sensitivity of currentsearches due to a veto on leptons from on-shell Z decays [456]. As a result, at present the limitson the interesting region of IDM parameter space are weak. In particular, the Higgs funnelregime, which accounts for both DM and a strongly first order electroweak phase transition (seeSec. 7.5), is completely unconstrained at the moment. Fortunately, Run 2 should begin to probethis regime [456]. Note that the above comments apply only to recasting of existing searches;dedicated IDM analyses should yield more stringent results. The existence of a charged scalaralso alters loop-induced Higgs decay rates. In particular, the h→ γγ signal strength is modified158by O(10%) for our benchmark scenarios. While this is consistent within error bars with currentmeasurements [462, 463], the LHC is expected to reach a precision of 4–8% for this branchingratio, thereby definitively testing our benchmark models [113].The inert doublet model addresses only one of Sakharov’s conditions for successful baryoge-nesis. The IDM does not introduce any new sources of CP violation. A more complete theoryof EWBG will also address this shortcoming of the SM as well. In this case, new sources ofCP violation can be probed by measurements of electric dipole moments of electrons, neutrons,atoms and molecules as described in Sec. 2.5.The imminent restart of the LHC promises to resolve several fundamental issues in particlephysics. As discussed above, the new collider data will be crucial for testing theories of darkmatter and baryogenesis; direct and indirect detection, along with precision measurementswill provide complementary probes of these models. Perhaps even more importantly, Run 2will determine the ultimate fate of naturalness, which has been a major motivator for physicsbeyond the Standard Model not far from the electroweak scale. The hierarchy and the closelyrelated naturalness issues are problems of low scale phenomena being extremely sensitive to highscale, ultraviolet physics. A century ago theoretical physics was also afflicted by an ultravioletcatastrophe in the classical black body radiation spectrum. The solution to that small probleminitiated the quantum revolution. The hierarchy problem may be the UV catastrophe of thetwenty first century. Its resolution will either lead to another revolution (supersymmetry, littleHiggs or something completely unexpected!) or to a fundamental realization that nature isfine-tuned, perhaps hinting at modification of naturalness or even environmental selection.159Bibliography[1] N. Blinov, J. Kozaczuk, A. Menon, and D. E. 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Jones, Bounds on scalar masses in two Higgs doublet models, JHEP0908 (2009) 069, [arXiv:0903.2856].188Appendix AMinimality of the Action UnderPath DeformationsIn this appendix we show that fixing a path in field space connecting two vacua and com-puting the one-dimensional bounce action along that path provides on upper bound on thebounce action for tunnelling between those vacua. Equivalently, the bounce solution of the Eu-clidean action is a minimum of the action with respect to deformations of fixed, one-dimensionalpaths in the field space. The implication of this result is that the path deformation methodof CosmoTransitions (CT) [335] is guaranteed to provide at least an upper bound on thetunnelling lifetime.Recall that the multi-field bounce solution φ¯(ρ), ρ =√t2 + x2, of the Euclidean action is anO(4)-symmetric solution of the classical equations of motion subject to the boundary conditions∂ρφ¯(ρ = 0) = 0 , limρ→∞ φ¯ = φ+ , (A.1)where φ+ is the metastable vacuum configuration. The bounce action is just the Euclideanaction evaluated on the bounce solution.Let us now restate our claim more precisely. The bounce solution is an element of the setof parametric curves on RF , where F is the number of scalar fields. Any path φ(ρ) in this setcan be written in terms of a unit speed curve γ(s):φ(ρ) = γ(s(ρ)) , where |γ˙(s)| = 1 . (A.2)The function s(ρ) is the solution ofdsdρ=∣∣∣∣dφdρ∣∣∣∣, (A.3)189and γ(s) = φ(ρ(s)). The Euclidean action in α spacetime dimensions becomesSE [γ, s] = Ωα∫dρρα−1[12(dsdρ)2+ V (γ(s(ρ)))], (A.4)where Ωα = 2piα/2/Γ(α2 ) is the surface area of a unit (α− 1)-sphere. Suppose we fix a path infield space γ connecting two vacua and extremize the action with respect to s(ρ) subject to theboundary conditions of the bounce along this one-dimensional trajectory. The correspondingsolution can then be used to obtain a restricted bounce action along the fixed trajectory. Thisis the procedure used by CT at each intermediate step of its deformation procedure. We claimthat the action obtained for any such fixed path is greater than or equal to the unconstrainedbounce action.To prove this claim, we use the fact that the bounce is a stationary point of the action. Fortunnelling configurations, however, it is not an extremum of the action. This coincides with thefact that the second variation of the action with respect to the fields has a negative eigenvalue.The corresponding operator is−δij∂2 +δVδφiδφj(φ¯) . (A.5)We assume that this operator has only a single negative mode [316, 348]. This has beenproved for a single field in the thin wall limit [316, 323]. If this assumption is false, the entireCallan-Coleman formalism does not apply. We show that this negative eigenvalue is associatedexclusively with the variation of s(ρ) using the argument of Ref. [348]. As a result, the bounceaction is an extremum with respect to variations in the orthogonal parameter γ, and can easilybe shown to be a minimum by explicit construction.Consider the scaling transformations(ρ) → s(ρ/λ). (A.6)The action of Eq. (A.4) transforms asS[γ, s] → λα−2ST [γ, s] + λαSV [γ, s], (A.7)whereST [γ, s] = Ωα∫dρρα−1 12(dsdρ)2(A.8)andSV [γ, s] = Ωα∫dρρα−1V (γ(s(ρ))). (A.9)190Requiring that S is stationary with respect to these scale variations yieldsδSδλ= 0 ⇒ ST = −αα− 2SV > 0. (A.10)We can also evaluate the second variation of Sδ2Sδλ2=−ST α = 3−2(α− 2)ST α > 3< 0. (A.11)This means that the bounce is a maximum of the action with respect to the scaling transfor-mation of Eq. (A.6). Thus the crucial negative eigenmode is due to scaling, and, since thistransformation does not involve the normalized path γ, it is due entirely to the functional vari-ation of s(ρ). The tunnelling action obtained by computing the bounce solution along a fixedone-dimensional path is therefore an upper bound on the true bounce action. This justifiesthe procedure of using a fixed normalized field path and computing s(ρ) as a way to check theCosmoTransitions results.191Appendix BAn Approximate Empirical Boundfor Charge and Colour-BreakingVacua in the MSSMIn this second appendix we describe an approximate empirical bound on metastability valid inthe parameter region r ≡ m2U3/m2Q3 ∼ 1, moderate tanβ, smaller µ, and larger mA. We beginby deriving a condition on absolute stability to motivate the functional form of the empiricalformula. Let us emphasize that our empirical bound is only an approximation, and is notguaranteed to work outside the limited regime we consider.To derive an improved bound on absolute stability of the SM-like (SML) vacuum, we imposeonly SU(3)C D-flatness and H0d = 0. Similar existing formulae typically also assume SU(2)Land U(1)Y flatness, which precludes the existence of a SML vacuum. For m2U3/m2Q3 ∼ 1,SU(3)C D-flatness should be a good approximation since the strong gauge coupling is largerthan the others [353]. Setting H0d = 0 is also well-justified for large tanβ near the SML vacuum;at the CCB minimum one typically finds |H0d | < |H0u| as well.Applying the SU(3)C D-flatness condition, we haveT ≡ t˜L = |t˜R| , (B.1)and the potential becomesV = m2TT 2 +m22(H0u)2 ± 2ytAtH0uT 2 + y2t[T 4 + 2T 2(H0u)2]+ g¯28[(H0u)2 − T 2]2 ,where m2T = m2Q3 +m2U3 , g¯ =√g2 + g′2 and m22 = m2Hu + |µ|2.Minimizing, we have0 = ∂V∂T= T[2m2T ± 4ytAtHu + 4y2tH2u −g¯22((H0u)2 − T 2) + 4y2t T 2]. (B.2)192The solutions are evidently T = 0 andT 2 =[∓2ytAtH0u −m2T − 2(y2t − g¯2/8)(H0u)2]/2(y2t + g¯2/8) . (B.3)Since we are restricting ourselves to H0u ≥ 0, the relative orientation of the stops in any potentialCCB minimum must be such that ∓ytAt = |ytAt|. Note as well that the A-term must overpowerthe others to make T 2 > 0. Under our given assumptions, this already provides a necessarycondition on the existence of a CCB vacuum,A2t > 2m2T (1− g¯2/8y2t ) . (B.4)This is a somewhat weaker requirement than the analytical formula Eq. (6.3).Minimizing with respect to Hu (and choosing the relative stop alignment as above) gives0 = ∂V∂Hu= 2m22H0u + 4(g¯2/8)(H0u)3 +[(2(y2t − g¯2/8)H0u − ytAt](2T 2) . (B.5)For T 2 = 0, this reproduces the SM-like minimum. On the other hand, we can also plug inour non-zero solution for T 2, which is quadratic in H0u. This generates a cubic equation for H0uthat can be solved analytically. A cubic equation has three roots, with at least one of themreal. The other two roots are either real, or complex conjugates of each other. We need at leastthree real roots to have both a SML vacuum and a CCB vacuum since there must also be atleast a saddle point between them.In this approximation, we can check for CCB vacua by simply scanning over stop parametersand computing cube roots, for which there exist analytical formulae. The EW vacuum is trivialto find, and corresponds to T = 0. The T 2 6= 0 solutions may correspond to CCB vacua.A necessary condition for this is that all the roots are real, and that at least two of themare positive. With the roots in hand, it is then straightforward to use them in the potentialto compare the relative depths of the minima. Fixing m22 = −m2Z/2 to get the correct SMLvacuum expectation value, we find numerically that A2t & (2.4)(m2T + m22) gives a very goodestimate of the condition for a CCB vacuum to be deeper than the SML vacuum for thissimplified potential.In our analysis of metastability, we find that the boundary between metastable and danger-ously unstable regions tends to track the boundary between SML and CCB regions. Motivatedby this and our previous result for CCB vacua, we will attempt to fit the boundary betweenmetastable and unstable regions by an expression of the formA2t = αm2T + β|m2Q3 −m2U3 |+ γm22 =(α+ β |1− r|1 + r)m2T + γm22 (B.6)The second term in the above expression is included to model the effect of small deviationsfrom SU(3)C D-flatness.1930.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]m2U3/m2Q3 = 0.1SMLCCBS < 400Analytic BoundOur Empirical BoundEmpirical Bound0.00.51.01.52.02.53.0-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0MS[TeV]Xt [TeV]m2U3/m2Q3 = 10.0SMLCCBS < 400Analytic BoundOur Empirical BoundEmpirical BoundFigure B.1: The deterioration of the empirical bound of Eq. (B.6) for m2U3/m2Q3  1 orm2U3/m2Q3  1. For these parameter ranges the assumption of SU(3)C D-flatnessthat motivated Eq. (B.6) breaks down and it cannot be used to reliably model theboundary between the metastable and unstable parameter regions.We estimate the parameters α and γ by using a least-squares fit to the lower boundaryof the metastable region in Fig. 6.2, without imposing the Higgs mass constraint. This is anarbitrary choice to fit to; different choices in Tab. 6.1 lead to variations in α on the order of15% and 100% in γ. The large variation in γ is not a big problem since it is multiplied by|m22| ∼ m2Z  m2T . We obtain α ' 3.4 and γ ' 60. With α and γ fixed, we fit β to modelswith r 6= 1. We again see that there is a significant variation O(20%) depending on what r is,indicating that the functional form of Eq. (B.6) is an oversimplification. With this in mind, wefind an average value of β ' −0.5 for r ∈ [0.3, 3]. We show the resulting bound in the resultsof Section 6.3.3. For r ' 1, Eq. (B.6) approximates the true boundary between metastable andunstable models well. However, we expect this constraint to deteriorate as one moves awayfrom the assumption of SU(3)C D-flatness by choosing soft masses with r  1 or r  1. Weshow an explicit example of this in Fig. B.1.We emphasize that this bound is a very rough guideline for metastability in the MSSM in aspecific corner of the parameter space and should only be used as a first order approximation.A full numerical analysis is required when any of the above assumptions are violated or betterprecision is required.194Appendix CRenormalization Group Equations inthe Inert Doublet ModelHere we list the beta functions for the inert 2HDM at one-loop order. The general form of theRG equations isdλdt= 116pi2βλ, (C.1)where t = lnQ/Q0 and Q0 is a reference scale. We take Q0 = MZ . The U(1)Y gauge couplinghas the GUT normalization: g1 =√5/3g′. The beta functions below have been checked withSARAH [438]. Partial one loop results can be found in Refs. [413, 421, 476] for dimensionlessparameters only. These agree with the formulae below.The gauge coupling evolution is determined byβg1 =215g31 (C.2)βg2 = −3g32 (C.3)βg3 = −7g33. (C.4)For the third generation Yukawas we haveβyt = −1720g21yt −94g22yt − 8g23yt +92y3t +32y2byt + yty2τ (C.5)βyb = −14g21yb −94g22yb − 8g23yb +32yby2t + yby2τ +92y3b (C.6)βyτ = −94g21yτ −94g22yτ + 3y2t yτ + 3y2byτ +52y3τ . (C.7)Next we consider the scalar potential parameters. The evolution of the dimensionless quarticcouplings λi is governed by195βλ1 = −95g21λ1 − 9g22λ1 +27200g41 +920g22g21 (C.8)+ 98g42 + 24λ21 + 2λ23 + λ24 + λ25 + 2λ3λ4+ 12λ1y2t − 6y4t + 12λ1y2b − 6y4b + 4λ1y2τ − 2y4τβλ2 = −95g21λ2 − 9g22λ2 +27200g41 +920g22g21 +98g42 + 24λ22 (C.9)+ 2λ23 + λ24 + λ25 + 2λ3λ4βλ3 = −95g21λ3 − 9g22λ3 +27100g41 −910g22g21 +94g42 (C.10)+ 4λ23 + 2λ24 + 2λ25 + 12λ1λ3 + 12λ2λ3 + 4λ1λ4+ 4λ2λ4 + 6λ3y2t + 6λ3y2b + 2λ3y2τβλ4 = −95g21λ4 − 9g22λ4 +95g22g21 + 4λ24 + 8λ25 (C.11)+ 4λ1λ4 + 4λ2λ4 + 8λ3λ4 + 6λ4y2t + 6λ4y2b + 2λ4y2τβλ5 = −95g21λ5 − 9g22λ5 + 4λ1λ5 + 4λ2λ5 + 8λ3λ5 (C.12)+ 12λ4λ5 + 6λ5y2t + 6λ5y2b + 2λ5y2τThe beta functions for the mass parameters are given byβµ21 = −910g21µ21 −92g22µ21 + 12λ1µ21 + 4λ3µ22 + 2λ4µ22 (C.13)+ 6µ21y2t + 6µ21y2b + 2µ21y2τβµ22 = −910g21µ22 −92g22µ22 + 12λ2µ22 + 4λ3µ21 + 2λ4µ21. (C.14)Finally, the anomalous dimensions for the Higgs and the inert scalar areγh = −920g21 −94g22 + 3y2t + 3y2b + y2τ (C.15)γφ = −920g21 −94g22. (C.16)196

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