Radiative Corrections to the Dark Matter Relic Density

Transcription

1 Radiative Corrections to the Dark Matter Relic Density Vincent Theeuwes Radboud University Institute for Mathematics, Astrophysics and Particle Physics Theoretical High Energy Physics Supervisor: Dr. Wim Beenakker September, 0

2 Contents Introduction 4. Conventions Dark Matter 5. Why Dark Matter Hot and Cold Dark Matter Measuring Dark Matter Direct Measurements Indirect Measurements Standard Model 8 3. Gauge Invariance Symmetry Breaking Feynman Rules Renormalization Supersymmetry 4 4. Why Supersymmetry Generators of supersymmetry Supersymmetric Lagrangian Minimal Supersymmetric Standard Model Relic Density 9 5. The Boltzmann Equation Summing Up the Boltzmann Equation Thermal Averaging Reformulation of the Boltzmann Equation Freeze-Out Approximation Relic Density Scenarios 5 6. Mixed Bino Light Wino Heavy Wino Higgsino Mixed Higgsino Bino Annihilating to Charged Leptons Bino Annihilating to Bottom Anti-Bottom Bino Coannihilating with Stau Corrections 8 7. Sommerfeld Corrections Effective Radiation Rules In Depth Treatment of Fermions Emitting a Vector Boson or CP-Odd Scalar Coulomb Corrections Massive Boson Exchange Summing Up the Correction Terms Electroweak Sudakov Corrections Results Heavy Wino Mixed Higgsino Conclusion 39 A Feynman Rules 40 B In Depth Calculation of Sommerfeld Corrections 40 B. Coulomb Corrections B. Heavy Boson Exchange

3 C A Special Logarithmic Integral 46 3

4 Introduction With the recent launch of the Planck satellite we expect to receive more precise results for the dark matter relic density. This dark matter is non-radiative weakly interacting matter that is needed to complete the cosmological model. Its relic density is the current day density of this relic of the early universe. The fact that there will be more precise experimental results means we will also need better theoretical predictions. In order to improve these predictions we need to find a manner in which to treat the higher order quantum corrections, usually referred to as radiative corrections. The two types of potentially large radiative corrections that will be looked at are Sommerfeld corrections, involving the exchange of a boson by the low velocity initial state particles, and Sudakov corrections, which involve the emission of virtual bosons by the final state particles if they are energetic enough. The goal of this research is to analyze which corrections are important in different scenarios and if higher order effects also will need to be taken into account. For a dark matter candidate the Minimal Supersymmetric Standard Model MSSM will be used. In the sections, 3 and 4 a rough background will be given on dark matter, the Standard Model and supersymmetry. Next there is an in depth look at the manner in which to calculate the dark matter relic density in section 5. In section 6 the different scenarios of the MSSM that are compatible with in the dark matter relic density are listed. Section 7 shows an assessment of how large the radiative corrections to this calculation can be, by considering the Sommerfeld and Sudakov corrections.. Conventions In this thesis natural units will be used = c =. Lorentz indices will be given by Greek letters and will run from 0 to 3. Roman letters are used to indicate the generators of gauge groups. Repeated indices are summed over. The Minkowski metric is defined as g µν = diag,,, and the Dirac gamma matrices are defined as: γ 0 = 0 0, γ i = 0 σi σ i 0 where is the identity matrix and σ i the Pauli matrices: 0 0 i 0 σ =, σ 0 =, σ i 0 3 = 0 The γ 5 matrix is defined by γ 5 iγ 0 γ γ γ 3 and the corresponding left- and right-handed projection operators by P L γ 5 and P R + γ 5. These projection operators are used to obtain the left- and right-handed part of a spinor. The abbreviation h.c. stands for Hermitian conjugate. The log function denotes the natural logarithm... 4

5 Dark Matter. Why Dark Matter In 933 Zwicky noticed that the rotation velocities of individual galaxies in the Coma cluster were larger than one would expect. In order for these galaxies to stay within this cluster the mass of the cluster needs to exceed the sum of the masses of the individual galaxies within this cluster. This conclusion was reached using the virial theorem, which states that if the cluster is in dynamical equilibrium the potential and kinetic energies are related as K = U. where U is the potential energy and K the kinetic energy. This allows us to find the average potential energy of the cluster if the kinetic energy is accurately known. This showed Zwicky that there is other matter that we do not observe as it emits no light, hence it was named dark matter. Other than this there is also a need for dark matter in galaxies. The velocity of stars within galaxies is also larger than one would expect. Through the use of Kepler s third law GM r v r =. r we can determine the velocity v at a distance r from the center of the galaxy with an enclosed mass of M. The moment that we go far enough outside the center of the galaxy, where there are hardly any luminous objects, we would expect M to be constant. This means the velocity will drop as v r /. This is not what we observe as can be seen in figure. Instead we observe that the velocity almost becomes constant implying that the mass still increases as M r. This implies that the mass still increases even if there are no luminous objects. In fact the outskirts of galaxies, the part where v r /, has never been convincingly observed in galaxies so we have no idea how much mass is present in individual galaxies. Figure : The observed and expected rotation curves of the dwarf spiral galaxy M33 [8]. An example of another way to explain this difference in rotation velocities is Modified Newtonian Dynamics MOND, but whereas dark matter is local and can be different in other circumstances MOND is a global change and needs to act the same way everywhere. We observe that in galaxies the dark matter appears to increase with distance, whereas in clusters the opposite is true. Gravitational lensing shows that the dark matter in clusters is mainly in the central region, while as we can see in figure inside a galaxy the dark matter is spread out more. Another observation supporting the dark matter hypothesis is the collision of clusters. As can be seen in figure, as the clusters pass through one another the gas inside heats up as it interacts with the gas of the other cluster and slows down while emitting x-rays, seen in pink. Dark matter interacts weakly with other matter and will pass straight through. This we can see as the majority of the mass in the clusters, seen in blue, passes through. The location of the majority of the mass is determined through gravitational lensing. The final evidence can be found in the cosmic microwave background. The way in which structure formed in our universe requires some extra matter which we do not see. There will be more mentioned on this in section.. 5

6 Figure : The collision of the bullet cluster, in pink is the hot gas emitting X-rays whereas blue is the location of most of the matter [9].. Hot and Cold Dark Matter We now know why we need dark matter and that it needs to interact very little with other matter, but what other properties do we know of the dark matter. We need dark matter to be of a non-baryonic nature as baryonic models have problems explaining the small density perturbations we have in our universe. The next question we can ask is what kind of dark matter particle we have. If most energy is in the form of kinetic energy at the time of decoupling, the dark matter type is called hot, whereas if most energy is in the form of mass at the time of decoupling, it is called cold. Everything in between is called warm. In the case of hot dark matter the particles have high relativistic velocities during structure formation. Due to this the structure formation will start at large scales, as any smaller structures will be wiped out by the dark matter particles. Typically these large structures will be clusters. The smaller structures, such as galaxies, will be formed by fragmentation in the larger structures. This is called the top-down scenario for structure formation. This theory is preferred less as it has problems forming small structures sufficiently early, to account for the galaxies that have been observed at very high redshifts. On the contrary cold dark matter has very low non-relativistic velocities during structure formation. It will start forming small structures due to density fluctuations. Those small structures will start to clump together forming the larger structures. This type of dark matter model is used most. In order to complete this model the cosmological constant, dark energy, also needs to be introduced. This final model is called Lambda Cold Dark Matter ΛCDM and is the model that is mostly used to explain our universe in its current state. Now that there is a model for the evolution of the universe the final parameters can be solved. The remaining parameters are how much dark matter and dark energy there is in the universe. The main focus in this study is dark matter, so the dark matter density is the variable of interest. The dark matter density cannot be found using the rotation velocities within galaxies or clusters as the density that needs to be found is a global density and these are all local effects. Instead we can simulate the evolution of the universe using different dark matter densities. In this manner the densities for which the simulation matches our observations can be found. The observations currently being used for this are made by the Wilkinson Microwave Anisotropy Probe WMAP, which looks at the cosmic microwave background radiation. The fluctuations in the cosmic microwave background are what is used to find the dark matter and dark energy densities. This result can then be compared to the predictions for a specific dark matter candidate. There will be more on this in section 5..3 Measuring Dark Matter In order to find dark matter we will need to measure it. There are two ways of measuring dark matter: direct measurements and indirect measurements. Direct measurements involve interaction with dark matter particles around us, whereas indirect measurements involve results of processes that require a dark matter particle..3. Direct Measurements In order to directly measure dark matter, experiments can look for an oscillating variation in the particle interactions with nuclei through nuclear recoils. This effect is due to the fact that at different moments of the year the velocity of earth through the "sea" of dark matter is different. Our earth travels around the sun at approximately 30 km/sec as our sun orbits the galaxy at approximately 0 km/sec. In addition the earth s orbit is inclined at a 60 degree angle to the galactic plane. Due to this the difference in velocity 6

7 between the earth and the dark matter depends on whether it is summer or winter, as can be seen in figure 3. As the cross section of dark matter with nuclei is dependent on the velocity of the dark matter, we expect a oscillating variation in the dark matter interactions with nuclei. The disadvantage of this method is that it can only be used if the cross section is large enough, because otherwise the oscillation will have far to low statistics. Figure 3: The difference in velocity of earth throughout the year [8]. This method is used in multiple detectors in order to find a dark matter signal. Some examples are: DAMA, CoGeNT and CRESST-II. These three have all found a signal: DAMA [7] at 8σ, CoGeNT [8] at.8σ and CRESST-II [9] at 4σ. However the signal of CRESST-II is incompatible with those of DAMA and CoGeNT. In addition all three are incompatible with XENON [0] and CDMS [], which do not rely on this method. The question as to what caused these signals in DAMA, CoGeNT and CRESST-II still remains. XENON and CDMS work with direct measurements without looking for an annual oscillation and as such can also work for lower cross sections..3. Indirect Measurements Dark matter can also be measured indirectly. This involves the annihilation of two dark matter particles, which can occur if the dark matter particle is its own anti-particle. What is actually measured is the resulting particles of this annihilation instead of the dark matter particles. The final state particles will eventually become electrons, neutrinos, photons and hadrons. This could produce a significant signal of high energetic particles from the galactic halo, which is the part of the galaxy where only dark matter resides. Some examples of experiments that are based on this method are: the Fermi satellite for the detection of gamma rays and electrons/positrons and Pamela for the detection of electrons/positrons. Alternatively it is possible to look at massive objects, such as the earth and sun. As dark matter can scatter off nucleons inside massive bodies, we expect them to slow down and a large amount of dark matter may gather gravitationally at the center of these massive objects. As the dark matter density is higher at the center of these objects the rate of annihilation will be higher, but only neutrinos would be able to escape these massive objects as it has very little interaction. These high energetic neutrinos should then be measurable here on earth as an excess of neutrinos. Experiments looking for these neutrinos are for example Icecube and Super-Kamiokande. The final manner in which we can detect dark matter is through production in collider experiments. These dark matter particles will be seen as missing energy. 7

8 3 Standard Model Right now the Standard Model is the most successful theory that describes elementary particle physics. This is the first stop on the journey in order to find the dark matter candidate. In this section there will be a brief overview of the basic concepts of the Standard Model, specifically focusing on the concepts important to this thesis. For a more detailed description please refer to other literature, for example []. 3. Gauge Invariance In nature there are certain symmetries, due to this the physics will remain invariant under transformations that belong to these symmetries. These symmetries are described by a group, for example the Standard Model uses the U SU SU 3 group. This means that the Standard Model is derived by assuming the theory is invariant under this group. This section will show how one would find a theory by assuming invariance under a transformation. We will start with a spin / particle, which is described by a spinor ψ x with mass dimension 3. This spinor has 4 degrees of freedom in general, from the particle and from the antiparticle. These degrees of freedom coincide with the helicity states, a left- and a right-handed part. We will start with the free Dirac Lagrangian, this describes a spin / particle: L = i ψ x γ µ µ ψ x m ψ x ψ x 3. Here m is the mass of the particle, γ µ are the Dirac gamma matrices and ψ = ψ γ 0. This is the most general renormalizable Lagrangian for a single fermion. This is because there cannot exist any terms with a negative mass dimension coupling and the mass dimension of each term needs to be 4. For more information on renormalization see section 3.4. This Lagrangian is invariant under a global U transformation: ψ x e iα ψ x 3. Where α is a real number independent of x. If we consider a local U transformation, α is now dependent of x and the Lagrangian is no longer invariant. The problem lies in the kinetic term, as i ψ x γ µ µ ψ x i ψ ] x e iαx γ µ µ [e iαx ψ x = i ψ x γ µ µ ψ x ψ x γ µ ψ x µ α x 3.3 The problem here lies in the derivative µ, this implies that for a local U invariant Lagrangian we need to introduce a covariant derivative that makes this term invariant under a local transformation: This will work if we define the covariant derivative as: D µ ψ x e iα D µ ψ x 3.4 D µ µ ig A µ x 3.5 Here we introduce a coupling constant g and a covariant vector field A µ that transforms as A µ x A µ x + g µ α x 3.6 If we assume A µ to be the electromagnetic field we can recognize this transformation as the gauge freedom in electrodynamics. In order to complete this Lagrangian a kinetic term for A µ also needs to added. This kinetic term has the form 4 F µνf µν, where the electromagnetic tensor is defined as F µν µ A ν ν A µ. This results in a final Lagrangian: L = i ψ x γ µ D µ ψ x m ψ x ψ x 4 F µνf µν 3.7 Note that this does not contain a mass term for the vector field as this would have the form M A µ A µ, which is not invariant under the local U transformation. As was stated before the Standard Model uses the U SU SU 3 group, where the group U represents the hypercharge interaction, SU represents the weak interaction and SU 3 represents the strong interaction. However, we have only looked at the U invariance up until now. The change that occurs when we look at a general SU N transformation is that generators of this group, indicated by T a, appear in certain places. The transformation will now be ψ x e iαa xt a ψ x 3.8 8

9 Particle representations Name Symbol SU 3, SU, U Quarks Left-handed quark doublet Q = u L, d L 3,, 6 Right-handed up-type quark u R 3,, 3 Right-handed down-type quark d R 3,, 3 Leptons Left-handed lepton doublet L = ν L, e L,, Right-handed charged lepton e R,, Gauge bosons Gluons g 8,, 0 W -bosons W, 3, 0 B-boson B,, 0 Higgs Higgs boson H,, Table : The representations of the Standard Model particles. Each of the quarks and leptons has heavier families in addition. The fermions are split into left- and right-handed parts as only the left-handed part couples through the weak interaction W. The W - and B-bosons are the interaction eigenstates, after symmetry breaking section 3. these will mix to form the mass eigenstates W ±, Z and the photon γ. Due to this the U representation labeled by the eigenvalue of the hypercharge generator Y is related to the electric charge Q = T 3 + Y, where T 3 is the third component of the SU isospin. Right-handed neutrinos can also be added to the Standard Model, but they do not couple to anything. The covariant derivative will now be defined as D µ µ ig N A a µ x T a 3.9 This means that there is now one vector field for each generator, for SU this is 3 and for SU 3 this is 8. These vector fields transform as A a µ x A a µ x + g N µ α a x + f abc A b µα c 3.0 Where f abc comes from the commutation relation of two generators: [ T a, T b] = if abc T c. Finally the field tensor is given by F a µν = µ A a ν ν A a µ + g N f abc A b µa c ν 3. The new term in the field tensor, g N f abc A b µa c ν, now adds self interactions among the vector fields. In order to fully describe the particle interactions in the Standard Model we need to give the representation of each particle. If we say something is in the representation 3 it means there are three different types of this particle, for example an up quark has a representation 3 under SU 3 since the up quark has three different colors. However if the representation is under SU 3 the particle is a singlet under the strong interaction. This means it does not feel the corresponding interaction. The representations of each particle can be seen in table. 3. Symmetry Breaking It is a great feat that we have derived the Standard Model gauge bosons from symmetries, however there is one problem. In this theory none of the particles are allowed to have a mass. As was mentioned before in section 3. the gauge bosons are not allowed to have mass as this term violates the symmetries. In addition fermions are also not allowed to have mass as the mass term has the form m ψ x ψ x = m ψ L x ψ R x+m ψ R x ψ L x. This term connects right- and left-handed particles to one another, but as can be seen in table left-handed particles are doublets of SU and right-handed particles are singlets. This means the fermion mass term violates the SU symmetry. The simplest way to allow a fermion mass term is to introduce a complex SU doublet φ. This doublet is able to couple the left-handed doublet to the right-handed singlet L = y ψ L φψ R = y ψu, ψ φu d ψ L φ R = y ψ u,l φ u ψ R y ψ d,l φ d ψ R 3. d Here u and d are used for the up and down components of the SU doublet and the coupling y is different for fermions with different masses. 9

10 This is not enough to obtain fermion mass terms. In order to produce mass terms the field φ needs to have a vacuum expectation value VEV. In that case it can be written as φ = φ + η where φ is the VEV and η contains the particle properties of φ quantum fluctuations. The η produces a coupling between three fields and the VEV term produces a fermion mass term. Using a SU transformation we can bring the VEV in the following form: φ = 0 v 3.3 Where v can be made positive and real using a U transformation. If this is inserted into the Lagrangian term we obtain L = y ψ L φψ R = yv ψd,l ψ R yη u ψu,l ψ R yη d ψd,l ψ R 3.4 This looks familiar the first term results in a fermion mass term with m = yv and the other terms result in interaction terms with new particles η u and η d. This generates a mass term for the down-type particle, the up-type particle mass term is obtained by using the charge conjugate of the field φ c. Here the charge conjugate is defined as φ c = iσ φ φ = d φ 3.5 u The Pauli matrix σ is responsible for allowing the upper component of φ c to have a VEV term, which results in a mass term for up-type particles. This means we are able to produce mass terms for all fermions. In order to obtain mass terms for the vector bosons we need expand the kinetic term for the scalar, D µ φ D µ φ, around the VEV, where D µ = µ i g B µ ig A a µt a with B µ the U field and A a µ the SU field the W. The relevant terms are L = 0, v g B µ + g A a µt a g Bµ + g A b,µ T b 0 v where T a = σa, V µ = A µ, A µ, A 3 µ, B µ and M = v 4 g g g g g 0 0 g g g = V µm V µ 3.6 This produces the mass eigenstates of the massless photon: A µ = g g A 3 +g µ + g B µ. In addition this also produces three massive boson eigenstates: the Z-boson, Z µ = g g A 3 +g µ g B µ, with mass m Z = v g + g and the W -bosons, W µ ± = A µ iaµ, with mass mw = v g. The next order in the expansion of φ is η, which will behave like a scalar particle known as the Higgs boson. This particle has one degree of freedom. The other three degrees of freedom will be used for the, now massive, gauge bosons. This is because a massless vector boson has two degrees of freedom, whereas a massive vector boson has three degrees of freedom. The only question that remains is how we can produce a VEV for our field. The fields we have used so far, fermions and gauge bosons, cannot obtain a VEV without breaking either Lorentz or gauge invariance. However a complex scalar field can obtain a VEV. In order to see this we will look at the most general Lagrangian for a complex scalar field that is gauge invariant and renormalizable: L = D µ φ D µ φ V φ = D µ φ D µ φ µ φ φ λ 4 φ φ 3.7 where λ, µ R. In order to make sure the potential is bounded from below we must have λ > 0. However for µ there are still two possibilities: If µ 0 the scalar potential has one minimum at φ = 0, this means there will be no vacuum expectation value resulting in no mass generation. However if µ µ < 0 the potential has a minimum at φ = λ, this means there will be a vacuum expectation value and masses will be generated This means that the last case can generate masses for both the fermions and the gauge bosons. This theory is called the Higgs mechanism after one of its discoverers. Another name is spontaneous symmetry breaking. This name refers to the fact that around the minimum the symmetry seems broken, but the potential itself is in reality still symmetric. 0

11 3.3 Feynman Rules We now have a Lagrangian, but this is not what we can measure in experiments. In experiments we measure cross sections. In order to get to the cross section from the Lagrangian the following steps need to be followed.. Go from the Lagrangian to the Hamiltonian in the standard manner.. Use the Hamiltonian to find the time evolution operator from initial time t i to final time t f in the following way: U t f, t i = T exp i ˆt f t i H I dt Where T is the time ordering operator and H I is the interaction Hamiltonian. 3. Define the S-matrix as S lim t U t, t. From this we can find the transition amplitude for a particular process using final state S initial state. 4. Combine the transition amplitude with the relativistic kinematics in order to give the cross section. It is quite cumbersome to do these steps for every cross section. Even worse the transition amplitude seems impossible to calculate. Luckily we can use perturbation theory to calculate a base term, tree level, and corrections to this base term. We can expand the exponential term using a Taylor expansion. This expansion can then be represented diagrammatically using propagators for propagating fields and vertices for interactions between fields. These propagators and vertices have Feynman rules that describe them in formula form. Once these rules are obtained all possible diagrams compatible with the initial and final state need to be drawn and calculated. As there are an infinite amount of diagrams we now make use of perturbation theory, the diagrams without loops are tree level, the base term. For each loop we add we go one order higher in the perturbation. To give an example, consider a real scalar field φ with Lagrangian L = µ φ µ φ m φ 3! gφ3. This theory is often called φ 3. The Feynman rules of this theory are: Propagator : Vertex : ph hi j i p m + iɛ ig where p is the momentum. A list of the used Feynman rules and conventions is given in appendix A. 3.4 Renormalization If calculations beyond tree level are performed we often find that these are divergent. As we cannot measure infinite cross sections, there must be something else going on here. As an example we can look at the φ 3 theory that was introduced in section 3.3. In every diagram containing a propagator we can add a loop by replacing this propagator with: i g = p m + iɛ ˆ d 4 k π k m + iɛ k p m + iɛ We can already see the divergence appear here. This integral has four powers of k in the numerator and four in the denominator. This means this integral is logarithmically divergent for k. This means we need extra tools to solve this integral. The most important step is to introduce a cut-off to the

12 momentum, so instead of integrating to infinity we will integrate all four components of k to a cut-off Λ. In the end you need to take the limit Λ. Here we define Σ,Λ p, m i g ˆΛ Λ d 4 k π k m + iɛ k p m + iɛ This one-loop contribution is not the only contribution to the propagator. We also have higher corrections: = = i p m iσ,λ p, m + iɛ i p m + iɛ i p m + iɛ + iσ,λ p, m Σ,Λ p, m / [p m + iɛ] = i p m iɛ i p m Σ,Λ p, m + iɛ 3.0 This looks as if the mass of the particle is altered. This effective propagator now has a new effective mass m eff = m + Σ,Λ p, m that has an energy dependence. This implies that we need to redefine our theory in terms of actual measurable parameters and not the parameters in our Lagrangian. In this manner the divergences can be absorbed. This implies that we need to eliminate the parameter m in our cross sections in favor of the physical parameter: m phys µ at a reference scale µ. This can be done by writing m as: m = m phys µ + δm 3. The extra term δm is called the counterterm as it is chosen so that it cancels the divergences of our theory. This can also specifically be done in the example of the φ 3 theory. Here we can isolate the divergent part in the following manner: Σ,Λ p, m = i g ˆΛ Λ +i g d 4 k π 4 k m + iɛ k p m + iɛ ˆΛ Λ d 4 k π 4 k µ + iɛ Σ fin,λ p, m, µ + Σinf,Λ µ k µ + iɛ Here the finite part Σ fin,λ p, m, µ contains all the physical information of the process p and m and the infinite part Σ inf,λ µ only contains the divergence and the reference scale. We now define the renormalized one-loop self energy as: Σ R p, m phys µ lim Σ,Λ p, m phys µ + δm = Σ fin p, m phys µ, µ + Λ δm + Σ inf µ 3. where δm is chosen in such a way that δm +Σ inf µ is finite, this is always possible for a renormalizable theory. The effective propagator will then take the form: i p m phys µ ΣR p, m phys µ + iɛ As long as the theory is fully renormalizable this procedure can then be repeated for the other masses, the coupling constants and even the fields themselves. A theory is fully renormalizable so long as no coupling constants have a negative mass dimension. This can also be repeated for the full standard model, but it is a bit more cumbersome.

13 The reference scale µ is an arbitrary parameter that we can choose. This means we could have redone everything with a different reference scale. Why would we want to choose a different value for this parameter? The answer is simple. As we try and find out more about the Standard Model we will try to reach higher energy scales. There are two options at these higher scales. The first is to calculate higher loop corrections to get a more precise result for our energy dependent parameters. On the other hand we can also choose a reference scale close to the energy of our process and expand around our reference scale. Using the second method we can insert this value back into our tree level calculation to gain a lot more precision. The equations that govern this reference scale dependence are the renormalization group equations. As this scale can be chosen arbitrarily, observables cannot depend on it. This can be used to find the renormalization group equations. Consider an observable O, which depends on some couplings g i and some masses m j. The fact that it does not depend on the reference scale implies: [ 0 = µ do dµ = µ µ + µdg i µ dµ g i µ + µdm j µ ] dµ m j µ 3.3 Using a set of observables, for example cross sections or decay widths, a set of coupled differential equations can be found. These will govern the scale dependence of the parameters. In practice these observables will only be calculated up to a certain loop order, this means the renormalization group equations will still depend on loop calculations. 3

14 4 Supersymmetry As was stated in section, we need a cold dark matter candidate. The Standard Model offers only one candidate, the neutrino, but this would result in hot dark matter. In other words we need an extension of the Standard Model. This section will give a brief explanation of a possible extension called supersymmetry. First the reason why to introduce this supersymmetric theory will be addressed and the things it can offer us. Next an explanation of the main concepts of supersymmetry will be given. Finally some general notes are given about the particle and mass spectrum. A more detailed description of supersymmetry can be found in [3]. 4. Why Supersymmetry A candidate for dark matter is not the main reason to introduce a supersymmetric theory. The main reason is because there is a problem in the Standard Model called the hierarchy problem. The hierarchy problem The Higgs particle couples to all massive particles, this means there are fermionic loop corrections of the following form: These corrections are proportional to y Λ, where y is the Yukawa interaction coupling constant and Λ is the cutoff scale. These quadratic divergences cannot be canceled by other diagrams. This means the corrections will need to be used to renormalize the Higgs mass. If we assume the Standard Model to hold until the Planck scale this will present a huge correction. We can renormalize this, but this will require us to fix the Lagrangian parameter µ very precisely. As an indication of how precise this needs to be done we will assume the Standard Model holds up to the Planck scale, Λ 0 9 GeV. This means renormalization needs to change the squared Higgs mass from 0 9 GeV to 0 GeV GeV requiring µ to be tuned to 34 digits. There is a way in which this problem can be fixed. This solution can be seen when looking at corrections for new scalar particles. These scalar particle loops take the form of: These corrections are proportional to λλ and can be used to cancel the fermion loops if we assume λ = y. However, this will only hold if there is a scalar particle for every fermion. This is where supersymmetry comes in. The stability bound There is another reason to suspect that there will be new physics. As a Higgs candidate has been found at the Large Hadron Collider LHC at a mass of approximately 5 GeV we can check if the theory will hold for all energy scales. What will happen, as can be seen in figure 4, is that the theory will no longer be stable above a scale of approximately 0 0 GeV where the λ of the Higgs potential becomes negative. In that case the real minimum of the Higgs potential will be at ± and the minimum we are in now is a local one and therefore no longer stable. Taking into account the theoretical uncertainties, this could imply that there is new physics before this energy scale. Dark matter As was stated before the dark matter candidate is still missing. However, under certain assumptions supersymmetry can offer this candidate. More on this assumption will be told in section 4.4. Coupling unification The fourth and last reason that will be listed here is the unification of coupling constants. At one loop order the inverse gauge couplings squared can be plotted against log µ resulting in straight lines. In the Standard Model these lines do not meet in one point. However if you where to use supersymmetry the coupling will meet in one point. This allows for these couplings to be unified at one scale, as predicted by Grand Unified Theories GUTs. 4

15 Figure 4: The stability bound at a scale Λ for a Higgs mass M H within the Standard Model. The current Higgs candidate mass is shown as a red line. This shows that the potential of the current Higgs candidate could have problems with a stable minimum []. Figure 5: The running coupling constants α = 4πg as a function of the logarithm of the energy scale. This shows that the Standard Model does not have the couplings unify at one scale, but the Minimal Supersymmetric Standard Model MSSM does [3]. 4. Generators of supersymmetry Now that some of the reasons to use supersymmetry are known we can move on to the next question. What is supersymmetry? Supersymmetry is a symmetry that relates fermions and bosons to one another in the following form: Q Boson = Fermion Q Fermion = Boson This implies that for every boson there exists a fermion and for every fermion a boson. As Q changes a boson into a fermion and a fermion into a boson it is fermionic transformation. This implies that the algebra of the generator Q uses anti-commutation relations. These anti-commutation relations will be proportional to the momentum: { Q, Q } P µ 4. In addition the supersymmetry generators also commute with the momentum and the generators of the gauge transformations. Due to this fact the bosons and fermions will have the same mass P and quantum numbers. In this manner these fermions and bosons can be grouped together into supermultiplets. Within such a supermultiplet the fermionic and bosonic degrees of freedom need to be the same. There are two types of supermultiplets: chiral supermultiplets and gauge supermultiplets. The first consists of a Weyl spinor and a complex scalar field and the second of a gauge boson field and a Majorana spinor. Weyl spinors are the left- or right-handed part of a spinor. Majorana spinors are real spinors. 5

16 As can be seen in table there are no Standard Model particles with the same quantum numbers. This means for every Standard Model particle we need to introduce a new particle, which will add a whole range of new particles. This adds another problem as we have not found any of these new particles yet, even though they should have the same mass. This implies supersymmetry will need to be broken. In order to name all these new particles, a simple convention is used. All supersymmetric partners of fermions are scalars, so they will get the prefix "s" for scalar in front of its Standard Model name. All supersymmetric partners of bosons receive the suffix "ino". 4.3 Supersymmetric Lagrangian In order to build a supersymmetric theory we need to create a Lagrangian. This Lagrangian will be made in the most general manner that is renormalizable and conserves supersymmetry and gauge symmetries. The first step is to look at our supermultiplets. These supermultiplets have equal degrees of freedom when the fields are on-shell satisfy the equations of motion. However when we look at the off-shell degrees of freedom we realize that they do not match. A Weyl spinor has four off-shell degrees of freedom whereas a complex scalar field only has two. This requires us to introduce a complex scalar auxiliary field F that has no degrees of freedom when on-shell. This means its equation of motion needs to be of the form F = 0, resulting in a Lagrangian of the form: L F = F F 4. The same can be done for the gauge supermultiplet. Here a gauge boson field has three off-shell degrees of freedom, whereas the Majorana spinor has four. In this case we need to introduce a real scalar auxiliary field D with a Lagrangian of the form: L D = D 4.3 These additional fields F and D do not have any kinetic terms as they do not propagate. This is because they are completely determined by the Lagrangian in the on-shell configuration and cannot change. Now that we have the field the next step is to find the interaction terms in this Lagrangian. If we write down the most general form that is allowed by renormalization, this means the mass dimension of the interaction terms cannot exceed four. This leads to an interaction Lagrangian of the form: L int = W ij ψ i ψ j + W i F i + h.c. 4.4 where ψ i is the left-handed Weyl spinor with inner product defined as ψ i ψ j = ψ T i iσ ψ j. The right-handed spinors first need to be conjugated to bring them into a left-handed form. The variables W ij and W i are polynomials defined as the functional derivatives of the superpotential in the following form: δ W W ij = δφ i δφ j W i = δw δφ i 4.5 with W the superpotential. This superpotential is an analytic function of the complex scalar fields φ i in the following way: W = L i φ i + M ij φ i φ j + 6 yijk φ i φ j φ k 4.6 If this is inserted into the interaction Lagrangian it can be seen that the M ij term contains a fermionic mass term and the y ijk term contains a Yukawa interaction. The linear term in φ i is only possible for a gauge singlet, which does not occur in the Minimal Supersymmetric Standard Model. The new equations of motion for the auxiliary field F and its complex conjugate become F i = W i and F i = W i. The final step required in order to complete the supersymmetric Lagrangian is to require gauge invariance. The fields φ i and ψ i form a multiplet in the fundamental representation of the gauge group. Introducing gauge invariance happens in a similar manner to the Standard Model, however in order to maintain supersymmetry we also need to introduce interactions between the chiral multiplet and the gaugino λ a and the auxiliary field D a. Here a takes values in the adjoint representation of the gauge group. In the end the auxiliary field can be eliminated using the equation of motion D a = gφ T a φ. This will finally result in a complete Lagrangian: L = D µ φ i D µ φ i + iψ i σ µ D µ ψ i 4 F a µνf µν,a + iλ a σ µ D µ λ a W ij ψ i ψ j + h.c. W i W i g φ T a φ g φ T a φ λ a + h.c. 4.7 where σ µ =, σ, σ, σ 3, λ a is the Majorana spinor with D µ λ a = µ λ a + gf abc A b µλ c and T a, D µ φ, D µ ψ and g defined as before. This Lagrangian was found only by specifying the supersymmetry and the superpotential, and by requiring renormalizability and gauge invariance. So in fact this theory only requires the superpotential and the same gauge invariance as the Standard Model. 6

17 4.4 Minimal Supersymmetric Standard Model We now have a general supersymmetric theory. This still leaves us freedom in the way in which to construct this theory. Any number of chiral and gauge supermultiplets can be introduced into this theory. For this thesis we will stick to the simplest expansion of the Standard Model. This Minimal Supersymmetric Standard Model MSSM is defined by the superpotential: W = ũ R y u Q H d R y d Q H ẽ R y e L H + µh H 4.8 where y i are 3 3 Yukawa coupling matrices in family space and the SU contractions are Q H = Q T iσ H. The sfermion fields correspond to their Standard Model counterparts listed in table. One thing to notice in this superpotential is that, while the Standard Model only has one Higgs doublet, this model requires two Higgs doublets. This is because in the Standard Model we used the charge conjugate of the Higgs doublet to give mass to the up type particles. This cannot be done here because the charge conjugate is not an analytic function of the field. This means it is not allowed to be a term in the superpotential. The second Higgs doublet introduced is a Higgs doublet with opposite hypercharge so that it is able to give mass to the up-type fermions. These two Higgs doublets are: H = H 0 H H = H + H 0 Just as in the Standard Model only the neutral components are allowed to get a vacuum expectation value. If this is not required the photon would also gain a mass. The Yukawa couplings in the MSSM can now be used to solve the hierarchy problem as these create quadratic interactions with the right coupling constant as stated in section 4.. R-Parity What still needs to be shown is that supersymmetry will offer a dark matter candidate. The superpotential of the MSSM is not the most general form that can be used. Other terms that can be introduced violate baryon and lepton number and will quickly result in an unstable proton. An example of this can be seen in figure Figure 6: An example of a process that could cause proton decay p π 0 e + in a general supersymmetric theory. The extra couplings this example requires are ud d and u de. In order to explain the absence of these terms we will introduce a new conserved quantum number. This quantum number is called R-parity and is defined as P R = 3B L s where B is the baryon number, L is the lepton number and s is the spin of the particle. For Standard Model particles this quantum number is always. However for the supersymmetric counterparts the spin is shifted by ±. This implies that R-parity is for the supersymmetric particles. If we demand R-parity to be conserved this will only allow interactions with an even number of supersymmetric particles. The proton decay of figure 6 has interactions with only one supersymmetric particle, which are not allowed under conservation of R-parity. This also means that any process has an even number of external supersymmetric particles, which results in the lightest supersymmetric particle LSP being stable. This is because it would need to decay to at least one supersymmetric particle but it cannot as all other supersymmetric particles are heavier. If this LSP is neutral it could offer a valid cold dark matter candidate. Spontaneous Symmetry Breaking The MSSM acts in the same way as the Standard Model. This means the particles that are massless in the unbroken Standard Model are also massless in the MSSM. So we also need spontaneous symmetry breaking in the MSSM. The squared Higgs mass terms in the Higgs potential are: µ H 0 + H + + H 0 + H

18 This term is positive, so it only has one global minimum at the origin. This means we cannot have a non-zero vacuum expectation value and also no spontaneous symmetry breaking. This is another reason why supersymmetry cannot be exact and must be broken. The mechanism of supersymmetry breaking will not be explained here, but what is important to note is that this will introduce a lot more degrees of freedom for the supersymmetric particle masses and couplings. Particle Spectrum As in the Standard Model see section 3. the interaction eigenstates and mass eigenstates do not necessarily need to be the same. In the Standard Model the W - and B-bosons are the interaction eigenstates, whereas W ±, Z and the photon γ are the mass eigenstates. In the MSSM we have their supersymmetric fermionic counterparts, but in addition the supersymmetric partners of the Higgs bosons can also mix. In this case W, B, H and H are the interaction eigenstates. These mix to form four neutral neutralino mass eigenstates denoted by χ 0 i i =,, 3, 4 and four charged chargino mass eigenstates denoted by χ ± j j =,. The lightest of the neutralinos is generally assumed to be the LSP and the dark matter candidate. The question that remains is what kind of interaction eigenstate will it be like. This depends on the parameters of the supersymmetric theory that is used. It mainly depends on the masses of the interaction eigenstate particles: M, M and µ. In addition it also depends on tanβ, which is defined as the division of the VEVs of the two Higgs doublets tan β = v /v. Generally these parameters are found by assuming certain values at the high scale GUT scale and use the renormalization group equations to find the behavior at lower scales. In this manner you can assume certain parameters to be unified at the GUT scale and still have a whole range of masses at low scale. However, if this is done M will generally be the lowest mass parameter. This will always result in a Bino-like neutralino as the LSP. This is very constraining and often results in problems, as will be explained in section 6. This is why this will not be assumed for this thesis. In addition, supersymmetric particles with a mass much higher than the LSP mass do not contribute much to the relic density. This is why we will use input values at the electroweak scale and choose a lot of the masses that will not matter at a value much higher than the LSP mass. 8

19 5 Relic Density In order to analyze dark matter and the dark matter candidates, we need to look at the main thing we know about dark matter, the relic density. The relic density is the density that the dark matter currently has. The derivations presented below will largely follow [] and parts of []. 5. The Boltzmann Equation In order to derive the relic density we start out with the Boltzmann equation, this describes the movement of particles in phase space and is written as ˆL [f i ] = C [f i ] 5. where ˆL is the Liouville operator, C is the collision operator and f i is the distribution function of particle i. The Liouville operator describes the evolution of phase space without collisions and the collision operator describes how interactions cause particles to change in phase space. The covariant, relativistic Liouville operator is [3] ˆL = p α x α Γα βγp β p γ p α 5. where p is the four momentum, x is the space-time coordinate and Γ α βγ is the Christoffel symbol of the second kind. In order to calculate the Christoffel symbol the Robertson-Walker metric will be used as this is an exact solution of Einstein s field equations of general relativity. It is given by ds = dt a t dr kr + r dθ + r sin θ dφ, where k describes the curvature and a t describes the radius of the curvature, which is also written as R. Assuming the phase space density to be homogeneous and isotropic, f i = f i E i, t, and using the Robertson-Walker metric the Liouville operator for particle i is given by ˆL [f i E, t] = E i f i t Ṙ R p i f i E i 5.3 Next we use the definition of the number density in terms of the phase space density: n i = g ˆ i π 3 d 3 p i f i 5.4 where g i are the internal degrees of freedom of particle i. By performing partial integration, the Boltzmann equation can be rewritten in the form dn i dt + 3Ṙ R n i = g ˆ i π 3 C [f i ] d3 p i E i 5.5 The type of processes that need to be looked at in order to calculate the relic density are χ+a B +C, χ A + B + C and A χ + B + C with χ the particle we want to look at and A, B and C being other particles. There are also decay processes into particles, but these will give the same result as can be seen further down. The most important process will be the annihilation of two supersymmetric particles, i and j, into standard model particles, B and C, so this will be the one that is looked at first. In the case that i j this process is called coannihilation. For this process the collision term will be g i π 3 ˆ ˆ C [f i ] d3 p i = dπ i dπ j dπ B dπ C π 4 δ 4 p i + p j p B p C E i [ ] M i+j B+C f if j ± f B ± f C M B+C i+j f Bf C ± f i ± f j 5.6 where + applies to bosons and - applies to fermions. These terms, ± f, are caused by bosons tending to be close together in phase space and fermions tending to be further apart due to the Pauli exclusion principle. The phase space elements are defined as dπ i g i π 3 d 3 p i E i 5.7 In order to simplify the collision term, two approximations will be used. The first is that we assume time reversal or CP invariance, which results in M i+j B+C = M B+C i+j = M 5.8 9