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Dark Matter Experiments A bachelor thesis by Niklas Grønlund Nielsen Supervised by Professor Francesco Sannino, PhD August 2013 Department of Physics, Chemistry and Pharmacy University of Southern Denmark Dark Matter Experiments Abstract Dark matter is an explanation for one of the most important problems in modern physics. It is a well established scientific paradigm that excess gravity is caused by a new and unobserved particle. Some of the most important experiments trying to detect dark matter is direct detection experiments, a method where one tries to measure recoils from collisions between dark matter particles and atomic nuclei. These collisions are very rare and hard to measure. In the first part of this bachelor thesis we will look at the theory behind direct detection, and we will test some of the astrophysical assumptions that must be made prior to any experiment. In the second part we will construct a model, where dark matter is a scalar field that interacts with a detector nucleus via two channels: through an exchange of the Higgs boson and through a small dipole interaction that allows a photon exchange. The dipole allows dark matter to feel the electric charge of the proton very weakly. As shown in e.g. [11] differentiating how dark matter interacts with protons and neutrons, can alleviate tension between prominent direct detection experiments. We will see that the correct differentiation can be achieved via the dipole and Higgs interactions, and find how the coupling parameters of our model must be tuned. In the end we will extend the possibilities of our model by considering which possible interactions can be included in a generic theory of scalar dark matter acting as a singlet under the symmetries of the Standard Model. 1/36 Dark Matter Experiments Contents 1 Introduction 2 Dark Matter Phenomenology and Appeal 2.1 Zwicky and the Coma Cluster . . . . . . . 2.2 Rubin, Ford and the Andromeda Galaxy . 2.3 The WIMP and Genesis of Dark Matter . 3 of the WIMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Dark Matter Detection Methods 3.1 Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Detector Kinematics . . . . . . . . . . . . . . . . . . . 3.1.2 The Event Rate . . . . . . . . . . . . . . . . . . . . . 3.1.3 Example of Sensitivity to Astrophysical Assumptions . 3.1.4 Annual Modulation . . . . . . . . . . . . . . . . . . . 4 4 5 6 . . . . . . . . . . 7 . 9 . 9 . 11 . 12 . 15 4 Modelling Scalar Dark Matter 4.1 Real Scalar Dark Matter under a Z2 Symmetry . . . . . . . . . . . . . . 4.1.1 Dark Matter-Nucleus Scattering Cross Section . . . . . . . . . . 4.1.2 Investigating whether Real Scalar DM is a good Candidate . . . 4.2 Upgrading to a Complex Field that Differentiates Protons and Neutrons . . . . . . . . 16 16 17 19 21 . . . . 25 25 26 26 27 5 An 5.1 5.2 5.3 5.4 5.5 Effective Dark Matter Theory for Scalar SM Singlets Self-interactions . . . . . . . . . . . . . . . . . . . . . . . . . . Interactions with the Higgs . . . . . . . . . . . . . . . . . . . Interactions with Fermions of the Standard Model . . . . . . Interactions with Gauge Bosons of the Standard Model . . . The General Scalar Dark Matter Model up to Dimension 6 Prospectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and Future . . . . . . . . 27 6 Summary and Conclusions A Appendices A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . A.2 Flat Friedmann Universe . . . . . . . . . . . . . . A.3 Spontaneous Electroweak Symmetry Breaking . . A.4 Scattering Cross Section for Scalar Higgs-coupled A.5 Scalar DM with Dipole Interference: Derivation with Photon Propagator . . . . . . . . Bibliography 29 . . . . . . . . . DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 30 30 31 31 . . . . . . . . . . . . . . . 33 36 2/36 Dark Matter Experiments 1 Introduction In 1933 astronomer Fritz Zwicky noted that the apparent gravitational mass of the Coma cluster far exceeded what the visible matter could provide [1]. This discovery is one of the greatest in the history of cosmology and arguably all of physics. The scientific significance was not fully appreciated, until Vera Rubin and Kent Ford in the 1970’s observed, that the radial velocity distribution in the Andromeda galaxy could only be explained by Newtonian dynamics if the bulk of gravitating mass was of a non-baryonic nature [2]. Since the 1970’s many other observations of large scale astrophysical phenomena have suggested the existence of dark matter distributed in halos within and around galaxies. These observations provide some of the strongest empirical evidence for physics beyond the Standard Model (SM) of particle physics. Today, some 80 years after Zwicky’s initial observations, the nature of dark matter is still undetermined. In fact the level of ignorance concerning dark matter is remarkable, it truly is one of the greatest problems in physics today. There has been large variety of credible explanations, ranging from new invisible particles to Modified Newtonian Dynamics (MoND) to dim baryonic matter. In this thesis we will look at the favoured paradigm: that dark matter consists of massive and nonbaryonic particles, which for some reason remain invisible to science. Usually a dark matter candidate known as the Weakly Interacting Massive Particle (the WIMP) is assumed, which as the name suggests mainly interacts via gravity (massive) and only weakly to the particles of the Standard Model through the other forces. The latest high precision cosmological survey from the Planck satellite (ref. [4]) estimate the energy content of our universe to be distributed as follows: 5 % ordinary matter, 26 % dark matter and 69 % dark energy (dark energy is an even deeper mystery than dark matter!), making dark matter much more abundant than ordinary matter. Detection of dark matter has naturally been a focus of a large experimental community. In the first part of this thesis we will focus on the theoretical background for direct detection, which is one of three major experimental types, that currently tries to detect dark matter (an perhaps already has). In the second part we will construct a dark matter model, where our WIMP is a scalar field that behaves as a singlet under the the Standard Model. This is arguably the simplest approach to dark matter modelling. First, we will construct the absolutely simplest model, which is only connected to the standard model via exchange of the Higgs boson. Secondly we will improve key features of the model ad hoc; such that experimental results can agree. The formulation will be done in the language of quantum field theory and treatment will be in the context of direct detection. In the last chapter an effective field theory will briefly be introduced, to illustrate a different model building approach and explore the possibilities to extend the already constructed model. Mathematical notation and lengthy derivations can be found in the appendices. 3/36 Dark Matter Experiments 2 Dark Matter Phenomenology and Appeal of the WIMP In cosmology the Big Bang nucleosynthesis (BBN) is the formation of all the lightest elements up to 7 Li. In the moments after Big Bang the universe underwent a series of chemical equilibria as expansion caused cooling. These processes are well understood and predictions of especially hydrogen and helium abundances (by far the most abundant constituents) are very close to observations. BBN’s predictions for the mass fraction of the less abundant materials turns out to be heavily dependent on the total baryonic mass density, this is especially true for deuterium (see [3] for ref.). Therefore the total baryonic mass density can be estimated from measuring the abundance of deuterium (and other light elements). It turns out that ΩB = 0.049, where Ω is the density in units of the critical density. Independently cosmologists have determined our universe to be very close to being flat, i.e. the curvature of space time k = 0. Assuming the Friedmann Equations (implicitly assuming the Robertson-Walker metric), k = 0 is equivalent to setting the total density parameter Ω = ΩB + ΩDark ≡ 1 (see Appendix A.2 for argument). Since the estimates from e.g. Planck sets dark energy at ΩDE = 0.696, there is a large dark matter component at ΩDM = 1 − ΩDE − ΩB = 0.265. So in terms of mass, dark matter is by far the largest component in our universe. 2.1 Zwicky and the Coma Cluster Fritz Zwicky (1898-1974; Bulgarian born and of Swiss nationality) is arguably one of the greatest astronomers of the 20th century, among his many original ideas and observations is his work on the Coma cluster of galaxies. Zwicky used the steady state virial theorem demanding for the cluster’s moment of inertia I that I¨ = 0. From this is obtained that hV i = −2hKi, (2.1) where hV i = −αGM 2 /rh is the average potential and α is a numerical factor depending on the average distance between galaxies; rh is the half radius of the cluster; and hKi = M hv 2 i/2 is the average kinetic energy. Zwicky’s expression for the gravitating mass of the Coma cluster was thus rh hv 2 i . (2.2) αG Amazingly the luminous matter only accounted for about 1/50 of the gravitational mass. Therefore Zwicky predicted the existence of a large amount of dunkle materie or dark matter. In reality most of the baryonic matter in galaxy clusters are interstellar gasses and other dim constituents, however ∼ 80% of the total mass is still unaccounted for by baryonic matter. Today we know Zwicky discovered one of our universe’s biggest mysteries; sadly in his time, Zwicky was not duly appreciated for the weight of his work. M= 4/36 Dark Matter Experiments 2.2 Rubin, Ford and the Andromeda Galaxy In 1978 Astronomers Vera Rubin and Kent Ford made a shocking discovery. They observed the rotational velocities of the Andromeda galaxy via redshifts and found a very unexpected result. On the galactic scale, the laws of Newtonian dynamics was thought to be firmly established, therefore rotational velocities should be found using the Newtonian gravity and the centripetal force GmM (r) mv 2 (r) = , r2 r s v(r) = GM (r) . r (2.3) (2.4) Approximating galaxies to be spherical one would expect M (r) ∝ r3 until the end of the luminous disk (r ≈ 15kpc for the Milky Way), so in this region one expects v(r) ∝ r. If M (r) roughly ended at the edge of the luminous disk, objects beyond this limit would behave as v(r) ∝ r−1/2 . However, this is not the behaviour observed! Figure 1: Rubin and Ford data from a 1978 article [2], clearly there is no Keplerian r−1/2 decrease. The discrepancy between expected and observed results can be explained in one of two ways: either there is a dark component such that M (r) = Mluminous (r) + Mdark (r), or dynamics at galactic scales need to be modified such that FG = FN ewton + Fgalactic or even a correction on FN ewton = GmM (r)/r2+ε . These explanations are called dark matter (DM) and Modified Newtonian Dynamics (MoND) respectively. Today MoND is almost totally abandoned for multiple reasons, for example the Bullet Cluster strongly suggests the existence of weakly interacting and gravitating dark matter. 5/36 Dark Matter Experiments It is also a problem that these galactic forces are not seen at even larger scales, so to be consistent with observations one must add new corrections for every length scale. MoND also requires corrections to General Relativity, while keeping intact the principle of general covariance, this proves difficult. Lastly, everybody agrees that neutrinos exist, and they are in fact weakly interacting massive particles and therefore dark matter. However, for reasons related to when and how galaxies formed and clustered, neutrinos cannot be more than a small fraction of the entire dark matter abundance, as they correspond to what has been coined hot dark matter. In ref. [9] the paradigm of hot vs. cold dark matter is treated. The point is, that matter invisible to electromagnetic and strong interactions is not inconceivable or exotic, in fact it exists all around us. Figure 2: The Bullet Cluster as seen by NASA’s Hubble Space Telescope (can be found on NASA’s web page, at the time of writing in the link http://apod.nasa.gov/apod/ap060824.html). Two galaxy clusters have crashed in a titanic collision, separating weakly interacting DM from luminous matter; the blue highlighted regions suggests weakly interacting dark regions that exhibit gravitational lensing of far away star light. As would be expected for weakly interacting particles, they are less likely to be captured in the collision compared to the purple highlighted baryonic matter. 2.3 The WIMP and Genesis of Dark Matter If we make the single assumption, that the missing gravity we observe in our universe is in fact a result of an invisible particle, what features would such a particle have? We would at least assume the particle to be electrically neutral and stable. That is, given that dark matter has been around since the early universe, we would require for the mean lifetime to be greater than the age of the universe. Furthermore, the fact that the mysterious dark matter particle has alluded attempts of 6/36 Dark Matter Experiments observation for 80 years, heavily limit how strongly DM can interact with atoms, hence the name Weakly Interacting Massive Particle. It should however be noted, that actual dark matter doesn’t care about scientific paradigms, and one could easily think of the scary but plausible scenario: that dark matter only interacts via gravity. Such a scenario would make all current observational attempts redundant. Hopefully this is not the case, and if one of the many shades of WIMP actually exists, an experimental group may soon claim its discovery. An important feature of the dark matter problem is the question of the origin. The non-exotic explanation is that dark matter is a thermal relic, frozen out as the universe cooled. This mechanism assumes some early equilibrium where the temperature of the universe exceeded the dark matter mass, as expansion caused the number density nχ to drop (and thus the annihilation cross section), the annihilation rate became smaller than the expansion rate of the universe (the Gamov condition) H ≥ nhσvi, at which point a large amount of sterile particles froze out. H being Hubble’s constant. This mechanism is well known from other thermal freeze outs such as the photons of the Cosmic Microwave Background. Another possible genesis is an early WIMP/anti-WIMP asymmetry. One can imagine that a large abundance of particles would still remain after the initial annihilation left an excess. This mechanism mimics the origin of baryonic matter and is termed asymmetric dark matter. For the possibility of an asymmetric origin, the dark matter particle should of course be different from its anti-particle, which is not true for all WIMP candidates. 3 Dark Matter Detection Methods Experiments for detection of a dark matter are very diverse and span a huge energy scale. The main experiments can be divided in the following methods: Direct Detection (DD) is a straight forward but technically difficult approach. Atomic nuclei with well known physical properties are shielded from cosmic background radiation and other contaminants (typically DD experiments are located deep under ground). Recoil energies are measured from collisions with possible WIMPs. The observed recoils can be expressed as a statistical region in the cross section/mass parameter space where a dark matter particle could possibly be found. The rate of measuring a recoil for a single nucleus is extremely low, therefore a large number of targets is used. The total exposure of a given experiment is measured in the exposure time times the size/mass of the target (kg days). This method probes the lowest energy region, which is in the 10’s of keV, corresponding to the energy of the recoil. This is a very low energy indeed and provides considerable challenges for construction of highly sensitive apparatus. Furthermore background radiation and other internal/external effects are difficult to exclude and could provide fake DM signals. There are many large DD experiments, among the most important is DAMA, CoGeNT, Xenon, CRESST and others. 7/36 Dark Matter Experiments Indirect Detection (ID) observes Standard Model products of dark matter decay or annihilation. The energy scale of ID is thus wide, from the smallest possible products up to the unknown mass of the possible dark matter candidate. Indirect experiments lacks many of the DD challenges, but for a specific indirect experiment, one could imagine other sources than dark matter annihilation if results are achieved. A current example of ID is the ongoing AMS-02 experiment on the International Space Station, which recently released data showing a e+ /e− -fraction larger than expected from a universe with no dark matter. This ratio could come from an astrophysical source such as a pulsar or possibly from dark matter annihilation (recent results and press releases are available on http://www.ams02.org/ at the time of writing). Collider Detection (CD) is in some sense the opposite of indirect detection. Particle accelerators attempt to create dark matter by colliding Standard Model particles. If successful such a detection, would be a detection of missing energy. The drawback is that energy could go missing in many ways! However if a theoretical dark matter candidate is suggested and its parameter space is constrained by direct detection, then observables such as the cross section for hadrons into dark matter can be tested at colliders if they are energetic enough to kinetically allow the process. This is the true strength of collider detection. Direct Detection (DD) Collider Detection (CD) Ideally the different methods should not be seen as competing, but complementary. Since results in each separate method could DM SM be observations of some unforeseen effect, a connection between different kinds of experiments operating at largely different energies is necessary to claim a discovery of a WIMP. For this thesis we will only focus on the theory of direct detection. Direct detection has the nice feature of being thoroughly non-relativistic, therefore all kinematics can be treated in this limit. Given a model, the observables we want to determine is therefore those related to the upward arrow in figure 3. In principle, dark matter can be treated DM SM independent of microscopic models by taking the approach of effective theories. This Indirect Detection (ID) can be done by writing down all allowed terms that can exist in a Lagrangian deFigure 3: Here is shown an unknown interscribing dark matter. For direct detection action between dark matter and a standard this is a particularly viable approach, bemodel particle. Different directions in the diacause DD is in the low energy regime, and gram correspond to specific search strategies. we always deal with two particles in the initial and final states, which limits how complicated the effective interactions can become. An effective DM model will be treated further in chapter 5, where we expand the number of possible interactions for the model build in chapter 4. 8/36 Dark Matter Experiments 3.1 Direct Detection Currently a large and varied effort is being made in the field of direct detection. Multiple experiments at many locations using different techniques have tried to limit the region of the cross section/mass parameter space where a WIMP could reside. However, considerable tension has risen between high profile experiments, as seen in figure 4. Figure 4: This plot is from reference [10] and shows the proton scattering cross section vs. the dark matter mass. There is assumed a standard contact interaction, i.e. equal proton and neutron interaction. The DM mass density is taken to be ρχ = 0.3/GeV/cm3 and the distribution is in an isothermal sphere with a velocity dispersion of 220 km/s. Lines from XENON and CDMS are upper bounds and contours are allowed regions. There is no common allowed region agreed upon by all experiments, so either some experiments are wrong or some assumptions must be changed. 3.1.1 Detector Kinematics In direct detection methods one can examine elastic and inelastic collisions. Elastic collisions preserve particle states and only exchanges momentum (p and k) between the 9/36 Dark Matter Experiments dark matter particle and the target nucleus. A general inelastic scattering is χ(p) + N (k) → χ0 (p0 ) + N 0 (k0 ). (3.1) In the following we will treat the kinematics of elastic scattering, because this is somewhat simpler. One should however keep in mind, that inelastic scattering is an interesting case to study, if one tries to make experiments agree (in chapter 4 we will go about this in a different way). Given a detector of perfect sensitivity and knowledge about the distribution of WIMP velocities in our galaxy, one could perform the integral dR = dQ Z ∞ dR dQ 0 f (v) dv, (3.2) where R is the averaged recoil count, Q is the recoil energy deposited in the nucleus and f (v) is the probability distribution in WIMP velocities. Unfortunately detectors are not perfect and the lowest detectable energy must be considered. This corresponds to a minimal velocity that a WIMP must have w.r.t. the detector before a scattering is observed. To find the minimal velocity with Qth as the threshold energy of the detector, we take the velocity of dark matter to be firmly non-relativistic; this is a fair assumption since the actual velocity is at least bounded from above by the galactic escape velocity vesc ∼ 10−3 c i. Furthermore the velocity in the laboratory frame is of equal magnitude to the relative velocity v in the center of mass frame. Take q = p − p0 = k0 − k to be the transferred momentum and p = µv in the center of mass frame. We have the c.o.m. condition p = −k and µ = mN mχ /(mN + mχ ) is the reduced mass of the WIMP-nucleus system. The recoil energy of the nucleus is Q= q2 . 2mN (3.3) Now demanding conservation of energy |p|2 = |p0 |2 ; (3.3) becomes Q= |p|2 +|p0 |2 −2|p||p0 |cos θ µ2 v 2 (1 − cos θ) = . 2mN mN (3.4) Taking a perfect head on collision (θ = π), one obtains the minimal velocity in the nonrelativistic limit. (If the collision is inelastic, where outgoing and incoming masses are unequal, then there is a small correction to the velocity) s vmin ' mN Qth . 2µ2 (3.5) The minimal velocity should of course be met by the actual WIMP velocity. This puts a direct requirement on how sensitive the detector must be. If we assume dark matter to be distributed in giant halos around our galaxies, and that there is no dark matter wind, then the average relative velocity would be thermally distributed around the Earth’s velocity in i vesc = 1012 M . p 2GM/R ≈ 500km/s ∼ 10−3 c with: G = 4.3 · 10−3 pc/M (km/s)2 , R = 100 pc, M = 10/36 Dark Matter Experiments the galactic frame. If very unlucky there could be a DM wind such that WIMP’s average velocity in the galactic frame matches the Earth’s, this case would require extremely sensitive detectors. 3.1.2 The Event Rate The most important observable in any direct detection experiments is the differential recoil rate with respect to the recoil energy. This is measured in units of [ #/kg/yr/energy] and is related to the differential cross section. The recoil for some specific velocity is R = nχ vσ, (3.6) where nχ = ρχ /mχ is the number density of DM, v is the local velocity and σ is the scattering cross section. The differential recoil rate is dR dσ = nχ v . dQ dQ This expression needs to be averaged over all velocities from vmin to vesc w.r.t. the velocity distribution f (v), (R = hRi) dR dσ = nχ v dQ dQ Z vesc = nχ f (v) d3 v f (v)v vmin dσ . dQ (3.7) Here a single dark matter candidate is assumed, but one could easily sum over different species that could be distributed differently. Here we consider only one dark matter candidate at a time, since we know of no a priori reason to do anything more complicated. Scattering Cross Section. The scattering cross section σ is an effective area. A large cross section corresponds to a large scattering probability and vice versa. Direct detection depends heavily on the differential cross section of the scattering. The differential cross section has separate contributions: spin dependent and independent dσSI dσSD dσ = + . dQ dQ dQ (3.8) In the model that we will build in chapter 4, we will take dσ dσSI ≈ . dQ dQ (3.9) Usually the differential cross section is written in terms of the scattering cross section and a function FN (Q) called the nuclear form factor containing information related to the fact, that the scattering is not point like dσ mN = FN2 (Q) 2 σ. dQ µv (3.10) Depending on the specific DM model and interaction type the scattering cross section behaves qualitatively different. Velocity Distribution. The local velocity distribution must be assumed so the integral in equation 3.7 can be performed. A reasonable assumption is that the distribution is Maxwell-Boltzmann, if self-interactions are taken to happen through elastic collisions. 11/36 Dark Matter Experiments f (v) = mχ 2πkT 3/2 ! mχ v 2 4πv exp − . kT 2 (3.11) The temperature of the distribution is bounded from above by the galactic escape velocity; it can however be lower depending on the mechanism for the synthesis of dark matter. Here is assumed no time dependence (that is neglecting annual modulation effects, discussed in section 3.1.4). If the nature of DM turns out to be different than expected, one can imagine completely different velocity distributions with the sole requirement that f (v) is normalized Z ∞ 1= d3 vf (v) ≈ Z vesc 0 d3 vf (v). (3.12) 0 One could for example play the game of having many components of dark matter, with distributions at different temperatures. Such possibilities makes the necessary assumptions highly tunable. In turn this makes the differential recoil rate deceptively simple. In fact it is a product of two large assumptions. From particle physics we must assume a cross section and from astrophysics we must assume the velocity distribution. So basically we have (Differential Recoil Rate) = (Astrophysical Assumptions)×(Particle Physics Assumptions) (3.13) Thus when two DD experiments have conflicting results, one cannot conclude that the paradigm of dark matter is wrong; because one of the assumptions could easily be incorrect (or even both!). In light of this one would require from a good dark matter model, that it at least explains the discrepancy between various DD experiments with reasonable assumptions. 3.1.3 Example of Sensitivity to Astrophysical Assumptions As a simple exercise in direct detection responses, here we consider a specific detector and assume a scattering cross section that behaves nicely; specifically we will assume it is constant w.r.t. velocities, the recoil energy and the dark matter mass σ = σ0 , (3.14) this is likely an oversimplification, but we take it to be correct to first order. The recoil rate becomes dR MN = nχ σ0 2 FN2 (Q) dQ µ Z vesc dv f (v)v. (3.15) vmin The form factor in this example includes all recoil energy dependence and is determined through nuclear physics. Here we assume a pure 70 Ge detector with an explicit spin independent form factor from ref. [8] p. 38 (here is only used a single of the many responses for illustrative purposes): FN = e−2y (1000 − 2800y + 2900y 2 − 1400y 3 + 350y 4 − 42y 5 + 1.9y 6 − 0.0027y 7 ), (3.16) 12/36 Dark Matter Experiments with y ≈ 105 GeV−1 × Q such that the minimal recoil around 10 keV corresponds to the dimensionless y to be minimal around 1. In the following figures on the right hand is shown the base 10 logarithm of the differential recoil rate. The differential recoil rate is in units of ρχ · σ0 ; so we are not looking for actual counts per year, but differences in qualitative behaviour from one assumption to another. The left figures are velocity distributions shown as contours in the DM velocity/mχ phase space, such that for a given DM mass, condition 3.12 holds. In the first plots we chose the thermal energy kT of the Maxwell velocity distributions, such that the lightest candidates < 10 GeV are distributed within the galactic escape velocity. In all plots we have assumed a good detector with a threshold of 10 keV recoil energy. Velocity distribution with kT= 10−6 GeV 2 dR/dQ 10 100 0 −2 90 −4 80 −6 mχ [GeV] mχ [GeV] 70 60 50 40 −8 −10 1 10 −12 −14 −16 30 −18 20 −20 10 0 −22 0 0 0.2 0.4 0.6 0.8 v [c] 1 1.2 1.4 1.6 −3 x 10 10 −5 10 −4 10 Q [GeV] Figure 5: The velocity distribution plot stretches to vesc ∼ 1.6 · 10−3 c and the recoil rate is shown for recoil energies between 10keV and 100keV. Every contour line is one power of 10 different from its neighbour with yellow being a high detection rate and green a low one. Clearly light candidates as close as possible to the energy threshold is most easily detectable. Interestingly this theoretical detector has a blind spot just above the mχ = 10 GeV line, which is not far from the (e.g.) DAMA and CoGeNT favoured region for scalar dark matter 13/36 Dark Matter Experiments Velocity distribution with kT= 10−5 GeV 2 0 90 −2 80 −4 70 −6 mχ [GeV] mχ [GeV] dR/dQ 10 100 60 50 40 −8 1 10 −10 −12 30 −14 20 −16 10 0 −18 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 v [c] 10 −5 10 1.6 −4 10 Q [GeV] −3 x 10 Figure 6: Here we have allowed the velocity distribution to stretch out from our estimate of the galactic escape velocity for dark matter of mass mχ ≤ 30 − 40 GeV. This is reasonable if dark matter actually has a higher mass. The main difference from before is, that an island of easier detectable candidates emerged beyond the blind spot. This is definitely a nice feature, although the region is rather small. Velocity distribution with kT=2 ⋅ 10−4 GeV 3 dR/dQ 10 1000 0 −2 900 −4 800 mχ [GeV] 700 mχ [GeV] −6 2 600 500 400 10 −8 −10 −12 1 10 300 −14 200 −16 100 0 −18 0 0 0.2 0.4 0.6 0.8 v [c] 1 1.2 1.4 1.6 −3 x 10 10 −5 10 −4 10 Q [GeV] Figure 7: For this last plot we increase the thermal energy kT all the way to 2 · 10−4 GeV and consider dark matter up to 1000 GeV. Here only distributions for very heavy DM of mass mχ ∼ 700 GeV or larger are contained in the allowed velocities within our galaxy. It seems qualitatively that such heavy candidates are difficult to detect, but there is clearly a stretching effect of the island beyond the mχ = 10GeV line, so this kind of recoil would not be completely suppressed. This is of course a theoretical detector, but it provides some illustrative points. First of all, if dark matter is Maxwell-Boltzmann distributed (especially if it’s cold) then light candidates are orders of magnitude more detectable. Secondly, in all cases low energy thresholds are very important for a detector. Say we had a detector with a threshold of 100 keV and dark matter turned out to weigh 15 GeV, then from our plots the recoil rate would be suppressed by anywhere between ∼ 10 − 20 orders of magnitude compared to a 10 keV detector. 14/36 Dark Matter Experiments 3.1.4 Annual Modulation Assuming the local DM velocity is constant in time is in many cases a fair approximation. However, dark matter is thought to be distributed as an aether in and around galaxies through which star systems move. The average velocity in Earth’s frame thus depends on whether Earth is moving parallel or anti-parallel to the the Sun’s galactic orbit. Earth’s velocity in the galactic frame is (values from ref. [10]) vE (t) = vG + vS + vmod (t) (3.17) 30 km/s ∼ 230 km/s −30 km/s Here vG and vS are our local system’s galactic velocity and the sun’s proper ve- Figure 8: Sun moving in the galactic frame. locity respectively, these are close to being aligned. vmod (t) is the modulation stemming from Earth’s revolution around the Sun and thus has a period of one year. The modulation term is of special interest, because increased dark matter signals should be observed when vmod (t) is aligned with vG (on the 2nd of June). the velocities are approximately vG = 220 ± 50 km/s, vS = 12 km/s and vmod = 30 km/s. One can think of the modulation effect as giving the velocity distribution a time dependence f (v(t)), the quantity measured is the integral from vmin to vesc of f (v(t)). When orbits are aligned more of the distribution lies within the integration borders and vice versa. This technique is used by the DAMA collaboration (among others) to observe a signal at a high confidence level. Figure 9: Partial results from the 2003 article Dark Matter Search [7] clearly showing an annual modulation of signals. 10 years ago DAMA already claimed a WIMP at 6.3σ and today the claim is even stronger statistically, unfortunately it is not entirely certain whether the signal is DM or something else. 15/36 Dark Matter Experiments 4 Modelling Scalar Dark Matter The Standard Model of particle physics is a very good (albeit finely tuned) quantum field theory for a small part of the energy content in our universe. One of the Standard Model’s shortcomings is the lack of a dark matter particle, therefore one must go beyond the Standard Model (BSM). Sticking to Occam’s Razor we will assume DM to be a scalar field protected under some global symmetry that ensures stability, this is certainly the most basic starting point we can imagine. Our goal in this chapter is to construct a model, and examine a general direct detection of our WIMP candidate. 4.1 Real Scalar Dark Matter under a Z2 Symmetry The simplest thing one can think of is a real scalar field χ which is a singlet under the symmetries of the Standard Model, we let χ be protected under a simple symmetry Z2 , such that the Lagrangian is invariant under χ → −χ. We want our dark matter candidate to have some weak interaction with the Standard Model. One place such a connection could still hide is in a coupling to the Higgs. The Z2 symmetry eliminates all odd power terms and we can write an extension to the SM Lagrangian density as 1 1 1 L = LSM + ∂µ χ∂ µ χ − m20 χ2 − gχ4 − λH † Hχ2 , (4.1) 2 2 4 where m0 is the mass of the DM candidate before spontaneous electroweak symmetry breaking, g and λ are the DM self coupling and Higgs coupling respectively, these are both assumed to be real and positive. H is the Higgs doublet which in unitary gauge is (see appendix A.3 for discussion) ! 1 0 H=√ , 2 h+v (4.2) here h is the Higgs scalar field and v ≈ 246 GeV is the vacuum expectation value (vev) of the mexican hat potential. After the spontaneous symmetry breaking the dark matter mass thus acquires an extra term, namely m2χ = m20 + λv 2 . One can easily verify that the vev for the potential of χ is 0 with m20 , g ≥ 0, if this was not the case the DM candidate would not be stable, so this is a requirement for the model. Equation 4.1 is arguably the simplest model assuming only one connection to the Standard Model. Other simple interactions include a mechanism where a hidden U(1) symmetry gives rise to a new dark photon that kinetically mixes with the SM photon. Writing out H † H, the Lagrangian reads 1 1 1 λ L = LSM + ∂µ χ∂ µ χ − m2χ χ2 − gχ4 − χ2 h2 + λvχ2 h . 2 2 4 |2 {z } (4.3) interaction terms The last two terms give rise to the following vertexes 16/36 Dark Matter Experiments χ χ h χ h h χ In this model, there are no other connections to the SM than through these two Higgs couplings. The three legged vertex is most important, as it allows a simple scattering with the Higgs as a propagator. We will not go further than three level in any of our calculations. Depending on the experiment employed to detect DM; different calculable quantities are of interest. For indirect detection cross sections for various annihilation channels to SM particles are naturally important. For direct detection the important quantity is the elastic scattering cross section between the DM scalar and a direct detector nucleus. 4.1.1 Dark Matter-Nucleus Scattering Cross Section In this section we calculate σ(N χ → N χ), the cross section for the DD direction of fig. 3 assuming spin independent scattering. At the tree level we have N χ χ n h h ≈ A× χ N χ n where time is upward and the approximation is, that the nucleus scattering is proportional to a scattering off a single nucleon times the the atomic number. n being the nucleons. Loop diagrams like χ h N χ h N are allowed but heavily suppressed. The general elastic scattering cross section for two initial and final states is (ref. [5]) 17/36 Dark Matter Experiments 1 σ= 4Eχ EN Z d3 pχ0 d3 pN 0 |M|2 (2π)4 δ (4) (pχ0 + pN 0 − pχ − pN ). (2π)6 4Eχ0 EN 0 (4.4) Working out the Feynman amplitude (see appendix A.4 for details in cross section derivation) one obtains |M|2 = 4m2χ m2p A2 fn2 . (4.5) where fn is given by fn = λf mp . m2H mχ (4.6) Solving the integral in equation 4.4 yields a simple cross section, which in the limit of zero momentum transfer is µ2 2 σ= ζ , (4.7) 4π where µ is the reduced mass of the DM-nucleus system, ζ = Zfp + (A − Z)fn = Afn , fn/p describes the strength of coupling to neutrons and protons, A is the number of protons and neutrons and Z is the number of protons in the nucleus. In this model protons and neutrons are indistinguishable in interactions i.e. fn = fp . Before we move on to explore how good this model fits to experiments, we take a closer look at the actual behaviour of this scattering cross section. This is interesting to us, since the scattering cross section corresponds to the assumption from particle physics, discussed in equation 3.13. Taking Amp ≈ mN and Ω to be a constant of energy dimension −4, the cross section is !2 mN µ σ= Ω. (4.8) mχ As of now we don’t know λ, but we guess the electroweak scale for m2H /λ = v 2 = (246GeV)2 ∼ 6.0 × 104 GeV2 so that we can plot the behaviour. We have in natural units 1 Ω= 4π λf m2H !2 ≈ 2.0 × 10−12 GeV−4 ≈ 3.0 × 10−67 cm4 , (4.9) and we can plot the scattering cross section in the parameter space of dark matter mass and nucleus mass. 18/36 Dark Matter Experiments log10 σ 3 2 [cm ] 10 −34 −35 [GeV] −36 2 −37 10 −38 m χ −39 1 −40 10 −41 −42 0 10 −43 0 20 40 60 mN [GeV] 80 100 Figure 10: Here contour lines are powers of tens of cm2 . Light nuclei have a strong advantage in this model, especially for detecting light DM. In this model a light dark matter particle clearly has a much smaller scattering cross section, and will consequently be harder to detect (if the assumption of the velocity distribution is not dominating). In the next section we will see that, assuming this model, a light candidate ∼ 8GeV is in fact favoured by some experiments; although they still disagree. Note here, that this this cross section behaviour is different from the constant cross section assumed in the example from section 3.1.3. 4.1.2 Investigating whether Real Scalar DM is a good Candidate The real scalar dark matter with a Higgs exchange is a nice model because it is very simple, but there are some problems; chiefly that the important direct detection experiments disagree under assumption of this model. Furthermore a real scalar cancels the possibility of an asymmetric genesis, which would be a nice possibility to include. To see disagreement between experiments under assumption of this model we introduce the DM-proton cross section by setting A = Z = 1 in equation 4.7 µ2p |fp |2 . (4.10) 4π For the moment we forget fn = fp and we factor equation 4.10 out of the recoil rate and sum over different isotopes. Now we can write σp = 19/36 Dark Matter Experiments X dR = σp κi dQ isotopes µi µp !2 2 fn IAi Z + (Ai − Z) , fp (4.11) where κi is the fraction of each isotope in the detector and Ai is the corresponding atomic number. IAi is the remaining integral from 3.15 IAi mi = nχ 2 FN2 (Q) µi Z vesc dv f (v)v. (4.12) vmin We will assume these integrals to be similar for different isotopes (as checked by ref. [11]). Setting equal interactions with protons and neutrons, the experimental recoil rate would be X dR = σp,exp κi dQ isotopes µi µp !2 IAi A2i . (4.13) Equations 4.10 and 4.13 now let us write the actual cross section in terms of the experimental one (letting IAi cancel) κi µ2i A2i . κi µ2i |Z + (Ai − Z)fn /fp |2 P σp = σp,exp P (4.14) Obviously σp = σp,exp if fn = fp . However it turns out, that by choosing fn /fp 6= 1, one can force agreement between experiments, although such a fraction cannot be obtained having only a Higgs exchange. 10-39 10-37 Σ p in cm2 Σ p in cm2 10-40 10-41 10-38 10-42 7.5 8.0 MΦ in GeV 8.5 9.0 7.5 8.0 8.5 9.0 MΦ in GeV Figure 11: Here is shown the DM/proton cross section vs. the dark matter mass. The blue contour is CoGent-favoured region at 90% confidence interval, the green region is the DAMA/LIBRA 3σ and the dashed line is one above which all is excluded by CDMSII. In the left figure fn /fp = 1 and the right fn /fp = −0.71. The right features an allowed region shown in red, for a dark matter candidate around 8 GeV and proton cross section around σp = 2 · 10−38 cm2 . This figure is recreated from plots in reference [11]. The model we have build in the previous section corresponds to the left plot of figure 11. To construct the effect from the right figure, we need an interaction term allowing fn /fp = −0.71. It should be noted that the allowed overlap at fn /fp = −0.71 is very sensitive, it could be a coincidence, but it seems like a big one. In the rest of this chapter we will examine a mechanism that can accommodate the allowed region. 20/36 Dark Matter Experiments 4.2 Upgrading to a Complex Field that Differentiates Protons and Neutrons A few motivated extensions to the previous model are in order. First of all, if the real scalar field is upgraded to a complex one and we simultaneously extend the Z2 symmetry to a general complex phase, i.e. a U(1) symmetry, then our candidate could have had an asymmetric origin, which is a possibility that we gladly include. We saw empirically that fn /fp = −0.71 would let experiments agree. An interaction we could ad to accommodate this, is one that feels the positive electric charge of the proton, since this is the main feature that distinguishes protons and neutrons. A weak interaction ← → that couples the dark matter current χ∗ ∂µ χ to the electromagnetic current ∂ν F µν can be imagined, we call this the dipole interaction. In the end of this chapter we will discuss how an otherwise electrically neutral scalar could interact via the dipole. If we only have the dipole, the WIMP would not feel the neutrons, so in order to have fn 6= 0 we keep the Higgs interaction. The Lagrangian for this upgraded model is thus 1 λ L = LSM + ∂µ χ∗ ∂ µ χ − m2χ χ∗ χ − g(χ∗ χ)2 − χ∗ χh2 + λvχ∗ χh + Ldipole , 2 |2 {z } (4.15) interaction terms → βe ∗ ← χ ∂µ χ∂ν F µν , (4.16) 2 Λ where e is the electromagnetic coupling, β contains the unknown coupling between the WIMP and the photon (in 4.1 β = 0), Λ is some relevant energy scale and is squared since the dipole is of dimension 6 and the Lagrangian is of dimension 4. F µν = ∂ µ Aν − ∂ ν Aµ is the standard electromagnetic field strength tensor, Aµ being the photon. Since the field is now complex the mass term is without a 1/2 pre-factor, so the acquired mass after spontaneous symmetry breaking is m2χ = m20 + λv 2 /2 instead. Ldipole = We make a similar approximation to earlier: that the dipole interaction with the whole nucleus is roughly the number of protons Z, times proton scattering: χ χ N χ γ ≈ Z× N p γ χ p Again time is upward. This photon exchange interferes with the coupling to the proton. If we call the interference to the coupling δ, we can write it as a function of the dipole coupling β. We write fp (λ, β) = fn (λ) + δ(β). (4.17) 21/36 Dark Matter Experiments Now we want to figure out how to tune λ and β to achieve fn /fp = −0.71, this condition can be written as 1 fn (λ0 ). 0.71 δ(β0 ) = − 1 + (4.18) Before we can figure out how β0 and λ0 must be tuned, we must evaluate the diagram. The evaluation of the Feynman diagram with the photon exchange can be done straight forwardly by reading off the Feynman rule in the Lagrangian for the vertex χ p0 χ p γ pγ and using the well known rules from quantum electrodynamics for the remainder (see this derivation in appendix A.5). A less tedious way is to treat the photon exchange as an effective contact interaction, corresponding to the diagram χ p γ χ p χ p −→ χ p This is usually only possible for heavy mediators where the transferred momentum is negligible, but in this case there are two derivatives in ∂ν F µν cancelling the 1/q 2 from the photon propagator, which makes it possible to consider the contact interaction. We use the non-homogeneous Maxwell equations to couple to the electromagnetic current, which in the units µ0 = ε0 = 1 are ∂ν F µν = J µ . (4.19) The conserved current can be written in terms of the charged protons. For fermionic conserved current we have (ref. [5]) J µ = eZ p̄γ µ p. (4.20) Our dipole interaction thus reads Zβe2 ∗ (χ ∂µ χ − χ∂µ χ∗ )p̄γ µ p. (4.21) Λ2 And now we can evaluate the diagram, we denote dark matter momenta p, p0 and proton momenta k, k 0 . Writing in momentum Fourier space Ldipole = 22/36 Dark Matter Experiments χ ∼ e−ip·x , χ∗ ∼ eip·x (4.22) we have χ p χ p Zβe2 (−ipµ − ip0µ )us γ µ ūr Λ2 2iZβe2 '− pµ us γ µ ūr = iM̃dipole , Λ2 = (4.23) where u and ū are spinors for initial and final protons and r, s is initial and final spins, and p − p0 = k 0 − k = q with −ipµ − ip0µ = −2ipµ + iqµ → −2ipµ in the limit of zero momentum transfer. We use g µν and gµν to raise and lower indices, and the identities for momenta pµ pµ = m2 . Summing over final and averaging over initial spins we get the amplitude 2βe2 Λ2 !2 1 = Z2 2 2βe2 Λ2 !2 2 2βe2 Λ2 1X 1 |Mdipole | = |M̃dipole |2 = Z 2 2 r,s 2 2 = 4Z X (pν us γ µ ūr ) × (ur γµ ūs pν ) r,s 0 m2χ tr[(k/ + mp )γ µ (k/ + mp )γµ ] !2 m2χ m2p . (4.24) We now have the total Feynman amplitude to have two contributions |MHiggs + Mdipole |2 = 4m2χ m2p (Afn + Zδ)2 , δ=− 2βe2 8πβα =− 2 , 2 Λ Λ (4.25) (4.26) where α = e2 /4π is the fine structure constant. Taking δ < 0 as the non-spin summed M̃dipole < 0 and δ is defined from Mdipole = 2mχ mp Zδ. If experiments are reconciled at fn /fp = −0.71 we get a constraint, √ assuming √ mχ = 8GeV as suggested by figure 11. Incidentally 1/0.71 is very close to 2 as 1/ 2 = 0.7071, it further adds to how incredible the coincidence would be if a famous irrational number just happens to be the correct fraction; even though we have no micro-physical explanation for 23/36 Dark Matter Experiments this, it seems likely that there is one. We also take mp = 0.938GeV, mH = 125 GeV and α ≈ 1/137 √ 1 δ =− ≈ − 2, 1+ fn 0.71 √ 8παβ0 mχ m2H = 2 + 1, 2 Λ λ0 f mp β0 = Λ2 √ 2 + 1 f mp λ0 ≈ 2.97 · 10−5 GeV−2 × λ0 . 8π αmχ m2H (4.27) If we fix σp = 2 · 10−38 cm2 we also get a constraint from the proton cross section in equation 4.10 which is λ0 = ! √ 2 σp π β0 mχ m2H + 8πα 2 . µp Λ f mp (4.28) In the end we find (up to a common sign) by plugging in all values β0 ∼ 2.8 · 10−4 GeV−2 , Λ2 λ0 ∼ 9.5. (4.29) This is how we fine tune the coupling parameters β/Λ2 and λ if experiments has to agree. Remember, while discussing the event rate, experimental reconciliation was decided to be a criterion for a good dark matter model. Luckily reconciliation can be achieved in our model. Now, if we tune the model as needed, is this dark matter? Well, probably not. The biggest reason being, that we for simplicity assumed dark matter to be a scalar particle, and there was no physical reason for doing so. In nature only the Higgs boson has ever been observed as a scalar, and the Higgs may even be composite and therefore not fundamental. However, if lucky, a scalar description of dark matter could be effectively correct. If there are no fundamental scalars in nature, but DM in some energy regime can be described by a composite scalar, then this model could be true. Although it does not describe any of the micro-physics composing the scalar. Another motivation for this scalar to be composite is to explain how the dipole interaction arises on the electrically neutral particle. A dipole can only exist on the scalar if it has an electrically charged substructure. In analogy, the substructure of the neutron (1 up and 2 down quarks) was discovered, by the existence of a small magnetic moment on the otherwise electrically neutral neutron indicating the existence of quarks. 24/36 Dark Matter Experiments 5 An Effective Dark Matter Theory for Scalar SM Singlets Building the dark matter model in the previous chapter we assumed many things. Besides assuming a scalar singlet protected under a U(1) symmetry, we also picked the Higgs interaction term and a dipole interaction term. The first was picked as a simple interaction, and the latter was introduced completely ad hoc to get fn /fp to equal −0.71 and thus make experiments agree as seen in figure 11. In this chapter we wish to relax the previous assumptions and explore the correspondingly more general model. This will illustrate the possible expansions and modifications that we can think of for future scalar DM modelling. The only assumption from now on is that dark matter can be described by a scalar field acting as a singlet under the symmetries of the Standard Model. In reference [12] the whole array of operators has been classified, not only for singlets w.r.t. the SM, but also for doublets and triplets. It would also be interesting to do a similar classification assuming dark matter to be fermionic, but we do not consider that option here. Now we look at all the relevant interaction terms for the general DM Lagrangian, this is the DM self-interaction and interactions with the Standard Model fields. To this end we (re-)introduce a cut-off energy scale Λ, in chapter 4 this was just a dimensionfull constant, here we set it to be some unknown (possibly large) energy scale above which a more fundamental theory of dark matter emerges, and the effective description (probably) breaks down. We will write operators in reciprocal powers of this cut-off and satisfy ourself with (energy) dimension = 6 interactions, such that possible higher dimensional terms are suppressed by Λ−k with k > 2. Remembering the energy dimensions [L] = 4, a scalar field [φ] = 1, [∂φ] = 2, a fermion [ψ] = 3/2 and the field strength tensor [F µν ] = 2. In this ← → chapter whenever Jµ appears, it is the DM current χ∗ ∂µ χ, not the EM current as in the previous chapter. 5.1 Self-interactions In chapter 4 the included self-interactions was the kinetic, mass and quartic terms g (5.1) ∂µ χ∗ ∂ µ χ − m2χ χ∗ χ − (χ∗ χ)2 . 2 If our only limitation is 6 energy dimensions we can write many more terms, including derivative interactions with non-derivatives, as long as all indices are properly summed over. Here all coefficients are real and made dimensionless by proper powers of Λ ∂µ χ∗ ∂ µ χ − m2χ χ∗ χ + 3 X i=2 ai (χ∗ χ)i + Λ2i−4 1 c (b1 ∂µ (χ∗ χ)∂ µ (χ∗ χ)+b2 ∂µ (χ∗ χ)J µ + b3 Jµ J µ ) + 2 (∂µ ∂ µ χ∗ )(∂ν ∂ ν χ). 2 Λ Λ (5.2) Of the 6 coefficient besides the kinetic and mass term, we only have a2 = − g2 6= 0. It is clear that many possibilities are left to consider, and some could influence how the velocity distribution behaves. 25/36 Dark Matter Experiments 5.2 Interactions with the Higgs In chapter 4 we included two terms with the Higgs scalar λ ∗ 2 χ χh . (5.3) 2 Similarly we can imagine many more terms up to dimension 6 if we simply count the powers and write all terms having both χ and h with or without derivatives λvχ∗ χh − χ∗ χ 4 X ai i=1 2 X (∂µ χ∗ )(∂ µ χ) 2 X hi hi ∗ 2 + (χ χ) b + i Λi−2 Λi i=1 1 X hi hi ∗ µ + ∂ (χ χ)(∂ h) d + µ i Λi Λi+1 i=0 ci i=1 Jµ (∂ µ h) 1 X i=0 ei hi (∂µ h)(∂ µ h) ∗ + f (χ χ) . Λi+1 Λ2 (5.4) In our microscopic model only coefficients a1 = λv and a2 = −λ/v are different from zero. 5.3 Interactions with Fermions of the Standard Model We take the interaction with the fermions of the standard model (ψ) to be of the forms χ∗ Oχ χψ̄Oψ ψ, (5.5) Oχ (χ∗ χ)ψ̄Oψ ψ, (5.6) where O are operators such that the interaction is a Lorentz scalar, and ψ̄, ψ are any two SM fermions that in combination are electrically neutral. Our operators are ← → Oχ ∈ {1, ∂µ , ∂ µ }, ← →µ Oψ ∈ {1, γ µ , D }, (5.7) →µ ← →µ ← where D ≡ ∂ − ieQAµ is the covariant derivative, where Aµ is the photon coming from the SM U(1). The possible interactions are thus 1 ∗ χ χψ̄ψ, Λ 1 ∂µ (χ∗ χ)ψ̄γ µ ψ, Λ2 1 Jµ ψ̄γ µ ψ, Λ2 ← → 1 ∗ χ χψ̄i D ψ; 2 Λ (5.8) (5.9) (5.10) (5.11) ← → in the last interaction Dµ is sandwiched between two spinors, therefore the index is suppressed and need not be summed over like γ matrices. Beyond these terms are 4 other, because letting Oψ −→ Oψ γ 5 still leaves the interaction as a Lorentz scalar. γ 5 = γ 0 γ 1 γ 2 γ 3 is the chirality matrix. 26/36 Dark Matter Experiments In the chapter 4 model we have no explicit fermion interactions. Although interestingly, the interaction 5.10 arose from the dipole when using the equation of motion from electromagnetism (Maxwell inhomogeneous: ∂ν F µν = p̄γ µ p) in the context of direct detection. In fact, this is also to be expected, since we effectively treated it as a contact scattering. 5.4 Interactions with Gauge Bosons of the Standard Model In the Standard Model gauge bosons are carriers of forces. The photon carries the electromagnetic force, the W and Z bosons carry the weak nuclear force and the gluons carry the strong force. In our model we only had the dipole interaction that included the photon. Gauge invariance under the electromagnetic U(1) part of the standard model only allows the photon to enter terms via the field strength tensor. In fact no terms of dimension lower than six appears, but we do find two new ones 1 ∗ χ χFµν F µν , Λ2 1 ∗ χ χFµν F̃ µν , Λ2 1 Jµ ∂ν F µν , Λ2 (5.12) (5.13) (5.14) where F̃ µν = µνρσ Fρσ and µνρσ is the Levi-Civita symbol. The interaction 1 ∂µ (χ∗ χ)∂ν F µν , (5.15) Λ2 vanishes identically, since one can perform an integration by parts in the Lagrangian density and use that ∂ν F µν is the conserved electromagnetic current. There are many more interactions with the Z and W bosons, the first of which emerge at dimension 4. The first interactions with the gluons are at dimension 6. Instead of listing them all here, we refer once again to [12] where all can be found. Beyond the possibilities listed and referred to in the sections of this chapter, one can imagine mixed type interactions for example including both χ, ψ and h through 1 ∗ χ χψ̄ψh, Λ2 (5.16) or one of many other possibilities. 5.5 The General Scalar Dark Matter Model up to Dimension 6 and Future Prospectives From the partial list of possible interactions in this chapter, it is clear that there is a wide variety of potential parameters to tune in a general theory of complex scalar dark matter. If we call the set of interactions that has been found I, then we can write the general Lagrangian density for complex scalar dark matter up to dimension 6 as a linear combination of i ∈ I 27/36 Dark Matter Experiments LSDM = #I X cn in . (5.17) n=1 In principle we can write all these terms and tune all the cn coefficients. However, it is more difficult to tune many coefficient simultaneously. The model in chapter 4 is obviously included in this general Lagrangian, where most of the coefficients are identically zero. In fact we only had two free parameters, and the fn /fp = −0.71 requirement gave us two conditions: a favoured proton cross section and favoured dark matter mass. If we include more interactions than two, we still only get the two constraints. Therefore our set of couplings will be under determined and we must scan the parameter space for values that gives us the desired experimental reconciliation (or find more constraints). In the future this could be very interesting to do with some of the other interactions that has been listed in this chapter. With the work from this thesis in mind, it would be especially interesting to examine some of the other terms that allow differentiation between protons and neutrons of direct detectors. 28/36 Dark Matter Experiments 6 Summary and Conclusions In this thesis we have investigated aspects of dark matter experiments, in particular we have focused on direct detection, and the theoretical background behind this type of dark matter experiment. We have constructed a quantum field theory to treat in this context. In chapter 2 key points of the dark matter problem was presented, and the dark matter paradigm was introduced. In chapter 3 the main strategies for detection of dark matter was introduced, and the theory behind direct detection was treated in greater depth. The only observables in direct detection are the recoils, these depend on both astrophysical and particle physics assumptions. Variations in these assumptions have been tested in section 3.1.3, finding that the assumed velocity distribution of dark matter and detector properties can change the observed recoil rate significantly. We conclude that detectors with a threshold energy as low as possible are preferable. Lastly we discuss exploitation of annual modulation effects, that originates from Earth’s revolution around the Sun, as this is a technique that yields a strong argument for dark matter. In chapter 4 we constructed the simplest dark matter extension to the Standard Model that we could think of. From our model we got a dark matter candidate that interacts with the nuclei in direct detectors. In this model, dark matter is a real scalar field interacting through a Higgs exchange. We saw in section 4.1.2, that experiments disagree when assuming the Higgs as the only mediator. This situation was remedied by treating protons and neutrons in detector nuclei differently, specifically having the ratio between the couplings fn /fp = −0.71 (this fraction was obtained from reference [11]). In section 4.2 we introduced a dipole interaction and tuned the couplings to the Higgs and dipole, such that this feature was accommodated. In the end we saw that a scalar model of dark matter can in fact alleviate tensions between experiments; if the free parameters are properly tuned. It is quite remarkable that experimental disagreement can √ be fixed in a relatively simple way. Furthermore we noted that the fraction 0.71 ≈ 1/ 2, which is probably a coincidence, although it is curious that the fraction√alleviating experimental disagreement, just happens to be the famous irrational number 2. In the last chapter we introduce an effective theory, where we try to think of all interactions for scalar dark matter up to energy dimension 6 in the low energy regime. This encompasses the model from chapter 4 and illustrates possible model extensions, and provides many interaction that could be interesting to investigate in the future. Acknowledgements First of all I thank my supervisor Francesco Sannino for providing great ideas, references and counselling for this thesis. I also thank Ole Svendsen and Mads Frandsen for help and for answering a lot of questions, and of course everybody who read and critiqued the thesis. 29/36 Dark Matter Experiments A Appendices A.1 Notation We use some notation from quantum field theory, here are some definitions. Covariant vectors are written with lower index and contravariant with upper index, the derivatives are ∂ , ∂xµ ∂ , ∂xµ (A.1) ← → φ† ∂ φ ≡ φ† ∂φ − (∂φ† )φ, (A.2) Ā = A† γ 0 , (A.3) ∂µ = / = γ µ Aµ , A ∂µ = where γ µ are the 4 × 4 γ-matrices satisfying the anti-commuting relation (Clifford algebra) {γ µ , γ ν } = 2g µν , (A.4) where g is the Minkowski metric with sign convention (+, −, −, −) and µ, ν ∈ {0, 1, 2, 3}. The chirality matrix is defined as (not called γ 4 for strange historical reasons) γ 5 = iγ 0 γ 1 γ 2 γ 3 . (A.5) We write Feynman amplitudes as M̃ and call them M after spin summation has been performed. Explicit values in this thesis are derived using the natural unit convention of k = c = h̄ = 1 A.2 Flat Friedmann Universe The Friedmann Equations come from Einsteins Field Equations in GR, assuming the Friedmann-Lemaître-Robertson-Walker metric, i.e. the cosmological principle and an ideal fluid universe. The first equation is H 2 (t) + kc2 8πGρ(x) = , 2 a (t) 3 (A.6) where H(t) = ȧ(t)/a(t) is Hubble’s constant, a(t) is the scale factor, and the critical density is ρc = 3H 2 (t)/8πG. Rewritten and taking the density fraction to be Ω = ρ/ρc we get kc2 = ȧ2 (t)(Ω − 1). (A.7) Since k ≈ 0 and we live in an expanding universe we have Ω ≈ 1. 30/36 Dark Matter Experiments A.3 Spontaneous Electroweak Symmetry Breaking The electroweak symmetry is SU(2)L × U(1)y where an SU(N ) means N × N matrices that are unitary U means unitary i.e. group elements u ∈ U(N ) have the property u† u = 1 and S means special and is the requirement that the determinant det(u) = 1. Since elements from SU(2) are 2 × 2 matrices, the Higgs is written as a doublet ! H= φ1 + iφ2 , φ3 + iφ4 (A.8) where φi ’s are real scalar fields. The potential is V = −m2H H † H − λH (H † H)2 , (A.9) where λH is the Higgs self coupling taken here to be negative. The minimal potential is at H † H = 0 if m2H < 0, if however m2H > 0 it is at 0= dV = −m2H − 2λH (H † H), d(H † H) −m2H . 2λH φ21 + φ22 + φ23 + φ24 = (A.10) Choosing some φi ’s correspond to picking a ground state and breaking the symmetry. In q √ 2 particular we take φ1 = φ2 = φ4 = 0 and φ3 = −m2H /(2λH ) = v/ 2 such that ! 1 0 H0 = √ . (A.11) 2 v √ √ Plugging in and rewriting φ1 + iφ2 = 1/ 2(ψ1 + iψ2 ) and φ3 + iφ4 = 1/ 2(v + h + iψ3 ) ! 1 ψ1 + iψ2 H=√ . 2 v + h + iψ3 (A.12) By a gauge transformation some fields can be transformed away, in particular choosing unitary gauge is ! 1 0 . H=√ v + h 2 A.4 (A.13) Scattering Cross Section for Scalar Higgs-coupled DM In this section we denote p, p0 as incoming and outgoing momentum for the DM scalar and k, k 0 for the nucleons. The dominant Feynman diagram is at the tree level. If the model only couples to the Standard Model through the Higgs exchange, we have for neutrons n and protons p 31/36 Dark Matter Experiments N χ χ n h h ≈ Z× χ N χ χ n p h + (A − Z)× χ p The Higgs couples almost equally to protons and neutrons with f mp /v. f is the dimensionless parameter containing the physics coupling the Higgs to the nucleon, the value is approximately f ∼ 0.3 (ref. [11]). So with a Higgs exchange here is really only one diagram with the Feynman amplitude iM̃Higgs " # f mp r i = A × ū (pN 0 ) i u (pN ) [iλv], v (pN 0 − pN )2 − m2H s (A.14) where mp is the nucleon mass and v ' 246GeV. We can take the limit of zero transferred momentum pN − p0N = q → 0 since it in any case is much smaller than the Higgs mass. Summing over final spins s and averaging over initial spins r at the amplitude level gives 1X 1 |M| = |M̃Higgs |2 = 2 s,r 2 2 = 1 2 Af mp λ m2H !2 A2 f mp λ m2H Af mp λ =2 m2H Af mp λ =4 m2H X ūs ur ūr us s,r !2 0 tr[(k/ + mp )(k/ + mp )] !2 (k · k 0 + m2p ) !2 m2p , (A.15) P where uū = k/ + mp , and taking that the trace of any odd number of γ matrices is zero. Again in the zero momentum transfer limit k · k 0 → k 2 = m2p . The cross section σ(N χ → N χ) is spin independent and can be calculated in the center of mass frame, where the center of mass energy is ECM = mχ + mN and p = −k. In the non-relativistic limit Eχ and EN is not much greater than the rest energy. We use A × M as the amplitude, because the way M is defined, we obtain 2 powers of mp and only multiply once with A, so we must multiply with A once more to take into account the mass of the whole nucleus. 32/36 Dark Matter Experiments d3 p0 d3 k0 1 |A · M|2 (2π)4 δ (4) (p0 + k 0 − p − k) σ= 4Eχ EN vr (2π)6 4Eχ0 EN 0 1 |p| = A2 |M|2 16π Eχ EN ECM |vχ − vN | Z m2χ m2N µ = 4π Eχ EN (mχ + mN ) ' Af mp λ m2H mχ !2 µ2 2 2 A fn , 4π (A.16) where fn = f mp λ/(m2H mχ ) is a factor that determines the coupling to the nucleons. A.5 Scalar DM with Dipole Interference: Derivation with Photon Propagator If the scattering could feel the electric charge of proton very weakly we would have fp = fn + δ. (A.17) A possible interaction could be: βe Jµ ∂µ F µν . Λ2 Writing out the interaction one immediately finds Ldipole = (A.18) → βe ∗ ← χ ∂µ χ [∂ µ ∂ ν + g µν ] Aν . (A.19) Λ2 Writing the dark matter fields in momentum Fourier space, we can choose the convention for positive and negative modes Ldipole = χ ∼ e−ip·x , 0 χ∗ ∼ eip ·x . (A.20) The Feynman diagram of interest is χ χ γ p p We treat the vertex by reading of Ldipole and using conservation of momentum p = p0 +pγ and defining the transferred momentum as p − p0 = k 0 − k = pγ ≡ q 33/36 Dark Matter Experiments χ p0 χ p γ pγ i h βe µ ν 2 µν 0 × −p p + p g −ip − ip γ γ γ µ Λ2 i h iβe = − 2 [2p − q]µ × −q µ q ν + q 2 g µν Λ i 2iβe h µ ν = − 2 pµ −q q + q 2 g µν , Λ = (A.21) using the pretty identity −qµ −q µ q ν + q 2 g µν = 0 in the last equality, courtesy of the photon. γ pγ k0 p k p = us [eZγ µ ]ūr , (A.22) where u, ū are spinors for incoming and outgoing protons respectively with spin indices s, r. m2γ Now we can write the entire diagram (γ is virtual and thus off shell meaning q 2 6= = 0) χ χ γ p p 34/36 Dark Matter Experiments i g 2iZβe2 h µ ν νρ 2 µν × 2 × ur γ ρ ūs −q q + q g p µ Λ2 q 2iZβe2 p·q r ρ s =− p − q ρ ρ ×u γ ū = iM̃dipole Λ2 q2 =− | {z Pρ (A.23) } 1X 1 |Mdipole |2 = |M̃dipole |2 = Z 2 2 r,s 2 2βe2 Λ2 !2 1 = Z2 2 2βe2 Λ2 !2 1 = Z2 2 2βe2 Λ2 !2 1 = Z2 2 2βe2 Λ2 !2 1 = Z2 2 2βe2 Λ2 !2 = 4Z 2βe2 Λ2 !2 2 = 4Z 2βe2 Λ2 !2 2 = 4Z 2βe2 Λ2 !2 2 X tr[P ρ ur γ µ ūs us γµ ūr Pρ ] r,s P2 X tr[ūr ur γ µ ūs us γµ ] r,s 0 P 2 tr[(k/ + mp )γ µ (k/ + mp )γµ ] 0 P 2 tr[(−2k/ + 4mp )(k/ + mp )] 0 P 2 tr[−2k/ k/ + 4m2p ] P 2 m2p m2p (p · q)2 p − q2 ! 2 m2p m2χ 1 − cos2 θ . (A.24) We could have rewritten P in terms of the proton momentum k, which should yield the same result. Since p and k are oriented anti-parallel in the center of mass frame, q is taken to be at a right angle to p that is cos2 θ = 0. This condition is the same as saying the second part of the P operator in equation A.23 is identically zero, which is also true when q is acting on the free field (see ref. [6] page 134). We now obtain the same result as treating the scattering as a contact interaction in section 4.2. 35/36 Dark Matter Experiments Bibliography [1] Fritz Zwicky Die Rotverschiebung von extragalaktischen Nebeln 1933. [2] Vera Rubin and Kent Ford Extended Rotation Curves for High-Luminosity Spiral Galaxies 1978 [3] Barbara Ryden Introduction to Cosmology 2006. [4] The Planck Collaboration Planck 2013 results. XVI. Cosmological Parameters 2013 [5] M. Peskin and D. Schroeder An Introduction to Quantum Field Theory 1995 [6] F. Mandl and G. Shaw Quantum Field Theory second edition 2010 [7] R. Bernabei, P. Belli, F. Cappella, R. Cerulli, F. Montecchia1, F. Nozzoli, A. Incicchitti, D. Prosperi, C.J. Dai, H.H. Kuang, J.M. Ma, Z.P. Ye Dark Matter search; arXiv:astro-ph/0307403 [8] A. L. Fitzpatrick, W. Haxton , E. Katz, N. Lubbers, Y. Xu The Effective Field Theory of Dark Matter Direct Detection 2012; arXiv:1203.3542 [hep-ph] [9] Nicolai Sandal Banke; SDU and CP3-origins Bachelor Thesis: Hot vs. Cold Dark Matter 2013 [10] E. Del Nobile, C. Kouvaris, P. Panci, F. Sannino, J. Virkajärvi Light Magnetic Dark Matter in Direct Detection Searches 2012 [11] E. del Nobile, C. Kouvaris and F. Sannino Interfering Composite Asymmetric Dark Matter for DAMA and CoGeNT 2011 [12] E. del Nobile and F. Sannino Dark Matter Effective Theory 2012 36/36