Cosmology and detection of the Dark Axion Portal

Cosmología y detección del Portal Axión Oscuro (Cosmology and detection of the Dark Axion Portal) Trabajo de Fin de Máster para acceder al MÁSTER EN FÍSICA DE PARTÍCULAS Y DEL COSMOS Autor: Juan Cortabitarte Gutiérrez Co-Director : Bradley Kavanagh Co-Directora : Núria Castelló Mor Julio - 2021 Cosmología y detección del Portal Axión Oscuro (Cosmology and detection of the Dark Axion Portal) Resumen El Fotón Oscuro y el axión son partículas ligeras y candidatos populares como Materia Oscura, por ser extensiones simples al Modelo Estándar. Cada una de ellas ha sido buscada actívamente a través de sus acoplamientos llamados portal vector y portal axión. El objetivo principal de esta tesis es presentar un modelo (basado en el trabajo de Kunio Kaneta, HyeSung Lee y Seokhoon Yun de 2017) que introduce un nuevo portal conectando el Fotón Oscuro y el axión, junto con su fenomenología, implicaciones para la cosmología y detectabilidad. Esto se hará realizando predicciones teóricas acerca de las propiedades del axión y del Fotón Oscuro tal que den cuenta de toda la Materia Oscura, siendo a su vez detectables en experimentos de búsqueda directa. La detectabilidad de los Fotones Oscuros será probada específicamente para la LBC, versión de prueba del experimento DAMIC-M, a través de simulaciones del ruido de fondo y cálculos de la señal esperada de los Fotones Oscuros en el detector de la LBC. Palabras clave: Materia Oscura, Fotones Oscuros, Axiones, radiopureza, búsqueda directa. Abstract The Dark Photon and the axion are light particles, popular candidates for Dark Matter for being simple extensions to the Standard Model. Each of them has been actively searched for through the couplings called the vector portal and the axion portal. The focal aim of this thesis is to present a model (based on the work of Kunio Kaneta, Hye-Sung Lee, and Seokhoon Yun in 2017) which introduces a new portal connecting the Dark Photon and the axion, together with its phenomenology, implications for cosmology and detectability. This will be done making theoretical predictions about the properties of the axion and Dark Photon in order for them to account for all Dark Matter while being detectable in direct search experiments. The detectability of Dark Photons will be proven in particular for the LBC, proof-of-concept for DAMIC-M experiment, through simulations of the background noise and calculations of the expected Dark Photon signal in the LBC detector. Key words: Dark Matter, Dark Photons, axions, radiopurity, direct searches. Contents 1 Introduction 1 1.1 DarkMatter......................................... 1 1.1.1 DarkPhoton .................................... 3 1.1.2 Axion ........................................ 4 1.2 EarlyUniverse ....................................... 5 1.2.1 Freeze-Out ..................................... 5 1.2.2 Freeze-In ...................................... 6 1.2.3 Misalignment mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 State of the art in direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 DAMIC-M and the use of CCDs as a Dark Matter detector . . . . . . . . . . 10 1.4 Aimofthisstudy...................................... 13 2 Dark Axion Portal Cosmology 14 2.1 Axions create Dark Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Standard Model photons create Dark Photons . . . . . . . . . . . . . . . . . . . . . . 22 2.3 DarkAxionPortal ..................................... 25 2.4 NoteonColdDarkMatter................................. 26 3 Detectability 29 3.1 Dark Photon absorption rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 Conclusions 38 4.1 Futurework......................................... 39 Bibliography 41 Appendix A Yield equation from scratch 47 Appendix B Minimum mass for Freeze-In 52 Appendix C Freeze-Out calculations 53 Appendix D Kinetic mixing Yield 55 1Introduction 1 1|Introduction Dark Matter accounts for 85% of all the matter content in the Universe, however the scientific community still don’t know its identity, nor its creation mechanism, nor its interactions. So it remains one of the biggest unsolved problems in physics. Many candidates have been named in the search for Dark Matter but there are only a few well-motivated interactions allowed by Standard Model symmetries that provide a connection (what in the following will be called a “portal”) from the Standard Model into Dark Matter [1]. The Dark Photon and the axion are light particles, popular candidates for Dark Matter for being simple extensions to the Standard Model, and in the case of axions, solving the strong CP problem. Each of them has been actively searched for through the couplings called the vector portal and the axion portal. The focal aim of this thesis is to present a model (based on the work of K. Kaneta, H. S. Lee and S. Yun [2]) which introduces a new portal connecting the Dark Photon and the axion, together with its phenomenology, implications for cosmology and detectability. This Dark Axion Portal allows for new couplings, not just from the vector portal and the axion portal individually. Understanding from the very beginning the development and implications of a particle physics/cosmological model and bringing it to the very end to its experimental detection has been the main goal of work in this thesis, which can be summarized as: i) development of the new Dark Axion Portal model based on [2] and exploration of its implications for cosmology (Chapter 2); ii) exploration of the detectability of Dark Photons for a direct search experiment: the LBC, proofof-concept for DAMIC-M (LBC stands for Low-Background Chamber, while DAMIC-M for Dark Matter In CCDs at Modane) (Chapter 3). 1.1 Dark Matter The search for Dark Matter is one of the main research lines among the scientific community, and it has astrophysicists, particle physicists and cosmologists working together. Strong evidence is held on the existence of this form of matter that accounts for approximately 85% of matter on the Universe and 27% of its total energy density [3]. Dark Matter started being investigated in the early twentieth century, with Fritz Zwicky in 1933 applying the virial theorem to infer the gravitational mass of the Coma Cluster, and finding evidence that Dark Matter exists and indeed, in much greater density than luminous matter [4]. This fact started to raise interest for astrophysicists, reaching a peak in the 1960s when Vera Rubin and Kent Ford measured the rotation curve of the Andromeda galaxy leading to new striking evidence on Dark Matter by comparing optical and radio observations [5]. While the optical measurements on galaxies showed a decaying mass distribution from the center, the rotation curves appeared flat, meaning there is some non-visible matter (which density increases towards the outskirts of the galaxy) adding 1Introduction 2 gravitational force. These two results (clusters and galaxy rotation curves) came together in the 1970s when astrophysicist started to look deeper into cosmology finding both problems were in the end, the same one [6]. This meant a point of no return in Dark Matter investigation, making it one of the main puzzles for modern physics. Nowadays, there is overwhelming astrophysical and cosmological evidence for Dark Matter as a major constituent of the universe. As mentioned previously, its gravitational influence is necessary to explain why galaxy clusters are bound together [7] and stars move faster than expected around their galaxy [8], the existence of a large-scale structure in the galaxies distribution in the universe [9] and the features in the Cosmic Microwave Background power spectrum [10]. Significant efforts have been made to understand the nature of Dark Matter and theories have been formulated to explain its existence: from Primordial Black Holes [11], to Dark Fluid theory [12], even alternatives such as modified Newtonian dynamics in the Einstein’s General Relativity [13], but in general terms, we can not talk about Dark Matter without getting into particle physics, where this work relies. Despite the success of the Standard Model, Dark Matter is a good reason to believe that there should exist more particles. Many particles have been proposed to solve the Dark Matter problem: some of them related to known Standard Model particles, such as sterile neutrinos [14]; some coming from new symmetries, such as Dark Photons [15]; some that try to explain other problems in particle physics, such as the axion [16] or supersymmetry particles (e.g. gravitino, neutralino...) [17]; and many more with broader properties, such as the WIMPs. WIMPs are Weakly Interacting Massive Particles, i.e. particles which interact via gravity and any other force, which is as weak as or weaker than the weak nuclear force; and have been the main paradigm in the search for Dark Matter in the last years [18]. However, as newer Dark Matter searches have been tightening the net around the WIMP paradigm and they still don’t appear, new wider searches are coming in the next few years [19]. Particles that do not interact much with the already known particles are usually known as the dark or hidden sector [20]. Given the complexity of the Standard Model, and knowing it only explains a subdominant component of the Universe, it wouldn’t be too surprising if the hidden sector (or sectors, if there were to be more than one) contains a rich structure itself, with Dark Matter making up only a part of it. This hidden sector may contain new light weakly-coupled particles, particles well below the electroweak-scale that interact only feebly with ordinary matter. These feebly interacting particles are the ones that could be detected in direct search experiments. Particles from the hidden sectors arise in many theoretical extensions to the Standard Model, such as moduli that are present in string theory [21] or new (pseudo-)scalars that appear naturally when symmetries are broken at high energy scales. Other powerful motivations include fine tuning problems, as the strong CP problem, and various experimental findings, including the discrepancy between the calculated and measured anomalous magnetic moment of the muon and puzzling results from astrophysics. Besides gravity, there are only a few well-motivated interactions allowed by Standard Model symmetries that provide a “portal” from the Standard Model into the dark sector [1]. The known portals are shown in Table 1.1. 1Introduction 3 Portal Particles Operator(s) Vector Dark Photon − 2cosθWBµνF0µν Axion Axion (Pseudoscalars) a faFµν ˜ Fµν, ... Higgs Dark scalars |S|2H†H, ... Neutrino Sterile neutrinos LHN Table 1.1: Known portals from the Standard Model to the hidden sector. [1] In this work our focus will be in the Vector and Axion portal (which will be summarized individually in Section 1.1.1 and Section 1.1.2 respectively), specifically in the emergence of a new portal when both are present (what we will call the Dark Axion Portal [2]), which will be put in a cosmological and phenomenological context in Chapter 2. Higgs and neutrino portal are better explored in high-energy colliders and neutrino detectors or astrophysical observations, respectively, which are out of the scope of this thesis. Unfortunately, no technique (scintillation crystals [22] [23], noble liquids [24] [25] [26], bubble chambers [27], cryogenic calorimeters [28] [29], among others) has been successful yet in the effort to detect directly these theorized particles. Therefore, its nature, so far elusive, constitutes one of the most exciting mysteries in science. 1.1.1 Dark Photon As already mentioned, Dark Matter suggest the possibility of a whole dark (hidden) sector connected feebly with some Standard Model particles, even existing particles of the hidden sector non-interacting with the Standard Model at all. A minimal extension to the Standard Model that allows for this dark sector is the existence of a dark gauge group, that includes an abelian U(1) gauge symmetry. The usual assumption is that the particle associated to such U(1)Dark are heavy, and decouple from the standard model particles, thereby avoiding all observational constraints. But it could be lighter, for example, if the hidden U(1) is broken by non-perturbative effects, the symmetry breaking scale and thus the photon mass is exponentially suppressed, and can be naturally light [15]. Said particle is called Dark (or Hidden) Photon because the dominant coupling to the Standard Model is via kinetic mixing with the Standard Model photon. But what do we mean when we talk about Dark Photon mixing? Similar to neutrino mixing, photons would have two states, and the propagation and the interaction eigenstates would be misaligned. So we have a massless state which couples electromagnetically (photon) and a massive state that doesn’t couple to the Standard Model in any other way than via mixing (Dark Photon). Kinetic mixing is allowed by all symmetries, and can be realized via a renormalizable coupling [30]. After a Dark Photon γ0gets a small mass, the vector portal well below the electroweak scale is given by: LV ector Portal =χ 2FµνZ0µν ,(1.1) where Fµν and Z0µν are the field strengths of the photon and Dark Photon, χis the kinetic 1Introduction 4 mixing parameter between the two U(1) gauge symmetries (χ21). The Dark Photon has been motivated from various Dark Matter related physics (such as the positron excess) and other physics (such as the gµ−2anomaly) [2]. This Dark Photon is one of the components in the Dark Axion Portal, and one of the Dark Matter candidates included in the theory of this thesis. 1.1.2 Axion The axion is a particle predicted in 1977 by Robert Peccei and Helen Quinn to solve the strong CP problem in the strong force [31] [32]. Axions have been widely studied both theoretically and experimentally, and there exist many different models for different types of axions. In this subsection we will try to explain a little about how the axions appear, how do they solve (and what is) the strong CP problem and what is their importance as a Dark Matter candidate. To see a further review on the physics of the axion, see [16], [33] or [34]. In very few words, there is no known reasons for the strong force (quantum chromodynamics, QCD) to preserve CP-symmetry (Charge-conjugation Parity symmetry). However, if QCD violated CP-symmetry it should give rise to a measurable neutron dipole moment, which would be comparable to 10−18 e·m, while the current best measured limit is (0.0±1.1)×10−28 e·m [35], meaning that QCD must conserve CP to a very high degree. This is a problem because at the end, there are natural terms in the QCD Lagrangian that are able to break the CP-symmetry. This fine tuning problem is known as the strong CP problem. Many solutions have been proposed to solve this problem, being one of the most important the so-called axion. In an effective field theory (EFT) description, the Standard Model is extended by introducing a single new pseudo-scalar particle a, the axion, for which only one coupling is mandatory, namely an effective coupling to the CP violating topological gluon density (a fa+θ)G˜ G, where fais the scale suppressing the effective operator, G=Gµν is the gluon field strength tensor, ˜ Gµν its dual, and we have added to the axion-gluon operator the CP violating θterm. With such a simple extension the strong CP problem is solved because the minimum of the vacuum energy occurs when the coefficient of G˜ Gvanishes. Thus, by acquiring a suitable vacuum expectation value, the axion disposes of the CP violating operator, therefore acquiring a tiny mass, and appearing as Dark Matter [16]. An important contribution of axions as Cold Dark Matter comes from the misalignment mechanism, explained in Section 1.2.3. Axions thus contribute to Dark Matter, in some theories they account alone for all Dark Matter, but for the theory in this thesis, axions will be the other key particle to fill the Dark Matter density in the Universe, together with the Dark Photon. 1Introduction 5 1.2 Early Universe To solve some of the already mentioned problems related with Dark Matter, such us the problem with large scale structure and the features in the CMB power spectrum, Dark Matter particles should be created (or exist) in the very early Universe, after inflation, just so their mass density could affect and alter the evolution of the Universe’s content. There are a lot of scenarios that could explain Dark Matter particles appearing in the Early Universe: the inflaton field decaying partly into Dark Matter particles; the latter existing before inflation but in a much greater number density; very feeble interactions that are big enough in the early Universe due to the extreme densities and temperatures creating all the known Dark Matter; or of course, the mixing of all these ideas and many more in some way or the other. This allows for a huge number of theories on many different Dark Matter particles and their possible origin. In this work we will aim primarily at the idea of a very feeble interaction that could generate the correct amount of Dark Matter in the Early Universe, while maintaining it until the present day. We will have a look at two different mechanisms: Freeze-Out, in which the Dark Matter particles are in thermal equilibrium in the Early Universe with the primordial plasma until their interaction with the Standard Model particles is so feeble that it is no longer efficient; Freeze-In, which works the other way around, starting with almost no Dark Matter and being created due to a feeble interaction but not entering thermal equilibrium. Many particles, including the Dark Photons, are expected to be created either by Freeze-In or Freeze-Out in many models. We will also explain the misalignment mechanism, by which a particle’s field oscillates around the potential minimum dissipating energy as particles in the process, the mechanism by which QCD axions are expected to be created. 1.2.1 Freeze-Out In the process of thermal Freeze-Out, Dark Matter particles have a substantial initial density. Particles in the thermal bath are in thermal equilibrium as long as they keep interacting with each other. This condition is met if the interaction rate between particles (Γ) is higher than the expansion rate of the Universe (H). As the Universe evolves, the number density decreases, temperature decreases and the expansion rate of the Universe increases. The idea can be better seen in the scheme in Figure 1.1. At temperatures greater than the mass of the particles, the particle abundance trace its equilibrium value, but when the temperature falls below the mass of the particles, they stop interacting and they reach the relic abundance, i.e. their comoving density freezes. An attractive feature of the Freeze-Out mechanism is that for renormalisable couplings the comoving number density is dominated by low temperatures with Freeze-Out typically occurring at a temperature factor of 20 −25 below the Dark Matter mass, and so is independent of the uncertain early thermal history of the universe eluding possible new interactions at high scales or times like 1Introduction 6 Figure 1.1: Scheme showing the Freeze-Out mechanism. The dashed line shows the behaviour of the particle number density in equilibrium, but as their interaction is too weak to keep them in thermal equilibrium in an expanding Universe, their relic density freezes [36]. the Grand Unification. [37] The Freeze-Out mechanism has been explored a lot in the literature (see for example [36]), being the genesis of many Standard Model particles, such as neutrinos for example, and being the main mechanism explored in the WIMP models. 1.2.2 Freeze-In Freeze-In, as opposed to Freeze-Out, starts with little or no Dark Matter density. Dark Matter particles are created by a feeble interaction, increasing the number density without entering thermal equilibrium. With the expansion of the Universe, the interaction stops being efficient, so the yield becomes fixed. The comparison between Freeze-Out and Freeze-In can be seen in Figure 1.2. Freeze-In has a simple explanation, and because of Dark Matter particles not entering thermal equilibrium, it allows the creation of cold and warm Dark Matter (important for large scale structure) at any mass scale, given that the interaction with the thermal bath is feeble enough. Dark Matter genesis through this mechanism could happen at very early times in the Universe, allowing for models related with any force, before any phase transition takes place, however it takes us closer to the uncertain early thermal history of the Universe. Freeze-In mechanism is growing in popularity for new Dark Matter models, such as the one presented in this work. For a detailed explanation on different Freeze-In models and candidates, check [37]. 1Introduction 13 1.4 Aim of this study This work is meant to present a Dark Matter direct search experiment in all its aspects, from evaluating a particle physics model and its implications for cosmology, to probing its detectability in a detector. In order to do that a model mixing axions and Dark Photons (two well motivated candidates for Dark Matter) known as the Dark Axion Portal has been studied. All its theoretical background along with its implications for cosmology will be explained and calculated in Chapter 2. As the final part of the modelling its detectability has been calculated (cross-section, event rate, recoil energy...) for a direct search experiment, in particular, LBC. In order to probe said detectability, simulations of the background noise on the LBC (proof-of-concept of DAMIC-M) experiment have been done using two different pieces of software: DAMICG4 and psimulCCDimg. A better explanation will be given in Chapter 3. This allows to study the background noise coming from radioisotopes and compare it to the event rate of the studied model, checking if the detection of, in this case, the Dark Photon is possible. The detection of axions is out of the scope of this work, although the axion search is a very active experimental field. The study of the Dark Photon detection allows to predict the detection limits of both the LBC and DAMIC-M, which will be presented in Chapter 4. 2Dark Axion Portal Cosmology 14 2|Dark Axion Portal Cosmology The theoretical model presented in this thesis relies on the work of Kunio Kaneta, Hye-Sung Lee, and Seokhoon Yun [2] and their idea of the Dark Axion Portal. In this section we will present the theoretical background and give a general idea of what the Dark Axion Portal is. The Dark Axion Portal connects the axion and the Dark Photon (two well motivated Dark Matter particles, as already explained in Section 1.1.2 and Section 1.1.1 respectively). In the presence of the Dark Photon, an axion can couple to two photons, two Dark Photons as well as a photon and a Dark Photon. These interactions are given by the nonrenormalizable Dark Axion Portal terms: LDark Axion P ortal =Gaγ0γ0 4aZ0 µν ˜ Z0µν +Gaγγ0 2aFµν ˜ Z0µν .(2.1) These couplings are shown in Figure 2.1 . For our theory we will always be talking about KSVZ axions (for the details of this axion model see [66, 67]), for them to solve the strong CP problem apart from being Dark Matter. The Dark Axion Portal could also be valid using ALPs and changing axions parameters, but we will stick to the QCD axion, KSVZ model in particular to keep it as simple, minimal extension to the Standard Model (a review of the KSVZ axion together with other axion models can be found at [16]). a γ γ a γ′ γ′ a γ′ γ Gaγγ Gaγγ′ Gaγ′γ′ Figure 2.1: Axion-Dark Photon Coupling. Note the fermions in the triangle loop can have both U(1)PQ and U(1)Dark charges as well as the electromagnetic charges, and the Dark Photon can couple to them directly. Figure from [2] The couplings are given by: Gagg =g2 S 8π2 PQΦ fa ,(2.2) Gaγγ =e2 8π2 PQΦ fa2NCQ2 ψ−2 3 4 + z 1 + z,(2.3) 2Dark Axion Portal Cosmology 15 Gaγγ0≃ee0 8π2 PQΦ fa2NCDψQψ+χGaγγ ,(2.4) Gaγ0γ0≃e02 8π2 PQΦ fa2NCD2 ψ+ 2χGaγγ0,(2.5) where eis the electromagnetic coupling, NC= 3 is the color factor, z=mu/md≃0.56 is the ratio between the mass of the up and down quark, gSis the SU(3)Ccoupling (related with the strong force coupling constant as αS=g2 S 4π≃0.12 at high energies), e0is the U(1)Dark coupling (which can be as sizable as the Standard Model gauge couplings), χis the kinetic mixing parameter of the Dark Photon, Dψis the U(1)Dark charge of the axion, P QΦthe U(1)P Q charge, and Qψthe electromagnetic anomaly of the axion. As e0can be as sizable as the Standard Model gauge couplings, it can be seen from these expressions how the Dark Axion Portal creates new relatively large couplings from the dark gauge symmetry (it could be as large as the axion coupling to the Standard Model photons), even when the connection with the axion-photon-photon coupling is greatly suppressed (χ21). Axions have electromagnetic and color anomaly to make them more interactive with photons and gluons respectively. These anomalies are model-dependent, so in our case, to keep things simple we will tune the electromagnetic anomaly Qψto 0, so that axions interaction with photons is minimized. All in all, our general charge assignment will be PQΦ= 1,e0= 0.1,Dψ= 3 and Qψ= 0. The model can get quite complicated and have multiple extensions if the parameters are changed, therefore we will use a careful approach to unveil the cosmological implications of this model step by step as it follows: •First, in Section 2.1, we analyze the model with the couplings mentioned before assuming there is no kinetic mixing between the Dark Photon and the Standard Model photon, i.e. χ= 0. •Then, in Section 2.2, we will forget for a while about the Dark Axion Portal and we will calculate the cosmological implications of just having Dark Photons kinetically mixed with the Standard Model photons. •Finally, in Section 2.3, we will merge both approaches and see how the Dark Axion Portal behaves with Dark Photons being kinetically mixed with Standard Model photons, and we will be able to check if the final model is detectable while accounting for all Dark Matter in the Universe summing the axions and Dark Photons contributions (this will be left for Chapter 3). 2.1 Axions create Dark Photons As an starting point to this theory we have production of Dark Photons through Freeze-In mechanism. The interaction leading the Freeze-In is gluon annihilation, with the axion mediating the interaction. The Feynman diagram of the interaction is: 2Dark Axion Portal Cosmology 16 g g γ0 γ0 a Figure 2.2: Gluon annihilation to Dark Photon pair production mediated by the axion. This annihilation happens at very early times in the Universe, right after inflation, so that the temperature and density of the primordial plasma is large enough to allow this interaction. One can relate the density parameter of Dark Photons with the properties of the axion and the Dark Photon as in equation 17 in [2]: Ωγ0h2≈S0mγ01080√10α2 sPQ4 Φe04D4 ψ36MPlT3 RH 256ρcg3/2π15f4 a ,(2.6) where S0= 2889.2cm−3is the entropy density at the present time, ρc= 1.05368 ×10−5GeV cm−3the critical density, MP l ≈2.4×1018 GeV the Planck Mass, Ωγ0h2the density parameter (0.12 to take account for all the Dark Matter in the Universe), g= 100 the number of degrees of freedom of the thermal bath of the Universe during the creation of the Dark Photons, αs= 0.12 the strong coupling constant, TRH the reheating temperature of the universe (temperature at which the Universe is at the end of inflation) and mγ0the mass of the Dark Photon. The calculations that lead to this equation are long, and in general do not provide vital information for the reading of the thesis. They are left out of the main body of this thesis but can be found in Appendix A, from obtaining the entropy density in the Early Universe, passing through the development of the Boltzmann equation for Dark Photons density, to the Equation (2.6) here presented. So this equation relates the mass of the Dark Photon with the axion decay constant and therefore to the mass of the axion, because after the QCD phase transition, the axion mass is given by: ma≃√z 1 + zfπ fa mπ,(2.7) where z=mu md≃0.56, and mπ≃135 MeV and fπ≃92 MeV are the mass and the decay constant of the pion, respectively, giving ma≈5957.87 MeV2 fa. Therefore, from eq. (2.6) we can get a plot for the mass of the axion in terms of the mass of the Dark Photon if we fix the Dark Matter relic abundance to fit all the Dark Matter in the Universe (Ωγ0h2= 0.12), for given values of the reheating temperature (as we have chosen the dark portal interaction charges), as illustrated in Figure 2.3, matching the plot in [2]. 2Dark Axion Portal Cosmology 17 Figure 2.3: The blue lines show Ωγ0h2= 0.12 for the given fa/TRH values. We choose e0= 0.1, Dψ= 3,PQΦ= 1 with g= 100. For low fa, the axion alone cannot explain the observed relic density, yet it can be accounted for with the Dark Photon. Figure 2.3 shows the linear correlation between the mass of the axion and the mass of the Dark Photon, for any value of the reheating temperature. For it to be more understandable, we set the solid lines to be only TRH dependent instead of the fa/TRH ratio, we extend the plot limits, and we set some known constrains. These constrains are: •Upper bound on ma: Peccei-Quinn axions cannot be heavier than ≈10−3GeV due to Supernova observations [33] •Lower bound on ma: They can’t also have a mass lower than ≈10−5GeV, because due to the misalignment mechanism, that would lead to a higher density parameter than the one for all Dark Matter in the Universe (0.12) [68]. •Lower bound on TRH : The reheating temperature “must be” higher than 109GeV due to leptogenesis [69]. •Lower bound on mγ0: If you want to use the Freeze-in mechanism, you need to avoid entering thermal equilibrium. This sets an upper constrain in the reheating temperature, because if the temperature is too high the Dark Matter annihilation takes place most frequently and it might be thermalized. Once entered this constraint in (2.6), it forces a lower bound on the mass of the Dark Photon (for Ωγ0h2= 0.12 that is mγ0>157 eV, however for smaller values 2Dark Axion Portal Cosmology 18 of Ωγ0h2the constraint relaxes and Dark Photon mass can be lowered, see Appendix Bfor the detailed calculation). This is an important constraint for Freeze-In model, not remarked by [2], that sets very valuable experimental boundaries. With the constraints and the same conditions as in the first figure (Figure 2.3) we can plot Figure 2.4. Figure 2.4: Figure 2.3 with the same parameters but putting the lines as a function of only the TRH and setting the mentioned constrains. Once we have the results for the Freeze-In model, we can try to fit in this parameter space the Freeze-Out model, which should give permitted values (if any) out of the boundaries of the Freeze-In model. After performing the calculations (check Appendix Cfor more information), the relativistic Freeze-Out model shows a single value for the Dark Photon mass (which is 1/3of the lower limit in the Freeze-In). This is because the Dark Photons are in thermal equilibrium while being highly relativistic in the Freeze-Out model, when Tmγ0, and the Yield becomes fixed from that moment on, fixing the mass and not depending on the axion parameters. The non-relativistic Freeze-Out model gives a single line of values, dependent on faand the density parameter Ωγ0h2. This is shown in Figure 2.5. The other change between Figure 2.4 and Figure 2.5 is the inclusion of the axion relic density to check if it is truly negligible. 2Dark Axion Portal Cosmology 19 As in this model we work with the QCD axion, some of its properties have been well explored. One is the axion contribution to the Universe’s energy density, done by Fox, Pierce and Thomas in [68] and quoted by Peccei in [33]: Ωah2= 0.5fa/ξ 1012 GeV7 6hθ2 i+σ2 θiγ , (2.8) where ξis the coefficient of the Peccei-Quinn anomaly (ξ= 1 for KSVZ models, the one we are using), γis the dilution factor (we assume no dilution, so γ= 1), θithe misalignment angle (average is Dθ2 iE=π2 3) and σθits fluctuations (we neglect them, so σθ= 0). This gives a constraint for the maximum value of fa(or what is the same: a minimum value for ma), for Ωah2= 0.12 if we only had axions as Dark Matter, but in our case, we also have Dark Photons. Then the constraint builds up as Ωγ0h2+ Ωah2= 0.12, and then both components have to share the parameter space. It can be seen how if we make the axion density parameter negligible, let’s say 0.001, then fa= 1.75×109, and thus it cannot fit into the parameter space, as it is forbidden by the lower bound on the reheating temperature. This forces us to take both density parameters into account, which is shown in Figure 2.5. Figure 2.5: Figure 2.4 including a constrain in the axion mass for its density parameter (in this case as exempli gratia Ωγ0h2= 0.09,Ωah2= 0.03 assuming all DM is axions +Dark Photons) in blue, and the Freeze-Out mass of the Dark Photon. It is important to remark how the shaded blue region in Figure 2.5 corresponds to example 2Dark Axion Portal Cosmology 20 values of the importance of the axion density parameter. It constraints even more the parameter space when taken into account, and it is not negligible. Only when the density parameter of the axion is large is the parameter space open, as the shaded region goes up (the mass of the axion can be smaller) and the minimum value for the Dark Photon mass because of Freeze-In gets reduced (so the mass of the Dark Photon can also be smaller). If the amount of axions as Dark Matter in the Universe is small, then the mass of the Dark Photon is strongly constrained. Now we can do some little approximations to see how easily can we detect this Dark Photons that couple to the Standard Model just through the axion. In order to try and direct detect this Dark Photons, we have to do at least a sloppy approximation to see the order of magnitude of: a) the recoil energy in the detector and b) the cross section of the interaction. With these two we can have an idea on how well will the direct detection be. For these calculations we will use DAMIC-M experiment as an approach, which uses silicon CCDs as detectors. Therefore we will calculate the energy recoil of silicon nuclei when scattering a Dark Photon, and we will check the cross section of a Dark Photon scattering with a gluon in said nuclei. Taken that the Dark Photons now have a relative velocity of ∼100 km/s (dispersion velocity in the Milky Way) the elastic scattering occurs in the extreme non-relativistic limit, and the recoil energy of the nucleon is easily calculated in terms of the scattering angle in the center of mass frame θ∗[70]: ER=µ2 Nv2(1 −cosθ∗) mN ,(2.9) where µN=mγ0mN mγ0+mN, the DP-Nucleus reduced mass. The recoil energy must be smaller than the scattering with the best conditions, i.e. opposite velocities for Earth and the Dark Photon, so v∼200 km/s ∼6.67 ×10−4c; perfect 180◦scattering angle. Knowing the mass of a silicon nucleus mN≈2.63 ×1010 eV, and assuming a heavy Dark Photon mγ0= 1 GeV: ER.32 eV.(2.10) And this approximation is for a very heavy Dark Photon (almost out of the parameter space of Figure 2.5) and without taking into account the efficiency of the detector to transform the energy recoil into a readable current. Due to the axion having a very little mass, there are corrections that would enhance the energy recoil, but very littly, so the direct detection within this model seems very hard, as typically CCD detectors have a threshold of a few tens eV. To calculate the cross section we can use the Feynman diagram of the interaction: 2Dark Axion Portal Cosmology 21 γ0γ0 gg a Figure 2.6: Dark Photon gluon scattering mediated through the axion. The upper vertex will have Gagg eq. (2.2) as coupling, while the bottom one will be Gaγ0γ0 eq. (2.5), so its cross section can be approximated as: σv ∼1 s|Gaggq21 q2Gaγ0γ0q2|2∼G2 aggG2 aγ0γ0 t2 s,(2.11) as q, the four-momentum of the exchanged particle is q∼√t, the energy in the center of mass reference frame. As all the initial momentum of the Dark Photon (p2) is transferred to the gluon (p3), which has 0 initial momentum (p1= 0), and remembering the Mandelstam variables (s= (p1+p2)2and t= (p1−p3)2) we have: σv ∼G2 aggG2 aγ0γ0 t2 s∼G2 aggG2 aγ0γ0 (−p2)4 (p2)2∼G2 aggG2 aγ0γ0p2 2.(2.12) If we take p2=mγ0vγ0, where v∼100 km/s and mγ0= 1 GeV, and we take a reasonable value for fasuch as 1011 GeV, then we find: σv ∼1.9×10−77 eV−2∼7.4×10−87 cm2.(2.13) This is a terrible result, not only because it is almost 50 orders of magnitude below any modern direct search experiment for that mass, but because it is even 40 remarkable orders of magnitude below the neutrino floor, cross section sensitivity at which the neutrinos would be detected in the same way as Dark Matter. Corrections on spin-dependent scattering are more important in this case than in the case of energy recoil calculations, however they won’t cover the 50 orders of magnitude that separate the theory from the experiments. So with this calculations it is demonstrated that an experiment like DAMIC-M wouldn’t be enough to detect this Dark Photons, and no experiment would be capable in the early future. At this point we will put a halt in the gluon-gluon-axion to axion-DP-DP model to focus on the 2Dark Axion Portal Cosmology 22 kinetic mixing between Dark Photons and Standard Model Photons. Later we will come back to mix both models into one and see the improvements in direct detection. 2.2 Standard Model photons create Dark Photons We now seek enlightenment in a new paper [15]. From this paper we can obtain the Yield equation for this new process (what we are looking for) and new constraints for the model, but we will do it slowly anyway. Dark Photon mixing, as explained in Section 1.1.1, gives a massless state which couples electromagnetically (photon) and a massive state that doesn’t couple to the Standard Model in any other way than via mixing (Dark Photon). Therefore we will denote γand γ0the flavor states, while γ1will be mostly a “photon-like” mass eigenstate and γ2the “Dark Photon-like”. This mixing leads to photon oscillations in vacuum, with a tiny mixing angle (i.e. a tiny probability of oscillation), however we are interested in the oscillation in the Early Universe, which should be greater to account for all the Dark Matter in the Universe today. In the Early Universe, the photons were in a thermal plasma, so matter effects should be taken into account. We can include the influence of the plasma in the photon propagation through the photon’s self energy, which acts a complex effective mass denoted by ωD. The real part of the mass encodes the refraction properties of the plasma and we will refer to it as the plasma mass or the photon mass denoted by mγ. The relevant definitions and calculations on the effective mass of photons in a plasma can be found on [15], for now we will just keep for ourselves the idea that in a plasma, the photons gain an effective mass that allows a resonance whenever the effective mass of the photons matches the mass of the Dark Photons, allowing a peak in Dark Photon production. This can be seen especially in the expression for the effective mixing angle in a damping dominated medium, given by [71]: χ2(ω, T)≃χ2 0 m4 γ0 m2 γ0−m2 γ2+ (ωD)2,(2.14) where χ0is the kinetic mixing parameter in vacuum. This expression shows how the effective mixing angle depends implicitly on the energy and the temperature through mγand ωD. The imaginary contribution to the photon mass, ωD, is typically smaller than the real part, mγ, so it only plays a role near the resonance mγ=mγ0where it acts as a cut off. There are several important mass regimes with different production mechanism. Whether the mass of the Dark Photon is greater than two times the mass of the electron is important, as it would lead to pair production of Dark Photons in electron-positron annihilation. In this work we will just focus in the idea of light Dark Photons mγ0<1MeV, so the production mechanism relevant to us is resonant production. Electrons are non-relativistic for such light Dark Photons, thus mγ∝T3/2e−me/T , so in the range 1eV< mγ0<1MeV the resonance happens not far from 3Detectability 29 3|Detectability In this chapter we will explore the detectability of the model explained in Chapter 2, first looking at the background noise in the LBC and then checking the Dark Photon absorption rate for the model. Simulations have been done in order to probe the detectability of the Dark Photons in the DAMIC-M’s proof of concept, the LBC. Impurity-induced background in the LBC is simulated using Monte Carlo methods by putting a lot of radioisotopes (5×107) in each part of the LBC geometry (see Figure 3.1) and mimicking the CCD response to the particles that get to it. Then we get a whole energy density spectrum, which we normalize to the known values of the radioactivity of the materials used to build the final Low Background Chamber. Figure 3.1: a) CCD module including the sensitive detector zone (light blue), kapton cable (green) and copper frame (orange). b) Copper box holding both CCD modules. c) Roman lead layer. d) “New" lead layer covering the Roman one. e) Whole LBC geometry including cryostat layer. In light blue are represented holder pieces and in green the electronic input for the cryostat. The LBC geometry consists in: •A sensitive detector: two 4000x6000 pixels (15x22.5 µm, 675 µm thick) CCDs composed of 3Detectability 30 silicon and polysilicon layers. •Kapton cables, for the electronic imput. •A copper frame for the CCDs. •Roman lead (taken from a spanish galleon) for protecting the CCDs from ionizing radiation while not radiating itself (because of being old, the decay time for lead nuclear chain has already passed and thus it does not induce more background noise). •“New" lead extra layer (new lead is cheaper and easier to find than the old one and radiopurity this far from the sensitive detector is unnecessary). •External cryostat to maintain the temperature at around 130 K. The final experiment also has a 30 cm polyethylene external layer in order to provide a shielding against high energy gammas, neutrons from cosmic rays and nuclear decays, preventing nuclear activation. The shielding works due to the high content of hydrogen atoms in polyethylene. It is not necessary for the purposes of this simulation, as its possible background noise hardly reaches the CCD, making it totally negligible. The software used to mimic the response of the DAMIC-M CCDs is composed of two packages: DAMICG4 and psimulCCDimg •DAMICG4 performs Geant4-based simulations of the detector; being Geant4 a C++ framework which relies on Monte Carlo (MC) modelling to emulate physical processes [76]; •psimulCCDimg is a Python3 module used to reconstruct the response of the detector [77]. A more extensive explanation on how Geant4 and psimulCCDimg work can be found in [78], here we will try to summarize each software’s functionality within this thesis. DAMICG4 is a Monte Carlo simulation software that, roughly speaking, simulates the passing of particles through the detector by simulating its fundamental interactions with the different materials, guided by a list of physical processes that are given probability values. The processes a particle undergoes are chosen according to said probabilities at every “step” (a short distance within the materials at which the software evaluates the properties of the particle and where it is). This simulated particles could create secondaries due to its interaction, which are also tracked. The output of the simulation is a collection of geometrical points inside the detector (in our case the CCDs) where the particles have lost energy. The energy loses inside the CCDs feed psimulCCDimg, a Python3 module that mimics the CCD response converting the energy losses in electron-hole pairs, drifting and diffusing the charges as if they were collected in a real CCD, pixelizing the signal and simulating the detector noise (dark current and readout noise, as mentioned in Section 1.3.1). 3Detectability 31 In our case, we are interested in simulating the radio-impurities in the LBC materials and the background noise they generate. For this purpose we start generating radioactive nuclei with no initial kinetic energy in each volume of the LBC geometry (check Section 1.3.1), so they start static in the geometry and then decay creating the decay products (electrons, positrons, alphas, neutrinos...) that move, interact and end reaching the CCDs. The simulated isotopes belong to several decay chains (238U, 226Ra, 210Pb and 232Th), but are simulated individually and decay one by one instead of simulating the whole chain. In Table 3.1 you can find all the radioactive isotopes simulated. Parent Chain Isotopes Considered Comments 238U234mPa 234Th 226Ra 214Pb 214Bi 210Pb 210Pb 210Bi 232Th 228Ac 228Ra 212Pb 212Bi 208Tl 40K40K Activation 56Co Only in Copper parts, and if it is not electroformed copper. All of them produced in the copper by spallation. 57Co 58Co 60Co 59Fe 54Mn 46Sc Table 3.1: Decay chains considered in this analysis. To find the energy spectrum these isotopes leave, we simulate 5×107of each one in each volume, distributed uniformly, so we have a large enough statistic. Then, in order to find the correct induced spectral background, we need to normalize said large statistic to the real number of events we would find in the experiment. The normalization is done with the following equation: nnorm clusters(Ei) = nsims clusters(Ei)1 Nsims AisotopeMsimvol 1 Mdetector Nbins Emax −Emin ,(3.1) where: 3Detectability 32 •nsims clusters(Ei)is the total number of clusters (individual energy deposits in the CCD) in the energy bin centered at Ei. •Nsims is the total number of simulations (5×107in this case). •Aisotope is the activity of the isotope at the simulated volume, in units of decays/day/kg of the material forming the simulated volume. •Msimvol is the total mass of the simulated volume, in units of kg. •Mdetector is the total mass of the simulated detector, in units of kg. •Emax −Emin is the full energy range evaluated in the spectrum, in keV. •Nbins is the total number of energy bins in the range [Emax, Emin], uniformly distributed. This normalization allows us to find the correct background noise generated by each isotope in each volume, in units of decays/kg/day/keV (the keV part is to eliminate the effects of the binning in the simulated spectrum and kg refer to the detector’s mass). In order to determine the induced spectral background, we need the activity of each isotope in each component of the detector (Aisotope in eq. (3.1)). For LBC simulation studies we will use the contamination levels obtained on the DAMIC analysis. Said levels can be found in Table 3.2. Part U-238 Ra-226 Pb-210 Th-232 K-40 CCD <0.53 <0.43 <33 <0.4 <0.04 Kapton cable 5013.8 ±423.4 420 ±490 420 ±490 276.5 ±42.0 2475.4 ±172.8 Copper <10.7 <10.7 2350 ±720 <3.5 <2.7 Module Screws 1400 ±3800 <138 2350 ±720 200 ±140 2400 ±1300 Ancient lead shield <10.7 <25.9 2850 ±285 <2.8 <0.5 Outer lead shield <1.1 <13 1560000 ±430000 <0.4 <19 Table 3.2: DAMIC activities used to constrain the amount of radioactivity in each component in units of decays/kg/day. In the case of the Copper, the measurements come from various sources (DAMIC100, Canfranc) as the new electroformed copper will be used. Now that it has been explained how the background simulations are done and normalized, it can be better understood with an example. We will use the Kapton cable, as all of the isotopes mentioned in Table 3.1 are important in this volume. First all isotopes are simulated individually, with 5×107isotopes equally distributed through the whole volume. Then the energy deposits of each one is plotted, see for example the one for 234mPa in Figure 3.2. Once we know the energy deposit we can normalize the histogram according to eq. (3.1), in this example with 234mPa, known that the Kapton cable mass is 5.81 grams, the mass of one CCD is 8.49 g, and using 100 bins between 0 and 20 keV. We scale Figure 3.2 resulting in Figure 3.3. 3Detectability 33 Figure 3.2: Simulated background noise induced by 5×107isotopes of 234mPa in the Kapton cable of the LBC. Figure 3.3: Simulated background noise induced by 234mPa in the Kapton cable of the LBC as in Figure 3.2, but rescaled for the values expected in the LBC, in units of decays/kg/day/keV. 3Detectability 34 At this point to know the background noise, we want the peaks to be excluded (as they can be subtracted knowing the radioisotopes typical emission lines), so the wanted data is the detection limit imposed by the Compton background (the flat noise). In order to do this, we fit the histogram to a constant as y=ain a zone with no emission peaks, in this case between 2and 6keV, finding for this particular case that the 234mPa from the Kapton cable induces 0.605±0.002 decays/kg/day/keV. Extrapolating this example to every isotope we can sum them all up to know the whole noise induced by the Kapton cable volume, which is shown in Figure 3.4. 0 2 4 6 8 10 12 14 16 18 20 Energy [keV] 5− 10 4− 10 3− 10 2− 10 1− 10 1 10 2 10 clusters [/day/kg/keV] [d.r.u.] Kapton Cable: 1.253 +/- 0.005 d.r.u. Kapton Cable: 1.253 +/- 0.005 d.r.u. Pa: 0.605 +/- 0.002 d.r.u. 234 Th: 0.405 +/- 0.002 d.r.u. 234 K: 0.1337 +/- 0.0007 d.r.u. 40 Pb: 0.0300 +/- 0.0001 d.r.u. 214 Tl: 0.0288 +/- 0.0001 d.r.u. 208 Ac: 0.02200 +/- 0.00009 d.r.u. 228 Pb: 0.01896 +/- 0.00008 d.r.u. 212 Co: 0.00382 +/- 0.00002 d.r.u. 57 Co: 0.00223 +/- 0.00001 d.r.u. 58 Pb: 0.00223 +/- 0.00001 d.r.u. 210 d.r.u. -4 Co: (6.93 +/- 0.02)x10 56 d.r.u. -4 Fe: (8.35 +/- 0.06)x10 59 d.r.u. -4 Ra: (1.34 +/- 0.09)x10 228 d.r.u. -4 Co: (4.11 +/- 0.02)x10 60 d.r.u. -5 Mn: (8.15 +/- 0.06)x10 54 d.r.u. -4 Sc: (1.274 +/- 0.008)x10 46 Figure 3.4: Background noise induced by every isotope and the total of the Kapton cable of the LBC in units of clusters/kg/day/keV. It couldn’t be unnoticed by a watchful eye how Figure 3.4 is missing all Bismuth isotopes included in Table 3.1. This was caused by a bug on the new version 10.06 of DAMICG4 and is intended to be corrected soon. Extending the work done with the Kapton Cable to all the volumes in the LBC geometry (Copper frame holding the CCDs named ColdCopper, Roman Lead called AncientLead and “‘New Lead” called LBLead), we can obtain the complete background noise in the LBC. This is shown in 3Detectability 35 Figure 3.5. 0 2 4 6 8 10 12 14 16 18 20 Energy [keV] 5− 10 4− 10 3− 10 2− 10 1− 10 1 10 2 10 clusters [/day/kg/keV] [d.r.u.] LBC: 1.543 +/- 0.009 d.r.u. Total: 1.543 +/- 0.009 d.r.u. Kapton Cable: 1.253 +/- 0.005 d.r.u. ColdCopper: 0.1838 +/- 0.0008 d.r.u. AncientLead: 0.106 +/- 0.003 d.r.u. LBLead: 0.0023 +/- 0.0016 d.r.u. Figure 3.5: Background noise induced by every volume and the total of the LBC in units of decays/kg/day/keV. Like this we have obtained the background noise induced by isotopes in the LBC. Because of the bug in Bismuth, and although at low energies (below 0.1keV) the background noise is reduced (there is less low energy ionization), we will keep a conservative approach by taking the background noise as 3 clusters/day/kg/keV. It is important also to remember the other noise sources, which include the readout noise and the dark current. As explained in Section 1.3.1 the readout noise is negligible. With DAMIC having the lowest dark current ever measured in a silicon detector, the dark current only constrains the measurement below two electrons (7.54 eV). 3Detectability 36 3.1 Dark Photon absorption rate Now that we know the expected background noise in the LBC, it has to be compared with the Dark Photons signal, especially looking at the Dark Photon absorption rate. Dark Photon absorption signal should be seen as a peak of electrons with an energy E∼mγ0. The absorption cross-section of a Dark Photon σγ0(mγ0)with velocity vis directly related with the photoelectric cross section for a photon with energy mγ0c2,σγ(mγ0c2)[55, 79, 80, 81] as: σγ0(mγ0)v=χ2σγ(mγ0c2)c . (3.2) This allows us to account for the rate of absorbed Dark Photons in a target as: R=ρDM mγ0 χ2σγ(mγ0c2)c , (3.3) where ρDM is the local density of Dark Matter composed by Dark Photons (ρDM = 0.3GeV c−2 cm−3if Dark Photons accounted for 100% of Dark Matter). Now we can get the values of the photoelectric cross section in silicon (Above 10 eV: [82] and below 10 eV: [83] [84]) and check the rate of absorbed Dark Photons (assuming they account for 100% of the Dark Matter of the Universe) per kg of detector and per day, as shown in Figure 3.6. Figure 3.6: Absorption rate of Dark Photons (Ωγ0h2= 0.12) in silicon as a function of mγ0. 3Detectability 37 At this point, we have a rough estimate of the detector sensitivity. We can determine the rate of Dark Photons absorbed in a detector for the Dark Axion Portal model (i.e. Dark Photons being 1% of the total Dark Matter), and compare it to the background noise to check if the absorption of Dark Photons will leave a peak in the detected spectra. This is shown in Figure 3.7. Figure 3.7: Absorption of Dark Photons (Ωγ0= 0.01ΩDM ) in silicon, compared with the background noise of the LBC ≈3kg−1day−1keV−1, and the expected from DAMIC-M ≈0.1kg−1day−1keV−1. The cyan vertical dotted lines represent 1, 2 and 3 electrons (3.77, 7.54 and 11.31 eV respectively), as because of the dark current, we can detect 3 electrons or more. It can be seen how the model can be detected even in the LBC (which was not first intended for detection but for R&D), as the signal is over the noise at an energy of 3 electrons (which is no longer obscured by dark current). This shows the detection power of the DAMIC-M experiment and especially the new skipper CCDs. 4Conclusions 38 4|Conclusions In this thesis we have studied direct Dark Matter searches end to end, from exploring a theoretical model for particle Dark Matter and its implications for cosmology, with Dark Photons and axions as the candidates, to examining the detectability of said Dark Photons in a particular direct search experiment, DAMIC-M (in particular, its proof-of-concept, the LBC). For the theoretical model, the Dark Axion Portal has been studied, exploring a connection between axions and Dark Photons. Pairs of Dark Photons are created via annihilation of gluons, mediated by the axion in the Early Universe. This happens through the Freeze-In mechanism, which opens a wider parameter space for the properties of both the axion and the Dark Photon than the Freeze-Out mechanism. Another connection of this portal to the Standard Model is the kinetic mixing of Dark Photons to Standard Model photons, which would create a negligible number density of Dark Photons, but would enhance the possibilities of direct detection. Thanks to the possibilities provided by the Dark Axion Portal, the model can account for all Dark Matter in the Universe, while being Cold Dark Matter, with 1% Dark Photons and 99% axions. This is a promising scenario involving multi-component Dark Matter, which has prospects of detectability in the near future, as many experiments work on the detection of Dark Photons, and many on axions. In order to illustrate the detectability of Dark Photons in a direct search experiment, simulations were done using the proof of concept of DAMIC-M, the LBC. The background noise induced by radioimpurities (from materials surrounding the detector) was simulated, while emulating the intrinsic detector noise at the same time. This allows us to know if we can discriminate the expected signal from Dark Photon absorption in the detector (see Figure 3.6) from the backgound noise in the LBC (see Figure 3.5). The result shows a possible detectability at low energies, which is very promising for the LBC, and is shown in Figure 3.7. Said detectability allows us to include the LBC and DAMIC-M in the constraints on the kinetic mixing done in [58] (see Figure 2.8), by tuning the mixing parameter until the background rate and the signal rate match. It can be noted how these curves match the ones obtained in Figure 3.7 but upside down because the kinetic mixing parameter is proportional to the rate of absorbed Dark Photons as stated in eq. (3.3). The constraints become even stronger if Dark Photons are set as all the Dark Matter in the Universe, instead of just a 1%. This is shown in Figure 4.1. 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By using the Second Law of Thermodynamics: TdS= dU+PdV−µdN , (A.1) and the mentioned values of Uand µin the radiation-dominated universe, we can find how: dS=1 T(d(ρV ) + PdV) = 1 T(ρdV+Vdρ+PdV) = 1 T(d [(ρ+P)V]−VdP).(A.2) Calculating the second derivatives: ∂S ∂V =1 T∂[(P+ρ)V] ∂V −VdP ∂V =1 T(ρ+P)−V∂P ∂V ⇒∂2S ∂V ∂T =−P+ρ T2,(A.3) ∂S ∂T =−V T ∂P ∂T −→ ∂2S ∂T∂V =−1 T ∂P ∂T ,(A.4) and using the equality of second derivatives ∂2S ∂T ∂V =∂2S ∂V ∂T : −1 T ∂P ∂T =−ρ+P T2−→ ∂P ∂T =ρ+P T.(A.5) Mixing equation eq. (A.2) and eq. (A.5): dS=d [(ρ+P)V] T−V(ρ+P) T2dT= d ρ+P TV.(A.6) If we try to find the changes of entropy with time: dS dt=d dtρ+P TV=V TdP dt+dρ dt+P+ρ T dV dt−(P+ρ)V T2 dT dt= V(ρ+P) T2 dT dt−V(P+ρ) T2 dT dt+V T dρ dt+P+ρ T dV dt= V T dρ dt+P+ρ T dV dt=V Tdρ dt+ (P+ρ)dV Vdt. (A.7) Taking the volume Vin terms of the scale factor of the Universe a,V∝a−3: V T"dρ dt+ (ρ+P)3a−4da dt a−3#=V Tdρ dt+ (ρ+P)3H= 0 .(A.8) Appendices 48 The last statement, eq. (A.8) being equal to 0, comes because that equation is exactly the same expression as the continuity equation for the universe. We have thus found how the entropy is conserved in the thermal bath of the early, radiationdominated Universe. As the Universe expansion is adiabatic (due to the definition of Universe), entropy must remain constant even beyond equilibrium. We can then define an important quantity, the entropy per comoving volume, i.e. Entropy density: s≡S V=ρ+P T.(A.9) We can find a useful expression for the entropy density, computing the density and pressure. We know that the radiation density is that of relativistic bosons: ρ=g (2π)3Zf(~p)E(~p) d3p , (A.10) with g the degrees of freedom of the gas and f(~p) = 1 e~p/T −1. Assuming an spherical symmetry E(~p) d3p= 4πp3dp, and thus: ρ=g (2π)3Z∞ 0 4πp3 ep/T −1dp=g 2π2T4Z∞ 0 x3 ex−1dx=g 2π2T4Γ(4)ζ(4) = π2 30gT 4.(A.11) Knowing that the pressure in a Bose-Einstein gas is P=ρ 3, then: s≡S V=ρ+P T= π2 30 +π2 90!gT 3=2π2 45 gT3.(A.12) Knowing the entropy density in the thermal bath, we can now explore non-equilibrium. First of all, let’s start from the Boltzmann equation: dn dt+ 3Hn =C[n],(A.13) where nis the number density of any particle, and C[n]is the collision term (all complex physics goes in there, and acts like a cross section for the collision of said nparticles). We can try to express the Boltzmann equation in terms of the Yield (Y=n s). d dtn s=dn dt 1 s+nd dt1 s=1 s dn dt−n s2 ds dt.(A.14) Knowing from the entropy conservation calculations that the entropy density is proportional to the scale factor as s∝a−3: 1 s dn dt−n s 3a−4da dt a−3=1 s dn dt+ 3HY . (A.15) All in all: Appendices 49 d dtn s=dY dt=1 s dn dt+ 3HY −→ dn dt=sdY dt−3HsY =sdY dt−3Hn . (A.16) By comparison with the Boltzmann equation eq. (A.13): C[n] = sdY dt=sdY dT dT dt=sT dY dT dT Tdt hT∝a−1i == −sT dY dT a−2da dt a−1.(A.17) And we finally arrive to equation (14) in [2], which in our calculations is eq. (A.18): −sTH dYγ0 dT=γ[nγ0].(A.18) We can find a useful expression for H working with the Friedmann equation: H2−8 3πGρ =−kc2 a2= 0 ,(A.19) as k= 0 because we have a flat Universe. Hence: H2=ρ 38πG =π2 90gT 41 M2 Pl ,(A.20) where MPl =1 √8πG ≈2.4×1018 GeV is the reduced Planck Mass. The problem now comes with the collision term. In general, the collision term is the production rate, minus the annihilation rate. This can include many physical processes depending on the particle we are looking at, but in the case of the Dark Photon production, mediated with the axion, and for Freeze-In production, we just have production of Dark Photons, and a negligible annihilation (pair annihilation of gluons create pairs of Dark Photons). This, in mathematical terms is hσvin2 g, that is the thermal averaged cross section multiplied by the number density of particles, in this case, gluons (squared as they annihilate by pairs). Guided by the calculations in [85] we arrive to the equation (19) in [86]: γ[nγ0] = T 64π4Z∞ 0 (σv)s3/2K1√s Tds . (A.21) To approximate the cross section, we can work with the Feynman diagram: g g γ0 γ0 a Appendices 50 The left vertex will have Gagg as coupling, while the right one will be Gaγ0γ0: Gagg =g2 s 8π2 PQΦ fa ,(A.22) Gaγ0γ0=e02 8π2 PQΦ fah2NCD2 ψi.(A.23) With gsthe strong coupling constant, e0the dark coupling constant, fathe axion decay constant, PQ the dark charge, and NC= 3 the color factor. The cross section can be approximated as: σv ∼1 s|M|2∼1 s|Gaggq21 q2Gaγ0γ0q2|2,(A.24) where q, the four-momentum of the exchanged particle is q∼2s, the energy in the center of mass reference frame, and thus: σv ∼4G2 aggG2 aγ0γ0s . (A.25) Then, going back to eq. (A.21): γ[nγ0] = T 64π4Z∞ 0 4G2 aggG2 aγ0γ0ss3/2K1√s Tds , (A.26) changing √s T=x, we have: γ[nγ0] = T 16π4Z∞ 0 G2 aggG2 aγ0γ0x5T5K1(x) 2xT dx= =T8 8π4G2 aggG2 aγ0γ0Z∞ 0 x6K1(x) dx=48 π4G2 aggG2 aγ0γ0T8.(A.27) Summarizing we can now enter in eq. (A.18) all the three terms eq. (A.12), eq. (A.20) and eq. (A.27), finding: Yγ0(T= 0) = −Z0 TRH γ[nγ0] SHT dT=ZTRH 0 48 π4G2 aggG2 aγ0γ0T8 2π2 45 gT3π √90g1/2T21 MP l T dT=1080√10 π7g3/2G2 aggG2 aγ0γ0T3 RHMP l . (A.28) We can turn this yield into a density parameter, as Ωi=ρi ρc, where ρc= 3H2M2 Pl is the critical density, and the density can be written as ρi=M V=mn =msY . Thus: Ωγ0h2=s0mγ0Yγ0(T= 0) ρc h2 ,(A.29) where the subscript 0means values at the present day: s0= 2889.2cm−3,ρc= 1.05368×10−5h2 GeV cm−3. We can expand this equation to have all terms, including the Yield from eq. (A.28), and inside it the couplings eqs. (A.22) and (A.23): Appendices 51 Ωγ0h2=s0 ρc h2 mγ0 1080√10 π7g3/2 g2 s 8π2 PQΦ fa!2 e02 8π2 PQΦ fah2NCD2 ψi!2 T3 RHMP l .(A.30) Fixing some parameters: the density parameter, so Dark Photons account for all Dark Matter, i.e. Ωγ0h2= 0.12;gbecause the degrees of freedom in the very Early Universe (before decoupling of the first species) were close to 100; and e0and Dψto typical values for this kind of models; we can get to Figure 2.3, relating the mass of the axion to that of the Dark Photon. Up to here, this was the work done in [2]. Appendices 52 B|Minimum mass for Freeze-In The only calculation between Figure 2.3 and Figure 2.4 is the lower bound on the mass of the Dark Photon. It comes from imposing that the reaction rate rγ0≡γ[nγ0] neq γ0 is smaller than the rate of expansion (Hubble rate H) at the reheating temperature, so we are in the Freeze-In regime, instead of Freeze-Out in which particles interact until the expansion rate is higher than the reaction rate and they cannot keep interacting. Let’s track this condition down. From Boltzmann Statistics we know that the number density at equilibrium is: neq γ0=3ζ(3) π2T3,(B.1) where ζis the Riemann zeta function, in particular ζ(3) = 1.202... is the Apéry’s constant. Imposing the reaction rate (remembering eq. (A.27)) smaller than the Hubble rate eq. (A.20) at the reheating temperature, we get: rγ0≡γ[nγ0] neq γ0 < H T=TRH −−−−−→ 48 3π2ζ(3)G2 aggG2 aγ0γ0T5 RH <π 3√10g1/2T2 RH MPl .(B.2) It can be seen how this constraints the reheating temperature to a cubic exponent with the couplings squared, same as happens in eq. (A.30). In fact, mixing both equations, it can be found how: 48 3π2ζ(3)Ωγ0h2 ρc h2 s0 π7g3/2 1080√10 1 mγ0MPl <π 3√10g1/21 MPl ,(B.3) so except for the density parameter, the degrees of freedom and the mass of the Dark Photon, the rest are constants. The degrees of freedom for T=TRH are known and constant: g= 100, so: mγ0>48π4ρc h2g 1080ζ(3)s0 Ωγ0h2→mγ0>1313.49 ·Ωγ0h2eV.(B.4) If Dark Photons were the one and only source of Dark Matter in the Universe, it would mean Ωγ0h2= 0.12 and thus mγ0>157.62 eV. This is an important constraint for Freeze-In model, not remarked by [2], that sets very valuable experimental boundaries. Appendices 53 C|Freeze-Out calculations Now let’s go for the calculations needed for figure 2.5 . The first change is the inclusion of the axion relic density, explained in the main text. The other change with respect to Figure 2.4 is the inclusion of the Freeze-Out model. Freeze-Out model works opposite to the Freeze-In model: In Freeze-Out we start with a large amount of Dark Photons, that are annihilating and turning into gluons until they reach thermal equilibrium. From that point on, the density of Dark Photons becomes fixed until at some point, the interaction is so feeble that it stops (becomes smaller than the expansion rate H), leaving the equilibrium value as the relic density for Dark Matter. So at some temperature TF O the Yield is that of equilibrium, and remains the same up to today: Yγ0(TFO) = Yeq γ0(TFO) = neq s=Y0 γ0.(C.1) Remembering eq. (B.1) (valid for relativistic particles) and eq. (A.12): Y0 γ0=neq s= 3ζ(3) π2T3 FO 2π2 45 gT3 FO =135ζ(3) 2π4g.(C.2) It can be seen how in the relativistic Freeze-Out model the Yield is independent on the axion properties, giving it a constant value. It is also independent on the Freeze-Out temperature, so it could happen at any time, but the Freeze-Out temperature is related to faso it is restricted to high values. Using eq. (A.29), we find: mγ0=Ωγ0h2ρc h2 s0Y0 γ0 = 437.83 ·Ωγ0h2eV.(C.3) Let’s call it a magical coincidence, that this value is exactly 1 3of the minimum value for the mass of the Dark Photon in the Freeze-In model. In the case Dark Photons are not relativistic when they Freeze-Out (necessary for Cold Dark Matter), the number density at equilibrium is: neq =gmγ0TFO 2π3/2 e−mγ0 TF O .(C.4) Therefore, with a similar approach to what was done with Freeze-In eq. (B.2), but keeping the reaction rate equal to Hat Freeze-Out temperature: rγ0≡γ[nγ0] neq γ0 =HT=TF O −−−−−→ 48 gπ4G2 aggG2 aγ0γ0T8 FO mγ0TF O 2π−3/2 e mγ0 TF O =π 3√10g1/2T2 FO MPl .(C.5) Appendices 54 Clearing TFO, we find: TFO =−2mγ0 9W−2 9m2/3 γ0k2/9,(C.6) where W() is the Lambert W function and ktakes all the constant values with it (including in this case fa): k=81√20 2π19/2g−3/2α2(e0Dψ)4f−4 aMPl .(C.7) We can also set another value for TF O with the new Yield: Y0 γ0=neq s=45m1/2 γ0e−mγ0 TF O 8π51/2T3/2 FO ,(C.8) by imposing the density parameter equation eq. (A.29), obtaining: TFO =−2mγ0 3W −2 3 Ωγ0h2ρc8π51/2 45s0!2/3  .(C.9) It can be clearly seen the correlation between both equations for TF O, and it’s worth pointing out how in the last case, there is no relation between TF O and fa, but there is with the density parameter Ωγ0h2, allowing us to plot this relation, encountering a single slope of permitted values. All in all, if we have a Freeze-In mechanism we have an open world of opportunities in the parameter space with the axions and Dark Photons together. However, for the relativistic Freeze-Out mechanism, we just have a single value for the mass of the Dark Photons, that is only dependent on its density parameter, almost leaving out the axion properties (there is still a connection between the DP density parameter and the axion density parameter as shown before). In the case of the nonrelativistic Freeze-Out, it only leaves open a tiny region with huge masses for both the axions (close to the upper limit in the axion mass) and the Dark Photons (close to the GeV). All this allows the creation of Figure 2.5.