Earth-stopping of millicharged dark matter

Facultad de Ciencias EARTH-STOPPING OF MILLICHARGED DARK MATTER (Efecto de frenado terrestre sobre Materia Oscura ‘Milicargada’) Trabajo de Fin de Grado para acceder al GRADO EN FÍSICA Autor: Joaquín Merino Areitio Director: Bradley James Kavanagh Septiembre - 2021 Agradecimientos Aqu´ı se acaban 4 a˜nos (+ un verano extra) de carrera. Es muy clich´e lo que voy a decir pero s´ı, se han pasado volando. Y sin embargo, en esta etapa tan aparentemente fugaz he conocido a grandes personas, a las que me gustar´ıa incluir en estos agradecimientos. Antes que nada, por supuesto, agradec´erselo a mis padres. Desde que entr´e a la carrera, dos personas que han sido siempre los m´aximos exponentes de las letras en casa, han comenzado a interesarse por la actualidad en f´ısica o incluso se han enganchado a ver v´ıdeos divulgativos. Y las conversaciones que surgen de ah´ı, y en general de cualquier lado, son siempre interesantes, y dejan un buen sabor de boca. Gracias por apoyarme todos estos a˜nos desde la distancia, aunque yo muchas veces no me haya dejado. Gracias tambi´en a Luc´ıa, que aunque tambi´en lejos, s´e que siempre estar´a ah´ı, tanto para hablar de banalidades como tambi´en de cosas m´as serias. En segundo lugar, gracias a los de la Meseta. Vine a Santander con miedo de no encontrar a un grupo de gente con la que llevarme bien y, en cambio, encontr´e a personas que, pese a ser bien distintas entre s´ı, siempre consiguieron que encontrara un hueco en el que estar a gusto. Gracias en especial a la ISS: Gracias a Senior y a sus inolvidables duchas por la ma˜nana con la m´usica a tope. Gracias a Unai por saber un poco de todo, pero de todo! Y, por supuesto, a Pablo, por las tan necesarias dosis de humor irreverente y... ejem. Quer´ıa agradec´ertelo en especial a t´ı, Elena. Si hay un motivo por el cual yo he podido seguir adelante todos estos a˜nos sin tirarme demasiado de los pelos, ha sido por que has estado siempre a mi lado, apoy´andome hasta en el momento m´as duro. Muchas gracias. And last but not least, I would like to thank Bradley for his tutoring. He has been very supportive from the beginning of the project, and has always helped me whenever I needed to with any question I had, while at the same giving me the freedom required to produce this kind of projects. Thanks a lot for your patience! i ii Abstract. One of the most prominent open problems in modern cosmology is the identification of the nature of Dark Matter (DM), a kind of matter which is apparently invisible and only manifests itself strongly through gravitational effects which have had the scientific community intrigued throughout the past few decades. Up to now, no Standard Model particle has been able to fully describe the properties of DM so, at the moment, there are plenty of beyond Standard Model particles proposed as candidates to be constituents of DM. In particular this project is focused on one of them: millicharge DM particles, which have a very small electric charge, much smaller than that of an electron. The existence of these particles within the DM Halo surrounding our Galaxy would mean that there exists a certain flux of millicharged DM (mcDM) which is going through the Earth, and right into our detectors. In its path, the DM particles may interact with the different nuclei species on Earth, which makes them slow down, or even stop. This ’Earth-stopping’ effect may give rise to an interesting new time-varying signals once the particles reach the detector so, in this project, the ’Earth-stopping’ effect is thoroughly studied in the framework of direct detection, by initially analyzing the situation for the conventional, electrically neutral DM, and then analyzing it for the millicharge case. Then, the time varying signals corresponding to both DM scenarios are calculated for detectors located at different places on Earth, and buried at different depths. The signals are compared, and it is discussed whether this is a valid comparison method. Key words: dark matter, direct detection, millicharged dark matter, earth-stopping effect, annual modulation. Resumen. Una de las cuestiones abiertas m´as relevantes en la cosmolog´ıa moderna es la identificaci´on de la naturaleza de la Materia Oscura, un tipo de materia que es aparentemente invisible y que tan solo se manifiesta fuertemente mediante effectos gravitatorios, que han tenido a la comunidad cient´ıfica intrigada durante d´ecadas. Hasta ahora, no ha habido nininguna propuesta v´alida de part´ıcula desde el Modelo Est´andar, con lo que existen numerosas propuestas fuera de este modelo y, particulamente, este proyecto se centra en una de dichas propuestas: la materia oscura milicargada, una part´ıcula con una carga el´ectrica mucho m´as peque˜na que la de un electr´on. La existencia de estas part´ıculas en el Halo de materia oscura que nos rodea significar´ıa que existir´ıa un cierto flujo de estas part´ıculas atravesando la tierra, y llegando a nuestros detectores. En su camino, la part´ıcula de materia oscura podr´ıa interactuar con los n´ucleos terrestres, de tal manera que pudiera ser frenada o incluso parada totalmente. Este ’efecto de frenado terrestre’ podr´ıa dar lugar a nuevas se˜nales dependientes del tiempo una vez que las part´ıculas frenadas llegaran a los detectores. En este proyecto este efecto de frenado se estudia concienzudamente en el marco de la detecci´on directa de materia oscura, analizando primero la situaci´on para materia oscura convencional, para luego pasar a la materia oscura milicargada. Luego, las se˜nales temporales asociadas a cada uno de los dos escenarios se calculan para detectores en distintas localizaciones y profundidades. Las se˜nales correspondientes se comparan, y se discute si esta manera es una forma v´alida de compararlas. Palabras clave: materia oscura, detecci´on directa, materia oscura milicargada, efecto de frenado terrestre, modulaci´on anual. Contents 1 Introduction 1 1.1 The dark matter hypothesis: First evidences .................... 2 1.2 Current DM status .................................. 4 1.2.1 Detection of DM ............................... 5 1.2.2 Alternatives to DM: Modified gravity .................... 6 1.3 Dark matter candidates ............................... 7 1.3.1 General properties .............................. 7 1.3.2 Proposed particles .............................. 8 1.4 Objectives and structure of this project ....................... 9 2 Direct detection of Dark matter 11 2.1 Direct detection formalism .............................. 11 2.1.1 Derivation of the differential recoil rate, dR dER................ 12 2.1.2 Derivation of the recoil energy ........................ 13 2.1.3 Spin-independent DM-nucleus differential cross section .......... 14 2.1.4 Form factor correction ............................ 15 2.2 Understanding the DM Halo ............................. 17 2.2.1 Local DM density, ρ0............................. 17 2.2.2 Velocity distribution: Standard Halo Model (SHM) ............ 17 2.2.3 SI recoil spectra for the Maxwellian distribution .............. 18 2.2.4 Earth’s motion and time dependence of the signal ............. 20 2.3 Nuclear stopping of DM on Earth .......................... 22 2.3.1 Coordinate system .............................. 25 2.3.2 Velocity transfer ............................... 25 2.3.3 Speed distribution at the detector ...................... 27 2.4 Standard DM velocity distributions ......................... 28 3 Direct detection of Millicharged Dark Matter 30 3.1 An overview on the candidate ............................ 30 3.1.1 Millicharged DM-nucleus differential cross section ............. 30 3.2 Millicharged recoil spectra for the Maxwellian distribution ............ 32 3.3 Nuclear stopping of millicharged DM on Earth ................... 33 3.3.1 Velocity transfer ............................... 34 3.4 Millicharged DM velocity distributions ....................... 35 4 Signal analysis. Constraints and proposed detectors 37 4.1 Time-dependence of the differential recoil rate ................... 37 4.2 Time-varying SI and millicharge signals ....................... 38 4.2.1 Signals for various detectors and analysis .................. 39 4.3 Constraints ...................................... 41 iii CONTENTS iv 5 Discussion and conclusions 43 Bibliography 44 Chapter 1 Introduction Understanding the Universe and the way in which it evolves is definitively one of the major open problems in modern physics. Within this gigantic problem it is of course particularly interesting to study the constituents that build up the Universe that we live in. Currently we can only identify approximately the 20% of its total matter content, with the remaining 80% [1] being an unknown substance whose origin is uncertain, which makes its existence apparent only via gravitational effects, but that apparently does not interact with conventional matter via the electromagnetic force, hence being invisible at any wavelength. To investigate the nature of this so-called Dark Matter (DM) has thus become one of the main priorities in the quest of characterizing the Universe, and it comes as no surprise that, given that no Standard Model (SM) particle could be found to describe a substance such as DM, many beyond SM particles have been proposed as DM candidates that would correlate with this rather strange, but yet abundant, matter. This work revolves around one particular proposal: Millicharged Dark Matter (mcDM) - particles of DM which have a very small electric charge, much smaller than the charge of an electron. Current constraints indicate that, if this proposal is in fact a reasonable candidate, it would only contribute a small fraction to the DM population (what is usually referred to as a sub-dominant contribution), though it is not yet discarded. It is thus necessary for new experiments to try and detect this kind of particles or at least probe in some way the currently available parameter space in which mcDM is still a valid candidate. It has to be remarked that no experiment today has been able to directly or indirectly detect a certain DM candidate; rather, they put limits in the possible values of different properties of the different candidates, such as their mass. As is to be expected, the list of proposed ways and ongoing experiments that try to detect DM candidates is not exactly short. In this project, direct detection techniques are explored to try and find a differentiating signature that would indicate towards the detection of one of these millicharged particles over any other type of DM candidate, in view of proposing new detectors that may exploit this signature. Particularly, one would expect these particles to be populating the galactic DM halo and - given the motion of the Earth relative to the galactic rest frame - to find them travelling through our home planet, leaving a possible signal at a sufficiently sensitive detector. However, we must take into account that these DM particles may interact with electrons and nuclei in the Earth in their path to the detector. These interactions can slow and even stop the incoming DM particles, modifying the signal in the process. This effect, which we will refer to as Earthstopping is sensible to differentiate a mcDM signal from an ’ordinary DM’ signal, due to the fact that the interaction cross section of each of the two types of DM has a different dependence 1 CHAPTER 1. INTRODUCTION 2 on the recoil energy imparted to the stopping nuclei. Naturally, this effect is time dependent, due to the aforementioned relative motion of the Earth, and this fact may open a window to explore differentiating properties of the DM signal left by a millicharged particle, as opposed to a conventional, electrically neutral, DM particle. In this context, the project explores the Earth-stopping effect for both ordinary and millicharged DM, in order to extract the form of the corresponding time-varying signals, and then compare them, in the hope of finding this characteristic signature that would differentiate them both. This study is going to be made in the low DM mass regime (sub-GeV to a few GeV range). The reasoning behind that decision is that, usually, the limits on the parameter space of heavy Dark Matter are already strong (their possible properties are already quite constrained), so there is not really much available interesting space for us to explore the Earth-scattering effect in. Once the signal analysis has been carried out, the goal would be to explore the existing literature to determine which constraints exist for mcDM and whether there exist regions of the currently available parameter space which can in fact be probed in future proposed detectors. The remaining pages of the introduction aim to contextualize Dark Matter historically and observationally, giving some early and also some more up-to-date evidences of its existence, as well as to talk about methods of detection, and about particles and other objects which have been presented as candidates to be constituents of Dark Matter. 1.1 The dark matter hypothesis: First evidences In 1933, Swiss astronomer Fritz Zwicky published a paper on the redshift of extragalactic nebulae [2] in which he particularly commented on the Coma cluster’s velocity dispersion. By making the assumption that the nebulae cluster had already reached a mechanically stationary state (following a process commonly referred to as virialization) he came to the conclusion that, for Coma to be stable, its average density would have to be ∼400 times greater than that derived from the observations based on luminous matter. The author then attributed this issue to the presence of a yet unseen, dark, matter (’dunkle Materie’), which greatly exceeded the luminous matter content of the galaxy cluster. He then proceeded to examine three other possibilities: 1. The cluster may have not been in a virialized state yet. This scenario still needed a great deal of dark matter to be consistent. 2. Analyze the high velocity dispersion measured for the cluster by assuming that its average density was only due to the observed luminous matter (and not due to a dark matter component). If that were the case, and the speeds were in fact real, the nebulae would be flying apart from the cluster at speeds on the order of O(103)km/s. One would thus expect that, at the evolutionary state of the Universe at that time, more individual nebulae would have been observed with such large speeds, but this was not at all the case, the typical speeds of these objects having been measured to be somewhere around 200 km/s. 3. Analyze the high velocity dispersion by assuming that measured speeds were not real, but rather a consequence of Einstein’s redshift. However, this required even more DM presence to be consistent. CHAPTER 1. INTRODUCTION 3 It was clear that the inconsistency found by Zwicky posed a staggering problem in physics: What was this dark matter? Why was it so apparently abundant, and yet no other signal or observation had led to a similar hypothesis? Despite the interesting nature of Zwicky’s proposal, serious scientific debate regarding the discrepancies found in an increasing number of galaxy clusters did not occur until the late 1950s [3]. These debates extended up until the early 1970s, and the possible explanations for the aforementioned discrepancies were numerous and varied in character; from a large presence of gravitational radiation to a need to change the law of gravity, never discarding the possibility of observational errors. Figure 1.1. Optically studied rotation curve of ionized hydrogen in the Andromeda galaxy (M31), Rubin and Ford [4]. In 1970, Vera Rubin famously published, together with Kent Ford, an study on the rotation curves that could explain the measured rotational velocities of the spiral galaxy M31 [4]. Spiral galaxies are basically rotating flat disks of matter. If this disk has a uniform distribution of matter, and we were to measure the velocity from the centre of the disk as a function of the radial distance, we would find that the velocity increases with the radius: the points furthest away from the centre and the ones next to it do a full rotation in the same time, so the former set of points must travel at higher orbital speeds. This can be understood in terms of the mass enclosed inside the radius of each orbit: more mass inside the orbit means more orbital speed. However, if we take a look at a spiral galaxy, we would see that the centre of the astronomical object is way more luminous than its exterior regions, so we would assume that most of the mass of the galaxy is concentrated at its centre region. If that is true, then there will be a radial distance beyond which, the mass enclosed by increasing orbital radius would remain almost constant, which means that the orbital velocity of external material necessarily decreases with distance. So, based on the distribution of luminous matter, a declining behaviour of the rotational velocities was expected for sufficiently large radial distances. However, they found that, in fact, the curve flattened as the radius increased (see figure 1.1), which meant that more gravity was present in the galaxy than that corresponding to the observed luminous matter. From that point on, other studies [3] which acknowledged this flattening issue appeared, and it was beyond doubt that it posed an open problem in the field. CHAPTER 1. INTRODUCTION 4 A change in the general research interests of astronomers and physicists directed the focus of the scientific community to extragalactic phenomena, Einstein’s general relativity and cosmology, which in turn made the correct determination of the mass of galaxies and galaxy clusters a very important issue [3]. This ultimately led to both of the aforementioned observations - anomalously high velocity dispersions and flat rotation curves at high radial distances in spiral galaxies - becoming the first independent evidences for the existance of DM. 1.2 Current DM status The rather slippery properties of DM have made it quite difficult to characterize. Numerous probes and constraints, which are later to be mentioned, have built up the general consensus that DM must be a non-baryonic kind of matter, with nearly neutral electromagnetical behaviour (otherwise, it would be able to somehow interact with light and this, in principle, should have been already detected) and which has negligible velocities (cold dark matter or CDM). Concerning the latter, it has long been excluded for all DM to have a large velocity dispersion [5], as this would entirely contradict the knowledge that we have on structure formation; objects such as the galaxies we observe, and live in, would have not been able to gravitationally collapse, and, thus, form, had the DM velocity dispersion been large enough. It is to be acknowledged that some degree of freedom is left to accept astrophysical / cosmological models which include modest velocity dispersions for DM but here we will focus on the benchmark assumption that the DM is, indeed, cold. The previous assumptions are encompassed within the ΛCDM cosmological model, which describes the universe as a perturbed Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) flat spacetime with dynamics satisfying Einstein’s equations [5]. It is the most accepted cosmological model given the goodness of its fit to the measured data. This kind of models present a certain set of ’density’ parameters, with each representing a particular property of the universe, such as: •Its matter content: baryonic matter (Ωb) and Dark Matter Ωc. •Its radiation density (Ωγ). •Its dark energy density (ΩΛ) (responsible for the accelerated expansion). ΛCDM is just the cosmological model which uses the least number of cosmological parameters, without compromising the goodness of its description regarding the observational data. According to the high precision measurements of the cosmic microwave background (CMB) anisotropies (see figure 1.2) made by the Planck collaboration [1], DM accounts for approximately the 25% of the critical density of the universe, which in turn means that it accounts for more than 80% of the total matter density. This large component of the total matter density of the universe is observed to be present in gravitationally collapsed structures, ranging in size from small galaxies to galaxies as big as the Milky Way, and also on bigger structures such as clusters. Backing up the previously mentioned early evidences for the existance of DM, a great deal of modern independent observations have made its presence in these collapsed structures even more apparent. Chapter 2 Direct detection of Dark matter Contents 2.1 Direct detection formalism ........................ 11 2.1.1 Derivation of the differential recoil rate, dR dER.............. 12 2.1.2 Derivation of the recoil energy ...................... 13 2.1.3 Spin-independent DM-nucleus differential cross section ......... 14 2.1.4 Form factor correction ........................... 15 2.2 Understanding the DM Halo ...................... 17 2.2.1 Local DM density, ρ0........................... 17 2.2.2 Velocity distribution: Standard Halo Model (SHM) ........... 17 2.2.3 SI recoil spectra for the Maxwellian distribution ............ 18 2.2.4 Earth’s motion and time dependence of the signal ........... 20 2.3 Nuclear stopping of DM on Earth ................... 22 2.3.1 Coordinate system ............................. 25 2.3.2 Velocity transfer .............................. 25 2.3.3 Speed distribution at the detector .................... 27 2.4 Standard DM velocity distributions .................. 28 In this chapter, a review on the basics of direct detection formalism is made, together with considerations regarding the speed distribution of DM in the galactic halo, the nuclear stopping of the DM particles in the Earth’s nuclei, which attenuates the detected signal, and the time dependence of this signal. The examples that are going to be shown and discussed in this chapter are for standard SI interacting DM, in order to review the basics of the formalism to be introduced, and for the sake of later comparison with the millicharged scenario. 2.1 Direct detection formalism Direct detection experiments aim to detect DM candidates by studying the recoil of the detector’s nuclei when interacting with the flux of DM particles coming from the galactic halo, which the Earth is traversing - this can be seen as a static DM halo with the Earth moving through it with a certain speed, ve. Here we will derive the expressions which will be relevant in the subsequent analysis. 11 CHAPTER 2. DIRECT DETECTION OF DARK MATTER 12 2.1.1 Derivation of the differential recoil rate, dR dER The ideas guiding the following calculation have been extracted from Mark Thompson’s Modern Particle Physics [21]. The calculation of the rate of interaction between particles starts by considering the incoming particle flux, defined as the number of particles crossing a unit area per unit time. Let φabe the incoming particle flux, traversing a region of space with nbtarget particles per unit volume. The interaction rate per target particle, rb, would be proportional to the incoming flux, φa, such that rb=σφa,(2.1) where σis the proportionality constant, with units of area, which contains the physical information of the interaction, and represents the underlying quantum mechanical probability that a certain interaction will occur. It is common to establish an analogy between this quantum mechanical parameter, σ, to the effective cross sectional area corresponding to each target particle. If the incoming particle crosses this tiny area around the target particle, they will interact. It has to be remembered that this is only an illustrative way of thinking about the problem, and should not be taken literally. With this picture in mind, the probability of interaction between incoming and target particles can be expressed as the ratio of the sum of all the effective cross sectional areas (Number of bparticles in the region ×σ) to the total area, A, parametrizing the region of interest. Figure 2.1. Illustration of a single particle of type a(velocity va) crossing a region of space with b particles (velocity vb). A is the area parametrizing the region while σis the effective cross-sectional area. Let us fix our attention into an aparticle, with velocity va, traversing the region characterized by the area A, which contains the bparticles, moving in the opposite direction as the former, with velocity vb(see fig. 2.1). In a differential time interval, δt, the aparticle would cross a region containing δN =nbvAδt, (2.2) particles of type b, where v=va+vb. Given the previous explanation, it is straightforward to see that the probability of interaction in that differential region of space is δP =σδN A,(2.3) This can be further developed using Eq. (2.2) into δP =σnbvAδt A=σnbvδt. (2.4) Thus, we have δP δt =σnbv=ra,(2.5) CHAPTER 2. DIRECT DETECTION OF DARK MATTER 13 which, as shown in the right hand side of the equation, is precisely the rate of interaction per incoming aparticle, ra. Now, for a beam of atype particles with number density naconfined within a volume V, the total rate of interaction, R, can be expressed as R=ranaV= (σnbv)naV=σ(nav)(nbV) = σφNb.(2.6) That is, R=cross section ×flux ×total number of target particles.(2.7) Taking this as the starting point, one can readily find the following to be true: dR dER =φdσ dER Nb.(2.8) If one wants to take into account the speed distribution that the incoming particles had in the first place, the following ’weighted sum’ shall be performed dR dER =Zv>vmin f(v)φdσ dER Nbdv3,(2.9) where the speed distribution would act as the weigh corresponding to each value of the velocity. The bounds of the integral are given by the minimum velocity that the incoming DM particle has to have in order to produce a recoil energy of value ER. In later sections we will learn that there is also an upper bound to this integral, given by the maximum velocity that DM particles can have without escaping the gravitational influence of our galaxy. Remembering that φ=nχv=ρχ mχv(now, the particles are the incoming DM ones), and evaluating in 2.9, we have: dR dER =ρχ mχZv>vmin vf(v)dσ dER Nbdv3.(2.10) In order for the previous expression to be more general, one can express it in terms of number of interactions per unit detector mass, mN·Nb, instead of in terms of total number of interactions: dR dER =ρχ mχmNZv>vmin vf(v)dσ dER dv3,(2.11) As can be deduced from Eq. (2.11), analysis carried out with this equation requires some previous assumptions regarding both parameters which are intrinsic to the candidate particles themselves, such as their mass and cross section of interaction with the nucleons; but also cosmological and dynamical parameters, such as their local density, ρ0, the galactic escape velocity or their speed distribution. Each assumption will be further developed and properly explained when examples are to be given. 2.1.2 Derivation of the recoil energy It is now convenient to derive the expression of the recoil energy, ER, so that, amongst other quantities, we can work out the lower limit of the velocity integral in equation Eq. (2.11). Consider a collision between a DM particle, of mass mχand velocity vχand a target nucleus of mass mNat rest in the laboratory frame. After the collision, the DM particle scatters elastically from the target nucleus with an angle θ∗in the center of mass frame (see figure 2.2). CHAPTER 2. DIRECT DETECTION OF DARK MATTER 14 Figure 2.2. Two-body center of mass collision of an incoming DM particle with a target nucleus. The center of mass velocity is given by vCM =mχ mχ+mN vχ=µχN mN vχ,(2.12) where we have defined µχN =mχmN mχ+mN .(2.13) After the collision, the velocity of the target nucleus (which was initially going at a velocity of vCM in the CM frame) resolves into two components, one parallel to the line of movement before the collision, and one perpendicular to it. Thus, and returning to the lab frame, we have (vx Nlab =vCM cos(θ∗)−vCM , vy Nlab =vCM sin(θ∗).(2.14) On the other hand, the kinetic energy transferred to the target nucleus in the lab frame is ER=1 2mN((vx Nlab)2+ (vy Nlab)2) =1 2mNv2 CM ((cos(θ∗)−1)2+sin2(θ∗)) =mNv2 CM (1 −cos(θ∗)). (2.15) Now, by evaluating Eq. (2.12) in ER, we get: ER=µχN2v2 χ(1 −cos(θ∗)) mN (2.16) Solving for vχin equation Eq. (2.16), it is clear that the minimum velocity that is able to produce a fixed recoil energy, ER, corresponds to a head-on collision, when the scattering angle in the CM frame, θ∗, is equal to π. That is: vmin =sERmN 2µ2(2.17) We have thus determined the lower bound of the velocity integral in Eq. (2.11). 2.1.3 Spin-independent DM-nucleus differential cross section For the problem that we are tackling, spin independent (SI) scattering formalism is the appropriate framework to work on. To go deep into this formalism is beyond our scope, as it would require for us to calculate the interaction strength between the DM particles and the internal components of the nucleus in terms of the effective Lagrangian describing the interaction. CHAPTER 2. DIRECT DETECTION OF DARK MATTER 15 We can however safely make use of the following general expression for the spin-independent differential cross section [22,23], dσ dER =mNσSI p 2µ2 χpv2A2F2(ER),(2.18) parametrised by σSI p, the DM-proton cross section at zero momentum transfer. The A2factor (atomic mass of the target nucleus squared) responds to the SI interaction behaving coherently across the entire nucleus (reasonable for the range of momentum exchange of interest, not large enough to probe the inner structure of the nucleus) plus the assumption that the coupling of DM to both protons and neutrons is equal. The latter can be understood in terms of A scattering amplitudes, one for each scattering centre, all adding in phase [23]. F2(ER) is the nuclear form factor, which takes the finite size of the nucleus into account. This will be further explained in the following sub-section. 2.1.4 Form factor correction As previously mentioned, the form factor is introduced to take into account the finite size of the nucleus. When the momentum transfer is such that the corresponding wavelength is on the order of magnitude, or below, the nuclear radius, the probability of interaction or, better, the interaction cross section, falls with increasing recoil energy (larger momentum transfer equals larger recoil energy). Within the First Born Approximation, the nuclear form factor is the Fourier Transform of a spherically symmetric mass distribution, normalized such that F(0) is equal to 1 [24]: F(q) = Zρmass(r)eiq·rd3r=Z2π 0 dφ Z∞ r rρmass(r)Z+1 −1 eiqrcos(θ)drd(cos(θ)) =4π qZ∞ 0 rsin(qr)ρmass(r)dr. (2.19) A problem arises here: the mass distribution in nuclei is hard to probe. To work around that, it is generally assumed that the distribution of mass in nuclei is approximately the same as that of its charge. However, instead of numerically integrating Eq. (2.19), most works use an already worked-out, analytical, form factor. Here, as in many other works, the Helm form factor [25], derived from combining the density of a uniform sphere with a Gaussian (which allows for the soft edges of the nuclei), will be used: |FSI(q)|2=3j1(qrn) qrn2 e−q2s2,(2.20) where j1(x) = sin(x) x2−cos(x) x(2.21) is the spherical Bessel function. Both the effective nuclear radius rnand the nuclear skin thickness sare fit parameters which are dependent on the target nucleus. In [23] Lewin and Smith performed a two-parameter least-squares fit to the Frickle et. al compilation of muon spectroscopy data, to find that the value for rnwhich best reproduced the Fourier transform of a two-parameter Fermi distribution was r2 n=c2+7 3π2a2−5s2,(2.22) CHAPTER 2. DIRECT DETECTION OF DARK MATTER 16 with c= 1.23A1/3−0.60 fm.(2.23) On their end, aand sare set to a= 0.52 fm, and s= 0.9 fm. The behaviour of the form factor for different target nuclei can be seen in figure 2.3, where the maximum recoil energies that different mass DM particles can produce in the target nucleus are also shown, in order to get a glance at the lowest value that the form factor can take for a given scattering scenario. It can be seen that for light DM the effect of the form factor on the scattering cross section is little to none, but for heavier DM, the effect is more than apparent. This is because the momentum transfer between the heavy nuclei and a light particle is much less energetically efficient than the same process with a DM particle with a mass more similar to the nucleus’. Also, it can be seen that the lighter the nucleus, the more momentum transfer (that is, the smaller the wavelength) is needed to lower substantially the value of F2, as expected. This can be better appreciated in figure 2.4, in which two nuclei with very different atomic masses are compared. Figure 2.3. Helm form factor for Ge (A=73, left) and Xe (A=131, right). Note the change in the limits of the x-axis. Maximum recoil energies for different DM masses are shown as vertical lines. Figure 2.4. Low energy close up of the Helm form factor. Xe (A=131, blue) and Na (A=23, black) CHAPTER 2. DIRECT DETECTION OF DARK MATTER 17 The bumps and dips present in the form factor have to do with the internal structure of the nucleus, and respond to the diffraction pattern arising from the interference of the incoming wave with itself along the scattering event: constructive or destructive interference will happen depending on the ’size’ of the wave or, equivalently, the magnitude of the momentum transfer, with respect to the nucleus it is interacting with. In this kind of procedure is easy to overlook the fact that the resulting analytical form factor is of good use only after the parameters sand rnhave been fitted to a particular scattering data set, by using a very specific nuclear density model (two-parameter Fermi, in this case). In [24], Duda et. al discuss the limits of this kind of methodology, which can lead to substantial errors when high momentum transfers are to be expected. However, according to the referenced study, for the low mass regime in which we are interested in these errors are negligible, so we should be fine in using this model dependent Helm form factor, without worrying of it introducing large uncertainties. Note that in most cases, the form factor will be neglected in the calculations. 2.2 Understanding the DM Halo 2.2.1 Local DM density, ρ0 The DM’s local density parameter is usually calculated by first modelling the mass distribution of the Milky Way, and then adjusting ρ0until finding a range of values which are consistent with current MW’s observational data, such as rotation curve measurements. Numerous works, using various observational data sets and including all kinds of MW models, some motivated by numerical simulations, agree that the local DM density lies in the [0.2−0.4] GeV/cm3 range [26]. In most works, the benchmark value used is just ρ0= 0.3[GeV/cm3] and it is thus convenient to maintain the value, for the sake of cross checking. 2.2.2 Velocity distribution: Standard Halo Model (SHM) At the moment, there are no direct measurements of the velocity distribution of DM in the MW, so it is usually derived from simulations. For the kind of work that we are doing, the standard way to proceed is to assume for the DM halo to be an isotropic and isothermal sphere with a density profile that falls with the square of the distance to its centre, ρ(r)∝r−2, and then solve the collisionless Bolzmann equation, thus finding the distribution f(v). This assumption is the SHM, or standard halo model, and it leads to a Maxwellian velocity distribution. The velocity dispersion, σv, corresponding to this distribution is related to the asymptotic value of the circular speed of objects orbiting the galactic centre, vc(r→ ∞) := v∞ c (at large radii the rotation curve of the SHM’s isothermal sphere flattens, tending to the value v∞ c), such that σv=p3/2v∞ c. Also, it is usually assumed that, at r=R0, with R0being the Solar radius as measured from the Galactic centre, the circular speed vc(r=R0) = vchas already reached its asymptotic value, so that vc=v∞ cand, finally, σv=r3 2vc(2.24) Though not in general, in the standard halo model the most probable speed ¯v0and the circular speed are identical, and can be used interchangeably. In this work, the vcparameter is fixed to be vc= 232km/s, a reasonable value backed by experiments that rely on measurements of the solar velocity with respect to an object at rest with respect to the Galactic centre, or even by CHAPTER 2. DIRECT DETECTION OF DARK MATTER 18 experiments concerning direct measurements of local radial force [5]. Note that the aforementioned velocity distribution, f(v), formally extends up to infinity, when, in reality, DM particles that exceed the Milky Way’s escape velocity, vesc(r) = p2|Φ(r)|, with Φ(r) being the gravitational potential, will stop being gravitationally bound to it and, thus, are not interesting to direct detection experiments. For the subsequent analysis, we will use the value vesc = 544 km/s [27]. This is usually dealt with by truncating the Maxwellian at some escape velocity, vesc(r=R0), such that [28]: ˜ f(v) =    1 Nesc 3 2πσ2 v3/2e−3v2/2σ2 v,|v|< vesc 0,otherwise (2.25) where Nesc = erf(z)−(2z/√π)e−z2(2.26) is a normalization factor which becomes unity for an untruncated maxwellian (vesc → ∞). The parameter z is: z=vesc/¯v0, and ¯v0is the most probable speed. As ad hoc as this approach is, there are relatively recent hydrodynamical simulations of galaxy formation, including baryonic physics (which has a non-negligible impact on galactic DM distribution) which conclude that, at least in the solar neighbourhood, the SHM is a good fit for the actual distribution of galactic DM, and it can be used confidently when analysing direct detection experiments [29]. It has to be noted that the halo has a bulk motion relative to us, so that the distribution that we are interested in is not Eq. (2.25), but rather: f(u) = ˜ f(vlab +u),(2.27) where uis the velocity of the DM halo, and vlab is just the velocity of the observer on Earth with respect to the galactic rest frame. 2.2.3 SI recoil spectra for the Maxwellian distribution Before continuing with the Earth’s motion discussion, and now that we have already introduced most of the concepts required to calculate the desired recoil rate, it may be interesting to take a step back and discuss how does the differential recoil rate behave as a function of the DM particle’s mass and target nuclei mass for the yet unperturbed (as it has not interacted with Earth nuclei for the moment) Maxwellian distribution. This way, we can ease later discussions regarding the behaviour of this quantity in the Earth-scattering scenario. For this preliminary analysis we need some specific information, particularly the DM mass, mxand the corresponding value of the SI DM-nucleon cross section. For this matter, it is important to have in mind which pairs of values are compatible, and have not yet been excluded by other experiments or other studies. CHAPTER 2. DIRECT DETECTION OF DARK MATTER 19 For light DM we shall use the recent study from E. Aprile et. al, [30], in which they probe the parameter space of light dark matter by studying the conventional elastic scattering scenario while adding the the irreducible inelastic processes that also occur along the scattering events, such as the Midgal effect or the Bremsstrahlung effect. In particular, they give updated limits to the DM-nucleus cross section as a function of the mass of the particle, see figure 2.5. Figure 2.5. Limits on the SI DM-nucleon interaction cross-sections at 90%C.L. Signal models from both Migdal and Bremsstrahlung effects in the XENON1T experiment have been used. Adapted from Ref. [30]. From the figure, we can see that for light DM we are safe to use σp≈10−30 for our examples. Now we will proceed on to examine the dependence of the differential recoil rate arising from the Maxwellian distribution of velocities, Eq. (2.25), on the DM mass and on the target nuclei mass. Note again that we are not taking into account the Earth stopping effect and, because of that, we just simply integrate the Maxwellian distribution over all angles (in a spherical fashion), and use it to perform the calculation of Eq. (2.11). (a) Fixed target (Germanium) + variable mass. (b) Fixed mass (0.5GeV) + variable target. Figure 2.6. Example plots for the recoil spectrum of the unperturbed Maxwellian distribution describing the DM halo. A log-log plot has been for the spectra to be seen nicely. In both plots, the maximum recoil energy corresponding to each curve is indicated with the appropriate color. CHAPTER 2. DIRECT DETECTION OF DARK MATTER 20 The calculations of the differential recoil rate for both fixed target / fixed DM mass scenarios are plotted in figure 2.6. The first thing to be noticed in the left panel is that the maximum value of the recoil rate is larger the smaller the DM mass, which has to do with the 1/mχfactor in Eq. (2.11) and the 1/µ2 χp factor in Eq. (2.18). In the right panel, we can see that the maximum recoil rates are reached for the heaviest target nuclei, which has to do with the A2factor in Eq. (2.18). Apart from these rather trivial scaling factors, we also note that the rate at which dR/dERfalls is different for different DM masses (or different target nuclei masses in the right panel). If we neglect the form factor, we can see that this dependence of how quickly does the differential recoil rate falls with mχand mNdoes not come directly from the differential cross section or any pre-factor but it actually comes from the velocity integral in 2.11. Note that the integral is performed for velocities greater than a certain vmin. The thing is that vmin depends on mχand mN(see Eq. 2.17), so that, for example, if we were to fix the target and then choose a light DM particle, the factor multiplying the ERinside the square root becomes large, thus making vmin grow quickly with growing ER. This in turn necessarily means that the integral in 2.11 must decrease, as the integration enclosure is also decreasing - remember that at some point, the velocity distribution is very small, as it tends to 0 with increasing speeds. A similar explanation can be given when considering the variation with µN, but the other way around: the larger the target nucleus, the quicker the decline of the differential recoil rate. Both of these effects can be clearly seen in figure 2.6. We shall expect a similar behaviour later on, when Earth-scattering is introduced - of course, we would expect smaller recoil rates, due to the attenuation caused by the DM-nucleon scattering. 2.2.4 Earth’s motion and time dependence of the signal Due to the motion of the Local Standard of Rest (vLSR), the Sun’s peculiar velocity (v,pec), the motion of the Earth around the Sun (v⊕), and the daily revolution of the Earth (v⊕,rot), a time dependence arises from vlab in equation Eq. (2.27). In section 2.3 we will study how does the interaction of the DM with the Earth’s nuclei modify the velocity distribution along the path which the DM particles traverse in their way to the detector. What is important for now is that the modification to the original velocity distribution depends on the path taken by the DM particles or, in other words, it depends on the position of the detector with respect to the average direction of the incoming DM flux. It is therefore useful to define an angle γ, which is defined as the angle between the direction of the mean DM flux coming from the Halo, and the position vector of the detector on Earth: γ=cos−1(hˆvχi·ˆrdet).(2.28) Here, the mean DM velocity is given by hˆvχi=−vlab(t).(2.29) Note that the time dependence in vlab is transferred on to the angle γ, which will be important later when discussing the annual modulation of the detected DM signal. CHAPTER 2. DIRECT DETECTION OF DARK MATTER 27 2.3.3 Speed distribution at the detector Provided that we know the initial distribution of velocities, f(vi), which in this case we assume it is a Maxwellian as the one presented in Eq. (2.25), it is easy to go from ˜ f(vi) to ¯ f(vf) - the distribution of velocities at the detector, after the particles go through the scattering process - by a simple change of variables: f(vi)d3vi=¯ f(vf)d3vf(2.41) Here we will again rely on the straight-line formalism, such that ˆvi=ˆvf:= ˆv. Thus, if we write Eq. (2.41) in terms of the corresponding unit vectors: f(vi)v2 idvidˆvi=¯ f(vf)v2 fdvfdˆvf ⇒¯ f(vf) = f(vi)vi vf2dvi dvf .(2.42) Now, as we are not interested in directional detection, we must integrate ¯ f(vf) over all incoming angles and, for that, we need to understand how vin Eq. (2.25) depends on the angular coordinates shown in figure 2.9. For that, let vbe the velocity of a DM particle in the galactic rest frame, vχthe velocity of a DM particle in the Earth’s frame and vlab the velocity of the lab (Earth) with respect to the DM halo (galaxy). They are related by: v=vχ−vlab (2.43) Squaring both sides of the equation, we have: |v|2=vχ2+v2 lab −2vχvlabcos(α) (2.44) With αbeing the angle between the velocity of a particular DM particle and the velocity of the Earth. Looking at figure 2.9, we can relate this angle to both θand γ, as well as to φ, the azimuthal angle (not shown on figure 2.9), such that: cos(α) = sin(γ)sin(θ)cos(φ) + cos(γ)cos(θ).(2.45) It is from the previous equation that the velocity dispersion gets the dependence on γand, thus, on time. Now that the form of the velocity dispersion in terms of (γ, θ, φ) has been fully determined, the integral over all angles can be performed. First, we have: ¯ f(vf) = Zv2 f¯ f(vf)d2ˆv =Zf(vi)v2 i dvi dvf d2ˆv (2.46) Note that we have distinguished between f(v), the velocity (v= (vx, vy, vz)) distribution and f(v), the speed (v=|v|) distribution, and we have used the following relation: f(v) = Rv2f(v)d2ˆv. Here, we are treating vias the initial velocity needed to obtain a given final velocity in the detector for a particle which is travelling in a straight line defined by the direction ˆv. Now, changing to the angular integrals, we have: ¯ f(vf, γ(t)) = Zf(vi, γ(t), φ, θ)v2 i dvi dvf dcos(θ)dφ =Z+1 −1Z2π 0 f(vi, γ(t), φ, θ)dφv2 i dvi dvf dcos(θ), (2.47) CHAPTER 2. DIRECT DETECTION OF DARK MATTER 28 where we have used that the only φdependence arises in the f(vi) function, from Eq. (2.45) - not in vior in dvi/dvf, as can be readily seen from figure 2.9. Note that this integral needs to be performed numerically, owing to the integral in Eq. 2.40 (which we need to solve in order to perform the numerical derivative dvi/dvf), which cannot be solved analytically because of the dependence of dv/dD on the DM velocity. 2.4 Standard DM velocity distributions With all the formalism at hand, we can discuss the way in which the Earth stopping process affects the initial DM velocity distribution, as a function of the mean DM flux. Then, we can study the dependence of the recoil spectra on time, which will ultimately give us the corresponding signal. As the signals will be further explained and compared in a separate chapter, we will just focus here in explaining the behaviour of the velocity distributions for a particular scattering scenario. In order to put forward specific examples, we will focus here on the SUF (CDMS-I) experiment. In figure 2.11a, various speed distributions at the SUF detector for given values of the DM mass, the DM-nucleon cross section and the angle γare shown. (a) Final velocity distribution as a function of γ, with evenly spaced γvalues. (b) Final velocity distribution as a function of time, one curve every 2h. Figure 2.11. Final speed distribution at SUF. Final speed distribution of light DM particles at the stanfordd Underground Facility (SUF), after propagating through the atmosphere, Earth and Pb shielding. The dashed curve is the unperturbed Maxwellian speed distribution. The distribution is shown as a function of evenly spaced values of the average DM flux, γ. Velocities below the dashed vertical line are not able to produce a 10keV recoil, which is the threshold corresponding to CDMS-I. In particular, the distribution for 11 different values of γis presented, from 0ºaverage DM particles directly coming from below, larger path to reach the detector - to 180ºaverage DM particles directly coming from overhead, shorter path to reach detector. In the right panel (2.11b) we have calculated the distribution of velocities for a set of γvalues corresponding to 12 time stamps distributed evenly throughout one arbitrary day - remember that we can calculate the angle γfor any time value that we want, through Eq. (2.28). The latter will be the set of curves used to calculate the time dependence of the differential and total recoil rates CHAPTER 2. DIRECT DETECTION OF DARK MATTER 29 later on, in Chapter 4. Note that γdenotes the average direction of the incoming DM, so that even for γ= 0º, there is not a total suppression of the population of particles arriving at the detector, as can be seen from the dark blue curve in 2.11a. Also note, looking at figure 2.11b, that gamma (which grows down-up) does not escalate linearly with time. Instead, it follows the dependence shown on figure 2.12, a calculation which has been performed in the same manner as the ones in 2.7. 223.0 223.2 223.4 223.6 223.8 224.0 Days from 1 January 2021 100 120 140 160 = cos 1( v rdet ) Figure 2.12. Daily variation of γ.The time stamps and corresponding gamma values used to plot figure 2.11b are marked with crosses. We can also see that the peaks of the distributions in 4.1 shift to the right for increasing γ. This is due to the fact that, even though the initial Maxwellian distribution is isotropic in the Galactic rest-frame, it is anisotropic in the Earth’s reference frame. This means that, for us, particles travelling parallel to the mean DM flux are the ones with larger velocities, while the ones travelling anti-parallel to the DM flux are the slower ones. For example, for γ= 0º, particles that travel with large velocities must also cross most of the Earth until reaching the detector, meaning that the large speeds region in the corresponding distribution (dark blue curve in 2.11a) is not very populated, as most of those particles have been attenuated before reaching it. On the other hand, slower particles would come mostly from above, having to cross less portion of the Earth and thus mostly surviving until reaching the detector. The situation is the other way around for γ=π, so that the curves in between just transition smoothly from one scenario to the other - that is, from a lower peak velocity to a higher one - thus the right-hand shift. Now that we are aware of the behaviour of these distributions, the next step would be to compute the recoil rates corresponding to the γcurves of interest, in order to make apparent the time dependence of dR/dER. However, this is going to be performed in Chapter 4, in which the signals of both standard DM and Millicharged DM are going to be compared and analyzed. Chapter 3 Direct detection of Millicharged Dark Matter Contents 3.1 An overview on the candidate ...................... 30 3.1.1 Millicharged DM-nucleus differential cross section ............ 30 3.2 Millicharged recoil spectra for the Maxwellian distribution . . . . 32 3.3 Nuclear stopping of millicharged DM on Earth ........... 33 3.3.1 Velocity transfer .............................. 34 3.4 Millicharged DM velocity distributions ................ 35 In this chapter, the formalism that has been gone through in Chapter 2 is adapted to the millicharge DM particles scenario, so that the velocity distributions at a given detector can be calculated, setting up the foundations for the comparison and analysis which is going to be done on Chapter 4. It has to be noted that the functional forms of the recoil rate and recoil energy extracted in Chapter 2, along with the considerations made for the velocity distribution associated to the DM halo and the motion of the Earth are all valid here. The differentiating quantity is the DM-nucleus interaction cross section, and it is this difference, and the unique effects that come with it, what is going to be discussed in this chapter. 3.1 An overview on the candidate We have to be aware that the current project does not aim to fully develop or study the specific theoretical framework in which millicharged DM is contemplated. We will now mention it briefly, but our focus is essentially the functional form of the differential cross section describing the interaction of our candidate particle with the nuclei on Earth (and atmosphere and detector), such that calculations similar to those performed in Chapter 2 with Spin Independent, electrically neutral, DM particles can also be done here. From the Lagrangian describing the millicharged DM model, a coupling between the DM field and the SM photon arises. As a consequence, the former obtains an electric millicharge,ε[37]. 3.1.1 Millicharged DM-nucleus differential cross section The aforementioned coupling introduces a Coulomb-like interaction between our DM candidate and electrically charged particles within the SM. 30 CHAPTER 3. DIRECT DETECTION OF MILLICHARGED DARK MATTER 31 Now, the DM-nucleus differential cross section, which is the one that’s interesting to us, given the problem at hand, is: dσmcDM N dER =2πZ2 Nε2α2 mNv2E2 R F2(ER),(3.1) where ZNand mNare the charge number and the atomic mass of the nucleus N, respectively, ε is the millicharge (in units of the electron charge) and αis the SM dimensionless fine structure constant (α≈1/137). The form factor appearing in Eq. (3.1) is the same form factor discussed in section 2.1.4. It is worth noting that the continuous and almost straight line trajectories formalism still holds true here, and it is enhanced even, due to the 1/E2 Rdependence of the cross section. If we take a look at the expression for the recoil energy, we note that ERbecomes smaller as the scattering angle decreases; and a smaller recoil energy means a higher cross section or, better, a higher scattering probability. In conclusion, millicharged DM is more likely to scatter in the forward direction, which furthe justifies the ’straight line’ scattering formalism described throughout section 2.4. It is also common to talk about the total cross section, which is obtained from integrating equation Eq. (3.1). For this, an important effect, specific for this kind of interaction, has to be introduced first: the screening effect. If the incoming millicharged DM particles are traveling at sufficiently low speeds, the momentum transfer with the interacting nucleus will not be large enough for the particle to probe the inside of the target, which means that the electron cloud of the nucleus screens its charge, so that the DM particle will see it as a neutral scattering centre, nullifying the electrical interaction and, thus, making the scattering probability drop to zero. Particularly, we can identify this transition with the instant in which the momentum transfer between the DM particle and the nucleus becomes comparable to the typical momentum of the electron, αme. Converting from the momentum to a recoil energy, we have the expression for the screening recoil energy, Escreen R=(αme)2 2mN (3.2) Now, for us to find the expression of the screening velocity, we shall match the screening energy with the maximum possible recoil energy. That way, we obtain the largest value for the velocity for which this screening effect arises, (αme)2 2mN =2µ2 χN v2 screen mN⇒vscreen =s(αme)2mN 4µ2 χN mN =αme 2µχN .(3.3) Here, me≈0.511MeV is the mass of the electron and muχN is the reduced mass of the DM-nucleus system. Below this velocity, the screening effect becomes apparent, and the corresponding DM particles will not be perturbed by Earth-scattering effects. Having this effect in mind, we can integrate Eq. (3.1) to obtain: σmcDM N(v) = ZEmax R Escreen R dσmcDM N dER dER=2πZ2 Nε2α2 mNv2ZEmax R Escreen R dER E2 R =2πZ2 Nε2α2 mNv21 Escreen R−1 Emax R=2πZ2 Nε2α2 mNv2"2mN α2m2 e−mN 2µ2 χN v2# =4πZ2 Nε2α2 v2"1 α2m2 e−1 4µ2 χN v2#, (3.4) CHAPTER 3. DIRECT DETECTION OF MILLICHARGED DARK MATTER 32 which is manifestly velocity dependent. Note that the form factor has been neglected here because in Coulomb-like interactions the momentum transfer is typically rather small [37]. The fact that we are focusing on light DM of course backs this decision. This dependence of the total cross section on the velocity prevents us from defining a total cross section in the same way as we can in the SI case, so we define a ’reference’ cross section, σmcDM ref , as the value of the DM-proton cross section evaluated at an arbitrary velocity, vref = √2meα/2µχN , such that: σmcDM ref =4πε2µ2 χN m4 eα2(3.5) This quantity will serve us later, in Chapter 4 when trying to compare the SI model with the millicharged one. Also, the code verne is prepared in such a way that whenever σSI pwere to be used internally in the SI scenario, the value for σmcDM ref is used instead, if the millicharged interaction is the one being studied. 3.2 Millicharged recoil spectra for the Maxwellian distribution Our goal here is to calculate the recoil spectrum without taking into account the Earthscattering effect for the millicharged interaction, for some particular values of the DM mass, the milli-charge εand the effective cross-section. For a reason that will become clear in the next sub-section, we set mχ= 5GeV. For later comparison with the SI case, it may be interesting to match the value of the SI DM-proton cross section with the millicharged reference cross section defined in Eq. (3.5). For the moment, we could set this value to be σmcDM ref = 1 ·10−30cm2, which converts back to ε= 3.5·10−11 (in units of the electron charge). According to [5], for this DM mass, the maximum milli-charge that one can have is εmax = 3.5·10−7(mχ= 5[GeV ])0.58 = 8.9·10−7, so our value for εseems safe enough. In Chapter 4, we will discuss whether these parameters are reasonable within the current millicharged paradigm. (a) Fixed target (Ge) + variable mass. (b) Fixed mass (5 GeV) + variable target Figure 3.1. Example plot for the millicharged (solid line) recoil spectrum of the unperturbed Maxwellian distribution describing the DM halo. A log-log plot has been used for the spectra to be seen nicely. In both plots, the maximum recoil energy corresponding to each curve is indicated with the appropriate color. The recoil spectra for SI interacting DM particles are also plotted (dashed line), for the same DM mass. CHAPTER 3. DIRECT DETECTION OF MILLICHARGED DARK MATTER 33 We can now plot the recoil spectra for various DM masses with a fixed target, see figure 3.1. Note that each curve will also have a unique εvalue, in order to preserve the value of the reference cross section. We can see that the behaviour is fairly similar to that discussed in section 2.2.3 for the case of SI, at least in what has to do with the maximum values of the recoil rates. However, we can see that the differential recoil rate falls with increasing recoil energy a lot faster for the millicharge interaction, which is ascribable to the dependence of the millicharge differential cross section on 1/E2 R. Of course, in a linear plot, the exponentially decaying behaviour becomes apparent. Also, note that the order of magnitude of the recoil rates is rather similar for both the millicharge and SI interactions, which is mostly due to the fact that the SI DM-proton cross section has been matched with the effective cross section. 3.3 Nuclear stopping of millicharged DM on Earth Here we will go through the same kind of calculations as in section 2.3, but now taking into account the corresponding form of the differential cross section, as well as the aforementioned screening effect. Following the already explained approach (see equations Eq. (2.30) and Eq. (2.31)), we write the average recoil energy transferred from the DM particle to the target nucleus as: σmcDM NhERii=ZEmax R,i Escreen R,i ER2πZ2 iε2α2 miv2E2 R F2(ER)dER =2πZ2 iε2α2 miZEmax R,i Escreen R,i 1 ERv2F2(ER)dER (3.6) Now, changing variables such that: (ER)→(xEmax R), x ∈[0,1] we have σmcDM NhERii=2πZ2 iε2α2 v2miZ1 (αme 2µv )2F2(xEmax R,i ) xdx (3.7) As we know, we are going to be dealing with light DM particles, so it is not that important here to analyze the coherence factor that would arise for this kind of interaction. Thus, we make F2(xEmax R,i ) tend to one and then solve the integral analytically, such that: σmcDM NhERii=4πZ2 iε2α2 v2mi log 2µv αme.(3.8) Now, combining Eq. (3.7) with Eq. (2.30), we have that: dhEχi dt =−X i ni(r)4πZ2 iε2α2 vmi log 2µv αme,(3.9) which, if we change variables as we did in Eq. 2.36, transforms into dv dD =−4πε2α2 v3X i ni(r)Z2 i mi log 2µv αme.(3.10) CHAPTER 3. DIRECT DETECTION OF MILLICHARGED DARK MATTER 34 Now, and in a similar manner as in the SI case, Eq. (3.10) gives us the necessary information to know how the millicharged particles traverse the different media until reaching the detector. The three layers to be considered are, naturally, the same (atmosphere, Earth and detector’s shielding) but, this time, the charge numbers of the different components are needed in order to perform calculations. 3.3.1 Velocity transfer Of course, the coordinate system is the same as the one described in section 2.3.1, and the relationship between the initial velocity at the top of the atmosphere, vi, and the final velocity at the detector, vfis still Eq. (2.40). Therefore, we can calculate the velocity transfer in the scattering process of millicharged particles. Given that, as it has already been stated, in this Coulomb-like interactions the momentum transfer is not large in general, we need to use a sufficiently heavy DM particle in order to clearly see the effect of the attenuation due to the interaction with nuclei on Earth. It has been found that a mass of about mx∼5GeV is able to give us some interesting enough curves. We also need to decide beforehand the value for the effective cross section, which, for the moment, is set to be σmcDM ref = 10−30cm2, and also the specific trajectory of DM particles, which is set to be θ=π/2. What we obtain is the plot shown in figure 3.2. 0 200 400 600 800 vi [km/s] 0 200 400 600 800 vf [km/s] vscreen ( Fe ) vscreen ( O 2) Atmosphere +Earth +Shield m = 5 GeV mcDM ref = 10 30 cm2 MPI (d = 0.3m) Figure 3.2. Example plot of the velocity transfer for millicharged DM particles (ε= 3.5·10−11) coming with an angle θ=π/2and being detected at MPI. Each curve represents the final velocity of the DM particles after going through the corresponding layer. We can readily see a few different effects already. First, we can see that the initial velocities are not at all modified in the first ∼120km/s, which is due to the already explained screening effect: Particles with speeds low enough will ’see’ nuclei as electrically neutral objects, therefore not interacting at all with them, as the interaction with which we are dealing is a coulombic one. The value of the speed at which velocities begin to be modified in the ’Atmosphere’ curve matches the screening velocity corresponding to Oxygen, which makes perfect sense, as this is the heaviest nucleus considered within this layer. In the ’+Earth’ curve, that value matches CHAPTER 3. DIRECT DETECTION OF MILLICHARGED DARK MATTER 35 the screening velocity corresponding to iron, the heaviest nucleus considered within Earth (and thus the lowest vscreen for Earth scattering). From vscreen on, we see that the velocities are in fact modified by the scattering effect. Particularly, the fact that the total DM-nucleus cross section depends on the DM velocity is very apparent. By looking at the functional form of the total cross section (Eq. (3.4)), we can see that, for small enough speeds (but greater than vscreen), the term in brackets enhances σmcDM N due to the −1/v2term inside it. The closer the speed to vscreen(O2), the greater the enhancing effect. However, the 1/v2term in the prefactor quickly takes over the expression for large speeds, suppressing the probability of interaction at high v, thus making vftend to the original velocities (dashed curve). A similar discussion can be done for the two remaining curves, bearing in mind that the subtracting term in Eq. (3.4) has a µ2 χN in the denominator, which means that the ’enhancing’ of the cross section is more persistent, and thus the curves begin to tend to vimuch later in the x-axis, but much more abruptly. Note that the curves for the Earth and shielding are almost identical. This is because the particles with speeds above vscreen(Fe) have already been slowed down to almost the vscreen of the detector’s material (Cu in the case of MPI), making both curves be almost the same. For larger velocities, which are not slowed down to vscreen(Cu), the cross section is already very suppressed, so that the shielding effect on the velocities is also really small. 3.4 Millicharged DM velocity distributions Now, we are finally able to plot the velocity distributions for millicharged particles, as a function of the mean DM flux (and, therefore, time), same as in figures 4.1. The example plots are calculated for the SUF (CDMS-I) experiment here as well. (a) Final velocity distribution of millicharged particles as a function of γ, with evenly spaced γvalues. (b) Final velocity distribution of millicharged particles as a function of time, one curve every 2h. Figure 3.3. Final speed distribution at SUF. Final speed distribution of moderate-mass millicharge DM particles at the Stanford Underground Facility (SUF), after propagating through the atmosphere, Earth and Pb shielding. The dashed curve is the unperturbed Maxwellian speed distribution. The distribution is shown as a function of evenly spaced values of the average DM flux, γ. The basic discussion regarding the behaviour of the velocity distribution has already been made in section 2.4, around figure 4.1, so it isn’t necessary to go through it all over again. CHAPTER 3. DIRECT DETECTION OF MILLICHARGED DARK MATTER 36 However, there are some very visual differences (apart from the different time dependence, which will be studied in the next Chapter) with the plots on figure 4.1, and they have all to do with the screening effect explained in section 3.1.1. First, we can see that velocities below the screening velocity of the largest nucleus on Earth (Iron, Fe) - thus the lowest screening velocity considered - are not at all affected by the Earth-stopping effect. Again, this has to do with our charged DM particles ’seeing’ the nuclei on Earth as neutral objects, with which they do not interact. The high-speed region of the plots seems rather similar to that of the SI distributions, but where we do see a difference is in the region just above vscreen. Particles in that region seem to be piling up, and that is because, at that point, they have been slowed so much, that they are losing energy very slowly, thus facilitating this ’bunching’ effect. The peaks that we see in that region correspond to the screening velocities for the different materials considered. In the plots, only the lowest value of vscreen for both the atmosphere and the Earth layers are shown. Also, there is another even more subtle effect playing out, and that is the fact that, precisely at vscreen, our formalism breaks, as the interaction pprobability quickly drops to zero. To save the normalization of the velocity distributions, we could add by hand a population of particles with the form of a Dirac’s delta distribution at v=vscreen. However, this is quite complicated to perform, and given that the velocities at which this effect happens are low enough, we might not need to worry about it when calculating the recoil spectrum, as will be shown in Chapter 4. Chapter 5 Discussion and conclusions In this project, we have proposed a novel way in which we could distinguish signals coming from two very different DM candidates, with very different differential cross sections. For that, we have learned some scattering theory, which is not given in the Baechelor’s degree, and we have derived relevant equations which have served us to analyze the whole scattering process, remarkably the two equations for dv/dD. Also, we’ve gone through relevant concepts, such as the efficiency of energy transfer between colliding particles or the suppression of the probability of interaction due to the coherence effect, which can be easily translated to other fields of physics. As we have already stated, the last analysis could have been more thorough in order for us to obtain more meaningful information; however, it is clear that with this method the two signals considered can be distinguished, which is already an achievement. Another way in which this kind of experiment could be carried out is by considering a directional detector, thus considering the three dimensional velocity distribution of the incoming DM particles, instead of the one-parameter distribution that we have used. This would most certainly give us much more information than the current study at the cost of becoming a much more advanced work. In general lines, the project has been useful for the student to introduce himself in a new fascinating field of physics, and also for him to learn programming in python, a versatile and directed to science programming language, so the pedagogical value of the project has been met. 43 Bibliography [1] N. Aghanim et al. “Planck 2018 results”. In: Astronomy Astrophysics 641 (Sept. 2020), A6. issn: 1432-0746. doi:10.1051/0004-6361/201833910.url:http://dx.doi.org/10. 1051/0004-6361/201833910. [2] F. Zwicky. “Die Rotverschiebung von extragalaktischen Nebeln”. 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