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CST Simulations of Dark Radio Experiment CST Simulations of Dark Radio Experiment Nathaniel MacFadden∗ Reed College and UC Davis Mentor: Mani Tripathiy and Mentor: Tony Tysonz UC Davis (Dated: October 8, 2018) This paper presents a calculation of the effective in situ antenna factor of an arbitrary antenna using simulation software CST. This work is focused on the `Dark Radio' dark matter search and thus focuses on simulating the feed voltage of an antenna in a Faraday cage excited by dark photons. Electrical and physical modeling of the surroundings and antenna were performed showing agreement to expectations, but more benchmarking should be done. Additionally, an excitation method was developed to emulate dark photon fields. This results in an antenna factor that can be used to determine the coupling, , from feed voltage. I. INTRODUCTION but the details of that discussion are not needed for this work. The nature of dark matter is one of the biggest open problems in physics. At a high level, our theory of grav- itation has been well tested and verified in high acceler- II. THEORY ation regions (e.g., the Earth, our solar system) but ob- servations in low acceleration regions (e.g., galaxies and Light is a traveling excitation of electromagnetic fields galaxy clusters) deviate from predictions. This discrep- whose motion is described by the wave equation: ancy is old, with modern investigation beginning with µ µ Lord Kelvin's lecture in 1904 [11], yet it is still unan- A = −µ0J (1) swered. (see appendix A for variable definitions and conventions). There are three possible resolutions to this discrep- This equation is effectively equivalent to Maxwell's equa- ancy: our observations are faulty, our understanding of tions in their standard differential form: dynamics is flawed, or our observations are correct but ρ there is unobserved mass (`dark matter'). Improved ob- r · E = e Gauss's Law, servations have made the first option unlikely and thus 0 the discrepancy exists and is either in faulty dynamics or r · B = µ0ρm Gauss's Law for magnetism, dark matter. These options are discussed in detail in ap- @B −∇ × E = + µ j Maxwell-Faraday eq., and pendix A 3 but, to summarize, modifications to dynamics @t 0 m have been unable to explain galaxy cluster observations @E and thus dark matter is the most promising avenue. r × B = µ + j Amp`ere'scircuital law, 0 0 @t 0 e While many dark matter theories are losing parameter (2) space with each generation of experiments, this paper focuses on a relatively unexplored dark matter theory: except that it lends itself to more advanced analysis. `dark photons'. See figure 1 for an overview of the avail- An alternative and modern description of light is as able dark matter parameter space and what is constrain- a massless particle traveling at c ≡ 299792458 m/s in ing it. This plot shows a large area of interest for dark a vacuum. We call this particle a `photon' and give it photons (`hidden photino'), a large motivation for their energy proportional to the frequency of traveling wave current search. that it corresponds to. These theories are unified in QFT Technically, dark photons are hypothesized low mass in which this particle is a quantized excitation of the vector bosons that interact with the standard model via electromagnetic fields. kinetic mixing. They are described by new massive vec- In a theoretical leap, new `dark' fields are hypothe- tor fields that can be thought of as a copy of electromag- sized that are analogs of the electromagnetic fields except netism with minor differences to be discussed in section they are massive[14] (mass m) and their interaction with II. This theory is discussed in more detail in appendix B charges is suppressed (coupling ). Then, just as pho- tons are quantized excitations of electromagnetic fields, dark photons are quantized excitations of the dark fields. These modifications are clear in the wave equation de- ∗ [email protected] scribing dark fields[8]: y http://tripathi.physics.ucdavis.edu/ z http://tyson.ucdavis.edu/ 2 0µ µ ( − m )A = −µ0J : (3) 2 FIG. 1. (taken from [10]) Dark matter parameter space of kinetic mixing χ ( in this writeup) to mass mγ0 (m in this writeup). Transparent tan regions indicate areas of interest for various dark matter theories; opaque colored regions exclude dark matter. The three dark photon parameters, then, are the field strength can be calculated: strength E0 (linked to B0), mass m, and coupling . 0 p E ≈ 2ρDM /0 ≈ 3300V/m (5) A. Field Strength using the observed local dark matter density ρDM = 0:3 ± 0:1 GeV cm−3[2] and assuming all dark matter is dark photons. In reality, even if dark photons are the pri- 2 Relativistically, mass and energy are equivalent (E = mary dark matter component, not all of dark matter will 2 2 2 (pc) + (mc ) ) and thus the observational discrepancies be dark photons. The corrections by other dark matter motivating dark matter are energy discrepancies. The components, however, are a finer detail than of current need for dark matter, then, is a need for a bound energy concern. density in galaxies and galaxy clusters that has so far been unseen[15]. To see if dark photons could provide such energy den- B. Mass and Momentum Distribution sity, we determine the energy density of a dark photon field known amplitude. Given a field described by La- Photons and dark photons are discrete excitations of grangian density L, the Hamiltonian H gives the energy the electromagnetic and dark fields respectively. These density[16]. For electromagnetism, for example, the en- excitations act as quantum harmonic oscillators at every ergy density is: point in space and thus, the energy levels for a given frequency are separated by hν for these oscillators, thus 1 E2 2 giving the energy of the particle E = hν. H = 2 + B (4) 2µ0 c On the other hand, Einstein's energy momentum re- lationship gives the energy of a particle in terms of mo- where the Legendre transform to get the Hamiltonian @L mentum and mass: is more complicated than just H = A_ ν − L since @A_ ν the Hamiltonian must be gauge-independent. Performing E = p(pc)2 + (mc2)2: (6) the same Legendre transform for the dark fields results 02 0 in H ≈ 0E =2 since the B component is velocity sup- These two views combine to give the relationship: hν = pressed (B0 / vE0 where v is the speed of dark matter[3]) p(pc)2 + (mc2)2. For photons this simplifies to p = hν=c and v small. since photons are massless. For dark photons, conversely, With the energy density of dark fields, the dark field this relationship is ν ≈ mc2=h since p mc by looking 3 at momentum distributions of dark matter. Thus the frequency of any dark photon signal gives the mass m. Despite p mc, the dark matter momentum spread is nonzero and this does affect the observed signal. Were dark photons all without momentum, then the observed 2 signal would be at a single frequency νsignal = mc =h with no spread. The nonzero momentum distribution, however, acts to spread the frequency of the signal[4]: mc2 mc2 ≤ ν (1 + 10−6) (7) h signal . h giving −6 ∆νsignal(ν) ∼ 10 ν (8) and thus a quality factor of the signal of ν (ν) Q = signal ≈ 106: (9) ∆νsignal(ν) This provides a serious constraint on possible dark pho- FIG. 2. CAD model of dark radio experiment. Green high- ton signals: they must have a very large quality factor. lighted object in the center of the room is the bicon antenna Given the high probability of systematic noise entering with transmission line feed. into detectors, constraints such as this are valuable in providing confidence that a measured signal is actually from dark photons. the detailed geometry of the room must be considered. The complexity of the geometry, especially in the actual experimental setup, requires numerical simulation to find C. Coupling such an (effective[18] since we take surroundings into ac- count) antenna factor. As described in section B 3, the dark photon field in- Defining the (effective) antenna factor as the ratio of teracts with charges like the electromagnetic field, except field strength (without the antenna there) to feed voltage suppressed by . In other words, a field E0 is equivalent to (with the antenna there): an electric field by E = E0. This allows search for dark photons through electromagnetic field detectors such as E(ν) AF (ν) = ; (10) antennae, which is used in this paper. V (ν) The search for dark photons is drastically different than that for high mass particles: low mass searches for we see that knowledge of the (effective) antenna factor weak signals through use of resonating structures; high and feed voltage would give the coupling by: mass searches for rare events through use of large de- E(ν) AF (ν) · V (ν) AF (ν) · V (ν) tection volume. An example of the high mass search is = = ≈ : (11) LZ: a cutting edge liquid xenon time projection cham- E0(ν) E0(ν) 3300V/m ber searching for WIMPs through waiting for WIMPs to interact with xenon nuclei and produce light[1]. Both Thus the antenna factor gives the coupling, the last un- detectors need to run for long times, but for different known. reasons. Dark photon searches need the dark photon sig- nal to `ring up' while other (noisy) signals average out; WIMP searches wait until the rare event (WIMP hitting III. EXPERIMENTAL SETUP a nucleus) occurs. The relationship between dark field strength and out- A COM-POWER AB-900A biconical antenna (or `bi- putted signal must be determined.
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