INTRODUCTION
1.4 Summary
The other type of modulation effect, daily modulation, is the focus of Ch. 4, and discussed in Ch. 2. While the magnitude of the Earth’s velocity will change very slightly, the more important effect here is that the DM wind will hit the target from different directions. For a target with an isotropic response function this does not matter. However for targets which have an anisotropic response this effect can cause O(1) fluctuations in the DM rate. This spectacular signal, if observed, would be a “smoking gun” signature that the effect is due to DM.
Setting Constraints Assuming No Backgrounds
In the absence of any events seen in the detector, and assuming negligible background events, the rate in Eq. (1.51) can directly used to place constraints on the coupling constants which govern the strength of the potential. While different statistical procedures exist for placing limits, perhaps the simplest is to hypothesis test with the null hypothesis being a Poisson distribution with expected number of events, N¯ =R×MT ×T,
P(n,N¯) =
N¯ne−N¯
n! , (1.54)
where T is the exposure time. The probability of seeing 0 events is then e−N¯. Therefore we can reject the null hypothesis at the 100×pth confidence level (C.L.) when e−N¯ = 1−p. Taking p = 0.95 implies N¯ ≈ 3. This is commonly reported in the figure captions as, e.g.,
“the 95% C.L. constraints (3 events) assuming no background”.
derived, and makes clear the assumptions being made in each calculation. Essentially, it is an extension of the general formalism discussed in Sec.1.3 focused on general spin independent (SI) DM scattering. Using this general formalism we illustrated how daily modulation effects could arise in electronic transitions in anisotropic targets, showed that Umklapp processes are important for phonon excitations in models with a heavy mediator, and highlighted the kinematic regimes each channel is likely to dominate in.
• Ch. 3: Multi-Channel Direct Detection of Light Dark Matter: Target Com- parison
Many different target materials have been utilized in direct detection experiments.
From noble liquids, Xenon, and Argon in large chambers, to Si and Ge in detectors focusing on electronic excitations. Different target materials can have dramatically different properties, and an important question to ask is whether the optimal target material is being used. In this chapter we compare and contrast the effectiveness of 26 different target materials as light DM detectors. This chapter is a companion to Ch. 2 and relies heavily on the formalism developed there. Specifically, we compute the electroninc, phononic, an nuclear recoil signals for a few benchmark DM models, including the light dark photon model discussed in Sec.1.2, in all 26 targets. In addition we give simple analytic expressions in terms of macroscopic material properties which can be optimized to further search for optimal targets.
• Ch. 4: Directional Detectability of Dark Matter With Single Phonon Exci- tations: Target Comparison
While alluded to in the previous two chapters, anisotropic materials will not only have a daily modulation, discussed in Sec. 1.3, in electronic excitations, but also phonon excitations. Understanding the daily modulation pattern of a given target can be crucial in rejecting backgrounds. This chapter is a detailed study of daily modulation in single phonon excitations in the same 26 target materials, and benchmark DM models, discussed in Ch.3. We find that a variety of anisotropic targets can haveO(1) modulation fractions for a variety of DM model space.
• Ch. 5: Detecting Light Dark Matter With Magnons
Phonon excitations have been the collective excitation mode focused on so far, but they are not the only ones to appear in target materials. In targets with magnetic order-
ing the low energy excitations appearing are known as magnons. These are quantized spin waves arising from a Heisenberg-like Hamiltonian of spin interactions. Similar to phonons, the energy of these excitations areO(1−100meV); guaranteeing kinematic matching between them and light DM candidates. Moreover these excitations are uniquely susceptible to DM models which couple to spin, and therefore offer a comple- mentary probe of light DM model space. In this chapter we study the light DM-single magnon scattering rate for a few spin-dependent light DM models, in a yttrium iron garnet (YIG) target.
• Ch. 6: Detectability of Axion Dark Matter With Phonon Polaritons and Magnons
As discussed in Sec.1.2, the QCD axion is a well motivated DM candidate with a unique production mechanism and ability to solve the Strong CP problem. Direct detection of the axion has a storied history, with many different detection ideas being utilized to search for it. Typically these rely on coupling the axion to an electromagnetic mode in a cavity and reading out the resultant electromagnetic field. While this is great for low, m . meV scale axions, there quickly becomes a fundamental problem that the energy of these modes is inversely proportional to the cavity size, limiting exposure. Furthermore since stellar cooling constraints only limit the QCD axion to m.100meV, there is open parameter space between O(1−100meV). This is ideally matched to phonon and magnons modes, and we show that such experiments could reach the QCD axion line. Technically, the absorption occurs on phonon-polariton modes which come from the mixing between the photon and phonon near the level crossing, clearly an important regime for DM absorption kinematics. We also show that changing the direction of the external magnetic field can give rise to different phonon modes, allowing for modulation effects.
• Ch. 7: Effective Field Theory of Dark Matter Direct Detection With Col- lective Excitations
In Ch. 2we discussed the general formalism for DM single phonon scattering via spin independent operators, and in Ch.5we discussed the formalism for DM single magnon scattering for operators depending on the electron spin. Here we unify and extend this formalism to account for general scattering potentials. The space of UV DM models is vast, and computing constraints for every single model is untenable. Thankfully this need not be done, there exists a basis of operators for which any UV DM model can be
mapped on to via an effective field theory (EFT) procedure. In this chapter we discuss the construction of this basis, and give examples of mapping some general DM model theories on to these scattering potentials. Moreover we show how this basis can be used for both DM-single phonon and single magnon scattering. We detail which response will be dominant in target materials which have both excitations. We also released an open source code,PhonoDark [178], which can compute the phonon excitation rate for any material and operator from density functional theory (DFT) input.
• Ch.8: Extended Calculation of Dark Matter Electron Scattering In Crystal Targets
As discussed previously there are many ongoing experiments looking for electronic excitations. Therefore it is important to have accurate calculations of the DM induced electronic excitation rate. Central to this calculation is an accurate description of the electronic wave functions and energy levels. Previous calculations approximated these states with a variety of methods. Initially with the simplest analytic forms solving the Hydrogen Schrödinger equation, to semi-analytic forms working with these functions as a basis, to using density functional theory (DFT) techniques which solve for them numerically. In this chapter we combine the best aspects of the previous calculations, by using them when they are appropriate. For example, the states tightly bound to the ionic sites can be well approximated as “core” electronic states in which semi- analytic approaches work well, and the more free valence states can be computed with DFT. This allows for the first complete calculation to be performed which takes in to account all kinematically allowed transitions. We also take in to account “all-electron”
reconstruction effects which correct the pseudo-potential calculated wave functions at small distances. We show that these effects can be important for heavy mediators, and more generally for transitions withω &10eV. We packaged this calculation in to the open source program, EXCEED-DM [179].
• Ch. 9: Dark Matter Absorption via Electronic Excitations
In addition to the DM-electron scattering rate calculations performed in Ch. 8 ab- sorption across the band gap can also take place. This ends up being a deceptively complex problem for kinematic reasons. In a perfectly vertical transition one needs to account for the fact that the energy eigenstates which the electron transitions between are orthogonal. Generally this leads to a suppression of the absorption rate, and since the DM momentum is smaller than the typical electron velocity, αme, electron veloc-
ity dependent effects can be important. To handle these subtleties we construct an non-relativistic EFT and carefully power count. This allows us to compute the DM absorption rate from first principles for any DM model. It has been previously shown that the vector DM and pseudoscalar DM absorption rates can be related to optical data, and we find good agreement between the data and our calculations. We also find that the scalar DM absorption rate cannot be related to optical data. Similar to Ch. 8we add an absorption module to EXCEED-DM to numerically compute these absorption rates.
• Ch. 10: Dark Matter Direct Detection In Materials With Spin-Orbit Cou- pling
While Chs.8and 9focused on DM scattering and absorption in the Si and Ge targets in use today, here we focus on more exotic targets, like ZrTe5, which have spin-orbit couplings. These targets can be special because they can have much smaller band gaps, O(meV), than the standard semiconductors used today. This allows for more DM model parameter space to be covered in experiments which read out electrons, without having to resort to phonon or magnon mode readout. These targets are also theoretically interesting because the presence of spin-orbit coupling means that the electronic spin is no longer a good quantum number, and the wave functions become two component. We show that these effects can be important for states close to the band gap. We also discuss the differences in the calculation presented here and those done previously for Dirac materials which assume a perfectly linear dispersion relation.
Lastly in Ch. 11 we look to the future and discuss some important extensions of this work done in this thesis.
C h a p t e r 2