INTRODUCTION
1.3 Direct Detection Preliminaries
The new force acts as the “portal” between the Standard Model and the “hidden sector” of DM particles which are also charged under this new force. As an example we will discuss the light “dark photon” model [170] which has proven to be the most popular benchmark hidden sector model (at least within the context of direct detection) since it simply adds a new U(1)0 gauge group. Minimally the model adds two terms to the Standard Model Lagrangian,
L ⊃ mA0
2 A02+κ
2FµνFµν0 , (1.21)
whereA0 is the dark photon,mA0 is its mass, andκis known as a “kinetic mixing” parameter since it mixes the kinetic terms of the photon and dark photon. This κ can be radiatively generated through loops of heavy particles charged under U(1)×U(1)0. To understand the phenomenology of this model it is typically useful to diagonalize the mass matrix of theA, A0 system via the field transformation,
Aµ→Aµ+κA0µ. (1.22)
Note that A0µ cannot be rotated since that would generate a mass for the photon. In this rotated basis all of the Standard Model fermions which have an electromagnetic charge are now charged underA0and therefore the field acts like an additional photon field with rescaled couplings, e→κe.
The dark photon itself can be the DM [171, 172], typically one adds a particle χ which is charged under the U(1)0 and takes this to be the DM. This χ field then serves as the light DM candidate, and can give rise to a direct detection signal by scattering off a target via mediating an A0.
While the motivation for any given dark photon model is perhaps weaker than the motivation for the axion, this class of models, with an extra gauge group and a DM candidate is a very natural explanation for DM. The Standard Model is a collection of seemingly random gauge groups so it is not unfeasible to have one more gauge group.Moreover, extra gauge groups are a fairly generic prediction of Grand Unified Theories. The dark photon model serves as a useful benchmark for these more general cases since it is easier to calculate for than extensions with an extra SU(N)group, or an MSSM-like model with many new parameters.
Light Dark Matter Kinematics
Before discussing the general interaction rate formalism it is useful to understand the kine- matics, or how energy and momentum conservation are satisfied in interactions involving light DM. Since DM is cold, as discussed in Sec. 1.1, it can be treated as a non-relativistic particle, i.e.,
Eχ ≈mχ+1
2mχv2χ, (1.23)
where χ is a DM particle, and vχ ∼ 10−3. There are two different processes we will focus on: scattering and absorption. In a scattering process a DM particle enters and leaves the detector, depositing some energy and momentum. Letpχ be the momentum of the incoming DM particle, andp0χ=p−q be the momentum of the outgoing DM particle,4 whereq is the momentum leaving the DM system (and entering the detector). In this scenario the energy deposited,ω, is determined by the momentum transfer,
ω=Eχ−Eχ0 = p2χ
2mχ − (mχvχ−q)2
2mχ =q·vχ− q2
2mχ, (1.24)
where Eχ is the initial energy of the DM particle and Eχ0 is the final energy of the DM particle.
From here, without any knowledge of the detector itself, we see that qmax= 2mχvχ,max∼keV mχ
MeV
, (1.25)
ωmax= 1
2mχvχ,max2 ∼eV mχ MeV
, (1.26)
where vχ,max is the maximum velocity of the incoming DM particle, qmax is the maximum momentum transfer, and ωmax is the maximum energy transfer. From this simple exercise we immediately understand the energy and momentum scales involved in DM scattering.
Moreover the experimental energy resolution necessary to see these interactions is clear; if the threshold energy of an experiment is aboveωmax then these events will not be visible.
In addition to the kinematics on the DM side the detector must be able to respond. Consider the canonical direct detection experiment, i.e., DM scattering off a stationary nucleus of mass mN. The final energy of the nucleus, q2/2mN, must be equal to the energy deposited by the DM,
q2
2mN =q·vχ− q2
2mχ. (1.27)
4Assuming that the outgoing particle is identical to the incoming one. If the DM is composite it can change its internal state, leaving a residual∆E=minχ −moutχ in Eq. (1.24). DM models with this behavior are sometimes referred to as “inelastic” DM [173] due to the inelastic nature of the collision.
Solving for q, ω gives
q = 2µN χvχ (1.28)
ω = 2µ2N χ
mNv2χ, (1.29)
whereµN χ is the reduced mass of the nucleus DM system. In the light DM limit, mχ mN, we can simplify the energy deposition,
ω = 2m2χ
mNv2χ∼meV mχ MeV
2
, (1.30)
where we have assumed a light nucleimN ∼GeV. For mχ =MeV we see that nuclear recoil can only extract a thousandth of the energy available in a DM scattering event. Therefore we see that nuclear recoil is not well suited to search for light DM candidates, and that to maximize detector sensitivity we need excitations which can kinematically match the incoming DM. As we will discuss in detail, electronic, phononic, and magnonic excitations are ideal for this purpose.
The kinematics of DM absorption is even simpler than scattering. Since there is no outgoing DM state all of the energy and momentum of the initial DM state must be absorbed,
q =mχvχ (1.31)
ω = mχ. (1.32)
This can be thought of as the opposite limit of a scattering event since q ω, whereas in a scattering event q ω. Because q is so small an absorption event can be thought of as a vertical transition in the q−ω plane.
General Formalism
The calculation of DM-target interaction rates is typically treated with standard perturba- tion theory since the DM-target coupling is small. In the absence of DM-target interactions, the Hamiltonian governing the behavior of the DM and target is simply the sum of their individual Hamiltonians,
H0 =HDM0 +Htarget0 , (1.33)
whereH0 has dimensions[eV]. Assuming the DM is a non-relativistic particle,HDM0 is simply HDM0 =mχ+ p2χ
2mχ. (1.34)
and the eigensystem is then simply a set of plane wave states
HDM0 |pχi=Eχ|pχi, (1.35)
Eχ =mχ+ p2χ
2mχ. (1.36)
While this is a great approximation for the cosmic cold DM, we make no such assumptions for the target states. This is because these states will take a vastly different form depending on the excitation, e.g. electrons or phonons. To be as general as possible we take the eigensystem of Htarget0 to be,
Htarget0 |Ii=ωI|Ii, (1.37)
for some set of states labelled byI. Therefore a DM-target state is then just a product state,
|Ii=|Ii ⊗ |pχi, (1.38)
H0|Ii=ωI|Ii, (1.39)
ωI =ωI+Eχ. (1.40)
We now add a coupling between the DM and target system via an interaction Hamiltonian,
H =H0+δH . (1.41)
Fermi’s Golden rule then dictates the transition rate, or number of transitions per unit time. Assuming the initial (final) DM-target state is |Ii(|F i) the transition rate between the states, ΓI→F, is
ΓI→F = 2π|hF |δH|Ii|2
hF |F ihI|Iiδ(ωF −ωI). (1.42) This formula should be familiar, except perhaps the denominator of state inner products.
Fermi’s Golden Rule is usually derived in the context of quantum mechanics where the states are implicitly unit normalized, and therefore the denominator is left out. However sometimes it is convenient to leave these factors in, since the DM states are sometimes normalized with QFT conventions. This formula simply generalizes Fermi’s Golden rule to account for different state normalizations. Here we assume all states are unit normalized, i.e, hI|Ii=hp|pi= 1.
Since the DM states are assumed to be plane waves, we can further simplify Eq. (1.42) by inserting an identity operator in the DM space,
1 = 1 V
Z
d3x|xihx|, (1.43)
where V is the target volume. Placing this inside of the matrix element gives hpχ|δH|p0χi= 1
V2 Z
d3xd3yhp0χ|xihx|δH|yihy|p0χi, (1.44)
= 1 V
Z
d3xeiq·xhx|δH|xiˆ , (1.45)
≡ 1
V V(−e q), (1.46)
assuming the interaction Hamiltonian is local,hx|δH|yiˆ =δxyhx|δH|xiˆ =V δ(3)(x−y)hx|δH|xi,ˆ hx|pi =eip·x, and q = pχ−p0χ is the momentum transferred to the target.5 We have intro- ducedVe which we will often refer to as the “scattering potential” which will depend on how the DM model couples to the target. Written in terms of the scattering potential, Eq. (1.42) is given by,
Γpχ,I→p0χ,F = 2π V2
hF|V(−e q)|Ii
2
δ(ωF −ωI−ω), (1.47) where ω≡Eχ−Eχ0 is the amount of energy the DM deposits on the target.
The total interaction rate for an incoming DM particle of momentum pχ is found by simply summing over all the other states. However, instead of summing over all possible p0χ with Pp0χ →V R d3p0χ
(2π)3, it is convenient to shift variables to q giving, Γpχ = 2π
V X
I
X
F
Z d3q (2π)3
hF|eV(−q)|Ii
2
δ(ωF −ωI−ω). (1.48) This is the number of events per unit time assuming one incoming DM particle with momen- tum pχ. The average excitation rate per incoming DM particle, Γ, can then be computed once the momentum distribution of DM particles is known. Usually this is given in terms of the DM velocity, vχ since pχ = mχvχ in the nonrelativistic limit. We will discuss the standard choice of velocity distribution below. Assuming for now that the DM velocity dis- tribution isfχ(vχ), the expected excitation rate per unit time, per incoming DM particle, is given by,
Γ = 2π V
Z
d3vχfχ(vχ)X
I
X
F
Z d3q (2π)3
hF|Ve(−q)|Ii
2
δ(ωF −ωI−ω). (1.49) The total number of interactions per unit time per detector mass, MT, is
R = Nχ
MTΓ = ρχV mχ
1
ρTV Γ = ρχ
mχρTΓ, (1.50)
= 2πρχ mχρTV
Z
d3vχfχ(vχ)X
I
X
F
Z d3q (2π)3
hF|Ve(−q)|Ii
2
δ(ωF −ωI−ω), (1.51)
5Another common choice for the definition ofqisp0χ−pχ, or simply the negative of the convention here.
While this would remove the minus sign in the scattering potential, the momentum would now be “coming out” of the target, which is, arguably, less intuitive.
where Nχ is the number of DM particles in the detector, ρχ is the local DM density, taken throughout this thesis to be ρχ = 0.4GeV cm−3 [174], and ρT is the target density. This rate, R, is dimensionless (in natural units), but usually converted to units of kg−1yr−1 such that one simply needs to multiply by the exposure to find the total number of interactions.
This is the central formula which will be used routinely in the coming chapters to compute the expected number of events in a detector.
A standard parameterization of the DM velocity distribution is a Maxwell-Boltzmann dis- tribution [174–176] cutoff at the galactic escape velocity,
fχ(v) = 1
N0e−v2/v02Θ(vesc− |v|), (1.52) N0 =π3/2v02
v0erf(vesc/v0)−2vesc
√π exp −vesc2 /v02
, (1.53)
where our conventions throughout this thesis will be to usev0 = 230km s−1,vesc = 600km s−1[177].
However this does not directly get substituted in to Eq. (1.51); this is the velocity distribu- tion in the galactic frame. Since the Earth is also moving through the galaxy we need to boost the Maxwell-Boltzmann distribution to the Earth’s frame, i.e., fχ(vχ+ve) is the ve- locity distribution we use. The Earth velocity vector, ve, introduces interesting modulation effects which will be the subject of the next subsection.
Modulation Effects
Since the interactions between the DM and any given target are weak, it is crucial to find ways to differentiate a DM signal from any background sources. This way, even if only a tiny signal is observed, we can still claim the signal is significant. The key differentiating factor between DM and the majority of other backgrounds is that Earth is moving through a cosmic background of DM. This means that as the Earth velocity changes, in the Earth frame, the direction of the incoming DM is changing.
The two main modulation effects are “annual modulation” and “daily modulation”. Since the Sun is also moving around the galactic center, the Earth’s motion relative to it on a yearly basis changes the magnitude of the velocity relative to the incoming DM. We take the central value of the Earth velocity is ve = 240km s−1, and annual modulation causes O(10km s−1) fluctuations. While seemingly a relative small effect, remember from Sec.1.3 that the maximum energy/momentum transfer is set by the maximum DM velocity, vmax =vesc+ve. As an example case where this might be important, consider an electronic transition across a band gap. If the DM mass is just barely kinematically able to drive the transition, then at different times of the year a transition may or may not be possible.
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”.