Detection of WIMP Dark Matter

Detection of WIMP Dark Matter Sunil Golwala, Dan McKinsey October 5, 2010 1 Introduction Dan Hooper has described in Chapter ?? the motivation for Weakly Interacting Massive Particle dark matter, some specific WIMP candidates from supersymmetry, universal extra dimensions, and generic TeV-scale new physics, and some aspects of their direct and indirect detection. In this chapter, we delve into direct detection in more detail, describing the generic experimental signatures and the experimental techniques used. 2 Direct Detection of WIMPs via WIMP-Nucleon Elastic Scattering “Direct detection” consists of observing WIMP-nucleus scattering events. The plausibility of direct detection was first demonstrated by Goodman and Witten [1] following the proposal of a neutral- current neutrino-detection method by Drukier and Stodolsky [2]. In the 25 years since, both the theoretical framework for calculating interaction rates and sufficiently sensitive experimental techniques have been developed. Once a WIMP-nucleus cross section has been calculated along the lines described in Chapter ??, it is straightforward to calculate the expected spectrum of recoil energies due to WIMP-nucleus scattering events. Lewin and Smith is the standard reference [3]. Particularly critical for the experimentalist is understanding the dependence of the event rate on target nucleus and recoil energy. Numerical results are provided at the end of the section. The WIMP velocity distribution is given by the WIMP phase-space distribution. For simplic- ity, it is standard to assume the WIMPs occupy an isothermal, isotropic phase-space distribution, appropriate for a fully gravitationally relaxed population of WIMPs. It is known that this is not completely accurate. However, these uncertainties are small compared to the theoretical uncer- tainty in the WIMP-nucleon cross section. Moreover, a simple standard model ensures that results presented by direct-detection experiments are easily interpreted. The full phase-space distribution function for an isothermal halo composed of particles of mass Mδ is M v2/2 + M φ(~x) f(~x,~v) d3x d3v ∝ exp − δ δ (1) kBT which is just a Boltzmann distribution with the gravitational potential φ(~x) included. Mδ is the mass of the WIMP δ. At a given point, the position-dependent factor is fixed and can be included in the normalization, leaving a simple Maxwellian velocity distribution: 2 (~v + ~vE) f(~v,~vE) = exp − 2 (2) v0 where ~v is the WIMP velocity relative to the Earth and ~vE is the Earth’s velocity relative to the nonrotating halo of the galaxy. Note that the definition of ~v has been changed relative to 2 2 Equation 1. The quantity v0 is characteristic of the WIMP kinetic energy, kBTδ = Mδv0/2 and −1 has value v0 ≈ 220 km s . It corresponds to the most probable velocity and is known as the 1 p velocity dispersion. It is related to the root-mean-square WIMP velocity vrms by vrms = v0 3/2, as holds for any Maxwellian distribution. The Maxwellian distribution is cut off at |~v + ~vE| = vesc by the halo escape velocity. Note that the cutoff is isotropic in the galactocentric WIMP velocity −1 ~v + ~vE, not in the Earth-centric ~v. In practice, the escape velocity vesc ≈ 550 km s [4] is so large compared to v0 that it has negligible effect at most WIMP masses of interest. It must be included, though, to obtain accurate recoil spectra for low-mass WIMPs. With the above velocity distribution, we can calculate the rate of scattering of this incoming “beam” of particles on nuclei in a detector, obtaining a recoil energy spectrum in the lab frame. The WIMP “beam” flux per unit WIMP energy has the form E exp(−E/E0) from phase space and the Boltzmann distribution. The differential cross section goes like v−2 ∝ 1/E and is independent of 2 momentum transfer q ∝ ER when details of nuclear structure are ignored. Therefore, the product of the beam flux and differential cross section is an exponential in E. This product is integrated from the minimum possible E that can yield ER to infinity. This lower limit is proportional to ER, leaving an exponential in ER with normalization factors set by the WIMP velocity (energy) distribution. With ~vE = 0 and vesc → ∞ to simplify this integral, one finds that the spectrum (in keV−1 kg−1 d−1) has the form dR R 0 −ER/E0r 2 2 (ER) = e F (q = 2MN ER) (3) dER ~vE = 0 E0r vesc → ∞ with 1 M M 2 N m2 2 δ N √ 0 r,δN 2 E0 = Mδv0 r = 4 2 R0 = n0 σ0 v0 σ0 = σδn 2 A (4) 2 (Mδ + MN ) π A mr,δn being the mean energy, a unitless mass ratio, the integral rate assuming F 2(q2) = 1, and the cross section on the nucleus. F 2(q2) is a “form factor” that parameterizes the breakdown of coherent scattering with the nucleus as the momentum transfer increases and the WIMP sees the nucleus’s internal structure. Mδ is the WIMP mass, MN is the nucleus mass, mr,δn and mr,δN are the WIMP-nucleon and WIMP-nucleus reduced masses. In the general case, with nonzero ~vE and finite vesc, the energy dependence becomes more complex. We quote the full formula, Equation 3.13 of [3]: √ dR k0 R0 π v0 vmin + vE vmin − vE −v2 /v2 (ER) = erf − erf − e esc 0 (5) dE k E r 4 v v v R ~vE ,vesc 1 0 E 0 0 p 2 3/2 where vmin = q/2mr = v0 ER/E0r and k0 = (πv0) are as defined before and vesc 2 vesc −v2 /v2 k1 = k0 erf − √ e esc 0 (6) v0 π v0 This form immediately leads us to annual modulation. ~vE is a function of the time of year with, to a good approximation for these purposes, t − 152.5 v = 232 + 15 cos 2π km s−1 (7) E 365.25 where t is in days (00:00 January 1 corresponds to t = 0). Because of the sinusoidal variation in vE, there is a small modulation of the recoil-energy spectrum on an annual basis, which is discussed in 2 −1 0 10 10 A = 28 A = 28 A = 73 A = 73 −2 A = 127 −1 A = 127 1 10 10 1 − − d d 1 1 − −3 − −2 10 10 keV 1 − −4 −3 10 10 Events (>E) kg Events kg −5 −4 10 10 −6 −5 10 10 0 50 100 150 200 0 50 100 150 200 Recoil Energy [keV] Recoil Energy [keV] Figure 1: Variation of recoil-energy spectrum with A. Left: differential spectrum. Right: integral of spectrum from ER to 200 keV (∼ ∞), indicating the integral rate above a given threshold. Spectra are shown for silicon (A = 28), germanium (A = 73), and iodine (A = 127). Note that xenon (A = 131) has a response very similar to iodine. some detail in the context of the DAMA experiment, below. The mean spectrum can be calculated −1 by setting vE = hvEi = 232 km s . We show the dependence of the differential and cumulative recoil energy spectra on A in Figure 1 for a 100-GeV c−2 WIMP. The change in the energy dependence is especially critical because detectors have thresholds in the few-to-tens-of-keV region. A germanium target offers a clear advantage over a silicon target, and iodine yields a benefit over germanium if the recoil-energy threshold is below 10 keV. The above formulae can be investigated numerically to get a feel for the energies and event rates involved. Lewin and Smith calculate a useful form for R0, 361 σ0 ρδ v0 −1 −1 R0 = −36 2 −2 −3 −1 kg d (8) MδMN 10 cm 0.3 GeV c cm 220 km s −2 where Mδ and MN are in GeV c . Recall that R0 is the integral rate with no form-factor correction and so is only indicative of the detectable event rate. A typical WIMP mass of interest today is −2 −2 −3 Mδ = 100 GeV c . Halo parameters typically used are ρδ = 0.3 GeV c cm and v0 = −1 −1 220 km s . This v0 corresponds to a rms WIMP velocity of vrms = 270 km s . An event rate detectable by present experiments is 0.1 kg−1 d−1. For germanium (A ≈ 73), this corresponds to −37 2 −43 2 σ0 ≈ 20 × 10 cm , or σδn ≈ 2 × 10 cm . The average energy deposition is given by Mδ v0 2 4MδMN E0r = −2 −1 2 × 26.9 keV (9) 100 GeV c 220 km s (Mδ + MN ) −2 For the standard halo parameters given above, Mδ = 100 GeV c , and a germanium target, this gives E0r ≈ 26 keV. For reference, R0 and E0r are calculated for a number of common targets and WIMP masses of 10, 100, and 1000 GeV c−2 in Table 1. 3 −1 −1 R0 [kg d ] E0r [keV] −2 −2 Mδ [GeV c ] = Mδ [GeV c ] = target 10 100 1000 10 100 1000 H(A = 1) 3.9×10−6 3.9×10−5 3.9×10−7 0.8 1.0 1.0 Si (A = 28) 7.8×10−2 5.5×10−2 8.1×10−3 2.2 17.6 26.6 Ge (A = 73) 3.0×10−1 5.4×10−1 1.3×10−1 1.2 25.9 64.1 I(A = 127) 5.8×10−1 1.7×100 6.4×10−1 0.8 26.7 101.7 Table 1: R0 and E0r for hydrogen, silicon, germanium, and iodine and for WIMP masses of 10, 100, and 1000 GeV c−2. 3 Background Sources Equal in importance to the expected WIMP interaction rate is the interaction rates of non-WIMP particles, termed “background” particles.

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