Phenomenological Introduction to Direct Dark Matter Detection

The dark matter of our galactic halo may be constituted by elementary particles that interact weakly with ordinary matter (WIMPs). In spite of the very low counting rates expected for these dark matter particles to scatter off nuclei in a laboratory detector, such direct WIMP searches are possible and are experimentally carried out at present. An introduction to the theoretical ingredients entering the counting rates predictions, together with a short discussion of the major theoretical uncertainties, is here presented. invited talk at the XXXI Rencontres de Moriond, “Dark Matter in Cosmology, Quantum Measurements, Experimental Gravitation,” Les Arcs, France, January 1996 This is a phenomenological introduction to the detection of dark matter through its scattering in a laboratory detector. For dark matter in the form of massive quasi-stellar objects, like brown dwarfs, which are much bigger and much heavier than the Earth, this type of detection is quite impracticable if not undesirable. I therefore consider dark matter in the form of elementary particles. Many particles, most of which hypothetical, are at present candidates for dark matter: neutrinos, neutralinos, axions, etc. The methods employed in hunting for these particles are very different. In this short note I focus on this meeting’s category of particle dark matter, viz. weakly interacting massive particles or WIMPs. WIMPs, in a broad sense, are particles with masses of the order of atomic masses or higher (m >∼ 10GeV/c) that interact with ordinary matter with cross sections typical of the weak interaction or smaller (σ <∼ 10cm off a proton). The presently most popular WIMP is the yet-undetected neutralino, the lightest supersymmetric particle in supersymmetric models. Other famous WIMPs are Dirac and Majorana neutrinos, which however, thanks to the on-going dark matter searches complemented by accelerator results, we know not to be the dominant component of our galactic halo. A general introduction to dark matter has been given by Olive at this meeting. Direct detection of WIMPs was first explored by Goodman and Witten. General reviews are Primack et al and Smith and Lewin. Engel et al. present the nuclear physics involved. At this meeting Cabrera discusses experimental aspects of direct dark matter detection, while I focus on the theoretical aspects. It is worth recalling some properties of the dark halo of our galaxy. Even if recent observations might change the details of our picture, the 1981 model by Caldwell and Ostriker is good for my purposes. The Sun lies at a distance of ≈ 8.5 kpc on the disk of our spiral galaxy, and moves around the center at a speed of ≈ 220 km/s. The luminous disk extends to ≈ 12 kpc, and is surrounded by a halo of ≈ 100 kpc where globular star clusters and rare subdwarf stars are found. Dynamical arguments suggest that the halo is filled with dark matter, whose local density in the vicinity of the Sun is estimated to be ρDM = 0.2–0.4GeV/c /cm = 0.7–1.4 10g/cm. Equilibrium considerations also give the root mean square velocity of halo consituents to be 200–400 km/s, not much different from the escape speed from the galaxy (500–700 km/s). Very little is known on the mean rotation speed of the halo, and we will assume it does not rotate. All in all, there is an optimistic factor of 2 uncertainty on the density and velocity of halo dark matter. Are there WIMPs in our galactic halo? The scientific way to answer this question is to detect them. Signals could come from WIMP annihilation (indirect detection) or WIMP scattering (direct detection). In the former we search for rare annihilation products like neutrinos, antimatter or gamma-ray lines. This is reviewed by Bergström at this meeting. In the latter, the basic philosophy is to build a target, wait and count. The WIMP scattering rate per target nucleus is the product of the WIMP flux φχ and of the WIMP-nucleus cross section σχi. For an order of magnitude estimate we take the effective WIMP-nucleon coupling constant to be Fermi’s constant GF = 2.3016 10 −19h̄c2 cm/GeV, which sets the scale of weak interactions. We distinguish two cases: (i) the WIMP couples to nucleon spin, σχi ≈ GFμi /h̄ <∼ 10cm ; and (ii) the WIMP couples to nucleon number, σχi ≈ GFμiAi /h̄ <∼ 10cm. Here μi = mχmi/(mχ+mi) is the reduced WIMP-nucleus mass, and Ai is the atomic number of the target (≈ 80 in the numerical examples) . The WIMP flux is φχ = vρχ/mχ ≈ 10cms/(mχc/GeV), for a WIMP density ρχ ≈ 10 g/cm and a typical WIMP velocity v ≈ 300 km/s. The resulting scattering rates, taking mχ ≈ 100 GeV/c, are of the order of <∼ 1/kg-day for spin-coupled WIMPs and of <∼ 10/kg-day for WIMPs coupled to nucleon number. These rates are quite small compared with normal radioactivity background. Therefore the common denominator of direct experimental searches of WIMPs is a fight against background. For this we get help from characteristic signatures that we do not expect for the background. For example, while the Earth revolves around the Sun, the mean speed of the WIMP “wind” varies periodically with an amplitude of 60 km/s. This leads to a ≈ 10% seasonal modulation in the detection rate, with a maximum in June and a minimum in December. As another example, the direction of the WIMP “wind” does not coincides with the Earth rotation axis, so the detection rate might present a diurnal modulation due to the diffusion of WIMPs while they cross the Earth (this however occurs for quite high cross sections). A final example of background discrimination is that the WIMP signal is directional, simply because most WIMPs come from the direction of the solar motion. WIMP-nucleus scattering Since the relative speed v ≈ 300km/s ≈ 10c, the process can be treated non-relativistically. The center of mass momentum is given in terms of the reduced WIMP-nucleus mass as k = μiv and is <∼ Ai MeV/c since μi ≤ mi. The corresponding de Broglie wavelength is >∼ 200 fm/Ai, and can be smaller than the size of heavy target nuclei, in which case nuclear form factors are important. In the laboratory frame, the nucleus recoils with momentum q = 2k sin(θcm/2) and energy ν = q/2mi. Here θcm is the center-of-mass scattering angle. The 4-momentum transfer is very small, Q <∼ Ai 10−6GeV/c2 (compare with a typical deep inelastic Q >∼ 1GeV/c2). The differential scattering rate per unit recoil energy and unit target mass is formally dR dν = ρχ mχ ∑ i fiηi(q) |Ti(q2)| 2πh̄ . (1) The sum is over the nuclear isotopes in the target, Ti(q ) is the scattering matrix element at momentum transfer squared q = 2miν, and fi is the mass fraction of isotope i. A sum over final and average over initial polarizations is understood in |Ti(q2)|2. The factor ηi(q) = ∫ ∞ q/2μi fχ(v) v dv, (2) with units of inverse velocity, incorporates the χ velocity distribution fχ(v). For a Maxwellian distribution with velocity dispersion vrms, seen by an observer moving at speed vO, ηi(q) = 1 2vO [