MCTP-03-33 The Phase of the Annual Modulation: Constraining the 4 0 0 WIMP Mass 2 n a J Matthew J. Lewis and Katherine Freese 0 3 2 Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, USA v 0 9 1 Abstract 7 0 The count rate of Weakly Interacting Massive Particle (WIMP) dark matter can- 3 didates in direct detection experiments experiences an annual modulation due to the 0 / Earth’s motion around the Sun. In the standard isothermal halo model, the signal h p peaks near June 2nd at high recoil energies; however, the signal experiences a phase - o reversal andpeaksin December atlow energy recoils. We showthat this phasereversal r may be used to determine the WIMP mass. If an annual modulation were observed t s a with the usual phase (i.e., peaking on June 2nd) in the lowest accessible energy recoil : v bins of the DAMA, CDMS-II, CRESST-II,EDELWEISS-II, GENIUS-TF, ZEPLIN-II, Xi XENON, or ZEPLIN-IV detectors, one could immediately place upper bounds on the r WIMP mass of [103, 48, 6, 97, 10, 52, 29, 29] GeV, respectively. In addition, detec- a tors with adequate energy resolution and sufficiently low recoil energy thresholds may determine the crossover recoil energy at which the phase reverses, thereby obtaining an independent measurement of the WIMP mass. We study the capabilities of various detectors, and find that CRESST-II, ZEPLIN-II, and GENIUS-TF should be able to observe the phase reversal in a few years of runtime, and can thus determine the mass of the WIMP if it is (100 GeV). Xenon based detectors with 1000 kg (XENON and O ZEPLIN-IV) and with energy recoil thresholds of a few keV require 25 kg-yr exposure, which will be readily attained in upcoming experiments. 1 Introduction Extensive gravitational evidence suggests that a dominant fraction of the matter in our Galaxy is nonluminous, or dark. Although the identity of this dark matter is currently unknown, it may consist of Weakly Interacting Massive Particles (WIMPs). Numerous 1 experiments aimed at the direct detection of these WIMPs are currently being developed. These experiments generically measure the nuclear recoil energy deposited in a detector when an incident WIMP interacts with a nucleus in the detector. An important signature of halo dark matter in direct-detection experiments is the annual modulation induced by the Earth’s motion with respect to the halo [1]. In the standard model of the dark halo, in which the velocity distribution of the WIMPs is a Maxwellian distribution truncated at the escape speed of the Galaxy, the modulation of the WIMP interaction rate at high nuclear recoil energies is in phase with the motion of the Earth with respect to the halo, peaking when the relative speed is maximal (June) and reaching a minimum when the relative motion is minimal (December). For low energy recoils, however, it is well known that that the interaction rate experiences a phase reversal, peaking in December [3, 4, 5, 9]. This phase reversal occurs below a particular crossover recoil energy, Q . In this paper we examine c what information about the mass of the WIMP can be obtained from this phase reversal. We emphasize that this phase reversal could constitute an important signature of a WIMP flux, andwealso pointout thatbecause thecrossover recoilenergy isafunctionoftheWIMP mass, the observation of the phase of annual modulation immediately places an upper limit on the allowed WIMP mass. Furthermore, if WIMP direct detection experiments with sufficient energy resolution determine the crossover energy Q , the WIMP mass could be independently measured using c only observations of the phase of the annual modulation. This paper is organized as follows. In section 2, we briefly review the basics of WIMP direct detection experiments and the isothermal model of the dark halo. In section 3, we explore the dependence of the annual modulation on energy recoil, and revisit the phase reversal that occurs at low recoil energies. Finally, we demonstrate how this observation may be exploited to place limits on the WIMP mass and to obtain estimates of the mass directly. Weemphasizethatobservationofthisphasereversalwillconstituteanunambiguous signature of an extraterrestrial WIMP flux. 2 WIMP Direct Detection Experiments More than twenty collaborations worldwide are presently developing detectors designed to search for WIMPs. Although the experiments employ a variety of different methods, the basic idea underlying WIMP direct detection is straightforward: the experiments seek to measure the energy deposited when a WIMP interacts with a nucleus in the detector [11]. If a WIMP of mass m scatters elastically from a nucleus of mass m , it will deposit a recoil χ N energy of Q = (m2v2/m )(1 cosθ), where r N − m m m /(m +m ) (1) r χ N χ N ≡ 2 is the reduced mass, v is the speed of the WIMP relative to the nucleus, and θ is the scattering angle in the center of mass frame. We may compute (following, e.g., [17, 13, 12]) the differential detection rate, per unit detector mass (i.e., counts/day/kg detector/keV recoil energy) associated with this process, dR σ ρ = 0 h F2(Q)T(Q,t) (2) dQ 2m2m r χ where ρ = 0.3 GeV/cm3/c2 is the halo WIMP density, σ is the total nucleus-WIMP inter- h 0 action cross section and F(Q) is the nuclear form factor for the WIMP-nucleus interaction that describes how the effective cross section varies with WIMP-nucleus energy transfer. We consider here only spin independent interactions, wherein the target nucleus can be approx- imated as a sphere of uniform density smoothed by a gaussian [14], and the resulting form factor is, 3[sin(Qr ) Qr cos(Qr )] F(Q) = 1 − 1 1 e−Q2s2/2 (3) q3r3 1 where r = (r2 5s2)1/2, s 1 fm, and, 1 − ≃ r [0.91(M/GeV)1/3 +0.3] 10−13cm (4) ≃ × is the radius of the nucleus. For spin-dependent interactions, the form factor is somewhat different but again F(0) = 1. In general, the form factor must be evaluated for each detector nuclei. For a more extensive discussion, see [11, 1, 15, 16, 17]. For purely scalar interactions, 4m2 σ = r[Zf +(A Z)f ]2. (5) 0,scalar p n π − Here Z is the number of protons, A Z is the number of neutrons, and f and f are the p n − WIMP couplings to nucleons. For purely spin-dependent interactions, σ = (32/π)G2µ2Λ2J(J +1). (6) 0,spin F Here J is the total angular momentum of the nucleus and Λ is determined by the expectation value of the spin content of the nucleus (see [11, 1, 15, 16, 17]). For the estimates necessary in this paper, we take the WIMP-nucleon cross section1 σ = 7.2 10−42 cm2, and take the total WIMP-nucleus cross section to be 2 p × 2 m σ = σ r A2 (7) 0 p mrp! 1 This is the cross section with nucleons at zero momentum transfer as discussed in Eq. (7.36) of [17]. 2In most instances, f f so that the following equation results from Eq.(5). n p ∼ 3 where the m is the proton-WIMP reduced mass, and A is the atomic mass of the target rp nucleus. Information about halo structure is encoded into the quantity T(Q,t), ∞ f (v) d T(Q,t) = dv, (8) v Zvmin where f (v) is the distribution of WIMP speeds relative to the detector, and where v d min ≡ (Qm /2m2)1/2 represents the minimum velocity that can result in a recoil energy Q. To N r determine T(Q,t), we must have a model for the velocity structure of the halo. 2.1 The isothermal halo In what follows we will consider dark matter detection experiments without directional ca- pabilities; such experiments are sensitive only to the WIMP flux integrated over the entire sky. The most frequently employed background velocity distribution is that of a simple isothermal sphere. In such a model, the galactic WIMP speeds with respect to the halo obey a Maxwellian distribution with a velocity dispersion σ , h 3 3/2 3v2 f (v)dv = 4π v2exp (9) h 2πσ2! −2σ2! h h where v is the WIMP velocity relative to the Galactic halo, and where we have performed an angular integration over the entire sky. We take the velocity dispersion of our local halo to be σ = 270 km/s. This is the local velocity distribution; it is expected to vary with spatial h position throughout the Galaxy. As pointed out by Drukier, Freese, and Spergel [1], and studied (in the context of an isothermal halo) by Freese, Frieman, and Gould [2], Earth-bound observers of the dark halo will see a different, time dependent velocity distribution as a result of the relative motion of the Earth with respect to the Galaxy. To take this into account, we estimate the velocity of the Earth with respect to the Galactic halo. Neglecting the ellipticity of the Earth’s orbit and the non-uniform motion of the Sun in right ascension, we may write the speed of the Earth with respect to the the dark WIMP halo as v = v2 +v2 +2v v cos(2πt φ ) (10) eh es sh es sh − h q where the phase φ = 2.61 0.02 corresponds to June 2nd 1.3 days (relative to φ = 0 h ± ± on January 1). The speed of the Earth with respect to the Sun is v = 29.8 km/s, and es the speed of the Sun with respect to the halo is v = 233 km/s. More precise expressions es for the motion of the Earth and Sun may be found in [4, 5]. Translating the distribution in Eq. 9 into the distribution as seen by an earthbound detector, f (v ), and integrating the d eh resulting distribution in Eq. 8, one obtains 4 1 √2(v +v (t)) √2(v v (t)) min eh min eh T (Q,t) = erf erf − . (11) h 2veh(t) " √3σh !− √3σh !# With T (Q,t), one can now compute the recoil energy spectrum associated with the isother- h mal halo, as well as the annual modulation of that spectrum that results from the motion of the Earth around the sun. For the case of an isothermal halo, one generally expects that, above a critical recoil energy, the annual modulation will be in phase with the motion of the Earth with respect to the halo, peaking near June 2nd. In the following section, we will demonstrate the dependence of the phase of the modulation on the observed recoil energy. 3 The Phase Reversal at Low Recoil Energies The annual modulation of the WIMP interaction rate [1, 2], and in particular the phase reversal at low recoil energies [3, 5, 4], has been studied in detail by several authors. Here Fig. 1 plotstheexpected differential event rateforaWIMP ofmass65 GeVatrecoilenergies of 35 keV and 10 keV. We note that there is a 180 degree phase shift between the two signals. The origin of this effect may be understood by considering a first order approximation to Eq. 11. We note that the velocity of the Earth with respect to the WIMP halo, v , given eh in Eq. 10 may be written, for v v , as es eh ≪ v η +ǫ (t). eh h h ∼ where we define, η = (v2 +v2 )1/2, (12) h es sh v v es sh ǫ (t) = cos(2πt φ ). (13) h (v2 +v2 )1/2 − h es sh Using this approximation, We may expand this expression about ǫ(t) = 0, an approximation that is well justified because ǫ (t)/η 1 for all t. With the following definitions, h h ≪ √2(v η ) ± min h X = erf ± , (14) h √3σ ! h 2(v η )2 ± min h E = exp ± , (15) h − 3σ2 ! h we find that, to first order in ǫ, 5 x 10−5 3.2 1 35 keV recoil energy − V ke 3.1 1 − y a 1 d 3 − g k nts 2.9 u o c 2.8 Jan Feb Mar April May Jun Jul Aug Sep Oct Nov Dec Time x 10−4 1.16 1 10 keV recoil energy − V ke 1.15 1 − y a d 1.14 1 − g k nts 1.13 u o c 1.12 Jan Feb Mar April May Jun Jul Aug Sep Oct Nov Dec Time Figure 1: The annual modulation of the WIMP differential detection rate for a WIMP with a mass of 65 GeV, in a 73Ge detector, for recoil energies of 35 and 10 keV. Note the 180 degree phase shift between the two signals. 1 X+ X− 8 E+ +E− X+ X− T (Q,t) h − h +ǫ (t) h h h − h . (16) h ≈ 2  ηh ! h s3π σhηh − ηh2     We see that the differential detection rate will be modulated by ǫ(t) cos(2πt φ ), as h ∝ − expected. We are particularly interested in the behavior of the amplitude of the oscillation. If we rewrite T (Q,t) of Eq. 16 as h T(Q,t) = B +Acos(2πωt φ ), (17) h − we identify the amplitude of oscillation as v v 8 E+ +E− X+ X− A(Q) = es sh h h h − h . (18) (v2 +v2 )1/2 s3π σ η − η2  es sh h h h   For A(Q) > 0, the modulation of the WIMP differential event rate is in phase with the motion of the Earth with respect to the halo, and will peak at June 2nd as usual. For A(Q) < 0, however, the modulation of the event rate moves 180 degrees out of phase with terrestrial motion, peaking in December rather than June. Values of A(Q) are plotted in 6 Fig. 2 for 0 Q 100 keV, for a WIMP mass of 30, 50 and 70 GeV in a 73Ge detector. ≤ ≤ In general, for a WIMP mass m and nuclear mass m there will exist a nonzero, crossover χ N recoil energy, Q , at which A(Q ) = 0, and below which A(Q < Q ) < 0. That is, for c c c Q < Q the WIMP interaction rate will peak in December rather than June. We emphasize c that these approximations, while illustrative, give rise to errors of a few percent; in practice, Eq. 11 should be used for all computations. x 10−5 1 30 GeV 50 GeV 70 GeV 0.5 0 ) N m m,X−0.5 Q, ( A −1 −1.5 −2 0 10 20 30 40 50 60 70 80 90 100 Recoil Energy Q (keV) Figure 2: The amplitude of modulation A(Q) as a function of recoil energy Q for WIMP masses of 30, 50, and 70 GeV in a 73Ge based detector. For each WIMP mass there exists a particular recoil energy Q , at which the first order annual modulation of the WIMP c differentialevent ratevanishes. ForQ < Q , wefindthatA(Q < Q ) < 0,andthedifferential c c WIMP interaction rate peaks in December, rather than June. The crossover recoil energy, Q is a function of the mass of the WIMP, as well as the mass c of the target nuclei in the detector. We can estimate the critical recoil energy, below which we observe the phase reversal. Fig. 3 plots the crossover recoil energy as a function of WIMP mass, for NaI, 73Ge, 131Xe, CaWO and Al O based WIMP direct detection experiments. 4 2 3 7 Collaboration Material Q (keV) M (kg) thresh D CDMS-II 73Ge 103 5 CRESST-II CaWO 1 10 4 ∼ DAMA NaI 22 100 EDELWEISS-II 73Ge 20 38 GENIUS-TF 73Ge 1 40 XENON LXe 4 1000 ZEPLIN-II LXe 10 40 ZEPLIN-IV LXe 4 1000 Table 1: Properties of various dark matter detection experiments including: the detector material, the detector energy recoil threshold, Q , and the total detector mass, M . In thresh D the case of future experiments, the quantities are projected. 3.1 Placing an Upper Bound on the WIMP Mass The observation of an annual modulation signal can be used to constrain the WIMP mass. For example, if a direct-detection experiment observes an annually modulated signal peaking in June at an energy recoil bin Q , we may place an upper limit on the WIMP mass because 0 we know that the crossover energy Q must have been less than Q , or a phase reversal would c 0 have been observed. Because the crossover energy is a monotonically increasing function of the WIMP mass, this places an upper limit on the WIMP mass. We will describe the upper bounds one can obtain in a variety of dark matter detectors. Of course, the experiments must have sufficient exposure to see the annual modulation in order to obtain these bounds. As a concrete example, consider the DAMA experiment, which reports an annual modu- lation in the event rate in its NaI detector in the 22-66 keV recoil energy bins [24]. Although other experiments rule out much of this region of parameter space [27], if one were to believe the DAMA results, the fact that annual modulation in DAMA peaks in June in this energy bin implies, from Fig. 3, that the maximum WIMP mass consistent with the DAMA data is m 103 GeV. ≤ This same logic may be applied to other dark matter experimental collaborations that may be sensitive to the phase reversal. In particular, in this paper we will consider the various versions of the CRESST [25], CDMS [26], EDELWEISS [27], XENON [29], GENIUS [30], and ZEPLIN [31] experiments. Table I collects the detector target materials, mass, and recoil energy thresholds associated with these experiments. In Table II, we present the maximum WIMP mass consistent with an observation of a June-peaking annual modulation in the lowest accessible recoil energy bins of the respective experiments, as determined using the arguments in the preceding paragraphs. 8 Collaboration Maximum allowed WIMP mass (GeV) CDMS-II 48 CRESST-II 6 DAMA 103 EDELWEISS-II 97 GENIUS-TF 10 XENON 29 ZEPLIN-II 52 ZEPLIN-IV 29 Table 2: The maximum WIMP masses consistent with observing a June-peaking annual modulation signal in the lowest accessible energy recoil bins of the respective dark matter experiments. 3.2 Detecting the Phase Reversal For WIMP direct detection experiments with sufficient energy resolution and sufficient ex- posure, it may be possible to determine the crossover recoil energy Q itself, thereby making c an independent measurement of the WIMP mass. If one determines the recoil energy at which the modulation phase changes sign, Fig. 3 may be used to estimate the WIMP mass. We now explore whether or not the phase reversal itself could be practically observed in the context of current and near-future direct detection experiments. In order to extract a small annual signal from the background, we require a large signal to noise ratio so that the fluctuations are small compared to the desired signal. This problem, in the context of WIMP direct detection experiments, has been discussed by Hasenbalg [3]. Because the count rate for direct detection experiments is so low, extended exposure times will be required to reliably detect a small modulation. We write the total signal as a function of time as, Qf dR S(t) = dQ (19) dQ ZQi = S (Q ,Q )+S (Q ,Q )cos(ωt)+ (S2 ) (20) 0 i f m i f O m where S is the amplitude of modulation, S is the unmodulated count rate, and dR/dQ is m 0 defined as in Eq. 2. These quantities will depend on the limits of integration, Q and Q , i f and may be be written in terms of the differential event rate evaluated at its maximum in June, S , and the rate at its minimum, in December, S , J D 9 70 131 Xe 60 keV)50 CaWO4 Q (c NaI y g er40 n E oil c Re30 er 73Ge v o s os20 Cr 10 Al O 2 3 0 0 50 100 150 200 250 300 350 400 450 500 WIMP Mass (GeV) Figure 3: The crossover recoil energy Q as a function of WIMP mass m for NaI, 73Ge, c χ 131Xe, CaWO , and Al O detectors, assuming a halo velocity dispersion of 270 km/s. The 4 2 3 observation of an annual modulation signal peaking in June at a given recoil energy neces- sarily places an upper limit on the allowed WIMP mass. For example, the observation of a June-peaking annual modulation in the 22 keV energy recoil bin implies an upper limit on the WIMP mass of about 103 GeV for a NaI based experiment. 1 1 S = [S S ] S = [S +S ]. (21) m J D 0 J D 2 − 2 If S S , the theoretical signal to noise ratio may be written as, m 0 ≪ S (Q ,Q ) (s/n) m i f √MT. (22) ≡ S (Q ,Q ) 0 i f q where M is the total detector mass and T is the total exposure time. As a reasonable criterion for distinguishing the modulation signal from the noise, we require that the s be at least 2σ greater than the statistical uncertainty. This amounts to requiring (s/n) = 2. For low count rates, large MT will be required to achieve this minimum signal to noise ratio. In order to observe a phase reversal in a particular WIMP detector with a given WIMP mass, an annual modulation peaking in December must be detected in the energy range Q to Q , and a modulation peaking in June must be observed in the energy range Q to thresh c c 10