Aspects of spin-dependent dark matter search V.A. Bednyakov Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, 141980 Dubna, Russia; E-mail: [email protected] The Weakly Interacting Massive Particle (WIMP) is the main candidate for the relic dark matter. A set of exclusion curves currently obtained for cross sections of the spin-dependent WIMP-proton and WIMP-neutron interaction is given. A two-orders-of-magnitude improvement of the sensitivity of the dark matter experiment is needed to reach the SUSY predictions for relic neutralinos. It is noted that near-future experiments with the high-spin isotope 73Ge can yield a new important constraint on the neutralino-neutron effective coupling and the SUSY parameter space. A. Introduction Nowadays the main efforts in the direct dark matter search experiments are concentrated in the field of so-called spin-independent (or scalar) interaction of a dark matter particle or the Weakly Interacting Massive Particle (WIMP), with a nucleus. The lightest supersymmetric (SUSY) partilce (LSP) neutralino is assumed here as a best WIMP candidate. It is believed that this spin-independent (SI) interaction of dark matter (DM) particles with nuclei makes a dominant contribution to the expected event rate of detection of these particles. The reason is the strong (proportional to the squared mass of the target nucleus) enhancement of SI WIMP-nucleus interaction. The results currently obtained in the field are usually presented in the form of exclusion curves (see for example Fig. 1). For the fixed mass of the WIMP the values of the cross section due to scalar elastic WIMP-nucleon interaction located above these curves are already excluded experimentally. There is also the DAMA closed contour which corresponds to the first claim for evidence for the dark matter signal due to the positive annual modulation effect [5]. In the paper we consider some aspects of the spin-dependent (or axial-vector) interaction of the DM WIMP with nuclei. There are at least three reasons to think that this spin-dependent (SD) interaction could also be very important. First, contrary to the only one constraint for SUSY models available from scalar WIMP-nucleus interaction, the spin WIMP-nucleus interaction supplies us with two such constraints (see for example [6] and formulas below). Second, one can notice [1, 7] that even with a very accurate DM detector (say, with sensitivity 10−5 events/day/kg) which is sensitive only to the WIMP-nucleus scalar interaction (with spinless target nuclei) one can, in principle, miss a DM signal. To safely avoid such a situation one should have a spin-sensitive DM detector, i.e. a detector with spin-non-zero target nuclei. Finally, there is a complicated (and theoretically very interesting) nucleus spin structure, which possesses the so-called long q-tail form-factor behavior for heavy targets and heavy WIMP. Therefore, the SD efficiency to detect a DM arXiv:hep-ph/0310041v1 3 Oct 2003 signal is much higher than the SI efficiency, especially for the heavy target nucleus and WIMP masses [8]. B. Zero Momentum Transfer A dark matter event is elastic scattering of a relic neutralino χ (orχ ˜) from a target nucleus A producing a nuclear recoil ER which can be detected by a suitable detector. The differential event rate in respect to the recoil energy is the subject of experimental measurements. The rate depends on the distribution of the relic neutralinos in the solar vicinity f(v) and the cross section of neutralino-nucleus elastic scattering 2 GENIUS-TF , pb (40 kg Ge) DAMA evidence 73 scalar (100 kg NaI) HDMS prototype HDMS (0.2 kg Ge) NAIAD, DRIFT W-N σ ZEPLIN GENIUS-TF EDELWEISS II CRESST CRESST GENIUS CDMS (100 kg nat Ge) (6.8 kg Ge BLIP) GENIUS CDMS Heidelberg-Moscow (2.5 kg enriched 76Ge) MWIMP , GeV FIG. 1: WIMP-nucleon cross section limits in pb for scalar (spin-independent) interactions as a function of the WIMP mass in GeV. Shown are contour lines of the present experimental limits (solid lines) and of projected experiments (dashed lines). Also shown is the region of evidence published by DAMA. The theoretical expectations are shown by scatter plots (circles and triangles are from [1, 2]) and by grey closed region [3]. From [4]. [9, 10, 11, 12, 13, 14, 15, 16]. The differential event rate per unit mass of the target material has the form vmax dR ρχ dσ 2 = N dvf(v)v 2 (v, q ). (1) dER mχ Zvmin dq 2 −6 The nuclear recoil energy E = q /(2MA) is typically about 10 mχ and N = /A is the number density R N of target nuclei, where is the Avogadro number and A is the atomic mass of the nuclei with mass MA. N The neutralino-nucleus elastic scattering cross section for spin-non-zero (J = 0) nuclei contains coherent 6 (spin-independent, or SI) and axial (spin-dependent, or SD) terms [8, 17, 18]: dσA 2 SA (q2) SA (q2) (v, q2) = |M| = SD + SI dq2 π vP2(2J + 1) v2(2J + 1) v2(2J + 1) A A σSD(0) 2 2 σSI(0) 2 2 = 2 2 FSD(q )+ 2 2 FSI(q ). (2) 4µAv 4µAv 3 The normalized non-zero-momentum-transfer nuclear form-factors A 2 2 2 SSD,SI(q ) 2 FSD,SI(q )= A (FSD,SI(0) = 1), (3) SSD,SI(0) are defined via nuclear structure functions [8, 17, 18] A 2 2 S (q) = J L(q) J J (q) J , (4) SI |h ||C || i| ≃ |h ||C0 || i| LXeven SA (q) = N el5(q) N 2 + N 5 (q) N 2 . (5) SD |h ||TL || i| |h ||LL || i| LXodd The transverse electric el5(q) and longitudinal 5(q) multipole projections of the axial vector current T L operator, scalar function L(q) are given in the form C i el5 1 a0 + a1τ3 (q) = √LML,L (q~ri)+ √L +1ML,L− (q~ri) , TL √ 2 − +1 1 2L +1 Xi h i 2 i 5 1 a0 a1mπτ3 (q) = + √L +1ML,L (q~ri)+ √LML,L− (q~ri) , LL √ 2 2(q2 + m2 ) +1 1 2L +1 Xi π h i L(q) = c jL(qri)YL(ˆri), (q)= c j (qri)Y (ˆri), C 0 C0 0 0 0 i, nucleonsX Xi L where a = a a (see (10)) and ML,L′ (q~ri) = jL′ (qri)[YL′ (ˆri)~σi] [8, 17, 18]. The nuclear SD and SI 0,1 n ± p cross sections at q = 0 (in (2)) have the forms 2 2 A 4µA SSI(0) µA 2 p σSI(0) = = 2 A σSI(0), (6) (2J + 1) µp 2 2 2 A 4µASSD(0) 4µA (J + 1) A A σSD(0) = = ap Sp + an Sn (7) (2J + 1) π J n h i h io µ2 (J + 1) 2 A p SA n SA = 2 σSD(0) p + sign(ap an) σSD(0) n . (8) µp,n 3 J q h i q h i mχMA Here µA = is the reduced neutralino-nucleus mass. The zero-momentum-transfer proton and mχ + MA neutron SI and SD cross sections 2 p µp 2 (p,n) (p,n) σ (0) = 4 c , c c = qf ; (9) SI π 0 0 ≡ 0 C q Xq 2 p,n µp,n 2 (p) (n) σ (0) = 12 a an = q∆ , ap = q∆ (10) SD π p,n A q A q Xq Xq depend on the effective neutralino-quark scalar q and axial-vector q couplings from the effective Lagrangian C A µ = ( q χγ¯ µγ χ qγ¯ γ q + q χχ¯ qq¯ ) + ... (11) Leff A · 5 · 5 C · · Xq (p,n) (p,n) (p,n) and on the spin (∆q ) and mass (fq ) structure of nucleons. The factors ∆q parametrize the quark (n,p) µ µ spin content of the nucleon and are defined by the relation 2∆ s p,s ψ¯qγ γ ψq p,s . The total q ≡ h | 5 | i(p,n) nuclear spin (proton, neutron) operator is defined as follows A Sp,n = sp,n(i), (12) Xi 4 where i runs over all nucleons. Further the convention is used that all angular momentum operators are evaluated in their z-projection in the maximal MJ state, e.g. S N S N J, MJ = J Sz J, MJ = J . (13) h i ≡ h | | i ≡ h | | i Therefore S is the spin of the proton (neutron) averaged over all nucleons in the nucleus A. The cross h p(n)i sections at zero momentum transfer show strong dependence on the nuclear structure of the ground state [19, 20, 21]. The relic neutralinos in the halo of our Galaxy have a mean velocity of v 300 km/s=10−3c. When the h i≃ product q R 1, where R is the nuclear radius and q =2µAv is the maximum momentum transfer in max ≪ max theχA ˜ scattering, the matrix element for the spin-dependentχA ˜ scattering reduces to a very simple form (zero momentum transfer limit) [20, 21]: = C N apSp + anSn N sχ = CΛ N J N sχ. (14) M h | | i · ˜ h | | i · ˜ Here sχ is the spin of the neutralino, and N apSp + anSn N N (apSp + anSn) J N Λ= h | | i = h | · | i. (15) N J N J(J + 1) h | | i It is seen that the χ couples to the spin carried by the protons and the neutrons. The normalization C involves the coupling constants, masses of the exchanged bosons and various LSP mixing parameters that have no effect upon the nuclear matrix element [22].