Aspects of Spin-Dependent Dark Matter Search*

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Aspects of Spin-Dependent Dark Matter Search* Physics of Atomic Nuclei, Vol. 67, No. 11, 2004, pp. 1931–1941. From Yadernaya Fizika, Vol. 67, No. 11, 2004, pp. 1957–1966. Original English Text Copyright c 2004 by Bednyakov. DARK MATTER Aspects of Spin-Dependent Dark Matter Search* V. A. Bednyakov ** Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received November 4, 2003 Abstract—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 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 new experiments with the high-spin isotope 73Ge can yield a new important constraint on the neutralino– neutron effective coupling and the SUSY parameter space. c 2004 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION 10−5 events/d/kg)which is sensitive only to the WIMP–nucleus scalar interaction (with spinless Nowadays, the main efforts in the direct dark mat- target nuclei)one can, in principle, miss a DM signal. ter search experiments are concentrated in the field of To safely avoid such a situation, one should have so-called spin-independent (or scalar)interaction of a a spin-sensitive DM detector, i.e., a detector with dark matter particle or the Weakly Interacting Mas- spin-nonzero target nuclei. Finally, there is a com- sive Particle (WIMP)with a nucleus. The lightest plicated (and theoretically very interesting)nucleus supersymmetric (SUSY)particle (LSP)neutralino spin structure, which possesses the so-called long q- is assumed here as the best WIMP candidate. It is tail form factor behavior for heavy targets and heavy believed that this spin-independent (SI)interaction WIMP. Therefore, the SD efficiency needed to detect of dark matter (DM)particles with nuclei makes a a DM signal is much higher than the SI efficiency, dominant contribution to the expected event rate of especially for the heavy target nucleus and WIMP detection of these particles. The reason for this is the masses [8]. strong (proportional to the squared mass of the target nucleus)enhancement of SI WIMP –nucleus inter- action. The results currently obtained in the field are 2. ZERO MOMENTUM TRANSFER usually presented in the form of exclusion curves (see, A dark matter event is elastic scattering of a relic for example, Fig. 1). For the fixed mass of the WIMP, neutralino χ (or χ˜)from a target nucleus A producing the values of the cross section due to scalar elas- a nuclear recoil E that can be detected by a suitable tic WIMP–nucleon interaction located above these R detector. The differential event rate, with respect to curves are already excluded experimentally. There is the recoil energy, is the subject of experimental mea- also the DAMA closed contour, which corresponds to surements. The rate depends on the distribution of the first claim for evidence for the dark matter signal the relic neutralinos in the solar vicinity f(v) and the due to the positive annual modulation effect [5]. cross section of neutralino–nucleus elastic scatter- In this paper we consider some aspects of the ing [9–16]. The differential event rate per unit mass spin-dependent (or axial-vector)interaction of the of the target material has the form DM WIMP with nuclei. There are at least three vmax reasons to think that this spin-dependent (SD)in- dR ρχ dσ 2 teraction could also be very important. First, contrary = N dvf(v)v 2 (v,q ). (1) dER mχ dq to the only constraint for SUSY models available vmin from scalar WIMP–nucleus interaction, the spin 2 WIMP–nucleus interaction supplies us with two The nuclear recoil energy ER = q /(2MA) is typically −6 N such constraints (see, for example, [6] and formulas about 10 mχ,andN = /A is the number den- below). Second, one can notice [1, 7] that even with sity of the target nuclei, where N is the Avogadro a very accurate DM detector (say, with sensitivity number and A is the atomic mass of the nuclei with mass MA. The neutralino–nucleus elastic scattering ∗This article was submitted by the author in English. cross section for spin-nonzero (J =0 )nuclei con- **e-mail: [email protected] tains coherent (spin-independent, or SI)and axial 1063-7788/04/6711-1931$26.00 c 2004 MAIK “Nauka/Interperiodica” 1932 BEDNYAKOV σ , pb SI GENIUS-TF DAMA evidence (40 kg Ge) (100 kg NaI) HDMS prototype HDMS (0.2 kg 73Ge) NAIAD, DRIFT ZEPLIN 10–6 GENIUS-TF EDELWEISS II CRESST CRESST GENIUS 10–8 (100 kg nat. Ge) CDMS (6.8 kg Ge BLIP) GENIUS 10–10 CDMS 10–12 Heidelberg–Moscow (2.5 kg enriched 76Ge) 102 103 MWIMP, GeV Fig. 1. WIMP–nucleon cross section limits for scalar (spin-independent)interactions as a function of the WIMP mass. Shown are contour lines of the present experimental limits (solid curve)and of projected experiments (dashed curve).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 the grey closed region [3]. (From [4].) 2 (spin-dependent, or SD)terms [8, 17, 18]: |J||C0(q)||J| , A |M|2 A 2 A | ||T el5 || |2 dσ 2 SSD(q ) SSD(q)= N L (q) N (5) 2 (v,q )= 2 = 2 (2) dq πv (2J +1) v (2J +1) L odd A 2 A A | ||L5 || |2 S (q ) σ (0) σ (0) + N L(q) N . + SI = SD F 2 (q2)+ SI F 2 (q2). v2(2J +1) 4µ2 v2 SD 4µ2 v2 SI A A The transverse electric T el5(q) and longitudinal L5(q) The normalized nonzero-momentum-transfer nu- multipole projections of the axial vector current oper- clear form factors ator, scalar function CL(q) are given in the form SA (q2) F 2 (q2)= SD,SI (3) 1 a + a τ i SD,SI SA (0) T el5(q)=√ 0 1 3 SD,SI L 2L +1 2 2 i (FSD,SI(0) = 1) √ √ × − LML,L+1(qri)+ L +1ML,L−1(qri) , are defined via nuclear structure functions [8, 17, 18] 2 i A 2 5 1 a0 a1mπτ3 S (q)= |J||CL(q)||J| (4) L (q)=√ + SI L 2L +1 2 2(q2 + m2 ) L even i π PHYSICS OF ATOMIC NUCLEI Vol. 67 No. 11 2004 ASPECTS OF SPIN-DEPENDENT DARK MATTER SEARCH 1933 √ √ × L +1ML,L+1(qri)+ LML,L−1(qri) , where i runs over all nucleons. Further, the conven- tion is used that all angular momentum operators are C L(q)= c0jL(qri)YL(ˆri), evaluated in their z projection in the maximal MJ i,nucleons state, e.g., ≡ | | ≡ | | C (q)= c j (qr )Y (ˆr ), S N S N J, MJ = J Sz J, MJ = J . 0 0 0 i 0 i (13) i ± Therefore, Sp(n) is the spin of the proton (neutron) where a0,1 = an ap (see (10)) and ML,L (qri)= averaged over all nucleons in the nucleus A.Thecross j (qr )[Y (ˆr )σ ]L [8, 17, 18]. The nuclear SD and L i L i i sections at zero momentum transfer show strong SI cross sections at q =0(in (2)) have the forms dependence on the nuclear structure of the ground 2 2 A 4µASSI(0) µA 2 p state [19–21]. σSI(0) = = 2 A σSI(0), (6) (2J +1) µp The relic neutralinos in the halo of our Galaxy have a mean velocity of v 300 km/s =10−3c.When 4µ2 S (0) 2 A A SD 4µA (J +1) the product qmaxR 1,whereR is the nuclear ra- σSD(0) = = (7) (2J +1) π J dius and qmax =2µAv is the maximum momentum 2 transfer in the χA˜ scattering, the matrix element for 2 µ (J +1) × A A A the spin-dependent χA˜ scattering reduces to a very ap Sp + an Sn = 2 µp,n 3J simple form (zero momentum transfer limit)[20, 21]: 2 M | | · × p A n A = C N apSp + anSn N sχ˜ (14) σSD(0) Sp + sgn(apan) σSD(0) Sn . = CΛN|J|N·sχ˜. Here, Here, sχ is the spin of the neutralino, and µ = m M /(m + M ) (8) A χ A χ A N|a S + a S |N Λ= p p n n (15) is the reduced neutralino–nucleus mass. The zero- N|J|N momentum-transfer proton and neutron SI and SD N|(a S + a S ) · J|N cross sections = p p n n . 2 J(J +1) p µp 2 σSI(0) = 4 c0, (9) π It is seen that the χ couples to the spin carried by c ≡ c(p,n) = C f (p,n); the protons and the neutrons. The normalization C 0 0 q q involves the coupling constants, the masses of the q exchanged bosons, and various LSP mixing param- µ2 eters, which have no effect upon the nuclear matrix σp,n(0) = 12 p,n a2 , SD p,n (10) element [22]. In the limit of zero momentum transfer π A (p) A (n) q =0, the spin structure function (5)reduces to an = q∆q ,ap = q∆q 2J +1 q q SA (0) = Λ2J(J +1). (16) SD π depend on the effective neutralino–quark scalar Cq and axial-vector Aq couplings from the effective La- Perhaps, the first model to estimate the spin con- grangian tent in the nucleus for the dark matter search was the independent single-particle shell model (ISPSM) µ Leff = (Aq · χγ¯ µγ5χ · qγ¯ γ5q + Cq · χχ¯ · qq¯ )+..
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