Searching for Dark Matter- Theoretical Rates and Exclusion Plots due to the Spin
aa r X i v : . [ h e p - ph ] D ec SEARCHING FOR DARK MATTER-THEORETICAL RATES AND EXCLUSION PLOTS DUE TO THE SPIN
J. D. Vergados
Theoretical Physics Division, University of Ioannina,Ioannina, Gr 451 10, GreeceandUniversity of Tuebingen, Tuebingen, Germany. (Dated: May 28, 2018)The recent WMAP data have confirmed that exotic dark matter together with the vacuum energy(cosmological constant) dominate in the flat Universe. The nature of the dark matter constituentscannot be determined till they are directly detected. Recent developments in particle physics providea number of candidates as constituents of dark matter, called Weakly Interacting Massive Particles(WIMPs). Since these interact weakly and are of low energy, they cannot excite the target and canonly be detected via measuring the recoiling nucleus. For all WIMPs, including the most popularcandidate, the lightest supersymmetric particle (LSP), the relevant cross sections arise out of thefollowing mechanisms: i) The coherent mode, due to the scalar interaction. ii) The charge coherentmode, with only proton contribution, as in the recent case of secluded dark matter scenario andiii) The spin contribution arising from the axial current. In this paper we will focus on the spincontribution, which maybe important, especially for light targets.
PACS numbers: 95.35.+d, 12.60.Jv
INTRODUCTION
Combining the recent WMAP data [1] with other experiments one finds that most of the matter in the Universe iscold dark matter (CDM):Ω b = 0 . ± . , Ω CDM = 0 . ± . , Ω Λ = 0 . ± . et al [15]for an update).Spin induced cross sections also arize in the case Kaluza-Klein (K-K) WIMPs in models with Universal Extra Dimen-sions (UED) [16]. This occurs regardless of whether the WIMP is a K-K boson or a K-K neutrino. They can also arisein technicolor theories [17]. In the Ultra Minimal Walking Technicolor model [18, 19, 20] there exist singlet compositeMajorana fermionic states. These, taken as dark matter candidates, can lead to spin induced cross sections.Knowledge of the spin induced nucleon cross section is very important since, for some special targets, it may leadto transitions to excited nuclear states, which provide the attractive signature of detecting the de-excitation γ raysin or without coincidence with the recoiling nucleus [21],[22]. Furthermore it may dominate in light systems like Heand F, which offer some attractive advantages [23],[24].In light of the above it is clear that the spin mechanism needs be considered. In this article we will discuss thetheoretical ingredients needed to obtain the WIMP-nuclear spin induced cross sections. Then we will give expressionsor and calculate the event rates, both modulated and unmodulated, in terms of the elementary proton ( σ p ) or neutron( σ n ) cross sections. After that we will provide exlusion plots in the ( σ p , σ n ) plane in terms of parameters relevant tothe experiments, for various targets of experimental interest. Our results will be presented in a way that will makethem useful in the analysis of the data of the odd mass targets. THE ESSENTIAL THEORETICAL INGREDIENTS OF DIRECT DETECTION.
Even though there exists firm indirect evidence for a halo of dark matter in galaxies from the observed rotationalcurves, it is essential to directly detect such matter. Such a direct detection, among other things, may also unravelthe nature of the constituent of cold dark matter, namely the Weakly Interacting Massive Particle (WIMP). Thepossibility of such detection, however, depends on the nature of its constituents. Our main conclusions apply to allheavy WIMPs. Since the WIMP is expected to be very massive, m χ ≥ GeV , and extremely non relativistic withaverage kinetic energy T ≈ KeV ( m χ / GeV ), it can be directly detected mainly via the recoiling of a nucleus(A,Z) in elastic scattering. The event rate for such a process can be computed from the following ingredients:1. An effective Lagrangian at the elementary particle (quark) level obtained in the framework of supersymmetryas described , e.g., in Refs [9, 25]. An analogous procedure can be found in the case of K-K WIMPs in UniversalExtra Dimension (UED) models [16] and technicolor theories [17].2. A well defined procedure for transforming the amplitude obtained using the above mentioned effective Lagrangianfrom the quark to the nucleon level where This step is not trivial, since the obtained results depend cruciallyon the content of the nucleon in quarks other than u and d. This is particularly true for the scalar couplings,which are proportional to the quark masses [26, 26, 27, 28] as well as the isoscalar axial coupling.3. Nuclear matrix elements.These must be obtained with as reliable as possible many body nuclear wave functions. Fortunately in the moststudied case of the scalar coupling the situation is quite simple, since then one needs only the nuclear formfactor. Some progress has also been made in obtaining reliable static spin matrix elements and spin responsefunctions [29, 30, 31, 32]4. A velocity distribution for WIMPsIn this article we will follow the standard practice and assume a M-B distribution, but other perhaps morerealistic velocity distributions have also recently been considered [33, 34]Since the obtained rates are very low, one would like to be able to exploit the modulation of the event rates due tothe earth’s revolution around the sun [35, 36, 37, 38]
THE WIMP NUCLEUS CROSS SECTIONS
The standard (non directional) differential rate can be written as dR = ρ (0) m χ mAm N dσ ( u, υ ) | υ | , (1)where m is the detector mass, ρ (0) = 0 . GeV /cm is the WIMP density in our vicinity, υ its velocity and m χ itsmass and dσ ( u, υ ) is given by dσ ( u, υ ) == du µ r bυ ) (cid:2) ¯Σ S F ( u ) + ¯Σ spin F ( u ) (cid:3) , (2)where u is the energy transfer Q in dimensionless units given by u = QQ , Q = [ m p Ab ] − = 40 A − / M eV, (3)with b is the nuclear (harmonic oscillator) size parameter. F ( u ) is the nuclear form factor and F ( u ) is the spinresponse function associated with the isovector channel. The scalar cross section is given by:¯Σ S ≈ σ SN,χ (cid:18) µ r µ r ( p ) (cid:19) A . (4)2 Q → keV FIG. 1: The spin response functions F ( Q ), F ( Q ) and F ( Q ) in the case of the target F as a function of theenergy transfer. In the region of interest for dark matter searches they are not distinguishable. σ SN,χ is the WIMP-nucleon scalar cross section. Note that, since the heavy quarks dominate, the isovector contributionis negligible, i.e. the proton and nucleon cross sections are the same. The spin Cross section is given by:¯Σ spin = ( µ r µ r ( p ) ) σ spinp,χ ζ spin , ζ spin = 13(1 + f A f A ) S ( u ) , (5) S ( u ) = "(cid:18) f A f A Ω (0) (cid:19) F ( u ) F ( u ) + 2 f A f A Ω (0)Ω (0) F ( u ) F ( u ) + Ω (0)) . (6)The spin response functions F ij , properly normalized to unity at momentum transfer zero, in the energy transfers ofinterest are almost the same for all isospin channels i, j ; i, j = 0 , F we get: F ( u ) = e − u (cid:18) . u − . u + 0 . u − u (cid:19) , (7) F = e − u (cid:18) . u − . u + 0 . u − u (cid:19) , (8) F = e − u (cid:18) . u − . u + 0 . u − u (cid:19) , (9)Thus they are indistinguishable (see Fig. 1) for the energy transfers of interest. Hence S ( u ) ≈ S (0) = (cid:18) f A f A Ω (0) + Ω (0) (cid:19) . (10)The nuclear matrix elements Ω (0) (Ω (0)) associated with the isovector (isoscalar) components are normalized sothat, in the case of the proton, they yield ζ spin = 1 at u = 0.The couplings f A ( f A ) are obtained by multiplying the corresponding elementary amplitudes obtained at the quarklevel by suitable renormalization factors g A and g A given in terms of the quantities ∆ q given by Ellis [39] g A = ∆ u + ∆ d + ∆ s = 0 . − . − .
15 = 0 . , g A = ∆ u − ∆ d = 1 .
26 (11)Thus, barring very unusual circumstances at the quark level, the isovector component is expected to be dominant. Itis for this reason that we started our discussion in the isospin basis.Heavy nuclei, however, are theoretically described in terms of protons and neutrons and the experiments are alsoanalyzed this way. So will present our results in this basis. The proton and neutron cross section are given by: σ spinp,χ = 3 σ | f A + f A | = 3 σ | a p | , σ spinn,χ = 3 σ | f A − f A | = 3 σ | a n | (12)3ABLE I: The static spin matrix elements for various nuclei. For He see Moulin, Mayet and Santos [41]. For theother light nuclei the calculations are from DIVARI [29]. For Ge and
I the results presented are from Ressel etal [30] (*) and the Finish group et al [31] (**). For
Pb they were obtained by the Ioannina team (+). [40], [32]. He F Si Na Ge I ∗ I ∗∗ Pb + Ω (0) 1.244 1.616 0.455 0.691 1.075 1.815 1.220 0.552Ω (0) -1.527 1.675 -0.461 0.588 -1.003 1.105 1.230 -0.480Ω p (0) -0.141 1.646 -0.003 0.640 0.036 1.460 1.225 0.036Ω n (0) 1.386 -0.030 0.459 0.051 1.040 0.355 -0.005 0.516 µ th µ exp µ th ( spin ) µ exp with a p and a n are the proton and neutron spin amplitudes, which, of course, depend on the model. In the case ofthe LSP [25] σ = 12 π ( G F m p ) = 0 . × − cm = 0 . × − pb. In extracting limits on the nucleon cross sections from the data we will find it convenient to write:¯Σ spin = ( µ r µ r ( p ) ) σ spinnuc , σ spinnuc = 13 | Ω p √ σ p + Ω n √ σ n e iδ | = 13 || Ω p |√ σ p + | Ω n |√ σ n e i ( δ + δ A ) | , (13)where Ω p (0) and Ω n (0) are the proton and neutron components of the static spin nuclear matrix elements, δ A is therelative phase between them (zero or π ) and δ the relative phase between the amplitudes a p and a n .The nuclear spin ME are defined as follows:Ω p (0) = r J + 1 J ≺ J J | σ z ( p ) | J J ≻ , Ω n (0) = r J + 1 J ≺ J J | σ z ( n ) | J J ≻ (14)where J is the total angular momentum of the nucleus and σ z = 2 S z . The spin operator is defined by S z ( p ) = P Zi =1 S z ( i ), i.e. a sum over all protons in the nucleus, and S z ( n ) = P Ni =1 S z ( i ), i.e. a sum over all neutrons.Furthermore Ω (0) = Ω p (0) + Ω n (0) , Ω (0) = Ω p (0) − Ω n (0) . (15)The spin ME can be obtained in the context of a given nuclear model. Some such matrix elements of interest to theplanned experiments are given in table I. The shown results are obtained from DIVARI [29], Ressel et al (*) [30], theFinish group (**) [31] and the Ioannina team (+) [40], [32].Before concluding this section we should emphasize that from the spin matrix elements of Table I those associatedwith F are the most reliable for the following reasons [29]: • The light s-d nuclei are very well described within the interacting shell model. • The magnetic moment of the ground state is dominated by the spin (the orbital part is negligible). • The calculated magnetic moment is quite large and in good agreement with experiment.To summarize: The proton and neutron spin cross sections can be obtained in a given particle model for the WIMP’s.As we have seen there is a plethora of such models to motivate the experiments. Some of them may yield as high asa few tens of events per kg of target per year [12]. But most of them depend on imput parameters that are not welldetemined. So none of them seems to be universally accepted. Thus in the present work, rather than following thestandard procedure of providing constrained parameter spaces, we will treat the proton and neutron cross sections asparameters to be extracted from the data. This can be done, once the nuclear spin matrix elements are known, forvarious values of the phase difference δ . The only particle parameter we will retain is the WIMP mass, which is themost important, since it enters not only in the elementary cross sections but the kinematics as well.4 FIG. 2: On the left the quantity c spin ( A, µ r ( A ) , m χ ) ζ spin for the A=19 system is shown for two cut off values Q min = 0, continuous curve, and Q min = 10 keV, dotted curve. On the right the same quantity is shown for theA=127 system. The advantages of the lighter target, especially for light WIMP, are obvious. EXPRESSIONS FOR THE RATES AND SOME RESULTS
To obtain the total rates one must fold the diffrential rate of Eq. (1) with WIMP velocity and then integrate theresultin expression over the energy transfer from Q min determined by the detector energy cutoff to Q max determinedby the maximum WIMP velocity (escape velocity, put in by hand in the M-B distribution), i.e. υ esc = 2 . υ with υ the velocity of the sun around the center of the galaxy(229 Km/s ).Ignoring the motion of the Earth the total (non directional) rate is given by R = ¯ R t ( a, Q min ) , ¯ R = ρ (0) m χ mAm p ( µ r µ r ( p ) ) p h v i (cid:2) σ Sp,χ A + σ spinnuc (cid:3) . (16)The WIMP parameters have been absorbed in ¯ R . The parameter t takes care of the nuclear form factor and thefolding with WIMP velocity distribution [12, 38, 42] (for its values see table II). It depends on Q min , i.e. the energytransfer cutoff imposed by the detector and a = [ µ r bυ √ − .In the present work we find it convenient to re-write it as: R = ˜ K ( σ ) " c coh ( A, µ r ( A ) , m χ ) σ Sp,χ σ + c spin ( A, µ r ( A ) , m χ ) σ spinnuc σ (17)For the spin cross section it is convenient to take σ = 10 − pb . Thus˜ K ( σ ) = ρ (0)100 GeV mm p p h v i σ ≃ .
60 10 − y − ρ (0)0 . GeV cm − m Kg p h v i kms − . (18)For the coherent mode it may be more convenient to pick σ = 10 − pb , which is close to the present experimentallimit. Furthermore c coh ( A, µ r ( A ) , m χ )) = 100 GeV m χ (cid:20) µ r ( A ) µ r ( p ) (cid:21) A t coh ( A ) , c spin ( A, µ r ( A ) , m χ )) = 100 GeVm χ (cid:20) µ r ( A ) µ r ( p ) (cid:21) t spin ( A ) A . (19)The parameters c coh ( A, µ r ( A ) , m χ ), c spin ( A, µ r ( A ) , m χ ), which give the relative merit for the coherent and the spincontributions in the case of a nuclear target compared to those of the proton, are tabulated in table II for energycutoff Q min = 0 ,
10 keV. Thus via Eq. (17) we can extract the nucleon cross section from the data.The quantity c spin ( A, µ r ( A ) , m χ ) ζ spin , when the isoscalar contribution is neglected and employing Ω = 1 .
22 (2 . I ( F ), is shown in Fig 2. In the case of the spin induced cross section, the light nucleus F has certainly anadvantage over the heavier nucleus
I (see Fig. 2). For the coherent process, however, the light nucleus is disfavored.(see Table II). 5ABLE II: The factors c
19 = c coh (19 , µ r (19) , m χ ), s
19 = c spin (19 , µ r (19) , m χ ), c
19 = c coh (73 , µ r (73) , m χ ), s
73 = c spin (73 , µ r (73) , m χ ) and c
127 = c coh (127 , µ r (127) , m χ ), s
127 = c spin (127 , µ r (127) , m χ ) for two values of Q min . Also given are the factors s c spin (3 , µ r (3) , m χ ) for Q min = 0. Q min m χ (GeV)keV 20 30 40 50 60 80 100 2000 t(3,s) 1.166 1.166 1.166 1.166 1.166 1.166 1.166 1.1660 c3 131 92.6 71.6 58.3 49.2 37.5 30.3 15.40 s3 14.6 10.3 7.95 6.48 5.47 4.16 3.36 1.710 t(19,c) 1.153 1.145 1.138 1.134 1.130 1.124 1.121 1.1120 t(19,s) 1.132 1.117 1.105 1.096 1.089 1.079 1.072 1.0560 c19 11500 10500 9420 8500 7700 6450 5540 32100 s19 31.2 28.3 25.4 22.8 20.6 17.2 14.6 8.400 t(23,c) 1.107 1.099 1.092 1.089 1.085 1.079 1.076 1.0680 t(23,s) 1.075 1.061 1.050 1.041 1.035 1.025 1.018 1.0030 c23 16100 15200 14100 1300 11900 10200 8830 52800 s23 29.5 27.8 25.6 23.4 21.4 18.2 13.8 9.450 t(73,c) 1.119 1.083 1.047 1.014 0.984 0.933 0.893 0.7800 t(73,s) 1.135 1.112 1.088 1.064 1.043 1.006 0.976 0.8860 c73 113000 131000 139000 143000 142000 187000 130000 9350000 s73 20.8 23.9 25.2 25.5 25.2 23.9 22.2 15.40 t(127,c) 0.984 0.844 0.721 0.621 0.542 0.430 0.358 0.2130 t(127,s) 0.948 0.796 0.671 0.574 0.501 0.401 0.340 0.2200 c127 206000 225000 223000 211000 197000 169000 145000 824000 s127 12.3 13.1 12.8 12.1 11.3 9.7 8.5 5.310 t(19,c) 0.352 0.511 0.592 0.639 0.667 0.710 0.720 0.77310 t(19,s) 0.340 0.489 0.563 0.606 0.631 0.669 0.676 0.72010 c19 3500 4676 4902 4789 4546 4075 3557 223310 s19 9.3 12.4 12.9 12.6 11.9 10.6 9.3 5.810 t(73,c) 0 0.020 0.119 0.246 0.363 0.539 0.651 0.84710 t(73,s) 0 0.0175 0.105 0.213 0.311 0.453 0.539 0.67710 c73) 0 2310 15300 32900 49600 73400 86300 8930010 s73 0 0.39 2.5 5.3 7.9 11.6 13.4 13.410 t(127,c) 0.000 0.156 0.205 0.222 0.216 0.191 0.175 0.10910 t(127,s) 0.000 0.135 0.177 0.192 0.190 0.174 0.165 0.12110 c127 0 41500 63200 75500 78500 74900 71000 4220010 s127 0. 2.2 3.4 4.0 4.3 4.2 4.1 2.9 TABLE III: The experimental sensitivity ratios (ESR) for various targets assuming a WIMP mass of 50 GeV. p, n and iv correspond to the elementary proton, neutron and isovector dominance respectively. He F Na Ge I p × × × × × n ×
10 1.2 × × × × iv ×
10 4.0 × × × × The experimental sensitivity ratios (ESR), i.e. the extracted from experiment nucleon cross section ratios satisfy:
ESR = σ spink,χ σ Sp,χ = (cid:20) c coh ( A, µ r ( A ) , m χ ) c spin ( A, µ r ( A ) , m χ ) (cid:21) k , k = p, n, iv, for proton, neutrom, isovector respectively (20)The quantity ESR for a WIMP mass of 50 GeV is shown in table III. It is clear from this table why the limits onthe spin cross section extracted from all targets is much bigger compared to that extracted for the coherent mode.6e should emphasize that the elementary cross sections do not depend on the target. It is only the values extractedfrom experiment that do so, giving a measure of the sensitivity of the various experiments. The elementary crosssections only depend on the particle model and the structure of the nucleon. Thus, e.g., in the case of K-K WIMPsthe coherent cross section dominates, if the WIMP is a K-K gauge boson, but the spin cross section is bigger, whenthe WIMP is a K-K neutrino [16].If the effects of the motion of the Earth around the sun are included, the total non directional rate is given by R = ˜ K ( σ ) " c coh ( A, µ r ( A ) , m χ ) σ Sp,χ σ (1 + h ( a, Q min ) cosα ) (coherent) , (21) R = ˜ K ( σ ) (cid:20) c spin ( A, µ r ( A ) , m χ ) σ spinnuc σ (1 + h spin ( a, Q min ) cosα ) (cid:21) (spin) , (22)where h ( h spin ) are the modulation amplitudes and α is the phase of the Earth, which is zero around June 2nd. Weare going to only briefly discuss the modulation amplitudes here since they depend only on the WIMP mass andare independent of the other particle parameters. In the case of the two very light targets, however, they are prettyindependent of the WIMP mass. In fact for the light systems : h = h spin = 0 . h = 0 . , h spin = 0 . He the quantity t is also essentially independent of the WIMP mass, since the WIMP isexpected to be much heavier than the nuclear mass. From table II we see that the coherent rate is quite small forthis light system, but the spin induced rate is only a factor of two smaller than that for F. As we have alreadymentioned this nucleus, has definite experimental advantages [41].In many instances the experiments are interested in the differential event rate. This is a function of two variables,the WIMP mass and the energy transfer Q . For the light systems, however, the dependence on the WIMP mass israther weak, especially for heavy WIMPs. Thus the presentation of the results is relatively simple and we are goingto present them here. One finds: dRdQ = ˜ K ( σ ) (cid:20) dR ( Q, A, m χ ) dQ (1 + H ( Q, A, m χ ) cosα ) σ spinnuc σ (cid:21) , (24)with an analogous expression for the coherent mode. The time average quantity dR ( Q, A, m χ ) /dQ and the relativemodulation amplitude H ( Q, A, m χ ) are shown in Fig. 3. The differential cross section is normalized so that the areaunder the corresponding curve gives the value c spin ( A, µ r ( A ) , m χ ) of Eq. (22). Note that the quantity H, being theratio of two amplitudes, the amplitude for modulation divided by the time independent amplitude, is independent ofthe nuclear model. So it is the same for the spin and coherent mode. Note also that, at relatively low energy transfers, H becomes negative, i.e. minimum in June and maximum in December. The negative value, however, for the lighttargets is small for all WIMP masses. For this reason for a light target the integrated modulation amplitude h isalways positive (maximum in June, minimum in December, as expected). The modulation curves H keep increasing asthe energy transfer increases, mainly because the time independent amplitude, coming in the denominator, decreases.Thus in spite of this increase of H , h remains constant. RESULTS FOR THE SPIN CONTRIBUTIONOne amplitude is dominant
This occurs in cases when the nuclear structure leads to a dominant spin ME, like F with a dominant protoncomponent. In this case, barring unusual circumstances at the quark level favoring the component not favored bynuclear physics, the analysis is simple. Thus, e.g., in the case of F (Ω p = 1 . n = − .
030 the event rate for anelementary cross section of 10 − pb is exhibited as a function of the WIMP mass in Fig. 4. From these plots, for agiven WIMP mass, one may extract limits on the relevant nucleon cross section from the experimental limits. Usingthe event rate of 13.75 Kg-d or 5020 Kg-y of PICASSO [24] and the most favorable WIMP mass of 30 GeV, fromFig. 4 we extract a proton spin cross section of 0.1 pb, to be compared with the value of 0.16 pb extracted there[24]. From Eq. (20) we extract a coherent cross section of 2 . × − pb for this system, which is poor compared tothe limits of CDMS [6] and XENON [7]. The PICASSO people are fighting with their new detector against the α background, with a flat plateau in the region of their signal, and their limit will soon substantially improve .7
10 15 20 25 300.51.01.52.02.5
50 100 150 2000.0050.0100.015 Q → keV FIG. 3: On the left the quantity dR ( Q,A,m χ ) dQ and on the right the quantity H ( Q, A, m χ involving the spin inducedprocess for the A=19 system as a function of the energy transfer Q in keV. The thick solid, the dotted , the dashedand the thin solid lines correspond to WIMP masses 10,30, 50 and 100 GeV respectively. On the left panel the rangeof Q is restricted to make the curve for low WIMP mass more visible. For masses heavier than 30 GeV thedifferential event rate has essentially a constant slope. So it is adequate to restrict ourselves to low Q . The fullrange of Q can be inferred from the right panel.
50 100 150 2000.150.200.250.300.350.40 m χ → GeV
50 100 150 2000.000060.000080.000100.000120.00014 m χ → GeV
FIG. 4: The event rate (kg-y) for the target F assuming a nucleon cross section of 10 − pb as a function of theWIMP mass in GeV. In the left panel the continuous curve takes into account only the proton component. Thedotted curve results when the proton and neutron cross sections are the same, but the corresponding amplitudes areopposite (the isoscalar amplitude is assumed to vanish). The difference is small. In the right panel we consider thecase that the elementary proton cross section vanishes. In this case the nuclear structure suppresses the rate. Exclusion plots in the √ σ p , √ σ n plane From the experimental data, using the nuclear spin matrix elements,one can extract a restricted region in the σ p , σ n plane[43]-[44]. The relevant relation is: || Ω p |√ σ p + | Ω n |√ σ n e i ( δ + δ A ) | = σ R ˜ K ( σ ) sA . (25)where sA is a short hand notation for c spin ( A, µ r ( A ) , m χ ). The extracted values, given the event rate and the spinME, depend on the WIMP mass and the relative phase of the two amplitudes.Since the procedure is much more complicated than that entering the analysis of the coherent node, a few explanationsregarding the presentation of our results (Figs 5-10) are in order: • We found it more convenient to present in the plots the extracted √ σ p and √ σ n rather than the cross sectionsthemselves. • For illustrative purposes the dependence on δ can be given in a simple graph whereby the cross sections can beexpressed in units containing all the parameters. The extracted shapes, which depend on δ A are shown in Fig.5. Such a plot, in principle, contains all the needed information, but it is too general to be practical.8 σ n → q σ R ˜ K ( σ ) | Ω n | s A √ σ p → r σ R ˜ K ( σ ) | Ω p | sA (a) √ σ n → q σ R ˜ K ( σ ) | Ω n | s A √ σ p → r σ R ˜ K ( σ ) | Ω p | sA (b) FIG. 5: A ”universal” exclusion plot in the ( √ σ p , √ σ n ) plane exhibiting the dependence on the phase δ . On teright panel the nuclear spins are of the same sign, while on the other of opposite signs. When the two amplitudesare relatively real, they are not bounded except when the the phase δ is the same with the relative phase of the twonuclear matrix elements. In both panels the dotted, the fine solid, the dashed, the dotted- dashed, and the thicksolid curve correspond to δ = 0 , π/ , π/ , π/ π respectively. • The contour for δ = 0 , π is in general an ellipse. For a given experimental bound, the allowed values of thecross sections are in the space enclosed by an ellipse. One can see that, depending on δ the maximum allowedcross sections can be quite a bit higher than those extracted assuming a single mode. If the two amplitudes arerelatively real, then the contours become straight lines and the cross sections may be constrained, but only ifthe relative phase of the two amplitudes is the same with δ A . If they differ by π , the individual cross sectionsare not bounded, they can be anywhere between the two lines. • For given nuclear spin ME the extracted √ σ p and √ σ n are presented in units q σ R ˜ K ( σ ) sA ) (see Fig. 6 -10 ).Once the experiment determines the rate R and the parameter sA , for the chosen WIMP mass, is read off fromtable II, one can immediately extract from the figures the cross sections in units of σ ( ˜ K ( σ ) is given by Eq.(18)). As an illustration we do this on the right panel of the the figures 6 -10 assuming an event rate of 1 eventper Kg target per year for the optimum value of m χ (maximum of sA )The following cases are of experimental interest:i) We first consider the case of nuclear spin matrix elements of opposite sign and | Ω n | > | Ω p | as is the case of the A=3system. The exclusion plots are shown in Fig. 6ii) Next comes the case of spin matrix elements of opposite sign and | Ω p | ≫ | Ω n | as is the case of the F target. Thiscase has already been analyzed above, considering only protons. Just in case the elementary proton cross section isvery suppressed, we present the relevant exclusion plots in Fig. 7.iii) After this we consider nuclear spin matrix elements of same sign and | Ω n | ≫ | Ω p | . This is the case of the Getarget. The relevant exclusion plots are shown in Fig. 8.iv We consider the case with both spin matrix elements being significant. Such may be the case of the
I target(Ω p = 1 . , Ω n = 0 . Na, which is present togetherwith
I in the target NaI (see Fig. 10). We notice that, since one has the same number of nuclei of each componentin a given mass of the target, Na competes well with the
I in the spin induced event rate.
CONCLUDING REMARKS
We have analyzed the spin induced WIMP nucleus elastic cross section and related event rates. Both dependrather sensitively on the spin structure of the nucleus. Barring unusual circumstances at the elementary level, thespin mode has no chance to compete with the coherent WIMP nucleus scattering in the case of heavy targets. Itcould, however, compete with it in the case of light targets. For light targets, and in particular for He and F,we believe the nuclear matrix elements are very accurate to allow reliable extraction of the nucleon cross sectionsfrom the data, if and when they become available. In the cases considered here, with the possible exception of
I,9 σ n → q σ R ˜ K ( σ ) s √ σ p → r σ R ˜ K ( σ ) s (a) √ σ n → p . × − pb √ σ p → √ . × − pb (b) FIG. 6: The exclusion plot in the ( √ σ p , √ σ n ) plane in the case of the target He for various values of the phase δ using the relevant spin ME of table I. The units depend on the parameters of table II, the experimental rate R aswell as and, for the chosen scale of σ , on ˜ K ( σ ) (a). The same exclusion plot in the case of a WIMP with a mass 20GeV normalized to 1 event per kg target per year in the indicated units, obtained using ˜ K ( σ ) = 1 . × y − and σ = 10 − pb (b) . Cross sections for other event rates can be trivially extracted by a simple rescaling of panel (b).When the two amplitudes are relatively real ( δ = 0 , π ), they are not bounded except when δ coincides with thephase difference δ A of the neclear matrix elements. For the labelling of the curves see Fig: 5. √ σ n → q σ R ˜ K ( σ ) s √ σ p → r σ R ˜ K ( σ ) s (a) √ σ n → p . × − pb √ σ p → √ . × − pb (b) FIG. 7: The same as in Fig. 6 in the case of F target. √ σ n → q σ R ˜ K ( σ ) s √ σ p → r σ R ˜ K ( σ ) s (a) √ σ n → p . × − pb √ σ p → √ . × − pb (b) FIG. 8: The same as in Fig. 6 in the case of Ge target. Now in panel (b) we exhibit the most sensitive case of aWIMP mass of 50 GeV.10 σ n → q σ R ˜ K ( σ ) s √ σ p → r σ R ˜ K ( σ ) s (a) √ σ n → p . × − pb √ σ p → √ . × − pb (b) FIG. 9: The same as in Fig. 6 in the case of
I target. Now in panel (b) we exhibit the most sensitive case of aWIMP mass of 30 GeV. √ σ n → q σ R ˜ K ( σ ) s √ σ p → r σ R ˜ K ( σ ) s (a) √ σ n → p . × − pb √ σ p → √ . × − pb (b) FIG. 10: The same as in Fig. 6 in the case of Na target. Now in panel (b) we exhibit the most sensitive case of aWIMP mass of 20 GeV.the nuclear structure tends to favour the proton or the neutron component. This allows a simple extraction of thecorresponding nucleon cross section. This is also true even if both components are present, but the isoscalar amplitudeat the nucleon level is suppressed. Finally, even if both the proton and the neutron amplitudes are important, wehave shown that knowledge of the nuclear matrix elements allows one to draw suitable exclusion plots. Unfortunately,then, the situation is technically a bit complicated by the fact that one must draw one exclusion plot for each WIMPmass. So, for targets with spin different from zero, exclusion plots should be drawn as more experimental data becomeavailable.
ACKNOWLEDGMENTS
The final stages of this work were completed while the author visited Tuebingen under a Humboldt Research Award.The Author is indebted for this opportunity to Alexander von Humboldt Foundation and Professor Amand Faessler. [1] E. Komatsu et al., Astrophys.J.Suppl. (2009), ;arXiv:0803.0547(atro-ph).[2] D. P. Bennett and et al , Phys. Rev. Lett. , 2867 (1995).[3] R. Bernabei and et al , Phys. Lett. B , 757 (1996).[4] R. Bernabei et al, Phys. Lett. B , 195 (1998).
5] A. Benoit et al , [EDELWEISS collaboration]: Phys. Lett. B , 43 (2002);V. Sanglar,[EDELWEISS collaboration] arXiv:astro-ph/0306233;D. S. Akerib et al ,[CDMS Collaboration]: Phys. Rev D , 082002 (2003); arXiv:astro-ph/0405033.[6] J. Yoo [CDMS Collaboration], arXiv:0810.3527 [hep-ex].[7] J. Angle and et al , Phys. Rev. Lett. , 021303 (2008), arXiv:0706.0039.[8] G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rep. , 195 (1996).[9] A. Bottino et al. , Phys. Lett B , 113 (1997).R. Arnowitt. and P. Nath,
Phys. Rev. Lett. , 4592 (1995); Phys. Rev. D , 2374 (1996); hep-ph/9902237;V. A. Bednyakov, H.V. Klapdor-Kleingrothaus and S.G. Kovalenko, Phys. Lett. B , 5 (1994).[10] U. Chattopadhyay and D. Roy, Phys. Rev. D , 033010 (2003), hep-ph/0304108.[11] B. Murakami and J. Wells, Phys. Rev. D p. 015001 (2001), hep-ph/0011082.[12] J. D. Vergados, J.Phys. G , 1127 (2004), 0406134.[13] J. Ellis, K. A. Olive, Y. Santoso, and V. C. Spanos, Phys.Rev. D , 055005 (2004).[14] J. Hisano, S. Matsumoto, M. M. Nojiri, and O. Saito, Phys.Rev. D , 015007 (2005).[15] J. Ellis, K. A. Olive, Y. Santoso, and V. C. Spanos, Phys.Rev. D , 095007 (2005), ;hep-ph/0502001.[16] V. Oikonomou, J. Vergados, and C. C. Moustakidis, Nuc. Phys. B 773 , 19 (2007).[17] C. Kouvaris, Phys. Rev. D , 015011 (2007), ;arXiv:hep-ph/0703266.[18] S. B. Gudnason, C. Kouvaris, and F. Sannino, Phys. Rev. D , 095008 (2006), arXiv:hep-ph/0608055.[19] M.Yu.Khlopov, Kouvaris, and F. Sannino, Pys. Rev. D , 065040 (2008), arXiv: 0806.1191 [astro-ph].[20] T. Ryttov and F. Sannino, Phys. Rev. D , 115010 (2008).[21] H. Ejiri, K. Fushimi, and H. Ohsumi, Phys. Lett. B , 14 (1993).[22] J. D. Vergados, P. Quentin, and D. Strottman, IJMPE , 751 (2005), hep-ph/0310365.[23] D. Santos et al , The MIMAC-He3 Collaboration, A New He Detector for non Baryonic Dark Matter Search, Invited talkin idm2004 (to appear in the proceedings).[24] Dark Matter Spin-Dependent Limits for WIMP Interactions on 19-F by PICASSO Archambault, F. Aubin, M. Auger, E.Behnke, B. Beltran, K. Clark, X. Dai, A. Davour, J. Farine, R. Faust, M.-H. Genest, G. Giroux, R. Gornea, C. Krauss,S. Kumaratunga, I. Lawson, C. Leroy, L. Lessard, C. Levy, I. Levine, R. MacDonald, J.-P. Martin, P. Nadeau, A. Noble,M.-C. Piro, S. Pospisil, T. Shepherd, N. Starinski, I. Stekl, C. Storey, U. Wichoski, V. Zacek ; arXiv:0907.0307[hep-ex].[25] J. D. Vergados, J. of Phys. G , 253 (1996).[26] M. Drees and M. M. Nojiri, Phys. Rev. D , 3843 (1993); Phys. Rev. D , 4226 (1993).[27] A. Djouadi and M. K. Drees, Phys. Lett. B , 183 (2000); S. Dawson,
Nucl. Phys. B , 283 (1991); M. Spira it et al,
Nucl. Phys.
B453 , 17 (1995).[28] T. P. Cheng,
Phys. Rev. D , 2869 (1988); H-Y. Cheng, Phys. Lett. B , 347 (1989).[29] P. C. Divari, T. S. Kosmas, J. D. Vergados, and L. D. Skouras, Phys. Rev. C , 054612 (2000).[30] M. T. Ressell et al. , Phys. Rev. D , 5519 (1993); M.T. Ressell and D. J. Dean, Phys. Rev. C , 535 (1997).[31] E. Homlund and M. Kortelainen and T. S. Kosmas and J. Suhonen and J. Toivanen, Phys. Lett B, ,31 (2004); Phys.Atom. Nucl. , 1198 (2004).[32] T. S. Kosmas and J. D. Vergados, Phys. Rev. D , 1752 (1997).[33] J. Vergados, S. N. Hansen, and O. Host, Phys. Rev. D 77 , 023509 (2008).[34] J. Vergados, AJ , 10 (2009), arXiv:08110382/astro-ph.[35] K. F. A. K. Drukier and D. N. Spergel, Phys. Rev. D , 3495 (1986).[36] K. Frese, J. A. Friedman, and A. Gould, Phys. Rev. D , 3388 (1988).[37] J. D. Vergados, Phys. Rev. D , 103001 (1998).[38] J. D. Vergados, Phys. Rev. D , 06351 (2001).[39] The Strange Spin of the Nucleon, J. Ellis and M. Karliner, hep-ph/9501280.[40] J. D. Vergados, Part. Nucl. Lett. , 74 (2001), hep-ph/0010151.[41] E. Moulin, F. Mayet, and D. Santos, Phys. Lett. B , 143 (2005).[42] J. D. Vergados, Phys. Rev. D , 023519 (2000).[43] C. Savage, P. Gondolo, and K. Freese, Phys. Rev. D , 123513 (2004).[44] F. Giuliani and T. A. Girard, Phys.Lett. B , 151 (2004)., 151 (2004).