Dark-matter detection by elastic and inelastic LSP scattering on 129Xe and 131Xe

Abstract We calculate the nuclear matrix elements involved in the elastic and inelastic scattering of the lightest supersymmetric particle (LSP) on the 129Xe and 131Xe dark-matter detector nuclei. This is the first time when both channels are addressed within the same unified microscopic nuclear framework, namely we perform large-scale shell-model calculations with a realistic two-body interaction to produce the participant nuclear wave functions. These wave functions successfully reproduce the spectroscopic data on the relevant magnetic moments and M1 decays. The tested wave functions are used to produce annual average detection rates for both the elastic and inelastic channels. It is found that the inelastic channel has great detection potential for 129Xe if the LSP is heavy and stems from a SUSY model that enhances the spin-dependent scattering.

We calculate the nuclear matrix elements involved in the elastic and inelastic scattering of the lightest supersymmetric particle (LSP) on the 129 Xe and 131 Xe dark-matter detector nuclei. This is the first time when both channels are addressed within the same unified microscopic nuclear framework, namely we perform large-scale shell-model calculations with a realistic two-body interaction to produce the participant nuclear wave functions. These wave functions successfully reproduce the spectroscopic data on the relevant magnetic moments and M1 decays. The tested wave functions are used to produce annual average detection rates for both the elastic and inelastic channels. It is found that the inelastic channel has great detection potential for 129 Xe if the LSP is heavy and stems from a SUSY model that enhances the spin-dependent scattering.
The recent observations on cosmic microwave background suggest that the universe is flat and a notable part of the energy density of the universe (some 30%) is in a form of cold dark matter (CDM). Additional information comes from the rotational curves of galaxies [1]. The most likely candidates for constituents of the CDM are the weakly-interacting massive particles (WIMPs). They could make up a major component of the dark matter in our own galactic halo.
The only way to access the nature of dark matter are the direct detection experiments in the laboratory. These experiments look for recoil signals of the nucleus on which the WIMP has scattered elastically or inelastically. Many experiments have reported on their results, like the DAMA [2], the CDMS [3], the NAIAD [4], the EDELWEISS [5], the CRESST [6], the ROSEBUD [7], the SIM-PLE [8], the PICASSO [9], the ZEPLIN I [10], the XENON10 [11] and the KIMS [12]. In particular, the ZEPLIN and XENON experimental programs are interesting from the point of view of the present Letter since they use 129 Xe and 131 Xe as stable detector material. Until now only the DAMA has reported positive evidence of the annual modulation signal of WIMP detection [2,13]. This data apparently contradicts results of the other experiments, and only a small window is left where the DAMA data could coexist with the * Corresponding author. E-mail address: suhonen@phys.jyu.fi (J. Suhonen).
other data [12]. In [14] an attempt was made to explain this discrepancy by the competition of the coherent and spin-dependent channels assuming a suitable SUSY scenario for the LSP (lightest supersymmetric particle) dark-matter candidate.
Supersymmetry naturally provides candidates for the constituents of the CDM [1]. In the minimal supersymmetric Standard Model (MSSM) the LSP is stable or almost stable and can be simply described as a Majorana fermion that can be formed as a linear combination of the neutral components of gauginos and higgsinos [15,16]. The LSP scatters off the nuclei by the neutral current exchanging, e.g., a Z boson or a squark. In this work we assume the LSP scenario for the WIMP. We further assume that the local LSP density is the standard ρ 0 = 0.3 GeVcm −3 and that the LSPs follow the Maxwell-Boltzmann velocity distribution with a characteristic velocity v 0 = 220 km/s in the galactic halo. The masses of the LSPs are assumed to cover the range m χ ≈ 100-300 GeV [16][17][18][19][20].
For the LSP-nucleus scattering we use the formalism of [14]. The event rate of an Earth-bound detector can be written as SUSY model [21]. The coefficients D n are folded with the velocity distribution of the LSPs and contain all the information about the nuclear structure. They are defined as Here F ρρ (u), ρ, ρ = 0, 1, are the usual spin structure functions, Ω ρ the static spin matrix elements and F (u) the nuclear form factor. 1 The expression for the modulation function G(ψ, ξ ) is given in [14]. The limits in the integrals of Eqs.
(2)-(5) are different for the elastic and inelastic channels. For the elastic channel they are given in [14] in terms of the threshold energy of the dark-matter detector. For the inelastic channel D 4 of (5) vanishes and we assume that the detector threshold is zero since the coincidence signal of the emitted gamma quantum can be used to reduce the threshold. For the inelastic channel the limits of the integrals (2)-(4) are given by ψ min = √ Γ , (6) ψ max = −λξ + λ 2 ξ 2 + 9.0891 + 0.135 cos a, Above E * is the nuclear excitation energy of the recoiling daughter nucleus, μ r is the reduced mass of the LSP-nucleus system, c is the light velocity, and v 0 = 220 km/s. Angle a represents the phase of the Earth [15], b = b( A) is the harmonic-oscillator size parameter for a target nucleus of mass A, where v E is given in [14]. The average kinetic energy T of the LSP can be obtained from the approximate expression [22] T = 40 keV m χ 100 GeV . (11) Hence, for heavy LSPs the scattering can well be inelastic, leading to the first excited states in 129 Xe and 131 Xe, at energies 39.6 keV and 80.2 keV, respectively. For 129 Xe the transition is 1/2 + g.s. → 3/2 + 1 and for 131 Xe it is 3/2 + g.s. → 1/2 + 1 . The corresponding nuclear matrix elements are not suppressed much relative to the elastic channel, as discussed later in this work. 1 The form factors F (u) and spin structure functions F ρρ (u) can be requested in numerical form from the corresponding author. Use of various theoretical methods has been made to compute nuclear matrix elements involved in the elastic WIMP-nucleus scattering. The most complete calculations have been done by using the nuclear shell model in [14,23,24]. In [14] the shell model results were compared with those calculated by the use of the microscopic quasiparticle-phonon model [25]. The shell model was found to better reproduce the magnetic moments of the ground states of 71 Ga, 73 Ge, and 127 I, thus being in a position of providing a more reliable description of the LSP-nucleus scattering process than the MQPM.
For the inelastic channel there are only few estimates [22,26,27] for the scattering cross sections or event rates. None of these works actually calculates the needed nuclear wave functions in a reliable microscopic nuclear framework. To our knowledge the present work is the first to address both the elastic and inelastic event rates within a unified and complete nuclear scheme.
In the present work we perform large-scale shell-model calculations in a realistic model space with realistic effective two-body interactions. The calculations were made using the shell model code eicode [28]. The ground states and the first excited states of 129 Xe and 131 Xe were computed in the valence space 2s1d0g 7/2 0h 11/2 . We have used effective nucleon-nucleon interactions based on the Bonn-CD G-matrix [29]. Due to the large number of active neutrons the neutron configurations had to be truncated. For 129 Xe we allowed at most one-particle-one-hole and for 131 Xe at most twoparticle-two-hole excitations from the full 1d 5/2 and 0g 7/2 shells. For 129 Xe the allowed neutron configurations were additionally restricted by using the energy centroid method [30].
Since the shell-model calculations have to be performed in a restricted single-particle valence space we have to renormalize the bare Hamiltonian, as mentioned above. To be consistent, all other operators, like the electromagnetic ones, should be renormalized also. Rigorous renormalization is usually considered too complex and simple approximations for it are used instead.
For the LSP scattering the relevant operators are the proton and neutron spin operators and the related magnetic dipole operator. In fact, to produce reliable matrix elements for both the elastic and inelastic LSP-nucleus scattering one needs to reproduce the data on the magnetic moments of the involved nuclear states and also M1 transitions between them. We produced the optimum computed magnetic moments by effective spin and orbital angularmomentum gyromagnetic factors (g factors in short), determined by a linear least squares (LLS) fit to 10 known magnetic moments of the nuclei 127 I, 129 Xe, 131 Xe, and 133 Cs. The thus obtained magnetic moments for the two lowest states in 129 Xe and 131 Xe are compared with the data in Table 1.
The effective gyromagnetic factors resulting from the LLS fit are g s,n = −3.370, g s,p = 3.189, g l,n = 0.01903, and g l,p = 1.119 in units of μ N /c. These are very close to the standard bare values g s,n = −3.826, g s,p = 5.586, g l,n = 0, and g l,p = 1. The use of fitted effective gyromagnetic factors improves the calculated magnetic moments substantially both for the ground states and the lowest excited states of the nuclei considered in the fit. For completeness, we also list the computed spin and orbital matrix elements in Table 1.  The static spin matrix elements (SSME), present in Eqs. (2)-(4), are reviewed in Table 2 for both the elastic and inelastic scattering channels. Variations in the values of the g factors induce variations in the values of the final computed SSMEs. To assess these variations we have included in Table 2 the results based on both the bare and effective g factors. It is seen that the SSMEs of the two calculations deviate some 12% from each other. Numerical calculations show that the differences in the final computed D coefficients (2)-(5) stem essentially from the Ω 2 factors and are thus of the order of 20%. One can thus say that the manipulation of the g factors causes a rough 20% variation in the values of the relevant observables listed later in this article.
From Table 2 we notice the interesting feature that for 129 Xe the SSMEs suppress by a factor of three the inelastic channel relative to elastic channel, whereas for 131 Xe there is only very little suppression. In [22] it was found within a very simplified nuclear model that the SSMEs would even enhance the inelastic channel for 127 I. It remains to be explored if a more complete shell-model calculation, like the present one, would reproduce this finding.
The spin structure functions of Eqs. For the elastic channel of Fig. 1 the F ρρ (u) are smooth functions whereas for the inelastic channel of Fig. 2 they behave more irregularly. The form factor F (u) of the coherent channel is absent from the inelastic channel. For the elastic channel it has a peaked structure as seen in Fig. 1 (the undulations beyond the first two peaks are masked by the scale of the figure).
There are not too many other calculations for the form factors of the Xe isotopes. In [31] a very rudimentary nuclear wave function for the ground state of 131 Xe was used to compute the structure functions related to the elastic LSP-nucleus scattering. We can compare our spin structure functions F ρρ (u) for the elastic scattering with the corresponding S ρρ (q) functions of [31] by using the conversion formula (18) of [25]. In [25] this formula was used to compare Fig. 2 of [25] with Fig. 4 of [23] for the elastic scattering of an LSP on 73 Ge. In the present case we obtain from the conversion formula and from the Ω factors of Table 2 for 131 Xe S 00 (q) = (6.5-7.8) × 10 −3 F 00 (u),  From the above conversion formulae and from Fig. 3 of [31] and Fig. 1, lower panel, we notice that the present shell-model calculation gives factors 3-6 smaller spin structure functions than the very much simplified approach of [31]. This is due to our quite small Ω 0 and Ω 1 factors of Table 2 for 131 Xe. The overall shape of the curves is similar for the two calculations.
The values of the D n coefficients in (2)-(5) depend on the LSP mass m χ , the detector threshold energy Q thr and the Earth's phase a. We can average D n over a to produce their annual average valuesD n . Let us denote by d n the Q thr = 0 values ofD n . Then d n depend only on m χ . In Table 3 we list these d n coefficients for the  Table 4 Computed annual averaged coefficientsD n (m χ ) of Eqs. (2) (15) for reasonably small values of Q thr . Here the reduced mass μ r of the nucleus-LSP system is given in units of GeV and Q thr in units of keV. The fitting was done for the ranges of 0 keV Q thr 30 keV. Our parametrization (15) enables an easy extraction ofD n for the wanted LSP mass and detector threshold energy. From Table 3 we notice that the coherent channel, represented byD 4 , is strong. The importance of this channel is enhanced further by the fact that it is proportional to A 2 , as seen from Eq. (1). However, as noticed in [14], for certain parametrizations of the SUSY models the spin-dependent channel, represented byD 1 -D 3 , can overwhelm the coherent channel. The spin-dependent channel seems to be more important for 129 Xe than for 131 Xe.
In Table 4 we preset the computed annual averaged coefficients D n (m χ ) for the inelastic scattering. As discussed earlier, the values computed by using the bare g factors are some 20% larger. Here we have assumed that the detector threshold is zero. As seen from Table 4, this channel is far more important for 129 Xe than for 131 Xe due to the two times higher first excited state of 131 Xe. Looking at the numbers of Tables 3 and 4 reveals that for 129 Xe the inelastic channel is suppressed only by a factor of about 10 relative to the spin-dependent elastic channel for heavy (m χ ≈ 300 GeV) LSPs.
In this case the inelastic channel could be well measurable if the LSP stems from a SUSY model that enhances the spin-dependent scattering.
In this Letter we predict annual average detection rates for elastic and inelastic LSP scattering on 129 Xe and 131 Xe. The cor-responding nuclear-structure calculations were done by using the nuclear shell model in a realistic single-particle space and using realistic nucleon-nucleon forces. This is the first time when both the elastic and inelastic channels are described within the same unified microscopic nuclear scheme.
We present a useful parametrization of the elastic detection rates in terms of the LSP mass and detector threshold energy. The inelastic rates are calculated assuming zero threshold. It is found that the inelastic channel has great detection potential for 129 Xe if the LSP is heavy and stems from a SUSY model that enhances the spin-dependent scattering. The obtained results are especially interesting for the ZEPLIN and XENON experimental programs.