Event rates for the scattering of weakly interacting massive particles from 23Na and 40Ar R. Sahu1∗, V.K.B. Kota2† 1 National Institute of Science and Technology, Palur Hills, Berhampur-761008, Odisha, India and 2Physical Research Laboratory, Ahmedabad 380 009, India (Dated: September 23, 2020) Detection rates for the elastic and inelastic scattering of weakly interacting massive particles (WIMP) off 23Na are calculated within the framework of Deformed Shell Model (DSM) based on Hartree-Fock states. First the spectroscopic properties like energy spectra and magnetic moments are calculated and compared with experiment. Following the good agreement for these, DSM wave functions are used for obtaining elastic and inelastic spin structure functions, nuclear structure coefficients etc. for the WIMP-23Na scattering. Then, the event rates are also calculated with a given set of supersymmetric parameters. In the same manner, using DSM wavefunctions, nuclear structure coefficients and event rates for elastic scattering of WIMP from 40Ar are also obtained. These results for event rates and also for annual modulation will be useful for the upcoming and future WIMP detection experiments involving detectors with 23Na and 40Ar. I. INTRODUCTION the NaI(Tl) detector. In these experiments, the predicted annual modulation was not yet confirmed [11]. Other re- There is now universal agreement among the cosmolo- lated experiments with NaI detectors are ANAIS [18] and gists, astronomers and physicists that most of the mass DM-Ice [19]. Also, there are the important DARKSIDE- 50 [20] and DEAP-3600 [21] experiments using lquid Ar- of the universe is dark [1–3]. There are overwhelming 40 evidences to believe that the dark matter is mostly non- gon (with Ar) as detector. baryonic. Also, data from the Cosmic Background Ex- Let us add that direct detection experiments are ex- plorer (COBE) [4] and Supernova Cosmology project [5] posed to various neutrino emissions. The interaction suggest that most of the dark matter is cold. The non- of these neutrinos especially the astrophysical neutrinos baryonic cold dark matter is not yet observed in earth- with the material of the dark matter detectors known as bound experiments and hence its nature is still a mys- the neutrino floor is a serious background source. Re- tery. Axions are one of the candidates for dark matter cently the coherent elastic scattering of neutrinos off nu- but they are not yet observed [1, 6]. However, the most clei (CEνNS) has been observed at the Spallation Neu- promising nonbaryonic cold dark matter candidates are tron Source at the Oak Ridge National Laboratory [22] the Weakly Interacting Massive Particles (WIMP) which employing the technology used in the direct detection of arise in super symmetric theories of physics beyond the dark matter searches. The impacts of the neutrino floor standard model. The most appealing WIMP candidate on the relevant experiments looking for cold dark matter is the Lightest Supersymmetric Particle (LSP) (lightest was investigated for example in [23]. neutralino) which is expected to be stable and interacts There are many theoretical calculations which de- weakly with matter [1, 7]. scribe different aspects of direct detection of dark mat- There are many experimental efforts [8–11] to detect ter through the recoil of the nucleus in WIMP-nucleus WIMP via their scattering from the nuclei of the detector scattering. For elastic scattering, we need to consider providing finger-prints regarding their existence. Some of spin-spin interaction coming from the axial current and these are Super CDMS SNOLAB project, XENON1T, also the more dominant scalar interaction. For inelastic arXiv:2009.10522v1 [nucl-th] 19 Sep 2020 PICO-60, EDELWEISS and so on; see for example part, scalar interaction practically does not contribute. [11–14]. Nuclei 23Na, 40Ar, 71Ga, 73Ge, 75As, 127I, The scalar interaction can arise from squark exchange, 133Cs and 133Xe are among the popular detector nu- Higgs exchange, the interaction of WIMPs with gluons clei; see [10, 11, 15] and references there in. Our fo- etc. Suhonen and his collaborators have performed a cus in this paper is on 23Na and 40Ar. The Sodium series of truncated shell model calculations for this pur- Iodide (NaI) Advanced Detector (NAID) array experi- pose [24–28]. In these studies, for example they have ment is a direct search experiment for WIMP operated by calculated the event rates for WIMP-nucleus elastic and 83 125 127 UK Dark Matter Collaboration in North Yorkshire [16]; inelastic scattering for Kr and Te [28] and also I, 129,131 133 the NaI contains 23Na. Similarly, the DAMA/NaI and Xe and Cs [26]. In addition, recently Vergados DAMA/LiBRA [17] experiments investigated the pres- et al [29] examined the possibility of detecting electrons ence of dark matter particles in the galactic halo using in the searches for light WIMP with a mass in the MeV region and found that the events of 0.5-2.5 per kg-y would be possible. Few years back full large-scale shell-model calculations are carried out in [15, 30] for WIMP scat- 129,131 127 73 19 23 27 29 ∗[email protected] tering off Xe, I, Ge, F, Na, Al and Si †[email protected] nuclei. Finally, using large scale shell model [31] and 2 75 coupled cluster theory [32] WIMP-nucleus and neutrino- 6 As 40 nucleus scattering respectively, with Ar, are studied. 9/2+ In recent years, the deformed shell model (DSM), 4 based on Hartree-Fock (HF) deformed intrinsic states with angular momentum projection and band mixing, 7/2+ 2 + has been established to be a good model to describe 9/2 5/2− the properties of nuclei in the mass range A=60-90 [33]. 5/2+ 0 − Among many applications, DSM is found to be quite suc- x x 1/2 + + 3/2 7/2 xx − cessful in describing spectroscopic properties of medium xx 3/2 x x 1/2+ heavy N=Z odd-odd nuclei with isospin projection [34], −2 double beta decay half-lives [35, 36] and µ e conversion 5/2+ − − 5/2 x x 3/2− in the field of the nucleus [37]. Going beyond these ap- −4 x x 1/2− Energy (MeV) plications, recently we have studied the event rates for 1/2−,3/2+ 73 + − WIMP with Ge as the detector [38]. In addition to the 1/2 ,3/2 −6 energy spectra and magnetic moments, the model is used x x 1/2− to calculate the spin structure functions, nuclear struc- 3/2− o −8 − ture factors for the elastic and inelastic scattering. Fol- 1/2 o o lowing this successful study, we have recently used DSM for calculating the neutrino-floor due to coherent elastic −10 neutrino-nucleus scattering (CEνNS) [23] for the candi- 1/2− o o date nuclei 73Ge, 71Ga, 75As and 127I. We found that E=−30.23 (MeV) Q=28.02 the neutrino-floor contributions may lead to a distortion K=3/2− of the expected recoil spectrum limiting the sensitivity of the direct dark matter search experiments. In [10], FIG. 1: HF single-particle spectra for 75As corresponding to DSM results for WIMP scattering from 127I, 133Cs and lowest prolate configuration. In the figure, circles represent 133Xe are described in detail. To complete these stud- protons and crosses represent neutrons. The HF energy E in ies that use DSM for the nuclear structure part, in the MeV, mass quadrupole moment Q in units of the square of the present paper we will present results for WIMP- 23Na oscillator length parameter and the total azimuthal quantum elastic and inelastic scattering and WIMP-40Ar elastic number K are given in the figure. scattering. Now we will give a preview. Section II gives, for completeness and easy reading of pleteness we give here a few important steps. In the case the paper, a brief discussion of the formulation of WIMP- of spin-spin interaction, the WIMP couples to the spin nucleus elastic and inelastic scattering and event rates. In of the nucleus and in the case of scalar interaction, the Section III the DSM formulation is described with exam- WIMP couples to the mass of the nucleus. In the ex- ples drawn from 75As spectroscopic results. In Section pressions for the event rates, the super-symmetric part is IV, spectroscopic results and also the results for elas- separated from the nuclear part so that the role played tic and inelastic scattering of WIMP from 23Na are pre- by the nuclear part becomes apparent. sented. Similarly, WIMP-40Ar elastic scattering results are presented in Section V. The results in Sections IV and V are the main results of this paper. Finally, concluding A. Elastic scattering remarks are drawn in Sect. VI. The differential event rate per unit detector mass for a WIMP with mass mχ can be written as [1], II. EVENT RATES FOR WIMP-NUCLEUS SCATTERING dσ dR = N φf d3v d q 2 (1) t d q 2 | | WIMP flux on earth coming from the galactic halo | | 5 2 is expected to be quite large, of the order 10 per cm Here, φ which is equal to ρ0v/mχ is the dark matter flux per second. Even though the interaction of WIMP with with ρ0 being the local WIMP density. Similarly, Nt matter is weak, this flux is sufficiently large for the galac- stands for the number of target nuclei per unit mass and tic WIMPs to deposit a measurable amount of energy in f is the WIMP velocity distribution which is assumed to an appropriately sensitive detector apparatus when they be Maxwell-Boltzmann type. It takes into account the scatter off nuclei. Most of the experimental searches of distribution of the WIMP velocity relative to the detector WIMP is based on the direct detection through their in- (or earth) and also the motion of the sun and earth.