Event rates for the scattering of weakly interacting massive particles from $^{23}$Na and $^{40}$Ar

Detection rates for the elastic and inelastic scattering of weakly interacting massive particles (WIMP) off $^{23}$Na 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-$^{23}$Na 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 $^{40}$Ar 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 $^{23}$Na and $^{40}$Ar.


I. INTRODUCTION
There is now universal agreement among the cosmologists, astronomers and physicists that most of the mass of the universe is dark [1][2][3]. There are overwhelming evidences to believe that the dark matter is mostly nonbaryonic. Also, data from the Cosmic Background Explorer (COBE) [4] and Supernova Cosmology project [5] suggest that most of the dark matter is cold. The nonbaryonic cold dark matter is not yet observed in earthbound experiments and hence its nature is still a mystery. Axions are one of the candidates for dark matter but they are not yet observed [1,6]. However, the most promising nonbaryonic cold dark matter candidates are the Weakly Interacting Massive Particles (WIMP) which arise in super symmetric theories of physics beyond the standard model. The most appealing WIMP candidate is the Lightest Supersymmetric Particle (LSP) (lightest neutralino) which is expected to be stable and interacts weakly with matter [1,7].
There are many experimental efforts [8][9][10][11] to detect WIMP via their scattering from the nuclei of the detector providing finger-prints regarding their existence. Some of these are Super CDMS SNOLAB project, XENON1T, PICO-60, EDELWEISS and so on; see for example [11][12][13][14]. Nuclei 23 Na, 40 Ar, 71 Ga, 73 Ge, 75 As, 127 I, 133 Cs and 133 Xe are among the popular detector nuclei; see [10,11,15] and references there in. Our focus in this paper is on 23 Na and 40 Ar. The Sodium Iodide (NaI) Advanced Detector (NAID) array experiment is a direct search experiment for WIMP operated by UK Dark Matter Collaboration in North Yorkshire [16]; the NaI contains 23 Na. Similarly, the DAMA/NaI and DAMA/LiBRA [17] experiments investigated the presence of dark matter particles in the galactic halo using * rankasahu@gmail.com † vkbkota@prl.res.in the NaI(Tl) detector. In these experiments, the predicted annual modulation was not yet confirmed [11]. Other related experiments with NaI detectors are ANAIS [18] and DM-Ice [19]. Also, there are the important DARKSIDE-50 [20] and DEAP-3600 [21] experiments using lquid Argon (with 40 Ar) as detector.
Let us add that direct detection experiments are exposed to various neutrino emissions. The interaction of these neutrinos especially the astrophysical neutrinos with the material of the dark matter detectors known as the neutrino floor is a serious background source. Recently the coherent elastic scattering of neutrinos off nuclei (CEνNS) has been observed at the Spallation Neutron Source at the Oak Ridge National Laboratory [22] employing the technology used in the direct detection of dark matter searches. The impacts of the neutrino floor on the relevant experiments looking for cold dark matter was investigated for example in [23].
There are many theoretical calculations which describe different aspects of direct detection of dark matter through the recoil of the nucleus in WIMP-nucleus scattering. For elastic scattering, we need to consider spin-spin interaction coming from the axial current and also the more dominant scalar interaction. For inelastic part, scalar interaction practically does not contribute. The scalar interaction can arise from squark exchange, Higgs exchange, the interaction of WIMPs with gluons etc. Suhonen and his collaborators have performed a series of truncated shell model calculations for this purpose [24][25][26][27][28]. In these studies, for example they have calculated the event rates for WIMP-nucleus elastic and inelastic scattering for 83 Kr and 125 Te [28] and also 127 I, 129,131 Xe and 133 Cs [26]. In addition, recently Vergados et al [29] examined the possibility of detecting electrons 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 scattering off 129,131 Xe, 127 I, 73 Ge, 19 F, 23 Na, 27 Al and 29 Si nuclei. Finally, using large scale shell model [31] and coupled cluster theory [32] WIMP-nucleus and neutrinonucleus scattering respectively, with 40 Ar, are studied.
In recent years, the deformed shell model (DSM), based on Hartree-Fock (HF) deformed intrinsic states with angular momentum projection and band mixing, has been established to be a good model to describe the properties of nuclei in the mass range A=60-90 [33]. Among many applications, DSM is found to be quite successful in describing spectroscopic properties of medium heavy N=Z odd-odd nuclei with isospin projection [34], double beta decay half-lives [35,36] and µ − e conversion in the field of the nucleus [37]. Going beyond these applications, recently we have studied the event rates for WIMP with 73 Ge as the detector [38]. In addition to the energy spectra and magnetic moments, the model is used to calculate the spin structure functions, nuclear structure factors for the elastic and inelastic scattering. Following this successful study, we have recently used DSM for calculating the neutrino-floor due to coherent elastic neutrino-nucleus scattering (CEνNS) [23] for the candidate nuclei 73 Ge, 71 Ga, 75 As and 127 I. We found that the neutrino-floor contributions may lead to a distortion of the expected recoil spectrum limiting the sensitivity of the direct dark matter search experiments. In [10], DSM results for WIMP scattering from 127 I, 133 Cs and 133 Xe are described in detail. To complete these studies that use DSM for the nuclear structure part, in the present paper we will present results for WIMP-23 Na elastic and inelastic scattering and WIMP-40 Ar elastic scattering. Now we will give a preview.
Section II gives, for completeness and easy reading of the paper, a brief discussion of the formulation of WIMPnucleus elastic and inelastic scattering and event rates. In Section III the DSM formulation is described with examples drawn from 75 As spectroscopic results. In Section IV, spectroscopic results and also the results for elastic and inelastic scattering of WIMP from 23 Na are presented. Similarly, WIMP-40 Ar elastic scattering results are presented in Section V. The results in Sections IV and V are the main results of this paper. Finally, concluding remarks are drawn in Sect. VI.

II. EVENT RATES FOR WIMP-NUCLEUS SCATTERING
WIMP flux on earth coming from the galactic halo is expected to be quite large, of the order 10 5 per cm 2 per second. Even though the interaction of WIMP with matter is weak, this flux is sufficiently large for the galactic WIMPs to deposit a measurable amount of energy in an appropriately sensitive detector apparatus when they scatter off nuclei. Most of the experimental searches of WIMP is based on the direct detection through their interaction with nuclei in the detector. The relevant theory of WIMP-nucleus scattering is well known as available in the papers by Suhonen and his group and also in our earlier papers mentioned above [24-26, 28, 38]. For com- pleteness we give here a few important steps. In the case of spin-spin interaction, the WIMP couples to the spin of the nucleus and in the case of scalar interaction, the WIMP couples to the mass of the nucleus. In the expressions for the event rates, the super-symmetric part is separated from the nuclear part so that the role played by the nuclear part becomes apparent.
A. Elastic scattering The differential event rate per unit detector mass for a WIMP with mass m χ can be written as [1], Here, φ which is equal to ρ 0 v/m χ is the dark matter flux with ρ 0 being the local WIMP density. Similarly, N t stands for the number of target nuclei per unit mass and f is the WIMP velocity distribution which is assumed to be Maxwell-Boltzmann type. It takes into account the distribution of the WIMP velocity relative to the detector (or earth) and also the motion of the sun and earth. If we neglect the rotation of the earth about its own axis, then v =| v | is the relative velocity of WIMP with respect to the detector. Also, q represents the momentum transfer to the nuclear target which is related to the dimensionless variable u = q 2 b 2 /2 with b being the oscillator length parameter. The WIMP-nucleus differential cross section in the laboratory frame is given by [24-26, 28, 38] dσ(u, v) du (3) where F Z (u) and F N (u) denote the nuclear form factors for protons and neutrons respectively. In Eq. (3), the first three terms correspond to spin contribution coming mainly from the axial current and the other three terms stand for the coherent part coming mainly from the scalar interaction. Here, f 0 A and f 1 A represent isoscalar and isovector parts of the axial vector current and similarly f 0 S and f 1 S represent isoscalar and isovector parts of the scalar current. The nucleonic current parameters f 0 A and f 1 A depend on the specific SUSY model employed. However, f 0 S and f 1 S depend, beyond SUSY model, on the hadron model used to embed quarks and gluons into nucleons. The normalized spin structure functions F ρρ ′ (u) with ρ, ρ ′ = 0,1 are defined as In the above equation ω 0 (j) = 1 and ω 1 (j) = τ (j); note that τ = +1 for protons and −1 for neutrons and j λ is the spherical Bessel function. The static spin matrix elements are defined as Ω ρ (0) = Ω (0,1) ρ (0). Now, the event rate can be written as In the above, G(ψ, ξ) is given by Parameters used in the calculation are the following: the WIMP density ρ 0 = 0.3 Gev/cm 3 , σ 0 = 0.77 × 10 −38 cm 2 , mass of proton m p = 1.67 × 10 −27 kg. The velocity of the sun with respect to the galactic centre is taken to be v 0 = 220 Km/s and the velocity of the earth relative to the sun is taken as v 1 = 30 Km/s. The velocity of the earth with respect to the galactic centre v E is given by where α is the modulation angle which stands for the phase of the earth on its orbit around the sun and γ is the angle between the normal to the elliptic and the galactic equator which is taken to be ≃ 29.8 • . Using the notations, X(1) = F 00 (u), per unit mass of the detector is given by where D i being the three dimensional integrations of Eq. (5), defined as (8) The lower and upper limits of integrations given in Eq. (5) and (8) have been worked out by Pirinen et al [28] and they are With the escape velocity v esc from our galaxy to be 625 km/s, the value of v 2 The values of u min and u max are Am p Q thr b 2 and 2(ψµ r bv 0 /c) 2 , respectively. Here, Q thr is the detector threshold energy and µ r is the reduced mass of the WIMP-nucleus system.

B. Inelastic scattering
In the inelastic scattering the entrance channel and exit channel are different. The inelastic scattering cross section due to scalar current is considerably smaller than the elastic case and hence it is neglected. Hence, we focus on spin dependent scattering. The inelastic event rate per unit mass of the detector can be written as where E 1 , E 2 and E 3 are the three dimensional integrations The limits of integration for E 1 , E 2 and E 3 are [26,28] u min(max) = where with E * being the energy of the excited state. ψ max is same as in the elastic case and the lower limit ψ min = √ Γ. The parameters like ρ 0 , σ 0 etc. have the same values as in the elastic case.

III. DEFORMED SHELL MODEL
The nucleonic current part has been separated from nuclear part in the expression for the event rates for elastic and inelastic scattering given by Eqs. (7) and (11) respectively with X(i) giving the nuclear structure part. However, the D i 's and E i 's depend not only on the nuclear structure part but also on the kinematics and assumptions on the WIMP velocity. The evaluation of X(i) depends on spin structure functions and the form factors. We have used DSM for the evaluation of these quantities. Here, for a given nucleus, starting with a model space consisting of a given set of single particle (sp) orbitals and effective two-body Hamiltonian (TBME + spe), the lowest energy intrinsic states are obtained by solving the Hartree-Fock (HF) single particle equation self-consistently. We assume axial symmetry. For example, Fig. 1 shows the HF single particle spectrum for 75 As corresponding to the lowest prolate intrinsic state. Used here are the spherical sp orbits 1p 3/2 , 0f 5/2 , 1p 1/2 , and 0g 9/2 with energies 0.0, 0.78, 1.08, and 3.20 MeV, respectively, while the assumed effective interaction is the modified Kuo interaction [39]. Excited intrinsic configurations are obtained by making particle-hole excitations over the lowest intrinsic state. These intrinsic states χ K (η) do not have definite angular momenta. Hence, states of good angular momentum are projected from an intrinsic state χ K (η) and they can be written as, (15) where N JK is the normalization constant. In Eq. (15), Ω represents the Euler angles (α, β, γ) and R(Ω) which is equal to exp(−iαJ z )exp(−iβJ y )exp( −iγJ z ) represents the general rotation operator. The good angular momentum states projected from different intrinsic states are not in general orthogonal to each other. Hence they are orthonormalized and then band mixing calculations are performed. This gives the energy spectrum and the eigenfunctions. Fig. 2 shows the calculated energy spectrum for 75 As as an example. In the DSM band mixing calculations used are six intrinsic states [23]. Let us add that the eigenfunctions are of the form The nuclear matrix elements occurring in the calculation of magnetic moments, elastic and inelastic spin structure functions etc. are evaluated using the wave function Φ J M (η). For example the calculated magnetic moments for the 3/2 1 , 3/2 2 and 5/2 1 states are (in nm units) 1.422, 1.613 and 0.312 compared to experimental values [40] 1.439, 0.98 and 0.98 respectively. The calculated values are obtained using bare gyromagnetic ratios and the results will be better for the excited states if we take g p ℓ = 0.5, g n ℓ = 0.7, g p s = 4 and g n s = −3. The neutron spin part is small and hence donot appreciably contribute to the magnetic moments of the above three states. Use of effective g-factors are advocated in [41].

IV. RESULTS FOR WIMP-23 NA SCATTERING
The nuclear structure plays an important role in studying the event rates in WIMP-nucleus scattering. Hence, we first calculate the energy spectra and magnetic moments within our DSM model for 23 Na. Agreement with experimental data will provide information regarding the goodness of the wave functions used. This in turn will give us confidence regarding the reliability of our predictions on event rates. These spectroscopic results are presented in Section IV-A. Let us add that in Sections IV-B and C the value of the oscillator length parameter b is needed and it is taken to be 1.573 fm for 23 Na. In our earlier work in the calculation of transition matrix elements for µ − e conversion in 72 Ge [37], we had taken the value of this length parameter as 1.90 fm. Assuming A 1/6 dependence, the above values of the oscillator parameter is chosen for 23 Na.

A. Spectroscopic results
In the 23 Na calculations, 16 O is taken as the inert core with the spherical single particle orbitals 0d 5/2 , 1s 1/2 and 0d 3/2 generating the basis space. "USD" interaction of Wildenthal with sp energies −3.9478, −3.1635 and 1.6466 MeV has been used in the calculation [42]. This effective interaction is known to be quite successful in describing most of the important spectroscopic features of nuclei in the 1s0d-shell region [42]. For this nucleus, the calculated lowest HF single particle spectrum of prolate shape is shown in Fig. 3. The odd proton is in the k = 3/2 + deformed single particle orbit. The excited configurations are obtained by particle-hole excitations over this lowest configuration. We have considered a total of five intrinsic configurations. As described above, angular momentum states are projected from each of these intrinsic configurations and then a band mixing calculation is performed. The band mixed wave functions S J Kη defined in Eq. (16) are used to calculate the energy levels, magnetic moments and other properties of this nucleus.
The calculated levels are classified into collective bands on the basis of the E2 transition probabilities between them. The results for lowest positive parity band for 23 Na are shown in Fig. 4. The experimental data are from Ref. [40]. For this nucleus, the ground state is 3/2 + which is reproduced in our calculation. A positive parity band built on 3/2 + has been identified for this nucleus. This band is quite well reproduced by the DSM calculation. An analysis of the wave functions shows that this band mainly originates from the lowest HF intrinsic configuration shown in Fig. 3. However, there are admixtures from the good angular momentum states coming from other intrinsic configurations. The wavefunction coming from the lowest HF intrinsic configuration slightly increases in value with increased angular momentum. This shows that the collectivity of this band does not change appreciably at higher angular momentum. Since we are considering WIMP-nucleus scattering from ground state and low lying positive parity states, the negative parity bands are not important for the present purpose. In the calculation of the event rates, spin plays an important role. Hence, magnetic moment of various lowlying levels in 23 Na are calculated. The result for the ground state of the lowest K = 3/2 + band is 2.393 nm and the corresponding available experimental data value is 2.218 nm. The contribution of protons and neutrons to the orbital parts are 0.957 and 0.262 and to the spin parts are 0.267 and 0.014, respectively. This decomposition gives better physical insight. The calculated value of magnetic moment for the ground state agrees quite well with experimental data [40]. Let us add that there are no experimental data for the magnetic moments of the excited states. An approach with state-dependent gyromagnetic moments, as advocated for example in [41] reproduces better the experimental magnetic moments. The DSM spectroscopic results are also in good agreement with the full shell model calculations reported in [15,30].

B. Results for elastic scattering
The DSM wave functions given by Eq. (16) are used to calculate the normalized spin structure functions given in Eq. (4) and also the squared nuclear form factors for these nuclei. Their values are plotted in Figs. 5 as a function of u. The static spin matrix elements Ω 0 and Ω 1 for the ground state of 23 Na have values 0.727 and 0.652 respectively. They compare well with other theoretical calculations for 23 Na given in [15,30]. An analysis of the normalized spin structure functions for 23 Na in Fig. 5 shows that the values of F 00 , F 01 and F 11 differ between u=0.4-3. Out side this region they are almost degenerate. The form factors for proton and neutron in 23 Na are almost identical up to u = 2. Afterwards they differ and beyond u=2.6 the neutron form factor becomes larger than proton form factor.
The nuclear structure dependent coefficients given in Eq. (8) are plotted in Fig. 6 for 23 Na, as a function   The peaks shift towards higher values of m χ as we go to larger threshold energy. The thickness of the graphs represents annual modulation. Annual modulation has largest value near the peaks of the graphs. Annual modulation provides strong evince regarding the observation of dark matter since the back ground does not exhibit such modulation; see [11] for a recent review on annual modulation measurements. As seen from Figs. 6 and 7, 23 Na shows larger modulation compared to heavier nuclei like 127 I, 133 Cs and 133 Xe [10]. The event detection rates for these nuclei have been calculated at a particular WIMP mass by reading out the corresponding values of D i s from the Fig. 6 and then evaluating Eq. (7) for a given set of supersymmetric parameters. The event detection rates for different values of m χ have been calculated using the nucleonic current parameters f 0 A = 3.55e−2, f 1 A = 5.31e−2, f 0 S = 8.02e−4 and f 1 S = −0.15 × f 0 S . These results are shown in Fig.  (7) for detector threshold energy Q th = 0, 10 keV for 23 Na. For 23 Na, the peak occurs at m χ ≃ 30 GeV. The event rate decreases at higher detector threshold energy but the peak shifts to the higher values of m χ occurring at ∼50 GeV.
C. Results for inelastic scattering 23 Na has 5/2 + excited state at 440 KeV above the ground state 3/2 + . Therefore, we consider inelastic scattering from the ground state for this nucleus to the 5/2 + state. The static spin matrix elements for the inelastic scattering to the J = 5/2 + are Ω 0 = −0.368, functions are given in Fig. (8). In the figures, F 00 , F 01 and F 11 are shown. The spin structure functions almost vanish above u=4. With the value of u lying between 1 to 4, the spin structure functions differ from each other. The nuclear structure coefficients E n are shown in Fig.  9 for this nucleus. The inelastic nuclear structure coefficients do not depend on the detector threshold energy. Hence the event rate can be calculated by reading the values of E i from the graph and using the nucleonic current parameters. The modulation for the inelastic scattering case is much smaller than the elastic case.

V. RESULTS FOR WIMP-40 AR ELASTIC SCATTERING
The event rates for WIMP-40 Ar elastic scattering are calculated using the nuclear wave functions generated through our DSM calculation. In our calculation, the active spherical single particles orbitals are taken as 0d 5/2 , 0d 3/2 , 1s 1/2 , 0f 7/2 , 0f 5/2 , 1p 3/2 and 1p 1/2 with 16 O as the inert core. An effective interaction named sdpf − u and developed by Nowacki and Poves [43] with single particle energies −3.699, 1.895, −2.915, 6.220, 11.450, 6.314 and 6.479 MeV, respectively for the above seven orbitals has been used. As discussed earlier, we first generate the lowest HF intrinsic state by solving the axially symmetric HF equation self-consistently. Then, we generate the excited configurations by particle-hole excitations. We have considered a total of 9 intrinsic states. Good angular momentum states are projected from each of these intrinsic states and then a band mixing calculation is performed. The band mixed wave functions defined in  Eq. (16) are used in calculating the elastic event rates and nuclear structure coefficients for the ground state of this nucleus. Note that the ground state is a 0 + state as 40 Ar is a even-even nucleus and inelastic scattering from ground needs excited 1 + state. However, the 1 + states lie very high in energy and hence only elastic scattering of WIMP from 40 Ar is important. The oscillator length parameter b for this nucleus is taken to be 1.725 f m. We have presented the nuclear structure dependent coef-ficients defined in Eq. (8) in Fig. 10 for this nucleus as a function of the WIMP mass for different values of the detector threshold. Since 40 Ar is a even-even nucleus, there is no spin contribution from the ground state. Hence, we have only D 4 , D 5 and D 6 corresponding to the proton, neutron and proton-neutron form factors as defined in Eq. (7). Their values are slightly larger compared to the corresponding quantities in 23 Na but the modulation is smaller. The peaks occur at around m χ = 35 GeV. However for larger values of Q thr , the peaks shift towards the larger m χ . The event rates for WIMP-40 Ar scattering is plotted as a function of the dark matter mass in Fig.  (11) for Q thr = 0 and 10 keV. The event rates are calculated using the same supersymmetric parameters as in 23 Na. The values are smaller than in 23 Na. This is because 40 Ar is a even-even nucleus and hence there is no spin contribution to the event rates in the ground state. At Q thr = 0, the peak occurs at 35 GeV. For Q thr = 10 keV, the peak shifts to 45 GeV.

VI. CONCLUSION
Deformed shell model is used to calculate first the event rates for the elastic and inelastic scattering of WIMP from 23 Na. Spectroscopic properties of this nucleus are calculated within DSM to check the suitability of the model. We have also calculated magnetic moments for the lowest level in this nucleus since spin plays an important role in the calculation of detection rates. Before 23 Na analysis, we have compared the DSM results also for 75 As for further confirmation of the goodness of DSM for spectroscopy. After ensuring the good agreement with experiment, we calculated the spin structure functions, form factors, nuclear structure coefficients and the event rates for WIMP-23 Na elastic and inelastic scattering. In addition, event rates for elastic scattering of WIMP from 40 Ar are also presented. Results in Figs. 7 and 11 for event rates and in Figs. 6,7 and 9-11 for the annual modulation should be useful for the upcoming and future experiments detecting WIMP involving detectors with 23 Na and 40 Ar. Let us add that the present study using DSM for the nuclear structure part is in addition to the results presented for WIMP scattering from 73 Ge in [38] and from 127 I, 133 Cs and 133 Xe in [10]. Finally, we hope that these and those obtained using other theoretical models for nuclear structure may guide the experimentalists to unravel the fundamental mysteries of dark matter particles.