Event Rates for the Scattering of Weakly Interacting Massive Particles from $^{23} $ Na and $^{40} $ Ar

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arXiv:2009.10522v1 [nucl-th] 19 Sep 2020 † ∗ IO6,EEWISads n e o example for see on; Nuclei so XENON1T, and project, [11–14]. SNOLAB EDELWEISS CDMS PICO-60, Super of are Some these detector existence. the their of regarding nuclei finger-prints the providing from scattering their via WIMP interacts and 7]. stable [1, be matter with to (lightestweakly expected (LSP) is Particle candidate which neutralino) Supersymmetric WIMP Lightest the appealing the beyond most is physics The of model. theories symmetric standard super which in (WIMP) most are arise Particles the candidates Massive However, Interacting matter matter Weakly 6]. the dark dark [1, cold for observed nonbaryonic mys- candidates yet promising a not the are still of they is one but earth- nature are in Axions its non- observed The hence yet tery. and not cold. experiments is is bound matter matter dark dark the cold [5]baryonic of Ex- most project Cosmology Background that Supernova suggest Cosmic and the non- [4] overwhelming mostly (COBE) from are is plorer data matter There Also, dark the baryonic. [1–3]. that mass believe dark to the is evidences of universe most the that physicists of and astronomers gists, 133 li e 1,1,1]adrfrne hr n u fo- Our in. on there is references paper and this 15] in 11, cus [10, see clei; neo akmte atce nteglci aousing halo galactic the in particles matter pres- dark the of investigated ence experiments [17] [16]; DAMA/LiBRA Yorkshire contains North NaI in the Collaboration by Matter operated WIMP experi- Dark for UK array experiment search (NAID) direct a Detector is ment Advanced (NaI) Iodide [email protected] [email protected] hr r ayeprmna ffrs[–1 odetect to [8–11] efforts experimental many are There cosmolo- the among agreement universal now is There sand Cs 133 1 vn ae o h cteigo ekyitrcigmassiv interacting weakly of scattering the for rates Event WM)off (WIMP) hs eut o vn ae n lofrana modulation wit annual detectors for involving also experiments and detection rates WIMP event future for results These tutr offiinsadeetrtsfreatcscatterin elastic manner, for same rates the event In and parameters. coefficients structure supersymmetric of WIMP- set the given for etc. coefficients ucin r sdfrotiigeatcadieatcspin inelastic and elastic obtaining for the Following used experiment. are with l functions compared properties and spectroscopic calculated the are First states. Hartree-Fock ainlIsiueo cec n ehooy au Hills, Palur Technology, and Science of Institute National eaeaogtepplrdtco nu- detector popular the among are Xe eeto ae o h lsi n nlsi cteigof scattering inelastic and elastic the for rates Detection .INTRODUCTION I. 23 23 Na, a iial,teDM/a and DAMA/NaI the Similarly, Na. 23 40 23 aaecluae ihntefaeoko eomdSelMod Shell Deformed of framework the within calculated are Na Ar, aand Na 2 hsclRsac aoaoy heaa 8 0,India 009, 380 Ahmedabad Laboratory, Research Physical 71 Ga, 40 r h Sodium The Ar. 73 Ge, 23 asatrn.Te,teeetrtsaeas acltdwit calculated also are rates event the Then, scattering. Na .Sahu R. Dtd etme 3 2020) 23, September (Dated: 75 from As, 127 23 1 ∗ aand Na I, ...Kota V.K.B. , 0[0 n EP30 2]eprmnsuigludAr- lquid using experiments (with [21] gon DARKSIDE- DEAP-3600 important and the [20] are 50 there and Also, [18] ANAIS [19]. re- are DM-Ice Other detectors NaI [11]. with confirmed experiments yet lated not predicted was the modulation experiments, annual these In detector. NaI(Tl) the h etioflo sasrosbcgon ore Re- nu- off source. neutrinos of (CE background scattering clei serious elastic a coherent as the known is cently detectors floor matter dark neutrino interaction the the neutrinos of The astrophysical material the the with especially emissions. neutrinos neutrino these of various to posed oe[42] nteesuis o xml hyhave they and example elastic for for WIMP-nucleus scattering for studies, inelastic rates these event pur- a the this In calculated performed for calculations have [24–28]. model collaborators pose shell his truncated gluons of and with series Suhonen WIMPs exchange, of squark interaction etc. from the arise exchange, contribute. can Higgs not interaction does scalar inelastic practically For The interaction interaction. scalar scalar and consider part, dominant current to more axial the need the also WIMP-nucleus from we coming in scattering, interaction nucleus elastic spin-spin the mat- For of dark recoil of scattering. detection the direct through of ter aspects different scribe [23]. in matter example dark floor for cold investigated for neutrino was looking the experiments of relevant impacts the of The on detection [22] direct searches. the Laboratory matter in National dark used Ridge technology Oak the the employing at Source tron 129 acltosaecridoti 1,3]frWM scat- WIMP for 30] [15, shell-model in off large-scale out tering full carried back are MeV years would calculations Few the kg-y in per 0.5-2.5 possible. mass of a events be the with that electrons WIMP found and light detecting region for of searches possibility the the in examined [29] al et uli ial,uiglresaeselmdl[1 and [31] model shell scale large using Finally, nuclei. e sadta ietdtcineprmnsaeex- are experiments detection direct that add us Let hr r aytertclcluain hc de- which calculations theoretical many are There , 131 k nrysetaadmgei moments magnetic and spectra energy ike 40 fWM from WIMP of g eand Xe Ar h tutr ucin,ncerstructure nuclear functions, structure ν ehmu-608 dsa ni and India Odisha, Berhampur-761008, odareetfrtee S wave DSM these, for agreement good 23 S a enosre tteSalto Neu- Spallation the at observed been has NS) ekyitrcigmsieparticles massive interacting weakly 2 129 ilb sflfrteucmn and upcoming the for useful be will 40 † aand Na sn S aeucin,nuclear wavefunctions, DSM using r sdetector. as Ar) , 131 133 Xe, s[6.I diin eetyVergados recently addition, In [26]. Cs 40 127 Ar. 40 83 I, raeas obtained. also are Ar rand Kr 73 l(S)bsdon based (DSM) el Ge, particles e 19 125 F, e[8 n also and [28] Te 23 Na, a h 27 land Al 127 29 Si I, 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. If we teraction with nuclei in the detector. The relevant theory neglect the rotation of the earth about its own axis, then of WIMP-nucleus scattering is well known as available in v = v is the relative velocity of WIMP with respect to the papers by Suhonen and his group and also in our the| detector.| Also, q represents the momentum transfer earlier papers mentioned above [24–26, 28, 38]. For com- to the nuclear target which is related to the dimensionless A Elastic scattering 3

of the scalar current. The nucleonic current parameters 75 0 1 4 fA and fA depend on the speciﬁc SUSY model employed. As 0 1 However, fS and fS depend, beyond SUSY model, on the hadron model used to embed quarks and gluons into nu- cleons. The normalized spin structure functions Fρρ′ (u) with ρ, ρ′ = 0,1 are deﬁned as 17/2−

− (λ,κ) (λ,κ) (17/2 ) Ωρ (u)Ωρ′ (u) 3 Fρρ′ (u)= ; ΩρΩρ′ Xλ,κ

(λ,κ) 4π (4) Ωρ (u)= 2Ji+1 A 13/2− q (13/2−) Jf [Yλ(Ωj ) σ(j)]κ jλ(√u rj )ωρ(j) Ji

Energy (MeV) ×h k ⊗ k i 2 j=1 X

In the above equation ω0(j) = 1 and ω1(j) = τ(j); note that τ = +1 for protons and 1 for neutrons and jλ is the spherical Bessel function.− The static spin matrix

− (0,1) 9/2 − elements are defined as Ωρ(0) = Ωρ (0). Now, the event (9/2 ) 1 rate can be written as 1 ψmax umax dσ (u) R = dξ dψ G(ψ, ξ) A du (5) h i du Z−1 Zψmin Zumin − − − − 1/2 5/2 3/2 5/2 − In the above, G(ψ, ξ) is given by 1/2 3/2− − 0 3/2 3/2− 2 2 ρ σ 1 c 2 2 EXPT. DSM G(ψ, ξ)= 0 0 ψe−λ e−ψ e−2λψξ m Am m b √πv χ p p 0 FIG. 2: Comparison of DSM results with experimental data (6) 75 for As for collective bands with negative parity. The exper- Here, ψ = v/v0, λ = vE/v0, ξ = cos(θ). Parameters imental values are taken from [40] used in the calculation are the following: the WIMP den- 3 −38 2 sity ρ0 = 0.3 Gev/cm , σ0 = 0.77 10 cm , mass −27 × of proton mp = 1.67 10 kg. The velocity of the 2 2 variable u = q b /2 with b being the oscillator length sun with respect to the× galactic centre is taken to be parameter. The WIMP-nucleus differential cross section v0 = 220 Km/s and the velocity of the earth relative to in the laboratory frame is given by [24–26, 28, 38] the sun is taken as v1 = 30 Km/s. The velocity of the 2 2 earth with respect to the galactic centre vE is given by dσ(u, v) 1 1 c dσA(u) 2 2 = σ ; (2) vE = v0 + v1 +2v0v1 sin(γ)cos(α) where α is the mod- du 2 0 m b v2 du p ulation angle which stands for the phase of the earth on its orbitp around the sun and γ is the angle between the with normal to the elliptic and the galactic equator which is ◦ taken to be 29.8 . Using the notations, X(1) = F00(u), ≃ 2 dσA(u) 0 2 0 1 1 2 X(2) = F01(u), X(3) = F11(u), X(4) = FZ (u) , =(fA) F00(u)+2fAfAF01(u) + (fA) F11(u) 2 | | du X(5) = FN (u) , X(6) = FZ (u) FN (u) the event rate 2 | | | || | + Z f 0 + f 1 F (u) 2 per unit mass of the detector is given by S S | Z | 0 1 2 2 0 2 0 1 1 2 + (A Z) f f FN (u) R el =(f ) D1 +2f f D2 + (f ) D3 − S − S | | h i 1 A A A 0 2 1 2 0 1 2 +2Z(A Z) (fS) (fS) FZ (u) FN (u) . + Z fS + fS D4 − − | || | (7) (3) 0 1 2 + (A Z) f f D5 where FZ (u) and FN (u) denote the nuclear form factors − S − S 0 2 1 2 for protons and neutrons respectively. In Eq. (3), the +2Z(A Z) (fS) (fS) D6 , first three terms correspond to spin contribution com- − − ing mainly from the axial current and the other three where Di being the three dimensional integrations of terms stand for the coherent part coming mainly from the Eq.(5), defined as scalar interaction. Here, f 0 and f 1 represent isoscalar A A 1 ψmax umax and isovector parts of the axial vector current and sim- D = dξ dψ G(ψ, ξ)X(i)du (8) ilarly f 0 and f 1 represent isoscalar and isovector parts i S S Z−1 Zψmin Zumin 4

The lower and upper limits of integrations given in Eq.(5) (11) respectively with X(i) giving the nuclear structure and (8) have been worked out by Pirinen et al [28] and part. However, the Di’s and Ei’s depend not only on they are the nuclear structure part but also on the kinematics and assumptions on the WIMP velocity. The evalua- c Am Q 1/2 tion of X(i) depends on spin structure functions and the ψ = p thr (9) min 2 form factors. We have used DSM for the evaluation of v0 2µr these quantities. Here, for a given nucleus, starting with a model space consisting of a given set of single particle 2 2 (sp) orbitals and effective two-body Hamiltonian (TBME 2 2 vesc v1 2v1 ψmax = λξ+ λ ξ + 1 sin(γ)cos(α) + spe), the lowest energy intrinsic states are obtained by − v2 − − v2 − v s 0 0 0 solving the Hartree-Fock (HF) single particle equation (10) self-consistently. We assume axial symmetry. For exam- With the escape velocity vesc from our galaxy to be 625 ple, Fig. 1 shows the HF single particle spectrum for 75As km/s, the value of v2 /v2 1 v2/v2 appearing in Eq. esc 0 − − 1 0 corresponding to the lowest prolate intrinsic state. Used (10) is 7.0525. Similarly, the value of (2v1/v0)sin(γ) 2 here are the spherical sp orbits 1p3/2, 0f5/2, 1p1/2, and is 0.135. The values of umin and umax are AmpQthrb 2 0g9/2 with energies 0.0, 0.78, 1.08, and 3.20 MeV, respec- and 2(ψµrbv0/c) , respectively. Here, Qthr is the detec- tively, while the assumed effective interaction is the modi- tor threshold energy and µr is the reduced mass of the fied Kuo interaction [39]. Excited intrinsic configurations WIMP-nucleus system. are obtained by making particle-hole excitations over the lowest intrinsic state. These intrinsic states χK (η) do not have definite angular momenta. Hence, states of good B. Inelastic scattering angular momentum are projected from an intrinsic state χK (η) and they can be written as, 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 J 2J +1 J ψ (η)= dΩD (Ω)R(Ω) χK (η) (15) the elastic case and hence it is neglected. Hence, we MK 8π2√N MK | i JK Z focus on spin dependent scattering. The inelastic event rate per unit mass of the detector can be written as where NJK is the normalization constant. In Eq. (15), Ω 0 2 0 1 1 2 represents the Euler angles (α, β, γ) and R(Ω) which is R in = (f ) E1 +2f f E2 + (f ) E3 (11) h i 1 A A A equal to exp( iαJ )exp( iβJ )exp( iγJ ) represents − z − y − z where E1, E2 and E3 are the three dimensional integra- the general rotation operator. The good angular mo- tions mentum states projected from different intrinsic states are not in general orthogonal to each other. Hence they 1 ψmax umax are orthonormalized and then band mixing calculations Ei = dξ dψ G(ψ, ξ)X(i)du . (12) −1 ψmin umin are performed. This gives the energy spectrum and the Z Z Z eigenfunctions. Fig. 2 shows the calculated energy spec- 75 The limits of integration for E1, E2 and E3 are [26, 28] trum for As as an example. In the DSM band mixing calculations used are six intrinsic states [23]. Let us add 2 2 that the eigenfunctions are of the form 1 2 2 v0 2 Γ umin(max) = b µr 2 ψ 1 1 2 (13) 2 c " ∓ s − ψ # J J J ΦM (η) = SKη(α) ψMK (α) . (16) where | i | i XK,α 2E∗ c2 Γ= 2 2 (14) µrc v0 The nuclear matrix elements occurring in the calculation of magnetic moments, elastic and inelastic spin struc- ∗ with E being the energy of the excited state. ψmax is ture functions etc. are evaluated using the wave function J same as in the elastic case and the lower limit ψmin = ΦM (η). For example the calculated magnetic moments √Γ. The parameters like ρ0, σ0 etc. have the same values for the 3/21, 3/22 and 5/21 states are (in nm units) as in the elastic case. 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 III. DEFORMED SHELL MODEL the results will be better for the excited states if we take p n p n gℓ = 0.5, gℓ = 0.7, gs = 4 and gs = 3. The neutron The nucleonic current part has been separated from spin part is small and hence donot appreciably− contribute nuclear part in the expression for the event rates for to the magnetic moments of the above three states. Use elastic and inelastic scattering given by Eqs. (7) and of effective g-factors are advocated in [41]. A Spectroscopic results 5

+ 23 23Na 3/2 6 Na

+ −2 (11/2 )

+ 3/2+ 11/2 5

1/2+ 1/2+ 4 −6

1/2+ + 5/2+ 5/2 3 9/2+

Energy [MeV] 9/2+

1/2+ −10 + 7/2 + 2 7/2 Energy [MeV]

+ + x x 3/2 1 3/2 o −14 5/2+ 5/2+

0 3/2+ 3/2+ EXPT. DSM

x x 1/2+ + FIG. 4: Comparison of deformed shell model results with ex- −18 1/2 o o 23 E=−66.19 (MeV) perimental data for Na for yrast band which is of positive Q=17.32 parity. The experimental values are taken from [40] K=3/2+

FIG. 3: HF single-particle spectra for 23Na corresponding to lowest configuration. In the figure, circles represent protons interaction is known to be quite successful in describing and crosses represent neutrons. The HF energy E in MeV, most of the important spectroscopic features of nuclei in mass quadrupole moment Q in units of the square of the os- the 1s0d-shell region [42]. For this nucleus, the calcu- cillator length parameter and the total azimuthal quantum number K are given in the figure. lated 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 23 are obtained by particle-hole excitations over this lowest IV. RESULTS FOR WIMP- NA SCATTERING configuration. We have considered a total of five intrinsic configurations. As described above, angular momentum The nuclear structure plays an important role in study- states are projected from each of these intrinsic configu- ing the event rates in WIMP-nucleus scattering. Hence, rations and then a band mixing calculation is performed. we first calculate the energy spectra and magnetic mo- The band mixed wave functions SJ defined in Eq. (16) 23 Kη ments within our DSM model for Na. Agreement with are used to calculate the energy levels, magnetic moments experimental data will provide information regarding the and other properties of this nucleus. goodness of the wave functions used. This in turn will The calculated levels are classified into collective bands give us confidence regarding the reliability of our pre- on the basis of the E2 transition probabilities between dictions on event rates. These spectroscopic results are them. The results for lowest positive parity band for 23Na presented in Section IV-A. Let us add that in Sections are shown in Fig. 4. The experimental data are from Ref. IV-B and C the value of the oscillator length parameter [40]. For this nucleus, the ground state is 3/2+ which is b is needed and it is taken to be 1.573 fm for 23Na. In reproduced in our calculation. A positive parity band our earlier work in the calculation of transition matrix built on 3/2+ has been identified for this nucleus. This elements for µ e conversion in 72Ge [37], we had taken band is quite well reproduced by the DSM calculation. the value of this− length parameter as 1.90 fm. Assum- An analysis of the wave functions shows that this band ing A1/6 dependence, the above values of the oscillator mainly originates from the lowest HF intrinsic configu- parameter is chosen for 23Na. ration shown in Fig. 3. However, there are admixtures from the good angular momentum states coming from other intrinsic configurations. The wavefunction com- A. Spectroscopic results ing from the lowest HF intrinsic configuration slightly in- creases in value with increased angular momentum. This In the 23Na calculations, 16O is taken as the inert core shows that the collectivity of this band does not change with the spherical single particle orbitals 0d5/2,1s1/2 and appreciably at higher angular momentum. Since we are 0d3/2 generating the basis space. ”USD” interaction of considering WIMP-nucleus scattering from ground state Wildenthal with sp energies 3.9478, 3.1635 and 1.6466 and low lying positive parity states, the negative parity MeV has been used in the calculation− − [42]. This effective bands are not important for the present purpose. A Spectroscopic results 6

1.0

F00 150 150 0.8 F 0 D 0 D 01 1 2

’ 0.6 F

ρρ 11 5

F 100 5 100 0.4 10 0.2 50 10 50 0.0 100

Proton 150 300 10−1 D D

2 0 3 4 Neutron 0 10−2 5 200

|F(u)| 100 5

−3 10 10 50 10 100 10−4 0 2 4 6 8 u 300 300 0 FIG. 5: Spin structure functions and squared proton and neu- D5 D6 23 0 tron form factors for Na for the ground state. 200 5 5 200

10 10 100 100 In the calculation of the event rates, spin plays an important role. Hence, magnetic moment of various low- 23 0 0 lying levels in Na are calculated. The result for the 0 50 100 150 50 100 150 ground state of the lowest K = 3/2+ band is 2.393 nm m and the corresponding available experimental data value χ is 2.218 nm. The contribution of protons and neutrons FIG. 6: Nuclear structure coeﬃcients plotted as a function to the orbital parts are 0.957 and 0.262 and to the spin of the WIMP mass in GeV for 23Na. The graphs are plotted parts are 0.267 and 0.014, respectively. This decomposi- for three values of the detector threshold Qthr namely Qthr = tion gives better physical insight. The calculated value 0, 5, 10 keV. The thickness of the graphs for each value of Qthr of magnetic moment for the ground state agrees quite represents the annual modulation. 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] 1.2 reproduces better the experimental magnetic moments. The DSM spectroscopic results are also in good agreement with the full shell model calculations reported in 1 [15, 30]. 0

0.8

B. Results for elastic scattering ]

−1 10 kg The DSM wave functions given by Eq. (16) are used −1 0.6 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 func- 0.4 Event rate [y tion of u. The static spin matrix elements Ω0 and Ω1 for the ground state of 23Na have values 0.727 and 0.652 respectively. They compare well with other theoretical 0.2 calculations for 23Na given in [15, 30]. An analysis of the normalized spin structure functions for 23Na in Fig. 5 0 shows that the values of F00, F01 and F11 diﬀer between 0 20 40 60 80 100 120 140 160 u=0.4-3. Out side this region they are almost degener- mχ ate. The form factors for proton and neutron in 23Na FIG. 7: The event rates in units of yr−1kg−1 as a function are almost identical up to u = 2. Afterwards they dif- 23 fer and beyond u=2.6 the neutron form factor becomes of dark matter mass in GeV for Na at detector threshold larger than proton form factor. Qth = 0, 10 keV The thickness of the curves represent the annual modulation. The nuclear structure dependent coeﬃcients given in Eq. (8) are plotted in Fig. 6 for 23Na, as a function B Results for elastic scattering 7

E1 1.0 50

F00

F01 0.8 F11

0 0.6 ] ’ ρρ −1 in F 50 E2 kg −1 0.4 [y i E

0.2 0

0.0 50 E3 0 2 4 6 8 10 u

FIG. 8: Spin structure function in the inelastic channel 5/2+ → 3/2+ for 23Na. 0 0 200 400 600 800 1000

mχ (GeV) of the WIMP mass for different values of the detector threshold. Since Ω0 and Ω1 are of same sign, Dis are all FIG. 9: Nuclear structure coefficients En in the inelastic chan- positive. The peaks of the nuclear structure coefficients nel 5/2+ → 3/2+ for 23Na. The thickness of the graphs rep- occur at around m 30 GeV at zero threshold energy. resents annual modulation. χ ∼ The peaks shift towards higher values of mχ as we go to larger threshold energy. The thickness of the graphs represents annual modulation. Annual modulation has functions are given in Fig. (8). In the figures, F00, F01 largest value near the peaks of the graphs. Annual mod- and F11 are shown. The spin structure functions almost ulation provides strong evince regarding the observation vanish above u=4. With the value of u lying between 1 of dark matter since the back ground does not exhibit to 4, the spin structure functions differ from each other. such modulation; see [11] for a recent review on annual The nuclear structure coefficients En are shown in Fig. modulation measurements. As seen from Figs. 6 and 7, 9 for this nucleus. The inelastic nuclear structure coef- 23Na shows larger modulation compared to heavier nuclei ficients do not depend on the detector threshold energy. like 127I, 133Cs and 133Xe [10]. Hence the event rate can be calculated by reading the val- The event detection rates for these nuclei have been ues of Ei from the graph and using the nucleonic current calculated at a particular WIMP mass by reading out parameters. The modulation for the inelastic scattering case is much smaller than the elastic case. the corresponding values of Dis from the Fig. 6 and then evaluating Eq. (7) for a given set of supersymmetric parameters. The event detection rates for different values 40 of mχ have been calculated using the nucleonic current V. RESULTS FOR WIMP- AR ELASTIC 0 1 0 parameters fA =3.55e 2, fA =5.31e 2, fS =8.02e 4 SCATTERING and f 1 = 0.15 f 0.− These results− are shown in Fig.− S − × S (7) for detector threshold energy Qth = 0, 10 keV for The event rates for WIMP-40Ar elastic scattering are 23Na. For 23Na, the peak occurs at m 30 GeV. The χ ≃ calculated using the nuclear wave functions generated event rate decreases at higher detector threshold energy through our DSM calculation. In our calculation, the ac- but the peak shifts to the higher values of mχ occurring tive spherical single particles orbitals are taken as 0d5/2, at 50 GeV. 16 ∼ 0d3/2,1s1/2,0f7/2,0f5/2,1p3/2 and 1p1/2 with O as the inert core. An effective interaction named sdpf u and developed by Nowacki and Poves [43] with single− parti- C. Results for inelastic scattering cle energies 3.699, 1.895, 2.915, 6.220, 11.450, 6.314 and 6.479 MeV,− respectively− for the above seven orbitals 23Na has 5/2+ excited state at 440 KeV above the has been used. As discussed earlier, we first generate the ground state 3/2+. Therefore, we consider inelastic scat- lowest HF intrinsic state by solving the axially symmet- tering from the ground state for this nucleus to the ric HF equation self-consistently. Then, we generate the 5/2+ state. The static spin matrix elements for the in- excited configurations by particle-hole excitations. We + elastic scattering to the J = 5/2 are Ω0 = 0.368, have considered a total of 9 intrinsic states. Good an- Ω = 0.462. These values are of the same order− of gular momentum states are projected from each of these 1 − magnitude as for the elastic scattering case. Again Ω0 intrinsic states and then a band mixing calculation is and Ω1 are of same sign. The inelastic spin structure performed. The band mixed wave functions defined in 8

ficients defined in Eq. (8) in Fig. 10 for this nucleus as a 300 0 0 300 D4 D5 5 function of the WIMP mass for different values of the de- 5 tector threshold. Since 40Ar is a even-even nucleus, there 200 200 10 10 is no spin contribution from the ground state. Hence, we have only D4, D5 and D6 corresponding to the proton, 100 100 neutron and proton-neutron form factors as defined in Eq. (7). Their values are slightly larger compared to the 0 0 23 50 100 150 corresponding quantities in Na but the modulation is 300 0 D6 smaller. The peaks occur at around mχ = 35 GeV. How- 5 mχ ever for larger values of Qthr, the peaks shift towards the 200 40 larger mχ. The event rates for WIMP- Ar scattering 10 is plotted as a function of the dark matter mass in Fig. 100 (11) for Qthr = 0 and 10 keV. The event rates are calculated using the same supersymmetric parameters as in 0 23Na. The values are smaller than in 23Na. This is be- 0 50 100 150 cause 40Ar is a even-even nucleus and hence there is no mχ spin contribution to the event rates in the ground state. FIG. 10: Nuclear structure coefficients plotted as a function At Qthr = 0, the peak occurs at 35 GeV. For Qthr = 10 40 of the WIMP mass mχ in GeV for Ar. The graphs are keV, the peak shifts to 45 GeV. plotted for three values of the detector threshold Qthr namely Qthr = 0, 5, 10 keV. The thickness of the graphs for each value of Qthr represents the annual modulation. VI. CONCLUSION

0.4 Deformed shell model is used to calculate ﬁrst the event rates for the elastic and inelastic scattering of WIMP 23

] 0 from Na. Spectroscopic properties of this nucleus are −1 calculated within DSM to check the suitability of the

kg 0.3

−1 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. Be- 10 fore 23Na analysis, we have compared the DSM results 75 Event rate [y 0.2 also for 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-23Na elastic and inelastic scat- 0.1 tering. In addition, event rates for elastic scattering of WIMP from 40Ar 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 0 future experiments detecting WIMP involving detectors 0 20 40 60 80 100 120 140 160 23 40 m with Na and Ar. Let us add that the present study χ using DSM for the nuclear structure part is in addition 73 FIG. 11: The event rates in units of yr−1kg−1 as a function of to the results presented for WIMP scattering from Ge 40 127 133 133 dark matter mass mχ in GeV for Ar at detector threshold in [38] and from I, Cs and Xe in [10]. Finally, Qth = 0, 10 keV The thickness of the curves represent the we hope that these and those obtained using other theo- annual modulation. retical models for nuclear structure may guide the exper- imentalists to unravel the fundamental mysteries of dark matter particles. 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 40Ar is a even-even nucleus and inelastic scattering from Acknowledgments ground needs excited 1+ state. However, the 1+ states lie very high in energy and hence only elastic scattering Thanks are due to Prof. T.S. Kosmas for his interest of WIMP from 40Ar is important. The oscillator length in this work. R. Sahu is thankful to SERB of Depart- parameter b for this nucleus is taken to be 1.725 fm. ment of Science and Technology (Government of India) We have presented the nuclear structure dependent coef- for financial support. 9

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