Spin-dependent WIMP-nucleus scattering off 125Te, 129Xe, and 131Xe in the microscopic interacting boson-fermion model

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Spin-dependent WIMP-nucleus scattering off 125Te, 129Xe, and 131Xe in the microscopic interacting boson-fermion model © 2019 The Authors Published version Pirinen, Pekka; Kotila, Jenni; Suhonen, Jouni Pirinen, P., Kotila, J., & Suhonen, J. (2019). Spin-dependent WIMP-nucleus scattering off 125Te, 129Xe, and 131Xe in the microscopic interacting boson-fermion model. Nuclear Physics A, 992, Article 121624. https://doi.org/10.1016/j.nuclphysa.2019.121624 2019 Available online at www.sciencedirect.com ScienceDirect Nuclear Physics A 992 (2019) 121624 www.elsevier.com/locate/nuclphysa Spin-dependent WIMP-nucleus scattering off 125Te, 129Xe, and 131Xe in the microscopic interacting boson-fermion model P. Pirinen a,∗, J. Kotila b,c, J. Suhonen a aUniversity of Jyvaskyla, Department of Physics, P. O. Box 35 (YFL), FI-40014, Finland bFinnish Institute for Educational Research, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland cCenter for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, CT 06520-8120, USA Received 19 August 2019; accepted 4 September 2019 Available online 10 September 2019 Abstract We perform calculations of structure functions for elastic and inelastic spin-dependent scattering of weakly interacting massive particles (WIMPs) off 125Te, 129Xe, and 131Xe. The nuclear structure calculations are performed in the microscopic interacting boson-fermion model (IBFM-2). In our calculations we employ one-body and leading long-range two-body WIMP-nucleus currents derived from chiral effective field theory. We demonstrate that the relevant matrix elements can be reliably computed in the IBFM-2, which will allow investigation of heavy deformed nuclei previously inaccessible to theoretical calculations. ©2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Dark matter; WIMP; Scattering; Interacting boson-fermion model; Nuclear structure 1. Introduction To this day, we do not know what most of our Universe is made of. Based on cosmic microwave background measurements [1,2], observations of galactic rotation curves [3–6], studies of structure formation [7,8], backed up by the Bullet Cluster observations, a nonbaryonic cold *Corresponding author. E-mail address: [email protected] (P. Pirinen). https://doi.org/10.1016/j.nuclphysa.2019.121624 0375-9474/©2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2P. Pirinen et al. / Nuclear Physics A 992 (2019) 121624 dark matter component that forms roughly 80% of all matter is required. The problem of dark matter still continues to puzzle scientists, and efforts to finally find the majority of matter in our Universe grow ever stronger. If dark matter consists mainly of weakly interacting massive particles (WIMPs), directly detecting the interaction of a WIMP with an atomic nucleus would be an excellent probe to the properties of dark matter [9]. We will assume the WIMPs have a nonzero spin (spin 1/2 in this work), and we focus on spin-dependent WIMP-nucleus scattering [10]. Analyzing detector signals of spin-dependent WIMP-nucleus scattering requires detailed information of the nuclear structure of the target nucleus in form of structure functions. Computing these structure functions requires a reliable microscopic model of the nucleus. Calculations of structure functions for WIMP-nucleus scattering have typically been performed in the nuclear shell model [11–22]. However, heavy target nuclei far away from closed shells present a challenge for the shell model. Here we consider a different approach in the microscopic interacting boson-fermion model (IBFM-2). The IBFM-2 models heavy nuclei and deformed nuclei well by design [23]. To our knowledge, only one calculation exists in the early years of WIMP-nucleus scattering studies, where ground-state spin expectation values for a range of nuclei were calculated in the IBFM [24]. The analysis of Ref. [24]was as such limited to zero momentum transfer. Our aim in this paper is to demonstrate that the IBFM-2 can be used to reliably compute structure functions for heavy nuclei used in dark matter direct detection by comparing calculations for 125Te, 129Xe, and 131Xe to earlier benchmark shell-model calculations in the literature. The IBFM-2 approach can then be used to gain information on deformed heavy nuclei used in dark matter detectors, such as 183W in the CRESST detector [25]. Axial-vector WIMP-nucleus currents were derived in the framework of chiral effective field theory (chiral EFT) in Refs. [11,12]. In addition to the conventional one-body currents [10], Refs. [11,12]were able to include the leading long-range two-body currents, which turned out to have a noticeable effect on the structure functions of spin-dependent WIMP-nucleus scattering. In Ref. [13]the analysis was extended to inelastic scattering of WIMPs off the odd-mass xenons. In the present work we employ the axial-vector WIMP-nucleus currents from chiral EFT, and show results for combined one and two-body currents along with results computed with one-body currents only. This article is organized as follows. In Section 2we outline the formalism for computing the axial-vector structure functions for spin-dependent WIMP-nucleus scattering. In Section 3the details of the performed IBFM-2 calculation are given. In Section 4we discuss our main results, and finally we summarize and draw conclusions in Section 5. 2. Structure functions The spin-dependent WIMP-nucleus cross section can be written as [10] dσ dq2=8G2 F (2Ji+1)v2SA(q), (1) where GFis the Fermi coupling constant, vis the speed of the WIMP in the laboratory frame, and qis the momentum transfer from the nucleus to the WIMP. SAis the axial-vector structure factor which can be expressed as a multipole decomposition as SA(q) = L≥0Jf||L5 L(q)||Ji 2 P. Pirinen et al. / Nuclear Physics A 992 (2019) 121624 3 + L≥1Jf||Tel5 L(q)||Ji 2+Jf||Tmag5 L(q)||Ji 2.(2) Here we have L5 L(q) =i √2L+1 A  i=1 1 2a0+a1τ3 i1+δa1(q) −2gπpnFπq2 2mpgA(q2+m2 π)+δaP 1(q) ×√L+1ML,L+1(qri)+√LML,L−1(qri),(3) Tel5 L(q) =i √2L+1 A  i=1 1 2a0+a1τ3 i1−2q2 2 A+δa1(q) ×−√LML,L+1(qri)+√L+1ML,L−1(qri),(4) and Tmag5 L(q) =i √2L+1 A  i=1 1 2a0+a1τ3 i1−2q2 2 A+δa1(q)ML,L(qri), (5) where Fπ=92.4 MeV is the pion decay constant, mπ=138.04 MeV the pion mass, mp= 938.27 MeV the proton mass, A=1040 MeV the axial mass scale, gA=1.26 the axial-vector coupling constant, and gπpn =13.05 the strong pion-nucleon coupling constant [12]. The operator ML,Lis defined as ML,L=jL(qri)[YL(ˆ ri)σi]L, where jLis a Bessel function, YLa spherical harmonic, and σa Pauli spin operator. The effect of two-body currents from chiral EFT enters the structure functions (3)–(5)in the coefficients δa1(q) and δaP 1(q). They are defined as [12] δa1(q) =− ρ F2 π1 3c4+1 4mp3Iσ 2(ρ, q) −Iσ 1(ρ, q) +1 3−c3+1 4mpIσ 1(ρ, q) −1+ˆc6 12mpIc6(ρ, q)(6) and δaP 1(q) =ρ F2 π−2c3q2 m2 π+q2+c3+c4 3IP(ρ, q) +1+ˆc6 12mp Ic6(ρ, q).(7) For details about the integrals Iσ 1, Iσ 2, Ic6, and IP, and the formalism in general, see Ref. [12]. We follow the choices of Ref. [12]for the values of low-energy couplings c3, c4, and ˆc6. We thus take ˆc6=5.83 [26], and a range of values of c3=−2.2 ... −4.78 GeV−1and c4= 2.4 ... 5.4 GeV−1combined from Refs. [27–30]. For the density we adopt the range of values ρ=0.10 ... 0.12 fm−3. It should be noted that the dependence of our results on variation of the parameter ˆc6is very mild and it only affects the structure functions at quite high q. Therefore we neglect the uncertainty in this parameter as the uncertainties in c3and c4are much more significant. 4P. Pirinen et al. / Nuclear Physics A 992 (2019) 121624 Table 1 Boson-fermion interaction parameters (MeV). Nucleus ρρAρ 125Te 1.2 0.04 −0.76 129Xe 0.85 0.28 −0.38 131Xe 0.51 0.32 −0.495 Fig. 1. Experimental and computed energy levels of (from left to right) 125Te, 129Xe, and 131Xe. 3. IBFM-2 calculation For the IBFM-2 calculation even-even 126Te, 130Xe and 132Xe nuclei were used as core to the odd 125Te, 129Xe and 131Xe nuclei, respectively. The parameters for the core Xe nuclei were taken from Ref. [31] with following modifications for A=130: ξ1=0.12, cν 2=0.00 and for A=132: κ=−0.10, χν=0.6, ξ1=0.12, cν 4=−0.131. For the Te core nucleus the starting parameters were taken from Ref. [32] and modified as =0.72, κ=−0.03, ξ1=ξ3=0.00, ξ2=0.25, cν 0=0.6. The valence space was chosen to span the 0g7/2, 1d5/2, 1d3/2, 2s1/2and 0h11/2proton and neutron orbitals with unperturbed single particle energies taken from [33], where the effect of single particle energies to occupation probabilities was studied. The used boson-fermion interaction parameters are listed in Table 1. The mapping of the single fermion creation operator onto the IBFM space follows the procedure introduced in Ref. [34] where evaluation of the relevant terms using exact values for the fermion matrix elements in the Generalized Seniority scheme was worked out and use of Number Operator Approximation (NOA) is avoided. For the even numbered nucleons, protons in the cases of interest here, the mapping procedure from the shell model into the microscopic IBM is described in detail in Refs. [35,36] and more recently in Ref. [37]in connection with studies of double beta decay and in Ref. [33]for calculating occupation probabilities. Basically, the shellmodel creation operators of collective pairs of angular momenta 0 and 2, the S and D pairs of interest here, are used to span the SD fermion space, which is a subspace of the full shell model space. The states of the SD subspace are then mapped onto boson states belonging to the IBM space. In Fig. 1we show the experimental and calculated low-lying energy spectra of 125Te, 129Xe, and 131Xe. The energy of the first excited state is fitted exactly to the experimental value by the P. Pirinen et al. / Nuclear Physics A 992 (2019) 121624 5 interaction parameters. The correspondence between calculated and experimental energy levels is quite good, especially for the positive-parity states. 4. Results We express our results for structure functions SAas a function of the momentum transfer in a dimensionless form u =b2q2/2, where bis the harmonic oscillator length. We use a decomposition to isoscalar and isovector parts: SA(u) =a2 0S00(u) +a0a1S01(u) +a2 1S11(u). (8) For the convenience of experiments, we present our results in form of so called proton-only (a0=a1=1) and neutron-only (a0=−a1=1) couplings: Sp(u) =S00(u) +S01(u) +S11(u), (9) Sn(u) =S00(u) −S01(u) +S11(u). (10) We will start by discussing some key magnetic properties of the target nuclei. The groundstate spin expectation values Spand Sndetermine the spin structure function SAfor elastic scattering at zero momentum transfer: SA(0)=(2J+1)(J +1) 4πJ (a0+a1+δa1(0))Sp+(a0−a1−δa1(0))Sn 2.(11) The magnetic dipole moment of the ground state and the M1 transition strength from the first excited state to the ground state involve the spin operator, and therefore these quantities can be used to give a rough idea of how the modeling error in our nuclear structure calculations might carry over to the WIMP-nucleus scattering results of elastic and inelastic scattering, respectively. In Table 2we present the computed spin expectation values for protons and neutrons in the ground state along with the computed and experimental ground-state magnetic moments for each of our target nuclei. We also compare our values with earlier shell-model calculations and the IBFM calculation of Ref. [24]. The spin expectation values for neutrons computed in the present work are in general smaller than in earlier shell-model calculations. It should be noted that the simplified IBFM calculation of Ref. [24]gives spin expectation values quite close to one-particle estimates, which accounts for the values being considerably larger than those computed in the present work. In Ref. [21]the computed ground-state magnetic moment for 125Te was overestimated much more than in the present IBFM-2 calculation. Therefore the magnetic properties of the ground state are likely to be better represented by the present calculation. Our calculation for 125Te is much more in line with the results of Ref. [15]. For the xenons our magnetic moments are in decent agreement with experiment, although the magnetic moment is slightly overestimated for 131Xe. In Ref. [11]the ground-state magnetic moments for the xenons were computed using effective gfactors, which yielded a good agreement with the experimental values. In the present work we used bare gfactors, which makes a direct comparison between the magnetic moment calculations difficult. In Table 3we show the calculated and experimental B(M1)transition strengths for the transition from the lowest excited state to the ground state for each of our target nuclei. For 131Xe the calculated value lies only slightly below the experimental lower limit. For 129Xe and especially 125Te the calculated values are significantly smaller than the experimental values. A somewhat similar difference between calculated and experimental B(M1) value for 125Te was obtained in 6P. Pirinen et al. / Nuclear Physics A 992 (2019) 121624 Table 2 Ground-state spin expectation values and ground-state magnetic moments for 125Te, 129Xe, and 131Xe. The calculations were made using bare magnetic gfactors, i.e., gs,n=−3.826, gs,p=5.586, gl,n=0, and gl,p=1. The results of the present calculation are compared to earlier shell model (SM) and interacting boson-fermion model (IBFM) calculations. The experimental data of column 6 was read from Ref. [38]. Nucleus Calculation SpSnμth gs (μN)μexp gs (μN) 125Te Present IBFM-2 −0.00008 0.266 −1.017 −0.8885051(4) SM [21]−0.00663 0.427 −1.598 SM [15]a) 0.001 0.287 −1.015 SM [15]b) −0.003 0.323 −1.134 IBFM [24]−0.0008 0.499 129Xe Present IBFM-2 −0.0078 0.216 −0.765 −0.7779763(84) SM [11,12]0.010 0.329 SM [19]−0.0019 0.273 −0.94 SM [15]a) 0.028 0.359 −0.983 SM [15]b) 0.0128 0.300 −0.701 IBFM [24]0.000 0.430 131Xe Present IBFM-2 −0.0222 −0.188 +0.896 +0.691862(4) SM [11,12]−0.009 −0.272 SM [19]−0.00069 −0.125 +0.72 SM [15]a) −0.009 −0.227 +0.980 SM [15]b) −0.012 −0.217 +0.979 IBFM [24]0.000 −0.277 Table 3 Calculated (column 3) and experimental (column 4) M1 transition strengths B(M1)for the transition from the first excited state to the ground state for 125Te, 129Xe, and 131Xe. The calculations were made using bare magnetic gfactors, i.e., gs,n=−3.826, gs,p=5.586, gl,n= 0, and gl,p=1. The experimental data was read from Ref. [38]. The values are given in Weisskopf units. Nucleus Calculation B(M1)(W.u.) th exp 125Te Present IBFM-2 0.0018 0.0226(4) SM [21]0.0056 129Xe Present IBFM-2 0.0068 0.0281(7) SM [19]0.023 131Xe Present IBFM-2 0.034 >0.057 SM [19]0.033 the shell-model calculation of Ref. [21]. In both this work and Ref. [21]the structure functions for inelastic WIMP-nucleus scattering might thus be underestimated for 125Te. Here it should be noted that the magnetic moment and B(M1)values do not probe exactly the same physics that is involved in WIMP-nucleus scattering. However, the involved operators are similar, and as possibilities for systematic error analysis are limited in these kinds of calculations, we take the opportunity to use whatever measure of accuracy that is available. One could in principle find effective gfactors to improve agreement with experiment for the magnetic moments and B(M1)values, and then renormalize the spin operator accordingly to deliver the effect into the axial-vector structure functions. This approach has been taken in Ref. [19]. P. Pirinen et al. / Nuclear Physics A 992 (2019) 121624 7 Fig. 2. (Color online.) Proton-only and neutron-only spin structure functions Spand Snfor elastic (upper panel) and inelastic (lower panel) scattering for 125Te. Results are shown for one-body currents only as solid (Sp) and dashed (Sn) lines as well as for two-body currents as red striped (Sp) and blue (Sn) bands. The thickness in the two-body current results represent the uncertainty in the low-energy couplings c3, c4, and the density ρ. However, we choose to use the bare gfactors due to the ambiguity related to choosing suitable effective gfactors and the fact that there is no guarantee that the renormalization should be the same for WIMP-nucleus interactions. The proton-only and neutron-only spin structure functions of Eqs. (9)and (10)for 125Te, 129Xe, and 131Xe are shown in Figs. 2, 3, and 4, respectively. We show results with and without the two-body current contributions. In our calculations we include the uncertainty arising from the unknown values of the c3and c4low-energy couplings as well as the density ρin the two-body contributions of Eqs. (6) and (7)as discussed in Section 2. The uncertainties are represented by colored error bands in our figures, and the bands correspond to values of c3=−2.2 ... −4.78 GeV−1and c4=2.4 ... 5.4 GeV−1, and ρ=0.10 ... 0.12 fm−3. For 125Te we can compare our results with the shell-model calculations of Ref. [21]. The formalism of Ref. [21]is somewhat different than the one used in this work in that only one-body axial-vector currents were included and the structure functions were normalized to unity at zero momentum transfer. Translated to the S-function formalism of the present work we note that the structure functions of [21]are larger in magnitude than the structure functions computed in the present work for both elastic and inelastic scattering. This is consistent with the larger computed ground-state magnetic moment and first excited state to ground state B(M1)values of Tables 2 and 3. The elastic scattering results of the present work could be considered more accurate than the results of [21]as the ground-state magnetic moment is in better agreement with experiment. The present calculation yields a worse agreement with experiment in the B(M1)values, though, 8P. Pirinen et al. / Nuclear Physics A 992 (2019) 121624 Fig. 3. (Color online.) Same as Fig. 2,butfor129Xe. Fig. 4. (Color online.) Same as Fig. 2,butfor131Xe.