First microscopic evaluation of spin-dependent WIMP-nucleus scattering off 183W

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 4.0 https://creativecommons.org/licenses/by/4.0/ First microscopic evaluation of spin-dependent WIMP-nucleus scattering off 183W © 2021 the Authors Published version Pirinen, P.; Kotila, J.; Suhonen, J. Pirinen, P., Kotila, J., & Suhonen, J. (2021). First microscopic evaluation of spin-dependent WIMP-nucleus scattering off 183W. Physics Letters B, 816, Article 136275. https://doi.org/10.1016/j.physletb.2021.136275 2021 Physics Letters B 816 (2021) 136275 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb First microscopic evaluation of spin-dependent WIMP-nucleus scattering off 183W P. Pirinen a,∗, J. Kotila b,c, J. Suhonen a aUniversity of Jyväskylä, 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 a r t i c l e i n f o a b s t r a c t Article history: Received 20 January 2021 Received in revised form 29 March 2021 Accepted 30 March 2021 Available online 2 April 2021 Editor: W. Haxton Keywords: Dark matter WIMP Direct detection Interacting boson-fermion model Nuclear structure Spin structure functions We perform the first consistent calculation of elastic-scattering and inelastic-scattering structure functions for spin-dependent WIMP-nucleus scattering off 183W in a microscopic nuclear-theory framework. The nuclear structure calculations are performed in the microscopic interacting bosonfermion model (IBFM-2). Our results show that while 183W is very insensitive to spin-dependent elastic scattering, the structure function for inelastic scattering is quite sizable at small momentum transfers. Moreover, to our knowledge 183W provides the first studied case where inelastic scattering can compete with elastic scattering as the primary detection signal. ©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction The hunt for dark matter intensifies, as detectors get progressively larger and more efficient at catching the most elusive of nature’s particles. The race to find the missing mass of the universe in the form of Weakly Interacting Massive Particles (WIMPs) is led by tonne-scale liquid-xenon detectors [1,2], attempting to detect interactions of WIMPs with atomic nuclei. Complementary direct-detection efforts can still probe different parts of the dark matter parameter space, and several such approaches exist. To evaluate the WIMP-nucleus scattering event rate in a detector, one must account for the structure of the nucleus via a structure function. Reliable estimates for structure functions require a realistic model for the nucleus, and such calculations have typically been performed in the nuclear shell model [3–14]. While most nuclei currently used in dark matter detectors have been examined in detail, tungsten, used currently in the CaWO4crystals of the CRESST detector, still lacks decent theoretical description. The heavy tungsten lies in a region of deformed nuclei, providing a challenge for nuclear models. Approximations of spin-dependent elastic scattering of WIMPs off 183W have been made before using a simple odd-group model [15]. Here we present the first calcula- *Corresponding author. E-mail address: pekka.a.pirinen@jyu.fi (P. Pirinen). tion of structure functions for 183W within a complete microscopic nuclear framework. We use the microscopic Interacting Boson- Fermion Model (IBFM-2), which is designed to handle deformed nuclei in a systematic way by using spectroscopic data to pin down the details of the model Hamiltonian for a given region of nuclei of interest. The formalism was benchmarked recently with wellestablished detector nuclei, and it was shown to have reasonable accuracy when compared with earlier calculations in the nuclear shell model [16]. Great progress has been made in recent years in evaluating spin structure functions based on axial-vector currents derived from chiral effective field theory [9,10,17,18]. Refs. [9,10]included for the first time the effect of the leading long-range two-body currents, which in general act to decrease the spin-dependent WIMP- nucleus cross section at low momentum transfers. In Ref. [17]all one-body and two-body currents relevant to WIMP-nucleus scattering were derived. Consequently, Ref. [18]very recently added all pion-exchange, pion-pole, and contact currents into the formalism of Refs. [9,10]. In this article, we utilize this formalism to perform the first calculation of spin-dependent structure functions for WIMPs scattering off 183W. In Section 2we summarize the essentials of the formalism required to compute spin structure functions for WIMP-nucleus scattering. In Section 3the IBFM-2 calculation is presented. Results of our calculations are given in Section 4, and in Section 5we report the conclusions of the present work. https://doi.org/10.1016/j.physletb.2021.136275 0370-2693/©2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. P. Pirinen, J. Kotila and J. Suhonen Physics Letters B 816 (2021) 136275 2. Structure functions We utilize the formalism derived in detail in Refs. [9,10] and used previously in conjunction with the IBFM in Ref. [16]. For convenience, we summarize the main parts again here with updates to the two-body currents from Refs. [17,18]. The spin-dependent WIMP-nucleus cross section can be written as [19] 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 function, expanded in a multipole decomposition as SA(q)= L≥0Jf||L5 L(q)||Ji 2 + L≥1Jf||Tel5 L(q)||Ji 2+Jf||Tmag5 L(q)||Ji 2. (2) Here the usual longitudinal, transverse electric, and transverse magnetic multipole operators are [10] L5 L(q)=i √2L+1 A  i=1 1 2a0+a1τ3 i1+δa1(q) −2gπpn Fπq2 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 parameter, gA=1.27641(56)[20]the axial-vector coupling constant, and gπpn =13.05 the strong pion-nucleon coupling constant [10]. The operator ML,Lis defined as ML,L= jL(qri)[YL(ˆ ri)σi]L, where jLis a spherical Bessel function, YL a spherical harmonic, and σa Pauli spin operator. The coefficients δa1(q)and δaP 1(q)contain the isovector contribution of two-body currents from chiral EFT at the normalordered one-body level. We use the updated two-body currents of Ref. [17] which include all pion-exchange, pion-pole, and contact terms leading to [18] 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+ˆ c6 12mpIc6(ρ,q) Table 1 Values of the low-energy constants (LECs) used in this work. Values of c1, c3, and c4were taken from Ref. [18], ˆ c6from [21], and the range of cDwas chosen to represent reasonable values used in other works (see text). LEC Value c1−1.20(17)GeV−1 c3−4.45(86)GeV−1 c42.69(70)GeV−1 ˆ c65.83 cD−8.0...2.0 −cD 4gAχ(6) and δaP 1(q)=ρ F2 π−2(c3−2c1)M2 πq2 (m2 π+q2)2+c3+c4 3IP(ρ,q) −1+ˆ c6 12mp−2 3 c1M2 π M2 π+q2Ic6(ρ,q) −q2 M2 π+q2c3 3[Iσ 1(ρ,q)+IP(ρ,q)]+1 3c4+1 4mp ×[Iσ 1(ρ,q)+IP(ρ,q)−3Iσ 2(ρ,q)] −cD 4gAχ q2 M2 π+q2.(7) Here we use χ=700 MeV for the chiral scale. For the density we adopt a range of values ρ=0.10...0.12 fm−1. Expressions for the integrals Iσ 1, Iσ 2, Ic6, and IPare derived in Ref. [10] and for a revision of the general formalism, we refer the reader to Ref. [18]. The coefficients ciand cDare low-energy constants (LECs) that arise from the chiral expansion. It is noteworthy that we use the convention of Ref. [10] where relativistic correction factors proportional to 1 mpin c4and ˆ c6are explicitly written out in Eqs. (6)–(7) and ˆ c6is written in a dimensionless form following Refs. [10,21]. This is different from the convention of Ref. [18] where the relativistic correction factors were absorbed into the LECs. Taking the difference in conventions into account, we use the values of Ref. [18]for c1, c3, and c4based on the Roy-Steiner equation analysis of Ref. [22]. The value of ˆ c6=5.83 was taken from Ref. [21]. We note that the results shown in this paper are essentially independent of changes in ˆ c6within any reasonable range of values and any uncertainty in ˆ c6is thus ignored. The value of cDis not as readily pinned down. An argument can be made from the notion that two-body currents should account for the quenching of Gamow-Teller beta decay matrix elements in the shell model led by axial-vector currents at zero momentum transfer [18]. However, we are not confident in expanding the idea to a heavy deformed nucleus in the IBFM-2, and opt to use a conservative range of values −8.0 ≤cD≤2.0 instead. This range contains most of the values obtained for cDin relevant literature [18,23–26]. In addition we also present all results with the contact terms turned off with cD=0. For recoil energies ER≥40 keV the uncertainty in cDbecomes the leading accountable uncertainty in our calculations for elastic scattering. The values of the low-energy constants used in our calculations are summarized in Table 1. 2 P. Pirinen, J. Kotila and J. Suhonen Physics Letters B 816 (2021) 136275 Fig. 1. Experimental and IBFM-2-computed energy spectra for 183W. Only negativeparity states were computed in the IBFM-2. 3. IBFM-2 calculation For the IBFM-2 calculation the even-even 184W nucleus was used as core for the odd 183W nucleus. The IBM-2 parameters for the core tungsten nucleus, described as four proton bosons and eight neutron bosons, were taken from Ref. [27]. The valence space was chosen to span 0g7/2, 1d5/2, 1d3/2, 2s1/2, and 0h11/2proton and 0h9/2, 1f7/2, 1f5/2, 2p3/2, 2p1/2, and 0i13/2neu- tron orbitals. The single-particle energies for protons were taken from [28], where the effect of single-particle energies on occupation probabilities was studied. The occupation probabilities and quasiparticle energies for neutrons were taken from [29] where odd tungsten isotopes were studied by means of IBFM-1. Finally, the used boson-fermion interaction parameters for negative-parity states read as ρ=−0.1285, ρ=0.004 and Aρ=−0.131. The obtained low-energy spectrum for negative-parity states is shown in Fig. 1, and it corresponds very well to the experimental spectrum. The mapping of the single-fermion creation operator onto the IBFM-2 space follows the procedure introduced in Ref. [30] where evaluation of the relevant terms using exact values for the fermion matrix elements in the Generalized Seniority scheme was worked out without using the Number Operator Approximation (NOA). For the even numbered nucleons, protons for 183W, the mapping procedure of the shell model into the microscopic IBM is described in detail in Refs. [31,32], and more recently in Ref. [33]in connection with double-beta-decay studies and in Ref. [28]for calculating occupation probabilities. Basically, the shell-model creation operators of collective pairs of nucleons of pair angular momenta 0 and 2, the Sand Dpairs, 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. Although it is generally accepted that the SD-pair approximation is valid in the spherical and vibrational regions where seniority is approximately conserved, the situation is more complicated when describing rotational states in the deformed region, as in the current case of 183W. Thus, the used Otsuka-Arima-Iachello (OAI) mapping [32]may not be optimal and we recognize the uncertainty this adds to our results. In particular, higher order interactions may be necessary since in principle any fermion operator in the fermion subspace can be mapped exactly onto the corresponding operator in the boson space at the expense of introducing higher order boson interactions. Alternatively, another type of boson mapping could be employed and tested. 4. Results In this section we will present the computed spin structure functions for WIMP-nucleus scattering off 183W. We will consider both elastic scattering and inelastic scattering populating the first 3/2−state. First, however, we will start by evaluating magnetic observables involving the spin operator to roughly assess the uncertainty in our nuclear model. The ground-state magnetic moment of 183W in our IBFM- 2 calculation is +0.315 μNwhile the experimental value is +0.11778476(9) μN[34]. The computed M1 transition strength of the gamma transition from the first 3/2−state to the 1/2− ground state is 0.047 W.u., and the measured one is 0.125(5)W.u. [34]. Our calculation appears to overestimate the neutron spin expectation value of the ground state which leads to the large ground-state magnetic moment, and is likely to cause the elasticscattering spin structure function to be overestimated by a similar factor of 2–3 at zero momentum transfer. Similarly, our calculation slightly underestimates the B(M1 :3/2− 1→1/2− gs)value which suggests that the predicted inelastic-scattering structure functions might also be somewhat smaller than in reality. We decompose the spin structure function SAof Eq. (2)to isoscalar and isovector parts: SA(q)=a2 0S00(q)+a0a1S01(q)+a2 1S11(q). (8) We present our results in the conventional form of so-called proton-only (a0=a1=1) and neutron-only (a0=−a1=1) couplings: Sp(q)=S00(q)+S01(q)+S11(q), (9) Sn(q)=S00(q)−S01(q)+S11(q). (10) The calculated spin structure functions for elastic and inelastic scattering of WIMPs off 183W are shown in Fig. 2as functions of the nuclear recoil energy ER=q2/(2mA), where mAis the mass of the nucleus. For convenience, we also give a function fit to Sp and Snin the Appendix. Here we immediately notice the usual enhancement of the structure function of the even species of nucleons (here protons) by the two-body currents at small momentum transfers. With two-body currents included in the elastic channel, both Snand Spat ER>40 keV quite closely follow the structure function computed with one-body currents only, especially when the contact terms in Eqs. (6)–(7)are ignored by setting cD=0. In the inelastic channel the two-body currents systematically decrease the value of the “neutron-only” Snat all ER. We note that the inelastic channel is very stable to variation in cD. Spin-dependent elastic scattering has been deemed an ineffective detection strategy for a detector using 183W due to the small spin expectation value of the ground state estimated in the odd group model [15]. Our analysis verifies this expectation, even when taking into account the fact that our calculation likely overestimates the expectation value by roughly a factor of 2. The structure function of 183W for elastic scattering is at ER=0 keV an order of magnitude smaller than that of 129,131Xe or 125Te computed in the IBFM-2 [16]. The structure function for inelastic scattering shows promise, however. In contrast to the inelastic scattering structure functions of 129,131Xe computed in the IBFM-2 [16] and the shell model [11], and 125Te computed in the IBFM-2 [16], here the structure function does not fall as ERapproaches zero, but it gets larger. This leads to the structure function being much larger at ER=0 keV than for any of the nuclei previously studied. Interestingly this is also the 3 P. Pirinen, J. Kotila and J. Suhonen Physics Letters B 816 (2021) 136275 Fig. 2. Elastic (left panel) and inelastic (right panel) spin structure functions for 183W as functions of the recoil energy ER. Solid and dashed lines represent the one-body- current contributions to the structure function, while the corresponding colored bands show the total structure function with two-body currents included. The thickness of the band is related to the uncertainty in the low-energy constants and the range of densities (see text). The solid colored bands were computed with cD=0, and the transparent bands were computed with a range of values −8.0 ≤cD≤2.0. Fig. 3. Differential event rate of elastic and inelastic WIMP-nucleus scattering off 183W for WIMP masses of 100 GeV (left panel) and 200 GeV (right panel). We have assumed “neutron-only” couplings, a WIMP velocity distribution following the standard halo model, and a WIMP-nucleon cross section of 10−40 cm2. The thickness of the curves represents the uncertainty in the low-energy constants involved in the calculation, and we use the large range of values for cD(see text). first case where the inelastic structure function at ER=0 keV is larger than that of the elastic channel. The wave functions of the ground state and first excited state of 183W in our IBFM-2 calculation are dominated by a neutron on either the 2p1/2or 2p3/2orbital coupled to a 0+or 2+boson state. This leads to strong single particle transitions between the neutron 2porbitals in our valence space for the L =1multipole. These strong single-particle transitions cause the structure function to be large at ER=0 keV. The experimentally observed M1 transition between the first excited state and ground state of 183W is also stronger than that for 125Te or 129,131Xe, which is likely made possible via the same single-particle transitions between the 2porbitals. In addition to the structure function, there are kinematical constraints which influence the magnitude and shape of the observed signal in a detector. To gain insight to this, we compute the differential event rate as a function of the recoil energy ERfor elastic and inelastic scattering of WIMPs off a 183W target as described in Ref. [11] (originally from Ref. [35]): dR dER=√πv0 2 R0 mWIMPmA g(vmin) E0r σ(ER) 10−36 cm2 ×ρ0 0.3GeVcm −3 v0 220 km s−1.(11) Here R0=361 events/kg/d, E0is the most probable kinetic energy of the WIMP, and ρ0=0.3 GeV/cm3is the local WIMP density. To evaluate the velocity integral g(vmin)= ∞  vmin f(v+vE) vd3v,(12) where vis the WIMP velocity in the galactic frame and vEis the Earth velocity in the galactic frame, we use a Maxwell-Boltzmann velocity distribution as parameterized in the standard halo model, i.e., v0=220 km/s, vE=232 km/s, vesc =544 km/s. The WIMP- nucleus cross section reads as σ(ER)=4 3 π 2Ji+1μ μnucleon σnucleon SA(ER), (13) with μand μnucleon the reduced masses of the WIMP-nucleus and WIMP-nucleon systems, respectively. For the unknown spindependent WIMP-nucleon cross section we employ a value of σnucleon =10−40 cm2. The resulting recoil spectra using “neutrononly” couplings, SA(ER) =Sn(ER), are shown in Fig. 3for WIMP masses of 100 GeV and 200 GeV. In Fig. 3we see that inelastic scattering competes with elastic scattering at recoil energies ER10 keV. We note that the scale of the differential event rate is set by the unknown spin-dependent WIMP-nucleon cross section, and therefore the shape and relative magnitude of the elastic and inelastic event rates are interesting here instead of absolute numbers. It is also noteworthy that the lowest nuclear recoil energies would in reality be cut off by the detector threshold energy. Therefore, inelastic scattering could be an interesting signal for a detector sensitive to spin-dependent scattering solely through 183W if the WIMPs are sufficiently heavy. The question whether building such a detector would be feasible or not, we leave for others to decide. As a final remark, the shell-model prediction of Ref. [11]for 129Xe gives a differential inelastic scattering event rate larger than that obtained for 183W in the present work over a large range of roughly 15 keV <ER<150 keV. The peak for 183W is quite narrow at ER≈10 keV whereas 129Xe has a broader peak around 4 P. Pirinen, J. Kotila and J. Suhonen Physics Letters B 816 (2021) 136275 Table A.1 Fits to the spin structure functions for elastic WIMP-nucleus scattering off 183W. The functions are fit in the form Sp/n(u) =e−u13 i=0˜ ciui, where u =q2b2/2 =mAb2ERis the recoil energy ERin a dimensionless form with b =2.4069 fm and mAthe mass of the nucleus. The fits are valid and accurate to within 1% up to u =8.0(ER=310 keV). Fits are given for lower and upper limits of the proton-only and neutron-only structure functions Spand Snboth for cD=0and the range cD=−8.0...2.0. Fit parameters ˜ cifor cD=0 iSpmin Spmax Snmin Snmax 0 4.0628026E-05 1.7043047E-04 6.9641478E-03 8.1206558E-03 1 3.2268168E-05 −5.9760629E-04 −2.3236565E-02 −2.8991097E-02 2 5.4206163E-04 2.9883154E-03 5.5688850E-02 7.3363724E-02 3−2.5496629E-05 −3.5671618E-03 −6.6981154E-02 −9.3658929E-02 4−8.4409679E-04 1.6828590E-03 4.4066291E-02 6.6632663E-02 5 7.8429897E-04 −1.2540494E-04 −1.7450277E-02 −2.9043286E-02 6−3.1543751E-04 −2.3265687E-04 4.5249081E-03 8.2291122E-03 7 5.9975831E-05 1.2755383E-04 −8.5989044E-04 −1.5639590E-03 8−1.9927959E-06 −3.5297595E-05 1.3915477E-04 1.9979421E-04 9−1.4461700E-06 6.1760977E-06 −2.0308508E-05 −1.6290587E-05 10 3.1606502E-07 −7.1272539E-07 2.3685540E-06 6.8150836E-07 11 −3.0801613E-08 5.2708400E-08 −1.8697783E-07 4.6628214E-09 12 1.5174806E-09 −2.2588652E-09 8.5216601E-09 −1.7823171E-09 13 −3.0682710E-11 4.2508593E-11 −1.6763274E-10 5.4729104E-11 Fit parameters ˜ cifor cD=−8.0...2.0 iSpmin Spmax Snmin Snmax 0 3.9311863E-05 1.7115940E-04 6.9595903E-03 8.1401654E-03 1−9.0549775E-06 −5.8434165E-04 −2.3338677E-02 −2.8572328E-02 2 4.8290690E-04 3.0629340E-03 5.5532621E-02 7.3129846E-02 3−2.5581378E-04 −3.6160702E-03 −6.6583326E-02 −9.2018840E-02 4−2.8136857E-04 1.6120871E-03 4.3759207E-02 6.2872488E-02 5 2.9930161E-04 −1.5767775E-05 −1.7332136E-02 −2.5234673E-02 6−9.0795222E-05 −3.0008894E-04 4.5017959E-03 6.0262095E-03 7−2.5072909E-06 1.5175386E-04 −8.5866339E-04 −7.5683664E-04 8 8.6938420E-06 −4.0897077E-05 1.3957384E-04 3.8411073E-06 9−2.5287693E-06 7.0425176E-06 −2.0412545E-05 1.5694760E-05 10 3.7027537E-07 −8.0245682E-07 2.3796742E-06 −2.7921324E-06 11 −3.0761458E-08 5.8705919E-08 −1.8762349E-07 2.4535925E-07 12 1.3863254E-09 −2.4932804E-09 8.5412640E-09 −1.1409277E-08 13 −2.6457518E-11 4.6581547E-11 −1.6787279E-10 2.2378186E-10 ER≈30 keV. Therefore advantages of using 183W over xenon in a detector are not obvious and would have to originate from the shape of the recoil spectrum or the elastic/inelastic scattering ratio. 5. Summary and conclusions We have reported the first calculation of spin-dependent structure functions for WIMP-nucleus scattering for the deformed 183W nucleus. Our results include the leading effects of two-body currents derived from chiral EFT, and the nuclear-structure calculations were performed in a reliable microscopic nuclear framework of IBFM-2. We confirm the earlier expectation of suppressed detection possibilities via spin-dependent elastic scattering. For inelastic scattering, however, we find the important “neutron-only” structure function to be sizable especially for small momentum transfers. Moreover, the inelastic-scattering structure function dominates over the elastic-scattering structure function for all momentum transfers. This makes 183W to our knowledge the first nucleus studied where inelastic scattering could be the dominant experimental signal. However, judging by the estimated recoil spectra, there seem to be no obvious advantages of using 183W instead of xenon, for example. Our results are of interest to the CRESST collaboration and possible future detectors using tungsten in one form or other. A study to estimate the spin-independent scattering response of all stable W isotopes will follow. The framework of the present work will also be applied to other heavy deformed nuclei that could be used in future detectors. Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: This work was supported by the Academy of Finland under Project No. 318043. The work of JK was supported by the Academy of Finland (Grant Nos. 314733 and 320062). Acknowledgements This work was supported by the Academy of Finland under Project No. 318043. The work of JK was supported by the Academy of Finland (Grant Nos. 314733 and 320062). Appendix A. Fits to spin structure functions In this appendix we provide fits to the spin structure functions shown in Fig. 2. The fit parameters for Spand Snfor elastic and inelastic scattering are given in Tables A.1 and A.2, respectively. References [1] E. Aprile, J. Aalbers, F. Agostini, M. Alfonsi, L. Althueser, F.D. Amaro, M. Anthony, F. Arneodo, L. Baudis, B. Bauermeister, M.L. Benabderrahmane, T. Berger, P.A. Breur, A. Brown, A. Brown, E. Brown, S. Bruenner, G. Bruno, R. Budnik, C. Capelli, J.M.R. Cardoso, D. Cichon, D. Coderre, A.P. Colijn, J. Conrad, J.P. Cussonneau, M.P. Decowski, P. de Perio, P. Di Gangi, A. Di Giovanni, S. Diglio, A. Elykov, G. Eurin, J. Fei, A.D. Ferella, A. Fieguth, W. Fulgione, A. Gallo Rosso, M. Galloway, F. Gao, M. Garbini, C. Geis, L. Grandi, Z. Greene, H. Qiu, C. Hasterok, E. Hogenbirk, J. Howlett, R. Itay, F. Joerg, B. Kaminsky, S. Kazama, A. Kish, G. Koltman, H. Landsman, R.F. Lang, L. Levinson, Q. Lin, S. Lindemann, M. Lindner, F. Lombardi, J.A.M. Lopes, J. Mahlstedt, A. Manfredini, T. Marrodán Undagoitia, J. Masbou, D. Masson, M. Messina, K. Micheneau, K. Miller, A. Molinario, K. Morå, 5 P. Pirinen, J. Kotila and J. Suhonen Physics Letters B 816 (2021) 136275 Table A.2 Same as Table A.1 but for inelastic scattering. Fit parameters ˜ cifor cD=0 iSpmin Spmax Snmin Snmax 0 1.1541016E-04 4.6107254E-04 1.8330374E-02 2.1347646E-02 1−7.4033802E-05 −1.3057754E-03 −6.2671503E-02 −7.4896303E-02 2 1.9196116E-03 5.3175913E-03 1.2610497E-01 1.5636725E-01 3−2.5643292E-03 −7.1539000E-03 −1.3866333E-01 −1.8379809E-01 4 1.1138779E-03 4.7599179E-03 8.8094267E-02 1.3152582E-01 5−2.1130042E-04 −2.1897267E-03 −3.3815932E-02 −6.2132540E-02 6 1.2440556E-04 9.0531722E-04 7.8796100E-03 2.0740850E-02 7−9.7358848E-05 −3.2171216E-04 −1.0083310E-03 −5.1246516E-03 8 3.8086483E-05 8.4068645E-05 2.5219665E-05 9.5350491E-04 9−8.3481302E-06 −1.4888691E-05 1.4258744E-05 −1.3160143E-04 10 1.0990465E-06 1.7207270E-06 −2.6421510E-06 1.2920317E-05 11 −8.6871347E-08 −1.2404284E-07 2.2850136E-07 −8.4304113E-07 12 3.8153744E-09 5.0621079E-09 −1.0358116E-08 3.2491605E-08 13 −7.1761082E-11 −8.9334987E-11 1.9801732E-10 −5.5667132E-10 Fit parameters ˜ cifor cD=−8.0...2.0 iSpmin Spmax Snmin Snmax 0 1.1171785E-04 4.6296577E-04 1.8318307E-02 2.1399771E-02 1−1.6148035E-04 −1.2616204E-03 −6.2921343E-02 −7.4037790E-02 2 2.2289516E-03 5.1051399E-03 1.2756169E-01 1.5120805E-01 3−2.9256834E-03 −6.7419319E-03 −1.4192336E-01 −1.7212785E-01 4 1.1941350E-03 4.3182923E-03 9.2023658E-02 1.1735971E-01 5−1.1204481E-05 −1.8983577E-03 −3.6706721E-02 −5.1648625E-02 6−9.8198636E-05 7.8053263E-04 9.2658241E-03 1.5685423E-02 7 1.5427874E-05 −2.8584467E-04 −1.4583603E-03 −3.4745391E-03 8 4.3647429E-06 7.7020025E-05 1.2598405E-04 5.8210777E-04 9−1.9769259E-06 −1.3939425E-05 −1.3183729E-06 −7.3904997E-05 10 3.2883753E-07 1.6347129E-06 −1.0104029E-06 6.8490646E-06 11 −2.9005877E-08 −1.1905059E-07 1.1804384E-07 −4.3037800E-07 12 1.3514010E-09 4.8955553E-09 −6.0043001E-09 1.6166542E-08 13 −2.6286672E-11 −8.6928771E-11 1.2222857E-10 −2.7157319E-10 M. 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