Nuclear spin-dependent interactions: Searches for WIMP, Axion and Topological Defect Dark Matter, and Tests of Fundamental Symmetries

We calculate the proton and neutron spin contributions for nuclei using semi-empirical methods, as well as a novel hybrid \emph{ab initio}/semi-empirical method, for interpretation of experimental data. We demonstrate that core-polarisation corrections to \emph{ab initio} nuclear shell model calculations generally reduce discrepancies in proton and neutron spin expectation values from different calculations. We derive constraints on the spin-dependent P,T-violating interaction of a bound proton with nucleons, which for certain ranges of exchanged pseudoscalar boson masses improve on the most stringent laboratory limits by several orders of magnitude. We derive a limit on the CPT and Lorentz-invariance-violating parameter $|\tilde{b}_{\perp}^p|<7.6 \times 10^{-33}$ GeV, which improves on the most stringent existing limit by a factor of 8, and demonstrate sensitivities to the parameters $\tilde{d}_{\perp}^p$ and $\tilde{g}_{ D\perp}^p$ at the level $\sim 10^{-29} - 10^{-28}$ GeV, which is a one order of magnitude improvement compared to the corresponding existing sensitivities. We extend previous analysis of nuclear anapole moment data for Cs to obtain new limits on several other CPT and Lorentz-invariance-violating parameters: $\left|b_0^p \right|<7 \times 10^{-8}$ GeV, $\left|d_{00}^p \right|<8 \times 10^{-8}$, $\left|b_0^n \right|<3 \times 10^{-7}$ GeV and $\left|d_{00}^n \right|<3 \times 10^{-7}$.


I. INTRODUCTION
The violation of the fundamental symmetries of nature is an active area of research. Atomic and molecular experiments, which probe P-odd and P,T -odd interactions, provide very sensitive tests of the Standard Model (SM) and physics beyond the SM [1][2][3]. Measurements and calculations of the Cs 6s-7s parity nonconserving (PNC) amplitude stand as the most precise atomic test of the SM electroweak theory to date, see e.g. [4][5][6][7][8][9][10][11]. Experimental searches for nuclear anapole moments are ongoing in Fr [12], Yb [13,14] and BaF [15,16]. At present, Hg provides the most precise limits on the electric dipole moment (EDM) of the proton, quark chromo-EDM and P,T -odd nuclear forces, as well as the most precise limits on the neutron EDM and quantum chromodynamics (QCD) θ term from atomic or molecular experiments [17,18], while ThO provides the most precise limit on the electron EDM [19]. Most recently, it was suggested that EDM measurements in molecules with P,T -odd nuclear magnetic quadrupole moments may lead to improved limits on the strength of P,T -odd nuclear forces, proton, neutron and quark EDMs, quark chromo-EDM and the QCD θ term [20].
Field theories, which are constructed from the principles of locality, spin-statistics and Lorentz invariance, conserve the combined CPT symmetry. The violation of one or more of these three principles, presumably from some form of ultra-short distance scale physics, opens the door for the possibility of CPT -odd physics. Some of the most stringent limits on CPT -odd and Lorentzinvariance-violating physics come from searches for the couplingb · s between a background cosmic field,b, and the spin of an electron, proton, neutron and muon, s * y.stadnik@unsw.edu.au [21][22][23][24][25][26][27][28][29][30][31][32]. For further details on the broad range of experiments performed and a brief history of the improvements in these limits, we refer the reader to the reviews of [33,34] and the references therein.
Many tests of the fundamental symmetries of nature and searches for axion dark matter involve couplings of the form X · s N between a field or operator X and the spin angular momentum s N of a proton (N = p) or neutron (N = n). We point out that in experiments, which measure nuclear spin-dependent (NSD) properties, the contribution of non-valence nucleon spins cannot be neglected, due to polarisation of these nucleons by the valence nucleon(s). Nuclear many-body effects have previously been considered in association with the interpretation of atomic clock experiments [58][59][60], nuclearsourced EDMs and NSD-PNC interactions mediated via Z 0 -boson exchange between electrons and the nucleus (see e.g. [60]), static spin-gravity couplings [61,62] and long-range dipole-dipole couplings [62].
In the present work, we calculate the proton and neutron spin contributions for a wide range of nuclei, which are of experimental interest in tests of the fundamental symmetries of nature and searches for dark matter, including axions, weakly-interacting massive particles (WIMPs) and topological defects, using semi-empirical methods. As an illustration of the importance of manybody effects in such studies, we revisit the experiments of Refs. [29,32], in which a 3 He/ 129 Xe comagnetometer was used to place constraints on the CPT and Lorentzinvariance-violating parameterb n ⊥ , which quantifies the interaction strength of a background field with the spin of a neutron. We show that, due to nuclear many-body effects, the 3 He/ 129 Xe system is in fact also particularly sensitive to proton interaction parameters. By reanalysing the results of [32], we derive a limit on the parameterb p ⊥ that is the world's most stringent by a factor of 35. We also extend our previous analysis of nuclear anapole moment data for Cs [63] to obtain new limits on several other CPT and Lorentz-invariance-violating parameters.

II. NUCLEAR THEORY
The nuclear magnetic dipole moment µ can be expressed (in the units of the nuclear magneton µ N = e /2m N ): where s z p and s z n are the expectation values of the total proton and neutron spin angular momenta, respectively, while l z p is the expectation value of the total proton orbital angular momentum. In the present work, we consider nuclei with either one valence proton or one valence neutron (even-even nuclei are spinless due to the nuclear pairing interaction).
We start by considering the contribution of the valence nucleon alone. Assuming all other nucleons in the nucleus are paired (and ignoring polarisation of the nuclear core for now), the spin I and nuclear magnetic dipole moment µ are due entirely to the total angular momentum of the external nucleon: I = j = l + s. In this case, the nuclear magnetic dipole moment is given by the Schmidt (singleparticle approximation) formula with The gyromagnetic factors are: g l = 1, g s = g p = 5.586 for a valence proton and g l = 0, g s = g n = −3.826 for a valence neutron. We present the values for s z 0 from Eq. (3) ("Schmidt model") in Tables I and II. Experimentally, the Schmidt model is known to overestimate the magnetic dipole moment in most nuclei. The simplest explanation for this is that the valence nucleon polarises the core nucleons, reducing the magnetic dipole moment of the nucleus. The degree of core polarisation can be estimated using experimental values of the mag-netic dipole moment, and improved estimates for s z p and s z n can hence be obtained. The reduction in nuclear magnetic dipole moment from the Schmidt value µ 0 to the experimental value µ can proceed by a number of mechanisms. The simplest and most efficient way is to assume that the internucleon spinspin interaction transfers spin from the valence proton (neutron) to core neutrons (protons): where s z p 0 and s z n 0 are the Schmidt model values (one of which is necessarily zero). In general, there is also polarisation of the proton (neutron) core by the valence proton (neutron), but transfer of valence proton (neutron) spin to core proton (neutron) spin does not change the result. Note that the denominator g p − g n = 9.412 in (5) is a large number, so the required change in s z p and s z n to obtain the experimental value µ is minimal. We present the values for s z p and s z n from Eq. (5) ("minimal model") in Tables I and II. It is also possible for a reduction in nuclear magnetic dipole moment to occur by different mechanisms, for instance, by transfer of the spin angular momentum of a valence proton (neutron) to core proton (neutron) orbital angular momenta, or in a more unlikely manner by transfer of valence proton (neutron) spin angular momentum to core neutron (proton) orbital angular momenta.
In the present work, we employ the preferred model of Refs. [59,60], which is intermediate to the two previously mentioned "extreme models". We assume that the total z projections of proton and neutron angular momenta, j z p and j z n , are separately conserved, and that the z projections of total spin and orbital angular momenta, s z p + s z n and l z p + l z n , are also separately conserved (which corresponds to the neglect of the spin-orbit interaction). In this case where j z p = I for a valence proton and j z p = 0 for a valence neutron, with s z 0 the Schmidt model value for the spin of the valence nucleon, given by (3). From Eqs. (1), (6) and (7), we find We present the values for s z p and s z n from Eqs. (8) and (9) ("preferred model") in Tables I and II. The difference between the minimal and preferred model values can be taken as a measure of the uncertainty in the preferred model values. For most of the isotopes considered, this difference is 15%.

III. 3 HE/ 129 XE COMAGNETOMETER
We revisit the experiments of Refs. [29,32], in which a 3 He/ 129 Xe comagnetometer was used to place constraints on the Standard Model Extension (SME) CPTand Lorentz invariance-violating parameterb n ⊥ [65,66], which quantifies the interaction strength of a background field with the spin of a neutron. The observed quantities are the amplitudes of sidereal frequency shifts, ε 1,X and ε 1,Y , which in the case of the 3 He/ 129 Xe system are related to the SME parameters via [33]: where J = X, Y , γ He and γ Xe are the gyromagnetic ratios of 3 He and 129 Xe, respectively, with γ He /γ Xe = 2.754, and χ = 57 • is the angle between the Earth's rotation axis and the quantisation axis of the spins. Within the Schmidt model, in which only valence neutrons participate in the spin-dependent coupling s ·b, it was determined that [32]: b n Y = (2.9 ± 6.2) × 10 −34 GeV.

IV. NUCLEAR ANAPOLE MOMENTS
Consider the following Lorentz-invariance-violating terms in the SME Lagrangian (in the natural units = c = 1) [33]: where b µ and d µν are background fields, ψ is the fermion wavefunction withψ ≡ ψ † γ 0 , γ 0 , γ 5 and γ µ are Dirac matrices, and the two-sided derivative operator ↔ ∂ ν is de- The first term in (17) is CPT -odd, while the second term is CPT -even. In the non-relativistic limit, the Lagrangian (17) gives rise to the following interaction Hamiltonian where m is the fermion mass, s is the fermion spin and p is the fermion momentum operator. In our previous work [63], we showed in the single-particle approximation that the first term in (18) gives rise to nuclear anapole moments associated with valence nucleons [67] (see also [68]). Experimentally, the nuclear anapole moment manifests itself as a NSD contribution to a PNC amplitude. Hence from the measured and calculated (within the SM) values of the anapole moments of Cs and Tl, we were able to extract direct limits on the parameter b p 0 . In the single-particle approximation, the nuclear anapole moment contribution from interaction (18) is where G F is the Fermi constant of the weak interaction, K = (I + 1/2)(−1) I+1/2−l , and the dimensionless constants κ N b and κ N d are given by where α = e 2 / c is the fine-structure constant, m N and µ N are the mass and magnetic dipole moment of the unpaired nucleon N (µ p = 2.8 and µ n = −1.9), respectively, and we take the mean-square radius r 2 = 3 5 r 2 0 A 2/3 , with r 0 = 1.2 fm, and A the atomic mass number. Combining the measured values for the nuclear anapole moment of κ a (Cs) = 0.364 (62) [5,69] and κ a (Tl) = −0.22 (30) [70,71], with the values κ a (Cs) = 0.19 (6) and κ a (Tl) = 0.17(10) from nuclear theory [72][73][74][75] (see also [2]), and with Eq. (21), we extract limits on the parameter d p 00 in the single-particle approximation (Table  IV).
We now leave the single-particle approximation and consider nuclear many-body effects. For a single-particle IV. New limits (1σ, in laboratory frame) on the SME parameters b p 0 , d p 00 , b n 0 and d n 00 . s.p. denotes single-particle (Schmidt model) limit and m.b. denotes many-body (preferred model) limit.
Ref. [63] This work Parameter Model Cs Tl Cs Tl state, the angular momenta factors in (19) can be rewritten as Hence, unlike NSD-PNC effects arising from Z 0 -boson exchange between electrons and the nucleus [2], we cannot simply average over the spins of the single-particle proton and neutron states without explicitly considering the angular momenta of each individual nucleon. To circumvent this difficulty, we make use of the following approximation. Note that for single-particle states with j > 1, the prefactors before s in Eq. (22) are ≈ −2.
For non-light nuclei, most nucleons have j > 1. Also, the deviations of the prefactors in (22) from −2 are of opposite sign for j = l ± 1/2. Thus for nuclei with valence nucleon(s), which have j > 1, we can approximately sum over the proton and neutron spin angular momenta that appear in (22) to give the many-body generalisation of formula (19): From Eq. (23), we extract limits on the parameters b p 0 , d p 00 , b n 0 and d n 00 for Cs, for which I = 7/2, in the preferred model (Table IV). For Tl, where I = 1/2, Eq. (23) is not a good approximation and so we do not present manybody model limits in this case.

V. DARK MATTER SEARCHES
We point out also that proton and neutron spin contents are important for interpretations of experimental data to be obtained from recently proposed dark matter detection schemes, which are based on effects involving couplings to nuclear spins. Axions can induce oscillating nuclear Schiff moments via hadronic mechanisms [68,76,77], which can be sought for either directly through nuclear magnetic resonance-type experiments (CASPEr) [78] or oscillating atomic EDMs [68]. Axions can interact directly with nuclear spins via the time-dependent spin-axion momentum coupling s N · p a cos(m a t), where m a is the axion mass [68,77,79], induce the time-dependent nuclear spin-gravity coupling s N · g cos(m a t) and oscillating nuclear anapole moments [68,80]. Magnetometry techniques can also be used to search for monopole-dipole and dipole-dipole axion exchange couplings [81,82]. Topological defect dark matter, which consists of axion-like pseudoscalar fields, can interact with nuclear spins via the time-dependent coupling s N · (∇a), where a is the pseudoscalar field comprising the topological defect [83], and can give rise to transient nuclear-sourced EDMs [84]. Both of these effects can be sought for using GNOME [85]. WIMP dark matter can undergo elastic, spin-dependent scattering off nuclei, see e.g. [86,87]. One may use Tables I and II for the interpretation of dark matter searches based on all of the mentioned schemes, as well as for tests of the fundamental symmetries of nature.