Nuclear spin content and constraints on exotic spin-dependent couplings

There are numerous recent and ongoing experiments employing a variety of atomic species to search for couplings of atomic spins to exotic fields. In order to meaningfully compare these experimental results, the coupling of the exotic field to the atomic spin must be interpreted in terms of the coupling to electron, proton, and neutron spins. Traditionally, constraints from atomic experiments on exotic couplings to neutron and proton spins have been derived using the single-particle Schmidt model for nuclear spin. In this model, particular atomic species are sensitive to either neutron or proton spin couplings, but not both. More recently, semi-empirical models employing nuclear magnetic moment data have been used to derive new constraints for non-valence nucleons. However, comparison of such semi-empirical models to detailed large-scale nuclear shell model calculations and analysis of known physical effects in nuclei show that existing semi-empirical models cannot reliably be used to predict the spin polarization of non-valence nucleons. The results of our re-analysis of nuclear spin content are applied to searches for exotic long-range monopole-dipole and dipole-dipole couplings of nuclei leading to significant revisions of some published constraints.

Experimental searches for exotic spin-dependent interactions utilize a wide variety of atoms and nuclei. In order to compare experimental sensitivities to new physics it is useful to estimate how constraints on atomic spindependent interactions relate to constraints on atomic constituents: protons, neutrons, and electrons. Here we parameterize the spin couplings to new physics in terms of an exotic atomic dipole moment χ = χ a F related to coupling constants χ e , χ p , and χ n for the electron, proton, and neutron, respectively (it is generally assumed that such couplings do not follow the same scaling as magnetic moments). In the following we also assume the new physics does not couple to orbital angular momentum. The nucleon coupling constants χ p and χ n can in turn be related to quark and gluon couplings via mea-surements and calculations based on quantum chromodynamics [21].
The relationship of the expectation value for total atomic angular momentum F to electron spin S e and nuclear spin I is a straightforward result of quantum mechanics based on the Russell-Saunders LS-coupling scheme: where L is the orbital angular momentum. It follows that for the exotic atomic dipole moment coupling constant χ a , χ a = χ e S e · F F (F + 1) + χ N I · F F (F + 1) , (2) where χ N is the exotic nuclear dipole coupling constant which can be expressed in terms of χ p and χ n . The projection of S e on F can be calculated in terms of eigenvalues of the system according to: where J = S e + L, and the projection of I on F is given by The relationship between χ N and the nucleon coupling constants, χ p and χ n , can be estimated using the nuclear shell model [22] in a similar way if we assume that the nuclear spin I is due to the orbital motion and intrinsic spin of one nucleon only and that the spin and orbital angular momenta of all other nucleons sum to zero [23]. This assumption is referred to as the Schmidt model [24]. In the Schmidt model the nuclear spin I is generated by a combination of the valence nucleon spin (S p or S n ) and the valence nucleon orbital angular momentum ℓ, so that we have = S p,n (S p,n + 1) + I(I + 1) − ℓ(ℓ + 1) where it is assumed that the valence nucleon is in a welldefined state of ℓ and S p,n . However, it is known that nuclear magnetic moments are only partially predicted by the Schmidt model, and consequently it is recognized that nuclear spin and magnetic moments are not solely due to the valence nucleon and furthermore that nucleons are generally not in welldefined states of ℓ and S p,n . Flambaum, Lambert, and Pospelov [13] introduced a simple refinement of the nuclear shell model (here denoted the FLP model) for oddneutron nuclei by incorporating experimental data on nuclear magnetic moments along with the assumption that the nuclear magnetic moment µ N arises from a linear combination of the spin magnetic moment of the valence neutron and core polarization of protons. The FLP model (for odd-neutron nuclei) neglects orbital contribution to the nuclear magnetic moment since the valence neutron is uncharged. According to the FLP model, the contribution to the nuclear spin I from neutron spin S n and proton spin S p can be obtained from simultaneous solution of equations for the magnetic moment (or equivalently, the nuclear g-factor g I ): g I µ n I = g n µ n S n + g p µ n S p , and the nuclear spin = S N (S N + 1) + I(I + 1) − ℓ(ℓ + 1) 2I(I + 1) I , (10) where S N = 1/2 in units of for nucleons, g n = −3.8, g p = 5.6, and µ n is the nuclear magneton. To coincide with the notation of Ref. [13], we define which gives us the relations: and σ N = S N (S N + 1) + I(I + 1) − ℓ(ℓ + 1) 2I(I + 1) .
Consequently, according to the FLP model, the exotic dipole moment coupling constant χ N is related to χ p and χ n via: The FLP model can also be applied to nuclei with valence protons so long as the contribution of the orbital motion of the proton to the magnetic moment is accounted for by replacing Eq. (13) with: where g ℓ = 1 is assumed and Table I lists the measured nuclear g-factors, nuclear spins, orbital angular momenta of valence nucleons (as determined from the Schmidt model), and estimates of σ n and σ p from the Schmidt model and from the FLP model for a variety of nuclei of interest in searches for exotic spin-dependent couplings. It is observed, as expected, that the closer the Schmidt model prediction for the nuclear magnetic moment is to the measured value, the smaller the discrepancy between the Schmidt model and FLP model predictions for σ n and σ p .
The majority of searches for exotic spin-dependent physics have been interpreted using the single-particle Schmidt model, where the sensitivity of nuclei to new physics is determined entirely by the valence nucleon. However, as Ref. [13] points out, description of the nuclear spin based on the FLP model allows reinterpretation of such experiments in light of refined estimates of the contributions of both protons and neutrons to the nuclear spin.
A critical question in the reinterpretation of existing constraints on exotic spin-dependent interactions is the reliability of the FLP model, or concretely, the estimated uncertainty in the calculated values of σ n and σ p listed in Table I. A well-studied case is 3 He, for which there have been both experimental determinations [25] and detailed shell model calculations [26] of the contribution of neutron and proton spin polarization to the total spin of the 3 He nucleus. The measurement of deep inelastic scattering of polarized electrons from a spin-polarized 3 He target reported in Ref. [25] determined a neutron spin polarization of 87 ± 2% and a proton spin polarization of −2.7 ± 0.4%. The calculation of Ref. [26] is in excellent agreement with the measurement of Ref. [25], and explains the departure from the expectation of the Schmidt model as a result of state mixing (there is some admixture of d-wave states in addition to the dominant s-wave states). It is apparent from these considerations that the accuracy of the FLP model is limited in part because it neglects the contribution of spin and orbital angular momentum of non-dominant wave functions to the nuclear spin, which in the case of the 3 He is a non-negligible effect. Nonetheless, the FLP prediction for the proton spin polarization is within a factor of 2 of the measured result for 3 He, and the neutron spin polarization prediction is within ≈ 20% of the measured result. There have also been detailed shell model calculations of σ n and σ p for 129 Xe and 131 Xe [27][28][29][30] (see Table II). These more sophisticated nuclear shell model calculations were tested by comparing the calculation results to a variety of experimental observables. The calculated σ n values from Refs. [27][28][29][30] are within ≈ 20% of the FLP values for 129 Xe and 131 Xe; whereas the calculated σ p values fall within a factor of ∼ 10 of the FLP values, but are systematically smaller in magnitude. This is again apparently due primarily to state-mixing and the importance of orbital angular momentum contributions to the nuclear spin.
Since the earliest version of this work was written, a re-examination of the FLP model was carried out by Stadnik and Flambaum [31]. The FLP model empirically accounts for the variance of the nuclear magnetic moment from the Schmidt model prediction by assuming that the internucleon spin-spin interaction transfers spin from the valence nucleon to the core nucleons. However, as pointed out in Refs. [32,33], there are other mechanisms that can couple valence nucleon spins to core nucleon spins and orbital angular momentum. A refinement of the FLP model based on the analysis of Refs. [32,33] that assumes separate conservation proton and neutron angular momenta (i.e., assumes no spin-orbit coupling) results in values for σ n and σ p that differ by no more than ≈ 15% from the original FLP estimates. Thus these values also disagree with detailed shell model calculations of σ n and σ p for non-valence nucleons by factors of between 2 and 10 [26][27][28][29][30].
Based on this analysis, we estimate that the FLP model's calculation of the contribution of the valence nucleon spin to the nuclear spin is reliable to within ≈ 20%, while the FLP estimate of the contribution of the nonvalence nucleon spin to the nuclear spin is only reliable to within an order of magnitude, and thus derived constraints for non-valence nucleons based on the FLP model should be relaxed by a factor of 10. Nonetheless, the important point raised in Ref. [13] that odd-neutron nuclei have some level of sensitivity to exotic proton spin couplings is certainly valid.
To extract constraints on exotic spin-dependent interactions from recent searches it is also crucial to examine the details of the measured experimental signature. As a first example, we re-analyze constraints on long-range monopole-dipole couplings. Assuming the monopole-dipole coupling originates from one-boson exchange within a Lorentz-invariant quantum field theory, a light scalar/pseudoscalar field generates a monopoledipole potential V 9,10 (r) of the form (the subscript is in reference to enumerated potentials in Ref. [12]): where g X p is the dimensionless pseudoscalar coupling constant for particle X, g Y s is the dimensionless scalar coupling constant for particle Y , m X is the mass of particle X, r is the displacement vector between X and Y , λ is the range of the new force, is Planck's constant, c is the speed of light, and S X is the intrinsic spin of particle X in units of . Assuming a long-range force (communicated by a massless boson) we may approximate λ → ∞, which gives for the monopole-dipole potential: If the new scalar/pseudoscalar field is considered to be an additional component of gravity, as suggested by certain scalar-tensor extensions of general relativity based on a Riemann-Cartan spacetime [34][35][36][37], the interaction could be considered a coupling of spins to gravitational fields. The dominant gravitational field in a laboratory setting is that due to the Earth, which generates a spindependent Hamitonian with the nonrelativistic form [13]: where k X is a dimensionless parameter setting the scale of the new interaction for particle X and g is the acceleration due to gravity. If the strength of the pseudoscalar coupling is the same as that of the usual tensor component of gravity, k X ≈ 1 [13], setting the scale for the energy difference between opposite spin orientations with respect to g at ≈ 4 × 10 −23 eV (corresponding to a spin precession frequency of ≈ 10 −8 Hz). The connection between Eqs. (20) and (21) is obtained by integrating the contribution of all the constituent particles making up the Earth (∼ M E /m p , where M E is the mass of the Earth and m p is the proton mass): where R E is the radius of the Earth. Note that in the above estimate there is an implicit assumption about scalar coupling to constituent particles in the Earth: we have assumed equal scalar coupling (g s ) to protons and neutrons, which we also assume to have nearly equal abundance in the Earth, and neglect scalar coupling to electrons. Of course other assumptions could be made which would change the extracted limits.
Presently the best constraints on long-range monopoledipole couplings of nuclear spins are obtained from the experiment of Venema et al. [38] comparing spin precession of mercury isotopes ( 199 Hg and 201 Hg) and the experiment of Wineland et al. [39] measuring hyperfine transitions in 9 Be + ions. Both experiments searched for the coupling of spins to the mass of the Earth.
The experiment of Venema et al. [38] explicitly constrains the quantity where A is the strength of the monopole-dipole interaction, and where g 201 /g 199 = −0.369139 expresses the ratio of the Hg Landé g-factors and ǫ 199 and ǫ 201 parameterize the exotic spin coupling of the Hg nuclei. Using the calculated σ n and σ p from the FLP model listed in Table I, we can compute the resultant ǫ ′ for the proton and the neutron: which (somewhat surprisingly compared to the expectation from the single-state nuclear Schmidt model based on which the experiment was assumed to constrain only neutron couplings) shows that the sensitivity to exotic spin couplings of the neutron and proton is comparable. However, due to the order-of-magnitude uncertainty of the FLP model in regards to calculation of σ p , we relax the constraint on exotic spin couplings of the proton by a factor of 10. The resulting constraints on long-range monopole-dipole couplings of the neutron and proton are listed in Table III and shown in Fig. 1. The experiment of Wineland et al. [39] constrains the frequency shift between the 9 Be + 2 S 1/2 |F = 1, M = 0 and 2 S 1/2 |F = 1, M = −1 states caused by a long-range monopole-dipole interaction. The expectation values of the nuclear and electron spin projections along the quantization axis for the |F = 1, M = 0 state are zero, and for the |F = 1, M = −1 state: The frequency shift when the leading magnetic field was reversed relative to g was constrained to be < 13.4 µHz (or in energy units < 5.5 × 10 −20 eV). Using the calculated σ n and σ p from the FLP model listed in Table I, this constraint is translated into the limits on longrange monopole-dipole couplings of the neutron and proton listed in Table III and shown in Fig. 1, both about an order of magnitude less stringent than those obtained from Ref. [38]. Again we relax the constraint on proton couplings by a factor of 10 to account for systematic uncertainty in the FLP estimate of σ p . Also of interest is the experiment by Youdin et al. [40] carried out to search for laboratory-range monopoledipole couplings between a 475-kg lead mass and the atomic spins of 133 Cs and 199 Hg atoms. In Ref. [40], the results of the experiment were interpreted to constrain electron and neutron spin couplings. Both the Schmidt and FLP models reliably predict that the experimental limit on couplings of the 133 Cs spins to the lead mass also constrains scalar-pseudoscalar couplings of the proton spin, leading to the limit for a range of 20 cm. The corresponding constraints are shown in Fig. 1. To our knowledge this is the first pub-lished constraint on scalar-pseudoscalar couplings of the proton spin at this length scale. Constraints on long-range monopole-dipole couplings of neutrons and protons can be compared to recent experiments searching for long-range monopole-dipole couplings of electrons using a spin-polarized torsion pendulum [2], which obtained the constraint k e < 10 from searching for a S e · g correlation. By searching for a long-range monopole-dipole interaction where the source mass was the sun, the constraint g e p g s /( c) < 2 × 10 −36 was established [2].
The above analysis is of particular relevance to our ongoing experiment to search for a long-range monopoledipole (spin-gravity) coupling of Rb spins to the mass of the Earth [7]. In our experiment, we measure the ratio of the difference between the 87 Rb and 85 Rb precession frequencies in the ground state F = 2 and F = 3 states divided by their sum, Measurement of the ratio R eliminates or reduces several common-mode sources of noise and systematic error. Taking the difference between R for a leading magnetic field parallel with g and anti-parallel with g yields a signal proportional to the spin precession frequency caused by nonmagnetic interactions. Ultimately the sensitivity of the experiment to long-range monopole-dipole couplings, δk, is related to the sensitivity to anomalous frequency shifts, δΩ. The sensitivity of the experiment to proton and neutron couplings can be estimated based on the σ p and σ n values listed in Table I, taking into account the fractional contribution of the nuclear spin to the total atomic spin given by Eq. (2). The valence nucleon of 87 Rb and 85 Rb is a proton, so the FLP model's estimate of σ p is reliable to ≈ 20%, whereas the FLP model's estimate of σ n is only reliable to within an order of magnitude. Therefore in this case we relax the sensitivity to exotic couplings of the neutron spin accordingly: δk n ≈ 1.4 × 10 4 × δΩ(µHz) .  [41]. The plot on the left shows constraints on proton couplings, the plot on the right shows constraints on neutron couplings. Note that based on the reanalysis carried out in this work, this plot is significantly different from the plot published in Ref. [7] resulting in less stringent constraints on |gpgs| / c.
If our experiment can achieve sub-µHz sensitivity to anomalous frequency shifts, our sensitivity to proton spin-gravity couplings would exceed existing constraints by two orders of magnitude and offer sensitivity to neutron spin-gravity couplings comparable to the strictest constraints of Ref. [38]. Note that because of these reconsiderations of the nuclear spin content, the updated parameter exclusion plot shown in Fig. 1 is significantly different from the parameter exclusion plot published in Ref. [7] (Fig. 1 in Ref. [7]). As a second example, we consider a long-range dipoledipole coupling V 3 (r) between nuclei, which can be generated by a pseudoscalar field (again the subscript is in reference to enumerated potentials in Ref. [12]): If the exchange boson is assumed to be nearly massless, the range of the interaction λ → ∞ and we obtain There have been two recent experiments [4,42] nominally searching for laboratory-scale dipole-dipole interactions between polarized neutrons. Based on the refinement of the estimate of nuclear spin content, both experiments are sensitive to long-range dipole-dipole couplings of the proton and between the proton and neutron, and establish constraints on such interactions that are more strict than existing limits reported in the literature.
The experiment of Glenday et al. [42] measured the spin precession frequencies of 3 He and 129 Xe in a dualspecies maser as the polarization of a nearby dense 3 He gas was reversed. The authors of Ref. [42] did assess the nuclear spin content using the FLP approach for 129 Xe (which had been applied to 129 Xe in an earlier work [43]) and measurements of deep inelastic scattering of electrons for 3 He [25], but reported only constraints on neutrons. Using the results from Table I and Ref. [25], we obtain the results shown in Table IV for neutron-proton and proton-proton couplings.
In the experiment of Vasilakis et al. [4] a spin-exchange relaxation free (SERF) comagnetometer [44,45] using 39 K and 3 He was used to search for exotic dipole-dipole couplings to a nearby dense 3 He gas. In the analysis carried out in Ref. [4], only the 3 He spins were considered and the analysis focused on neutron-neutron couplings (again using the nuclear spin content determined by Ref. [25]). In the SERF regime, the valence electron of 39 K is rapidly kicked between the ground state F = 2 and F = 1 hyperfine levels by spin-exchange collisions and therefore the effective fraction of atomic spin due to IV: Constraints on long-range dipole-dipole couplings of neutrons and protons, using the parameterization of Eq. (35). Constraints on proton spin couplings based on 129 Xe are relaxed by an order of magnitude as compared to the direct calculation of the limit based on the FLP model to account for the uncertainty of the FLP estimate of σp (constraints on neutron spin couplings based on 39 K are similarly relaxed due to uncertainty in σn). For calculations related to 3 He, the directly measured [25] and calculated [26] values for σp and σn are used, which are reliable to within a few percent and require no correction. nuclear spin polarization is given by the weighted average of the nuclear spin polarization in the two ground state hyperfine levels: σ N = 5/8. The SERF comagnetometer has similar sensitivity to magnetic field couplings for 39 K and 3 He (in the experiment described in Ref. [4] the magnetometric sensitivity reached ∼ 0.5 aT). However, it is crucial to note that the similar magnetometric sensitivity translates into different energy sensitivities for the two species due to the different magnetic moments of 39 K and 3 He: the sensitivity of 39 K to exotic spin couplings is reduced by a factor of ∼ 130 as compared to 3 He, the ratio of the magnetic moments taking into account the statistical weighting between 39 K ground-state hyperfine levels due to the rapid spin-exchange collisions. Because of this factor, in spite of the larger σ p for 39 K as compared to 3 He, constraints from this experiment on exotic couplings to 3 He spins are more sensitive to proton couplings than those derived from exotic couplings to 39 K. The derived constraints on long-range dipole-dipole couplings for neutrons and protons are shown in Table IV.
To put these constraints into context, the previous best limit reported in the literature for a long-range exotic dipole-dipole coupling of the form expressed in Eq. (35) between protons was that obtained by Ramsey using spectroscopy of molecular hydrogen [46]: g p p g p p /(4π c) < 2.3 × 10 −5 . The results obtained by Vasilakis et al. [4] combined with straightforward re-analysis of the nuclear spin content improve these constraints by a factor of ∼ 50. These new constraints on exotic proton dipoledipole couplings are quite reliable since they are derived from the well-measured and understood σ p of 3 He. Note that the constraints on long-range exotic dipole couplings of electron spins are far more stringent: the spinpolarized torsion pendulum experiment of Ref. [3] reports g e p g e p /(4π c) < 2.2 × 10 −16 .
As a final point, we consider a new experimental effort being initiated to search for exotic spin-dependent interactions that produce transient signals [47,48]. While a single comagnetometer system, such as the SERF comagnetometer described in Refs. [4,44,45], could detect such transient events, it would be exceedingly difficult to confidently distinguish a true signal generated by hereto-fore undiscovered physics from "false positives" induced by occasional abrupt changes of comagnetometer operational conditions (e.g., magnetic-field spikes, laser-mode jumps, electronic noise, etc.). Effective vetoing of false positive events requires an array of comagnetometers. Furthermore, there are key benefits in terms of noise suppression and event characterization to widely distributing the comagnetometers geographically. The Laser Interferometer Gravitational Wave Observatory (LIGO) collaboration has developed sophisticated data analysis techniques [49] to search for similar correlated "burst" signals from a worldwide network of gravitational wave detectors, and we have recently demonstrated that these data analysis techniques can be applied to data from synchronized comagnetometers [48]. Our proposed comagnetometer array, the Global Network of Optical Magnetometers to search for Exotic physics (GNOME), would be uniquely sensitive, for example, to cosmic events generating coherent bursts or waves of a heretofore undiscovered field [16], to correlated noise produced by a fluctuating [50] or oscillating [51] background field whose timeaveraged value is zero, or passage through topological defects such as pseudoscalar domain walls [47]. Eventually, the GNOME will consist of at least five dedicated atomic comagnetometers located at geographically separated stations.
Construction and testing of a prototype sensor for the GNOME is presently underway. The design is based on the SERF comagnetometer scheme developed by Romalis, Kornack, and colleagues [4,6,44,45]. Our prototype sensor will utilize coupled 3 He and 87 Rb spins, which, according to the analysis presented here, offers sensitivity to exotic electron, neutron, and proton spin couplings. Based on Eq. (2) and the surrounding discussion combined with the results of the FLP model for 87 Rb (Table I) and measurements of σ p and σ n for 3 He [25], the exotic dipole moments for 3 He and 87 Rb in terms of χ e , χ n , and χ p are given by: = −0.25χ e + 0.09χ n + 0.33χ p , (39) Due to the averaging over 87 Rb hyperfine levels due to rapid spin-exchange collisions in the SERF regime, the effective exotic atomic dipole moment for 87 Rb is given by χ 87 Rb = 0.06χ e + 0.07χ n + 0.25χ p .
As noted previously in our discussion of the SERF comagnetometer scheme, in order to determine the relative sensitivity of 87 Rb and 3 He to new physics the reduction in sensitivity of 87 Rb relative to 3 He by the ratio of the magnetic moments must also be taken into account. As a result, the GNOME will be most sensitive to exotic spin-dependent couplings to neutrons, with sensitivity to proton couplings (primarily arising from the σ p of 3 He) reduced by a relative factor of ∼ 30 and sensitivity to electron couplings reduced by a relative factor of ≈ 2000.
In conclusion, we have re-analyzed nuclear spin content for several nuclei of interest in searches for exotic spindependent interactions according to the model presented in Ref. [13] and assessed the reliability of the model. Applying revised nuclear spin content to results of previous experiments, we have re-derived constraints on longrange monopole-dipole (spin-gravity) and dipole-dipole interactions. In particular, we established new laboratory constraints on exotic monopole-dipole couplings between protons at a range of ≈ 20 cm based on the work of Youdin et al. [40] and found that the experiment of Vasilakis et al. [4] constrains long-range exotic dipoledipole couplings between protons over an order of magnitude more stringently than previously reported in the literature. The analysis of this work would greatly benefit from more detailed nuclear theory calculations that could more reliably predict nuclear spin content.
The author is sincerely grateful to Dmitry Budker, Maxim Pospelov, Volker Koch, Feng Yuan, and Michael Romalis for enlightening discussions. This work was supported by the National Science Foundation under grant PHY-1307507. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation. This research was also supported in part by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.