Using the Sun and the Moon as Source masses and the Earth's Rotation as a Modulation to Search for Exotic Spin-Dependent Interactions at Astronomical Distances

Exotic spin-dependent interactions mediated by new light particles led to solutions to several important questions in modern physics. Such interactions involving a scalar coupling $g_S^N$ at one vertex and a pseudo-scalar coupling $g_P^n$ at the polarized neutron vertex can be induced by the exchange of spin-0 bosons, or a vector/axial-vector coupling $g_V^N$/$g_A^N$ at one vertex and an axial-vector coupling $g_A^n$ at the polarized neutron vertex can be induced by the exchange of spin-1 bosons. If such new interactions exist, the Sun and the Moon can induce sidereal variations of effective fields along the direction perpendicular to the Earth's rotation axis. We derived new experimental upper limits on such exotic spin-dependent interactions at astronomical interaction ranges by analyzing existing data from laboratory measurements on the Lorentz and CPT violation. We set the most stringent experimental limits on $g_S^Ng_P^n$ ranging from $\sim 2\times 10^{10}$m to $\sim 10^{14}$m. Previously, the best limit on $g_S^Ng_P^n$ at this range is from astrophysics. The result is the first time laboratory limits surpass the astrophysical ones on the scalar-pseudoscalar type interaction, to our best knowledge. We report new constraints on vector-axial-vector and axial-axial-vector type interaction at the range of astronomical scales. The new limits on vector-axial-vector are improved by as much as $\sim$12 orders of magnitude. We also apply the analysis to the Hari-Dass interactions and obtain corresponding new constraints on the interactions. We discuss the possibilities of using the beam method to further search the interaction involving other particles, such as electrons, muons, etc., based on the same idea.

Introduction.− Axions, predicted by the PQ (Peccei-Quinn) mechanism [1-3], can induce spin-dependent interactions [4]. The originally proposed axions were quickly ruled out due to the broken energy scale of the electroweak scale; however, new models with higher broken energy scales were subsequently proposed. Axions can have arbitrarily small mass and weak couplings to ordinary matter because the scale at which the PQ symmetry is broken can be arbitrarily large [5]. Thus, axions might mediate interactions in ranges from nanometers to astronomical distances. Though the PQ mechanism was originally proposed to solve the strong CP problem, the axions, which are light, weakly interacting, and pseudoscalar, are also considered as possible candidates for cold dark matter. New interactions might also be mediated by vector particles such as the paraphoton (dark, hidden, heavy or secluded photon) [6,7], Z ′ boson [8], graviphoton [9], etc., or even unparticles [10]. Reference [11] proposed 16 different types of new interactions, 15 of which are spin-dependent. Non-Yukawa exotic interactions due to the dark or hidden sector were also proposed recently [12,13]. As early as 1980, Fayet [14,15] pointed out that the new U(1) vector bosons,characterized by small masses and weak couplings to ordinary matter, could be generated through the spontaneous breaking of supersymmetric theories. Searching for the new interactions mediated by the new particles is related to the strong CP problem, dark matter, dark energy, and finding evidence of supersymmetry, which is among the most important unsolved problems in modern physics [16].
Particles with similar properties as axions, predicted by various generalized theories, are usually called ALPs (Axion Like Particles) [8]. If exist, ALPs (φ) can generate a new interaction through the coupling to a fermion ψ, L φ =ψ(g S + ig P γ 5 )ψφ, where g S (g P ) is the scalar (pseudoscalar) coupling constant [4]. The SP (scalarpseudoscalar) interaction between a polarized neutron and an unpolarized nucleon can be expressed as where λ = /m φ c is the interaction range, m φ is the mediator mass, σ is the spin operator of the polarized neutron, m n is the neutron mass, N (n) represents the nucleon (neutron), and r is the distance between the two interacting particles. This interaction could also be generated from Lφ = g Sψ ψφ+g P /(2m)ψγ µ γ 5 ψ∂ µ φ [4,17,18], which includes a derivative term. Such a term appears in the axion models [19]. Notably, the SP interaction generated from the derivative coupling between axions and fermions is discussed in Ref. [20]. Thus, studying this interaction can be used to investigate not only ALPs but also axions. Recently, the SP interaction has begun to attract more attention [21][22][23][24][25][26][27]. For example, Wei et al. [28] proposed a laboratory experiment scheme that could surpass the astrophysical limit for the SP interaction. Laboratory limits become closer and closer to the limits derived by combining g N S from the torsion balance experiment and g n P from SN1987A; however, all the existing limits are less stringent than the combined astrophysicallaboratory ones at astronomical distances.
VA (vector-axial-vector) and AA (axial-axial-vector) interactions can be derived from a general Lagrangian L X = X µψ (g V γ µ + g A γ 5 γ µ )ψ in the nonrelativistic limit, where X is a new vector particle and g V (g A ) is the vector (axial-vector) coupling constant. The interactions between a polarized neutron and an unpolarized nucleon can be expressed as where v is their relative velocity. Many studies have been carried out to look for the new interactions, detecting ei-ther the macroscopic forces or the torques exerted on the polarized probe spins. For example, Leslie et al. [29] proposed experimental schemes to detect the new spindependent interaction between a polarized source and a mechanical oscillator. For another example, Ding et al. [30] used a micro-fabricated magnetic structure as a polarized source, and then tried to detect the AA interaction at the range of ∼ µm sensed by a gold-spherecantilever. Many groups have been searching for the new interaction through its rotating effects as a effective magnetic field on the polarized spin. The VA and AA interactions between different combinations of fermions have been investigated already, such as electron-nucleon [30][31][32][33], neutron-nucleon [34][35][36], electron-electron [37], and electron-antiproton [38]. Studies on these exotic interactions involving muons were performed very recently [39]. Since the signal induced by new interactions is tiny, using a large mass source and modulating the signal to a high frequency are crucial for the detection. A mass source with vast constituents could make the signal measurable. By modulating the signal, on the one hand, the SNR (Signal-to-Noise Ratio) can be increased with the decrease of the noise bandwidth. On the other hand, the 1/f noise can be significantly reduced at high frequencies. In this Letter, treating the Sun and the Moon as mass sources and using the Earth's rotation as a modulation, we obtain new limits on exotic spin-dependent SP (1), VA (2), and AA (3) interactions at astronomical distances.
The basic idea.− All three types of spin-dependent interactions are in the form of s · B ′ , where B ′ can be viewed as a kind of effective magnetic field. Searching for these interactions becomes a problem probing the effective field acting on polarized spins. We first illustrate the basic idea using the Sun as the source mass. In the Sun-centered frame shown in Fig. 1 (a), the aforementioned new interactions can generate effective magnetic fields at the Earth center as [40] where R is the distance from the Earth to the Sun, γ n is the gyromagnetic ratio of the neutron, Ω ⊕ is the Earth's orbital angular frequency, and N ⊙ is the total nucleon number of the Sun. For a laboratory frame on the Earth as shown in Fig. 1 (b), the effects of the Earth's rotation can be taken into account via the Euler rotations. We first rotate the frame by the angle ω ⊕ t about theẐ axis where ω ⊕ is the Earth's rotation frequency, and then rotate about theŶ axis by an angle η which is the Earth's obliquity. In the laboratory frame, we will observe effective time-varying fields as where v ⊕ = Ω ⊕ R is the orbital speed of the Earth. As the most straightforward case, b AA clearly shows effective magnetic fields rotating in the laboratory frame at the Earth's rotation frequency. Although b SP and b VA appear more complicated due to their mixture with the Earth's orbital rotation, the situation can be greatly sim- plified by considering the fact that ω ⊕ ≫ Ω ⊕ .
In summary, if the exotic spin-dependent interactions exist, the perpendicular components of the effective fields induced by the Sun are modulated by the Earth's rotations; thus, we could observe its signal in the laboratory. Although the frequency of the Earth's rotation is not high, its frequency modulation effect on the nuclear precession in the comagnetometer makes precision measurements on these new interactions possible.
Constraining the exotic spin-dependent interactions at astronomical distances.−Dual-species comagnetometers are convenient for detecting the tiny signals caused by new spin-dependent interactions, since the two components occupy the same space and common-mode background field noise can be mostly canceled. In principle, we can separate the sidereal modulated signal from the noisy background during precise measurements. The ultrahigh sensitivity of the comagnetometer to the magnetic field changes has an extensive implementation in new physics detection, including electric dipole moments, CPT and Lorentz violation, spin-gravity interaction, and so on [41][42][43][44]. This comagnetometer method has been used to search for the constant cosmic background field due to Lorentz violation by detecting the sidereal variants of the field observed in the laboratory frame on the Earth in Refs. [45,46], where a 129 Xe+ 3 He comagnetometer and a K+ 3 He one were respectively adopted. Stringent constraints on the components of the constant field perpendicular to the Earth's rotation axis at a similar level were obtained.
We find that the limits on exotic spin-dependent interactions induced by the Sun can be obtained by using the Lorentz violation searching results. For example, for the experiment described in Ref. [45], the Ω ⊕ t in Eqs. (5), (6) and (7) approximately equals to π/2, given that the experiment was performed for ∼10 days when the Earth was around the vernal equinox. The sidereal oscillating effective field b ⊥ perpendicular to the Earth's rotation axis can be detected as [40] b SP⊥ = g N S g n P N ⊙ 4πm n γ n ( 1 λR Using the result of Ref. [45], at the 95% confidence level (95% C.L.), we could derive |b ⊥ | < 0.023 fT. (9) Plugging in all the known parameters such as η = 23.4 • , N ⊙ ≈ 1.2 × 10 57 , R = 1.5 × 10 11 m, and v ⊕ ≈ 3.0 × 10 4 m/s, we can obtain the constraints on the SP (1), VA (2), and AA (3) interactions between polarized neutrons and unpolarized nucleons. The derived constraint on |g N S g n P | is shown in Fig. 2. For λ 2 × 10 10 m, it gives the most stringent limit on |g N S g n P |. For λ > 10 12 m, our bounds |g N S g n P | < 1.6×10 −35 (95% C. L.). Previously, the most stringent constraints  [18,47], which was derived by combining g N S of weak equivalence with g n P from SN1987A. The dark grey area is excluded by the result of this work and the light gray one is excluded by the result of Refs. [18,47]. on |g N S g n P | were astrophysical-laboratory limits that combined astrophysical constraints on g n P from SN1987A with the laboratory ones on g N S from the weak equivalence principle experiment. This work improves the existing upper bound by as much as ∼70 times. In particular, the new limits on the scalar-pseudoscalar coupling combination exceed the combined astrophysical-laboratory limits for the first time.
Our constraint on |g N V g n A | is shown in Fig. 3. For λ 3×10 7 m, it gives the most stringent limit on |g N V g n A |. For λ > 10 12 m, our bounds |g N V g n A | < 7.1 × 10 −59 (95% C. L.). Previously, the most stringent constraint on |g N V g n A | near the interaction range under consideration was given in Ref. [36]. If the previous result can be extended to the range of ∼ 10 12 m, the present work improves the existing upper bound by as much as 12 orders of magnitude.
The obtained constraint on |g N A g n A | is shown in Fig. 4. For λ 10 7 m, it gives the first limit on |g N A g n A |. For λ > 10 12 m, our bounds |g N A g n A | < 8.1×10 −31 (95% C.L.). It is the only known constraint on |g N A g n A | to us at the astronomical ranges.
We can apply the same analyzing method by using the Moon as a source. In this case, we shall consider errors due to several systematic effects. We show details of the analysis in the Supplementary Material [40]. We also plotted the results using the Moon in Figs. 2, 3, and 4.
Furthermore, the spin-gravity interaction proposed by Leitner and Okubo [48] and later generalized by Hari Dass [49] can also be strictly constrained. Assuming CPT invariance, two types of discrete symmetry violation spingravity potentials are constructed as where G N is the Newton constant of gravitation, M is the mass of the gravity source, and α 1 (α 2 ) is a dimensionless constant. These potentials are the starting point of many low-energy experiments [50,51]. They also provide a direct way to test symmetry violation and the equivalence principle in General Relativity [49,52]. Using the Sun as the mass source, we derive the limits on α as |α 1 | < 2.2 × 10 2 (95% C.L.), |α 2 | < 2.4 × 10 6 (95% C.L.).
When comparing with the results of Ref. [53], our limit on α 1 improves the existing one by ∼11 times, and on α 2 we get an improvement of ∼4 orders of magnitude. Conclusion and discussion.−By using the Sun and the Moon as sources, the Earth's rotation as modulation, and the existing laboratory limits on the Lorentz and CPT violation at distances of astronomical scales, we have constrained three types of possible new interactions between polarized neutrons and unpolarized nucleons. We derived new limits on the SP interaction with ranges from ∼ 2 × 10 10 m to ∼ 10 14 m. At the distance of ∼ 10 12 m, the limit is improved by ∼70 times compared to the previous combined astrophysical-laboratory limit. This result is the first time the limits from a single laboratory experiment exceed the combined astrophysical-laboratory ones for the SP interaction. We obtained new limits on the VA interaction with ranges from ∼ 3 × 10 7 m to ∼ 10 14 m. At the distance of ∼ 10 12 m, the limit is improved by ∼12 orders of magnitude in comparison with the previous result of 3 He spin relaxation experiment. We derived the first limits on the AA interaction with ranges from ∼ 10 7 m to ∼ 10 14 m. We also constrained the Hari-Dass spin-dependent interactions and obtained new limits on them.
How can we extend the current work to include other particles, such as electrons and muons? One possibility is to employ the beam method proposed in Refs. [39,54], which uses superconducting magnetic shielding to create a region with zero background field. By directing spinpolarized particle beams through this region and detecting sidereal variations in polarization along the direction perpendicular to the Earth's rotation axis, we can investigate spin-dependent new interactions induced by the Sun for the probing electrons, muons, etc.
Note added.−After this work was submitted, Ref. [55] reports improved limits on g N S g n P in the interaction range of 10 6 ∼ 10 10 m, and Ref. [56] reports similar limits on g N A g n A in the interaction range of ∼ 10 12 m. We acknowledge supports from the National Natural Science Foundation of China under grants U2230207 and U2030209. This work was also supported by the National Key Program for Research and Development of China under grants 2020YFA0406001 and 2020YFA0406002. We thank Dr. C. Fu and Y. M. Ma for their helpful discus-  40, 223 (1978).