Constraints on ultralight axions, vector gauge bosons, and unparticles from geodetic and frame-dragging measurements

The geodetic and frame-dragging effects are the direct consequences of the spacetime curvature near Earth which can be probed from the Gravity probe B (GP-B) satellite. The satellite result matches quite well with Einstein's general relativistic result. The gyroscope of the satellite which measures the spacetime curvature near Earth contains lots of electrons and nucleons. Ultralight axions, vector gauge bosons, and unparticles can interact with these electrons and nucleons through different spin-dependent and spin-independent operators and change the drift rate of the gyroscope. Some of these ultralight particles can either behave as a long range force between some dark sector or Earth and the gyroscope or they can behave as a background oscillating dark matter fields or both. These ultralight particles can contribute an additional precession of the gyroscopes, limited to be no larger than the uncertainty in the GP-B measurements. Compared with the experimental results, we obtain bounds on different operator couplings.


I. INTRODUCTION
The gravity probe B (GP-B) is a satellite-based experiment that was launched in April 2004 to test the general relativity (GR) phenomena such as the geodetic effect or the de-Sitter effect [1] and the Lense-Thirring precession [2] or the gravitomagnetic frame-dragging effect predicted by Einstein's GR theory. This Earth satellite contained four gyroscopes and it was orbiting at 650 km altitude. The spacetime near the Earth is changed due to the presence of the Earth and its rotation which modifies the stress-energy tensor near the Earth [3]. When the gyroscope's axis of the satellite is parallel transported around the Earth then after one complete revolution, the tip does not end up pointing exactly in the same direction as before. The geodetic effect measures the drift rate of the gyroscope due to the presence of the Earth which wraps the spacetime near it and the frame-dragging effect measures the drift rate due to the rotation of the Earth which drags the spacetime around it. The GR predicts the geodetic drift rate as −6606.1 mas/yr whereas the frame-dragging drift rate is −39.2 mas/yr. In 2011, the GP-B experiment published the data for the geodetic drift rate as −6601.8 ± 18.3 mas/yr and the frame-dragging drift rate as −37.2 ± 7.2 mas/yr, where "mas" stands for milliarcsecond [4]. The GP-B result matches quite well with the GR predicted value. If there are interactions of electrons and nucleons in the gyroscope with the ultralight axion, gauge bosons, and unparticles through spin dependent and spin independent couplings then these interactions exert a new force on the gyroscope. This new force can contribute an additional precession of the gyroscopes. However, the contribution is within the measurements of experimental uncertainty. Comparing with the experimental results we obtain the constraints on axions, gauge bosons, and unparticles mediated forces.
In the following, we have discussed what are axions and axion like particles, gauge bosons, and unparticles.
Axion is a pseudo-Nambu Goldstone Boson (pNGB) which was first postulated by  to solve the strong CP problem [5][6][7][8]. The most stringent probe of the strong CP problem is the measurement of neutron electric dipole moment (nEDM). From chiral perturbation theory, one can obtain the nEDM as d n 10 −16θ e.cm, whereθ is related with the QCD θ angle byθ = θ + argdet(M) [9,10], where M is the general complex quark mass matrix. A natural choice ofθ ∼ O(1) violates the current experimental bound on nEDM which is d expt n < 10 −26 e.cm. Hence,θ is as small asθ < 10 −10 [11]. The smallness ofθ is called the strong CP problem. Quantum Chromodynamics (QCD) is the theory that can calculate the nEDM and its Lagrangian is where q denotes the quark field, D denotes the covariant derivative,G µν = 1 2 µναβ G αβ is the dual gluon field tensor, θ is the QCD theta angle and h.c. denotes the Hermitian conjugate.
The last term in Eq.1 is a CP violating term. The last term comes from the symmetry of the Lagrangian and it must be present since all the quark masses are non-zero. However, QCD is good CP symmetric. To solve this strong CP problem, Peccei and Quinn came up with an idea thatθ is not just a parameter but a dynamical field that goes to zero by its classical potential. They postulated a global U (1) P Q symmetry which spontaneously breaks at a symmetry breaking scale called the axion decay constant f a and explicitly breaks due to non-perturbative QCD effects at a scale Λ QCD , where the pseudo-Nambu Goldstone bosons called the QCD axions to get mass. The mass of the QCD axion is related to the axion decay constant as m a 5.7 × 10 −12 eV 10 18 GeV fa [12]. Also, there exist other pseudoscalar particles which are not exactly the QCD axions but have similar kinds of interactions to the QCD axions. These are called Axion Like Particles or ALPs. These particles are motivated by string/M theory [13][14][15][16]. We assume the ALPs do not get any instanton induced mass and remain naturally light. For ALPs, m a and f a are independent of each other whereas, for QCD axions, m π f π ∼ m a f a , where m π is the pion mass and f π is the pion decay constant.
The Lagrangian which describes the interaction between the ALPs and the standard model where g's are the coupling constants. The first term denotes the kinetic term of ALP, the second term denotes the coupling of ALP with the gluon field G µν , the third term denotes the coupling of ALP with the electromagnetic photon field F µν and the fourth term denotes the derivative coupling of ALP with the fermion field f . Note, g ag 1 to avoid any instanton induced mass for generic ALPs. The generic ALP is a pNGB, however, the scale at which the ALP gets mass need not be Λ QCD . If the associated symmetry corresponding to a gauge group has mixed anomaly and has strong dynamics then the Goldstone boson acquires a The ALPs can also be generated from string theory. They arise as Kaluza-Klein zero modes of antisymmetric tensor fields in ten dimensions [13,16,18]. The string axions inherit the shift symmetry from gauge invariance. They can also be generated from the clockwork mechanism [19]. Hence, for the QCD axion, the mass only depends on one free parameter f a whereas, for ALPs, the mass depends on additional parameters of the theory. The couplings of ALPs with SM particles are small since all the couplings are proportional to 1 fa and f a generally takes a larger value. The axions or ALPs can also couple with the nucleons or quarks through the electric and magnetic dipole moment operators described by the terms g EDM aN σ µν γ 5 N F µν and g M DM aN σ µν N F µν respectively.
The axion is a promising non thermal dark matter candidate which can solve some of the small scale structure problems in the universe [20][21][22][23]. The axion field can oscillate with time as a(t) ∼ √ 2ρ DM ma sin(m a t), where ρ DM is the dark matter energy density. Axion can also form topological defects like cosmic strings and domain walls [24][25][26]. They can also behave as dark radiation [27][28][29][30][31][32]. If the mass of the axion is very small then it can also mediate long range forces and the corresponding potential is Yukawa type 1 r e −mar . The axions can also contribute to the monopole-monopole, monopole-dipole, and dipole-dipole interactions between the visible sector particles [33][34][35][36].
There is no direct evidence of axions so far. However, there are bounds on the axion mass and decay constant from the laboratory, astrophysical, and cosmological experiments.
Constraints on axion mediated force from torsion pendulum experiment are discussed in [83].
The light gauge bosons can also mediate long range forces or can also serve as a background oscillating dark matter fields. The SM of particle physics is a SU (3) c × SU (2) L × U (1) Y gauge theory. However, in the leptonic sector one can construct three symmetries, L e − L µ , L e − L τ , and L µ − L τ in an anomaly free way and they can be gauged. The L e − L µ,τ type of gauge force can be constrained from neutrino oscillation experiments [84], and perihelion precession of planets [85]. The L µ −L τ type of gauge force can be constrained from the orbital period loss of the binary systems [86]. The other bounds on L i − L j force are discussed in [87,88]. B − L symmetry can also be gauged in an anomaly free way and it can mediate long range force. The bounds on ultralight B − L gauge bosons are discussed in [89]. Constraints on long and short range forces mediated by scalars and vectors are discussed in [90,91] . The fields of the UV theory become scale invariant below a scale Λ u (typically Λ u ∼ 1 TeV) and it acquires a dimension d u different from the canonical one by the dimensional transmutation. Thus, the unparticle operator O u coupled with the SM operator as Λu Exchange of scalar, pseudoscalar, vector, pseudovector unparticles can give rise to long range forces which are discussed in [98][99][100][101]. Unparticles can couple to energy stress tensor and mediate ungravity [102]. The unparticle coupling with Higgs, gauge bosons, and other SM particles are discussed in [103][104][105][106][107][108][109][110][111]. Unparticle mediated long range force can also be tested from the perihelion precession of Mercury [112]. Also, dark matter and dark energy can interact with unparticles which are discussed in [113][114][115][116][117].
Axions, vector gauge bosons, and unparticles can mediate long range forces and contribute to the precessional velocity of the gyroscope of GP-B satellite within the experimental uncertainty limit. The axions, dark photons, vector gauge bosons, and unparticles can interact with the gyroscope and change the precession of the gyroscope. Using the GP-B results, we obtain bounds on the coupling and mass of these particles.
The paper is organized as follows. In Sec.II we calculate the fraction of polarized spins in the GP-B gyroscope to constrain spin dependent coupling. In Sec.III, we consider axion mediated long range Yukawa type of potential between a visible sector and a dark sector.
In Sec.IV, we discuss the interaction of the GP-B gyroscope with the background oscillating axionic dark matter field. In Sec.V, we consider that the time dependent oscillating axionic field can interact with the nucleons of the gyroscope through the electric dipole moment operator and change its drift rate. The dark photon can also behave as a background oscillating dark matter field and can interact with the gyroscope's nucleons through electric and magnetic dipole moment operators which are discussed in Sec.VI. In Sec.VII, we discuss the mediation of L e − L µ,τ type of gauge bosons which gives rise to long range force and changes the precession rate of the gyroscope. In Sec VIII, we discuss the unparticle mediated long range force which can be constrained from the GP-B result. Finally, in Sec.IX, we discuss our results.
We have used the natural system of units throughout the paper.

ROSCOPE
Unless any spin alignment mechanism, macroscopic objects with randomly oriented spins would not induce any spin dependent force. In [101], Earth has been treated as a polarized electron source to search for long range spin-spin interaction. A fraction of the total number of electrons in Earth can be polarized due to the presence of Earth's dipolar magnetic field. However, for the measurements of frame-dragging and geodetic effects, GP-B satellite explicitly shielded external magnetic fields (e.g., the Earth field) to < 0.1 nT. If there is no shielding in GP-B, the spin alignment would presumably have to adiabatically track the Earth's field direction as the satellite orbited in its polar orbit through the Earth's mostly dipolar (with higher moments) field. This alone would place a torque on the gyroscopes that, if a large number of polarized spins were achieved, may have been problematic from the perspective of the GR mission. The precession of the gyroscope would then take contributions both from spacetime curvature as well as background magnetic field. However, the quartz gyroscopes are coated with superconducting niobium. When the niobium coated gyroscope rotates, it can produce a small magnetic field. The London moment induced field for the GP-B gyros is likely on the order of B London ∼ 2M Q ω = 1.08 × 10 −8 T, assuming M ∼ 2m e , Q ∼ 2e, and with ω being the angular frequency corresponding to the ∼ 9000 rpm gyro spin rate achieved in GP-B. Hence, the London-moment field is the larger possible field and it is aligned with the gyro rotational axis. The quartz spheres that form the GP-B gyros are silica (SiO 4 ) which consists of 14 Si and 17 O isotopes. The nuclear moments are respectively, µ14 Si ∼ 0.6µ N and µ17 O ∼ 1.9µ N , where µ N = (e/2m p ) is the nuclear magneton (m p is the mass of proton). Here, we consider the oxygen spins to be the relevant ones, both because there is more oxygen than silicon in silica and because it has a larger moment.
For the T B = 2.5 K ∼ 2.15 × 10 −4 eV temperature environment in GP-B, the energy splitting for the maximally spin-aligned and maximally anti-aligned states for the 17 kT B , so the estimate for spin polarization for the oxygen nuclei is of the There is some small correction (O(1) factors) for the presence of the Si, but this is a rough order of magnitude estimate of spin polarization fraction due to London-moment induced magnetic field. The value of α is tiny, although larger than the accidental polarization estimate of Note, the accidental net spin in absence of an alternative alignment mechanism would quite likely vary stochastically in both magnitude and direction over time on something like the spin coherence time, governed by how the randomly aligned spins in the quartz gyro interact. This would lead to an additional suppression of any new physics precession on the gyro, as the effect would execute a random walk instead of a coherent linear growth. This will cause an additional suppression by ∼ T spin /T 1 (with T spin the spin coherence time and T the mission duration) and α accidental α. In the following, we will constrain the spin dependent coupling due to the presence of London-moment induced magnetic field which is the larger possible field for which the fraction of polarized spins is α ∼ 5.6 × 10 −12 .

III. CONSTRAINTS ON AXION MEDIATED LONG RANGE YUKAWA TYPE OF POTENTIAL
The generic ultralight axion like particles (ALPs) which arise in string/M theory has a shift symmetry a → a + θ (θ is a real number) and transforms as a → −a under CP symmetry. Since CP is violated in nature, we consider CP violating coupling of axions with the dark sector. However, in the QCD sector, CP is conserved and we consider CP conserving coupling of axions with nucleons. Due to very light mass, the ultralight ALPs can mediate a long range Yukawa type of potential (∼ 1 r e −mar , m a is the mass of the ALP) between the visible sector which consists of standard model particles and the dark sector which consists of dark matter particles. The dark sector can also consist of some compact objects formed by axion dark matter, topological defects such as domain walls, cosmic strings or even it can be primordial black holes (PBH) [118][119][120]. The mass of the ALP is typically constrained by the distance between the dark sector and the visible sector. The range of the ALP mediated force is λ 1 ma which makes the ALPs ultralight. The equation of motion of the ALP field (a) from the CP violating coupling with the dark sector is [81] where the source term J(t, x) denotes the current density in the CP violating dark sector.
Such CP violating coupling of axions with the dark sector arises in a dark QCD model where the dark nucleon couples with the axion by a monopole force (L eff ⊃ g aN N aN N , where N is the dark nucleon) if the strong CP phase is not tuned [33,34,81]. The axion DM can also emit monopole force in the axiverse scenario with several axions and the vacuum of the axion potential is CP violating [18,121]. The axionic topological defects can also emit monopole force [24][25][26]. Since, J(t, x) is the current density, it should be proportional to the energy density of the CP violating dark sector, ρ D . From the dimensional analysis, it is also inversely proportional to the energy scale in the theory, here it is f a , the axion decay constant. Hence, one can parametrize the current density as . Hence, to solve Eq.3, we expand the axion field in a perturbative way with the perturbation parameter GM D R D (where G denotes Newton's gravitation constant, M D denotes the mass of the dark object and R D denotes its radius) and keeping the leading order term with its Yukawa behaviour. Hence, we can write the homogeneous solution for the axion field as where the leading order term a 1 (r) = 1 r e −mar has an Yukawa behaviour. Putting Eq.4 in Eq.3 in the Schwarzschild spacetime background, we obtain the resultant homogeneous axion field solution as where t dt is called the exponential integral function. Since we are not considering any particular dark sector object therefore, we do not have any information about M D and R D . Hence, we keep ourselves in the regime GM D R D 1 and the axion field solution (basically, the Green's function) becomes a 0 (r) ≈ 1 r e −mar . Hence, the axion field for the stationary non relativistic dark source is where we have assumed κ(t) = 1.
The derivative coupling of ALP with the nucleons in the CP conserving SM sector is governed by the Lagrangian where c f denotes the dimensionless constant for the SM fermions (f = p, n). In the nonrelativistic limit, the Dirac bilinears take the following form Hence, from Eq.7, we can write the interacting Hamiltonian in the non relativistic limit of the fermions as Also, the Hamiltonian for a particle having spin in a magnetic field ( B) has a magnetic Comparing H with H in Eq.9, we obtain the induced magnetic field due to the mediation of long range axionic Yukawa field as where µ n ≈ −1.9µ 0 , µ p ≈ 2.8µ 0 and µ 0 = e 2m N = 1.5 × 10 −10 eV −1 in natural units. Due to the primordial density fluctuations and the structure formations, the dark matter of the CP violating dark source is not spatially homogeneous and we have a non zero value of B a .
But if the dark matter distribution is spatially homogeneous then B a is zero even if c f = 0.
The energy density of the dark matter ρ D in Eq.6 can be decomposed into the galactic (ρ gal ) and the extragalactic parts (ρ egal ). We take the galactic contribution of dark matter ρ gal as simply the Navarro-Frenk-White (NFW) profile defined as [122,123] where ρ s = 0.184 GeV/cm 3 = 1.428 × 10 −6 eV 4 , r ≈ 8.33 kpc = 1.297 × 10 27 eV −1 , and r s = 24.43 kpc = 3.807 × 10 27 eV −1 . Hence, around the galactic centre, the contribution from ρ gal is important and the axion induced magnetic field is approximately r s e −mar r s + 2r + 2m a e ma(rs+r) (r s + r) 2 E i (−m a (r s + r)) The extragalactic contribution becomes dominant for the region where the Compton wavelength of the ALP ( 1 ma ) is much larger than the size of the galaxy. The dark matter energy density from the extragalactic part within the horizon is ρ c × Ω DM , where ρ c = 1.1 × 10 −5 h 2 GeV/cm 3 = 3.83 × 10 −11 eV 4 , is the critical density of the universe and Ω DM h 2 0.12 is the relic density of the dark matter. h = 0.67 denotes the reduced Hubble parameter. Due to the primordial density fluctuations (O(10 −5 )) from inflation, an inhomogeneity exists in ρ egal which is parametrized as Hence, the induced magnetic field mediated by long range axion force due to the extragalactic contribution of dark matter is The

IV. CONSTRAINTS ON OSCILLATING AXIONIC FIELD FROM GP-B RESULT
The axionic field can also oscillate with time and behave as a background. The oscillating field can interact with the spin of the gyroscope of GP-B satellite and affects the precession rate. The oscillating axionic field can be defined as where ρ DM = 0.3 GeV/cm 3 = 2.33 × 10 −6 eV 4 is the local dark matter density. The axions can have a derivative coupling with the nucleons as described by the Lagrangian Eq.7. In the rest frame of the gyroscope, the corresponding Hamiltonian is where σ is the spin of the nucleon that precesses in presence of a magnetic field characterised by ∇a and g aN N ∝ 1 fa is the axion nucleon coupling in the unit of energy −1 . Hence, using Eq.15, the time dependent effective magnetic field becomes where µ N = −1.9µ 0 (2.8µ 0 ) for neutron (proton), µ 0 = e 2m N ≈ 0.1 e.fm is the nuclear magneton, and v is the relative velocity of the gyroscope. This time dependent induced magnetic field can exert a force on the gyroscope as a whole and the precessional velocity of the gyroscope due to the oscillating axionic field becomes Eq.18 can contribute to the precessional velocity of the gyroscope and its contribution is within the experimental uncertainties in the measurements of frame-dragging and geodetic effects. In Fig.2

V. CONSTRAINTS ON AXION EDM COUPLING FROM THE GP-B RESULT
The axions can couple with the nucleons through the electric dipole moment operator described by the Lagrangian where g d is the coupling constant. Hence, in the non relativistic limit, the precessional velocity of the gyroscope due to the electric dipole moment operator becomes where a is the time oscillating axionic field, E is the electric field induced by the Earth's magnetic field in the rest frame of the satellite, and it is E ∼ v × B. We take the Earth's magnetic field as B ∼ 0.1 Gauss and we assume that it is attenuated by a factor of δ ∼ where the first term denotes the magnetic dipole moment operator, the second term denotes the electric dipole moment operator, and In the frame of the gyroscope travelling with a velocity v, the dark magnetic field is is the amplitude of the oscillating dark electric field in the lab frame. Since, in the non relativistic limit < F µν σ µν >∼ σ N . B A µ and < F µν γ 5 σ µν >∼ σ N . E A µ , the precessional velocity of the gyroscope due to the magnetic dipole moment operator is where cos θ picks the normal component of the dark electric field with respect to the gyroscopic plane and m A is the mass of the dark photon. Similarly, for the electric dipole moment operator, the precessional velocity due to the dark electric field is v EDM The background vector dark photon field changes the precessional velocity of the gyroscope through Eq.24 and Eq.25 and its contribution is within the uncertainties in the measurements of the precessional velocity of the gyroscope as obtained from GP-B result. In Fig.4 The electrons inside the Earth can generate a potential V (r) at the quartz sphere surface as where N ≈ 3.35 × 10 51 is the number of electrons inside the Earth, M Z is the mass of the gauge boson and a = 7027.4 km = 3.5 × 10 13 eV −1 is the distance between the Earth and the gyroscope. The mass of the gauge boson is constrained by the inverse of the distance between Earth and the gyroscope which gives M Z 2.82 × 10 −14 eV. Hence, the electric field at the gyroscope due to the long range Yukawa potential is Now the orbital velocity of the satellite in the polar orbit is v ∼ 2.51 × 10 −5 . The magnetic field in the boosted frame tied with the GP-B satellite is B Z ∼ − v × E Z and its magnitude is Frame-dragging Now the rate of change of angular momentum is d L dt = τ = µ × B with d L dt = Ω × L, whereˆ Ω is the precession axis and Ω denotes the precession rate. For all the previous calculations, we have taken L ∼ = 1 since the particles were all microscopic fermions and that gives Ω = µB. Now we have a macroscopic system for which L = Iω, where I ∼ 2 5 M R 2 is the moment of inertia of the sphere about its centre of mass. Hence, the precessional velocity due to the exchange of L e − L µ,τ gauge boson is The contribution in gyroscope precession rate due to L e − L µ,τ gauge bosons must be within the uncertainties that give bounds on g as g 2.33 × 10 −20 from geodetic measurement and g 1.42 × 10 −20 from frame-dragging measurement. In Fig.5, we have shown the variation of gauge coupling with the gauge boson mass from geodetic (blue line) and framedragging measurements (red line). The regions above these lines are excluded. The framedragging puts the stronger bound on the gauge coupling for the gauge bosons of mass M Z 2.82 × 10 −14 eV. Though the bound on the gauge coupling is 10 5 times weaker than that obtained from neutrino oscillation experiment [84] or the perihelion precession of planets [85], it is important to conclude the fact that such type of particle physics model can be constrained from GP-B experiment as well. Moreover, we suggest that the interaction between the gauge boson and the electron, giving rise to a long range fifth force in a gauged L e − L µ,τ scenario can be constrained from geodetic and frame-dragging measurements from GP-B satellite that was initially built for gravity experiments.

VIII. CONSTRAINTS ON UNPARTICLE MEDIATED LONG RANGE FORCE FROM GP-B RESULT
The vector unparticle couples to the leptonic or the baryonic current through the effective where O u µ is the unparticle operator, J µ is the baryonic or leptonic current, c u is the vector coupling, Λ u is the scale where the fields become conformally invariant, and d is the scaling dimension. We assume that O u and the fermion field ψ obey the following U (1) gauge Hence, the unparticle remains massless below the scale Λ u . One can write the unparticle propagator as where By taking the Fourier transform of propagator in the static limit, we obtain the long range unparticle mediated potential at the surface of the quartz sphere due to the presence of SM particles in the Earth as where N ≈ 3.35×10 51 is the number of electrons/nucleons in Earth. For the vector coupling, the forces add coherently. Hence, from Eq.35 the electric field at the surface of the quartz sphere is From Eq.36, we can write the corresponding magnetic field in the frame tied with the GP-B satellite is where v is the velocity of the satellite at the polar orbit which is v ∼ 2.51 × 10 −5 . We have previously calculated the magnetic moment of the rotating charged gyroscope as µ = 4.037 × 10 22 rad.eV −1 . Hence, for the macroscopic system, the precessional velocity due to the exchange of unparticle is The unparticle mediated long range force can contribute to the precessional velocity of the gyroscope and its contribution is within the experimental uncertainty in the measurements of geodetic and frame-dragging effects. Here, we have chosen r as the Earth-satellite distance.
In For the spin-independent vector couplings of ultralight particles, the forces add coherently. However, for spin-dependent operator couplings, a fraction of spins of SM particles in the gyroscope needs to be polarized. Here, we show a small fraction of polarization exists in the gyroscope due to the London-induced magnetic moment. The spin-dependent long range forces have an explicit dependence on the spin polarization fraction in the gyroscope. Alternatively, instead of a quartz sphere, one can make spin polarized ferromagnetic gyroscope or single domain magnetic needle [128,129] in the laboratory to obtain stronger bounds for the spin dependent couplings of axions, dark photons, and unparticles. Such a laboratory made spin polarized gyroscope can be used in GP-B satellite (future mission, if any) and one can constrain spin dependent long range forces for astrophysical distances which was made to test Einstein's theory of gravity. Other than GP-B, laser ranging network [130][131][132][133][134], interferometry method [135,136] can also measure the geodetic and frame-dragging effects. Future experiments like LARES2 [137,138] which can measure the frame-dragging with better sensitivity can significantly improve our bounds.