Quantum Chromodynamics (QCD) describes the strong interactions among quarks and gluons and offers definite predictions at the high energy-perturbative domain. At low energies, the non-linear nature of the theory introduces a non- trivial vacuum which violates the CP symmetry. The CP violating term is parameterized by θ and experimental bounds indicate that θ ≤ 10−10. The smallness of θ is known as the strong CP problem.
An elegant solution has been offered by Peccei-Quinn    . A global U(1)PQ symmetry is introduced, the spontaneous breaking of which provides the cancellation of the θ-term. As a byproduct, we obtain the axion field, the Nambu- Goldstone boson of the broken U(1)PQ symmetry. There are extensive reviews covering the theoretical aspects and the experimental searches for the axion  -  .
A general feature of the axion is its two-photon coupling
where α is the axion field, the (dual) electromagnetic field strength tensor and g the photon-axion coupling constant. Accordingly, in the presence of a magnetic field, a photon may oscillate into an axion and vice-versa. A prototype experiment in the search for solar axions is CAST experiment, which set the limit for g < 10−10 GeV−1  . The CAST experiment involves a magnetic field B = 9T and a magnetized region L=9.3 m. Therefore, the relevant scale (BL)2 is (BL)2 ≈ 7000 T2m2. Always in the search for solar axions, a space-based experiment has been proposed, where use is made of the Earth’s magnetic field  . The weakness of the geomagnetic field B is somehow compensated by the larger L value.
In the present work, we suggest using the geomagnetic field in order to study the inverse process, photon-axion transition. We consider a GPS signal travelling from one satellite to another. In the presence of the Earth’s magnetic field, the photon may oscillate to an axion and vice-versa. Our proposal involves an increased (BL)2 scale, a higher accuracy and the exploration of a new range of g, ma (coupling constant and axion mass respectively). To further increase the accuracy of our evaluation, we include the effect of Earth’s gravitational field on the whole process.
2. GPS Signal and the Influence of Earth’s Gravitational Field
Global Positioning System (GPS) offers to geodesy position measurements with a millimeter to centimeter-level precision. GPS contributed also to significant advances in geophysics, seismology, atmospheric science and natural hazard science. Besides accurate positioning, all disturbances in the propagation of the transmitted GPS signal from satellite to receiver are mined for information  . The GPS system is in effect a realization of Einstein’s view of space and time. Indeed, the system cannot function properly without taking into account fundamental relativistic principles  .
In the presence of the geomagnetic field we may envisage the transition of the GPS signal into an axion. To reach the highest accuracy in the evaluation of the probability, we must include also the effect of Earth’s gravitational field.
The geometry outside a spherical star like the Earth is provided by the Schwarzschild metric
where Rs = 2GM with M the mass of the star. The relevant scale in our problem, with Rg the radius of the GPS satellite from the Earth’s center, is
Given the smallness of the gravitational strength, we adopt from the very start the metric of a weak gravitational field
The metric is independent of the coordinate t and this implies that the energy is conserved
where we defined.
The quantum mechanical phase accumulated by a particle propagated in space-time is given by the invariant quantity  - 
Using the relation
and the relation (5), we obtain
The quantum phase acquires the form
Equation (7) allows to rewrite
The above expression is accurate within the weak gravity approach. Notice that in the absence of gravity we obtain
which is the well-known established result for the Minkowski spacetime.
Working always in the weak gravity limit and ignoring terms (Rs/r)2, we evaluate, using Equation (7)
We conclude that
In our case we consider a light signal travelling from a satellite at r = Rg to another satellite at r = Rg.
The trajectory is almost a straight line and the closest distance to the Earth is denoted by b. Then the traveled distance is and
For the second contribution we have to evaluate the integral.
Defining we obtain
We obtain finally
It should be noted that the energy E is the energy measured at infinity. The energy E and the energy Eg measured at distance r = Rg (the position of the satellites) are connected by
The above relation represents the well-known gravitational red shift. Expressing everything in terms of the measured Eg and considering the case we find the compact expression
We conclude that the influence of the Earth’s gravitational field can be absorbed into a definition of an effective mass μ
3. Photon-Axion Oscillations
Consider a GPS signal travelling from a satellite at r = Rg to another satellite at r = Rg, both moving at low altitude around the Earth. Let us define as z axis the direction of photon’s propagation. The polarization of the photon lies then at the x-y plane. The photon is moving in the presence of the geomagnetic field. The component of parallel to the direction of motion does not induce photon-axion mixing. Following Equation (1), the transverse magnetic field couples to, the photon polarization parallel to and decouples from, the photon polarization orthogonal to.
The photon-axion mixing is governed by the following equation:
The 2-dimensional matrix M is
where μ2 is defined in Equation (22). For a photon, moving in a medium with number density of electrons ne, the photon mass mγ is given by
The axion mass mα is not experimentally known. Matrix M is diagonalized through the angle θ with
we obtain for the probability that a photon converts into an axion after travelling a distance s
When the oscillatory term in Equation (29) is small, i.e., we obtain the behavior (L ≡ s)
thus the relevant scale for an experimental setup is.
Imagine a photon scratching the Earth at and becoming an axion at this position.
The probability is
The axion reemerging from the Earth travels to the other satellite. The probability of being detected there like a photon is
Therefore the probability for light shining through the Earth is given by
4. Discussion and Overall Conclusions
We suggest a new experimental technique, where thanks to the Earth’s magnetic field, a GPS signal is transformed to an axion particle. There are clear advantages in our proposal.
First, the high accuracy. The distance L is measured with a precision at the millimeter-centimeter level. Second, the large value of the (BL)2 scale. The Earth’s magnetic field is of a dipole form with a mean value on the Earth’s surface. The magnetic field is falling off like, where RE is the radius of the Earth (approximately 6370 km) and r the radial distance from the center of the Earth. To obtain the best available values for the geomagnetic field (≈B0), the satellites should remain in a low orbit around the Earth. Actually, rather than working with present GPS, we should use inexpensive microsatellites, known as cubesats, orbiting close to the Earth’s surface. The smallness of the magnetic field, compared say, to the CAST experiment, is overbalanced by the much larger L value, which may reach L = 2RE. The scale (BL)2 becomes then 140,000 T2m2, orders of magnitude above existing or proposed values. Third, the energy range of the photons and the possibility to search in an unexplored domain of g, mα values. GPS photons travel with a frequency of approximately 1 GHz. It is appropriate, for our experimental needs, to use a higher frequency of 1 THz, so that we can reach lower values for mα, down to μeV  and explore even smaller g values. Finally, our emitter and receiver are in constant motion and therefore the parameter L may vary offering plentiful information. Relying on Einstein’s view of space and time, GPS has been established as the ideal tool for geodesy. Next to the relativistic conceptions we included a quantum approach, and we obtained the probability for the transition of a GPS photon to an axion particle in the presence of the Earth’s magnetic field. This transition will result in a dimming of the photons and further to light shining through the Earth phenomenon. We may envisage that in the future the long list of scientific disciplines served by GPS will be enriched by particle physics.
The present work was initiated while I was a visiting scholar at the Center for Axion and Precision Physics, Institute for Basic Science (CAPP-IBS) in Korea. I would like to thank CAPP’s director Prof. Yannis Semertzidis for the kind invitation and many enlightening discussions. Aspects of GPS related technology were clarified during a visit at the Laboratoire Astroparticule et Cosmologie (APC) in Paris. I am appreciative of this help provided by APC’s director Prof. Stavros Katsanevas. Mr. Dimitris Evangelinos assisted in the typesetting.
Our proposal involves the combination of relativity theory and quantum theory, in order to carry out a quantum particle experiment in space. One might question the prevalence of quantum principles in light propagation in outer space over a long distance. But the recently announced, after our paper appeared, Chinese achievement, with the Micius satellite sending entangled photons over thousands of kilometers  , proves the feasibility of the proposal.