X-Ray Polarization Signals from Magnetars with Axion-Like-Particles

Axion-like-particles (ALPs) produced in the core of a magnetar can convert to photons in the magnetosphere, giving rise to novel features in the X-ray spectrum. Since ALPs only mix with the parallel mode of the photon, the polarization of the soft and hard X-ray spectra is predicted to have an O-mode component, in addition to the mainly X-mode component given by most astrophysical models. The relative strength of the O-mode component depends on the intensity of ALPs produced in the core and the probability of conversion. We quantify our results by considering X-ray emission produced both by astrophysical processes and by ALP-photon conversion, in an uncorrelated fashion, and in different relative proportions, which we parametrize by the angle $\chi_0$. We then define a normalized astrophysics-subtracted Stokes parameter $R$ which only acquires non-zero values in the presence of ALP-photon conversion. We find, remarkably, that the parameter $R$ factorizes into a product of the ALP-to-photon conversion probability and $\cos(2\chi_0)$ and display $R$, as well as the usual Stokes parameter $Q$, as a function of the photon energy and relative fractions of ALP and photon intensities. For benchmark points currently allowed by the CAST experiment, the O-mode prediction can be tested in future X-ray polarimeters and used either to constrain ALPs or find evidence for them.


Introduction
X-ray polarimetry is a nascent field that can explore different astrophysical sources, ranging from compact objects to pulsar wind nebulas, supernova remnants, and molecular clouds. For a recent review of the astrophysical processes that can lead to polarized X-rays, as well as a description of current and planned missions, we refer to [1].
The emission from compact objects, which are the focus of this article, can exhibit polarization due to the different opacities of the surface plasma to different components of the photon electric field. However, polarization of X-ray radiation from neutron stars can also arise due to fundamental physics whose origin is distinct from astrophysics, and it is this possibility that we will entertain in our work. It is worthwhile to remember that polarization experiments probe physical anisotropies, and there is an important anisotropy in the photon Lagrangian if axion-like-particles (ALPs) [2] exist. Namely, ALPs mix only with the parallel and not the perpendicular component of the electric field in the presence of an external magnetic field. This anisotropy of the photon-ALP Lagrangian can thus be probed by X-ray polarimetry if the magnetic field near a compact object is strong enough.
Magnetars are an interesting subclass of neutron stars characterized by extremely strong magnetic fields, generally exceeding the quantum critical value B c = m 2 e /e = 4.414 × 10 13 G [3], and constitute the natural target for our investigations.
In a previous paper [4], we considered the production of ALPs from the core of magnetars and their subsequent conversion into photons (we refer to [5,6] for previous work in this direction).
The relevant terms in the ALP-photon Lagrangian are L ⊃ − g 4 aF µνF µν + g aN (∂ µ a)N γ µ γ 5 N, (1.1) where a denotes the ALP and the coupling constants g ≡ g aγ and g aN have mass dimension −1.
The first term is responsible for ALP-to-photon conversion in an external magnetic field [7,8], while the second term in (1.1) is the coupling between the ALP and nucleons N that leads to ALP production in the core of neutron stars. As mentioned before, ALPs mix with the component of the electric field that lies in the plane containing the external magnetic field B and the radial direction of motion, while the perpendicular component of the electric field propagates unaffected.
This implies that the polarization pattern of the observed spectrum will be affected by the presence of ALPs.
The purpose of this paper is to investigate the changes in polarization patterns in the observed X-ray spectra of magnetars in the presence of ALPs. Our approach consists of the following steps. Firstly, we assume that for every frequency ω ∼ 1 − O(few hundred) keV, both photons and ALPs are produced near the surface of the magnetar, in an uncorrelated manner. The important observation is that the astrophysical processes leading to photon production both in the soft as well as the hard X-ray regime are completely independent of the processes that give rise to ALPs, mainly by nucleon-nucleon bremsstrahlung [9].
The evolution of the ALP-photon system in the magnetic field of the magnetar is conveniently described in terms of a set of first-order differential equations coupling the amplitudes and phase difference of the ALP and photon fields. This parametrization has been shown to drastically simplify numerical analyses of the system [4]. The uncorrelated production mechanism of ALPs and photons alluded to above allows us to average over the initial phase differences. On the other hand, we parametrize the initial relative amplitudes by the angle χ 0 . For example, χ 0 = 0 signifies a pure ALP initial state. The introduction of χ 0 allows us to remain agnostic about astrophysical models of production of X-ray photons near a magnetar's surface.
Our second step is to define an astrophysics-subtracted (or surface-subtracted) normalized Stokes parameter R. This is done as follows. The sum (I) and difference (Q) of the phase-averaged photon intensities in the parallel and perpendicular planes are first computed, and then the quantities ∆I and ∆Q are defined, which are the values of the respective Stokes parameters away from the magnetar, minus their values at the magnetar's surface r 0 . This is very useful, since at the surface (which we take to also include the plasma) astrophysical effects due to differential opacities of the plasma lead to polarization, which should be subtracted away to extract the contribution to the polarization coming purely from ALP-photon mixing in the magnetosphere. In the absence of ALPs, and assuming that astrophysical processes in the magnetosphere do not lead to substantial polarization, the surface-subtracted Stokes parameters should vanish.
We find, remarkably, that the surface-subtracted Stokes parameters ∆I and ∆Q factorize into two pieces: the ALP-to-photon conversion probability and a factor that encodes the composition of the initial state. Specfically, normalizing the surface-subtracted Stokes parameter and defining the quantity where A is related to the initial amplitudes of the ALP and parallel photon states, we find that R = P a→γ cos(2χ 0 ). (1.2) This is our main result. It has several implications. Firstly, the spatial dependence of the Stokes parameters, as well as their dependence on the properties of the magnetar and the ALP parameters like the mass and coupling constant, are all encoded in the ALP-to-photon conversion probability P a→γ . Secondly, given the conversion probability, one can obtain the value of R simply by scaling with the appropriate initial condition cos(2χ 0 ). We utilize the methods of our previous paper [4] to display the dependence of R on the photon energy as well as ALP parameters.
The next generation of X-ray polarimeters [10,11] and increasingly sophisticated modeling of the astrophysics of magnetars [5,6,12] provide an opportunity to investigate ALPs using our methods. In the absence of ALPs, R should be uniformly zero at all frequencies ω since both P a→γ as well as cos(2χ 0 ) vanish. A non-zero R, on the other hand, would require both non-zero ALP-to-photon conversion as well as an initial ALP-parallel photon mixture which is not halfand-half (i.e. χ 0 = π/4). Energy-dependent modulations in the polarisation of magnetar spectra displayed in the left panel of Fig. 2 can be used to identify or constrain ALPs.
Our paper is organized as follows. In Section 2, we recapitulate the ALP-photon coupled system and the results of [4]. In Section 3, we perform the calculation of the Stokes parameters.
In Section 4, we display the parameter R as a function of the photon energy, ALP mass, and ALP-photon coupling. We also briefly describe the observational possibilities and contrast our results with previous work. We end with our conclusions in Section 5. Finally, Appendix A demonstrates the factorization property of the normalized surface-subtracted Stokes parameter R.

Oscillations
In this section we introduce the evolution equations for the photon and the ALP relevant to magnetars. We use the general formalism of [4] for oscillations in the limits where the space variations of the magnetic field are large compared to the particle wavelength and dispersion is weak. The reader interested in the details of this formalism is referred to [4].

Evolution Equations
For particles with energies in the 1 to 200 keV range (corresponding to soft and hard X-ray photons) propagating radially outwards from a magnetar, the system is in the appropriate limits (as long as the magnetic field is not too large) with negligible plasma contributions to the evolution equations [5,7], leading to Here a(x), E (x) and E ⊥ (x) are the ALP and parallel and perpendicular photon electric fields respectively while x = r/r 0 with r the distance from the center of the magnetar and r 0 the magnetar's radius. Moreover, ω is the energy of the particles, m a is the ALP mass, g is the ALP-photon coupling constant and θ is the angle between the magnetic field and the direction of propagation of the particle. q and q ⊥ are dimensionless functions of the magnetic field B given by [5,7] q = 7α 45π b 2q ,q = 1 + 1.2b 1 + 1.33b + 0.56b 2 , e /e = 4.414 × 10 13 G is the critical QED field strength. Since the plasma contributions are negligible, the three-state system (2.1) effectively decomposes into two independent oscillation systems, a two-state system for the ALP and the parallel photon, and a one-state system for the perpendicular photon.
Due to the probability conservation property d dx [|a(x)| 2 + |E (x)| 2 ] = 0 discussed in [4], the different states can be expressed as where A a = A cos[χ(x)], A = A sin[χ(x)] and A ⊥ are the amplitudes at position xr 0 of the ALP field, the parallel photon field and the perpendicular photon field respectively. It is important to note that A and A ⊥ are constants which can always be chosen real and positive while χ(x), φ a (x), φ (x) and φ ⊥ (x) are real functions. Hence, the intensities at position xr 0 are given by the respective amplitudes squared, i.e. I a (x) = A 2 cos 2 [χ(x)], I (x) = A 2 sin 2 [χ(x)] and I ⊥ (x) = A 2 ⊥ for the ALP field, the parallel photon field and the perpendicular photon field. Probability conservation then implies that I a (x) + I (x) = A 2 and I ⊥ (x) = A 2 ⊥ are constants.
Using (2.2) in (2.1), the evolution equations become where ∆φ(x) = φ a (x) − φ (x) is the phase difference between the ALP field and the parallel photon is the sum of the phases of the ALP field and the parallel photon field. In addition to showing that the differential equation for φ ⊥ (x) decouples, the evolution equations (2.3) imply also that the differential equation for Σφ(x) decouples in the sense that it is completely determined once the solutions to the coupled χ(x) and ∆φ(x) differential equations are known. Hence both Σφ(x) and φ ⊥ (x) are irrelevant in computing the intensities.
The initial states are determined at the surface of the magnetar from the boundary conditions at x = 1. For the two-state system described by χ(x) and ∆φ(x), pure initial states satisfy χ(1) = nπ/2 with n ∈ Z, and the boundary condition for ∆φ(1) must satisfy ∆φ(1) = mπ with m ∈ Z to avoid singularities. The two different choices of phase difference for pure initial states lead to the same intensities.
Indeed, it is straightforward to verify that the transformation leaves the evolution equations (2.3) invariant. Therefore, for boundary conditions given by χ(1) = χ 0 and ∆φ(1) = ∆φ 0 , the intensities verify Thus, for the pure ALP initial state χ 0 = 0, the two different choices of phase difference, say ∆φ 0 = 0 and ∆φ 0 = π, give the same intensities. For the pure parallel photon initial state χ 0 = π/2, the two different choices of phase difference, say ∆φ 0 = 0 and ∆φ 0 = π again, give the same intensities since (2.3) are also invariant under the transformation χ(x) → χ(x) ± π. By using such arguments, more can be said about the intensities.
Indeed, two other transformations play an important role in the following. First, the transformation leaves the evolution equations (2.3) invariant, which implies that the intensities satisfy the following relations, Second, in general in the definitions (2.2), one can always take χ(x) ∈ [0, π/2]. With that interval in mind, it is interesting to note that π/2 − χ(x) ∈ [0, π/2]. Hence, applying the transformation leaves the evolution equations of the coupled χ(x) and ∆φ(x) invariant while keeping the boundary condition on χ(x) in the appropriate interval. Therefore, it is easy to conclude that if χ(x) and ∆φ(x) are solutions to (2.3), then from (2.6) π/2 − χ(x) and ∆φ(x) ± π are also solutions to (2.3).
As before, this implies some conditions on the intensities which are here given by It is important to note that (2.7)

Mixed Initial States
As discussed in the introduction, in the X-ray regime, production mechanisms in magnetars have very different origins for ALPs and photons. The former comes predominantly from nucleon-nucleon bremsstrahlung of ALPS [9] while the latter originates from the magnetar itself and interactions with the plasma in the magnetar's atmosphere (e.g. parallel-perpendicular mode conversion in inhomogeneous magnetar's atmosphere) [5].
Hence, it is expected that at the magnetar's surface, where the ALP and parallel photon fields start oscillating into one another as they travel outwards, their amplitudes can differ substantially depending on the actual values of the ALP parameters. Moreover, their phase difference is effectively random.
Thus, for given initial amplitudes corresponding to specific ALP parameters, i.e. for a specific A and χ(1) = χ 0 , it is natural to average over the initial phase difference ∆φ(1) = ∆φ 0 to determine the intensities away from the magnetar. This implies that, at a distance xr 0 from the magnetar's surface, the averaged intensitiesĪ(x) for initial Due to the singularities appearing for pure initial states and the equality of the intensities for the two different choices of phase difference, there is no averaging for pure initial states and only one initial phase difference (say ∆φ 0 = 0) is necessary to determine the averaged intensities at position xr 0 .
Since the intensities transform as in (2.5) under the transformation (2.4) and (2.7) under the transformation (2.6), the averaged intensities (2.8) also behave as (2.10) Moreover, from the invariance under the transformation χ(x) → χ(x) ± π, the averaged intensities are periodic functions of χ 0 with period π. Hence from (2.9) the averaged intensities are even functions of χ 0 with respect to π/2 while from (2.10) the averaged intensities for some initial amplitudes are related to the averaged intensities for the inverted initial amplitudes. As a corollary of (2.10), it is straightforward to see thatĪ a (π/4, x) =Ī (π/4, x) = A 2 /2. Therefore, for a half-and-half initial state, the averaged intensities are exactly 1/2 of the sum of the ALP and parallel photon initial intensities. Thus, it is only necessary to compute the averaged intensities for initial χ(1) ∈ [0, π/4) to determine the averaged intensities for all possible initial states. In fact, the next section demonstrates that we can do better.
Obviously, averaging over the perpendicular photon initial phase is of no consequence for the averaged perpendicular photon intensity, henceĪ ⊥ (x) = A 2 ⊥ and the averaged perpendicular photon intensity remains constant.

Photon Polarizations
This section introduces the Stokes parameters describing the polarization state of the photon signal. The effect of ALP-photon coupling is discussed qualitatively, with a more quantitative analysis for a particular magnetar presented in the next section.

Stokes Parameters
At a given position xr 0 from the magnetar, the relevant Stokes parameters are I and Q which correspond respectively to the sum of and the difference between the averaged perpendicular photon intensity and the averaged parallel photon intensity, i.e.
As pointed out before, since the perpendicular photon field does not mix with the ALP-parallel photon two-state system, its averaged intensity is constant. Therefore, all modifications to the Stokes parameter are driven by the averaged parallel photon intensity.
Moreover, since plasma contributions are negligible outside the magnetar's atmosphere, in the absence of ALPs the evolution equations (2.1) correspond to two independent one-state systems, one for the parallel photon field and one for the perpendicular photon field. Hence without ALPs the Stokes parameters at a distance xr 0 from the magnetar would be the same as the Stokes parameters at the magnetar's surface. The effects of the existence of ALPs and the possible photon oscillations with them on the Stokes parameters can therefore be conveniently analysed by studying the differences between the Stokes parameters at xr 0 (3.1) and the Stokes parameters at ( 3.2) The relation ∆Q(χ 0 , x) = −∆I(χ 0 , x) demonstrates perfect anti-correlation between the surfacesubtracted Stokes parameter ∆I(χ 0 , x) and the surface-subtracted Stokes parameter ∆Q(χ 0 , x).
Therefore, in our scenario, there is only one independent quantity to keep track of, which we choose as the surface-subtracted Stokes parameter ∆I(χ 0 , x).
It is clear from this analysis and especially (3.2) that an astrophysical understanding of soft and hard X-ray polarized emission from magnetars can constrain or even discover ALPs. Indeed, on the one hand standard astrophysical considerations dictate the expected Stokes parameters I exp and Q exp at the magnetar's surface. On the other hand, observations on Earth lead to observed Stokes parameters I obs and Q obs . Thus theoretical and observational astrophysics determine the differences of the Stokes parameters ∆I astro = I obs − I exp and ∆Q astro = Q obs − Q exp . Nonvanishing values for ∆I astro and ∆Q astro suggest either a misunderstanding of the astrophysical processes at play in magnetars or observational errors. However, from (3.2) non-vanishing values for ∆I astro and ∆Q astro such that ∆Q astro = −∆I astro strongly suggest that ALPs exist and ALPphoton oscillations occur in the magnetic field of the magnetar. The relation ∆Q astro = −∆I astro originating from (3.2) can thus be seen as a smoking gun signal for ALPs.
Before turning to an analysis of the transformation properties of the surface-subtracted Stokes parameters and their consequences, it is of interest to define a new quantity, the normalized surface-subtracted Stokes parameter R(χ 0 , x).

Normalized Surface-Subtracted Stokes Parameter
Since we do not rely on any specific models for the ALP and photon production mechanisms, the values of A, χ 0 and A ⊥ which determine the initial ALP and photon amplitudes at the magnetar's surface are not fixed. However, from (3.2) it is clear that the differences of the Stokes parameters at infinity and at the magnetar's surface does not depend on A ⊥ . Moreover, sinceĪ (χ 0 , x) ∝ A 2 , their dependence on A is simple. It is thus convenient in the following to study the normalized surface-subtracted Stokes parameter Alternatively, one can set A = 1 in (3.2) to compute the normalized surface-subtracted Stokes parameter (3.3). Apart from the implicit dependence on the ALP and magnetar parameters, the normalized surface-subtracted Stokes parameter (3.3) is thus a function of the boundary condition χ 0 only instead of the boundary conditions A, χ 0 and A ⊥ .
The transformation properties of the normalized surface-subtracted Stokes parameter (and its non-normalized counterpart) are easily obtained. They are from the invariance of (2.3) under the transformations χ(x) → χ(x)±π, (2.4) and (2.6) respectively.
Therefore (3.4) implies that the normalized surface-subtracted Stokes parameter is a periodic function of χ 0 with period π which is even in χ 0 with respect to π/2 and odd in χ 0 with respect to π/4. As such, it can be expanded in Fourier modes as with the modes given by The latter can be computed directly from (3.3) since the integral is χ 0 -independent. Indeed, averaging over ∆φ 0 eliminates all dependence on the boundary conditions (this statement is highly non-trivial, for a proof the reader is referred to Appendix A). Then the Fourier modes (3.6) are which are χ 0 -independent and thus the normalized surface-subtracted Stokes parameter (3.5) is It can be written in a more familiar form by setting χ 0 = 0. Indeed, since (3.7) is χ 0 -independent, we can set χ 0 = 0 and replace the average over ∆φ 0 by ∆φ 0 = 0. Analogously, we can set χ 0 = 0 directly in (3.8) and compare both sides. Both approaches lead to our final result R(χ 0 , x) = R(χ 0 = 0, x) cos(2χ 0 ) = sin 2 χ(x)| χ(1)=0,∆φ(1)=0 cos(2χ 0 ) = P a→γ (x) cos(2χ 0 ), (3.9) where P a→γ (x) = sin 2 χ(x)| χ(1)=0,∆φ(1)=0 , (3.10) is the ALP-to-photon conversion probability at a distance xr 0 for pure ALP initial state. Therefore, the normalized surface-subtracted Stokes parameter factorizes into two parts, the ALP-to-photon conversion probability for pure ALP initial state P a→γ (x) that is a function of the ALP and magnetar parameters, and cos(2χ 0 ) which encodes the dependence on the mixed initial state. In addition, from (3.9) it is clear that −1 ≤ R(χ 0 , x) ≤ 1 with R(π/4, x) = 0 as proven before.
The behavior on the mixture of the initial state has been completely uncovered in (3.9), (3.11), (3.13) and (3.12). Indeed, the dependences on A, χ 0 and A ⊥ are exhibited explicitly. Hence it is easy to determine the behavior of the averaged intensities or the Stokes parameters under changes of the mixed initial state. For example, the Stokes parameter Q(χ 0 , x) at a distance xr 0 is always smaller (larger) than or equal to the Stokes parameter Q(χ 0 , 1) at the magnetar's surface if χ 0 ∈ [0, π/4] (χ 0 ∈ [π/4, π/2]), i.e. if the initial state is a mixture dominated by ALP (parallel photon).
Moreover, the ALP-to-photon conversion probability for pure ALP initial state is the only necessary quantity to compute in order to determine all the remaining quantities of interest. This important observation simplifies greatly the problem since P a→γ (x) is independent of χ 0 and no averaging on ∆φ 0 is necessary.
The remarkable factorization of the dependence on the initial state in the averaged intensities is a consequence of the averaging over the initial phase difference. If the average had not been performed, there would have been no such factorization. Hence, if there exists a common origin for the production of ALPs and photons, then the initial phase difference would be fixed and the dependence on χ 0 would not be as straightforward as in (3.13).
Finally, since the argument leading to the results above relies only on the probability conservation property of the system, it is important to note that relations analogous to (3.13) for averaged intensities exist for any two-state oscillation system satisfying the probability conservation property. Although we have not investigated it further, we expect that relations analogous to (3.13) for averaged intensities can be found for any n-state oscillation system as long as probability is conserved.

Results
In this section we solve numerically the evolution equations (2.3) with dipole magnetic field (valid a few r 0 away from the magnetar) for magnetar parameters corresponding to SGR 1806-20 in the soft and hard X-ray range. As mentioned in the introduction, we stay agnostic with respect to the different production mechanisms for ALPs and photons. Our results are therefore conveniently expressed with the help of the normalized surface-subtracted Stokes parameter (3.3) far away from the magnetar where the magnetic field is effectively turned off.
In principle, (3.3) forces us to compute R by averaging over the initial phase difference for several mixed initial states. However, the relation (3.9) allows us to focus only on the ALP-tophoton conversion probability for pure ALP initial state. Nevertheless, to verify the relation (3.9), we compute (for various ALP and magnetar parameters) R the hard way using (3.3) and compare with the ALP-to-photon conversion probability for pure ALP initial state through (3.9), validating the argument of the previous section.
For SGR 1806-20, the ALP-to-photon conversion probability for pure ALP initial state in the soft and hard X-ray range was studied in [4]. For completeness, we include here the relevant plots of [4] as well as a discussion of the relevant features of P a→γ ≡ P a→γ (∞).

Dependence on the ALP and Magnetar Parameters
As mentioned previously, standard astrophysical considerations dictates how the magnetar and its atmosphere behave. Hence, theoretical astrophysics determines the initial amplitudes of the particles at the magnetar's surface. The particles then travel through the plasma in the magnetar's atmosphere, which is of the order of centimeters, where they can encounter a resonance in the parallel-perpendicular photon space [5]. Since we remain agnostic about the astrophysical processes at play, we can start evolving the evolution equations (2.3) once the particles exit the magnetar's atmosphere, which is the only region where plasma effects are non-negligible. Therefore, our initial amplitudes at the magnetar's surface correspond more precisely to the amplitudes outside the magnetar's atmosphere. However, since the magnetar radius is of the order of kilometers while the atmosphere's thickness is of the order of centimeters, we can choose x = 1 as our initial point to solve the evolution equations (2.3) without consequences. From this point on, the plasma is completely negligible and the analysis presented before is valid.
However, the dipolar approximation holds only several radii away from the magnetar. Hence, to assume a dipole magnetic field, it is necessary to redefine the initial amplitudes at the magnetar's surface by the amplitudes several radii away from the magnetar where the dipolar approximation is valid. The evolution equations (2.3) are thus evolved numerically from there and the average in (3.3) is performed on the phase difference at that position. Since the conversion radius, where most of the ALP-parallel photon conversion occurs, is a few hundred radii away from the magnetar for our benchmark point, the dipolar approximation is clearly justified.
The particles thus evolve through an astrophysical sector consisting of the production region (either inside or closely by the magnetar), the plasma region and the region where the dipolar approximation is not valid. The amplitudes at the end of this sector are then used as initial amplitudes for the particles evolving through a vacuum sector consisting of the dipolar magnetic field of the magnetar with negligible plasma effects. It is now clear that our work focuses on the vacuum sector.
On the one hand, the ALP-to-photon conversion probability P a→γ for pure ALP initial state in the limit of dipolar magnetic field depends on six ALP and magnetar parameters corresponding to the ALP energy ω, the ALP mass m a , the ALP-photon coupling constant g, the magnetar's radius r 0 , the (dimensionless) magnetar magnetic field at the surface b 0 = B 0 /B c and the angle between the direction of propagation and the magnetic field θ. On the other hand, the normalized surface-subtracted Stokes parameter R(χ 0 ) ≡ R(χ 0 , ∞) depends also on the mixed initial state through χ 0 . Fig. 1 shows the normalized surface-subtracted Stokes parameter (3.3) for two points in the ALP and magnetar parameter space in function of χ 0 . A comparison with P a→γ cos(2χ 0 ) is also shown, confirming the validity of (3.9) (they are undistinguishable).
As explained in [4], the fact that the conversion probability peaks in the X-ray range (as shown in the first panel of Fig. 2) has interesting consequences since the ALP nucleon-nucleon bremsstrahlung emission spectrum peaks in the hard X-ray range for our benchmark model (for a degenerate medium relevant to magnetars). The second panel shows R as a function of the ALP mass m a , which interpolates between two plateaus with a quick transition between them. The plateau at vanishing conversion probability occurs for large m a while the plateau at non-vanishing conversion probability corresponds to small m a .  Fig. 1: Normalized surface-subtracted Stokes parameter R as a function of χ 0 . The blue lines correspond to R(χ 0 ) obtained from averaging over ∆φ 0 as dictated in (3.3) while the red lines correspond to P a→γ cos(2χ 0 ) following (3.9). The ALP and magnetar parameters are ω = 1 keV, m a = 10 −9 keV, g = 10 −15 keV −1 , r 0 = 9 km, B 0 = 10 14 G and θ = π/4 for the left panel while they are ω = 10 keV, m a = 10 −8 keV, g = 10 −14 keV −1 , r 0 = 11 km, B 0 = 10 13 G and θ = −π/8 for the right panel. Variations with respect to the other parameters can be understood qualitatively from the evolution equations (2.3) as explained in [4]. For example, for small ALP-photon coupling constant g, there is effectively no oscillations and the conversion probability vanishes. At larger g, the conversion probability increases and the oscillatory nature of the problem appears in small variations around some decreasing mean value for the conversion probability. Moreover, the θ-dependence of the conversion probability verifies P a→γ (−θ) = P a→γ (θ), as expected from the invariance of the evolution equations (2.3) under the combined transformations ∆φ → ∆φ ± π and θ → −θ.
The differences between the conversion probabilities for ω = 1 and 200 keV exhibited in Fig. 3 show that the precise locations of the transitions in the ALP mass and the ALP-photon coupling constant change as the ALP energy changes. Moreover, as expected from the qualitative discussion above, R is negligible everywhere except for small m a and large g. In these regions, the conversion probability, which can be quite large, behaves in agreement with the oscillatory nature of the problem.

Observational Outlook
We now turn to some comments on the observational outlook of our methods.
The field of X-ray polarimetry is currently very active, with several new instrument designs being proposed and detectors that have either been launched or are in the planning stage [1]. The theoretical targets include understanding the emission from white dwarfs and neutron stars in binary systems, and the coupling of the plasma to the magnetic field in accreting X-ray pulsars [13].
The targets most germane to our work are neutron stars and magnetars. For these stellar objects, there are two sources of polarization that have been explored theoretically: the polarization due to non-linear QED effects [14] and the polarization due to the anisotropic opacities of the surface plasma [15]. While polarization due to ALP-photon mixing has been studied, this has mainly been in the context of soft thermal emission [5].
The main new feature of our analysis has been to show that polarization is expected in both the soft as well as the hard X-ray regime of magnetars. There are several missions that are poised to explore these features. Both the Imaging X-ray Polarimetry Explorer (IXPE) [11] and the Enhanced X-Ray Timing and Polarimetry Mission (eXTP) [16] missions will launch in the next few years, and look for signals in the 2 − 10 keV range. Among the missions that study hard X-rays, most are focused on solar flares and gamma-ray bursts. However, X-Calibur and PolariS will be sensitive to signals in the 20 − 60 keV and 10 − 80 keV range from neutron stars, respectively [17].

Conclusion
X-ray polarimetry is an emerging field in astronomy that is capable of probing both extended and compact objects. Polarimetry is particularly suited to probe anisotropic couplings of the photon.
From the perspective of fundamental physics, one particularly important coupling that can be probed by polarimetry is the ALP-photon coupling. Magnetars, with their extreme magnetic fields, are natural laboratories in this context.
In this paper we investigated oscillations between ALPs and photons in the magnetosphere of a magnetar. Since ALP and photon production mechanisms in the soft and hard X-ray regimes are independent, we remained agnostic with respect to the production mechanisms and studied the normalized surface-subtracted Stokes parameter averaged over the initial phase difference.
We then found that the normalized surface-subtracted Stokes parameter R, which is the normalized Stokes parameter I at infinity minus the normalized Stokes parameter I at the magnetar's surface, factorizes into the ALP-to-photon conversion probability for pure ALP initial state times a simple function of the ALP-parallel photon mixture of the initial state. This highly non-trivial statement was then proved mathematically and checked numerically for several ALP and magnetar parameters. The averaged intensities as well as the Stokes parameters at infinity were then obtained in terms of the parameters describing the initial state and the ALP-to-photon conversion probability.
Apart from the factorization property, R has several important properties. Firstly, it vanishes when ALPs do not exist or ALP-parallel photon oscillations do not occur. Moreover, it is non-zero if ALP-parallel photon oscillations occur and the initial state is not an equal mixture of ALPs and parallel photons. For benchmark values corresponding to SGR 1806-20, we exhibited the behavior of R in the ALP mass and ALP-photon coupling plane for different ALP energies. Our results show that R can be quite large for magnetars, implying large deviations from the no-ALP case. Since our general approach is applicable to other oscillation systems with the probability conservation property, one could investigate changes in polarizations for neutrino oscillation problems.
Finally, we note that polarization signals in the soft X-ray regime emanating from neutron stars have been proposed before in the context of the astrophysics of the surface plasma, vacuum polarization effect, and even the oscillation of thermal soft X-ray photons into ALPs. Our work shows that polarization signals could exist in the hard X-ray regime as well. Overall, magnetars are rich targets for understanding fundamental physics using X-ray polarimetry.

A. Proof of the Factorization Property
To prove the factorization property, we focus on a generic two-state oscillation system given by To simplify the notation in (A.3), we introduced primes to denote derivatives with respect to x.
It is interesting to note that if the average over ∆φ 0 is not performed, the differential equation for the non-averaged quantity is the same than (A.3) but the boundary conditions (A.4) are modified such that the first and second derivatives at x = 1 of the non-averaged quantity depend on χ 0 . The average is thus necessary for the factorization property to hold. Moreover, it might be curious at first to see that P (x, χ 0 ) satisfies a third-order differential equation instead of a second-order differential equation. The reason why this occurs is most clearly understood from the non-averaged quantity, since without the average on ∆φ 0 , one needs a third-order differential equation for the non-averaged quantity to have two genuine boundary conditions, the boundary condition of the non-averaged quantity being zero for any χ 0 and ∆φ 0 .
Finally, we verified numerically that the differential equation (