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 $B_c=m_e^2/e=4.414\times10^{13}\,\text{G}$ [@Turolla:2015mwa; @Beloborodov:2016mmx; @Kaspi:2017fwg], and constitute the natural target for our investigations.

In a previous paper [@Fortin:2018ehg], we considered the production of ALPs from the core of magnetars and their subsequent conversion into photons (we refer to [@Lai:2006af; @Chelouche:2008ta; @Jimenez:2011pg; @Perna:2012wn] for previous work in this direction). The relevant terms in the ALP-photon Lagrangian are \[EqnL\] where $a$ denotes the ALP and the coupling constants $g\equiv g_{a\gamma}$ and $g_{aN}$ have mass dimension $-1$. The first term is responsible for ALP-photon conversion in an external magnetic field [@Raffelt:1987im; @Marsh:2015xka; @Graham:2015ouw], while the second term in 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 