Photon conversion to axions and dark photons in magnetized plasmas: a finite-temperature field theory approach

Some of the most stringent constraints on physics beyond the Standard Model (BSM) arise from considerations of particle emission from astrophysical plasmas. However, many studies assume that particle production occurs in an isotropic plasma environment. This condition is rarely (if ever) met in astrophysical settings, for instance due to the ubiquitous presence of magnetic fields. In anisotropic plasmas, the equations of motion are not diagonal in the usual polarization basis of transverse and longitudinal modes, causing a mixing of these modes and breaking the degeneracy in the dispersion relation of the two transverse modes. This behavior is captured by a $3\times3$ mixing matrix $\pi^{IJ}$, determined by projecting the response tensor of the plasma $\Pi^{\mu\nu}$ into mode space, whose eigenvectors and eigenvalues are related to the normal modes and their dispersion relations. In this work, we provide a general formalism for determining the normal modes of propagation that are coupled to axions and dark photons in an anisotropic plasma. As a key part of this formalism, we present detailed derivations of $\Pi^{\mu\nu}$ for magnetized plasmas in the long-wavelength limit using the real-time formalism of finite-temperature field theory. We provide analytic approximations for the normal modes and their dispersion relations assuming various plasma conditions that are relevant to astrophysical environments. These approximations will allow for a systematic exploration of the effects of plasma anisotropy on BSM particle production.