Mixing dynamics of dimension-five interactions (scalar/pseudoscalar-photon) in magnetized medium

In many extentions of standard model, dimension-5 scalar di-photon ($g_{\gamma\gamma\phi}\phi$ $F^{\mu\nu}F_{\mu\nu}$ ) or pseudoscalar di-photon ($g_{\gamma\gamma a}a\tilde{F}^{\mu\nu} F_{\mu\nu}$, ) interaction materializes due to scale symmetry breaking or $U_A(1)$ symmetry breaking. In a magnetized vacuum (i.e., in an external background field $\bar{F}_{\mu\nu}$) the transverse degrees of freedom of the photons-- for such systems-- can be described in terms of the form factors constructed out of the background field strength tensor ($\bar{F}_{\mu\nu}$) and the same for dynamical photon ($f^{\mu\nu}$); they happen to be $\bar{F}_{\mu\nu}f^{\mu\nu}$ and ${\tilde{\bar{F}}_{\mu\nu}f^{\mu\nu}}$. These form factors transform differently under CP transformation. While $\bar{F}_{\mu\nu}f^{\mu\nu}$ (describing polarization orthogonal to B ($|\gamma_{||} >$)) is CP even, the other one, ${\tilde{\bar{F}}_{\mu\nu}f^{\mu\nu}}$( describing polarization along B ($|\gamma_\bot >$), is CP odd. In the interaction Lagrangian, if the scalar is interchanged with the pseudoscalar, the role of the two form factors just gets interchanged. Thus for nearly degenerate strengths of the coupling constants ( $g_{\gamma\gamma\phi}$ and $g_{\gamma\gamma _a}$ ) and masses ($m_\phi$ and $m_a$ ) of the respective candidates, proper identification of one from the other may become very difficult in laboratory or astrophysics based experiments. The basic motivation of this investigation is to reduce this uncertainty through incorporation of parity violating ({\it originating through magnetized medium effects }) part of the photon self-energy in the effective Lagrangian. This step, in turn affects the (Pseudo) Scalar Photon mixing dynamics drastically and brings out a significant change in the spectrum of the electromagnetic beam undergoing such interaction.