Lorentz symmetry violating low energy dispersion relations from a   dimension-five photon scalar mixing operator
  Dimension-five photon $(\gamma )$ scalar $(\phi)$ interaction terms usually appear in the bosonic sector of unified theories of electromagnetism and gravity. In these theories the three propagation eigenstates are different from the three field eigenstates. The dispersion relation in an external magnetic field shows that, for a non- zero energy $(\omega)$, out of the three propagating eigenstates one has superluminal phase velocity $v_p$. During propagation, another eigenstate undergoes amplification or attenuation, showing signs of an unstable system. The remaining one maintains causality. In this paper, using techniques from optics as well as gravity, we identify the energy $(\omega)$ interval outside which $v_p \le c$ for the field eigenstates $|\gamma_{\parallel} > $ and $ |\phi > $, and stability of the system is restored. The behavior of group velocity $v_g$ is also explored in the same context. We conclude by pointing out its possible astrophysical implications. 
Avijit K. Ganguly and Manoj K. Jaiswal
hep-ph astro-ph.HE