Collective modes of gluons in an anisotropic thermomagnetic medium

We study the collective modes of gluons in an anisotropic thermal medium in the presence of a constant background magnetic field using the hard-thermal loop (HTL) perturbation theory. The momentum space anisotropy of the medium has been incorporated through the generalized $`$Romatschke-Strickland' form of the distribution function, whereas, the magnetic modification arising from the quark loop contribution has been taken into account in the lowest Landau level approximation. We consider two special cases: (i) a spheroidal anisotropy with the anisotropy vector orthogonal to the external magnetic field and (ii) an ellipsoidal anisotropy with two mutually orthogonal vectors describing aniostropies along and orthogonal to the field direction. The general structure of the polarization tensor in both cases are equivalent and consists of six independent basis tensors. We find that the introduction of momentum anisotropy ingrains azimuthal angular dependence in the thermo-magnetic collective modes. Our study suggests that the presence of a strong background magnetic field can significantly reduce the growth rate of the unstable modes which may have important implications in the equilibration of magnetized quark-gluon plasma.