Collective modes of an Anisotropic Quark-Gluon Plasma II

We continue our exploration of the collective modes of an anisotropic quark gluon plasma by extending our previous analysis to arbitrary Riemann sheets. We demonstrate that in the presence of momentum-space anisotropies in the parton distribution functions there are new relevant singularities on the neighboring unphysical sheets. We then show that for sufficiently strong anisotropies that these singularities move into the region of spacelike momentum and their effect can extend down to the physical sheet. In order to demonstrate this explicitly we consider the polarization tensor for gluons propagating parallel to the anisotropy direction. We derive analytic expressions for the gluon structure functions in this case and then analytically continue them to unphysical Riemann sheets. Using the resulting analytic continuations we numerically determine the position of the unphysical singularities. We then show that in the limit of infinite contraction of the distribution function along the anisotropy direction that the unphysical singularities move onto the physical sheet and result in real spacelike modes at large momenta for all "out-of-plane" angles of propagation.