Propagation of gauge fields in hot and dense plasmas at higher orders

Thermal field theory is indispensable for describing hot and dense systems. Yet perturbative calculations are often stymied by a host of energy scales, and tend to converge slowly. This means that precise results require the apt use of effective field theories. In this paper we refine the effective description of slowly varying gauge field known as hard thermal loops. We match this effective theory to the full theory to two-loops. Our results apply for any renormalizable model and fermion chemical potential. We also discuss how to consistently define asymptotic masses at higher orders; and how to treat spectral functions close to the lightcone. In particular, we demonstrate that the gluon mass is well-defined to next-to-leading order.