  We study the propagation of non-relativistic bound states moving at constant
velocity across a homogeneous thermal bath and we develop the effective field
theory which is relevant in various dynamical regimes. We consider values of
the velocity of the bound state ranging from moderate to highly relativistic
and temperatures at all relevant scales smaller than the mass of the particles
that form the bound state. In particular, we consider two distinct temperature
regimes, corresponding to temperatures smaller or higher than the typical
momentum transfer in the bound state. For temperatures smaller or of the order
of the typical momentum transfer, we restrict our analysis to the simplest
system, a hydrogen-like atom. We build the effective theory for this system
first considering moderate values of the velocity and then the relativistic
case. For large values of the velocity of the bound state, the separation of
scales is such that the corresponding effective theory resembles the soft
collinear effective theory (SCET). For temperatures larger than the typical
momentum transfer we also consider muonic hydrogen propagating in a plasma
which contains photons and massless electrons and positrons, so that the system
resembles very much heavy quarkonium in a thermal medium of deconfined quarks
and gluons. We study the behavior of the real and imaginary part of the static
two-body potential, for various velocities of the bound state, in the hard
thermal loop approximation. We find that Landau damping ceases to be the
relevant mechanism for dissociation from a certain "critical" velocity on in
favor of screening. Our results are relevant for understanding how the
properties of heavy quarkonia states produced in the initial fusion of partons
in the relativistic collision of heavy ions are affected by the presence of an
equilibrated quark-gluon plasma.
