  The production of heavy quarkonium in heavy ion collisions has been used as
an important probe of the quark-gluon plasma (QGP). Due to the plasma screening
effect, the color attraction between the heavy quark antiquark pair inside a
quarkonium is significantly suppressed at high temperature and thus no bound
states can exist, i.e., they "melt". In addition, a bound heavy quark antiquark
pair can dissociate if enough energy is transferred to it in a dynamical
process inside the plasma. So one would expect the production of quarkonium to
be considerably suppressed in heavy ion collisions. However, experimental
measurements have shown that a large amount of quarkonia survive the evolution
inside the high temperature plasma. It is realized that the in-medium
recombination of unbound heavy quark pairs into quarkonium is as crucial as the
melting and dissociation. Thus, phenomenological studies have to account for
static screening, dissociation and recombination in a consistent way. But
recombination is less understood theoretically than the melting and
dissociation. Many studies using semi-classical transport equations model the
recombination effect from the consideration of detailed balance at thermal
equilibrium. However, these studies cannot explain how the system of quarkonium
reaches equilibrium and estimate the time scale of the thermalization.
Recently, another approach based on the open quantum system formalism started
being used. In this framework, one solves a quantum evolution for in-medium
quarkonium. Dissociation and recombination are accounted for consistently.
However, the connection between the semi-classical transport equation and the
quantum evolution is not clear. In this dissertation, I will try to address the
issues raised above. As a warm-up project, I will first study a similar
problem: $\alpha$-$\alpha$ scattering at the $^8$Be resonance inside an
$e^-e^+\gamma$ plasma. By applying pionless effective field theory and thermal
field theory, I will show how the plasma screening effect modifies the $^8$Be
resonance energy and width. I will discuss the need to use the open quantum
system formalism when studying the time evolution of a system embedded inside a
plasma. Then I will use effective field theory of QCD and the open quantum
system formalism to derive a Lindblad equation for bound and unbound heavy
quark antiquark pairs inside a weakly-coupled QGP. Under the Markovian
approximation and the assumption of weak coupling between the system and the
environment, the Lindblad equation will be shown to turn to a Boltzmann
transport equation if a Wigner transform is applied to the open system density
matrix. These assumptions will be justified by using the separation of scales,
which is assumed in the construction of effective field theory. I will show the
scattering amplitudes that contribute to the collision terms in the Boltzmann
equation are gauge invariant and infrared safe. By coupling the transport
equation of quarkonium with those of open heavy flavors and solving them using
Monte Carlo simulations, I will demonstrate how the system of bound and unbound
heavy quark antiquark pairs reaches detailed balance and equilibrium inside the
QGP. Phenomenologically, my calculations can describe the experimental data on
bottomonium production. Finally I will extend the framework to study the
in-medium evolution of heavy diquarks and estimate the production rate of the
doubly charmed baryon $\Xi_{cc}^{++}$ in heavy ion collisions.
