  Integrating out high energy degrees of freedom to yield a low energy
effective field theory leads to a loss of information with a concomitant
increase in entropy. We obtain the effective field theory of a light scalar
field interacting with heavy fields after tracing out the heavy degrees of
freedom from the time evolved density matrix. The initial density matrix
describes the light field in its ground state and the heavy fields in
equilibrium at a common temperature $T$. For $T=0$, we obtain the reduced
density matrix in a perturbative expansion, it reveals an emergent mixed state
as a consequence of the entanglement between light and heavy fields. We obtain
the effective action that determines the time evolution of the \emph{reduced}
density matrix for the light field in a non-perturbative Dyson resummation of
one-loop correlations of the heavy fields. The Von-Neumann \emph{entanglement
entropy} associated with the reduced density matrix is obtained for the
non-resonant and resonant cases in the asymptotic long time limit. In the
non-resonant case the reduced density matrix displays an \emph{incipient}
thermalization albeit with a wave-vector, time and coupling dependent
\emph{effective temperature} as a consequence of memory of initial conditions.
The entanglement entropy is time independent and is the \emph{thermal entropy}
for this effective, non-equilibrium temperature. In the resonant case the light
field fully \emph{thermalizes} with the heavy fields, the reduced density
matrix looses memory of the initial conditions and the entanglement entropy
becomes the \emph{thermal entropy} of the light field. We discuss the relation
between the entanglement entropy ultraviolet divergences and renormalization.
