  In complex materials various interactions play important roles in determining
the material properties. Angle Resolved Photoelectron Spectroscopy (ARPES) has
been used to study these processes by resolving the complex single particle
self energy $\Sigma(E)$ and quantifying how quantum interactions modify bare
electronic states. However, ambiguities in the measurement of the real part of
the self energy and an intrinsic inability to disentangle various contributions
to the imaginary part of the self energy often leave the implications of such
measurements open to debate. Here we employ a combined theoretical and
experimental treatment of femtosecond time-resolved ARPES (tr-ARPES) and show
how measuring the population dynamics using tr-ARPES can be used to separate
electron-boson interactions from electron-electron interactions. We demonstrate
the analysis of a well-defined electron-boson interaction in the unoccupied
spectrum of the cuprate Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+x}$ characterized by an
excited population decay time constant $\tau_{QP}$ that maps directly to a
discrete component of the equilibrium self energy not readily isolated by
static ARPES experiments.
