  The quantum master equation is an important tool in the study of quantum open
systems. It is often derived under a set of approximations, chief among them
the Born (factorization) and Markov (neglect of memory effects) approximations.
In this article we study the paradigmatic model of quantum Brownian motion of
an harmonic oscillator coupled to a bath of oscillators with a Drude-Ohmic
spectral density. We obtain analytically the \emph{exact} solution of the
Heisenberg-Langevin equations, with which we study correlation functions in the
asymptotic stationary state. We compare the \emph{exact} correlation functions
to those obtained in the asymptotic long time limit with the quantum master
equation in the Born approximation \emph{with and without} the Markov
approximation. In the latter case we implement a systematic derivative
expansion that yields the \emph{exact} asymptotic limit under the factorization
approximation \emph{only}. We find discrepancies that could be significant when
the bandwidth of the bath $\Lambda$ is much larger than the typical scales of
the system. We study the \emph{exact} interaction energy as a \emph{proxy} for
the correlations missed by the Born approximation and find that its dependence
on $\Lambda$ is similar to the \emph{discrepancy} between the exact solution
and that of the quantum master equation in the Born approximation. We quantify
the regime of validity of the quantum master equation in the Born approximation
with or without the Markov approximation in terms of the system's relaxation
rate $\gamma$, its \emph{unrenormalized} natural frequency $\Omega$ and
$\Lambda$: $\gamma/\Omega \ll 1$ and \emph{also} $\gamma \Lambda/\Omega^2 \ll
1$. The reliability of the Born approximation is discussed within the context
of recent experimental settings and more general environments.
