  Recent measurements of the $pp\to t\bar{t}W^\pm$ process in multi-lepton
final states, as performed by the ATLAS collaboration in the context of the
Higgs boson studies in the $t\bar{t}H$ channel, have shown discrepancies
between theoretical predictions and experimental data. Such discrepancies have
been observed both in the overall normalisation as well as in the modelling of
the $t\bar{t}W^\pm$ process. With the goal of understanding and resolving the
modelling issues within the SM $t\bar{t}W^\pm$ process we report on the
state-of-the-art NLO QCD computation for this process. Specifically, we
calculate higher-order corrections to the $e^+ \nu_e \,\mu^-\bar{\nu}_\mu \,
e^+ \nu_e \, b\bar{b}$ and $e^- \bar{\nu}_e \, \mu^+ {\nu}_\mu \, e^-
\bar{\nu}_e \, b\bar{b}$ final state at the LHC with $\sqrt{s}=13$ TeV. In the
computation off-shell top quarks are described by Breit-Wigner propagators,
furthermore, double-, single- as well as non-resonant top-quark contributions
along with all interference effects are consistently incorporated at the matrix
element level. Results at NLO QCD accuracy are presented in the form of
fiducial integrated and differential cross sections for two selected
renormalisation and factorisation scale choices and three different PDF sets.
The impact of the top quark off-shell effects on the $t\bar{t}W^\pm$ cross
section is also examined by an explicit comparison to the narrow-width
approximation.
