  Di-lepton searches for Beyond the Standard Model (BSM) Z' bosons that rely on
the analysis of the Breit-Wigner (BW) line shape are appropriate in the case of
narrow resonances, but likely not sufficient in scenarios featuring Z' states
with large widths. Conversely, alternative experimental strategies applicable
to wide Z' resonances are much more dependent than the default bump search
analyses on the modelling of QCD higher-order corrections to the production
processes, for both signal and background. For heavy Z' boson searches in the
di-lepton channel at the CERN Large Hadron Collider (LHC), the transverse
momentum q_T of the di-lepton system peaks at q_T \ltap 10^{-2} M_{ll}, where
M_{ll} is the di-lepton invariant mass. We exploit this to treat the QCD
corrections by using the logarithmic resummation methods in M_{ll} / q_T to all
orders in the strong coupling constant \alpha_s. We carry out studies of Z'
states with large width at the LHC by employing the program {\tt reSolve},
which performs QCD transverse momentum resummation up to
Next-to-Next-to-Leading Logarithmic (NNLL) accuracy. We consider two benchmark
BSM scenarios, based on the Sequential Standard Model (SSM) and dubbed `SSM
wide' and `SSM enhanced'. We present results for the shape and size of Z' boson
signals at the differential level, mapped in both cross section (\sigma) and
Forward-Backward Asymmetry (A_{\rm FB}), and perform numerical investigations
of the experimental sensitivity at the LHC Run 3 and High-Luminosity LHC
(HL-LHC).
