  This paper presents the impact of lepton transverse momentum $p^{l}_{T}$
threshold on the $W$ boson charge asymmetry predictions in perturbative QCD for
the inclusive $W^{\pm}+X \rightarrow l^{\pm} \nu +X$ production in
proton-proton ($pp$) collisions. The predictions are obtained with various
low-$p^{l}_{T}$ thresholds $p^{l}_{T} >$ 20, 25, 30, and 40 GeV in a fiducial
region encompassing both central and forward detector acceptances in terms of
the lepton pseudorapidty 0 $\leq \eta_{l} \leq$ 4.5. The predicted
distributions for the lepton charge asymmetry, which is defined by means of the
$\eta_{l}$ ($A_{\eta_{l}}$), at next-to-next-to-leading order (NNLO) accuracy
are compared with the CMS and LHCb data at 8 TeV center-of-mass collision
energy. The 8 TeV predictions are observed to reproduce the data fairly well
within the quoted uncertainties. The 13 TeV predictions by using various
$p^{l}_{T}$ thresholds are reported for the $A_{\eta_{l}}$ and also the charge
asymmetries that are defined in terms of the differential cross sections in
bins of the $W$ boson rapidity $y_{W}$ ($A_{y_{W}}$) and transverse momentum
$p^{W}_{T}$ ($A_{p^{W}_{T}}$). The NNLO predictions for the $A_{\eta_{l}}$,
$A_{y_{W}}$, and $A_{p^{W}_{T}}$ distributions are assessed to be in close
correlation with the $p^{l}_{T}$ value. The $A_{\eta_{l}}$ and $A_{y_{W}}$
distributions are particularly shown to be more correlated at a higher
$p^{l}_{T}$ threshold. The $A_{p^{W}_{T}}$ distributions are also reported from
the merged predictions with improved accuracy by the inclusion of
next-to-next-to-next-to-leading logarithm (N$^{3}$LL) corrections, i.e., at
NNLO+N$^{3}$LL. The predicted distributions from various $p^{l}_{T}$ thresholds
represent also a finer probe to provide more constraints on the ratio of $u$
and $d$ quark distribution functions in the parton momentum fraction range
$10^{-4} < x < 1$.
