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This paper presents the impact of lepton transverse momentum $p^{l}_{T}$threshold on the $W$ boson charge asymmetry predictions in perturbative QCD forthe inclusive $W^{\pm}+X \rightarrow l^{\pm} \nu +X$ production inproton-proton ($pp$) collisions. The predictions are obtained with variouslow-$p^{l}_{T}$ thresholds $p^{l}_{T} >$ 20, 25, 30, and 40 GeV in a fiducialregion encompassing both central and forward detector acceptances in terms ofthe lepton pseudorapidty 0 $\leq \eta_{l} \leq$ 4.5. The predicteddistributions 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) accuracyare compared with the CMS and LHCb data at 8 TeV center-of-mass collisionenergy. The 8 TeV predictions are observed to reproduce the data fairly wellwithin the quoted uncertainties. The 13 TeV predictions by using various$p^{l}_{T}$ thresholds are reported for the $A_{\eta_{l}}$ and also the chargeasymmetries that are defined in terms of the differential cross sections inbins 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 closecorrelation 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 fromthe merged predictions with improved accuracy by the inclusion ofnext-to-next-to-next-to-leading logarithm (N$^{3}$LL) corrections, i.e., atNNLO+N$^{3}$LL. The predicted distributions from various $p^{l}_{T}$ thresholdsrepresent also a finer probe to provide more constraints on the ratio of $u$and $d$ quark distribution functions in the parton momentum fraction range$10^{-4} < x < 1$.

Impact of lepton $p_{T}$ threshold on the charge asymmetry predictions for the inclusive $W$ boson production in $pp$ collisions at 13 TeV