Reduction of PDF uncertainty in the measurement of the weak mixing angle at the ATLAS experiment *

We investigate the parton distribution function (PDF) uncertainty in measurement of effective weak mixing angle $\sin^2\theta_{\text{eff}}^{\ell}$ at CERN Large Hadron Collider (LHC). The PDF-induced is large proton-proton collisions LHC due to dilution effect. Drell-Yan forward-backward asymmetry ($A_{FB}$) can be used reduce PDF measurement. However, when including full mass range lepton pairs $A_{FB}$ data analysis, correlation between updating procedure and extraction leads a sizable bias obtained value. From our studies, we find that significantly reduced by removing events with invariant around $Z$ pole region, while most sensitivity reducing remains. Furthermore, charge $W$ boson as rapidity charged leptons, $A_\pm(\eta_\ell)$, known another observable which constraint from $A_\pm(\eta_\ell)$ complementary $A_{FB}$, thus no affects extraction. studies are performed using Error Updating Method Package ({\sc ePump}), based on Hessian methods. In this article, CT14HERA2 set an example.