Impact of the PDFs on the $Z$ and $W$ lineshapes at LHC

The parton distribution functions (PDFs) of the proton play a role in determining the lineshape of $W$ and $Z$ bosons produced at the LHC. In particular, the mode of the distribution of the gauge boson virtuality gets shifted with respect to the boson mass due to the dependence of the partonic luminosity on the boson virtuality itself. This shift contributes to the systematic uncertainty on the direct measurement of the boson mass. A detailed study of the shift and of its systematic uncertainty due to the limited knowledge of the PDFs is obtained using a tree-level model of $W$ and $Z$ boson production in proton-proton collisions at $\sqrt{s}$ = 13TeV. For the special case of W boson production, a Monte Carlo simulation is further used to validate the tree-level model and study the dependence of the shift on the transverse momentum of the $W$ boson. The tree-level calculation is found to provide already a good description of the shift. The systematic uncertainty due to the PDFs is estimated to be below one MeV in the phase-space relevant for a future high-precision measurement of the $W$ and $Z$ boson masses at the LHC.