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Proton-PDF uncertainties in extracting nuclear PDFs from $W^\pm$ production in p+Pb collisions

We discuss the recent CMS Collaboration measurement of $W^\pm$ boson production in p+Pb collisions at 8.16 TeV in terms of the constraining power on nuclear parton distribution functions (PDFs). The impact of the free-proton PDF uncertainties on the nuclear PDF extraction is quantified by using a theoretical covariance-matrix method and Hessian PDF reweighting. We discuss different ways to mitigate these theoretical uncertainties, including self-normalization, forward-to-backward ratios and nuclear-modification ratios. It is found that none of these methods offer perfect cancellation of the free-proton PDFs but, with the present data uncertainties, the residual free-proton-PDF dependence has, conveniently for the global analyses, little effect on the extraction of the nuclear modifications. Based on a simple estimate of obtainable statistics at the LHC Run 3, we argue that this will change in the near future and it becomes more important to propagate the proton-PDF uncertainties accordingly. Using the obtained information on the correlations of the free-proton uncertainties, we also identify a new charge asymmetry ratio, where the cancellation of the proton-PDF uncertainties is found to be extremely good.