Far forward tau neutrinos at the LHC: parton distribution function uncertainties in theoretical predictions

Tau neutrinos and antineutrinos, predominately from $D_s^\pm$ production and decay, are produced in the far-forward region at the Large Hadron Collider (LHC). Two experiments, FASER$\nu$ and SND@LHC, will take data during Run 3, and proposed experiments for a Forward Physics Facility would operate during the high luminosity era. These experiments will measure neutrino pseudorapidities in the range of $\eta>6.9$ and higher, where both very small and large parton-$x$ values in the parton distribution functions (PDF) are important in theoretical predictions. We show the sensitivity of a next-to-leading order (NLO) QCD evaluation of $D_s^\pm$ production, followed by decay to tau neutrinos on the 40 PDF sets of the PROSA19 group, on QCD scale uncertainties and on regions of $x$ in the PDF. We compare results with other 3-flavor NLO PDF sets of the CT14, ABMP16 and NNPDF3.1 collaborations. The Forward Physics Facility in the high luminosity LHC era will  provide data capable of constraining NLO QCD evaluations with these PDF sets.