The high collision energies available at the Large Hadron Collider (LHC) at CERN allow for an abundant production of hard probes, such as quarkonia, high \( p_T \) jets and intermediate vector bosons (\(\mathrm{W^\pm}\), \(\mathrm{Z^0}\)). The W bosons are produced in initial hard parton scattering processes, with a formation time of the order of \(1/\mathrm{M_W}\sim\)0.003 fm/c and, having a lifetime of 0.09 fm/c, they decay before the formation of the quark-gluon plasma (QGP), which is a deconfined phase of QCD matter produced at 1 fm/c after high-energy heavy-ion collisions. Furthermore, their leptonic decay products do not interact strongly with the hot and dense QCD medium. In proton-nucleus and nucleus-nucleus collisions, precise measurements of W-boson and Z-boson production can constrain the modication of the Parton Distribution Functions (PDFs) in the nucleus and they can be used to test the scaling of hard particle production with the number of binary nucleon-nucleon collisions. The analysis can be test-benched in proton-proton collisions, where the W boson production is theoretically known with good precision.

The production of W bosons in p-p collisions at \(\sqrt{s}\)=7 TeV, 8 TeV and p-Pb collisions at \(\sqrt{\mathrm{s_{NN}}}\)=5.02 TeV are measured via the contribution of W decays to the inclusive \(p_T\)-differential muon yield reconstructed with the ALICE muon spectrometer. In p-p collisions the rapidity covered by muon spectrometer is -4<\(y^\mu_{cms}\)<-2.5 and in p-Pb collisions it separates into 2.03<\(y^\mu_{cms}\)<3.53 (forward) and -4.46<\(y^\mu_{cms}\)<-2.96 (backward) via changing beam direction. After a presentation of the motivation for the study of W bosons and a description of the ALICE muon spectrometer, the latest results on W-boson production in p-p and p-Pb collisions will be shown and discussed.