Precision physics at the muon collider: $m_W$ and CKM matrix elements

We examine the potential for a 10~TeV lepton collider to carry out precision measurements of the W boson mass and W boson couplings strength, i.e. the CKM matrix elements. We consider the several W boson production mechanisms and focus on the most copious at 10~TeV, that is effective $\gamma W \to W$, a process viable at both opposite sign and same-sign leptonic colliders. We find that the leptonic W decay channel can hardly be competitive with present determinations, due to lack of rate. The hadronic channel has potential to improve over the current $\simeq$10~MeV from measurements at hadron colliders, motivating detector developments towards high-precision hadronic energy measurements. We find that the precision understanding of the detector response to hadrons can also lead to a determination of the CKM matrix elements. We expect determination of CKM matrix elements surpassing by far the present precision for couplings involving heavy quarks, notably $V_{cb}$, avoiding the present bottle-necks due to poor knowledge of hadronic matrix elements needed in low energy extractions of CKM matrix elements. Our findings motivate detector developments towards high-precision hadronic energy measurements and flavor tagging.