We study the top-Higgs coupling with a CP violating phase ξ at a future multi-TeV muon collider. We focus on processes that are directly sensitive to the top quark Yukawa coupling: \( t\overline{t}h \) , tbhμν, and \( t\overline{t} h\nu \overline{\nu} \) with \( h\to b\overline{b} \) and semileptonic top decays. At different energies, different processes dominate the cross section, providing complementary information. At and above an energy of \( \mathcal{O}(10) \) TeV, vector boson fusion processes dominate. As we show, in the Standard Model there is destructive interference in the vector boson fusion processes \( t\overline{t} h\nu \overline{\nu} \) and tbhμν between the top quark Yukawa and Higgs-gauge boson couplings. A CP-violating phase changes this interference, and the cross section measurement is very sensitive to the size of the CP-violating angle. Although we find that the cross sections are measured to \( \mathcal{O}\left(50\%\right) \) statistical uncertainty at 1σ, a 10 and 30 TeV muon collider can bound the CP-violating angle |ξ| ≲ 9.0° and |ξ| ≲ 5.4°, respectively. However, cross section measurements are insensitive to the sign of the CP-violating angle. To determine that the coupling is truly CP violating, observables sensitive to CP-violation must be measured. We find in the \( t\overline{t}h \) process the azimuthal angle between the \( t+\overline{t} \) plane and the initial state muon+Higgs plane shows good discrimination for ξ = ±0.1π. For the tbhμν and \( t\overline{t} h\nu \overline{\nu} \) processes, the operator proportional to \( \left({\overrightarrow{p}}_{\mu}\times {\overrightarrow{p}}_h\right) \) · \( {\overrightarrow{p}}_t \) is sensitive to the sign of CP phase ξ. From these observables, we construct asymmetry parameters that show good distinction between different values and signs of the CP violating angle.
