Probing the CP Structure of the Top Quark Yukawa at the Future Muon Collider

We study the top-Higgs coupling with a CP violating phase $\xi$ at a future multi-TeV muon collider. We focus on processes that are directly sensitive to the top quark Yukawa coupling: $t\bar{t}h$, $tbh\mu\nu$, and $t\bar{t}h\nu\bar{\nu}$ with $h\rightarrow b\bar{b}$ 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 $t\bar{t}h\nu\bar{\nu}$ and $tbh\mu\nu$ 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 $\mathcal{O}(50\%)$ statistical uncertainty at $1\sigma$, a 10 and 30 TeV muon collider can bound the CP-violating angle $|\xi|\lesssim9.0^\circ$ and $|\xi|\lesssim5.4^\circ$, 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 $t\bar{t}h$ process the azimuthal angle between the $t+\bar{t}$ plane and the initial state muon+Higgs plane shows good discrimination for $\xi=\pm0.1\pi$. For the $tbh\mu\nu$ and $t\bar{t}h\nu\bar{\nu}$ processes, the operator proportional to $\left(\vec{p}_\mu\times\vec{p}_h\right)\cdot \vec{p}_t$ is sensitive to the sign of CP phase $\xi$. From these observables, we construct asymmetry parameters that show good distinction between different values and signs of the CP violating angle.