Is there room for CP violation in the top-Higgs sector?

We discuss direct and indirect probes of chirality-flipping couplings of the top quark to Higgs and gauge bosons, considering both CP-conserving and CP-violating observables, in the framework of the Standard Model effective field theory. In our analysis we include current and prospective constraints from collider physics, precision electroweak tests, flavor physics, and electric dipole moments (EDMs). We find that low-energy indirect probes are very competitive, even after accounting for long-distance uncertainties. In particular, EDMs put constraints on the electroweak CP-violating dipole moments of the top that are two to three orders of magnitude stronger than existing limits. The new indirect constraint on the top EDM is given by $|d_t| < 5 \cdot 10^{-20}$ e cm at $90\%$ C.L.