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We discuss direct and indirect probes of chirality-flipping couplings of thetop quark to Higgs and gauge bosons, considering both CP-conserving andCP-violating observables, in the framework of the Standard Model effectivefield theory. In our analysis we include current and prospective constraintsfrom collider physics, precision electroweak tests, flavor physics, andelectric dipole moments (EDMs). We find that low-energy indirect probes arevery competitive, even after accounting for long-distance uncertainties. Inparticular, EDMs put constraints on the electroweak CP-violating dipole momentsof the top that are two to three orders of magnitude stronger than existinglimits. The new indirect constraint on the top EDM is given by $|d_t| < 5 \cdot10^{-20}$ e cm at $90\%$ C.L.

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