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The conjoined production at the LHC of single top and Higgs boson via$t$-channel weak boson exchange is ideal to probe the top-quark Yukawacoupling, due to a delicate cancellation between the amplitudes with the $htt$and the $hWW$ couplings. We find that the top quark is produced with $100\%$polarization in the leading order, and its quantum state is determined by thespin-vector direction in the $t$-quark rest frame. We relate the spin directionto the four-momenta of the top, Higgs and a jet in the helicity amplitudeframework. We identify a polarization asymmetry that is sensitive to CPviolation, even after partial integration over the forward jet momentum. ThisCP violating asymmetry may be observed at the LHC via the component of thetop-quark polarization that is perpendicular to the $th$ scattering plane.

Probing the Higgs Yukawa coupling to the top-quark at the LHC via single top + Higgs production