CP-violating top-Higgs coupling in SMEFT

The total cross section of the process $\mu^- \mu^+ \to \nu_\mu \bar{\nu}_\mu t \bar{t} H$ has strong dependence on the CP phase $\xi$ of the top Yukawa coupling, where the ratio of $\xi=\pi$ and $\xi = 0$ (SM) grows to 670 at $\sqrt{s}$ = 30 TeV, 3400 at 100 TeV. We study the cause of the strong energy dependence and identify its origin as the $(E/m_W^{})^2$ growth of the weak boson fusion sub-amplitudes, $W_L^- W_L^+ \to t \bar{t} H$, with the two $W$s are longitudinally polarized. We repeat the study in the SMEFT framework where EW gauge invariance is manifest and find that the highest energy cross section is reduced to a quarter of the complex top Yukawa model result, with the same energy power. By applying the Goldstone boson (GB) equivalence theorem, we identify the origin of this strong energy growth of the SMEFT amplitudes as associated with the dimension-6 $\pi^- \pi^+ ttH$ vertex, where $\pi^\pm$ denotes the GB of $W^\pm$. We obtain the unitarity bound on the coefficient of the SMEFT operator by studying all $2\to2$ and $2\to3$ cross sections in the $J=0$ channel.