subprocess) in Fig. \[fig:helicity-sm\], where $\lambda$ ($\bar \lambda$) is the helicity of the incoming $W^-$ ($W^+$) boson and the mass of the Higgs boson is set to be $500$ GeV. The polarization vector for the $W$ boson with the momentum $p^\mu=(E_W^{},0,0,p_W^{})$ ($E_W^2=m_W^2+p_W^2$) and the helicity $\lambda$ is defined by $$\begin{aligned}
\epsilon_W^{\mu}(p,\lambda) = \left\{
\begin{array}{cc}
 \frac{1}{m_W^{}}(p_W^{},0,0,E_W^{})& ({\rm for\;\;} \lambda=L) \\
                 (0,1, \pm i,0) & ({\rm for\;\;} \lambda=\pm) \\
\end{array} \right. .\end{aligned}$$

It can be seen that the contribution from the longitudinally polarized $W$ bosons, $\hat{\sigma}_{LL}^{\rm SM}$ dominates over those from the other polarization sets in a wide region of $\sqrt{\hat{s}}$. In particular, around the scale of the Higgs boson mass, $\hat{\sigma}_{LL}^{\rm SM}$ is about one or two orders of magnitude greater than $\hat{\sigma}_{L-}^{\rm SM}$, the second largest mode, because of the $s$-channel Higgs-boson resonance.

Next, we consider the the cross section in the model with the dimension-six operators ${\cal O}_{t1}^{}$ and ${\cal O}_{Dt}^{}$. The Feynman diagrams are the same as those in the SM in Fig. \[fig:diagram-sm\], but ${\cal O}_{t1}$ and ${\cal O}_{Dt}$ change the coupling strength of $t\bar t H$, $Z t \bar t$ and $W^\pm t b$ from their SM values. The diagrams for the Higgs boson mediation, the 