Maximum value of the spin-independent cross section in the THDM+a

We investigate the maximum value of the spin-independent cross section ($\sigma_\text{SI}$) in a dark matter (DM) model called the two-Higgs doublet model + a (THDM+a). This model can explain the measured value of the DM energy density by the freeze-out mechanism. Also, $\sigma_\text{SI}$ is suppressed by the momentum transfer at the tree level, and loop diagrams give the leading contribution to it. The model prediction of $\sigma_\text{SI}$ highly depends on values of $c_1$ and $c_2$ that are the quartic couplings between the gauge singlet CP-odd state ($a_0$) and Higgs doublet fields ($H_1$ and $H_2$), $c_1 a_0^2 H_1^\dagger H_1$ and $c_2 a_0^2 H_2^\dagger H_2$. We discuss the upper and lower bounds on $c_1$ and $c_2$ by studying the stability of the electroweak vacuum, the condition for the potential bounded from the below, and the perturbative unitarity. We find that the condition for the stability of the electroweak vacuum gives upper bounds on $c_1$ and $c_2$. The condition for the potential to be bounded from below gives lower bounds on $c_1$ and $c_2$. It also constrains the mixing angle between the two CP-odd states. The perturbative unitarity bound gives the upper bound on the Yukawa coupling between the dark matter and $a_0$ and the quartic coupling of $a_0$. Under these theoretical constraints, we find that the maximum value of the $\sigma_\text{SI}$ is $\sim 5\times 10^{-47}$ cm$^2$ for $m_A =$ 600 GeV, and the LZ and XENONnT experiments can see the DM signal predicted in this model near future.