Freeze-in Production of Pseudo-Nambu-Goldstone Dark Matter Model with a Real Scalar

In this work, we study a pseudo-Nambu-Goldstone boson (pNGB) dark matter model extended with a real scalar. The dark sector is assumed to be feebly coupled with the standard model (SM) via a Higgs portal, so that the pNGB dark matter is produced by the freeze-in mechanism. Since the production happened in a very high energy era, we introduce an extra scalar field which is weakly coupled to the SM for stablizing the electroweak vacuum. Our model can reproduce the correct relic abundance of dark matter favored by observations. In addition, we determine the evolution of couplings in higher energy scale by solving the renormalization group equations, and show that the self coupling of the real scalar, $\lambda_S$, and the mixing coupling between the Higgs field and the real scalar, $\lambda_{HS}$ are stringently constrained by the conditions of vacuum stability and couplings perturbativity up to the Planck scale. We also find that the relic abundance of DM is insensitive to the values of $\lambda_S$ and $\lambda_{HS}$ unless the dominant production processes are $\overline{H}+H(S+S)\to\phi+\phi$ via t- and u-channels.