UV stability of 1-loop radiative corrections in higher-derivative scalar field theory

We consider the theory of a higher-derivative (HD) real scalar field $\phi$ coupled to a complex scalar $\sigma$, the coupling of the $\phi$ and $\sigma$ being given by two types, $\lambda_{\sigma\phi}\sigma^\dagger \sigma\phi^{2}$ and $\xi_{\sigma\phi}\sigma^\dagger \sigma\left(\partial_{\mu}\phi\right)^{2}$. We evaluate $\phi$ one-loop corrections $\delta V(\sigma)$ to the effective potential of $\sigma$, both the contribution from the positive norm part of $\phi$ and that from the {\it negative norm part} (ghost). We show that $\delta V(\sigma_{\rm cl})$ at $\sigma_{\rm cl}\to \infty$, where $\sigma_{\rm cl}$ is a classical value of $\sigma$, is positive, implying the stability of $\delta V(\sigma_{\rm cl})$ by the HD 1-loop radiative corrections at high energy.