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 $ϕ$ coupled to a complex scalar $σ$, the coupling of the $ϕ$ and $σ$ being given by two types, $λ_{σϕ}σ^\dagger σϕ^{2}$ and $ξ_{σϕ}σ^\dagger σ\left(\partial_μϕ\right)^{2}$. We evaluate $ϕ$ one-loop corrections $δV(σ)$ to the effective potential of $σ$, both the contribution from the positive norm part of $ϕ$ and that from the {\it negative norm part} (ghost). We show that $δV(σ_{\rm cl})$ at $σ_{\rm cl}\to \infty$, where $σ_{\rm cl}$ is a classical value of $σ$, is positive, implying the stability of $δV(σ_{\rm cl})$ by the HD 1-loop radiative corrections at high energy.

💡 Research Summary

The authors study a simple quantum field theory consisting of a higher‑derivative (four‑derivative) real scalar field ϕ coupled to an ordinary complex scalar σ. The Lagrangian contains the usual kinetic terms for ϕ with two mass parameters m₁ and m₂, a quartic self‑interaction λ_ϕ ϕ⁴, the kinetic and potential terms for σ, and two renormalizable interaction operators: λ_{σϕ} σ†σ ϕ² and ξ_{σϕ} σ†σ (∂μϕ)². Because of the higher‑derivative kinetic term, ϕ can be rewritten in terms of two second‑order fields ψ₁ (positive‑norm) and ψ₂ (negative‑norm or ghost). The ghost field carries the opposite sign in its kinetic term, a situation reminiscent of Lee‑Wick theories and of the massive spin‑2 ghost in quadratic gravity.

The paper focuses on the one‑loop correction to the effective potential V(σ) generated by integrating out ϕ while keeping σ at a constant classical background σ_cl (taken real without loss of generality). The quadratic fluctuation operator for ϕ in momentum space is O_ϕ(k,σ_cl)=k⁴+(m₁²+m₂²−ξ_{σϕ}σ_cl²)k²+(m₁²m₂²+λ_{σϕ}σ_cl²), which can be parametrised by α(σ_cl)=(m₁²+m₂²−ξ_{σϕ}σ_cl²)/2 and β(σ_cl)=m₁²m₂²+λ_{σϕ}σ_cl²−α(σ_cl)². The one‑loop contribution is δV(σ_cl)=½ Tr ln