Vacuum stability and nature of additional minima in the Elko–scalar effective potential

Determine whether the additional minima of the renormalized effective potential for a four-dimensional model consisting of a mass-dimension-one Elko fermionic field η interacting quadratically with a real scalar field φ (including a quartic scalar self-interaction λφ φ^4/4!) under Dirichlet boundary conditions between two parallel plates separated by distance L correspond to physically realized vacua (such as distinct phases or metastable states) or are artifacts of the approximation scheme, by conducting a systematic vacuum stability analysis of the full effective potential depending on the background variables (the Elko bilinear \bar{Ψ}Ψ, the scalar field Φ, and their interaction).

Background

The paper studies a four-dimensional interacting system with a mass-dimension-one Elko fermionic field coupled quadratically to a real scalar field and subject to Dirichlet boundary conditions on parallel plates. Using the effective potential framework, the authors compute vacuum energy and topological mass corrections, noting that the effective potential depends on three background variables associated with the Elko bilinear, the scalar field, and their interaction.

This richer structure implies the possibility of multiple competing vacua and boundary-induced phase transitions. The authors explicitly state that determining whether additional minima are physically meaningful or artifacts of the approximation requires a systematic vacuum stability analysis, which they defer.

References

Understanding whether these additional minima carry physical consequences, or are merely artifacts of the approximation scheme, requires a systematic vacuum stability analysis, which we leave for future investigation.

— Vacuum Energy and Topological Mass in Interacting Elko and Scalar Field Theories  (2512.08750 - Junior et al., 9 Dec 2025) in Introduction, final paragraph

Finally, in the massless case, $\Psi=0$ corresponds to the global minimum of the effective potential and is the unique stable vacuum configuration in the perturbative regime. In contrast, the massive case may exhibit additional local minima; however, a more detailed analysis of the corresponding integrals would be required to determine whether a global minimum exists.

— Quantum corrections to the Casimir effect in a scalar Hořava-Lifshitz theory with rough plates at low temperature  (2609.40081 - Bórquez et al., 30 Sep 2026) in Section 4, subsection “One-loop correction,” discussion following Eq. (\ref{vacuum-solutions})