Vacuum Energy and Topological Mass in Interacting Elko and Scalar Field Theories

TL;DR

The paper analyzes Casimir-like vacuum effects for a four-dimensional Elko fermion interacting with a real scalar under Dirichlet confinement between parallel plates. It employs the effective potential via path integrals and generalized zeta-function regularization to obtain the vacuum energy and one-loop corrections, expressed through Epstein-Hurwitz sums and Bessel functions. A key result is that the Elko contribution to the vacuum energy is eight times larger than the scalar one, and first-order couplings generate boundary-dependent mass shifts (topological masses) for both fields. The work also reveals possible boundary-induced instabilities in the scalar sector and suggests avenues for extending the framework to complex scalars, higher-loop effects, finite temperature, and curved spacetime, with implications for Elko-based phenomenology and dark matter scenarios.

Abstract

In this paper, we consider a four-dimensional system composed of a mass-dimension-one fermionic field, also known as Elko, interacting with a real scalar field. Our main objective is to analyze the Casimir effects associated with this system, assuming that both the Elko and scalar fields satisfy Dirichlet boundary conditions on two large parallel plates separated by a distance $L$. In this scenario, we calculate the vacuum energy density and its first-order correction in the coupling constants of the theory. Additionally, we consider the mass correction for each field separately, namely the topological mass that arises from the boundary conditions imposed on the fields and which also depends on the coupling constants. To develop this analysis, we use the mathematical formalism known as the effective potential, expressed as a path integral in quantum field theory.

Figures (4)

• Figure 1: A schematic representation of two perfectly reflecting parallel plates lying in the $x$-$y$ plane and separated by a distance $L$ along the $z$-axis is shown. Both the Elko and the real scalar fields satisfy Dirichlet boundary conditions at the plates, located at $z=0$ and $z=L$.
• Figure 2: Graph of the dimensionless energy $\mathcal{E}\left( m_{\mathrm{R}}L\right)$, Eq. (\ref{['de0.1']}), as a function of $m_{\mathrm{R}}L$ and fixed values of $m_{\mathrm{E}}L$.
• Figure 3: Graph of $M_{\mathrm{T}}^{2}\left( m_{\mathrm{R}}L\right)$, Eq. (\ref{['dtm0.1']}), as a function of $m_{\mathrm{R}}L$, with $g=10^{-3}$.
• Figure 4: The graph on the left shows the $M^{2}\left( m_{\mathrm{R}}L\right)$, Eq. (\ref{['dtm0.2']}), with $g=10^{-3}$ and $\lambda _{\varphi }=10^{-1}$, as a function of $m_{\mathrm{R}}L$, while the graph on the right takes the values $g=10^{-1}$ and $\lambda _{\varphi }=10^{-3}$.