Casimir energy and topological mass for a massive scalar field with Lorentz violation

A Lorentz symmetry violation aether-type theoretical model is considered to investigate the Casimir effect and the generation of topological mass associated with self-interacting massive scalar fields obeying Dirichlet, Newman, and mixed boundary conditions on two large and parallel plates. By adopting the path integral approach, we find the effective potential at one- and two-loop corrections, which provides both the energy density and the topological mass when taken in the ground state of the scalar field. We then analyze how these quantities are affected by the Lorentz symmetry violation and compare the results with previous ones found in literature.

Figure 1

Schematic configuration for the two parallel plates with area $L^2$ separated by a distance $a$ ($a\ll L$).

Figure 2

The behavior of the Casimir energy per unit area $E(am)=\frac{a^3}{L^2}{E}_C$ given by Eq. (2.31) as a function of $am$ is exhibited in the left panel, and the two-loop contribution to the Casimir energy per unit area $E_{\lambda }(am)=\frac{a^3}{L^2}{E}_C^{(\lambda )}$ in Eq. (2.40) as a function of $am$ is presented in the right panel, considering $\lambda ={10}^{-3}$. For both plots, different values for the parameter $\chi$ are considered.

Figure 3

The behavior of the ratio of the topological mass to the scalar field mass as a function of $am$ is plotted assuming $\lambda ={10}^{-3}$ and different values for the Lorentz-violating parameter $\chi$.

Figure 4

The behavior of the Casimir energy per unit area $E(am)=\frac{a^3}{L^2}{E}_C$ given by Eq. (2.66) as a function of $am$ is presented in the left panel, and the two-loop correction $E_{\lambda }(am)=\frac{a^3}{L^2}{E}_C^{(\lambda )}$ given by Eq. (2.70) as a function of $am$ is in the right one. In the latter, we have taken $\lambda ={10}^{-3}$ and considered different values for $\chi$.

Figure 5

Plot exhibiting the behavior of the ratio of the topological mass by the mass of the field as a function of $am$. The plot considers $\lambda ={10}^{-3}$ and different variations of $\chi$.

Figure 6

The behavior of the Casimir energy per unit area $E(am)=\frac{a^3}{L^2}{E}_C$ given by Eq. (2.91) as a function of $am$ is exhibited in the left plot. The behavior of the two-loop contribution $E_{\lambda }(am)=\frac{a^3}{L^2}{E}_C^{(\lambda )}$, given by Eq. (2.96), is exhibited in the right plot. For the latter, we assume $\lambda ={10}^{-3}$, and consider different values for the parameter, $\chi$.

Figure 7

The ratio of the topological mass to the field mass as a function of $am$. The plot considers $\lambda ={10}^{-3}$ and different values of $\chi$.

Figure 8

The left plot presents the behavior of Casimir energy per unit area $E(am)=\frac{a^3}{L^2}{E}_C$, given by Eq. (2.123), as a function of $am$. The two-loop contribution $E_{\lambda }(am)=\frac{a^3}{L^2}{E}_C^{(\lambda )}$, given by Eq. (2.128), is exhibited in the right panel as a function of $am$, fixing $\lambda ={10}^{-3}$. In both plots, we have considered various values of $\chi$.

Figure 9

Graph presenting behavior of the ratio of the topological mass to the mass of the field as a function of $am$. The plot considers $\lambda ={10}^{-3}$ and different values for the Lorentz-violating parameter, $\chi$.