On the Stability of Topologically Non-Trivial Vacuum Bubbles in a Three Form Gauge Sector

Abstract

We study a three-form gauge sector in four spacetime dimensions coupled to electrically charged spherical membranes whose worldvolume dynamics are governed by a Dirac--Born--Infeld action. The associated four-form field strength has no local propagating degrees of freedom and contributes a branch-dependent vacuum energy. Motivated by the Hartle--Hawking--Wu selection argument, we restrict attention to the semiclassically admissible four form flux window for which the Hartle-Hawking wave function has support. We then endow the bubble wall with a worldvolume $U(1)$ gauge field carrying quantized monopole flux $n \in \mathbb{Z}$ and evaluate the full DBI energy of the resulting spherical configurations. We show that the energetically preferred branch collapses toward a microscopic core rather than stabilizing at finite radius, but for nonzero monopole flux the energy does not vanish in the collapsed limit. Instead, the bubble relaxes to a finite-energy remnant whose mass is set by the wall scale and the conserved flux. We interpret these objects as stable flux-supported particle-like states, which we call topolons. Within the admissible sector, the effective energy analysis distinguishes stable collapsed remnants from the contrasting runaway vacuum-decay channel, thereby isolating the sector relevant for cosmological relic formation. At macroscopic distances, topolons behave as heavy localized states and provide a concrete microphysical realization of a dark relic candidate. The detailed cosmological abundance and phenomenology are left for future work.

Figures (5)

• Figure 1: Spherical membrane separating two constant four-form flux sectors. The interior and exterior are characterized by constant values $q_{\rm in}$ and $q_{\rm out}$. The membrane carries charge $q_{b}$, enforcing the jump condition $q_{\rm out} - q_{\rm in} = q_{b}$.
• Figure 2: Patchwise description of the worldvolume gauge potential for a monopole flux on the bubble ($n = 1$). The light-green azimuthal circulation indicates the local form of the potential $A$ in the north and south patches (clockwise in the north patch; anticlockwise in the south patch), which are related on the overlap by a gauge transformation, which can be written schematically as $A_S = A_N - n \, d \phi$. The physical magnetic flux is the globally defined 2-form $F = dA$ with quantized total flux through the bubble.
• Figure 3: Comparison of the thin-wall energy and the full DBI energy of a flux-carrying bubble as a function of radius $R$. Both axes are shown on logarithmic scales. The thin-wall contribution $E_{\rm wall}(R) = 4 \, \pi T_{\rm eff} \, R^2$ vanishes as $R\to 0$, whereas the DBI completion $E_{\rm DBI}(R) = 4 \, \pi \, T_{\rm eff} \, \sqrt{R^4 + \dfrac{n^2}{4 \, g_{\rm YM}^2 \, T_{\rm eff}}}$ approaches a finite constant set by the quantized flux. The plot uses dimensionless, order-one parameter choices and arbitrary overall units. Its purpose is purely illustrative and is to highlight the different small-$R$ behavior of the two energy functions.
• Figure 4: Illustrative behavior of the total bubble energy $E_{t}(R)$ as a function of the bubble radius $R$ for the two physically admissible cases with $\Delta \rho \geq 0$. Panel (a) shows the case $\Delta \rho = 0$, for which the energy has a stable minimum at $R = 0$ and rises quartically about the collapsed configuration. Panel (b) shows the case $\Delta \rho > 0$, for which the collapsed state remains the global minimum and the additional positive cubic term further strengthens stability. Note that in both cases, the energy of the bubble at $R = 0$ is not zero but is finite and is equal to $2 \, \pi$. The plots are generated using the benchmark values listed in Table \ref{['tab:benchmark_energy_plot']}, and are intended only to display the qualitative dependence of the energy profile on the sign of $\Delta \rho$.
• Figure 5: Illustrative behavior of the total bubble energy $E_{t}(R)$ for the contrast case $\Delta \rho < 0$, generated using the same benchmark values as in Table \ref{['tab:benchmark_energy_plot']}. For the illustrative choice $\Delta \rho = -1$, the energy initially decreases away from the collapsed configuration at $R = 0$, showing that the collapsed state is not locally stable in this branch. The profile instead develops a local minimum at $R \approx 0.252$, corresponding to a metastable finite-radius bubble, and a local maximum at $R \approx 1.984$, corresponding to the barrier beyond which the negative vacuum term drives runaway expansion. The energy of the bubble at $R = 0$, $R \approx 0.252$ and $R \approx 1.984$ is $6.28$, $6.266$ and $17.149$ respectively.