Bogomol'nyi equations for Dirac-Born-Infeld cosmic string

TL;DR

This paper demonstrates that Dirac–Born–Infeld cosmic strings can possess Bogomol'nyi–Prasad–Sommerfield configurations if the scalar potential is chosen self-consistently. Using both the BPS Lagrangian method and a stressless energy–momentum approach, it derives exact first-order BPS equations and a trigonometric potential, with a tension that scales linearly with the winding number and deforms with the DBI parameter α. In the α → 0 limit, the results recover the Nielsen–Olesen vortex, confirming consistency with the Abelian–Higgs framework. The study reveals a sine-Gordon–like structure underlying the BPS DBI string and identifies regularity bounds that constrain α relative to β, providing a nuanced extension of BPS theory to nonlinear DBI dynamics.

Abstract

We revisit the question of whether Dirac-Born-Infeld (DBI) cosmic strings can admit Bogomol'nyi-Prasad-Sommerfield (BPS) configurations. Earlier work by Babichev et al. arXiv:0809.2013 concluded that DBI strings with the standard Mexican-hat potential possess no BPS limit, implying an unavoidable nonzero binding energy. In contrast, using the BPS Lagrangian method, we show that DBI strings do admit BPS solutions, provided the potential is chosen self-consistently. Imposing the existence of Bogomol'nyi equations uniquely determines the admissible potential and yields exact first-order BPS equations for DBI vortices. We independently verify the consistency of these equations using the stressless (vanishing-pressure) condition on the energy-momentum tensor. The resulting solutions saturate the Bogomol'nyi bound, exhibit zero binding energy, and smoothly recover the Nielsen-Olesen string in the limit $α\to 0$. Regularity of the gauge-field equation requires $α< π^2$. A notable outcome of the construction is that the BPS-compatible potential takes a trigonometric form closely related to the sine-Gordon potential, revealing a natural correspondence between the sine-Gordon string and the BPS DBI string. The BPS tension scales linearly with the winding number $n$ but acquires an $α$-dependent deformation.

Figures (4)

• Figure 1: Plot of the potential in Eq. \ref{['dbipotentialbps']} for for different values of $\alpha$. For $\alpha < 4\pi^2$, the potential maintains the desired symmetry-breaking structure with minima at $|f|=1$. As $\alpha$ increases beyond the critical value $4\pi^2 \simeq 39.478$, local minimum appears in the interval $-1<f<1$, destabilizing the topological sector.
• Figure 2: Plot of the BPS tension from Eq. \ref{['dbitensionbps']} for various winding number $n$. The tension decreases with increasing $\alpha$ and vanishes at the critical value $\alpha = 4\pi^2 \simeq 39.478$. Beyond this point, the tension formally becomes negative .
• Figure 3: Numerical solutions of Eqs. \ref{["a'"]}–\ref{["f'"]} for winding number $n=1$ at various values of $\alpha$. As $\alpha$ increases, the scalar and gauge field profiles become more localized around the string core. At the upper bound $\alpha = \pi^2$, the solutions collapse, and no regular vortex configuration exists beyond this bound.
• Figure 4: Numerical solutions of Eqs. \ref{["a'"]}–\ref{["f'"]} for winding numbers $n=1$ and $n=2$ at $\alpha=1$. The numerically computed tensions are in excellent agreement with the analytic BPS expression \ref{['dbitensionbps']}, confirming the linear scaling with $n$ and the vanishing of the binding energy at the BPS state.