Exact Wall Solutions in 5-Dimensional SUSY QED at Finite Coupling

TL;DR

The paper constructs exact BPS wall solutions in five-dimensional $ \mathcal{N}=2$ SUSY QED at finite gauge coupling, yielding single and multi-wall configurations with massive vector fields localized on the walls. By promoting moduli to dynamical four-dimensional fields, the authors derive a finite-coupling low-energy effective theory in which massless Nambu-Goldstone modes assemble into a four-dimensional $\mathcal{N}=1$ chiral multiplet, while the vector sector remains massive; inter-wall forces are shown to be stronger at finite coupling than in the infinite-coupling limit. They obtain a family of exact solutions for couplings satisfying $g^2\zeta = 8m^2/k^2$ with $k\in\{0,2,3,4\}$, including explicit $S_k$ and $D_k$ forms, and analyze the moduli dynamics of two walls via a Kahler metric $K_{rr^*}\propto F(mR)$. A key finding is that the conventional expansion in inverse powers of $g^2$ is non-convergent and wildly oscillatory, prompting a new asymptotic large-$y$ scheme that depends on the coupling parameter $k$, offering a robust route for finite-coupling dynamics. Overall, the work provides concrete exact finite-coupling solutions and a controlled framework for wall moduli dynamics with implications for brane-world scenarios and localized gauge fields.

Abstract

A series of exact BPS solutions are found for single and double domain walls in N=2 supersymmetric (SUSY) QED for finite gauge coupling constants. Vector fields are found to be massive, although it is localized on the wall. Massless modes can be assembled into a chiral scalar multiplet of the preserved N=1 SUSY, after an appropriate gauge choice. The low-energy effective Lagrangian for the massless fields is obtained for the finite gauge coupling. The inter-wall force is found to be much stronger than the known infinite coupling case. The previously proposed expansion in inverse powers of the gauge coupling has pathological oscillations, and does not converge to the correct finite coupling result.

Figures (10)

• Figure 1: The exact BPS wall solution $S_2(m)$ for finite gauge coupling $g={\sqrt2 m}/{\sqrt{\zeta}}$ and the tension $T_{\rm w}=2m\zeta$. a) Scalar field $\Sigma(y)$ of vector multiplet in Eq.(\ref{['eq:single-wall-sigma']}) divided by the mass parameter $m$ as a function of the coordinate $y$ times $m$. b) Hypermultiplet scalar field $H^{11}(y)$ in Eq.(\ref{['eq:single-wall-H1']}) as a function of the coordinate $y$ times $m$ (solid line), and $H^{12}(y)$ in Eq.(\ref{['eq:single-wall-H2']}) as a function of the coordinate $y$ times $m$ (dotted line).
• Figure 2: The potential $V(x)$ of Eq.(\ref{['5DW-1.20']}) for $S_2(m)$ (solid line), $S_3(m)$ (dotted line), and $S_{4}(m)$ (dashed line) in the eigenvalue equation for the vector fluctuation (\ref{['5DW-3.16']}) as a function of extra dimension coordinate $y$. The mass parameter is taken to be $m=1$.
• Figure 3: Relation between $R$ defined in Eq.(\ref{['tong-16']}) and $\tilde{R}$. The solid line represents the relation defined in Eq.(\ref{['5DW-3.21']}) for the finite coupling solution $D_4(m)$ in Eq.(\ref{['5DW-3.19']}). The dashed line represents the relation defined in Eq.(\ref{['5DW-3.20']}) for the infinite coupling solution $D_0(m)$ in Eq.(\ref{['5DW-3.18']}).
• Figure 4: Comparison of the vector multiplet scalar $\Sigma$ as a function of $my$ for exact solutions with various gauge couplings. a) Single wall solutions $S_0(m)$ (dotted line), $S_2(m)$ (solid line), $S_3(m)$ (short dashed line), and $S_4(m)$ (dashed line). b) Double wall solutions $D_0(m)$ (dotted line), and $D_4(m)$ (solid line).
• Figure 5: Comparison of the hypermultiplet scalars $H^{1A}$ as a function of $my$ for exact solutions with various gauge couplings. a) Single wall solutions $S_0(m)$ (dotted line), $S_2(m)$ (solid line), $S_3(m)$ (short dashed line), and $S_4(m)$ (dashed line). b) Double wall solutions $D_0(m)$ (dotted line), and $D_4(m)$ (solid line).