Massive $U(1)$ gauge field and their accompanying scalars in brane world

TL;DR

The paper addresses how a bulk U(1) gauge field in brane-worlds yields massive vector KK modes accompanied by scalar KK modes, and proposes a bulk term $ (\\nabla^M X_M)^2 $ to provide a clear, Stückelberg-like mechanism for these scalars. By performing KK reductions in 5D and 6D brane models, the authors derive Schrödinger-like equations for the KK towers, establish a residual gauge symmetry, and show that scalar KK modes acquire geometry-induced masses with clear mass hierarchies (scalars lighter than vectors in 5D). In 6D, two scalar types emerge and mix, leading to oscillations whose physical masses come from diagonalizing a block mass matrix $M^2$. Across two 6D geometries (thick brane and football-shaped), the scalar sectors exhibit distinct but related spectra, highlighting the geometric origin of scalar degrees of freedom and their potential phenomenology while preserving gauge consistency. The work thus provides a geometrically grounded, gauge-consistent framework for vector–scalar interactions in higher-dimensional brane models and opens avenues for further quantum and phenomenological exploration of these geometric scalar modes.

Abstract

In brane-world scenarios, the effective action of a massless bulk \(U(1)\) gauge field preserves gauge invariance via couplings between massive vector Kaluza-Klein (KK) modes and scalar KK modes. In this work, we extend this framework by introducing a term \((\nabla^M X_M)^2\) into the massless bulk \(U(1)\) gauge action. This modification explicitly breaks the full gauge redundancy while preserving a residual gauge symmetry both in the bulk and on the brane. In this setup, the scalar KK modes can acquire masses from the background geometry. Notably, we find that on the 5D brane, these scalar KK modes are lighter than the vector KK modes. In contrast, on the 6D brane, two types of scalar modes emerge; the mixed interactions between them give rise to oscillations among these scalar modes.

Figures (1)

• Figure 2: Effective potentials for the KK modes in a 6D brane with line element \ref{['line-element1']}.