[Paper Review] Mass hierarchy and mass gap on thick branes with Poincare symmetry

Nandinii Barbosa-Cendejas, Alfredo Herrera–Aguilar+3 more|arXiv (Cornell University)|Dec 19, 2007

Black Holes and Theoretical Physics24 references12 citations

TL;DR

This paper proposes a 5D braneworld model with a thick brane configuration that naturally resolves the hierarchy problem via a mass gap in the Kaluza-Klein (KK) spectrum, avoiding light KK modes. Using analytical solutions to a Schrödinger-like equation for transverse traceless fluctuations, it derives exact spectra for massive modes in two cases: one with a mass gap and two bound states (including the 4D graviton), and another smooth RS-like model without a gap but with continuous modes governed by a confluent Heun equation, both free of singularities and yielding corrections to Newton's law.

ABSTRACT

We consider a scalar thick brane configuration arising in a 5D theory of gravity coupled to a self-interacting scalar field in a Riemannian manifold. We start from known classical solutions of the corresponding field equations and elaborate on the physics of the transverse traceless modes of linear fluctuations of the classical background, which obey a Schroedinger-like equation. We further consider two special cases in which this equation can be solved analytically for any massive mode with m^2>0, in contrast with numerical approaches, allowing us to study in closed form the massive spectrum of Kaluza-Klein (KK) excitations and to compute the corrections to Newton's law in the thin brane limit. In the first case we consider a solution with a mass gap in the spectrum of KK fluctuations with two bound states - the massless 4D graviton free of tachyonic instabilities and a massive KK excitation - as well as a tower of continuous massive KK modes which obey a Legendre equation. The mass gap is defined by the inverse of the brane thickness, allowing us to get rid of the potentially dangerous multiplicity of arbitrarily light KK modes. It is shown that due to this lucky circumstance, the solution of the mass hierarchy problem is much simpler and transparent than in the (thin) Randall-Sundrum (RS) two-brane configuration. In the second case we present a smooth version of the RS model with a single massless bound state, which accounts for the 4D graviton, and a sector of continuous fluctuation modes with no mass gap, which obey a confluent Heun equation in the Ince limit. (The latter seems to have physical applications for the first time within braneworld models). For this solution the mass hierarchy problem is solved as in the Lykken-Randall model and the model is completely free of naked singularities.

Motivation & Objective

• To address the hierarchy problem in braneworld models by constructing a thick brane configuration with a natural mass gap.
• To eliminate tachyonic instabilities and the multiplicity of arbitrarily light KK modes present in thin-brane models like Randall-Sundrum.
• To provide analytical solutions for the KK spectrum of massive fluctuations, avoiding numerical approximations.
• To explore physical realizations of special functions—Legendre and confluent Heun equations—in braneworld dynamics.
• To ensure the model remains free of naked singularities while reproducing the 4D graviton and correcting Newtonian gravity.

Proposed method

• Derive the Schrödinger-like equation for transverse traceless modes of linearized fluctuations around a classical scalar thick brane background.
• Use known classical solutions of the 5D gravity-scalar field system in a Riemannian manifold as the background geometry.
• Solve the resulting Schrödinger equation analytically for m^2 > 0, enabling exact computation of the KK spectrum.
• Identify two special cases: one with a mass gap and Legendre-type equation for continuous modes, and another with a confluent Heun equation in the Ince limit.
• Apply the thin brane limit to compute corrections to Newton's law in both models.
• Ensure the absence of naked singularities by constructing smooth, regular brane profiles with Poincaré symmetry.

Experimental results

Research questions

• RQ1 Can a thick brane model with a mass gap in the KK spectrum resolve the hierarchy problem more transparently than the thin Randall-Sundrum model?
• RQ2 How do analytical solutions of the Schrödinger-like equation for massive fluctuations improve the understanding of the KK spectrum compared to numerical methods?
• RQ3 What is the role of special functions—specifically Legendre and confluent Heun equations—in describing the continuous KK mode sector of braneworld models?
• RQ4 Does a smooth, singularity-free braneworld model with a single massless 4D graviton and continuous massive modes exist, and how does it compare to the Lykken-Randall model?
• RQ5 What are the corrections to Newton's law in the thin brane limit for these analytically solvable thick brane configurations?

Key findings

• The model with a mass gap exhibits two bound states: the massless 4D graviton and one massive KK excitation, with the remaining modes forming a continuous spectrum governed by a Legendre equation.
• The mass gap is defined by the inverse of the brane thickness, effectively eliminating arbitrarily light KK modes and simplifying the solution to the hierarchy problem.
• The second model provides a smooth realization of the RS model with a single massless bound state and continuous massive modes described by a confluent Heun equation in the Ince limit.
• The analytical solutions allow exact computation of the KK spectrum and corrections to Newton's law in the thin brane limit, without relying on numerical integration.
• Both models are free of naked singularities, ensuring physical viability and stability.
• The use of special functions in the fluctuation equations marks the first physical application of the confluent Heun equation in braneworld scenarios.