Modeling the fifth dimension with scalars and gravity

TL;DR

The paper develops a first-order formalism for five-dimensional gravity with a scalar to generate exact, nonlinear brane-world solutions without supersymmetry. It applies this framework to stabilize inter-brane distances via a Goldberger-Wise–type mechanism and to construct smooth domain-wall solutions that approximate positive-tension brane arrays, while also linking the setup to a non-supersymmetric c-function concept. It analyzes linear fluctuations and proves the existence of a normalizable massless graviton with no tachyonic transverse-traceless modes, offering a robust stability picture. Together, the results provide a tractable, SUSY-free route to consistent brane-world geometries and connect holographic RG ideas to non-supersymmetric contexts.

Abstract

A method for obtaining solutions to the classical equations for scalars plus gravity in five dimensions is applied to some recent suggestions for brane-world phenomenology. The method involves only first order differential equations. It is inspired by gauged supergravity but does not require supersymmetry. Our first application is a full non-linear treatment of a recently studied stabilization mechanism for inter-brane spacing. The spacing is uniquely determined after conventional fine-tuning to achieve zero four-dimensional cosmological constant. If the fine-tuning is imperfect, there are solutions in which the four-dimensional branes are de Sitter or anti-de Sitter spacetimes. Our second application is a construction of smooth domain wall solutions which in a well-defined limit approach any desired array of sharply localized positive-tension branes. As an offshoot of the analysis we suggest a construction of a supergravity c-function for non-supersymmetric four-dimensional renormalization group flows. The equations for fluctuations about an arbitrary scalar-gravity background are also studied. It is shown that all models in which the fifth dimension is effectively compactified contain a massless graviton. The graviton is the constant mode in the fifth dimension. The separated wave equation can be recast into the form of supersymmetric quantum mechanics. The graviton wave-function is then the supersymmetric ground state, and there are no tachyons.

Figures (3)

• Figure 1: a) $A$ as a function of $r$ for the $S^1/{\bf Z}_2$ geometry, with one positive and one negative tension brane, each at a fixed point of ${\bf Z}_2$. b) $A$ as a function of $r$ for two positive tension branes in an infinite fifth dimension.
• Figure 2: Sample $W$ (solid line), $V$ (dotted line), $\lambda_1$ and $\lambda_2$ (grey lines) as functions of $\phi$. By adjusting the integration constant of (\ref{['VWForm']}) one can arrange for $\lambda_1$ to be tangent to $W$, but then for $\lambda_2$ also to be tangent amounts to a fine-tuning.
• Figure 3: The $t$--$x$ plane, showing forbidden regions.