Stabilization of Sub-Millimeter Dimensions: The New Guise of the Hierarchy Problem

TL;DR

The paper proposes a brane-world scenario with a TeV-scale fundamental gravity $M_*$ and $n$ sub-millimeter extra dimensions, placing Standard Model fields on a 3-brane to address the hierarchy problem while linking it to the cosmological constant problem. Radius stabilization is achieved via a small bulk cosmological constant $\bar{\Lambda}$—technically natural if SUSY is broken on the brane—and two UV mechanisms: (i) brane-lattice crystallization with a large brane-number $N_{\rm wall}$, and (ii) topological stabilization using conserved charges such as monopole numbers or Ramond-Ramond invariants; Casimir effects are also considered. The analysis yields a light radion with mass $m^2_{\rm radial} \sim \bar{\Lambda}/M_*^{2+n}$ and provides concrete scaling relations for stabilized radii, e.g. $r_0 \sim N_{\rm wall}^{1/(n-2)}/M_*$ and $N_{\rm wall} \sim (M_{\rm pl}/M_*)^{2(n-2)/n}$, while ensuring compatibility with laboratory, astrophysical, and cosmological constraints. Overall, the work offers a non-supersymmetric or non-technicolor route to the hierarchy problem, substituting the challenge of large extra dimensions and a small bulk cosmological constant for the traditional naturalness problem, and it presents testable implications for sub-mm gravity and radion phenomenology.

Abstract

A new framework for solving the hierarchy problem was recently proposed which does not rely on low energy supersymmetry or technicolor. The fundamental Planck mass is at a $\tev$ and the observed weakness of gravity at long distances is due the existence of new sub-millimeter spatial dimensions. In this picture the standard model fields are localized to a $(3+1)$-dimensional wall or ``3-brane''. The hierarchy problem becomes isomorphic to the problem of the largeness of the extra dimensions. This is in turn inextricably linked to the cosmological constant problem, suggesting the possibility of a common solution. The radii of the extra dimensions must be prevented from both expanding to too great a size, and collapsing to the fundamental Planck length $\tev^{-1}$. In this paper we propose a number of mechanisms addressing this question. We argue that a positive bulk cosmological constant $\barΛ$ can stabilize the internal manifold against expansion, and that the value of $\barΛ$ is not unstable to radiative corrections provided that the supersymmetries of string theory are broken by dynamics on our 3-brane. We further argue that the extra dimensions can be stabilized against collapse in a phenomenologically successful way by either of two methods: 1) Large, topologically conserved quantum numbers associated with higher-form bulk U(1) gauge fields, such as the naturally occurring Ramond-Ramond gauge fields, or the winding number of bulk scalar fields. 2) The brane-lattice-crystallization of a large number of 3-branes in the bulk. These mechanisms are consistent with theoretical, laboratory, and cosmological considerations such as the absence of large time variations in Newton's constant during and after primordial nucleosynthesis, and millimeter-scale tests of gravity.