Renormalisation Group for Gravity and Dimensional Reduction

TL;DR

This paper applies the functional renormalisation group to Einstein-Hilbert gravity with extra dimensions, exploring how the effective Planck scale can be lowered, which could address the hierarchy problem.

Contribution

It extends the renormalisation group analysis to include one compact extra dimension, revealing trajectories that lower the Planck scale to the electroweak scale.

Findings

Renormalisation group flows show a crossover from four to five dimensions.

Trajectories suggest the true Planck scale can be reduced to the electroweak scale.

Behavior of gravitational coupling at the dimensional crossover is characterized.

Abstract

The functional renormalisation group for the Einstein-Hilbert action is investigated for the case of four infinite (or large) and one compact dimension. The motivation for this study is given by the suggestion that gravity in more than four dimensions might be able to solve the so-called hierarchy problem of the Standard Model of elementary particle physics because the assumed extra dimensions allow to unify the true Planck scale with the electroweak scale. In a first step the Einstein-Hilbert theory for the functional renormalisation group treatment of Quantum Gravity is introduced. To set the scene for the following investigation including a compact dimension first some corresponding calculations for four and more large dimensions are performed and the results are found to be in agreement with already known ones. The equations determining the renormalisation group flows are then derived for the case of one compact dimension. These dimensionally reduced renormalisation group equations are then numerically solved for the case of four large and one compact dimension. Results for the four- to five-dimensional crossover are then discussed employing two forms of the background field flow in the Einstein-Hilbert theory for the case of one extra compact dimension. Renormalisation group trajectories allowing for a significant lowering of the true Planck scale to the electroweak scale are identified. The behaviour of the running gravitational coupling at the four- to five-dimensional crossover and the true Planck scale is displayed. Potential phenomenological implications are briefly discussed.