Pole Solution in Six Dimensions and Mass Hierarchy

Abstract

A solution of the 6D gravitational model, which has the pole configuration, is found. The vacuum setting is done by the 6D Higgs potential and the solution is for a family of the vacuum and boundary parameters. The boundary condition is solved by the $1/k^2$-expansion (thin pole expansion) where $1/k$ is the {\it thickness} of the pole. The obtained analytic solution is checked by the numerical method. This is a dimensional reduction model from 6D to 4D by use of the soliton solution (brane world). It is regarded as a higher dimensional version of the Randall-Sundrum 5D model. The mass hierarchy is examined. Especially the {\it geometrical see-saw} mass relation is obtained. Some physical quantities in 4D world such as the Planck mass, the cosmological constant, and fermion masses are focussed. Comparison with the 5D model is made.