The renormalization group and spontaneous compactification of a higher-dimensional scalar field theory in curved spacetime

TL;DR

This paper applies the renormalization group to a higher-dimensional scalar field theory in curved spacetime, analyzing vacuum energies and spontaneous compactification effects with RG improvements.

Contribution

It formulates RG equations for the effective potential in curved spacetime and explores their implications for spontaneous compactification in Kaluza-Klein backgrounds.

Findings

RG improved vacuum energies differ significantly from classical estimates.

Spontaneous compactification behavior changes with RG improvements.

Explicit series expressions enable numerical analysis of compactification effects.

Abstract

The renormalization group (RG) is used to study the asymptotically free $\phi_6^3$-theory in curved spacetime. Several forms of the RG equations for the effective potential are formulated. By solving these equations we obtain the one-loop effective potential as well as its explicit forms in the case of strong gravitational fields and strong scalar fields. Using zeta function techniques, the one-loop and corresponding RG improved vacuum energies are found for the Kaluza-Klein backgrounds $R^4\times S^1\times S^1$ and $R^4\times S^2$. They are given in terms of exponentially convergent series, appropriate for numerical calculations. A study of these vacuum energies as a function of compactification lengths and other couplings shows that spontaneous compactification can be qualitatively different when the RG improved energy is used.