Can effective four-dimensional scalar theory be asymptotically free in a spacetime with extra dimensions?

TL;DR

This paper investigates whether a four-dimensional scalar theory derived from a higher-dimensional space with compactified extra dimensions can exhibit asymptotic freedom, analyzing the effects of Kaluza-Klein modes on the running coupling.

Contribution

It demonstrates how compactification influences the asymptotic freedom of an effective 4D scalar theory with infinite KK modes, revealing modified running behavior.

Findings

Effective 4D coupling runs to zero at high scales

Compactification modifies the asymptotic freedom behavior

Physical implications depend on the size of the compact manifold

Abstract

We trace what happens with asymptotically free behavior of the running coupling in $\phi^{3}$ theory in six-dimensional space-time if to compactify two spatial dimensions on a 2D closed manifold. The result can be considered as an effective 4D theory of infinitely many KK-type scalar fields with triple interactions. The effective \emph{dimensional} coupling inherits running to zero at high mass scales in a modified form depending on the size of the compact manifold. Some physical implications are discussed.