Asymptotic freedom for $\lambda \phi^4_{\star}$ QFT in Snyder-de Sitter space

TL;DR

This paper investigates a scalar field theory in Snyder-de Sitter space, demonstrating asymptotic freedom in certain regimes and revealing a curvature parameter transition from IR de-Sitter to UV anti-de Sitter space.

Contribution

It provides the first analytical and numerical analysis of the beta functions for a $_{}$ scalar field in Snyder-de Sitter space, highlighting asymptotic freedom and curvature behavior.

Findings

The model exhibits at least one regime of asymptotic freedom.

The curvature parameter can change sign, indicating a transition from IR de-Sitter to UV anti-de Sitter space.

Numerical solutions support the analytical beta function analysis.

Abstract

We analyze the model of a self-interacting $\phi^4_{\star}$ scalar field theory in Snyder-de Sitter space. After analytically computing the one-loop beta functions {in the small noncommutativity and curvature limit}, we solve numerically the corresponding system of differential equations, showing that in this limit the model possesses at least one regime in which the theory is asymptotically free. Moreover, in a given region of the parameter space we also observe a peculiar running of the parameter associated to the curvature, which changes its sign and therefore can be interpreted as a transition from an IR de-Sitter space to and UV anti-de Sitter one.