UV/IR Mixing in Nonassociative Snyder phi^4 Theory

TL;DR

This paper analytically investigates UV and IR divergences in nonassociative Snyder phi^4 scalar field theory, revealing partial regularization of UV divergences and the emergence of IR divergences linked to UV-IR mixing due to nonassociativity.

Contribution

It provides exact formulas for one-loop two-point functions in Snyder phi^4 theory and explores how nonassociativity induces UV-IR mixing and IR divergences.

Findings

Snyder deformation partially regularizes UV divergences.

Different nonassociative products yield different momentum integrals.

IR divergences are linked to UV-IR mixing in the theory.

Abstract

Using a quantization of the nonassociative and noncommutative Snyder phi^4 scalar field theory in a Hermitian realization, we present in this article analytical formulas for the momentum-conserving part of the one-loop two-point function of this theory in D-, 4-, and 3-dimensional Euclidean spaces, which are exact with respect to the noncommutative deformation parameter beta. We prove that these integrals are regularized by the Snyder deformation. These results indicate that the Snyder deformation does partially regularize the UV divergences of the undeformed theory, as it was proposed decades ago. Furthermore, it is observed that different nonassociative phi^4 products can generate different momentum-conserving integrals. Finally most importantly, a logarithmic infrared divergence emerges in one of these interaction terms. We then analyze sample momentum nonconserving integral qualitatively and show that it could exhibit IR divergence too. Therefore infrared divergences should exist, in general, in the Snyder phi^4 theory. We consider infrared divergences at the limit p -> 0 as UV-IR mixings induced by nonassociativity, since they are associated to the matching UV divergence in the zero-momentum limit and appear in specific types of nonassociative phi^4 products. We also discuss the extrapolation of the Snyder deformation parameter beta to negative values as well as certain general properties of one-loop quantum corrections in Snyder phi^4 theory at the zero-momentum limit.