On the UV/IR mixing of Lie algebra-type noncommutatitive $\phi^4$-theories

TL;DR

This paper investigates how UV divergences in noncommutative $$-theories lead to UV/IR mixing phenomena, analyzing their properties and specific cases like Moyal and -Minkowski spaces.

Contribution

It provides a general framework linking UV divergences to UV/IR mixing in Lie algebra-type noncommutative $$-theories and discusses specific applications.

Findings

UV divergence of propagator integral implies UV/IR mixing at one-loop

Two-point function is finite in -Minkowski space

General properties of UV/IR mixing in Lie algebra-type noncommutative spaces

Abstract

We show that a UV divergence of the propagator integral implies the divergences of the UV/IR mixing in the two-point function at one-loop for a $\phi^4$-theory on a generic Lie algebra-type noncommutative space-time. The UV/IR mixing is defined as a UV divergence of the planar contribution and an IR singularity of the non-planar contribution, the latter being due to the former UV divergence, and the UV finiteness of the non-planar contribution. Some properties of this general treatment are discussed. The UV finiteness of the non-planar contribution and the renormalizability of the theory are not treated but commented. Applications are performed for the Moyal space, having a UV/IR mixing, and the $\kappa$-Minkowski space for which the two-point function at one-loop is finite.