Renormalization of Noncommutative Quantum Field Theories

TL;DR

This paper demonstrates that noncommutative  scalar field theories on the GM plane are renormalizable, free of UV/IR mixing, and share fixed points and -functions with their commutative equivalents, with a review of S-matrix computation methods.

Contribution

It proves the renormalizability and UV/IR mixing absence of noncommutative  scalar theories and extends these results to generic polynomial matter field theories.

Findings

Scalar field theories are renormalizable on the GM plane.

These theories are free of UV/IR mixing.

They share fixed points and -functions with commutative theories.

Abstract

We report on a comprehensive analysis of the renormalization of noncommutative \phi^4 scalar field theories on the Groenewold-Moyal (GM) plane. These scalar field theories are twisted Poincar\'e invariant. Our main results are that these scalar field theories are renormalizable, free of UV/IR mixing, possess the same fixed points and \beta-functions for the couplings as their commutative counterparts. We also argue that similar results hold true for any generic noncommutative field theory with polynomial interactions and involving only pure matter fields. A secondary aim of this work is to provide a comprehensive review of different approaches for the computation of the noncommutative S-matrix: noncommutative interaction picture and noncommutative LSZ formalism.