Perturbative Wilsonian Formalism for Noncommutative Gauge Theories in the Matrix Representation

Abstract

We study the perturbative approach to the Wilsonian integration of noncommutative gauge theories in the matrix representation. We begin by motivating the study of noncommutative gauge theories and reviewing the matrix formulation. We then systematically develop the perturbative treatment of UV states and calculate both the leading and next to leading order one- and two-loop corrections to the quantum effective action. Throughout, we discuss how our formalism clarifies problems associated with UV-IR mixing, a particular emphasis being placed on the dipole structure imposed by noncommutative gauge invariance. Ultimately, using the structural understanding developed in this work, we are able to determine the exact form of perturbative corrections in the UV regime defined by $θΛ^2\gg 1$. Finally, we apply our results to the analysis of the divergence structure and show that 3+1 and higher dimensional noncommutative theories that allow renormalization beyond one-loop are not self-consistent.