A non-perturbative approach to non-commutative scalar field theory

TL;DR

This paper introduces a non-perturbative matrix model approach to analyze non-commutative scalar field theories, revealing phase transitions and potential fixed points in four dimensions.

Contribution

It develops a matrix model framework for non-commutative scalar fields, enabling non-perturbative analysis and identifying phase transitions and fixed points.

Findings

Eigenvalue density dominates the eigenvalue sector.

Identifies a phase transition to a striped or matrix phase.

Provides evidence for a non-trivial fixed point in 4D NC  model.

Abstract

Non-commutative Euclidean scalar field theory is shown to have an eigenvalue sector which is dominated by a well-defined eigenvalue density, and can be described by a matrix model. This is established using regularizations of R^{2n}_\theta via fuzzy spaces for the free and weakly coupled case, and extends naturally to the non-perturbative domain. It allows to study the renormalization of the effective potential using matrix model techniques, and is closely related to UV/IR mixing. In particular we find a phase transition for the \phi^4 model at strong coupling, to a phase which is identified with the striped or matrix phase. The method is expected to be applicable in 4 dimensions, where a critical line is found which terminates at a non-trivial point, with nonzero critical coupling. This provides evidence for a non-trivial fixed-point for the 4-dimensional NC \phi^4 model.