Noncommutative Linear Sigma Models

TL;DR

This paper investigates noncommutative linear sigma models with U(N) symmetry at one-loop, showing that Goldstone's theorem holds in certain models due to noncommutative commutator interactions, contrasting with previous O(N) results.

Contribution

It demonstrates that U(N) noncommutative linear sigma models can preserve Goldstone's theorem at one-loop, highlighting the role of noncommutative commutator terms and comparing different symmetry representations.

Findings

Goldstone's theorem holds in U(N) models with proper ordering.

Violations occur in O(N) models with N>2 and in certain representations.

Noncommutative commutator interactions vanish in the commutative limit.

Abstract

We examine noncommutative linear sigma models with U(N) global symmetry groups at the one-loop quantum level, and contrast the results with our previous study of the noncommutative O(N) linear sigma models where we have shown that Nambu-Goldstone symmetry realization is inconsistent with continuum renormalization. Specifically we find no violation of Goldstone's theorem at one-loop for the U(N) models with the quartic term ordering consistent with possible noncommutative gauging of the model. The difference is due to terms involving noncommutative commutator interactions, which vanish in the commutative limit. We also examine the U(2), and O(4) linear sigma models with matter in the adjoint representation, and find that the former is consistent with Goldstone's theorem at one-loop if we include only trace invariants consistent with possible noncommutative gauging of the model, while the latter exhibits violations of Goldstone's theorem of the kind seen in the fundamental of O(N) for N>2.