Nonuniform symmetry breaking in noncommutative $\lambda \Phi^4$ theory

TL;DR

This paper investigates how noncommutative $^4$ theory exhibits nonuniform symmetry breaking, revealing that the broken phase involves a spatially varying background rather than a uniform vacuum, with the mass gap influenced by noncommutativity.

Contribution

It demonstrates that noncommutative $^4$ theory undergoes nonuniform symmetry breaking, a novel insight into phase structure in such noncommutative field theories.

Findings

No phase transition to a uniform vacuum occurs.

Broken phase involves a nonuniform background with spatial modulation.

Mass gap depends on momentum angles and noncommutativity parameter.

Abstract

The spontaneous symmetry breaking in noncommutative $\lambda\Phi^4$ theory has been analyzed by using the formalism of the effective action for composite operators in the Hartree-Fock approximation. It turns out that there is no phase transition to a constant vacuum expectation of the field and the broken phase corresponds to a nonuniform background. By considering $<\phi(x)>=A \cos(\vec Q \cdot \vec x)$ the generated mass gap depends on the angles among the momenta $\vec k$ and $\vec Q$ and the noncommutativity parameter $\vec\theta$. The order of the transition is not easily determinable in our approximation.