Noncommutative Scalar Solitons at Finite $\theta$

TL;DR

This paper explores how noncommutative scalar soliton solutions behave at finite noncommutative scale, revealing a hierarchy of existence depending on the potential's properties and identifying a universal critical value for the ^4 potential.

Contribution

It provides a detailed analysis of the existence and singularity of noncommutative scalar solitons at finite  and identifies a universal critical ^4 potential parameter.

Findings

Fewer solitons exist as  decreases.

A finite critical  exists below which solitons vanish for bounded potentials.

Universal critical ^4 parameter m^2  13.92 for symmetric  potentials.

Abstract

We investigate the behavior of the noncommutative scalar soliton solutions at finite noncommutative scale $\theta$. A detailed analysis of the equation of the motion indicates that fewer and fewer soliton solutions exist as $\theta$ is decreased and thus the solitonic sector of the theory exhibits an overall hierarchy structure. If the potential is bounded below, there is a finite $\theta_c$ below which all the solitons cease to exist even though the noncommutativity is still present. If the potential is not bounded below, for any nonzero $\theta$ there is always a soliton solution, which becomes singular only at $\theta = 0$. The $\phi^4$ potential is studied in detail and it is found the critical $(\theta m^2)_c =13.92$ ($m^2$ is the coefficient of the quadratic term in the potential) is universal for all the symmetric $\phi^4$ potential.