Noncommutative Scalar Solitons at Finite $θ$

Abstract

We investigate the behavior of the noncommutative scalar soliton solutions at finite noncommutative scale $θ$. A detailed analysis of the equation of the motion indicates that fewer and fewer soliton solutions exist as $θ$ 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 $θ_c$ 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 $θ$ there is always a soliton solution, which becomes singular only at $θ= 0$. The $ϕ^4$ potential is studied in detail and it is found the critical $(θm^2)_c =13.92$ ($m^2$ is the coefficient of the quadratic term in the potential) is universal for all the symmetric $ϕ^4$ potential.