In quantum field theory, a non-topological soliton (NTS) is a soliton field configuration possessing, contrary to a topological one, a conserved Noether charge and stable against transformation into usual particles of this field for the following reason. For fixed charge Q, the mass sum of Q free particles exceeds the energy (mass) of the NTS so that the latter is energetically favorable to exist. The interior region of an NTS is occupied by vacuum different from the ambient vacuum. The vacuums are separated by the surface of the NTS representing a domain wall configuration (topological defect), which also appears in field theories with broken discrete symmetry. Infinite domain walls contradict cosmology, but the surface of an NTS is closed and finite, so its existence would not be contradictory. If the topological domain wall is closed, it shrinks because of wall tension; however, due to the structure of the NTS surface, it does not shrink since the decrease of the NTS volume would increase its energy.
Introduction Quantum field theory has been developed to predict the scattering probability of elementary particles. However, in the mid 1970s it was found out that this theory predicts one more class of stable compact objects: non-topological solitons (NTS). The NTS represents an unusual coherent state of matter, called also bulk matter. Models were suggested for the NTS to exist in forms of stars, quasars, the dark matter and nuclear matter. A NTS configuration is the lowest energy solution of classical equations of motion possessing a spherical symmetry. Such a solution has been found for a rich variety of field Lagrangians. One can associate the conserved charge with global, local, Abelian and non-Abelian symmetry. It appears to be possible that the NTS configuration exists with both bosons as well as with fermions. In different models either one and the same field carries the charge and binds the NTS, or there are two different fields: charge carrier and binding field.
The spatial size of the NTS configuration may be elementary small or astronomically large, depending on the model fields and constants. The NTS size could increase with its energy until the gravitation complicates its behavior and finally causes the collapse. In some models, the NTS charge is bounded by the stability (or metastability) condition.
Simple examples
One field For a complex scalar field with the U(1) invariant Lagrange density
L = | ∂ μ Φ | 2 − U ( | Φ | ) {\displaystyle {\mathcal {L}}=|\partial _{\mu }\Phi |^{2}-U(|\Phi |)\,}
the NTS is a ball with radius R filled with the field Φ = ( ϕ 0 / 2 ) e i ω t {\displaystyle \Phi =(\phi _{0}/{\sqrt {2}})e^{i\omega t}} . Here ϕ 0 {\displaystyle \phi _{0}} is a constant inside the ball except for a thin surface coat where it sharply drops to the global U(1) symmetrical minimum of U ( | Φ | ) {\displaystyle U(|\Phi |)} . The value ϕ 0 {\displaystyle \phi _{0}} is adjusted so that it minimises the energy of the configuration
E = ( ϕ 0 2 ω 2 2 + U ( ϕ 0 2 ) ) 4 3 π R 3 . ( 1 ) {\displaystyle E={\Big (}{\frac {\phi _{0}^{2}\omega ^{2}}{2}}+U({\frac {\phi _{0}}{\sqrt {2}}}){\Big )}{\frac {4}{3}}\pi R^{3}.\,\,\,\,\,\,\,\,\,\,\,\,\,(1)}
Since the U(1) symmetry gives the conserved current j μ = − i ( Φ ∗ ∂ μ Φ − ∂ μ Φ ∗ Φ ) , {\displaystyle j_{\mu }=-i(\Phi ^{*}\partial _{\mu }\Phi -\partial _{\mu }\Phi ^{*}\Phi ),\,}
the ball possesses the conserved charge
Q = ∫ j 0 d 3 x = ω φ 0 2 4 3 π R 3 . {\displaystyle Q=\int j_{0}d^{3}x=\omega \varphi _{0}^{2}{\frac {4}{3}}\pi R^{3}.}
The minimization of the energy (1) with R gives
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