page_title: Non-topological soliton
section_title: Q-star
hierarchical_section_title: Non-topological soliton / Soliton-stars / Q-star
caption_attribution_description: English: Upper limit for the boson-field soliton star energy.
context_page_description: In quantum field theory, a non-topological soliton 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 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, 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.
context_section_description: As the charge Q grows and E(Q) the order of , the gravitation becomes important for NTS. A proper name for such an object is a star. A boson-field Q-star looks like a big Q-ball. The way gravity changes E(Q) dependence is sketched here. It is the gravity what makes  for Q-star — stabilize it against the fission.
  Q-star with fermions has been described by Bahcall/Selipsky. Similar the NTS of Friedberg & Lee, the fermion field carrying a global conserved charge, interacts with a real scalar field.
  The  inside Q-star moves from a global maximum of the potential changing the mass of fermions and making them bound.
  But this time Q is not the number of different fermion species but it is the large number of one and the same kind particles in the Fermi gas state. Then for the fermion field description one has to use  instead of  and the condition of pressure equilibrium instead of the Dirac equation for . Another unknown function is the scalar field  profile which obeys the following motion equation : . Here is the scalar density of fermions, averaged on statistical ensemble:
  Fermi energy of the fermion gas .
  Neglecting the derivatives of  for large Q, that equation together with the pressure equilibrium equation , constitute a simple system which gives  and  inside the NTS. They are constant since we have neglected the derivatives. The fermion pressure
  For example, if  and , then  and . That means fermions appear to be massless in the NTS. Then the full fermion energy . For an NTS with the volume  and the charge , its energy is proportional to the charge: .
  The described above fermion Q-star has been considered  as a model for neutron star in the effective hadron field theory.
caption: The gravitation upper limit for the boson field Q-star energy
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