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Review of Nontopological Solitons in Theories with U(1)-Symmetry

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Abstract

We provide a review of nontopological solitonic solutions arising in theories with a complex scalar field and global or gauge U(1)-symmetry. It covers Q-balls, homogeneous charged scalar condensates, and nonlinear localized holes and bulges in a classically stable condensate. A historical overview is followed by the discussion of properties of solutions, including their stability, from different perspectives. Solitons in models with additional massive degrees of freedom are also revisited, and their relation to one-field Q-balls is showed. We also discuss theories with a gauge field giving rise to gauged Q-balls and theories with dynamical gravity giving rise to boson stars.

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Notes

  1. The exact solution they found was already known by that time.

  2. Such solutions are also referred to as solitary waves.

  3. Although it may not be absolutely stable, see Section 2.3.

  4. By spatially homogeneous we understand a solution both magnitude and phase of which are independent of position in space. Note that it is possible to construct configurations breaking spatial translations spontaneously but with the linear combination of translation and U(1) generators remain unbroken; see, e.g., [33].

  5. There can be only a finite number of solutions which differ by the angular velocity but have coincident E and Q.

  6. If the requirement for f to be monotonic is omitted, the profile develops nodes at finite r, and one speaks of “excited states” of Q-balls (see, e.g., [45]).

  7. Note that the charge and energy defined by Eqs. (13) make sense not only for solutions of the equation of motion, but for general configurations of the form (6). Hence, they depend on infinite number of parameters. We choose ω to parameterize one-dimensional sets of Q-balls and other solutions. Then, ∂Q/∂ω and ∂E/∂ω are understood as directional derivatives of Q and E along a set of solutions. On this set, the relations (14) and (15) hold.

  8. If one permits unbounded from below potentials, it is enough to have a quartic nonlinearity, V = m2ϕ*ϕ – λ(ϕ*ϕ)2 [51]. However, the Q-balls in this potential are all unstable.

  9. To authors’ knowledge, the relevance of the shear force to understanding the properties of Q-balls was first pointed out in [54].

  10. There is no general proof of the uniqueness of a zero mode for classically unstable Q-balls, but it happens to be unique in all explicit calculations, see [55] for further discussion.

  11. It is interesting to note that the condition analogous to (24) was obtained earlier for solutions of the nonlinear Schrödinger equation [64, 65].

  12. It is possible to make L much larger than the size of the Q-ball with the charge equal to Qc.

  13. The similar question is discussed in [56].

  14. The fundamental solution of the NSE describing the dark soliton was first obtained in [85].

  15. The Bose–Einstein condensate is not to be confused with the scalar condensate discussed in Section 2. In our notations, the latter is relativistic and homogeneous, while the former is nonrelativistic and, in general, nonhomogeneous.

  16. Such complex kinks can be identified with sphalerons in the abelian Higgs model, see Ch. 11 of [100].

  17. In finite volume the requirement |Erel| < ∞ is unnecessary, and the angular velocity of a Q-hole (Q-bulge) does not, in general, coincide with the angular velocity of a condensate.

  18. The integrability of the model is due to the fact that it corresponds to the problem of motion of classical particles in the Hénon–Heiles potential [103].

  19. Note that, contrary to the case of one scalar field, a two-field potential allowing for Q-balls can be renormalizable.

  20. Interestingly, both global [13] and gauged [67] Q-balls were proposed in a successive order in the same issue of the journal.

  21. Although this fact has not been proved rigorously, it happens to be true in numerical calculations.

  22. It is worth noting that stationary spherically symmetric scalar field configurations with the horizon do not exist in theories satisfying the weak energy condition, with the canonical scalar field kinetic term, and with the minimal coupling of the scalar field to gravity [115]. Such solutions (black holes with scalar hair) were obtained in the class of axially symmetric, rotating configurations; see [116] for a review.

  23. What matters in this case is the response of the mass distribution of a compact object to a tidal field (see, e.g., [124, 125]). This is an example of the “effacement” of an internal structure of bodies in respect of the external problem of their motion [126].

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ACKNOWLEDGMENTS

The authors are grateful to Adrien Florio, Alexander Panin, D. Levkov, Mikhail Shaposhnikov, Yakov Shnir, Mikhail Smolyakov, Inar Timiryasov and Sergey Troitsky for useful discussions and critical comments on the manuscript. The work of E.N. (Sections 1, 2, 3.1) was supported by the Russian Science Foundation Grant no. RSF 16-12-10-494. The work of A.S. (Sections 3.2, 3.3, 4, 5) was supported by the Swiss National Science Foundation.

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Authors and Affiliations

  1. Institute for Nuclear Research, Russian Academy of Sciences, 117312, Moscow, Russia

    E. Ya. Nugaev

  2. Institute of Physics, Ecole Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

    A. V. Shkerin

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Correspondence to E. Ya. Nugaev or A. V. Shkerin.

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Nugaev, E.Y., Shkerin, A.V. Review of Nontopological Solitons in Theories with U(1)-Symmetry. J. Exp. Theor. Phys. 130, 301–320 (2020). https://doi.org/10.1134/S1063776120020077

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