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Q-ball - Wikipedia, the free encyclopedia

Q-ball

From Wikipedia, the free encyclopedia

For the ball used in billiards, see cue ball.
"Q ball" is also the name of the Q sensor module in the Apollo spacecraft launch escape system.

In theoretical physics, Q-ball refers to a type of non-topological soliton. A soliton is a localized field configuration that is stable—it cannot spread out and dissipate. In the case of a non-topological soliton, the stability is guaranteed by a conserved charge: the soliton has lower energy per unit charge than any other configuration. (In physics, charge is often represented by the letter "Q", and the soliton is spherically symmetric, hence the name.)

Contents

[edit] Constructing a Q-ball

In its simplest form, a Q-ball is constructed in a field theory of a complex scalar field φ, in which Lagrangian is invariant under a global U(1) symmetry. The Q-ball solution is a state which minimizes energy while keeping the charge Q associated with the global U(1) symmetry constant. A particularly transparent way of finding this solution is via the method of Lagrange multipliers. In particular, in three spatial dimensions we must minimize the functional

 E_{\omega} = E + \omega \left[ Q - \frac{1}{2i} \int d^{3} x(\phi^{*} \partial_{t} \phi - \phi \partial_{t} \phi^{*}) \right],

where the energy is defined as

 E = \int d^{3} x \left[ \frac{1}{2} \dot{\phi}^{2} + \frac{1}{2} |\nabla \phi|^{2} + U(\phi, \phi^{*}) \right],

and ω is our Lagrange multiplier. The time dependence of the Q-ball solution can be obtained easily if one rewrites the functional Eω as

 E_{\omega} = \int d^{3} x \left[ \frac{1}{2} |\dot{\phi} - i \omega \phi|^{2} + \frac{1}{2} |\nabla \phi|^{2} + \hat{U}_{\omega}(\phi, \phi^{*}) \right]

where  \hat{U}_{\omega} = U - \frac{1}{2} \omega^{2} \phi^{2} . Since the first term in the functional is now positive, minimization of this terms implies

 \phi(\vec{r},t) = \phi_{0}(\vec{r}) e^{i\omega t}.

We therefore interpret the Lagrange multiplier ω as the frequency of oscillation of the field within the Q-ball.

The theory contains Q-ball solutions if there are any values of φ * φ at which the potential is less than m2φ * φ. In this case, a volume of space with the field at that value can have an energy per unit charge that is less than m, meaning that it cannot decay into a gas of individual particles. Such a region is a Q-ball. If it is large enough, its interior is uniform, and is called "Q-matter". For a review see (Lee et al 1992 [1]).

[edit] Thin-wall Q-balls

The thin-wall Q-ball was the first to be studied, and this pioneering work was carried out by Sidney Coleman in 1986 [2]. For this reason, Q-balls of the thin-wall variety are sometimes called "Coleman Q-balls."

We can think of this type of Q-ball a spherical ball of nonzero vacuum expectation value. In the thin-wall approximation we take the spatial profile of the field to be simply

φ0(r) = θ(Rr0.

In this regime the charge carried by the Q-ball is simply  Q = \omega \phi_{0}^{2} V . Using this fact we can eliminate ω from the energy, such that we have

 E = \frac{1}{2} \frac{Q^{2}}{\phi_{0}^{2} V} + U(\phi_{0}) V.

Minimization with respect to V gives

 V = \sqrt{\frac{Q^{2}}{2 U(\phi_{0}) \phi_{0}^{2}}}.

Plugging this back into the energy yields

 E = \sqrt{ \frac{2 U(\phi_{0})}{\phi_{0}^{2}}}~ Q .

Now all that remains is to minimize the energy with respect to φ0. We can therefore state that a Q-ball solution of the thin-wall type exists if and only if

 min = \frac{2 U(\phi)}{\phi^{2}},  for φ > 0.

When the above criterion is satisfied the Q-ball exists and by construction is stable against decays into scalar quanta. The mass of the thin-wall Q-ball is simply the energy

M(Q) = ω0Q.

It should be pointed out that while this kind of Q-ball is stable against decay into scalars, it is not stable against decay into fermions if the scalar field φ has nonzero Yukawa couplings to some fermions. This decay rate was calculated in 1986 by Andrew Cohen, Sidney Coleman, Howard Georgi, and Aneesh Manohar [3]

[edit] History

Configurations of a charged scalar field that are classically stable (stable against small perturbations) were constructed by Rosen in 1968 [4]. Stable configurations of multiple scalar fields were studied by Friedberg, Lee, and Sirlin in 1976 [5]. The name "Q-ball" and the proof of quantum-mechanical stability (stability against tunnelling to lower energy configurations) come from Sidney Coleman (Coleman 1986 [6]).

[edit] Occurrence in nature

It has been theorized that dark matter might consist of Q-balls (Frieman et al 1988 [7], Kusenko et al 1997 [8]) and that Q-balls might play a role in baryogenesis, i.e. the origin of the matter that fills the universe (Dodelson et al 1990 [9], Enqvist et al 1997 [10]). Interest in Q-balls was stimulated by the suggestion that they arise generically in supersymmetric field theories (Kusenko 1997 [11]), so if nature really is fundamentally supersymmetric then Q-balls might have been created in the early universe, and still exist in the cosmos today.

[edit] Fiction

  • In the movie Sunshine, the Sun is undergoing a premature death. The movie's science adviser, CERN scientist Brian Cox, proposed "infection" with a Q-ball as the mechanism for this death, but this does not appear in the movie itself.
  • The character Quinn Mallory in the TV series Sliders has the nickname Q-ball.

[edit] External links

  • Cosmic anarchists, by Hazel Muir. A popular account of the proposal of Alexander Kusenko.

[edit] References

  1. ^ T.D. Lee, Y. Pang, "Nontopological solitons", Phys. Rept. 221:251-350 (1992)
  2. ^ S. Coleman, "Q-Balls", Nucl. Phys. B262:263 (1985); erratum: B269:744 (1986)
  3. ^ Andrew Cohen, Sidney Coleman, Howard Georgi, and Aneesh Manohar, "The Evaporation of Q-balls", Nucl. Phys. B272:301 (1986)
  4. ^ G. Rosen, J. Math. Phys. 9:996 (1968)
  5. ^ R. Friedberg, T.D. Lee, A. Sirlin, Phys. Rev. D13:2739 (1976)
  6. ^ S. Coleman, "Q-Balls", Nucl. Phys. B262:263 (1985); erratum: B269:744 (1986)
  7. ^ J. Frieman, G. Gelmini, M. Gleiser, E. Kolb "Solitogenesis: Primordial Origin Of Nontopological Solitons", Phys. Rev. Lett. 60:2101 (1988)
  8. ^ A. Kusenko, M. Shaposhnikov, "Supersymmetric Q balls as dark matter", [Phys. Lett. B418:46-54 (1998)]
  9. ^ S. Dodelson, L. Widrow, "Baryon Symmetric Baryogenesis", Phys. Rev. Lett. 64:340-343 (1990)
  10. ^ K. Enqvist, J. McDonald, "Q-Balls and Baryogenesis in the MSSM", Phys.Lett. B425 309-321 (1998)
  11. ^ A. Kusenko, "Solitons in the supersymmetric extensions of the Standard Model", [Phys. Lett. B405:108 (1997)]


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