Q-ball

"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 ofnon-topological soliton.A soliton is a localized field configurationthat 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 solitonhas lower energy per unit charge than any other configuration.(In physics, charge is often represented by the letter "Q", and the solitonis spherically symmetric, hence the name.)

Constructing a Q-ball

In its simplest form,a Q-ball is constructed in a field theory of a complex scalar field$phi$,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

$E\_\{omega\}\; =\; E\; +\; omega\; left\; [\; Q\; -\; frac\{1\}\{2i\}\; int\; d^\{3\}\; x(phi^\{*\}\; partial\_\{t\}\; phi\; -\; phi\; partial\_\{t\}\; phi^\{*\})\; ight]\; ,$

where the energy is defined as

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

and $omega$ is our Lagrange multiplier. The time dependence of the Q-ball solution can be obtained easily if one rewrites the functional $E\_\{omega\}$ as

$E\_\{omega\}\; =\; int\; d^\{3\}\; x\; left\; [\; frac\{1\}\{2\}\; |dot\{phi\}\; -\; i\; omega\; phi|^\{2\}\; +\; frac\{1\}\{2\}\; |\; abla\; phi|^\{2\}\; +\; hat\{U\}\_\{omega\}(phi,\; phi^\{*\})\; ight]$

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

$phi(vec\{r\},t)\; =\; phi\_\{0\}(vec\{r\})\; e^\{iomega\; t\}.$

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

The theory contains Q-ball solutions if there are any values of$phi^*phi$ at which the potential is less than$m^2phi^*phi$. In this case, a volume of spacewith the field at that value can have an energy per unit charge that isless than $m$, meaning that it cannot decay into a gasof individual particles. Such a region is a Q-ball. If it is largeenough, its interior is uniform, and is called "Q-matter".For a review see (Lee et al 1992 [T.D. Lee, Y. Pang, "Nontopological solitons", Phys. Rept. 221:251-350 (1992)] ).

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 [S. Coleman, "Q-Balls", Nucl. Phys. B262:263 (1985); erratum: B269:744 (1986)] . 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

$phi\_\{0\}(r)\; =\; heta(R-r)\; phi\_\{0\}.$

In this regime the charge carried by the Q-ball is simply $Q\; =\; omega\; phi\_\{0\}^\{2\}\; V$. Using this fact we can eliminate $omega$ 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 $phi\_\{0\}$. 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 $phi\; >\; 0$.

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

$M(Q)\; =\; omega\_\{0\}\; Q.$

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 $phi$ has nonzero Yukawa couplings to some fermions. This decay rate was calculated in 1986 by Andrew Cohen, Sidney Coleman, Howard Georgi, and Aneesh Manohar [Andrew Cohen, Sidney Coleman, Howard Georgi, and Aneesh Manohar, "The Evaporation of Q-balls", Nucl. Phys. B272:301 (1986)]

History

Configurations of a charged scalar field that are classically stable(stable against small perturbations) were constructed by Rosenin 1968 [G. Rosen, J. Math. Phys. 9:996 (1968)] .Stable configurations of multiple scalar fields were studied byFriedberg, Lee, and Sirlin in 1976 [R. Friedberg, T.D. Lee, A. Sirlin, Phys. Rev. D13:2739 (1976)] .The name "Q-ball" and the proof of quantum-mechanical stability(stability against tunnelling to lower energy configurations)come from Sidney Coleman (Coleman 1986 [S. Coleman, "Q-Balls", Nucl. Phys. B262:263 (1985); erratum: B269:744 (1986)] ).

Occurrence in nature

It has been theorized that dark matter might consist of Q-balls(Frieman et al 1988 [J. Frieman, G. Gelmini, M. Gleiser, E. Kolb"Solitogenesis: Primordial Origin Of Nontopological Solitons", [http://lss.fnal.gov/archive/test-preprint/fermilab-pub-88-013-a.shtml Phys. Rev. Lett. 60:2101 (1988)] ] , Kusenko et al 1997 [ A. Kusenko, M. Shaposhnikov, "Supersymmetric Q balls as dark matter", [http://www.arxiv.org/abs/hep-ph/9709492 Phys. Lett. B418:46-54 (1998)] ] )and that Q-balls might play a role in baryogenesis, i.e. the originof the matter that fills the universe (Dodelson et al 1990 [S. Dodelson, L. Widrow, "Baryon Symmetric Baryogenesis", Phys. Rev. Lett. 64:340-343 (1990)] ,Enqvist et al 1997 [K. Enqvist, J. McDonald, "Q-Balls and Baryogenesis in the MSSM", [http://www.arxiv.org/abs/hep-ph/9711514 Phys.Lett. B425 309-321 (1998)] ] ). Interest in Q-balls was stimulated by the suggestion that they arise generically in
supersymmetric field theories (Kusenko 1997 [A. Kusenko, "Solitons in the supersymmetric extensions of the Standard Model", [http://www.arxiv.org/abs/hep-ph/9704273 Phys. Lett. B405:108 (1997)] ] ), so ifnature really is fundamentally supersymmetric then Q-ballsmight have been created in the early universe, and still existin the cosmos today.

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.
*A parody of the Q-Ball, the 8-Ball Doomsday Device, appears in The Tick cartoon series as the ultimate weapon of the alien race Fey, which is later revealed to simply be a hovering 8-ball.

External links

* [http://www.physics.ucla.edu/~kusenko/new_sci_qballs.html Cosmic anarchists] , by Hazel Muir. A popular account of the proposal of Alexander Kusenko.

References