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

$E\_\; =\; E\; +\; omega\; leftQ\; -\; frac\; int\; d^\; x(phi^\; partial\_\; phi\; -\; phi\; partial\_\; phi^)\; right,$

where the energy is defined as

$E\; =\; int\; d^\; x\; leftfrac\; dot^\; +\; frac\; |nabla\; phi|^\; +\; U(phi,\; phi^)\; right,$

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

$E\_\; =\; int\; d^\; x\; leftfrac\; |dot\; -\; i\; omega\; phi|^\; +\; frac\; |nabla\; phi|^\; +\; hat\_(phi,\; phi^)\; right$

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

$phi(vec,t)\; =\; phi\_(vec)\; e^.$

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 VEV. In the thin-wall approximation we take the spatial profile of the field to be simply

$phi\_(r)\; =\; theta(R-r)\; phi\_.$

In this regime the charge carried by the Q-ball is simply $Q\; =\; omega\; phi\_^\; V$. Using this fact we can eliminate $omega$ from the energy, such that we have

$E\; =\; frac\; frac\}^\; V\}\; +\; U(phi\_)\; V.$

Minimization with respect to $V$ gives

$V\; =\; sqrt\})\; phi\_^\}\}.$

Plugging this back into the energy yields

$E\; =\; sqrt)\}^\}\}~\; Q\; .$

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

$min\; =\; frac\},$ 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

$M(Q)\; =\; omega\_\; Q.$

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 1976R. 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", Phys. Rev. Lett. 60:2101 (1988),Kusenko et al 1997 A. Kusenko, M. Shaposhnikov, "Supersymmetric Q balls as dark matter", 46-54 )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", Phys.Lett. B425 309-321 (1998)). Interest in Q-balls was stimulated by thesuggestion that they arise generically insupersymmetric field theories (Kusenko 1997A. Kusenko, "Solitons in the supersymmetric extensions of the Standard Model", 108 ), so ifnature really is fundamentally supersymmetric then Q-ballsmight have been created in the early universe, and still existin the cosmos today.

Fiction

• The plot of Sunshine , set in the near future is reported to use the Q-Ball as the basis for the impending death of the Sun, though it is not actually mentioned in the film.
• The character Quinn Mallory in the TV series Sliders has the nickname Q-ball.

External links

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

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