Q-Ball
Encyclopedia
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.)
ic particles, when there is an attraction between the particles. Loosely speaking, the Q-ball is a finite-sized "blob" containing a large number of particles. The blob is stable against fission into smaller blobs, and against "evaporation" via emission of individual particles, because, due to the attractive interaction, the blob is the lowest-energy configuration of that number of particles. (This is analogous to the fact that Nickel-62
is the most stable nucleus because it is the most stable configuration of neutrons and protons. However, Nickel-62 is not a Q-ball, in part because neutrons and protons are fermion
s, not bosons.)
For there to be a Q-ball, the number of particles must be conserved (i.e. the particle number is a conserved "charge", so the particles are described by a complex-valued field
), and the interaction potential 
of the particles must have a negative (attractive) term.
For non-interacting particles, the potential would be just a mass term
, and there would be no Q-ball. But if one adds an attractive 
term (and positive higher powers of 
to ensure that the potential has a lower bound) then there are values of 
where 
, i.e. the energy of these field values is less than the energy of a free field. This corresponds to saying that one can create blobs of non-zero field (i.e. clusters of many particles) whose energy is lower than the same number of individual particles far apart. Those blobs are therefore stable against evaporation into individual particles.
, in which Lagrangian is invariant under a global 
symmetry. The Q-ball solution is a state which minimizes energy while keeping the charge Q associated with the global 
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
where the energy is defined as
and
is our Lagrange multiplier. The time dependence of the Q-ball solution can be obtained easily if one rewrites the functional 
as
where
. Since the first term in the functional is now positive, minimization of this terms implies
We therefore interpret the Lagrange multiplier
as the frequency of oscillation of the field within the Q-ball.
The theory contains Q-ball solutions if there are any values of
at which the potential is less than 
. In this case, a volume of space with the field at that value can have an energy per unit charge that is less than 
, 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).
in 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
In this regime the charge carried by the Q-ball is simply
. Using this fact we can eliminate 
from the energy, such that we have
Minimization with respect to
gives
Plugging this back into the energy yields
Now all that remains is to minimize the energy with respect to
. We can therefore state that a Q-ball solution of the thin-wall type exists if and only if
for 
.
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
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
has nonzero Yukawa couplings to some fermions. This decay rate was calculated in 1986 by Andrew Cohen, Sidney Coleman, Howard Georgi, and Aneesh Manohar.
in 1968. Stable configurations of multiple scalar fields were studied by Friedberg, Lee, and Sirlin in 1976. The name "Q-ball" and the proof of quantum-mechanical stability
(stability against tunnelling to lower energy configurations) come from Sidney Coleman
.
might consist of Q-balls (Frieman et al.. 1988, Kusenko et al.. 1997) 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, Enqvist et al.. 1997). Interest in Q-balls was stimulated by the
suggestion that they arise generically in supersymmetric
field theories (Kusenko 1997)), 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.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, Q-ball refers to a type of non-topological soliton
Non-topological soliton
In quantum field theory, a non-topological soliton is a 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...
. 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.)
Intuitive explanation
A Q-ball arises in a theory of bosonBoson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
ic particles, when there is an attraction between the particles. Loosely speaking, the Q-ball is a finite-sized "blob" containing a large number of particles. The blob is stable against fission into smaller blobs, and against "evaporation" via emission of individual particles, because, due to the attractive interaction, the blob is the lowest-energy configuration of that number of particles. (This is analogous to the fact that Nickel-62
Nickel-62
Nickel-62 is an isotope of nickel having 28 protons and 34 neutrons.It is a stable isotope, with the highest binding energy per nucleon of any known nuclide . It is often stated that 56Fe is the "most stable nucleus", but actually 56Fe has the lowest mass per nucleon of all nuclides...
is the most stable nucleus because it is the most stable configuration of neutrons and protons. However, Nickel-62 is not a Q-ball, in part because neutrons and protons are fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s, not bosons.)
For there to be a Q-ball, the number of particles must be conserved (i.e. the particle number is a conserved "charge", so the particles are described by a complex-valued field
), and the interaction potential of the particles must have a negative (attractive) term.For non-interacting particles, the potential would be just a mass term
, and there would be no Q-ball. But if one adds an attractive term (and positive higher powers of to ensure that the potential has a lower bound) then there are values of where , i.e. the energy of these field values is less than the energy of a free field. This corresponds to saying that one can create blobs of non-zero field (i.e. clusters of many particles) whose energy is lower than the same number of individual particles far apart. Those blobs are therefore stable against evaporation into individual particles.Constructing a Q-ball
In its simplest form, a Q-ball is constructed in a field theory of a complex scalar field, in which Lagrangian is invariant under a global symmetry. The Q-ball solution is a state which minimizes energy while keeping the charge Q associated with the global symmetry constant. A particularly transparent way of finding this solution is via the method of Lagrange multipliersLagrange multipliers
In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
. In particular, in three spatial dimensions we must minimize the functional
where the energy is defined as
and
is our Lagrange multiplier. The time dependence of the Q-ball solution can be obtained easily if one rewrites the functional aswhere
. Since the first term in the functional is now positive, minimization of this terms impliesWe therefore interpret the Lagrange multiplier
as the frequency of oscillation of the field within the Q-ball.The theory contains Q-ball solutions if there are any values of
at which the potential is less than . In this case, a volume of space with the field at that value can have an energy per unit charge that is less than , 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).Thin-wall Q-balls
The thin-wall Q-ball was the first to be studied, and this pioneering work was carried out by Sidney ColemanSidney Coleman
Sidney Richard Coleman was an American theoretical physicist who studied under Murray Gell-Mann.- Life and work :Sidney Coleman grew up on the Far North Side of Chicago...
in 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
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
. In the thin-wall approximation we take the spatial profile of the field to be simply
In this regime the charge carried by the Q-ball is simply
. Using this fact we can eliminate from the energy, such that we haveMinimization with respect to
givesPlugging this back into the energy yields
Now all that remains is to minimize the energy with respect to
. We can therefore state that a Q-ball solution of the thin-wall type exists if and only if for .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
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
has nonzero Yukawa couplings to some fermions. This decay rate was calculated in 1986 by Andrew Cohen, Sidney Coleman, Howard Georgi, and Aneesh Manohar.History
Configurations of a charged scalar field that are classically stable (stable against small perturbations) were constructed by Rosenin 1968. Stable configurations of multiple scalar fields were studied by Friedberg, Lee, and Sirlin in 1976. The name "Q-ball" and the proof of quantum-mechanical stability
(stability against tunnelling to lower energy configurations) come from Sidney Coleman
Sidney Coleman
Sidney Richard Coleman was an American theoretical physicist who studied under Murray Gell-Mann.- Life and work :Sidney Coleman grew up on the Far North Side of Chicago...
.
Occurrence in nature
It has been theorized that dark matterDark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
might consist of Q-balls (Frieman et al.. 1988, Kusenko et al.. 1997) and that Q-balls might play a role in baryogenesis
Baryogenesis
In physical cosmology, baryogenesis is the generic term for hypothetical physical processes that produced an asymmetry between baryons and antibaryons in the very early universe, resulting in the substantial amounts of residual matter that make up the universe today.Baryogenesis theories employ...
, i.e. the origin of the matter that fills the universe (Dodelson et al.. 1990, Enqvist et al.. 1997). Interest in Q-balls was stimulated by the
suggestion that they arise generically in supersymmetric
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
field theories (Kusenko 1997)), 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.
Fiction
- In the movie SunshineSunshine (2007 film)Sunshine is a 2007 British science fiction film directed by Danny Boyle and written by Alex Garland about the crew of a spacecraft on a dangerous mission to the Sun. In 2057, with the Earth in peril from the dying Sun, the crew is sent to reignite the Sun with a massive stellar bomb with the mass...
, the SunSunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
is undergoing a premature death. The movie's science adviser, CERNCERNThe European Organization for Nuclear Research , known as CERN , is an international organization whose purpose is to operate the world's largest particle physics laboratory, which is situated in the northwest suburbs of Geneva on the Franco–Swiss border...
scientist Brian CoxBrian Cox (physicist)Brian Edward Cox, OBE , is a British particle physicist, a Royal Society University Research Fellow and a professor at the University of Manchester. He is a member of the High Energy Physics group at the University of Manchester, and works on the ATLAS experiment at the Large Hadron Collider at...
, proposed "infection" with a Q-ball as the mechanism for this death, but this is mentioned only in the commentary tracks and not in the movie itself. - In the fictional universe of Orion's ArmOrion's ArmOrion's Arm, is a multi-authored online science fiction world-building project, first established in 2000 by M. Alan Kazlev, Donna Malcolm Hirsekorn, Bernd Helfert and Anders Sandberg and further co-authored by many people since...
, Q-balls are one of the speculated sources for the large amounts of antimatter used by certain groups. - In the Television show SlidersSlidersSliders is an American science fiction television series. It was broadcast for five seasons, beginning in 1995 and ending in 2000. The series follows a group of travelers as they use a wormhole to "slide" between different parallel universes. The show was created by Robert K. Weiss and Tracy Tormé...
, the character Rembrandt BrownRembrandt BrownRembrandt Lee Brown is a fictional character played by Cleavant Derricks on the science fiction television show Sliders. In 1994, Rembrandt was living in San Francisco. Rembrandt is a musician, whose stage name is The Crying Man because of his ability to "cry real tears" on stage...
refers to his fellow main character Quinn MalloryQuinn MalloryQuinn R. Mallory is a fictional character on the science fiction television show Sliders, played by Jerry O'Connell.-Childhood:Quinn was born in 1973 and raised in San Francisco, California, the son of Michael and Amanda Mallory...
as Q-ball.
External links
- Cosmic anarchists, by Hazel Muir. A popular account of the proposal of Alexander Kusenko.