Sphaleron
Encyclopedia
A sphaleron is a static (time independent) solution to the electroweak field equations of the Standard Model
of particle physics
, and it is involved in processes that violate baryon
and lepton
number. Such processes cannot be represented by Feynman diagram
s, and are therefore called non-perturbative. Geometrically, a sphaleron is simply a saddle point of the electroweak potential energy (in the infinite dimensional field space), much like the saddle point
of the surface
in three dimensional analytic geometry.
In the standard model, processes violating baryon number convert three baryon
s to three antileptons
, and related processes. This violates conservation of baryon number and lepton number
, but the difference B−L is conserved. In fact, a sphaleron may convert baryons to anti-leptons and anti-baryons to leptons, and hence a quark may be converted to 2 anti-quarks and an anti-lepton, and an anti-quark may be converted to 2 quarks and a lepton.
A sphaleron is similar to the midpoint (
) of the instanton
, so it is non-perturbative. This means that under normal conditions sphalerons are unobservably rare. However, they would have been more common at the higher temperatures of the early universe
.
Sphalerons and baryogenesis
Since a sphaleron may convert baryons to anti-leptons and anti-baryons to leptons, if the density of sphalerons was at some stage high enough, they would wipe off any baryon assymetry (i.e. any net excess of baryons or anti-baryons). This has two important implications in any theory of baryogenesis
within the Standard Model
In some theories of baryogenesis
an imbalance of the number of leptons and antileptons is formed first by leptogenesis
and sphaleron transitions then convert this to an imbalance in the numbers of baryons and antibaryons.
, we have the following equations for the gauge field and the Higgs
field in the gauge A0 = Ar = 0
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, and it is involved in processes that violate baryon
Baryon
A baryon is a composite particle made up of three quarks . Baryons and mesons belong to the hadron family, which are the quark-based particles...
and lepton
Lepton
A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...
number. Such processes cannot be represented by Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s, and are therefore called non-perturbative. Geometrically, a sphaleron is simply a saddle point of the electroweak potential energy (in the infinite dimensional field space), much like the saddle point
Saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
of the surface
in three dimensional analytic geometry.In the standard model, processes violating baryon number convert three baryon
Baryon
A baryon is a composite particle made up of three quarks . Baryons and mesons belong to the hadron family, which are the quark-based particles...
s to three antileptons
Lepton
A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...
, and related processes. This violates conservation of baryon number and lepton number
Lepton number
In particle physics, the lepton number is the number of leptons minus the number of antileptons.In equation form,so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0...
, but the difference B−L is conserved. In fact, a sphaleron may convert baryons to anti-leptons and anti-baryons to leptons, and hence a quark may be converted to 2 anti-quarks and an anti-lepton, and an anti-quark may be converted to 2 quarks and a lepton.
A sphaleron is similar to the midpoint (
) of the instantonInstanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
, so it is non-perturbative. This means that under normal conditions sphalerons are unobservably rare. However, they would have been more common at the higher temperatures of the early universe
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
.
Sphalerons and baryogenesisBaryogenesisIn 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...
Since a sphaleron may convert baryons to anti-leptons and anti-baryons to leptons, if the density of sphalerons was at some stage high enough, they would wipe off any baryon assymetry (i.e. any net excess of baryons or anti-baryons). This has two important implications in any theory of baryogenesisBaryogenesis
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...
within the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
-
- Any baryon net excess existing before the electroweakElectroweak interactionIn particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different...
symmetry breakingHiggs mechanismIn particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
was wiped off because sphalerons were abundant due to the high temperatures existing in the universe. - A baryon net excess can be created during the electroweakElectroweak interactionIn particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different...
symmetry breakingHiggs mechanismIn particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
can be later preserved only if this phase transition was first-order. This is because in a second-order phase transition sphalerons would wipe off any baryon assymetry as it is created, while in a first-order phase transition sphalerons would wipe off baryon assymetry only in the unbroken phase.
- Any baryon net excess existing before the electroweak
In some theories of 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...
an imbalance of the number of leptons and antileptons is formed first by leptogenesis
Leptogenesis (physics)
In physical cosmology, leptogenesis is the generic term for hypothetical physical processes that produced an asymmetry between leptons and antileptons in the very early universe, resulting in the dominance of leptons over antileptons...
and sphaleron transitions then convert this to an imbalance in the numbers of baryons and antibaryons.
Equations
For an SU(2) gauge theory, neglecting, we have the following equations for the gauge field and the HiggsHiggs
The term Higgs appears in:* Alan Higgs, English businessman and philanthropist* Sir Derek Higgs, an English business leader and merchant banker* Eric Sidney Higgs, English archaeologist*Griffin Higgs...
field in the gauge A0 = Ar = 0
-
where
, 
, the σ-s are the SU(2) generators, g is the electroweak coupling constant ν is the HiggsHiggsThe term Higgs appears in:* Alan Higgs, English businessman and philanthropist* Sir Derek Higgs, an English business leader and merchant banker* Eric Sidney Higgs, English archaeologist*Griffin Higgs...
VEVVacuum expectation valueIn 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...
absolute value.
h(ξ) and f(ξ) are functions going from 0 to 1 as ξ goes from 0 to
. These functions are found numerically.
For a sphaleron in the background of a non-broken phase, the Higgs field must obviously fall off eventually to zero as ξ goes to infinity.
Note that in the limit
, the gauge sector approaches one of the pure gauge transformation 
, which is the same as the pure gauge transformation to which the BPST instantonBPST instantonThe BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU Yang-Mills theory in Euclidean space-time , meaning it describes a transition between two...
approaches as
at t=0, hence establishing the connection between the sphaleron and the instanton.