![Saddle point](https://upload.wikimedia.org/wikipedia/commons/1/1e/Saddle_point.svg)
A sphaleron ( el, σφαλερός "slippery") is a static (time-independent) solution to the
electroweak field equations of the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
of
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, and is involved in certain hypothetical processes that violate
baryon
In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classif ...
and
lepton numbers. Such processes cannot be represented by
perturbative method
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for whic ...
s such as
Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s, and are therefore called
non-perturbative
In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function
: f(x) = e^,
which does not have a Taylor series at ''x'' = 0. Every coefficient of the Taylor ...
. Geometrically, a sphaleron is a
saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
of the electroweak potential (in infinite-dimensional field space).
[003.09625Sphaleron in the first-order electroweak phase transition with the dimension-six Higgs operator">][910.04761On the phenomenology of sphaleron-induced processes at the LHC and beyond">][910.00234Probing the Electroweak Sphaleron with Gravitational Waves">][005.03125The Electroweak Sphaleron in a strong magnetic field">]
This saddle point rests at the top of a barrier between two different low-energy equilibria of a given system; the two equilibria are labeled with two different baryon numbers. One of the equilibria might consist of three baryons; the other, alternative, equilibrium for the same system might consist of three antileptons. In order to cross this barrier and change the baryon number, a system must either
tunnel
A tunnel is an underground passageway, dug through surrounding soil, earth or rock, and enclosed except for the entrance and exit, commonly at each end. A pipeline is not a tunnel, though some recent tunnels have used immersed tube cons ...
through the barrier (in which case the transition is an
instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
-like process) or must for a reasonable period of time be brought up to a high enough energy that it can classically cross over the barrier (in which case the process is termed a "sphaleron" process and can be modeled with an eponymous sphaleron particle).
In both the instanton and sphaleron cases, the process can only convert groups of three baryons into three antileptons (or three antibaryons into three leptons) and vice versa. This violates conservation of
baryon number
In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as
::B = \frac\left(n_\text - n_\bar\right),
where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Baryo ...
and
lepton number
In particle physics, lepton number (historically also called lepton charge)
is a conserved quantum number representing the difference between the number of leptons and the number of antileptons in an elementary particle reaction.
Lepton number ...
, but the difference
B − L is conserved. The minimum energy required to trigger the sphaleron process is believed to be around 10 TeV; however, sphalerons ''cannot'' be produced in existing
LHC collisions, because although the LHC can create collisions of energy 10 TeV and greater, the generated energy cannot be concentrated in a manner that would create sphalerons.
A sphaleron is similar to the midpoint of the instanton, so it is
non-perturbative
In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function
: f(x) = e^,
which does not have a Taylor series at ''x'' = 0. Every coefficient of the Taylor ...
. 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
The chronology of the universe describes the history and future of the universe according to Big Bang cosmology.
Research published in 2015 estimates the earliest stages of the universe's existence as taking place 13.8 billion years ago, wit ...
.
Baryogenesis
Since a sphaleron may convert baryons to antileptons and antibaryons to leptons and thus change the baryon number, if the density of sphalerons was at some stage high enough, they could wipe out any net excess of baryons or anti-baryons. This has two important implications in any theory of
baryogenesis
In physical cosmology, baryogenesis (also known as baryosynthesis) is the physical process that is hypothesized to have taken place during the early universe to produce baryonic asymmetry, i.e. the imbalance of matter (baryons) and antimatter (a ...
within the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
:
* Any baryon net excess arising before the
electroweak
In particle physics, the electroweak interaction or electroweak force 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 differe ...
symmetry breaking
In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observe ...
would be wiped out due to abundant sphalerons caused by high temperatures existing in the early universe.
* While a baryon net excess can be created during the electroweak symmetry breaking, it can be preserved only if this phase transition was
first-order
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...
. This is because in a second-order phase transition, sphalerons would wipe out any baryon asymmetry as it is created, while in a first-order phase transition, sphalerons would wipe out baryon asymmetry only in the unbroken phase.
In absence of processes which violate B − L it is possible for an initial baryon asymmetry to be protected if it has a non-zero projection onto B − L. In this case the sphaleron processes would impose an equilibrium which distributes the initial B asymmetry between both B and L numbers.
In some theories of baryogenesis, an imbalance of the number of leptons and antileptons is formed first by
leptogenesis __notoc__
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 present-day dominance of leptons over ...
and sphaleron transitions then convert this to an imbalance in the numbers of baryons and antibaryons.
Details
For an
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
gauge theory, neglecting
, we have the following equations for the gauge field and the
Higgs field
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field,
one of the field (physics), fields in particl ...
in the gauge
::
where
,
, the symbols
represent the generators of
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
,
is the electroweak coupling constant, and
is the
Higgs VEV
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
absolute value. The functions
and
, which must be determined numerically, go from 0 to 1 in value as their argument,
, goes from 0 to
.
For a sphaleron in the background of a non-broken phase, the Higgs field must obviously fall off eventually to zero as
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 instanton
In theoretical physics, the 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(2) Yang–Mills the ...
approaches as
at
, hence establishing the connection between the sphaleron and the instanton.
Baryon number violation is caused by the "winding" of the fields from one equilibrium to another. Each time the weak gauge fields wind, the count for each of the
quark families and each of the lepton families is raised (or lowered, depending on the winding direction) by one; as there are three quark families, baryon number can only change in multiples of three. The baryon number violation can alternatively be visualized in terms of a kind of
Dirac sea
The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by th ...
: in the course of the winding, a baryon originally considered to be part of the vacuum is now considered a real baryon, or vice versa, and all the other baryons stacked inside the sea are accordingly shifted by one energy level.
Energy release
According to physicist
Max Tegmark
Max Erik Tegmark (born 5 May 1967) is a Swedish-American physicist, cosmologist and machine learning researcher. He is a professor at the Massachusetts Institute of Technology and the president of the Future of Life Institute. He is also a scienti ...
, the theoretical energy efficiency from conversion of baryons to antileptons would be orders of magnitude higher than the energy efficiency of existing power-generation technology such as nuclear fusion. Tegmark speculates that an extremely advanced civilization might use a "sphalerizer" to generate energy from ordinary baryonic matter.
See also
*
Chiral anomaly
In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have mor ...
*
Instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...
*
Theta vacuum
In quantum field theory, the theta vacuum is the semi-classical vacuum state of non- abelian Yang–Mills theories specified by the vacuum angle ''θ'' that arises when the state is written as a superposition of an infinite set of topologicall ...
References and notes
;Notes
;Citations
{{reflist, 25em
Electroweak theory
Anomalies (physics)