formulasearchengine - User contributions [en]

Inversion temperature

2015-01-12T16:11:04Z

106.51.134.109:

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Lie algebra bundle

2014-11-06T10:00:00Z

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Simultaneous perturbation stochastic approximation

2014-06-18T14:47:42Z

106.51.241.231: Added external link to a reference

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Talk:Stochastic matrix

2014-02-18T04:52:21Z

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Dilemma

2014-01-31T14:43:55Z

106.51.252.234: /* Constructive dilemmas */

A '''sphaleron''' ({{lang-el|σφαλερός}} "weak, dangerous") 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 z(x,y)=x2−y2 in three dimensional analytic geometry.

In the standard model, processes violating baryon number convert three [[baryon]]s to three [[lepton|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 (<math>\tau=0</math>) 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 [[Big Bang|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 asymmetry (i.e. any net excess of baryons or anti-baryons). This has two important implications in any theory of [[baryogenesis]] within the [[Standard Model]] <ref name=FarrarShaposhnikov1993>
{{cite journal
| last1 = Shaposhnikov |first1=M. E.
| last2 = Farrar |first2=G. R.
| year = 1993
| title = Baryon Asymmetry of the Universe in the Minimal Standard Model
| journal = [[Physical Review Letters]]
| volume = 70 | pages = 2833–2836
| doi = 10.1103/PhysRevLett.70.2833
| arxiv = hep-ph/9305274
|bibcode = 1993PhRvL..70.2833F }}</ref><ref name=KRS1985>
{{cite journal
| last1 = Kuzmin |first1=V. A.
| last2 = Rubakov |first2=V. A.
| last3 = Shaposhnikov |first3=M. E.
| year = 1985
| title = On anomalous electroweak baryon-number non-conservation in the early universe
| journal = Physic Letters B
| volume = 155 | pages = 36–42
| doi = 10.1016/0370-2693(85)91028-7
|bibcode = 1985PhLB..155...36K }}</ref>
:
* Any baryon net excess existing before the [[Electroweak interaction|electroweak]] [[Higgs mechanism|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 [[Electroweak interaction|electroweak]] [[Higgs mechanism|symmetry breaking]] can be later preserved only if this phase transition was [[Phase transition#Ehrenfest classification|first-order]]. This is because in a second-order phase transition sphalerons would wipe off any baryon asymmetry as it is created, while in a first-order phase transition sphalerons would wipe off baryon asymmetry only in the unbroken phase.

In some theories of [[baryogenesis]] an imbalance of the number of leptons and antileptons is formed first by [[leptogenesis (physics)|leptogenesis]] and sphaleron transitions then convert this to an imbalance in the numbers of baryons and antibaryons.

== Equations ==
For an [[SU(2)]] gauge theory, neglecting [[Electroweak interaction#Formulation|<math>\theta_W</math>]], we have the following equations for the gauge field and the [[Higgs]] field in the gauge A0 = Ar = 0 <ref name=ArnoldMcLarren1987>
{{cite journal
| last1 = Arnold |first1=P.
| last2 = McLarren |first2=L.
| year = 1987
| title = Sphalerons, small ﬂuctuations, and baryon-number violation in electroweak theory
| journal = [[Physical Review D]]
| volume = 36 | pages = 581–596
|bibcode = 1987PhRvD..36..581A |doi = 10.1103/PhysRevD.36.581 }}</ref>
:
::<math> \bold{A} = \nu \frac{f(\xi)}{\xi}\hat{\bold{r}}\times\bold{\sigma} \, , \,
\phi = \frac{\nu}{\sqrt{2}}h(\xi)\hat{\bold{r}}\cdot\bold{\sigma}\phi_0</math>

where <math>\xi = rg\nu</math>, <math>\phi_0 = \begin{bmatrix}1 \\ 0\end{bmatrix}</math>, the σ-s are the [[SU(2)]] generators, g is the electroweak coupling constant ν is the [[Higgs]] [[Vacuum expectation value|VEV]] absolute value.

h(ξ) and f(ξ) are functions going from 0 to 1 as ξ goes from 0 to <math>\infty</math>. 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 <math>\xi\rightarrow\infty</math>, the gauge sector approaches one of the [[pure gauge transformation]] <math> \frac{\bold{r}\cdot\bold{\sigma}}{r} </math>, which is the same as the pure gauge transformation to which the [[BPST instanton]] approaches as <math>r\rightarrow\infty</math> at t=0, hence establishing the connection between the sphaleron and the instanton.

==See also==
*[[Instanton]]
*[[Theta vacuum]]

==References==
*{{cite journal | last1= Klinkhamer |first1=F. R. |last2=Manton |first2=N. S. | title=A saddle-point solution in the Weinberg-Salam theory |journal=Phys. Rev. D |year=1984 |volume=30 |issue=10 |pages=2212–2220 |doi=10.1103/PhysRevD.30.2212 |bibcode=1984PhRvD..30.2212K}}

{{Reflist}}

[[Category:Quantum field theory]]
[[Category:Quantum chromodynamics]]
[[Category:Anomalies in physics]]