Sphaleron

A sphaleron (Greek: σφαλερός "slippery") 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 numbers. Such processes cannot be represented by perturbative methods such as Feynman diagrams, and are therefore called non-perturbative. Geometrically, a sphaleron is simply a saddle point of the electroweak potential (in 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, the anomaly violating baryon number can convert three baryons to three antileptons, and related processes, with a change in baryon number of 3. This violates conservation of baryon number and lepton number, but the difference B−L is conserved. A sphaleron may convert three baryons to anti-leptons and three anti-leptons to baryons, in multiples of 3.

A sphaleron is similar to the midpoint ( $τ$ =0) 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.

Baryogenesis

Since a sphaleron may convert baryons to antileptons and antibaryons to leptons, if the density of sphalerons was at some stage high enough, they would wipe out any net excess of baryons or anti-baryons. This has two important implications in any theory of baryogenesis within the Standard Model:^[1]^[2]

Any baryon net excess arising before the electroweak symmetry breaking 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. 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 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.

Details

For an SU(2) gauge theory, neglecting $\theta _{W}$ , we have the following equations for the gauge field and the Higgs field in the gauge $A_{0}=A_{r}=0$ ^[3]

{\mathbf {A}}=\nu {\frac {f(\xi )}{\xi }}{\hat {{\mathbf {r}}}}\times {\mathbf {\sigma }}\,,\,\phi ={\frac {\nu }{{\sqrt {2}}}}h(\xi ){\hat {{\mathbf {r}}}}\cdot {\mathbf {\sigma }}\phi _{0}

where $\xi =rg\nu$ , $\phi _{0}={\begin{bmatrix}1\\0\end{bmatrix}}$ , the $\sigma$ -s are the SU(2) generators, $g$ is the electroweak coupling constant, $\nu$ is the Higgs VEV absolute value.

$h(\xi )$ and $f(\xi)$ are functions going from 0 to 1 as $\xi$ goes from 0 to $\infty$ . 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 $\xi$ goes to infinity.

Note that in the limit $\xi \rightarrow \infty$ , the gauge sector approaches one of the pure-gauge transformation ${\frac {{\mathbf {r}}\cdot {\mathbf {\sigma }}}{r}}$ , which is the same as the pure gauge transformation to which the BPST instanton approaches as $r\rightarrow\infty$ at $t=0$ , hence establishing the connection between the sphaleron and the instanton.

References

Klinkhamer, F. R.; Manton, N. S. (1984). "A saddle-point solution in the Weinberg-Salam theory". Phys. Rev. D. 30 (10): 2212–2220. Bibcode:1984PhRvD..30.2212K. doi:10.1103/PhysRevD.30.2212.

↑ Shaposhnikov, M. E.; Farrar, G. R. (1993). "Baryon Asymmetry of the Universe in the Minimal Standard Model". Physical Review Letters. 70: 2833–2836. Bibcode:1993PhRvL..70.2833F. arXiv:hep-ph/9305274 . doi:10.1103/PhysRevLett.70.2833.
↑ Kuzmin, V. A.; Rubakov, V. A.; Shaposhnikov, M. E. (1985). "On anomalous electroweak baryon-number non-conservation in the early universe". Physic Letters B. 155: 36–42. Bibcode:1985PhLB..155...36K. doi:10.1016/0370-2693(85)91028-7.
↑ Arnold, P.; McLerran, L. (1987). "Sphalerons, small ﬂuctuations, and baryon-number violation in electroweak theory". Physical Review D. 36: 581–596. Bibcode:1987PhRvD..36..581A. doi:10.1103/PhysRevD.36.581.

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Sphaleron

Baryogenesis

Details

See also

References