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 is involved in certain hypothetical 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 a saddle point of the electroweak potential (in infinite-dimensional field space). A sphaleron (Greek: σφαλερός 'slippery') is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and is involved in certain hypothetical 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 a saddle point of the electroweak potential (in infinite-dimensional field space). 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 through the barrier (in which case the process is a type of instanton 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 and 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 while 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 (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. 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 within the Standard Model: 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 and sphaleron transitions then convert this to an imbalance in the numbers of baryons and antibaryons. For an SU(2) gauge theory, neglecting θ W {displaystyle heta _{W}} , we have the following equations for the gauge field and the Higgs field in the gauge A 0 = A r = 0 {displaystyle A_{0}=A_{r}=0} where ξ = r g ν {displaystyle xi =rg u } , ϕ 0 = [ 1 0 ] {displaystyle phi _{0}={egin{bmatrix}1\0end{bmatrix}}} , the σ {displaystyle sigma } -s are the SU(2) generators, g {displaystyle g} is the electroweak coupling constant, ν {displaystyle u } is the Higgs VEV absolute value. h ( ξ ) {displaystyle h(xi )} and f ( ξ ) {displaystyle f(xi )} are functions going from 0 to 1 as ξ {displaystyle xi } goes from 0 to ∞ {displaystyle infty } . These functions are found numerically.