Sphalerons

Prerequisites: Instantons, Standard Model of Particle Physics. Full path (38 pages)

Why is it interesting?
Intuitive
Concrete
Abstract

Why is it interesting?

Sphalerons are an important part of several proposed solutions of the Baryon asymmetry problem such as Leptogenesis.

Intuitive

The electroweak theory has infinitely many classical lowest-energy states. They all have the same energy and differ in how the gauge field is "wound up" in space, which is counted by an integer. Neighbouring vacua are separated by an energy barrier. A picture is a ball rolling in a long row of identical valleys separated by hills of equal height.

There are two ways to get from one valley to the next. At zero temperature the system can only tunnel through the hill. This is the process described by instantons, and in the electroweak theory it is so strongly suppressed that it has never been observed and plays no practical role today. At high temperature, as in the early universe, thermal fluctuations can instead carry the system over the top of the hill. The field configuration at the top of the hill, at the lowest pass between two valleys, is the sphaleron. It is a static solution of the field equations, but an unstable one: a small push sends it down into one valley or the other. The name comes from a Greek word meaning "ready to fall". Its energy is of order 10 TeV.

The reason sphalerons matter is that moving from one valley to the next changes the number of baryons and leptons. Because of a quantum anomaly, each step creates or destroys three baryons and three leptons together (one of each per generation), while the difference between baryon and lepton number stays fixed. In the hot early universe these transitions were frequent, so they can convert a lepton asymmetry into a baryon asymmetry, which is the mechanism behind leptogenesis, or wash out an asymmetry created earlier.

Concrete

A sphaleron is a maximum of the energy functional in the direction of changing winding number, and a minimum in the orthogonal direction. Thus it is a saddle point. (page 336 in An introduction to gauge theories and modern particle physics by E. Leader and E. Predazzi.)

Abstract

Consider the bosonic part of the electroweak theory, or for simplicity the $SU(2)$ gauge theory with a Higgs doublet ( $g' = 0$ ). In the temporal gauge $W_0 = 0$ , finite energy requires the Higgs field at spatial infinity to lie in the vacuum manifold $\{\Phi^\dagger\Phi = v^2/2\} \cong S^3$ , and the classical vacua are pure gauge configurations $W_i = -\frac{i}{g}(\partial_i U)U^{-1}$ (with $W_i = W_i^a \tau^a/2$ and $D_\mu = \partial_\mu - i g W_\mu$ ), $\Phi = U\,(0, v/\sqrt2)^T$ with $U:\mathbb{R}^3\cup\{\infty\} \cong S^3 \to SU(2)$ . These are classified by $\pi_3(SU(2)) = \mathbb{Z}$ , and the integer is the Chern-Simons number $N_{CS}$ of the vacuum. For a general static configuration $N_{CS}$ is a real number, and a path from one vacuum to the next changes it by one.

The fermion number currents are anomalous. With $N_f$ generations,

$\partial_\mu j^\mu_B = \partial_\mu j^\mu_L = \frac{N_f}{32\pi^2}\left( g^2\, W^a_{\mu\nu}\widetilde{W}^{a\,\mu\nu} - g'^2\, B_{\mu\nu}\widetilde{B}^{\mu\nu} \right), \qquad \widetilde{W}^{a\,\mu\nu} = \tfrac12\epsilon^{\mu\nu\rho\sigma}W^a_{\rho\sigma},$

(with the overall sign fixed by the convention for $\epsilon^{0123}$ ), so $B - L$ is conserved and a transition between vacua whose Chern-Simons numbers differ by $\Delta N_{CS}$ changes $B$ and $L$ by $N_f\,\Delta N_{CS}$ each ('t Hooft). The $B\widetilde{B}$ term is also a total derivative, but the abelian hypercharge field has no topologically distinct vacua, and its contribution vanishes for transitions between vacua.

Manton showed that the configuration space of the $SU(2)$ -Higgs theory contains non-contractible loops of field configurations that begin and end at the vacuum. Along each such loop the energy has a maximum, and minimizing this maximum over all loops in the class (a minimax procedure) yields a static saddle point of the energy functional. This is the sphaleron of Klinkhamer and Manton. It has $N_{CS} = \frac12$ , exactly one unstable direction (along the path between neighbouring vacua) and energy

$E_{\text{sph}} = \frac{4\pi v}{g}\, f\!\left(\lambda/g^2\right),$

where $\lambda$ is the Higgs quartic coupling and $f$ is a slowly varying function of order 2 (between about 1.5 and 2.7 over the full range of $\lambda/g^2$ ). With $v \approx 246$ GeV this is of order 10 TeV.

Because the sphaleron is a saddle point with one negative mode, it sets the rate of thermal transitions. At zero temperature, tunnelling between vacua is suppressed by $e^{-16\pi^2/g^2} = e^{-4\pi/\alpha_W}$ . In the Higgs phase at temperature $T$ below the electroweak crossover, the rate per unit volume is proportional to $e^{-E_{\text{sph}}(T)/T}$ , with the temperature-dependent sphaleron energy computed with $v(T)$ . In the symmetric phase there is no barrier set by $v$ , and the rate per volume scales as $\alpha_W^5 T^4$ up to logarithms and a numerical constant obtained from lattice simulations. This makes baryon and lepton number violation fast in the early universe above the electroweak scale and negligible today.

Sphalerons

Why is it interesting?

Intuitive

Concrete

Abstract

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