# Topological effects in Particle Physics

1. Jul 16, 2015

### ShayanJ

I've been checking a university's descriptions of its research groups and their interests, where I encountered the phrase "Topological effects in Particle Physics" which had no explanation. I searched in the internet, but I couldn't find anything. Could anyone explain about such effects and point to a reference? Is it related to topological quantum field theory?
Thanks

2. Jul 16, 2015

### DrDu

Words like "instantons" come to my mind.

3. Jul 16, 2015

### cosmik debris

Maybe do a search on TQFT (Topological Quantum Field Theory). Also look at stuff on the Fractional Hall effect.

4. Jul 16, 2015

### fzero

I doubt they mean topological quantum field theory, since those are very specially defined QFTs that usually don't even have states that you could identify as particles.

As DrDu says, instantons are a possibility. Basically, the electroweak theory and QCD are both Yang-Mills theories, and YM theories have a rich mathematical structure. One aspect is that the classical equations of motion for the gauge field have non-trivial solutions that have a topological charge. I don't want to get into much mathematical detail, but will try to explain a bit and give some keywords that might help finding more complete discussions. Some concrete references are Coleman's lectures in Aspects of Symmetry and the more recent book by Manton and Sutcliffe, Topological Solitons. However, some searching should turn up more easily obtained reviews on the arxiv.

The classical solutions that tend to be interesting are stable, have finite energy, and are localized in space, so they are examples of solitions. They are particularly interesting because they usually have a dependence on the gauge coupling like $1/g^2$ or $1/g$, which is not something that you can obtain by considering a finite number of perturbative corrections. So we call these objects nonperturbative and hope that we can use them to shed light on physics of gauge theories beyond perturbation theory, for example at strong-coupling, like in the QCD vacuum.

The first such solution in YM theory is the Wu-Yang monopole, which already exists for $U(1)$ gauge theory (a very readable review by Yang is http://physics.unm.edu/Courses/Finley/p495/handouts/CNYangonMonopoles.pdf [Broken]). This is a pointlike-solution with a magnetic charge that is integrally quantized by a condition related to a topological invariant, which is the first Chern class of the gauge bundle. As I implied, it would take a lot to explain in detail, but searching on the key terms should generate lots of references.

Another type of monopole solution is the 't Hooft-Polyakov monopole, which arises in gauge theories that include a Higgs-type scalar field. If the gauge group $G$ is broken to the group $H$ by the Higgs effect, then the charge of the monopole is related to the second homotopy group $\pi_2(G/H)$. These monopoles are particularly interesting in the context of GUT physics and cosmology. Basically, one would have expected lots of monopoles to be formed in the very early universe if a GUT gauge group was spontaneously broken. The absence of magnetic monopoles today is one of the motivations for inflationary cosmology.

Related to the monopole is an unstable (really metastable) solution of the YM+Higgs called the sphaleron. In the electroweak theory, these solutions have 3 units of baryon number charge and 3 units of lepton number charge, so they can mediate interactions which convert baryons to antileptons and antibaryons to leptons. Such processes are forbidden in perturbation theory, hence the comments about nonperturbative physics earlier. The latter case is very intriguing as a possible explanation of the baryon-antibaryon asymmetry that is observed in the universe. One type of what's called electroweak baryogenesis is the hypothesis that sphalerons produced during the electroweak breaking in the cosmological evolution converted most of the antibaryons to leptons, leaving the observed asymmetry. So far this hypothesis has not been very promising, but has not been completely ruled out either.

Finally we have the instantons, which are solutions to the Euclidean YM equations. While the monopoles were static solutions, in Euclidean theory, time is not singled out, so the instanton is actually localized in space and time, if we Wick rotate back to Lorentian spacetime. Furthermore, instantons have an interpretation as solutions which describe tunneling from one vacuum state to another. Such an effect cannot be seen in perturbation theory, which is performed around a single vacuum state. In 4D, instantons are characterized by a topological invariant known as the second Chern class of the gauge bundle. In particle physics, 't Hooft explained how instantons could be used to explain the mass difference between the $\eta$ and $\eta'$ mesons. They are also thought to be important in the physics of the QCD vacuum, through their interpretation as tunneling solutions.

Last edited by a moderator: May 7, 2017
5. Jul 17, 2015