The Confinement of Monopoles

What is the connection of the basic physics of the Higgs phase of electromagnetism with confinement?

To see the connection, we need to think about what would happen if we place a Dirac monopole inside a superconductor.

To get some grounding, let’s first consider a monopole and anti-monopole in vacuum.

Their magnetic field lines spread out in a pattern that is familiar from the games we played with iron filings and magnets when we were kids. This is sketched in the left-hand figure.

These field lines result in a Coulomb-like force between the two particles,

V (r) ⇠ 1/r.

Now what happens when we place these particles inside a superconductor? The magnetic flux lines can no longer spread out, but instead must form collimated tubes.

This is sketched in the right-hand figure. This tube of flux is the vortex that we described above As we have seen, happily the magnetic flux carried by a single vortex coincides with the magnetic flux emitted by a single Dirac monopole.

Figure 17: A simulation of a separated quark anti-quark pair in QCD. Figure 18: A simulation a separated baryon state in QCD.

The energy cost in separating the monopole and anti-monopole by a distance r is now

V (r) = r

where is the energy per unit length of the vortex string. In other words, inside a superconductor, magnetic monopoles are confined!

What lesson for Yang-Mills can we take away from this?

1. The confinement of quarks in Yang-Mills is again due to the emergence of flux lines, this time (chromo)electric rather than magnetic flux lines.

However, in contrast to the Abelian Higgs model, the Yang-Mills flux tube is not expected to arise as a semi-classical solution of the Yang-Mills equations.

Instead, the flux tube should emerge in the strongly coupled quantum theory where one sums over many field configurations.

Such flux tubes are seen in lattice simulations where they provide dominant contributions to the path integral. An example is shown in the figure5 .

It is less obvious how these flux tubes form between N well separated quarks which form a baryon. Simulations suggest that the flux tubes emitted by each quarks can join together at an N -string vertex.

The picture for a well separated baryon in QCD, with G = SU (3) gauge group, is shown in the figure.

We might also wish to take away another lesson from the superconducting story. In the Abelian Higgs model, the electrically charged field condenses, resulting in the confinement of monopoles.

These simulations were created by Derek Leinweber. You can find a host of beautiful QCD animations on his webpage.

Duality then suggests that to confine electrically charged objects, such as quarks, we should look to condense magnetic monopoles. This idea smells plausible, but there has been scant progress in making it more rigorous in the context of Yang-Mills theory.

The idea can be shown to work in certain supersymmetric theories.

Nonetheless, it encourages us to look for magnetic objects in non-Abelian gauge theories. We will describe these in Sections 2.6 and 2.8.

Regge Trajectories

The idea that quark anti-quark pairs are held together by flux tubes has experimental support.

Here we’ll provide a rather simplistic model of this set up. Ignoring the overall translational motion, the energy of two, massless relativistic quarks, joined together by a string, is given by:

E =p+ r

with p = p1 p2 the relative momentum. We’ll embrace the spirit of Bohr, and require that the angular momentum is quantised: J = pr 2 Z. We can then write the energy as

For a fixed J, this is minimized at r = J/ , which gives us the relationship between the energy and angular momentum of the states, E2 ⇠ J

We can now compare this to the data for hadrons.

A plot of the mass2 vs spin is known as a Chew- Frautschi plot. It is shown on the right for light vector mesons6.

We see that families of meson and their resonances do indeed sit on nice straight lines, referred to as Regge trajectories.

The slope of the lines is determined by the QCD string tension, which turns out to be around ⇠ 1.2 GeV 2 .

Perhaps more surprisingly, the data also reveals nice straight Regge trajectories in the baryon sector.

Figure 19: