Understanding Line Vortex(Strings) & Problems in Vacuum

[ChrisVer]

I am just trying to understand a little bit the concept behind this, but I feel lost.

If we have the lagrangian:
$L= (D_{μ}Φ)(D^{μ}Φ^{*}) - \frac{1}{4} F_{μν}F^{μν}+ V(|Φ|^{2})$
with V being the Higg's Potential we can try to put as a solution of the dynamic system the relation:
$Φ(x)= ρ(r) e^{iθn2π}$
This solution causes "problems" in the vacuum, because each winding of Φ stores up some energy.

One question I have about it, is why is this the case? I understand that in the covariant derivatives, terms of $∇_{θ}$ will give additional energy proportional to the winding number n. But isn't the Φ field U(1) invariant? So I can make U(1) transformations which will cancel out the exponential factor giving me the Goldstone boson dof as longitudial dof of the gauge bosons?

Also, any good book from which I can look up for Domain walls, these cosmic strings and GUT monoples would be appreciated :)

Thanks

 

[AndyW]

for your question! Let me try to explain the concept behind this in a more straightforward way.

First, let's break down the Lagrangian you mentioned. The first term, (D_{μ}Φ)(D^{μ}Φ^{*}), describes the kinetic energy of the scalar field Φ. The second term, - \frac{1}{4} F_{μν}F^{μν}, represents the kinetic energy of the gauge field A_{μ}, which is responsible for mediating the interactions between the scalar field Φ. The last term, V(|Φ|^{2}), is the Higgs potential, which gives mass to the gauge bosons and the scalar field Φ.

Now, let's focus on the solution you mentioned, Φ(x)= ρ(r) e^{iθn2π}. This is a special type of solution called a "vortex" or a "soliton". It is characterized by the winding number n, which is related to the number of times the phase θ winds around the origin. This solution causes "problems" in the vacuum because it has a non-zero energy density. In other words, there is energy stored in the space where the scalar field Φ is non-zero. This is similar to the potential energy stored in a stretched rubber band.

You are correct in saying that the Φ field is U(1) invariant, meaning it is unchanged by rotations in the complex plane. However, this does not mean that the energy associated with the winding number n can be cancelled out by U(1) transformations. The energy is still there, and it is a result of the non-zero value of the scalar field in the vacuum.

As for your question about Goldstone bosons, they arise from the spontaneous breaking of a continuous symmetry, such as the U(1) symmetry in the Higgs potential. The longitudinal degrees of freedom of the gauge bosons do not cancel out the energy associated with the winding number n.

As for resources, I would recommend "Gauge Theories in Particle Physics" by Ian J.R. Aitchison and Anthony J.G. Hey. It has a comprehensive discussion on domain walls, cosmic strings, and GUT monopoles. I hope this helps clarify the concept for you. Keep exploring and asking questions!