Using the Goldstone theorem

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In summary, the Goldstone theorem predicts that there will be eight massless scalars (corresponding to the eight broken generators of the gauge group SU(3)) and no massive gauge bosons in the spectrum of this gauge theory with SU(3) gauge group and four complex scalar matter fields.
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Homework Statement


Consider a gauge theory with gauge group SU(3). Suppose that the matter field content consists of four different complex scalars, all transforming as triplets of SU(3). Suppose that the potential is the most general possible potential such that the only symmetry of the complete Lagrangian is the gauge SU(3).
On the basis of the Goldstone theorem, how many massive/massless real scalars are going to be present in the spectrum? How many massive/massless gauge bosons?


The Attempt at a Solution


I'm honestly at a loss here, I'm trying to revise for an exam, and this is a past paper problem. I know how to use the generalized Goldstone theorem if I know what the gauge symmetry is, and what it goes to. e.g. if if SO(3) -> SO(2) then there are 2 Goldstone bosons, but I don't know how to use the theorem in the case mentioned above. I assume the massless gauge bosons it asks for are the Goldstone bosons, and the massive ones would be 'normal' bosons, is this true?
 
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Thank you for your post. In this case, the Goldstone theorem tells us that for every broken generator of the gauge group SU(3), there will be one massless scalar (Goldstone boson) in the spectrum. Since the gauge group SU(3) has eight broken generators, there will be eight massless scalars in the spectrum. These correspond to the eight gluons in the SU(3) gauge theory.

As for the massive scalars, they correspond to the four complex scalars in the matter field content. However, since the potential is chosen to preserve the gauge symmetry, there will be no massive gauge bosons in the spectrum.

I hope this helps. Good luck with your exam!
 

1. What is the Goldstone theorem?

The Goldstone theorem is a fundamental result in particle physics that states that if a symmetry is spontaneously broken, a massless particle, called the Goldstone boson, must exist. This theorem is important in understanding the behavior of elementary particles and their interactions.

2. How is the Goldstone theorem used in particle physics?

The Goldstone theorem is used to explain the origin of mass in particles, as well as to predict the existence of new particles based on broken symmetries. It also plays a crucial role in the Standard Model of particle physics, which describes the fundamental particles and their interactions.

3. Can you give an example of the Goldstone theorem in action?

One example of the Goldstone theorem in action is in the theory of electroweak interactions, which describes the behavior of particles such as the electron, proton, and neutrino. In this theory, the symmetry between the electromagnetic and weak forces is broken, and as a result, the W and Z bosons acquire mass while the photon remains massless.

4. Is the Goldstone boson a real particle?

Yes, the Goldstone boson has been observed in experiments and is considered a real particle. The most well-known example is the Higgs boson, which was predicted by the Goldstone theorem and was finally discovered in 2012 at the Large Hadron Collider.

5. What are the implications of the Goldstone theorem for the study of the universe?

The Goldstone theorem has significant implications for our understanding of the universe, as it helps to explain the structure and behavior of particles and their interactions. It also provides a framework for exploring new theories and predicting the behavior of particles in extreme conditions, such as the early universe or black holes.

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