Spontaneous Symmetry Breaking of SU(3)

The resulting terms will correspond to the mass terms for the gauge bosons.In summary, the conversation discusses the generators of SU(3) and symmetry breaking in an SU(3) theory generated by a triplet of complex scalar fields. The kinetic term and mass term of the gauge bosons are derived by expanding the covariant derivative and kinetic term around the vacuum expectation value of the scalar fields.
  • #1
jazznaz
23
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Homework Statement



The generators of SU(3) are the Gell Mann matrices, [tex]\lambda_a[/tex]. Consider symmetry breaking of an SU(3) theory generated by a triplet of complex scalar fields [tex] \Phi = \left(\phi_1, \phi_2, \phi_3\right)[/tex]. Assuming the corresponding potential has a minimum at [tex]\Phi_0 = \left(0,0,v\right)[/tex], write down the kinetic term of the scalar fields and extract the mass term of the gauge bosons.

Homework Equations



The covariant derivative is,

[tex] D_\mu \phi = \left(\partial_\mu - ig\frac{\lambda_a}{2} G^{a\nu}_{\mu} \right) \phi [/tex]

(I think)

The Attempt at a Solution



Started by writing the kinetic term as [tex] \|D_\mu \phi\|^2 [/tex], but I'm having trouble getting to anything that looks vaguely like a mass term. :(

Any suggestions would be fantastic!
 
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  • #2
The covariant derivative is similar to

[tex] D_\mu \phi = \left(\partial_\mu - ig\frac{\lambda_a}{2} G^{a\nu}_{\mu} \right) \phi [/tex]

but you might want to put in the rest of the indices to understand the structure. Also, you want to expand around the VEV, so let

[tex]\phi_i = \langle \phi_i \rangle + \varphi_i[/tex]

and expand the kinetic term, keeping track of all gauge index structure.
 

FAQ: Spontaneous Symmetry Breaking of SU(3)

1. What is Spontaneous Symmetry Breaking of SU(3)?

Spontaneous Symmetry Breaking of SU(3) is a phenomenon in particle physics where the symmetry of a system is broken spontaneously, resulting in the appearance of new, lower energy states. This concept is based on the SU(3) symmetry group, which describes the behavior of fundamental particles such as protons, neutrons, and mesons.

2. How does Spontaneous Symmetry Breaking of SU(3) occur?

Spontaneous Symmetry Breaking of SU(3) occurs when a system is in a symmetric state, but due to small fluctuations, it transitions to a lower energy state that is not symmetric. This can happen in systems with a large number of particles, such as in the early universe or in high-energy collisions.

3. What is the significance of Spontaneous Symmetry Breaking of SU(3)?

Spontaneous Symmetry Breaking of SU(3) is a key concept in understanding the behavior of subatomic particles and the fundamental forces of nature. It helps explain the origins of mass and the differences between particles, as well as the fundamental interactions that govern the universe.

4. Can Spontaneous Symmetry Breaking of SU(3) be observed in experiments?

Yes, Spontaneous Symmetry Breaking of SU(3) has been observed in various experiments, particularly in high-energy particle accelerators. One notable example is the discovery of the Higgs boson, which was predicted by the theory of Spontaneous Symmetry Breaking and confirmed by experiments at CERN's Large Hadron Collider.

5. Are there any real-world applications of Spontaneous Symmetry Breaking of SU(3)?

While the study of Spontaneous Symmetry Breaking of SU(3) primarily has theoretical and fundamental implications in particle physics, it also has some practical applications. For example, the understanding of Spontaneous Symmetry Breaking has led to the development of new materials with unique properties, such as superconductors, which have potential uses in technology and energy production.

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