Spontaneous symmetry breaking in the SM

In summary, the conversation discusses a Lagrangian and its invariance under certain transformations. The question asks to show that the solution to the classical equations of motion with minimal energy leads to a vacuum that breaks the symmetry spontaneously. To find the solution, one can construct the Hamiltonian and set certain terms to zero in the classical field equations. This will lead to a simple solution and show that the fermion field acquires a mass proportional to g.
  • #1
AlphaNumeric
290
0
[tex]\mathcal{L} = \frac{1}{2}(\partial_{\mu}\underline{\phi}).(\partial^{\mu}\underline{\phi}) + \frac{1}{2}\mu^{2}\underline{\phi}.\underline{\phi} - \frac{\lambda}{4}(\underline{\phi}.\underline{\phi})^{2} + \bar{\psi}(i\gamma . \partial )\phi - g\bar{\psi}(\phi_{1}+i\gamma^{5}\phi_{2})\psi [/tex]

where [tex]\underline{\phi} = \left( \begin{array}{c} \phi_{1} \\ \phi_{2} \end{array} \right) [/tex]

I've shown this Lagrangian is invariant under [tex]\phi_{1} \to \cos \alpha \phi_{1} - \sin \alpha \phi_{2}[/tex] [tex]\phi_{2} \to \sin \alpha \phi_{1} + \cos \alpha \phi_{2}[/tex] [tex]\psi \to \exp\left( -\frac{i \alpha \gamma^{5}}{2} \right)\psi[/tex]

The question then asks to show that the solution to the classical equations of motion with minimal energy lead to a vacuum which breaks the symmetry spontaneously. Then, to pick a suitable vacuum solution, and use it to show the fermion field acquires a mass proportional to g.

If someone could give me pointers in the right direction I've be very grateful. I've tried mucking about with the equations of motion for the phi's and psi's, but seem to going round in circles. Thanks :smile:
 
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  • #2
Hi AlphaNumeric,

Since you are asked to find the classical lowest energy field configuration it makes sense to construct the Hamiltonian for your system. This step is essentially trivial since your Lagrangian is nice and simple in its time derivatives (note that I think you have a typo in the kinetic term for the dirac field, that [tex] \phi [/tex] should be a [tex] \psi [/tex]). I won't spoil it for you, but you will find that some terms in the Hamiltonian should obviously be zero in the lowest energy configuration. Once you set these terms to zero in your classical field equations, everything becomes simple and the solution will pop right out.
 

1. What is spontaneous symmetry breaking in the Standard Model?

Spontaneous symmetry breaking is a phenomenon in particle physics where the symmetries of a system are not present in its ground state, leading to the emergence of new properties and interactions. In the Standard Model, this breaking of symmetries is responsible for the mass of fundamental particles and the differences between the electromagnetic and weak nuclear forces.

2. How does spontaneous symmetry breaking occur in the Standard Model?

In the Standard Model, spontaneous symmetry breaking is achieved through the Higgs mechanism. This involves the Higgs field interacting with particles to give them mass, and the Higgs particle is the manifestation of this field. As the universe cooled after the Big Bang, the Higgs field settled into its lowest energy state, breaking the symmetries of the system and giving rise to the masses of particles.

3. What evidence supports the concept of spontaneous symmetry breaking in the Standard Model?

The discovery of the Higgs boson at the Large Hadron Collider in 2012 provided strong evidence for the existence of the Higgs field and the mechanism of spontaneous symmetry breaking in the Standard Model. Additionally, the predictions of the Standard Model, such as the masses of particles, have been confirmed through experiments and observations.

4. Can spontaneous symmetry breaking occur in other theories beyond the Standard Model?

Yes, spontaneous symmetry breaking is a concept that can occur in various theories beyond the Standard Model, such as in theories attempting to unify the four fundamental forces or in models of dark matter. However, the mechanism and effects may differ from those in the Standard Model.

5. How does spontaneous symmetry breaking impact our understanding of the universe?

Spontaneous symmetry breaking is a fundamental concept in the Standard Model and has greatly contributed to our understanding of the universe. It explains the origin of particle masses and the interactions between particles, and it has been confirmed through experiments and observations. Furthermore, the concept of spontaneous symmetry breaking has been extended to other areas of physics, such as condensed matter physics, offering insights into the behavior of complex systems.

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