[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](adsbygoogle = window.adsbygoogle || []).push({});

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

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# Homework Help: Spontaneous symmetry breaking in the SM

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