- #1
welcomeblack
- 13
- 0
I've been thinking about chapter 11 of Griffiths' Introduction to Elementary Particles. In section 11.7, he gives the Lagrangian density
[itex]\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)+\frac{1}{2}\mu^{2}\phi^{2}-\frac{1}{4}\lambda^{2}\phi^{4}[/itex]
and shows that the minimum occurs at
[itex]\phi=\pm\frac{\mu}{\lambda}[/itex]
and thus that the shifted potential picks up a mass term
[itex]m=\sqrt{2}\mu[/itex]
in natural units. Now, I'm wondering if you can do the same thing with a spin-half particle. For example, if we have the spinor Lagrangian density
[itex]\mathcal{L}=i\bar{\psi}\gamma^{\nu}\partial_{\nu}\psi+\mu\bar{\psi}\psi-\frac{1}{2}(\bar{\psi}\psi)^{2}[/itex]
can we shift the spinors in such a manner that we can read off the mass term? Even the simple task of minimizing the potential is confusing. If we take the partial w.r.t. ψ, then
[itex]\frac{\partial\mathcal{L}}{\partial\psi}=\mu\bar{\psi}-\lambda\bar{\psi}(\bar{\psi}\psi)[/itex]
which when set to zero gives
[itex]\bar{\psi}\psi_{min}=\frac{\mu}{\lambda}[/itex].
That's as far as I can really get without confusing myself entirely. Any insight would be much appreciated.
[itex]\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)+\frac{1}{2}\mu^{2}\phi^{2}-\frac{1}{4}\lambda^{2}\phi^{4}[/itex]
and shows that the minimum occurs at
[itex]\phi=\pm\frac{\mu}{\lambda}[/itex]
and thus that the shifted potential picks up a mass term
[itex]m=\sqrt{2}\mu[/itex]
in natural units. Now, I'm wondering if you can do the same thing with a spin-half particle. For example, if we have the spinor Lagrangian density
[itex]\mathcal{L}=i\bar{\psi}\gamma^{\nu}\partial_{\nu}\psi+\mu\bar{\psi}\psi-\frac{1}{2}(\bar{\psi}\psi)^{2}[/itex]
can we shift the spinors in such a manner that we can read off the mass term? Even the simple task of minimizing the potential is confusing. If we take the partial w.r.t. ψ, then
[itex]\frac{\partial\mathcal{L}}{\partial\psi}=\mu\bar{\psi}-\lambda\bar{\psi}(\bar{\psi}\psi)[/itex]
which when set to zero gives
[itex]\bar{\psi}\psi_{min}=\frac{\mu}{\lambda}[/itex].
That's as far as I can really get without confusing myself entirely. Any insight would be much appreciated.