- #1
bomanfishwow
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I'm taking 5 mins (hours) during some down-time to remind myself of some theory. Taking a simple Abelian Higgs model, where the Lagrangian is given by:
[tex]\mathcal{L} = |D_\mu\Phi|^2 - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} - V(\Phi)[/tex]
With the covariant derivative, field strength tensor and potential given by:
[tex]D_\mu = \delta_\mu - ig_\mu[/tex],
[tex]F_{\mu\nu} = \delta_\nu A_\mu - \delta_\mu A_\nu[/tex],
[tex]V(\Phi) = \lambda|\bar{\Phi}\Phi|^2 - \mu^2\bar{\Phi}\Phi[/tex].
I'm working in the unitary gauge, such that [tex]\Phi[/tex] is given by:
[tex]\Phi = \frac{1}{\sqrt{2}}\left(v + H\right)[/tex].
Taking the expanded potential after symmetry breaking, and plugging into [tex]|D_\mu\Phi|^2[/tex] like:
[tex]|D_\mu\Phi|^2 = D_\mu\Phi^*D^\mu\Phi = \frac{1}{2}\left[\left(\delta_\mu +igA_\mu\right)\left(v+H\right)\left(\delta^\mu - igA^\mu\right)\left(v+H\right)\right][/tex]
yields the expected interaction and mass terms. Some of the 'extra' terms trivially cancel as they contain derivatives of constants such as [tex]\delta_\mu v[/tex]. However, there are some extra terms which I don't see mentioned in the standard texts:
[tex]-ig\left[\delta_\mu H\right] HA^\mu[/tex]
[tex]-igv\delta_\mu HA^\mu[/tex]
[tex]igHA_\mu\delta^\mu H[/tex]
[tex]igvA_\mu\delta^\mu H[/tex].
Can anyone suggest a) if I've done something wrong b) if these terms also disappear c) Something else...
Thanks!
[tex]\mathcal{L} = |D_\mu\Phi|^2 - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} - V(\Phi)[/tex]
With the covariant derivative, field strength tensor and potential given by:
[tex]D_\mu = \delta_\mu - ig_\mu[/tex],
[tex]F_{\mu\nu} = \delta_\nu A_\mu - \delta_\mu A_\nu[/tex],
[tex]V(\Phi) = \lambda|\bar{\Phi}\Phi|^2 - \mu^2\bar{\Phi}\Phi[/tex].
I'm working in the unitary gauge, such that [tex]\Phi[/tex] is given by:
[tex]\Phi = \frac{1}{\sqrt{2}}\left(v + H\right)[/tex].
Taking the expanded potential after symmetry breaking, and plugging into [tex]|D_\mu\Phi|^2[/tex] like:
[tex]|D_\mu\Phi|^2 = D_\mu\Phi^*D^\mu\Phi = \frac{1}{2}\left[\left(\delta_\mu +igA_\mu\right)\left(v+H\right)\left(\delta^\mu - igA^\mu\right)\left(v+H\right)\right][/tex]
yields the expected interaction and mass terms. Some of the 'extra' terms trivially cancel as they contain derivatives of constants such as [tex]\delta_\mu v[/tex]. However, there are some extra terms which I don't see mentioned in the standard texts:
[tex]-ig\left[\delta_\mu H\right] HA^\mu[/tex]
[tex]-igv\delta_\mu HA^\mu[/tex]
[tex]igHA_\mu\delta^\mu H[/tex]
[tex]igvA_\mu\delta^\mu H[/tex].
Can anyone suggest a) if I've done something wrong b) if these terms also disappear c) Something else...
Thanks!