- 27

- 0

[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!