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'Extra' terms in Abelian Higgs model

  1. Mar 7, 2009 #1
    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...

  2. jcsd
  3. Mar 7, 2009 #2
    You did it right, and those 4 terms cancel one another.

    [tex]|D_{\mu}\phi|^2 = \frac{1}{2} [(\partial_{\mu} - ieA_{\mu})(v+H)] [(\partial^{\mu} + ieA^{\mu})(v+H)] = \frac{1}{2}[\partial_{\mu}H\partial^{\mu}H + e^2A_{\mu}A^{\mu}(v+H)^2 + ieA^{\mu}(v+H)\partial_{\mu}H - ieA_{\mu}(v+H)\partial^{\mu}H][/tex]

    Because of the summation over [tex]\mu[/tex], we have for the third term [tex]ieA^{\mu}(v+H)\partial_{\mu}H = ieA_{\mu}(v+H)\partial^{\mu}H[/tex], cancelling the last term.

  4. Mar 8, 2009 #3
    Ah yes of course, thank you.
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