# 'Extra' terms in Abelian Higgs model

#### bomanfishwow

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:

$$\mathcal{L} = |D_\mu\Phi|^2 - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} - V(\Phi)$$

With the covariant derivative, field strength tensor and potential given by:

$$D_\mu = \delta_\mu - ig_\mu$$,
$$F_{\mu\nu} = \delta_\nu A_\mu - \delta_\mu A_\nu$$,
$$V(\Phi) = \lambda|\bar{\Phi}\Phi|^2 - \mu^2\bar{\Phi}\Phi$$.

I'm working in the unitary gauge, such that $$\Phi$$ is given by:

$$\Phi = \frac{1}{\sqrt{2}}\left(v + H\right)$$.

Taking the expanded potential after symmetry breaking, and plugging into $$|D_\mu\Phi|^2$$ like:

$$|D_\mu\Phi|^2 = D_\mu\Phi^*D^\mu\Phi = \frac{1}{2}\left[\left(\delta_\mu +igA_\mu\right)\left(v+H\right)\left(\delta^\mu - igA^\mu\right)\left(v+H\right)\right]$$

yields the expected interaction and mass terms. Some of the 'extra' terms trivially cancel as they contain derivatives of constants such as $$\delta_\mu v$$. However, there are some extra terms which I don't see mentioned in the standard texts:

$$-ig\left[\delta_\mu H\right] HA^\mu$$
$$-igv\delta_\mu HA^\mu$$
$$igHA_\mu\delta^\mu H$$
$$igvA_\mu\delta^\mu H$$.

Can anyone suggest a) if I've done something wrong b) if these terms also disappear c) Something else...

Thanks!

#### Buzzy^

You did it right, and those 4 terms cancel one another.

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

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

Cheers,
Sebas

#### bomanfishwow

Ah yes of course, thank you.

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving