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I'm trying to understand the way that the Higgs Mechanism is applied in the context of a U(1) symmetry breaking scenario, meaning that I have a Higgs complex field [itex]\phi=e^{i\xi}\frac{\left(\rho+v\right)}{\sqrt{2}}

[/itex] and want to gauge out the [itex]\xi[/itex] field that is causing my off-diagonal term, in normal symmetry breaking. I present the following transformation rules that hold in order to preserve local gauge invariance in Spontaneous Symmetry breaking non 0 vev for [itex]\rho[/itex] :

$$

\begin{cases}

\phi\rightarrow e^{i\theta}\phi\\

A_{\mu}\rightarrow A_{\mu}-\frac{1}{q}\partial_{\mu}\theta\\

D_{\mu}=\partial_{\mu}+iqA_{\mu}

\end{cases}

$$

As I understand, the Higgs gauge fixing mechanism is used to specify the transformations that gauge [itex]\xi[/itex] away. The idea is that we want to look for the angle that gives us a Higgs field with one real degree of freedom, as in

$$

\phi\rightarrow e^{i\theta}\phi=e^{i\theta}e^{i\xi}\frac{\left(\rho+v\right)}{\sqrt{2}}=e^{i\theta^{'}}\frac{\left(\rho+v\right)}{\sqrt{2}}

$$

so

$$\begin{cases}

\phi\rightarrow e^{i\theta}\phi=e^{i\theta}e^{i\xi}\frac{\left(\rho+v\right)}{\sqrt{2}}=e^{i\theta^{'}}\frac{\left(\rho+v\right)}{\sqrt{2}}\\

A_{\mu}\rightarrow A_{\mu}-\frac{1}{q}\partial_{\mu}\theta^{'}

\end{cases}

$$

where $$\theta^{'}=\theta+\xi$$. I omit some factors of v on the exponential for the time being. That's what I see my books doing. For $$\theta^{'}=0$$ this becomes

$$

\begin{cases}

\phi\rightarrow\frac{\left(\rho+v\right)}{\sqrt{2}}\\

A_{\mu}\rightarrow A_{\mu}

\end{cases}

$$

and the rest is the derived desired interactions and terms in general. I note that this does not preserve local gauge invariance because

$$\begin{cases}

\phi\rightarrow e^{i\theta^{'}}\frac{\left(\rho+v\right)}{\sqrt{2}}\neq e^{i\theta^{'}}\phi\\

A_{\mu}\rightarrow A_{\mu}-\frac{1}{q}\partial_{\mu}\theta^{'}

\end{cases}

$$

So is this transformation the one that we do or have I wronged somewhere and it can be done correctly via a legit gauge transformation?

My problem is that invariance of U(1) transformations means that you end up with the same field in the end, the gradient of the exponent in [itex]U\phi [/itex] is necessary to cancel with the corresponding term of the field 4-vector transformation rule. Any light shed is welcome!

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# Is the Higgs mechanism a gauge transformation or not?

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