Higgs mechanism

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Main Question or Discussion Point

Dear @ll,

the central point (for the unitary gauge) in the higgs-mechanism is the equality

Φ = (v + η + iξ) = (v + η)ei(ξ/v) (see for example Halzen, Martin: Quarks and Leptons, eq. 14.56)

Φ = complex scalar Field
v = vacuum that breaks the symmetry spontaneously
η,ξ = shifted fields.

I am unhappy that i did not found a derivation of this equation. Is it a triviality? Some people say that this is an exact equation (polar representation of the field ?). Other (Halzen,Martin) say that it is an approximation up to the lowest order only.

Can somebody help me and post a derivation of this equation ?

Ohs

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mfb
Mentor
Taylor approximation to first order: $(v + η)e^{i(ξ/v)} \approx (v + η)(1+iξ/v) = v + η + iξ + iηξ/v$. I guess the last term is neglected.

It is certainly not exact in general.

I am glad that this point is now clear for me: it is only an approximation!!

But what about the physics of this approximation?
Does it mean, that the unitary gauge ist an approximation also ?
And that the Goldstone-particles does vanish from the lagrangian not exactly but approximated only?

Questions over questions.

mfb
Mentor
I don't have the book so I don't know the context, but typically you take the limit of shifts->0 at some point, in that case higher orders do not matter and the result is exactly true.

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vanhees71