Deriving the Unitary Gauge of the Higgs Mechanism

[ohs]

Dear @ll,

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

Φ = (v + η + iξ) = (v + η)e^i(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 ?

Thanks in advance

Ohs

 

[mfb]

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.

 

[ohs]

Thank you very much for your reply.
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]

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.

 

[ohs]

In general: yes.
But i am feeling, that the unitary gauge must be exact without neglections.
The context: http://ajbell.web.cern.ch/ajbell/Documents/eBooks/Quarks & Leptons.pdf, eq. 14.56, page 327

 

[vanhees71]

The trick with the unitary gauge is to write the Higgs field in the polar form and then observing that you can gauge away the exponential factor. In this way you see the particle content and unitarity of the S-matrix explicitly, i.e., three of the Higgs field degrees of freedom (which would be the massless Goldstone modes if the symmetry was global) are absorbed into the gauge fields, providing the additional field degree of freedom necessary to make them massive. This must be so, because massless vector fields have two helicity degrees of freedom, massive vector fields have three spin degrees of freedom.

This also shows that a local gauge symmetry cannot be spontaneously broken, as proven by Elitzur (although that's usually the sloppy jargon used in all textbooks and papers ;-)): There are no massless Goldstone bosons, which necessarily should occur if there was real spontaneous symmetry breaking, which occurs when a global symmetry is spontaneously broken (e.g., the pions are the approximate Goldstone modes of spontaneous chiral-symmetry breaking in the light-quark sector of QCD).