Deriving the Unitary Gauge of the Higgs Mechanism

In summary, the Higgs mechanism involves writing the Higgs field in polar form and using the unitary gauge to gauge away the exponential factor. This allows for the explicit observation of particle content and unitarity of the S-matrix. The unitary gauge must be exact without any approximations, and it shows that local gauge symmetry cannot be spontaneously broken. This is proven by Elitzur's theorem, which states that there are no massless Goldstone bosons, a necessary occurrence for real spontaneous symmetry breaking.
  • #1

ohs

3
0
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 ?

Thanks in advance

Ohs
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #6
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).
 

What is the Higgs Mechanism?

The Higgs Mechanism is a theory in particle physics that explains how particles acquire mass. It proposes the existence of a field called the Higgs field, and a corresponding particle called the Higgs boson, which interact with other particles to give them mass.

What is the Unitary Gauge of the Higgs Mechanism?

The Unitary Gauge is a mathematical representation of the Higgs Mechanism that simplifies the equations and makes it easier to understand and work with. It is a specific gauge choice that eliminates the imaginary component of the Higgs field, resulting in a real scalar field.

Why is the Unitary Gauge important in the Higgs Mechanism?

The Unitary Gauge is important because it helps to make the Higgs Mechanism more intuitive and easier to calculate. It also allows for the prediction of measurable quantities, such as the mass of the Higgs boson, which was later confirmed by experimental evidence.

How is the Unitary Gauge derived?

The Unitary Gauge is derived by applying a gauge transformation to the Higgs field, which transforms it into a real scalar field. This transformation simplifies the equations and removes the imaginary component, making it easier to work with and understand.

What are the implications of the Unitary Gauge for the Higgs Mechanism?

The Unitary Gauge has several implications for the Higgs Mechanism. It helps to explain the origin of mass in particles, it allows for the prediction of measurable quantities, and it provides a simpler mathematical framework for understanding the Higgs field and its interactions with other particles.

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