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In many Quantum Field Theory text books they says these about Spontaneous Breaking and Higgs mechanism like this

In unitary gauge, the Goldstone Bosons are eaten by $W^\pm$ and Z and become their longitude degrees of freedom.

I don't understand how do we know that Goldstone Boson actually become their longitude degrees of freedom. How do we identify the longitude degrees of freedom for a boson mathematically? What kind of term can be viewed as longitude degrees of freedom? I am always confused by these unclear expressions in text books.

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A massless spin-1 gauge boson (such as a photon) has two degrees of freedom (m=+1 and m=-1) which are called "transverse" (Think of an electromagnetic wave.) A massive spin-1 gauge boson has an additional "longitudinal" degree of freedom (m=0). Saying that the W+, W- and Z bosons get a longitudinal degree of freedom is just saying that they acquire a mass term in the Lagrangian after symmetry breaking.

Physical systems have a fixed number of degrees of freedom. So where does this extra degree of freedom "come from"? From the Goldstone bosons, which can be made to disappear from the Lagrangian by a gauge transformation of the W+, W-, and Z. This allows the total number of degrees of freedom to remain the same. This is what "The gauge bosons eat the Goldstone bosons" means.

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We can identify the longitudinal degrees of freedom by looking at the angular momenta (often composed primarily of spin, although there is always a potential orbital contribution) by of the particles that a short-lived gauge boson interacts with. This can mean the particles produced when the intermediate vector boson decays, or it can be particles that are produced at the same time as the gauge boson. This is just like how we know, from looking at the angular momenta of other particles involved in their interactions, that neutrinos are always produced with left chirality. So it is (comparatively) straightforward to pick out events in which the intermediate vector boson involved was longitudinally polarized (meaning, it has zero angular momentum along the direction of its motion).

That these longitudinal polarization states are "eaten" elements of the Higgs multiplet can be demonstrated by looking at the strength with which the longitudinal states interact. The couplings of the longitudinal polarizations to other fields are determined not primarily by the gauge boson couplings, but mostly by the Higgs Yukawa couplings. That makes sense if the mode is really a Higgs component that has been "eaten." So you can confirm the identity of the longitudinal modes by comparing the rates for processes in which they can appear as intermediates to other observables (such as the fermion masses) that depend on the Yukawa couplings.

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