Higgs mechanism and Angular momentum

[fermi]

There is an aspect of the Higgs mechanism I am troubled about. This applies to any theory with Higgs, but for sake of definiteness I will restrict the discussion to the Standard Model. The (unbroken) theory starts with a complex Higgs doublet, which corresponds to four degrees of freedom. Then Higgs develops a vacuum expectation value, and through the usual well known machinery three of these four degrees of freedom are absorbed in the longitudinal components of the gauge bosons. The two (oppositely) charged Higgs get absorbed in W+ and W-, and one combination of the neutral Higgs gets absorbed in Z. Before the symmetry breakdown, the gauge bosons were massless, and they had only two physical (transverse) components. After the symmetry breakdown, they acquire a new degree of freedom from the Higgs field, which shows up as the longitudinal component.

Now, why does that not break the conservation of Angular momentum J? To be sure, the third component of the angular momentum, J₃ is OK in this case: J₃ = 0, and it is conserved. But J² appears not to be conserved. How did a scalar degree of freedom in Higgs became a part of the three degrees of freedom in a spin-1 particle without violating J² conservation? I had believed all these years that the longitudinal component of a spin-1 boson is not equivalent to a scalar. Is this belief wrong? Is it not possible to distinguish it from a scalar? And yet in Higgs mechanism it appears to have come from a scalar.

 

[Vanadium 50]

What exactly is J² operating on? (Before and after)

 

[fermi]

What exactly is J² operating on? (Before and after)

Good question, and I am not sure what the answer is (perhaps that's the problem.) There are three possibilities: (1) The vacuum (of course the vacuum is different before and after the symmetry breakdown.) (2) One particle states of the theory (and these are different too before and after.) (3) Something more complicated (I don't know what that might be.)

(1) sounds trivial, the vacuum must be in zero angular momentum state before and after both: <0|J²|0>=0. If this were not true, I would have to rethink what spontaneous symmetry breakdown means.

(2) is what I have been thinking, I suppose. For example <fi|J²|fi>=0, and <W|J²|W>=j(j+1)<W|W>=2<W|W>, and <e_L|J²|e_L>= (3/4)<e_L|e_L>, etc...

 

[fermi]

Hmmm? Did I ask a very hard and complicated question? Or a rather boring one? Is the question unclear? (I guess it was to some degree, and I had to make a clarification for Vanadium 50.) Never before I had to wait so many days for an answer to a question I asked. I either stumped everybody, or I put everybody to sleep with this question.

I still hope to have some answers soon. At least a discussion... Thank you all who try.

 

[Vanadium 50]

I think you are thinking about "before and after" the symmetry breaking. That's not looking at a single physical system as it evolves in time, so I don't think conservation laws apply. Instead, it's looking at "before and after" a particular step in the derivation.

 

[fermi]

I think you are thinking about "before and after" the symmetry breaking. That's not looking at a single physical system as it evolves in time, so I don't think conservation laws apply. Instead, it's looking at "before and after" a particular step in the derivation.

I think conservation laws associated with unbroken symmetries should apply at all times, even while the vacuum is in the process of changing and settling to a different one. I find it rather unsatisfactory to say that angular momentum is conserved "before" and "after" separately, but not during the process of symmetry breakdown which must have taken place some 13 billion years ago.

 

[Vanadium 50]

But you haven't shown any violation of the conservation of angular momentum yet. You keep asserting it, but you haven't shown any.

 

[hamster143]

It is an interesting question, but I don't think that there really is a problem.

First of all, I'm not sure if J^2 is really conserved in the expanding universe. Expansion (and GR effects in general) can do all sorts of crazy things, e.g. particle creation.

Symmetry breaking can occur even without expansion. Your concern seems to be along the line that J^2 was zero because the system was in the vacuum state, and now it's not zero because we have spin-1 particles all around. But the initial state was not vacuum, so the expectation value of J^2 was not zero to begin with.

 

[Vanadium 50]

But Hamster (and maybe fermi), it's simply not true that in the past EWK symmetry was unbroken. "Before" refers to a step in the derivation, not at some historical time.

 

[hamster143]

But Hamster (and maybe fermi), it's simply not true that in the past EWK symmetry was unbroken. "Before" refers to a step in the derivation, not at some historical time.

Yes, it was unbroken. Electroweak symmetry breaking is analogous to a phase transition, which occurs when the universe is cooled to a certain critical temperature.

 

[Vanadium 50]

Not exactly - what that means is that at high enough temperature either the broken or unbroken symmetric representations give the same predictions.

But you still haven't shown a system where you have the same system before-and-after where J is not conserved.

 

[hamster143]

Not exactly - what that means is that at high enough temperature either the broken or unbroken symmetric representations give the same predictions.

Well, it *IS* the same field and the same Lagrangian, after all. The difference is that before breaking, the expectation value of the Higgs is zero everywhere, and after breaking it's not.