Electroweak Bosons: Examining W1, W2, W3

[Lucifuge Rofocale]

This question is in regards to the electroweak force itself before its symmetry is broken into the weak and electromagnetic force. Let's say hypothetically that the ambient temperature of the universe or of a certain volume of space was hot enough to restore the electroweak symmetry like it was during the electroweak epoch. What are the actual bosons that would mediate this force in this environment. I mean clearly the photon, Z, W+, and W- bosons wouldn't exist right? I read somewhere about a B boson and three other W bosons labeled W1, W2, and W3. Would those be the actual bosons mediating the electroweak force under such conditions or are they merely a mathematical figment. If they do have a tangible existence, are those three W bosons unique from each other or are they the exact same thing labeled to show that three come out of the math?

 

[Orodruin]

The photon A and the Z-boson are linear combinations of the hypercharge boson B and the W3 boson. After symmetry breaking, it is more convenient to use a description in terms of A and Z since the propagators are diagonal in that basis. Before symmetry breaking, all of those bosons have the same mass (0) and it is more convenient to use the interaction basis.

It is not a matter of a type of particle "disappearing" - it is a matter of what basis you use.

 

[Lucifuge Rofocale]

Okay soo the B bosons for example doesn't actually have a tangible existence and is just a mathematical curiosity?

 

[Orodruin]

What do you mean by tangible existence?

I would definitely not call it a mathematical curiosity. It is a matter of what basis you use to span a given vector space.

 

[Lucifuge Rofocale]

I guess what I'm asking is that similar to how the photon and weak force bosons exist currently under the present conditions of the universe, do the B, W1, W2, and W3 bosons "tangibly" exist at the temperatures/conditions that unify EM and weak force into the electroweak force?

Lets say extremely hypothetically that some physicist somehow existed in the universe during the electroweak epoch. Would he actually see the B and three W bosons in his experiments? And if so is there an actual difference between the properties of the W1, W2, and W3 bosons or are they exactly the same entities just labeled differently for mathematical or other reasons.

I apologize for my seemingly tenuous grasp of this concept, but I very much appreciate the help.

 

[PeterDonis]

do the B, W1, W2, and W3 bosons "tangibly" exist

We can't answer this question until you tell us what you mean by "tangibly" exist.

Lets say extremely hypothetically that some physicist somehow existed in the universe during the electroweak epoch. Would he actually see the B and three W bosons in his experiments?

Yes.

is there an actual difference between the properties of the W1, W2, and W3 bosons

If you mean the W1, W2, and W3 (and B) as compared to the W+, W-, and Z (and photon), yes. The W1, W2, and W3 are all massless; the W+, W-, and Z all have mass. The B and photon are both massless, but they are not the same; the difference requires a fair bit of background in quantum field theory.

 

[bhobba]

If you want to understand this stuff get the following book:
https://www.amazon.com/dp/3319192000/?tag=pfamazon01-20

See section 7.2 where the full detail is given.

Now what you do is start with 2 massless spin 1/2 fields. You notice it has SU(2) global symmetry but you want it to be local. To do this we add three massless spin 1 fields Wi to the Lagrangian. But massless spin 1/2 fields have a global U(1) symmetry so we want that to be local so in a similar way to EM so you add B another spin 1 field. The Lagrangian is both locally SU(2) and U(1) invariant. All the fields are massless. But let's say you want the fields massive, at least some them anyway. It turns out if you look at it and you want mass then you will loose SU(2) symmetry - bummer. But you insist. Why - well we have some experimental evidence at least some of these fields have mass but ignoring that and purely as an intellectual exercise how would you go about it? The only ingredient we haven't used so far is a spin 0 field. So let's try this.

You find a Lagrangian with two terms p and λ and a spin 0 field if you want it to have local SU(2) symmetries. The spin 0 field is related to the Higgs. It is in fact two complex spin 0 fields called a doublet - the original spin 1/2 fields are a doublet as well. This has 4 numbers but since it has local SU(2) gauge freedom similar to the EM U(1) gauge changes to them make no difference so those fields can't be measured - no experiment can detect them. It is in that sense we have these 'unreal' fields appear - due to all these symmetries their values can't be measured in experiments. The 4th field though has experimental consequences so is real, in that sense. They are all real, in the sense they are there but have no measurable consequences. This gauge freedom, again similar to EM, means it all depends on what gauge you choose. But regardless some fields don't have physical consequences - they just appear in our equations to ensure symmetries.

During very high temperatures like during the early universe the values of the p term is such the potential energy term from the spin 0 field Lagrangian has a single minimum and particles doing what particles do generally go the minimum.

Now as temperature decreases the value of p changes and the potential energy plot develops the famous Mexican hat potential you have probably read about. The particle naturally wants to still go to a minimum so 'rolls down' so to speak to a minimum on the hat rim. This is the so called spontaneous symmetry breaking you have heard about. Once this happens you have a look at what the Lagrangian now is (see equation 7.97 in the book above) and behold you get the EM Lagrangian and other terms for other particles (the W and such) with terms in front of some that are interpreted as mass terms.

As you can see its quite complex and you, like me, will need to go through it a couple of times. It all fits together elegantly and beautifully and confirms the very striking fact - symmetry is the key thing here. This is all very strange and IMHO a deep deep mystery.

Thanks
Bill