We do we enlarge the gauge group of the electroweak theory?

[AuraCrystal]

Hello,

I've been reading about the weak interaction.

Basically, the weak interaction couples to particles that are left-handed, and we introduce the electron-electron neutrino as a (left-handed) SU(2) doublet. So, the gauge bosons (W⁺, W^-, and Z) transform SU(2) triplet. Am I right?

So...my question is...why do we enlarge the gauge group to SU(2)×U(1)? And what transforms in U(1)?

 

[Dickfore]

So...my question is...why do we enlarge the gauge group to SU(2)×U(1)? And what transforms in U(1)?

What transforms in su(2)?

 

[AuraCrystal]

The left handed electron-electron neutrino doublet and the right-handed electron singlet? (And similarly for the other leptons.)

 

[Dickfore]

So, are you saying quarks don't transform in su(2)?

 

[AuraCrystal]

Haha oops! I forgot about the quark sector. I think they do transform in SU(2) though.

 

[Dickfore]

The name of the unification "electroweak" reveals the significance of the gauge groups. The weak isospin tells you in what representation of the su(2) (singlet, doublet, triplet) does the particle transform (together with its partners).

u(1) is an abelian group with only one generator. it has only one-dimensional representations. The weak hypercharge tells you the representation in which the particle transforms.

Due to electroweak symmetry breaking, T₃ and B mix to give Z⁰, and the photon , which, as a Nambu-Goldstone boson, remains massless. The charge is a combination of the corresponding quantum numbers
<br /> Q = T_{3} + \frac{Y_{W}}{2}<br />

 

[AuraCrystal]

OK.

So...what is B?

And also, why do we enlarge the gauge group?

 

[Dickfore]

B is the boson coupling to the weak hypercharge (the u(1)). We do not enlarge the gauge group. It is a proposed model (Weinberg, Salam, Glashow) for unifying the weak and electromagnetic interaction. It is incorporated in the current Standard Model of elementary particles and fundamental interactions.

 

[AuraCrystal]

Oh I see. So we use it because it works?

So the particle that couples to the U(1) is not the photon but another boson? And when mixed, it gives the photon and the Z boson?

Does that have any relation to the fact that U(1) corresponds to weak hypercharge and not electric charge?

 

[jtbell]

So the particle that couples to the U(1) is not the photon but another boson? And when mixed, it gives the photon and the Z boson?

Correct. The "original" U(1) gauge boson is usually called the B⁰. The three "original" SU(2) gauge bosons are the W^-, W⁰, and W⁺. The B⁰ and W⁰ mix to produce the photon and Z⁰.

http://en.wikipedia.org/wiki/Electroweak_interaction

 

[Dickfore]

So the particle that couples to the U(1) is not the photon but another boson? And when mixed, it gives the photon and the Z boson?

Does that have any relation to the fact that U(1) corresponds to weak hypercharge and not electric charge?

Yes, it has everything to do with it. And, mind you, it's not simply a change in name, but, because of this mixing, we can't observe t'Hooft-Polyakov magnetic monopoles due to topological defects that break a simple su(2) symmetry.


BTW, this alternative theory was also considered a viable candidate for electroweak unification. Unfortunately, it does not predict the Z⁰ boson.

 

[AuraCrystal]

OK, I think I understand now.

Thank you! :)