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

In summary, 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?
  • #1
AuraCrystal
68
0
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)?
 
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  • #2
AuraCrystal said:
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)?
 
  • #3
The left handed electron-electron neutrino doublet and the right-handed electron singlet? (And similarly for the other leptons.)
 
  • #4
So, are you saying quarks don't transform in su(2)?
 
  • #5
Haha oops! I forgot about the quark sector. I think they do transform in SU(2) though.
 
  • #6
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
[tex]
Q = T_{3} + \frac{Y_{W}}{2}
[/tex]
 
  • #7
OK.

So...what is B?

And also, why do we enlarge the gauge group?
 
  • #8
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.
 
  • #9
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?
 
  • #10
AuraCrystal said:
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
 
  • #11
AuraCrystal said:
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.
 
  • #12
OK, I think I understand now.

Thank you! :)
 

What is the electroweak theory?

The electroweak theory is a theory in physics that explains the relationship between two of the four fundamental forces of nature - the electromagnetic force and the weak nuclear force. It is a unified theory that combines the theories of electromagnetism and weak interactions.

Why do we need to enlarge the gauge group of the electroweak theory?

The original electroweak theory, proposed in the 1960s, only had a gauge group (a mathematical concept that describes the symmetry of a system) of SU(2) x U(1). However, this theory was incomplete as it did not account for the observed masses of certain particles. Enlarging the gauge group allows for the inclusion of additional fields, such as the Higgs field, which gives particles mass.

What are the advantages of enlarging the gauge group of the electroweak theory?

Enlarging the gauge group allows for a more complete and accurate description of the interactions between particles. It also helps to unify the laws of nature and provides a deeper understanding of the fundamental forces of nature.

What are the consequences of enlarging the gauge group of the electroweak theory?

One consequence is the prediction of the existence of additional particles, such as the Higgs boson, which was confirmed by experiments in 2012. This also leads to the prediction of new phenomena, such as electroweak symmetry breaking, which is responsible for the mass of particles.

What ongoing research is being done on the enlarged gauge group of the electroweak theory?

Scientists continue to study the electroweak theory and its enlarged gauge group in order to better understand the fundamental nature of the universe. Ongoing research includes experiments at the Large Hadron Collider (LHC) and theoretical studies to further refine and improve the theory.

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