Standard model + symmetry questions

[aoner]

Homework Statement

1) Which U(1), SU(2) and SU(3) gauge invariances are implemented in nature according to the Standard Model? What are the related quantum numbers?

2) The SU(2) symmetry is referred to as a non-abelian symmetry. What does this imply for the interactions between the force carriers?

3) Give the expression for a SU(3) local gauge transformation acting on a quark spinor triplet.

2. The attempt at a solution

1) Gauge symmetries in the SM, used describe three of the fundamental interactions, are based of the SU(3)xSU(2)xU(1) group. Roughly speaking the symmetries of the SU(3) group describe the stronge force, the SU(2) group the weak interaction and U(1) the electromagnetic force.

2) The electroweak model breaks parity maximally. All its fermions are chiral Weyl fermions, which means that the charged weak gauge bosons only couple to left-handed quarks and leptons (making this a non-abelian symmetry).

3) ?

 

[vanhees71]

Ad 1) Think again about the SU(2) x U(1) in the weak sector. Is this (!) U(1) really the em. U(1)?

Ad 2) What's the point about non-Abelian gauge invariance compared to an Abelian one? Think about the interactions of the non-Abelian gauge bosons in contrast to that of Abelian ones!

Ad 3) How does the color gauge group act on the quark fields? It's pretty "fundamental" ;-)).

 

[aoner]

Okay, thank you! Another shot, still not perfect though:

1) U(1) -> Photon (QED). Quantum number Q
SU(2) - > W⁺, W^- and Z-boson (electroweak interaction), Quantum number Y
SU(3) -> 8 gluons (QCD) (Gell-Mann Matrices), Quantum number T₃

2) The force carriers self-interact so W^-W⁺ $\otimes$ Z

3) ψ = Uψ where ψ is the quark field, a dynamical function of space-time, in the fundamental representation of SU(3) and U the gell-mann matrices.

 

[vanhees71]

The electroweak part of (1) is not yet correct. Note that SU(2) X U(1) is the symmetry group before "spontaneous symmetry breaking" of the gauge group. It's spontaneously broken to another (1) U(1), which latter is the em. U(1) associated with electric charge! The former U(1) is the group associated with weak hypercharge, Y!

 

[aoner]

Thanks!