- #1

Aethaeon

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I'm trying to understand the standard model, and I'm confused in a few places. Correct me please, if I seem confused somewhere. I'll give my understanding first, and then a few questions. I'm certain I have a couple of things not straight in my head.

Alright, so the standard model has three gauge fields,

SU(3) gluons.

SU(2) W/Z

U(1) photon

The covariant derivative for, for concreteness, the SU(3) gauge field are

[tex]D_i_j = \delta_i_j\partial-igA^aT_i_j^a[/tex]

with [itex]T^a[/itex] some generators for SU(3).

There are 8 gauge bosons, and their fields are the 8 [itex]A^a[/itex] vector fields.

Now suppose I want to couple these to a little cluster of fermion fields, [itex]\Psi_i[/itex]. For concreteness, say I want 3 of them.

First I find a representation of SU(3), [itex]T^a[/itex] that has dimension 3.

Next, I try to write down a term for my lagrangian. I want to do this in 4 dimensions, and I want my theory to be renormalizable, so I want something of mass dimension <=4. The only such lorentz invariant terms are

[tex]\overline{\Psi_i}\gamma^0D_i_j\Psi_j[/tex]

Alright...

**Question 0**: Was that understandable? Were there points where I was obviously confused?

**Question 1**: What does singlet/double/triplet refer to? I initially thought that it referred to the number of fields in the little cluster, or the dimension of the representation.. But Wikipedia mentions SU(3) singlets, and SU(3) doesn't have a dimension 1 representation (unless [itex]T^a=1[/itex] counts? But it doesn't satisfy the SU(3) lie algebra..) More recently I'm coming to believe that it's a way of saying that two or three different clusters have the same coupling constant for their interactions with a particular gauge field. Am I sort of on the right track?

**Question 2**: Um, so if I'm not mistaken, my little cluster of fields represent the different possible charges of the associated fermion.. So for example, the red up-quark, blue up-quark, and green up-quark form a little 3-dimensional SU(3) cluster. Is that correct? From my notes, e-neutrinos and left-handed e's carry an SU(2) charge. Are there two different possible weak charges for each of e-neutrinos and e's? Or...?

**Question 3**: Is there a nice, clear reference for this somewhere? Srednicki seems to be very terse on the subject, and I'm having trouble finding anything relating to this in P&S.. Should I hunt down a copy of Weinberg?Thanks so much for reading! - I really appreciate any help understanding this.