Coupling to gauge fields - charge, doublets, etc.

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Hello,
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.
 

Answers and Replies

  • #2
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Hello,
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?
Your identification of the gauge fields isn't quite correct. It should be:

SU(3): gluons
SU(2): [itex]W_i[/itex], {i=1,2,3}
U(1): B

Where [itex]Z = \cos \theta_W W_3 - \sin \theta_W B[/itex], [itex]A = \cos \theta_W B + \sin \theta_W W_3[/itex], and [itex]\tan \theta_W = \frac{g'}{g}[/itex], where [itex]g'[/itex] and [itex]g[/itex] are the coupling constants of the U(1) and SU(2) forces respectively. This mixing of the neutral gauge bosons is on of the hallmarks of the pattern of electroweak symmetry breaking.

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?
Every group has a trivial representation given simply by the identity. ([itex]T^a=0[/itex] satisfies the algebra, and leads to every element of the group acting as the identity. I said it was trivial, didn't I?)

In physics terms, being in the trivial representation of a group means being unaffected by the symmetry transformation. So, in gauge theory, fields in the trivial representation of a group are uncharged under it. Thus, for instance, the lepton doublets are in the trivial representation of SU(3).

Hopefully, in these terms it's clear that your initial understanding was basically correct. The singlet/doublet/triplet terminology does, in fact, tell you the number of fields being grouped together under which transform into each other under the symmetry transformations in question. Certainly, for this to work as a symmetry, these fields must have the same coupling to the gauge field, or the symmetry transformation would not leave the system's dynamics unchanged.

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...?
Strictly, the up quarks of different color are distinct particles. However, because the SU(3) symmetry is unbroken, the definitions of blue, green, and red are ambiguous. So, we just talk about up quarks with the understanding that they have one unit of color charge.

As for the SU(2) charges, it is exactly this charge that distinguishes the e-neutrino from the left-handed electron. Similarly, it is this which distinguishes the (gauge-basis; but, that's an added complication) left-handed down-quark from the left-handed up-quark of the same color. This charge is sometimes referred to as "weak isospin;" and, the reason that we can distinguish the states related under it is that the symmetry it's associated with is broken. The symmetry breaking also has the effect of violating the conservation of weak isospin.

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.
P&S cover this is the later chapters. Howard Georgi has a book available on his person webpage which is mostly about the electroweak theory and covers a lot of these ideas.
 

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