# Weak isospin and types of weak charge

My understanding is that QCD has three color charges that are conserved as a result of global SU(3) invariance. What about SU(2) weak? Does it have two types of charges? What I'm getting at is:

U(1) --> 1 type of charge

SU(2) --> ?

SU(3) --> 3 types of charge

Does SU(2) have two types? If not, what is the relation between SU(N) invariance and the number of charge types?

Idea: Maybe both I and I_3 (weak isospin and its third component) are conserved before electroweak symmetry breaking? Is that true? If so, then that would answer my question.

• Actually each of the symmetry generators commutes with the Hamiltonian and gives a conserved charge. For an SU(N) group the number of generators is $N^2 - 1$. The total number of generators in the SM is $1 + (2^2 - 1) + (3^2 -1) = 12$. Commented Mar 14, 2013 at 13:41
• @Michael Brown, I am aware that local invariance of SU(N) leads to N^2-1 gauge fields, however I think this is a different question from what I am asking. SU(3) has 8 gauge fields (quantized becomes 8 gluons), but 3 colors. 3 conserved charges, due to global invariance of SU(3), not local. Commented Mar 14, 2013 at 13:54
• Look up Noether's theorem again. There is a conserved current $j_\mu^A$ for every generator $T^A$ of the group, i.e. the currents live in the adjoint rep. Therefore there are $N^2 - 1$ conserved charges. The number of components in the fundamental rep is a different matter. For SU(N) that is $N$. Commented Mar 14, 2013 at 14:45
• @Michael Brown, I thought that color charge was conserved, and that SU(3) implied three such colors. Am I mistaken that the "3" colors are related to the "3" in SU(3), and that their conservation is related to Noether's theorem? Any help would be appreciated. Commented Mar 14, 2013 at 15:24
• The "3" is the same 3 as in the SU(3). There are indeed three colours. I didn't say gluons are conserved. But there are 8 conserved quantities in an SU(3) theory, which are the generators of the symmetry group. If it helps you can think of these formally as colour-anticolour pairs (look up 't Hooft double line formalism) since the charges live in the adjoint rep which is isomorphic to fundamental $\otimes$ antifundamental. There is always a conserved quantity associated to any continuous symmetry via Noether's theorem. # of symmetries = # of conserved quantities. Commented Mar 14, 2013 at 16:39

## 1 Answer

@Michael Brown is right. The SM has 12 exactly conserved currents.

• All local invariances, a fortiori also imply global invariances, if you ignore (for the sake of argument) the spacetime variability of transformation parameters/angles. So SU(3) has 8, not 3 conserved charges, RG, BG, .... The group has 8 generators. Likewise, SU(2) has 3, not 2 conserved currents: you know this from spin, where each of the 3 projections of a rotationally invariant system is conserved. U(1) has one conserved charge.

• SSB does not affect the number of conserved currents, in sharp contrast to explicit symmetry breaking: The currents are still conserved, except they have a special nonlinear form (their leading term is linear, not bilinear in the fields, so the corresponding charges shift the fields "nonlinearly"). The symmetry is hidden, and much less apparent, but it is still there, which is why these symmetries are so powerful: they control systematically the divergences of the corresponding QFT. (Actually, though, the 3 charges corresponding to the 3 broken generators are ill-defined/divergent themselves, although their corresponding currents are conserved: the symmetry is still there.)

• There are further approximate symmetries in the SM, meaning that their charges are violated by a "small" amount (a technical characterization), or even quantum anomalies (collective quantum action of the Dirac sea of fermions coupling chirally to them).

Further see 149324