# I Why is quark charge exactly 1/3 or 2/3 the lepton charge?

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1. Oct 8, 2017

### harrylentil

Even with the charge screening effect it is exact. Is there an explanation in the standard model? Can the fact there are 3 families of particles be involved?

2. Oct 8, 2017

### Orodruin

Staff Emeritus
With the given particle content of the Standard Model SU(2) and SU(3) groups, there are not many assignments of a U(1) charge that are free of quantum anomalies. The charge assignments of the hypercharge is one anomaly free option that after spontaneous symmetry breaking results in the quarks having electric charges in multiples of a third of the lepton charge.

The anomaly cancelation occurs generation by generation and so the number of generations is not really relevant for this.

3. Oct 8, 2017

Staff Emeritus
One implication of this is that a world where each family had a different unit charge - say 1.0e, 1.1e and 1.2e, would be completely internally consistent.

4. Oct 8, 2017

### nikkkom

How would muon decay look in such a world?

5. Oct 8, 2017

Staff Emeritus
It wouldn't. I said it would be internally consistent, not that it would match our world.

6. Oct 8, 2017

### Orodruin

Staff Emeritus
Wouldn't it? I have not thought much about it until you raised the issue here, but at face value it would seem to me that the muon would have a charge of 1.1e and the muon neutrino a charge of 0.1e. The decay $\mu \to e + \nu_\mu + \bar \nu_e$ would then still be possible. The different quantum numbers of the families would of course prevent any kind of mixing by forbidding Yukawa couplings between families.

Of course I also agree that it clearly does not match our world.

7. Oct 9, 2017

Staff Emeritus
I was thinking in terms of a scale. If you multiply all charges by a common factor, your theory is still anomaly-free. But you're right, one can also shift the lepton charges, providing one also shifts the quark charges by the correct corresponding amount.

8. Oct 9, 2017

### Orodruin

Staff Emeritus
Can you do that for electric charge? What you can play with should be the hypercharge. For example, the left-handed leptons forming $SU(2)_L$ doublets invariably requires a charged lepton-neutrino-W-coupling. If the neutrino was electrically neutral and the charged lepton had a different charge from the W, this would not happen.

9. Oct 9, 2017

Staff Emeritus
Sure. There is nothing in the SM that sets what the electron's elementary charge is. So a scale works just fine. An offset is trickier, since you need to also alter the relation between weak and electric charges. If you just want your theory to be anomaly-free, as opposed to have SM-like EWK unification, this is also trivial.

10. Oct 9, 2017

### Orodruin

Staff Emeritus
While true for one generation, I do not see how you can give different generations different charges while keeping neutrinos neutral as they have to couple to the Ws and the Ws will have a fixed electric charge. Based on the Gell-Mann-Nishijima formula (which essentially just depends on how you break electroweak symmetry with the Higgs), the electric charge difference between the charged lepton and the neutrino should be the same in all generations (one if you normalise to the charge difference between the upper and lower part of the $SU(2)$ doublets - this should also set the normalisation of hypercharge). The electromagnetic current should be of the form
$$j^\mu_{\rm EM} = \bar{\psi} \left(\tau_3 + \frac{Y}{2}\right) \gamma^\mu \psi.$$
Whatever hypercharge you decide to give to a certain doublet, the charge difference should be one between the eigenstates of the $SU(2)$ generator. All the anomaly cancellations contain the hypercharges to the same powers (within each cancellation) so multiplying all hypercharges simultaneously should still preserve anomaly cancellation. It would therefore seem to me that it is the values of the hypercharges, not the electric charges, that can be scaled freely.

11. Oct 9, 2017

Staff Emeritus
If your only requirement is that your theory be free from anomalies, you can have different generations have different charges. If you require that this theory match reality - such as the difference between the lepton and neutrino charges be exactly the same as the W charge - that's a different kettle of fish. (A theory with different "multipliers" per generation would have very different weak interactions)

The reason this works for electric charges is because I can simultaneously scale the strength of weak isospin and weak hypercharge to make it so.

12. Oct 10, 2017

### Orodruin

Staff Emeritus
But unlike hypercharge you cannot scale the SU(2) coupling constant arbitrarily between generations. It has to be the same coupling constant that appears in the non-linear kinetic gauge term or you break gauge invariance. Of course it does not have to match reality, but I am assuming that the theory is internally consistent and, that SU(2) is broken by a single Higgs so that we can still talk about electric charge. Since the EM U(1) remains unbroken, electric charge must be conserved.

Furthermore I am making the assumption about the particle content of each generation having the same fields in terms of SU(2) doublets and SU(3) triplets. The only way you can get different weak interactions is if you also have several different SU(2) groups, but then your fields will not have the same quantum numbers between generations.

13. Oct 10, 2017

Staff Emeritus
I am out of energy. You are right in absolutely all things, even what I haven't read, and I regret having ever posted in this thread.

14. Oct 11, 2017

### Meir Achuz

Most of the answers to your question were theoretical. A simple phenomenological reason for the quark charge is
that the $\Delta^{++}$ baryon, composed of three identical u quarks has charge +2.