# Could quarks and leptons mix if they carried flavor charges?

If quarks and leptons carried flavor charges that differed across generations (as they do in some theories), then could mixing take place?

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It can happen if this flavor charge is broken at high enough energies. The symmetry has to be broken, otherwise we would either see long-range gauge forces different between different families, or else a confinement of the family charge at some accessible energies.

The breaking of family symmetry will allow the families to mix with each other by interacting with the Higgs field breaking the symmetry. However, this would mean that to mix different families using the Higgs would require both the standard model Higgs and the new Higgs, and so it would raise the dimension of the term. This would suppress the family mixing naturally by the ratio of the GUT scale to the new Higgs scale, as compared to the Higgs interaction within a family. This can be a natural explanation of the mysterious fact that the off diagonal CKM matrix elements are much smaller than they could be, naturally.

The easiest way to make this happen is to have a U(1) charge which is different for the different families. Assign one family charge 0 and the other two charge -1 and 1. Then assuming there is a Higgs field $G$ for this U(1), with mass scale $M_G$, around the GUT scale, the interaction of the standard model Higgs field H with fields in the standard model is suppressed when it requires using G by the ratio $M_g/M_{pl}$. The term in the low energy theory giving mass to the quarks would be

$$H G Q_i q_j {M_G\over M_{pl}}$$

Where $Q_i$ is a quark SU(2) doublet and $q_j$ is a quark SU(2) singlet in a different family differing by one unit of charge. For families differing by two units of charge, you would need two G fields, and another power of ${M_G\over M_{pl}}$. The interactions within a given family would be unchanged (if you gave all the members of the family the appropriate cancelling charges).

This mechanism is like a low-energy effective theory caricature of string models proposed about a decade ago by Vafa, to explain the strange suppression of inter-family mixing. Vafa's models are based on gauge theories from branes, where the breaking can allow supression of certain off diagonal mixings based on the distances of the branes. But in the low-energy theory, this is equivalent to a Higgs mechanism on the gauge theory of the branes, and the suppressions are essentially just by the mechanism you suggest.

It might be worthwhile to see if there is a simple charge assignment which can reproduce qualitative patterns in the standard model from an idea of this sort. The biggest clue is that unlike the quark mixing, the inter-family neutrino mixing doesn't seem to be suppressed at all.

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Dear Ron,thanx for the answer. – Somdatta Bhattacharya Jul 14 '12 at 11:50