The Standard Model

The Standard Model (SM) of Leptons and Quarks is the most successful physics model of all in terms of testing its concepts. The model has six leptons in pairs in three lepton families and six quarks in pairs in three quark families, with the quarks in three different colors. Aesthetically, the matching of three to three is pleasing. Mathematically, this matching of numbers of lepton and quark families cancels out infinities in quantum field theory calculations, such as the infinities that would arise from the famous triangle anomaly. However important this family matching may be, can you provide a fundamental physics argument for the specific matching of the first lepton family to the first quark family, of the second lepton family to the second quark family, and so on?

Answer

As far as we know, no such definitive argument for matching specific families exists in the Standard Model of Leptons and Quarks and their interactions. As long as six leptons cancel out the anomalies of the six quarks, for example, all is well! Indeed, one can use the second family of quarks to cancel the anomaly contributions from the first family of leptons, the third family of quarks to cancel the second family of leptons, and the first family of quarks to cancel the third family of leptons. In fact, any permutation of the traditional lineup of cancellations would succeed. This ambiguity in the cancellations probably indicates that the Standard Model as understood is incomplete. One would expect the traditional scheme, but the conceptual understanding provided by the Standard Model does not dictate uniqueness.

One of us (F. P.) has proposed an interesting mathematical argument for matching lepton families to quark families based on correlations among finite rotational subgroups of the Standard Model gauge group for the leptons and quarks. In this scheme, each lepton family and each quark family is in a unique subgroup, and the one-to-one correlations are dictated by the

mathematics. Although the proposed scheme successfully predicted the mass of the top quark, this geometrical basis for the Standard Model awaits confirmation of other specific predictions for collisions, which are under way at Fermilab and soon to be done using the Large Hadron Collider.