Charge quantization and the Standard Model from the CP2CP2 and CP3CP3 nonlinear σ-models

We investigate charge quantization in the Standard Model (SM) through a CP2CP2 nonlinear sigma model (NLSM), SU(3)G/(SU(2)H×U(1)H)SU(3)G/(SU(2)H×U(1)H), and a CP3CP3 model, SU(4)G/(SU(3)H×U(1)H)SU(4)G/(SU(3)H×U(1)H). We also generalize to any CPkCPk model. Charge quantization follows from the consistency and dynamics of the NLSM, without a monopole or Grand Unified Theory, as shown in our earlier work on the CP1CP1 model (arXiv:1309.0692). We find that representations of the matter fields under the unbroken non-abelian subgroup dictate their charge quantization under the U(1)HU(1)H factor. In the CP2CP2 model the unbroken group is identified with the weak and hypercharge groups of the SM, and the Nambu–Goldstone boson (NGB) has the quantum numbers of a SM Higgs. There is the intriguing possibility of a connection with the vanishing of the Higgs self-coupling at the Planck scale. Interestingly, with some minor assumptions (no vector-like matter and minimal representations) and starting with a single quark doublet, anomaly cancellation requires the matter structure of a generation in the SM. Similar analysis holds in the CP3CP3 model, with the unbroken group identified with QCD and hypercharge, and the NGB having the up quark as a partner in a supersymmetric model. This can motivate solving the strong CP problem with a vanishing up quark mass.