Charge Quantization in the $\mathbb{CP}(1)$ Nonlinear Sigma-Model

Abstract

We investigate the consistency conditions for matter fields coupled to the four-dimensional ($\mathcal{N}=1$ supersymmetric) $\mathbb{CP}(1)$ nonlinear sigma model (the coset space $SU(2{)}_G/U(1{)}_H$). We find that consistency requires that the $U(1{)}_H$ charge of the matter be quantized, in units of half of the $U(1{)}_H$ charge of the Nambu-Goldstone (NG) boson, if the matter has a nonsingular kinetic term and the dynamics respect the full group $SU(2{)}_G$. We can then take the linearly realized group $U(1{)}_H$ to comprise the weak hypercharge group $U(1{)}_Y$ of the Standard Model. Thus we have charge quantization without a Grand Unified Theory (GUT), completely avoiding problems like proton decay, doublet-triplet splitting, and magnetic monopoles. We briefly investigate the phenomenological implications of this model-building framework. The NG boson is fractionally charged and completely stable. It can be naturally light, avoiding constraints while being a component of dark matter or having applications in nuclear physics. We also comment on the extension to other NLSMs on coset spaces, which will be explored more fully in a followup paper.

Comments: 6 pages; v2 minor updates (anti-holomorphic details, typos), matches published version