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

TL;DR

This paper demonstrates that in a supersymmetric $ ext{CP}(1)$ nonlinear sigma model, matter charge quantization arises naturally without GUTs, leading to stable, fractionally charged Nambu-Goldstone bosons with potential dark matter applications.

Contribution

It shows that matter charge quantization can occur in the $ ext{CP}(1)$ model without GUTs, avoiding typical GUT-related problems, and explores phenomenological implications of fractionally charged NG bosons.

Findings

Charge quantization condition derived for matter fields.

NG boson is fractionally charged and stable.

Potential role of NG boson as dark matter candidate.

Abstract

We investigate the consistency conditions for matter fields coupled to the four-dimensional (${\cal N} = 1$ 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.