Semidirect product gauge group $[SU(3)_{\rm c} \times SU(2)_{\rm L}]\rtimes U(1)_{\rm Y}$ and quantization of hypercharge

TL;DR

This paper explores a semidirect product gauge group structure in the Standard Model, showing it constrains hypercharges to specific fractional values and ensures anomaly cancellation, reproducing standard charge assignments and providing insights into exotic matter hypercharges.

Contribution

It demonstrates that adopting a semidirect product gauge group uniquely determines hypercharge ratios and guarantees anomaly cancellation, unlike the conventional direct product group.

Findings

Hypercharges are quantized to specific fractions under the semidirect product group.

Anomaly cancellation conditions strongly constrain hypercharge ratios.

Standard model charge assignments are consistent within this framework.

Abstract

In the Standard Model the hypercharges of quarks and leptons are not determined by the gauge group $SU(3)_{\rm c} \times SU(2)_{\rm L} \times U(1)_{\rm Y}$ alone. We show that, if we choose the semidirect product group $[SU(3)_{\rm c} \times SU(2)_{\rm L}] \rtimes U(1)_{\rm Y}$ as its gauge group, the hyperchages are settled to be $n/6 \mod {\mathbb{Z}}\;(n = 0,1,3,4) $. In addition, the conditions for gauge-anomaly cancellation give strong constraints. As a result, the ratios of the hypercharges are uniquely determined and the gravitational anomaly is automatically canceled. The standard charge assignment to quarks and leptons can be properly reproduced. For exotic matter fields their hypercharges are also discussed.