Right-handed neutrino gauge singlet

Definition

A right-handed neutrino gauge singlet is a right-handed Weyl fermion $\nu_R$ carrying the trivial representation

(1,1)_0

of $SU(2)_L\times SU(3)_C\times U(1)_Y$ . Equivalently, every element of the Standard Model internal gauge group acts as the identity on its one-dimensional internal state space.

Physics interpretation

Because $\nu_R$ has no color, weak isospin, or hypercharge, it has no direct coupling to the Standard Model gauge bosons. It is therefore called a gauge singlet and often “sterile.” A gauge-invariant Majorana mass term is permitted, and a neutrino Yukawa coupling may be formed after including the Higgs and lepton doublets. These dynamical possibilities go beyond the definition.

The minimal Standard Model does not contain $\nu_R$ . It is a common extension motivated by neutrino-mass models, but “one generation” must say whether it is included.

Role in the exterior-algebra model

In the $\Lambda\mathbb C^5$ model, the two trivial lines $\Lambda^0\mathbb C^5$ and $\Lambda^5\mathbb C^5$ represent a right-handed neutrino and its antiparticle. Removing them leaves the $30$ -dimensional representation for a generation and its antiparticles without $\nu_R$ .

In the $E_7$ construction, the six neutrino and antineutrino lines for three generations arise from root spaces of a commuting generation $\mathfrak{sl}_3$ , unlike the other $90$ fermionic directions. Their gauge-singlet property says that $\mathfrak g_{\mathrm{SM}}$ acts trivially on them.

References

John C. Baez and John Huerta, “The Algebra of Grand Unified Theories,” Bulletin of the American Mathematical Society 47 (2010), 483–552. arXiv record. Relevant: §§2.3 and 3.1.
John C. Baez, “Three Generations in $E_7$ ,” 2026. arXiv record. Relevant: §10.