Weak hypercharge - Knowlpedia

Definition

Weak hypercharge $Y$ is the weight labeling a one-dimensional representation of the $U(1)_Y$ factor of the Standard Model internal symmetry group. In the convention used here,

Q=T_3+\frac{Y}{2},

where $T_3$ is the third weak-isospin generator. Since Standard Model hypercharges lie in $\frac13\mathbb Z$ , one may let $z\in U(1)$ act on hypercharge $Y$ as multiplication by $z^{3Y}$ , whose exponent is integral.

Values in one generation

Y(Q_L)=\frac13,\quad Y(L_L)=-1,\quad Y(u_R)=\frac43,\quad Y(d_R)=-\frac23,\quad Y(e_R)=-2,\quad Y(\nu_R)=0.

Antiparticle or charge-conjugate representations carry the opposite hypercharge.

Normalization conventions

Another widespread convention writes $Q=T_3+Y_{\mathrm{half}}$ , with

Y_{\mathrm{half}}=Y/2.

Thus the left-handed quark doublet has $Y=1/3$ here and $Y_{\mathrm{half}}=1/6$ there. A bare numerical hypercharge is ambiguous unless the convention is given.

Generator in the

SU(5)

embedding

For $\mathbb C^5=\mathbb C^2\oplus\mathbb C^3$ , the defining-representation hypercharge operator is

\widehat Y=\operatorname{diag}\left(1,1,-\frac23,-\frac23,-\frac23\right).

The integral generator $3\widehat Y=\operatorname{diag}(3,3,-2,-2,-2)$ exponentiates to

z\longmapsto\operatorname{diag}(z^3I_2,z^{-2}I_3).

Changing block order permutes the entries; dualizing reverses their signs.

Hypercharge and the central quotient

Compatibility of these weights with weak isospin and color makes the central $\mathbb Z_6\subset U(1)\times SU(2)\times SU(3)$ act trivially on every Standard Model multiplet. Thus the assignments factor through the effective quotient.

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: §§2.3.2 and 3.1.
David Tong, “The Standard Model,” lecture notes, 2017. arXiv record.
John C. Baez, “Three Generations in $E_7$ ,” 2026. arXiv record. Relevant: §§2–3.