Worked Gauge-Mapping Micro-Example
Explicitly mapping a few
N,w,T
tuples to (Q, T3, Y) and the color class c (mod 3).Hypercharge functional
Electric charge is $Q=T_3+\tfrac{1}{2}Y$. The color class is $c=(N+2w+T)\bmod 3$.
| Field | (N,w,T) | Chirality | T3 | Y | Q | c mod 3 | Notes |
|---|---|---|---|---|---|---|---|
| ν_L | (1,0,2) | L | 0.5 | -1.000000 | -0.000000 | 0 | lepton (colorless) |
| e_L | (1,1,1) | L | -0.5 | -1.000000 | -1.000000 | 1 | lepton (colorless) |
| e_R | (1,1,1) | R | 0.0 | -2.000000 | -1.000000 | 1 | lepton (colorless) |
| u_L | (0,2,2) | L | 0.5 | 0.333333 | 0.666667 | 0 | quark (color via c mod 3) |
| d_L | (0,1,3) | L | -0.5 | 0.333333 | -0.333333 | 2 | quark (color via c mod 3) |
| u_R | (1,3,4) | R | 0.0 | 1.333333 | 0.666667 | 2 | quark (color via c mod 3) |
| d_R | (2,2,2) | R | 0.0 | -0.666667 | -0.333333 | 2 | quark (color via c mod 3) |