Standard Model representations

Write the local gauge algebra using $SU(3)_c\times SU(2)_L\times U(1)_Y$, leaving possible global quotient choices outside this lesson. A label $(3,2,1/6)$ means a color fundamental, a weak doublet, and hypercharge $1/6$. It describes six internal components before spinor components are counted. The weak doublet contains two fields whose third weak-isospin eigenvalues are $+1/2$ and $-1/2$, while their hypercharge is the same.

Our convention is $Q=T_3+Y$; another common convention rescales hypercharge and writes $Q=T_3+Y/2$. Never combine the numerical labels from one convention with the charge equation from the other. A color singlet carries no color index, and a weak singlet has $T_3=0$. Representation dimension is not electric charge: the color triplet number does not mean a quark has charge three or that three separate quark flavors are present.

A gauge-invariant local term is a singlet under every factor, not merely electrically neutral after symmetry breaking. The candidate $\overline q_L H d_R$ contracts its color indices between the barred quark and the right-handed quark, and its weak index between the barred doublet and the Higgs. Its hypercharges add to $-1/6+1/2-1/3=0$. The bar reverses the abelian charge, which is an easy sign to overlook.

The up-type coupling instead uses $\widetilde H=i\sigma_2H^*$, whose hypercharge is $-1/2$. The antisymmetric weak tensor converts the conjugate doublet into an object transforming as a doublet. A direct $\overline q_L d_R$ mass term cannot contract the unmatched weak index and also has nonzero hypercharge. When the Higgs acquires its conventional vacuum expectation value, the invariant Yukawa interaction supplies a mass; the original interaction did not need an explicitly gauge-violating mass coefficient.

Quantum gauge anomalies are a further consistency test. The chiral fermion charges must make the relevant anomaly sums cancel, including color multiplicities and the fact that a right-handed field can be described as a left-handed conjugate with reversed abelian charge. Classical invariance of each written term is necessary but does not by itself establish quantum consistency. This is one reason an arbitrary new charged chiral particle cannot simply be added without examining the rest of the spectrum.

Three repeated generations share the same gauge representations, but their Yukawa matrices need not be diagonal in the same flavor basis. Gauge labels therefore do not predict the observed fermion mass hierarchy or mixing angles. Neutrino masses also require an extension of the minimal renormalizable field-and-interaction content. Distinguish the well-tested interaction framework from a claim that its representation table explains every parameter or every observed phenomenon, including gravity and the cosmological dark matter abundance.