Chemistry - SU(3) × SU(2) × U(1)

In mathematics and mathematical physics, the Lie group SU(3) × SU(2) × U(1) is the formulation of the Standard Model as a gauge theory with the gauge group SU(3) × SU(2) × U(1) or SU(3) × SU(2) × U(1)/Z₆ with a couple of fermion fields and a Higgs field, which is a $(1,2)_{\frac{1}{2}}$ and/or a $(1,2)_{-\frac{1}{2}}$ . SU(3) describes quantum chromodynamics, SU(2) describes the weak interaction* and U(1) describes hypercharge.

*Technically speaking, the Z and W bosons are described by a field which is really a linear combination of SU(2) and U(1). See electroweak.

There are three families of fermions, each consisting of the representations, $(3,2)_{\frac{1}{6}}$ (q for left-handed quark), $(\bar{3},1)_{\frac{1}{3}}$ (d^c for the left-handed anti d-quark), $(\bar{3},1)_{-\frac{2}{3}}$ (u^c for the left handed up antiquark), $(1,2)_{-\frac{1}{2}}$ (l for the left handed leptons), $(1,1) 1$ (e^c for the left-handed positron) and $(1,1) 0$ (ν^c for the left-handed antineutrino, which is now known to exist. See Neutrino oscillation.).

The Higgs field acquires a VEV, resulting in a spontaneous symmetry breaking from $[SU(2)\times U(1)]/\mathbb{Z}_2$ or $SU(2)\times U(1)$ to $U (1) e m$ .

Of course, calling the representations things like $(3,2)_{\frac{1}{6}}$ is purely a physicist's convention, not a mathematician's convention, where representations are either labelled by Young tableaux or Dynkin diagrams with numbers on their vertices, but still, it is standard among high energy physicists.

Since the homotopy group

$\pi_2\left(\frac{[SU(2)\times U(1)]/\mathbb{Z}_2}{U(1)_{em}}\right)=0$

this model predicts no monopoles associated with the electroweak breaking scale. See Hooft-Polyakov monopole.

The Yukawa couplings of the scalar Higgs fields with the fermions produces the fermion masses after the Higgs field acquires a VEV.