Mathematical formulation of the Standard Model

Definition

Work on Minkowski space $\R^{1,3}$. Let $G_{\mathrm{SM}} = SU(3) \times SU(2) \times U(1)$, built from the special unitary groups and the unitary group $U(1)$. Write an irreducible representation of $G_{\mathrm{SM}}$ as $(a, b)_Y$. Here $a$ is the dimension of the $SU(3)$ factor, and $\bar 3$ is the dual of the defining representation. The number $b$ is the dimension of the $SU(2)$ factor. The hypercharge $Y$ says that $e^{i\theta} \in U(1)$ acts by $e^{iY\theta}$, up to a fixed normalization. The Standard Model is the gauge theory with gauge group $G_{\mathrm{SM}}$ and the following fields.

1. A connection with values in the Lie algebra $\mathfrak{su}(3) \oplus \mathfrak{su}(2) \oplus \mathfrak{u}(1)$: the gluon, $W$ and $B$ fields.
2. Three generations of left-handed Weyl spinor fields, each generation consisting of $Q \in (3, 2)_{1/6}$, $u^c \in (\bar 3, 1)_{-2/3}$, $d^c \in (\bar 3, 1)_{1/3}$, $L \in (1, 2)_{-1/2}$ and $e^c \in (1, 1)_{1}$.
3. A complex scalar field, the Higgs field $H \in (1, 2)_{1/2}$.

The Lagrangian density is the most general one that is Lorentz invariant, gauge invariant and renormalizable. That is, it is a polynomial in the fields and their covariant derivatives of mass dimension at most $4$. It consists of Yang–Mills terms for the three factors and kinetic terms $i \bar\chi \bar\sigma^\mu D_\mu \chi$ for each Weyl field $\chi$. It also has the term $(D_\mu H)^\dagger D^\mu H - V(H)$ with $V(H) = -\mu^2 H^\dagger H + \lambda (H^\dagger H)^2$, and Yukawa terms that couple $H$ to pairs of fermion fields.