Fermions in the Standard Model

This post is more like a note-to-self for future reference.

Elementary spinors in the Standard Model appear as fundamental representations of the symmetry $SU(3) \times SU(2)_L \times U(1)_Y$, whereas the gauge bosons appear in the adjoint representation. Basically, there are three generations of left-handed Weyl fields that are assumed to exist as color-triplets, isospin-doublets and hypercharge-singlets in the Standard Model. The Higgs scalar additionally exists as a doublet under the weak interaction $SU(2)_L$, namely in the $(\textbf1, \textbf2, -\frac12)$ representation, the same as leptons but with spin $0$.

The following table summarizes the electroweak charges of the assumed fermions in the Standard Model. Sadly the weak force only couples to left-handed fermions. Therefore, the right-handed fermions are weak-singlets. Note that the quarks come as color-triplets and so there are eighteen of them (three color for each flavour). The electric charge appears when the electroweak symmetry is spontaneously broken $SU(2)_L \times U(1)_Y \to U(1)_{EM}$. Of course, all of these particles have their antiparticles in the conjugate representation. For example, charge conjugating a right-handed electron $e \in (\textbf1, \textbf1, -1)$ would produce a left-handed positron $\overline e \in (\textbf1, \textbf1, +1)$.

To summarise, in terms of left-handed Weyl fields, there are three copies of leptons $L$, positrons $\overline e$, quark doublets $Q$ and antiquarks, namely $3 \times \{ (\nu_e, e), \overline e, (u,d), \overline u, \overline d \} $ living in the representation $(\textbf1, \textbf2, -\frac12) \oplus (\textbf1, \textbf1, +1) \oplus (\textbf3, \textbf2, +\frac16) \oplus (\overline{\textbf3}, \textbf1, -\frac23) \oplus (\overline{\textbf3}, \textbf1, +\frac13) $, plus a complex scalar field in $(\textbf1, \textbf2, -\frac12)$ (otherwise, a mass term for these in the trivial representation $(\textbf1, \textbf1, 0)$ would be impossible). This information is not predicted, but assumed in the Standard Model.

Beware of an ambiguity in the assignment of the hypercharge. Some books define the electric charge using $Y =: 2(Q-T_3)$, and so you might find twice the value of the hypercharge listed above. Note that the weak isospin $T_3$ is fixed by whether or not a particle is a doublet or singlet under $SU(2)$, whereas the electric charge $Q$ comes from empirical data. The Higgs mechanism then fixes the weak hypercharge $Y$ up to an overall constant.

The following is an image borrowed from Wikipedia showing all the particles in the Standard Model along with the hypothetical graviton.