On page 527 of Srednicki's textbook "Quantum Field Theory", the Standard Model is described as follows:
It can be succinctly specified as a gauge theory with gauge group $SU(3) \times SU(2) \times U(1)$, with left-handed Weyl fields in three copies of the representation $(1, 2, -\frac{1}{2}) \oplus (1, 1, +1) \oplus (3, 2, + \frac{1}{6}) \oplus (\overline{3}, 1, -\frac{2}{3}) \oplus (\overline{3}, 1, +\frac{1}{3})$, and a complex scalar field in the representation $(1, 2, -\frac{1}{2})$. Here the last entry of each triplet gives the value of the $U(1)$ charge, known as $\it{hypercharge}$.
I am puzzled by the group representation $(1, 2, -\frac{1}{2}) \oplus (1, 1, +1) \oplus (3, 2, + \frac{1}{6}) \oplus (\overline{3}, 1, -\frac{2}{3}) \oplus (\overline{3}, 1, +\frac{1}{3})$. How does it come about? What are the steps (if any) to get this representation?
Best Answer
Your text assumes you are familiar with the quantum number content of the elementary particle fermions, determined by the Millikan oil-drop experiment, structure functions of the light quarks, V-A structure of the weak currents, etc. These are experimental inputs and they come from out there, your world.
It helps you summarize the self-evident logic of their apparently diverse quantum numbers so you could write a compact QFT for them, that's all. I assume you seek an appreciation of the manifest logic involved.
It gives you the SU(3) color rep, singlet for leptons, color 3 for quarks, or color anti triplet for antiquarks. Likewise, their SU(2) weak isospin, vanishing for right handed singlets, and doublet for left-handers. (No separate 2-bars, of course, as SU(2) is pseudoreal.) And, of course, mutatis mutandis for their CPT conjugates. You only have singlets and fundamental reps, since these are fundamental fermion building blocks of our world.
Thus,
The hypercharge in the third entry is dross -- an error-correction number, if you wish, given by $Y_W\equiv Q-T_3$, once you input the charge, in the "minority usage", but actually modern mainstream definition, so you might have to multiply it by 2 to agree with hidebound historical listings, like those linked here. It is the eigenvalue of U(1), as your particles are all singlets, of course, under it, and multiplies the B coupling charge of the fermion currents. The sooner you get used to its Golden Mnemonic, the better: It is the average charge of isomultiplets.
There is nothing more to it. Given these numbers you may completely, and concisely specify the fermion sector of the SM QFT.