Notation for Standard Model Charges?

Does anybody know what these following numbers describing an electron $(1, 1, -1)$ represent in $SU(3) \times SU(2) \times U(1)$? Or, these numbers that describe an up quark: $(3, 1, 2/3)$? I'm really confused!

Answer

This is a standard notation for the (reducible) representation the field transforms under.

Usually, the first number is the dimension of the representation for $SU(3)_c$. So a $\mathbf 1$ represents a Lepton (the one-dimensional representation is the trivial representation), a $\mathbf 3$ is a quark. In GUT physics one needs the right-handed fields to transform like left-handed, so you the right-handed paricles by left-handed antiparticles. Then a $\mathbf{\overline{3}}$ is an antiquark.

The second number is the dimension of the $SU(2)_L$ representation. A $\mathbf 1$ represents a right-handed field that does not interact via the $SU(2)_L$. A $\mathbf 2$ represents an isospinor.

The third number is the Hypercharge, not the electrical charge! Here we have a freedom to rescale the numbers, which is why there is no unique notation. There are two conventions, that differ by a factor of 2. For the right handed fields, hypercharge and electrical charge coincide in one of the conventions, leading to the confusion in the comments to the question. The Gell-Mann Nishijima formula that links Hypercharge and electrical charge reads $$ Q = I_3 + \left( \frac{1}{2} \right) Y$$ Where the factor of 1/2 is present, depending on the convention. $I_3$ is the third component of Isospin, i.e. $I_3 = 0$ for right handed particles and $I_3 = \pm \frac{1}{2}$ for Isospin 1/2 up (+) or down (-) states.

The notation you gave does NOT use the factor of 1/2 that I put in brackets!

Let's take a look at the examples you gave:

• (1, 1, -1): first 1 means that this state has no color charge. It is a lepton. The second 1 means it carries no weak isospin, i.e. it is right-handed. So it can be either the right handed electron or the right-handed neutrino. The electical charge is $Q = 0 + (-1) = -1$ so it clearly is the right handed electron


• (3, 1, 2/3): the 3 means this is a quark (comes in three colors!). The second 1 again indicates a right-handed particle. Its electric charge is $Q = 0 + 2/3 = 2/3$. We found the up quark

And since this is fun, let's do one more example:

• (3, 2, 1/6): the $3$ indicates a quark. The 2 sais we are in the doublet (so up and down). The 1/6 then tells us that the charges are $Q = \pm 1/2 + 1/6 = 2/3$ or $-1/3$, so exactly what we'd expect for a pair of quarks.