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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!

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I'm not a QCD physicist, but I do know that the third number is the electric charge –  Jim Jun 18 '13 at 18:23
    
OK, that helps! Thanks! –  curiousGeorge119 Jun 18 '13 at 18:25
    
I think that you are working with right-handed leptons, so you have (SU(3) representation : triplet or singlet, SU(2) representation : singlet, U(1) representation: 1/2 hypercharge (Y/2)). In your case, because you have a singlet SU2, the electric charge equals half the hypercharge Y. –  Trimok Jun 18 '13 at 18:26
    
Thanks, Trimok... appreciate that info. Should I look at the electric charge for U(1), or the hypercharge? –  curiousGeorge119 Jun 18 '13 at 18:44
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Btw, what does your question have to do with the beta funcions? –  Neuneck Jun 20 '13 at 14:32
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1 Answer

up vote 3 down vote accepted

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 $Q = 0 + (-1) 0 -1$ 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.
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EXCELLENT! Thank you so much, Neuneck! I am finding the beta function for U(1) and SU(2) and SU(3). Can you help me fugure out SU(3)? Only the quarks, so I'd have three contributions, right? The upR, the downR, and the doublet quarkL. If each contribution was (1/2), then would it be (1/2)*3*3 for the left handed and (1/2+1/2)*3*3 for the right-handed??? Thank you again –  curiousGeorge119 Jun 21 '13 at 14:51
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