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How to determine quantum numbers for beta functions?

  1. Jun 18, 2013 #1
    I'm trying to understand the notation (3, 1, 2/3) for the up quark and (3, 2, 1/6) for the left-handed up and down quarks.... Is the first number related to SU(3), the second SU(2) and the third I believe is the hyper charge... Not sure what the significance is of the first two numbers...

    I think the 1 in (3, 1, 2/3) means the up quark doesn't interact with the weak force. But what would the 3 mean? A triplet in SU(3)? If so, how would I find it?

    Possible equations:

    Q = I3 + Y
    Q = I3 - Y
    Q = T3 + Y
  2. jcsd
  3. Jun 18, 2013 #2


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    The 3 means the quarks are members of a color triplet. Otherwise they are characterized by weak isospin T and weak hypercharge Y. The relationship connecting T and Y to Q is Q = T3 + Y/2.

    Left-handed fermions are weak isodoublets. Right-handed ones are weak isosinglets. This means they don't interact with the W boson, but they still do interact with the Z. Quark values:

    uL: T = 1/2, T3 = +1/2, Y = 1/3, Q = 2/3
    dL: T = 1/2, T3 = -1/2, Y = 1/3, Q = -1/3
    uR: T = 0, T3 = 0, Y = 4/3, Q = 2/3
    dR: T = 0. T3 = 0, Y = -2/3, Q = -1/3
  4. Jun 18, 2013 #3
    OK, that makes sense. Thank you, Bill K!

    ...Do right-handed quarks couple to the W or Z bosons? My guess is no, which might be why there is not a right-handed quark doublet? Is this because their weak isospin is 0?
  5. Jun 18, 2013 #4
    Can anyone tell me what this process is called, so maybe I can read a bit more about it? Any suggestions would be great
  6. Jun 18, 2013 #5


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    Some authors (e.g., Srednicki) normalize hypercharge so that Q = T3 + Y. In the OP's notation of (3,1,2/3) for the right-handed up quark, this is the normalization that is used.

    For the basics of how the various fields interact, see "After electroweak symmetry breaking" in

    For more details, see any good book on particle physics or QFT of the Standard Model.
  7. Jun 18, 2013 #6
    Thank you, Avodyne... Helpful!!! Thanks for pointing out it's normalized... I don't think I could have figured that out on my own :)
    Last edited: Jun 18, 2013
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