How to determine quantum numbers for beta functions?

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Discussion Overview

The discussion revolves around understanding the quantum numbers associated with quarks, specifically the notation used for the up quark and left-handed quarks in the context of the Standard Model of particle physics. Participants explore the significance of the numbers in the notation, their relation to SU(3) and SU(2), and the implications for weak interactions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that the first number in the notation (3, 1, 2/3) indicates membership in a color triplet under SU(3), while the second number relates to weak isospin under SU(2), and the third represents hypercharge.
  • One participant suggests that the "1" in the up quark's notation indicates it does not interact with the weak force, questioning the meaning of the "3".
  • Another participant clarifies that left-handed fermions are weak isodoublets and right-handed ones are weak isosinglets, noting that right-handed quarks do not couple to the W boson due to their weak isospin being zero.
  • There is a mention of different normalizations for hypercharge, with some authors using Q = T3 + Y and others using Q = T3 + Y/2.
  • One participant requests further reading on the process related to the discussion, seeking suggestions for resources.

Areas of Agreement / Disagreement

Participants express varying interpretations of the quantum numbers and their implications for weak interactions, indicating that multiple competing views remain. The discussion does not reach a consensus on the significance of the numbers or the normalization of hypercharge.

Contextual Notes

There are unresolved aspects regarding the normalization of hypercharge and the implications of weak isospin for right-handed quarks. The discussion also reflects differing interpretations of the relationships between quantum numbers and particle interactions.

lonewolf219
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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
 
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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
 
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?
 
Can anyone tell me what this process is called, so maybe I can read a bit more about it? Any suggestions would be great
 
Bill_K said:
The relationship connecting T and Y to Q is Q = T3 + Y/2.
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
http://en.wikipedia.org/wiki/Electroweak_interaction

For more details, see any good book on particle physics or QFT of the Standard Model.
 
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:

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