How to determine quantum numbers for beta functions?

In summary, the notation (3, 1, 2/3) and (3, 2, 1/6) refer to the up quark and left-handed up and down quarks respectively. The first number represents the quark's membership in a color triplet, the second number represents its weak isospin, and the third number represents its weak hypercharge. The relationship connecting these values to the quark's electric charge is Q = T3 + Y/2. Right-handed quarks do not interact with the W boson, but still interact with the Z boson. The process of understanding these interactions is called electroweak symmetry breaking and can be studied in books on particle physics or QFT of
  • #1
lonewolf219
186
2
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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  • #2
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
 
  • #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?
 
  • #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
 
  • #5
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.
 
  • #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:

Related to How to determine quantum numbers for beta functions?

1. What are quantum numbers and why are they important in determining beta functions?

Quantum numbers are a set of numerical values that describe the properties and characteristics of a quantum system, such as an electron. They are important in determining beta functions because they help to identify the energy levels and orbital configurations of particles, which are essential in understanding their behavior in quantum mechanics.

2. How many quantum numbers are needed to fully describe an electron's state?

There are four quantum numbers needed to fully describe an electron's state: the principal quantum number, orbital quantum number, magnetic quantum number, and spin quantum number. These numbers specify the energy level, shape, orientation, and spin of an electron, respectively.

3. How can the principal quantum number be determined for an electron?

The principal quantum number can be determined by the electron's energy level. It is represented by the letter "n" and can have any positive integer value, with higher values corresponding to higher energy levels.

4. What is the relationship between the orbital quantum number and the shape of an electron's orbital?

The orbital quantum number, represented by the letter "l", determines the shape of an electron's orbital. It can have integer values from 0 to n-1, where n is the principal quantum number. The value of l corresponds to different orbital shapes, such as s, p, d, and f orbitals.

5. How does the magnetic quantum number contribute to the determination of beta functions?

The magnetic quantum number, represented by the letter "m", specifies the orientation of an electron's orbital in space. It can have integer values ranging from -l to +l, and each value corresponds to a different orientation. This information is important in determining the spatial distribution of an electron's probability density, which is a crucial factor in calculating beta functions.

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