How to treat quark color pairs mathematically

In summary, the conversation discusses the difficulty of working through a problem involving quark color pairs in the textbook "Particle Physics in a Nutshell." The problem involves rotating a meson color singlet by θ using the first Gell-Mann matrix as the generator. The expert suggests expressing the representation of the group element in the appropriate basis or expanding it in terms of eigen-vectors. The action of the Gell-Mann generator on the given state is also discussed, and the process of expanding the exponential as a power series is explained.
  • #1
winstonboy
4
0
I am trying to work through a problem in the textbook "Particle Physics in a Nutshell." However, I am realizing how little I actually understand about working through problems involving quark color pairs.

Given in the problem is the meson color singlet [tex]1/\sqrt{3}(r\bar{r}+g\bar{g}+b\bar{b})[/tex] which I understand. The problem asks to rotate this state by θ with the first Gell-Mann matrix as the generator.

I understand that I should be operating with a rotation operator [tex]e^{-i\theta\lambda_1}[/tex]
but I am not sure what I should be operating on, exactly.

I have not been able to find anything that might point me in the right direction in the textbook or online. Once I know how to work with these color/anticolor components I should be able to progress.

Thanks,
W
 
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  • #2
Your particle state is in a tensor product space. You will need to express the representation of the group element (the exponential of the Gell-Mann matrix) in the appropriate basis, or you could try to see the component parts and expand in terms of eigen-vectors starting with the representation of the Gell-Mann Matrix itself.

Here's the action of the Gell-Mann generator on the state you gave:
[tex]
\delta \psi = 1/\sqrt{3}\left( [\lambda_1r]\otimes\bar{r} + r\otimes[\lambda_1^\dagger \bar{r}] +[\lambda_1g]\otimes\bar{g} + g\otimes[\lambda_1^\dagger \bar{g}] +[\lambda_1b]\otimes\bar{b} + b\otimes[\lambda_1^\dagger \bar{b}] \right) [/tex]
Where the [itex]\lambda_1[/itex] is the fundamental rep. Now since the first Gell-Mann generator basically swaps r and g colors and maps b to zero (if the color ordering is the same as I recall) then this infinitesmal action will become:
[tex]\delta \psi = 1/\sqrt{3}\left( g\otimes\bar{r} + r\otimes \bar{g} +r\otimes\bar{g} + g\otimes \bar{r} \right) = 2/\sqrt{3}\left( g\otimes\bar{r} + r\otimes \bar{g}\right) [/tex]
(Ok I'll now stop being explicit with the tensor products and just use the same dyadic notation you began with...)
Now you'll be expanding the exponential as a power series so you need to see the next term by acting on this again with [itex]\lambda_1[/itex].
[tex]\delta^2\psi = 4/\sqrt{3}\left(r\otimes\bar{r} + g\otimes\bar{g} + g\otimes\bar{g}+r\otimes\bar{r} \right) = 4/\sqrt{3} \left(r\otimes\bar{r} + g\otimes\bar{g}\right)[/tex]
So we're repeating some of the terms. So separate out the b component and then work out the power expansion of the exponential action. You should see real and imaginary terms that resolve as cosine times some terms plus i sine times others plus the blue terms that go away after the 0th order term in the expansion.
 

Question 1: What is a quark color pair?

A quark color pair refers to the combination of two quarks, which are fundamental particles that make up protons and neutrons. Quarks have a property called "color charge" which can be red, green, or blue. A quark color pair consists of one quark with a certain color charge and another quark with the opposite color charge.

Question 2: How are quark color pairs used in mathematical calculations?

Quark color pairs are used in the mathematical framework of quantum chromodynamics (QCD) to describe the interactions of quarks and gluons, which are particles that mediate the strong nuclear force. In QCD, quark color pairs are represented by mathematical objects called "color matrices" that act on quark fields.

Question 3: What is the mathematical expression for a quark color pair?

The mathematical expression for a quark color pair is written as a combination of two quark fields, one with a certain color charge and the other with the opposite color charge. In QCD, this is represented by a "color singlet" state, which is a linear combination of the three possible color charges that is color neutral overall.

Question 4: How do quark color pairs contribute to the overall color charge of a particle?

In QCD, the total color charge of a particle is determined by the combination of its quark color pairs and any additional gluons that are present. The color charges of the quarks and gluons cancel out in a way that ensures the overall color charge of the particle is neutral.

Question 5: Can quark color pairs be observed experimentally?

No, quark color pairs cannot be directly observed experimentally. This is because quarks are confined within particles and cannot exist as free particles. However, their effects can be observed through high-energy collisions and other experiments that probe the strong nuclear force and the structure of matter.

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