How Do You Calculate Flavor-Spin Wavefunctions in Quantum Chromodynamics?

[Hepth]

I'm having a little trouble recreating some things from a paper and it is due to my lack of knowledge of working with Flavor-Spin wavefunctions.
I'm trying to show that :
<br /> \left\langle \Lambda \left|b_s^{\dagger }b_b\right|\Lambda _b\right\rangle =\frac{1}{\sqrt{3}}<br />
and
<br /> \left\langle p \left|b_u^{\dagger }b_c\right|\Lambda _c^+\right\rangle =\frac{1}{\sqrt{2}}<br />

These two papers both take their operation from the same text, but none of it explicitly shows how they get it or what wavefunctions they use. I assume this is because it is ELEMENTARY but while searching around, many books/articles give different wavefunctions using different approaches.

I'm wondering where I should start. I feel like this should be a simple multiplication but everytime I try it I don't get their answers.

From you guys, do any of you know offhand know the flavor-spin states for a lambda and p (both in s=1/2)? One paper gives both octet proton flavor wavefunctions as :
p&#039;=\frac{1}{\sqrt{2}}(\text{udu}-\text{duu})
p\text{&#039;&#039;}=\frac{1}{\sqrt{6}}(2\text{uud}-\text{duu}-\text{udu})

Which is right? Or is it that only in a linear combination with some spin states to make the total flavor-spin function symmetric :
\left.|56,S=\frac{1}{2},8\right\rangle =\frac{1}{\sqrt{2}}(p&#039;\chi &#039;+p\text{&#039;&#039;}\chi \text{&#039;&#039;})

where the spinors are pretty much the same as the protons function but with spin arrows up and down instead of "u u d".

Am I on the right track?

 

[Hepth]

Also, if I get something like:

u^{\dagger } u^{\dagger } d^{\dagger } u c^{\dagger } d u c

and I believe its the trace of this (since i summed over spin states?)
Is there a way to simplify? I know I can cyclically permutate traces to simplify, but that doesn't help this one.
Since I'm summing over spin, and this is merely the flavor representation, can I interchange any two operators freely? Or it would add the (-1) because its not symmetric as just the flavor representation in s=1/2?

If that's the case, then this would simplify to:
- u^{\dagger} u u^{\dagger} u d^{\dagger} d c^{\dagger} c

Which is just -1?

If I follow that method, I get something that isn't what they have. Or am I way off?

 

[Hepth]

Ok, so no responses. Let me clean up my questions then.

From Appendix A (Page 25&26) of http://arxiv.org/PS_cache/hep-ph/pdf/9304/9304286v1.pdf

I have, in their choice of wavefunctions for a proton and a lambda_c:

p=\frac{1}{\sqrt{3}}\left[u u d \chi _s +(13)+(23)\right]
\Lambda _c^+=\frac{-1}{\sqrt{6}}\left[(u d c- d u c) \chi _A +(13)+(23)\right]
\chi _s=\frac{1}{\sqrt{6}}\left[a^{\uparrow } b^{\uparrow } c^{\downarrow }-a^{\uparrow } b^{\downarrow } c^{\uparrow }-a^{\downarrow } b^{\uparrow } c^{\uparrow }\right]
\chi _A=\frac{1}{\sqrt{2}}\left[a^{\uparrow } b^{\downarrow } c^{\uparrow }-a^{\downarrow } b^{\uparrow } c^{\uparrow }\right]

Where (13) is just the permutation.

I'm trying to show that:N_{\text{fi}}=\, _{\text{flavor} \text{spin}}\left\langle \Lambda \left|b_s^{\dagger }b_c\right|\Lambda _c^+\right\rangle {}_{\text{flavor} \text{spin}}=\frac{1}{\sqrt{3}} (in http://arxiv.org/PS_cache/hep-ph/pdf/9502/9502391v3.pdf, bottom of page 13, same author)

and
N_{\text{fi}}=\, _{\text{flavor} \text{spin}}\left\langle p\left|b_u^{\dagger }b_c\right|\Lambda _c^+\right\rangle {}_{\text{flavor} \text{spin}}=\frac{1}{\sqrt{2}} (page 18, same article)I've tried working it out by hand, I don't htink I'm doing it right because I keep getting 3/Sqrt[6] for the proton one, and I get 1 for the lambda_c to lambda.

I know I shouldn't get one, I think I'm doing the operator wrong.

What would I do for the 2 quark flavor spin operator here? Does it replace all c's with u's? Or does it affect spin in any way? Etc. I know its killing off the c, and creating a u, but how does that individually affect "u c d" parts of the wavefunction?

Please help guide me, I just can't seem to do it correctly. ANY help will be amazing.

-hepth