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"Quix" particle (6 under color su(3)) problem
I am working on the following problem from Georgi's book in Lie Algebras (independent study), but I am stuck:
1. Suppose that a "quix", Q, a particle trasnforming like a 6 under color SU(3) exists. What kinds of bound states would you expect, both of the quiz by itself and the quiz with ordinary quarks? How would each set of states transform under Gell-Mann's SU(3) (flavor)?
So far I have solved the first question. Since the quix is a 6 in color su(3), it has two upper symmetric indices Qij. The bound states correspond to color singlets. From this I can build the following stable (i.e. that cannot be factored into color singlets) states:
(to avoid unnecessary labels for the quix and the quarks here the order is important: [\itex]q^{i}q^{j}\neq q^{j}q^{i}[/itex] )
[\itex]Q\bar{Q}: Q^{ij}\bar{Q}_{ij}[/itex]
[\itex]QQQ: \epsilon_{ikm}\epsilon_{jln}Q^{ij}Q^{kl}Q^{mn}[/itex]
[\itex]Qqqqq: \epsilon_{ikl}\epsilon_{jmn}Q^{ij}q^{k}q^{l}q^{m}q^{n}[/itex]
[\itex]Q\bar{q}\bar{q}: Q^{ij}\bar{q}_{i}\bar{q}_{j}[/itex]
[\itex]Qqq\bar{q}: \epsilon_{jkm}Q^{ij}q^{k}q^{m}\bar{q}_{i}[/itex]
I think I understand that the quix is a flavor singlet, and thus [\itex]Q\bar{Q}[/itex] and [\itex]QQQ[/itex] are singlets with respect to SU(3) flavor. However, it is not clear to me how to find the SU(3) properties of the other states.
For instance, I think that [\itex]Q\bar{q}\bar{q}[/itex] must be a [\itex]3[/itex] or [\itex]\overline{6}[/itex] because it contains two antiquarks. To decide which one one must check that the wave function is antisymmetric. It is not clear to me how to do this. Should I invoke SU(18) and find the antisymmetric representation that leads to a color singlet? Do I need to assign a spin to Q? Is there a different way?
I am working on the following problem from Georgi's book in Lie Algebras (independent study), but I am stuck:
1. Suppose that a "quix", Q, a particle trasnforming like a 6 under color SU(3) exists. What kinds of bound states would you expect, both of the quiz by itself and the quiz with ordinary quarks? How would each set of states transform under Gell-Mann's SU(3) (flavor)?
So far I have solved the first question. Since the quix is a 6 in color su(3), it has two upper symmetric indices Qij. The bound states correspond to color singlets. From this I can build the following stable (i.e. that cannot be factored into color singlets) states:
(to avoid unnecessary labels for the quix and the quarks here the order is important: [\itex]q^{i}q^{j}\neq q^{j}q^{i}[/itex] )
[\itex]Q\bar{Q}: Q^{ij}\bar{Q}_{ij}[/itex]
[\itex]QQQ: \epsilon_{ikm}\epsilon_{jln}Q^{ij}Q^{kl}Q^{mn}[/itex]
[\itex]Qqqqq: \epsilon_{ikl}\epsilon_{jmn}Q^{ij}q^{k}q^{l}q^{m}q^{n}[/itex]
[\itex]Q\bar{q}\bar{q}: Q^{ij}\bar{q}_{i}\bar{q}_{j}[/itex]
[\itex]Qqq\bar{q}: \epsilon_{jkm}Q^{ij}q^{k}q^{m}\bar{q}_{i}[/itex]
I think I understand that the quix is a flavor singlet, and thus [\itex]Q\bar{Q}[/itex] and [\itex]QQQ[/itex] are singlets with respect to SU(3) flavor. However, it is not clear to me how to find the SU(3) properties of the other states.
For instance, I think that [\itex]Q\bar{q}\bar{q}[/itex] must be a [\itex]3[/itex] or [\itex]\overline{6}[/itex] because it contains two antiquarks. To decide which one one must check that the wave function is antisymmetric. It is not clear to me how to do this. Should I invoke SU(18) and find the antisymmetric representation that leads to a color singlet? Do I need to assign a spin to Q? Is there a different way?
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