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Is there a straightforward way to see what the spin of the recently-discovered pentaquark states should be, from the representation theory of $SU(3)\times SU(2)\subset SU(6)$? I can see that from the representations of color $SU(3)$, the five quark tensor product $qqqq\bar{q}$ has three singlets, one of which is presumably just a baryon-meson pair, the other two I'm guessing correspond to the recently discovered pentaquarks. I'm just wondering how to include spin into the picture and whether there is a prediction of the spin.

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    $\begingroup$ Note: Although it is for some arcane reason usual among physicists to write $\otimes$ for the direct product of groups, the proper mathematical sign is $\times$. We take the tensor product of representation spaces, but the direct product of groups. $\endgroup$ – ACuriousMind Jul 15 '15 at 18:34
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    $\begingroup$ I think $\otimes$ is used because physicists usually work with the Lie algebra, not the Lie group, in which case it is correct to call it a tensor product (and they generally don't bother with the fraktur script). I can switch it though if it is bothersome. $\endgroup$ – asperanz Jul 15 '15 at 18:50
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if it's a singlet then its spin is 0. the SU(3) rep 3x3x3x3x3' (dim=243) decomposes into a bunch of 1,8,10,27, and 35 multiplets with many possible spins (mathematically speaking at least)

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I can't answer your question per se, but I hope I can offer something that's of some use: See this report:

"The LHCb team is confident that the particles are indeed pentaquarks that comprise two up quarks, one down quark, one charm quark and one anticharm quark. "Benefitting from the large data set provided by the LHC, and the excellent precision of our detector, we have examined all possibilities for these signals, and conclude that they can only be explained by pentaquark states," explains LHCb physicist Tomasz Skwarnicki of Syracuse University in the US."

Whilst the pentaquark is not just a close-coupled baryon-meson pair, IMHO you can glean something by looking at the charmed Lambda baryon $Λ^+_c$ and the $D^0$ meson. Together these have the same "partons" as the pentaquark, and they are spin ½ and spin 0 particles respectively. So this/these pentaquark/s should be spin ½. Other pentaquarks might not be. After all the Delta baryon is a triquark with spin 3/2. And other pentaquarks might be stable.

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