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Could a $uuuu\bar{u}$ pentaquark exist or is it forbidden due to Pauli exclusion principle?

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I don't think there's any obstruction due to Pauli Exclusion to writing down a local operator that looks like it may excite a $uuuu\overline{u}$-type pentaquark, for example you could write something like: $$ \mathcal{O}^{\mu,i,j} = \epsilon_{abc} u_a^\mu (u_b^T C \gamma_i u_c) (\overline{u} \gamma_j u)$$ where an appropriate projection will give you something that lives in a spin-5/2 representation. This operator is not identically zero by Pauli exclusion, so it seems like a candidate. Even if you asked for a $uuuuuuuu\cdots uuuuu$ type hadron - you could still write down an operator that seemed to excite such a state; the operator would necessarily have to be nonlocal (to avoid Pauli Exclusion principle issues), but that's ok.

There's no reason to expect that there is an actual pentaquark state with those quantum numbers though. More likely - this operator will just overlap onto $\rho \Delta^{++}$ two-particle states.

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