# A question on the strong interaction and charm number

I know that the particle $$P_c(4380)^+$$, has quark content $$\bar{c}cuud$$. Furthermore I know that the reaction $$K^-+P_c(4380)^+\rightarrow K^-+J/\psi+p$$, is strong and the quark content on the left hand side is $$s\bar{u}+c\bar{c}+uud$$.

I have two questions:

1) Intuitively one would expect that as this is a strong interaction that all the quarks on the right hand side should match up with all the quarks on the left hand side , correct ? Which makes it obvious (say we didn't know and we were trying to figure it out ) that the quark content of the $$P_c(4380)^+$$ particle is $$\bar{c}cuud$$.

2) The definition of the charm number is $$C=n_c-n_{\bar{c}}$$, so on the right hand side this gives zero . How can we use this number for determining how many charm quarks are involved considering , you could have zero charm, anticharm quarks or any number of them ( as long as it's the same amount for both) and it will still give C=0.

The part you're missing is that the $$P_c(4380)$$ was observed in $$\Lambda_b$$ decays: $$\Lambda_b \to P_c(4380) K^- \to J/\psi p K^-$$
The presence of a kaon and a $$J/\psi$$ in the final state tells you this decay involves a $$b \to c\overline{c}s$$ transition. The $$\Lambda_b$$ is the lightest $$b$$-baryon and there is no quark with a mass between that of the $$c$$ and $$b$$ quarks, so any intermediate state decaying to the $$J/\psi$$ must contain $$c\overline{c}$$. Therefore any $$X \to J/\psi p$$ observed in this system must be a $$c\overline{c}uud$$ state and any $$X \to J/\psi K^-$$ must be $$c\overline{c}s\overline{u}$$ (the $$B^+$$ is too heavy).
So to answer your questions: 1) yes, but it's the other way around: we know this is allowed as a strong decay because the quark content doesn't change and 2) yes, all that $$C=0$$ tells you is that there are an equal number of $$c$$ and $$\overline{c}$$ quarks.
• As a side note, since the discovery that the $P_c(4450)$ is actually two narrow states, the motivation for the $P_c(4380)$ has disappeared. We need to perform an amplitude analysis with the current dataset to see whether it (or any other broad pentaquarks) are actually there. Commented Apr 23, 2019 at 7:02