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Considering, for example, the baryon octet, we have then that for $L = 0$, $J = 1/2$, as it is made of spin-1/2 particles. Hence, the isospin $I = 1/2$ and so, some of the wavefunctions of the octet are:

  • For the proton: $|1/2; 1/2>$
  • For the netron: $|1/2; -1/2>$

So far it all makes sense. However, for the sigma particles in the S = -1 row, $I = 1$ since the strange quark doesn't make any contribution to isospin. The question is: if I didn't know what the sigma baryons' spin is, how could I deduce it by just knowing their valence quarks or their isospin?

Thanks in advance.

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  • $\begingroup$ Due diligence. Please review the wavefunction at the bottom of p 12 and ask your question appreciative of the wavefunction provided and explained. Your teacher should have taught you that. $\endgroup$
    – Cosmas Zachos
    Commented Dec 14, 2022 at 22:56
  • $\begingroup$ It would be a mercy to the reader if you simply wrote down the wavefunction of the Σ, putting your finger on it for what it is about it that discomfits you... $\endgroup$
    – Cosmas Zachos
    Commented Dec 14, 2022 at 23:04
  • $\begingroup$ Thanks @CosmasZachos. The document you've provided me is being really useful. Once I have assimilated the theory behind I will reformulate the question. $\endgroup$
    – user9867
    Commented Dec 14, 2022 at 23:08

1 Answer 1

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Before the quark model of hadrons was proposed, people saw that the proton and neutron were very similar particles, and drawing inspiration from spin, we introduced the notion of isospin and defined the proton to be isospin-up and the neutron to be isospin-down. With the development of the quark model, we now understand that isospin really just describes the up/down flavor inside of the hadrons. That is, the up and down quarks are the isospin states $| \frac{1}{2}, \frac{1}{2} \rangle$ and $| \frac{1}{2}, -\frac{1}{2} \rangle$ respectively. The other quarks do not have isospin, so using this you can essentially count up the number of up/down quarks inside a hadron and weight with the appropriate isospin projection to determine the isospin of the hadron in question, so for the $\Sigma^{+} = uus$ we have $I_3 = 2(1/2) = 1$. I will leave the other sigma baryons for you to do. Hope this helps!

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  • $\begingroup$ Thanks very much! Yes, I understand it now. I'm sorry, just by the time you answered I edited the question. So, ok, for example, for the sigma plus I know that it's made of uus. So, from there, how could I deduce the spin (not isospin) of the particle? $\endgroup$
    – user9867
    Commented Dec 14, 2022 at 19:34
  • $\begingroup$ You can use the spin addition rules. For example, the total spin of a baryon can be $1/2$ or $3/2$. The $\Sigma^{+}$ is defined to be the particle made up of $uus$ quarks with spin-1/2. I know it may not be entirely satisfying, but you cannot really take the derivation much further. What you'll see is that the bound states can take on different combinations of properties, and the particles are empirically defined to have a specific combination of spin, isospin, etc. $\endgroup$
    – Richard Whitehill
    Commented Dec 14, 2022 at 19:50
  • $\begingroup$ So, for example, if we were dealing with a pentaquark, let's say $(\bar{b}suud)$, as it has five quarks, would it have a spin of $5/2$? If I'm not wrong its isospin would be $I=3/2$ $\endgroup$
    – user9867
    Commented Dec 14, 2022 at 20:00
  • $\begingroup$ @conradDell Why? it looks like a proton merged with a $\bar b s$ meson. If that meson were spinless, you'd have a spin 1/2 isodoublet just like the proton. What are you asking, and why? Clarify your question. $\endgroup$
    – Cosmas Zachos
    Commented Dec 14, 2022 at 22:48
  • $\begingroup$ Thanks! I just want to understand the theory. If that meson had spin 1, would the spin of that pentaquark be 5/2? Is what I said above correct? $\endgroup$
    – user9867
    Commented Dec 14, 2022 at 22:54

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