Why is there no Omega-minus with spin 1/2?

[Eise]

I am reading 'Particle Hunters', written by Yuval Ne'eman, and he describes of course about the prediction of the Omega-minus, the baryon with three s-quarks. (He predicted it himself, together with Murray Gell-Mann.) But I would not expect that the 3 s-quarks have the same direction of spin. Wouldn't have 3 s-quarks, where one quark has an opposite spin as the two others, less energy than all spins parallel? The Pauli exclusion principle seems to be no problem, as the quarks have different colours. What am I missing?

[Genady]

IIUC, the spin quantum number of the particle is 3/2, which means that the z-component of the spin has 4 possible values, i.e., -3/2, -1/2, 1/2, and 3/2.

[swansont]

Not my area of expertise, but I recall the “eightfold way” discussion, and the octet for spin 1/2 has strangeness of 0, 1 or 2. A spin 1/2 version of the omega would have S=3

 

[Eise]

51 minutes ago, Genady said:

IIUC, the spin quantum number of the particle is 3/2, which means that the z-component of the spin has 4 possible values, i.e., -3/2, -1/2, 1/2, and 3/2.

I am not sure if I understand this. Do you mean that the Omega-minus can have any of these 4 spin directions? That, so to speak, max(abs(spin)) for the Omega-minus = 3/2?

16 minutes ago, swansont said:

Not my area of expertise, but I recall the “eightfold way” discussion, and the octet for spin 1/2 has strangeness of 0, 1 or 2. A spin 1/2 version of the omega would have S=3

I think that is the statement of fact. One octet (spin 1/2), and one decuplet (spin 3/2). I just do not understand why a baryon with 3 strange quarks cannot principally belong to the octet. What is the connection between S = 3 and spin 3/2?

In the book is the story that Feynman would look into the office of Ne'eman, jokingly saying 'Did you hear? They found the Omega-minus! It has a spin of 1/2'. Meaning that the Eightfold Way would be wrong.

[Genady]

4 minutes ago, Eise said:

I am not sure if I understand this. Do you mean that the Omega-minus can have any of these 4 spin directions? That, so to speak, max(abs(spin)) for the Omega-minus = 3/2?

Yes, this is it.

Any particle with a spin number s has 2s+1 possible spin values.

[joigus]

You have to distinguish total spin J=3/2 from spin projection. A particle of total spin J has 2J+1 possible spin projections. Eg, a particle of spin 1/2 has 2*(1/2)+1=2 spin projections, which are -1/2, +1/2. In the case of omegas, we have 2*3/2+1=4 possible spin projections, which are -3/2, -1/2, +1/2, and +3/2. If omegas lasted long enough, we would be able to perfom a Stern-Gerlach experiment and separate them into 4 distinct beams, I'm sure.

Omega^- has spin 3/2 for the reason that these are isospin multiplets, so all the particles in the n-plet have the same spin. The ultimate reason for that is the concept of approximate symmetry iso-spin='same spin'. IOW, baryons with the same spin have approximately the same mass.

 

2 minutes ago, Genady said:

Any particle with a spin number s has 2s+1 possible spin values.

Exactly. I wouldn't call it S, as that's reserved for strangeness. I particle physics it's traditionally called J.

[Genady]

I guess that the OP question is answered. As a personal OT note I wanted to mention that I don't have a positive vibe regarding Y. Ne'eman because he is rather familiar to me as a right-wing politician that he was when I lived in Israel.

Edited July 13, 2023 by Genady

[joigus]

Yeah I knew Ne'eman was related to the higher echelons of Israeli politics, or the military. I do remember an interview with Gell-Mann in which he mentioned Y. Ne'eman's interests really lay in general relativity, but he was somehow forced into particle physics.

I do consider this era of physics kind of a heroic one. It's not for everybody to find your bearings in this terrain of approximate symmetries and mass formulas, empirically guessed relations and the like. Great respect on my part.

[swansont]

3 hours ago, Eise said:

I think that is the statement of fact. One octet (spin 1/2), and one decuplet (spin 3/2). I just do not understand why a baryon with 3 strange quarks cannot principally belong to the octet. What is the connection between S = 3 and spin 3/2?

I think it’s from Group theory and the SU(3) symmetry - a spin 1/2 sss particle is not one of the permutations of the symmetry - but that’s where my understanding gets very fuzzy. When they got to isospin and hypercharge and Lie algebra in class, my eyes glazed over.

[StringJunky]

8 minutes ago, swansont said:

I think it’s from Group theory and the SU(3) symmetry - a spin 1/2 sss particle is not one of the permutations of the symmetry - but that’s where my understanding gets very fuzzy. When they got to isospin and hypercharge and Lie algebra in class, my eyes glazed over.

Lie Algebra was ajb's area of interest, wasn't it?

[swansont]

3 hours ago, swansont said:

I think it’s from Group theory and the SU(3) symmetry - a spin 1/2 sss particle is not one of the permutations of the symmetry - but that’s where my understanding gets very fuzzy. When they got to isospin and hypercharge and Lie algebra in class, my eyes glazed over.

I found this:

https://www.physicsforums.com/insights/a-beginners-guide-to-baryons/

“For all baryons, nature demands that the combination of flavor and spin must be completely symmetric”

Spin 3/2 is completely symmetric, and spin 1/2 is antisymmetric. Three identical quarks is symmetric. 

I notice that the uuu and ddd particles are also spin 3/2, for the same reason.

3 hours ago, StringJunky said:

Lie Algebra was ajb's area of interest, wasn't it?

I don’t have a clear recollection, since I didn’t dive into such discussions.

[Mordred]

5 hours ago, StringJunky said:

Lie Algebra was ajb's area of interest, wasn't it?

Both Ajb and I regularly discussed lie algebra. It's a very useful tool to understand physics in particular the standard model. However it's used in every major physics theory in general.

I'm a little tied up atm but I will add more detail later on with regards to isospin and hypercharge.

 

[Genady]

Here is why the total spin J for the sss baryon cannot be 1/2, AFAIU:

The wavefunction for a baryon (a fermion) has to be anti-symmetric. In the ground state, this wavefunction is a product of the wavefunctions for spin * flavor * color. This product has to be anti-symmetric.

Since all three quarks are of the same flavor, the flavor wavefunction is symmetric. Thus, the product of the wavefunctions for spin * color has to be anti-symmetric.

Next,

(Phy489_Lecture9a_2013.ppt (utoronto.ca))

Since the color wavefunction is anti-symmetric, for the product of wavefunctions for spin * color to be anti-symmetric, the wavefunction for spin has to be symmetric.

The three 1/2-spin wavefunction is fully symmetric only in the case of all three spins being the same, i.e., J=3/2. It is not fully symmetric if one spin is different from the other two, i.e., if J=1/2.

QED  

Edited July 14, 2023 by Genady

[Eise]

Thanks all. That helps. I will dive into the references given by @swansont and @Genady, in the hope that it even increases my understanding a bit more. 

15 hours ago, joigus said:

Ne'eman's interests really lay in general relativity, but he was somehow forced into particle physics.

Yes, he was 'forced' by the distance between the Israel embassy in London, where he worked as a military attaché, and the university where GR was taught: it was too far. So he went into quantum physics, which was taught in a university close by.

From here:

Quote

We find out that a British physicist advised Ne’eman not to study general relativity as it was stagnant and too mathematical – advice he followed only because London’s traffic made it much too complicated for him to reach King’s College. He instead studied the rich spectrum of new particles produced by accelerators (may the future bring new particles upon us as well) at Imperial College under Abdus Salam. This was nearer to the Israeli embassy, where Ne’eman worked as military attaché buying things such as British submarines. We find out that the very same Imperial College felt that he could not get a PhD in physics for using algebraic methods to classify the hadrons, so awarding him a PhD in mathematics was deemed an appropriate solution

16 hours ago, Genady said:

As a personal OT note I wanted to mention that I don't have a positive vibe regarding Y. Ne'eman because he is rather familiar to me as a right-wing politician that he was when I lived in Israel.

I can imagine that:

Quote

In the late 1970s, Ne'eman founded Tehiya, a right-wing breakaway from Likud, formed in opposition to Menachem Begin's support for the Camp David talks that paved the way for peace with Egypt and the evacuation of Yamit.

What? More right than Likud? Oh boy.