Direct product of space and spin in hadrons

[geoduck]

The uuu hadron doesn't violate Pauli's exclusion principle presumably because there is color.

But even without color, can't the uuu exist if spatial wavefunctions are different? Suppose one u quark is located at r₁, another at r₂, and another at r₃, and say that all three u quarks have spin up.

Then you can construct the antisymmetric wavefunction:
$$(|r_1 r_2 r_3 \rangle - |r_2 r_1 r_3 \rangle+...)\otimes |\uparrow \uparrow \uparrow \rangle
\otimes | u u u \rangle
$$
where r₁, r₂, and r₃ are permuted with negative sign if the permutation is odd.

If you swap any two particles, you get a negative sign.

Also in general, must we assume that the wavefunction is a direct product? This seems to say that spin is uncorrelated with flavor. If a baryon is made of a u, d, and s quark, why must each of these different quarks have the same amplitude of being found spin up or spin down (as implied by a direct product)?

 

[Einj]

If you exclude color, the quantum numbers of each quark are spin up and u-flavor, which means that the wave function for the i-th quark is \psi_{\uparrow, u}(r_i). This means that, no matter of the position of the quarks the wave function is always going to be:
$$
\psi_{\uparrow, u}(r_1)\psi_{\uparrow, u}(r_2)\psi_{\uparrow, u}(r_3).
$$
You can't symmetrize/anti-symmetrize anything (if you don't admit the presence of color).

 

[ChrisVer]

Also in general [...] :
Because that's the most natural thing to say. Why would being an u quark imply which spin polarization it'll have?
The same then for d and s...

 

[geoduck]

If you exclude color, the quantum numbers of each quark are spin up and u-flavor, which means that the wave function for the i-th quark is \psi_{\uparrow, u}(r_i). This means that, no matter of the position of the quarks the wave function is always going to be:
$$
\psi_{\uparrow, u}(r_1)\psi_{\uparrow, u}(r_2)\psi_{\uparrow, u}(r_3).
$$
You can't symmetrize/anti-symmetrize anything (if you don't admit the presence of color).

I was thinking a spatial wavefunction along the lines of:
$$
\psi(r_1,r_2,r_3)=r_1 r_2^2-r_2 r_1^2+r_2r^2_3-r_3r_2^2+r_3r_1^2-r_1r_3^2
$$
It is already antisymmetric in r₁, r₂, r₃.

That is, the probability amplitude for finding a spin up, u-quark, at locations r₁, r₂, r₃, is given by the above probability amplitude, which is antisymmetric when you swap two of the quarks.

Such wavefunctions might seem odd, but adding color is a drastic step.

 

[Orodruin]

The requirement to be able to (completely) anti-symmetrize the spatial part of the wave function is simply that all of the quarks are in different eigenstates of the system. This means more energy (they cannot all be in the lowest energy state) and thus larger mass (and typically shorter life-time).

 

[geoduck]

The requirement to be able to (completely) anti-symmetrize the spatial part of the wave function is simply that all of the quarks are in different eigenstates of the system. This means more energy (they cannot all be in the lowest energy state) and thus larger mass (and typically shorter life-time).

It's true that color allows all quarks to be in the lowest one-particle eigenstate.

But if you take the analogy of atomic physics, wouldn't it be like saying that there must be an additional quantum number in addition to the spin, so that lithium can have all three electrons in a 1s state? You have Fermi particles, so they can't all occupy lowest single particle states, but you introduce color to do so. This seems ad hoc.

 

[Orodruin]

But Lithium does not occur with all three electrons in the 1s state for precisely this reason. The logic is that if you observe the uuu baryon with spin 3/2 and all valence quarks in the lowest spatial energy state, then there must be an additional quantum number - not the other way around.

 

[ChrisVer]

Also the existence of uuu hadron you are talking about, wasn't the experimental proof of the existence of color. It was just an evidence...

 

[Orodruin]

I am not sure that is a good formulation. You never prove anything in empirical sciences ... You collect evidence and reject theories that do not make correct predictions.

 

[geoduck]

The logic is that if you observe the uuu baryon with spin 3/2 and all valence quarks in the lowest spatial energy state, then there must be an additional quantum number - not the other way around.

Do we really observe the energy state of each individual quark in uuu? I thought we can only see the composite particle and not the individual constituents, so a uuu spin 3/2 in an excited state looks like a different baryon. In that case there can be a spin 3/2 uuu baryon with the lowest energy that comprises antisymmetrizing 3 different energy eigenstates, and we would see it as the lowest energy baryon, but this would not imply that all u quarks are in the lowest 1-particle energy eigenstates (which is automatically symmetric and hence needs color).

 

[geoduck]

Also in general [...] :
Because that's the most natural thing to say. Why would being an u quark imply which spin polarization it'll have?
The same then for d and s...

From what I understand (for example these notes, slide 222: http://www.hep.phy.cam.ac.uk/~thomson/partIIIparticles/handouts/Handout_7_2011.pdf )

the proton wavefunction seems to say that flavor and spin are a bit entangled. That is, assuming the color part is antisymmetric, and the spatial wavefunction is symmetric, the proton has a wavefunction:
$$\left[\phi(M_s)\chi(M_s)+\phi(M_a)\chi(M_a)\right] \otimes \psi(r_1)\psi(r_2)\psi(r_3) \otimes RGB_a$$
where \psi is the spatial wavefunction of the ground state (and this is symmetric), and RGB are the colors antisymmetrized. \phi is the flavor/isospin and \chi is the spin, and M_s stands for mixed symmetry in the first two particle indices while M_a stands for mixed antisymmetry in the first two particle indices.

So it seems if you measure the spin of your 3 quarks and determine it's \chi(M_s), then your flavor wavefunction is \phi(M_s), so in some sense flavor does determine spin and vice versa.

 

[ChrisVer]

No there is no entanglement due to that.
There you still don't know what spin goes to which flavor.
For example if you tell me that you found x to be up,up,down spin
then what do you know about phi? suppose you have uds quarks... you can assign the spins to each quark...

 

[Orodruin]

Do we really observe the energy state of each individual quark in uuu? I thought we can only see the composite particle and not the individual constituents, so a uuu spin 3/2 in an excited state looks like a different baryon. In that case there can be a spin 3/2 uuu baryon with the lowest energy that comprises antisymmetrizing 3 different energy eigenstates, and we would see it as the lowest energy baryon, but this would not imply that all u quarks are in the lowest 1-particle energy eigenstates (which is automatically symmetric and hence needs color).

We do not observe the individual quarks, no. However, my point was just that - in order to have the mass corresponding to all quarks in the lowest energy state, the lowest energy state must be degenerate in order to allow anti-symmetrization. Hence the need for a new quantum number to accommodate this.