How are meson spins worked out?

[genloz]

How are meson spins worked out? I can't understand why the pion has spin 0 when it is made up of one up quark and one anti upquark, while the rho meson is made up of the same two quarks but has spin 1...

 

[malawi_glenn]

How are meson spins worked out? I can't understand why the pion has spin 0 when it is made up of one up quark and one anti upquark, while the rho meson is made up of the same two quarks but has spin 1...

Its how you combine two spin half particles. In rho-meson you have parallel spin, they add upp to ½+½ = 1. And in pi-meson you have them anti-parallel; ½-½ = 0.

 

[genloz]

Thanks! but how do you know if the spin is parallel or not?

 

[malawi_glenn]

There are rules for adding angular momenta, that arises from symmetry properties etc.

Experimentally you measur deflection in magnetic fields and so on.

Some mesons also have orbital angular momenta (relative the two quarks) and then you have to add three angular momenta.

 

[genloz]

can you work out these properties or do you need to look them up? If so, where?

 

[malawi_glenn]

can you work out these properties or do you need to look them up? If so, where?

In almost all intermediate and intro QM books, see Sakurai Modern Quantum mechanics for instance. Or just google a bit about addition of angular momenta in QM.

http://en.wikipedia.org/wiki/Clebsch-Gordan_coefficients as a start perhaps.

This thing with mesons in quarks is the same as you have in positronium; electron and positron in bound state.

 

[blechman]

This thing with mesons in quarks is the same as you have in positronium; electron and positron in bound state.

Perhaps I should be satisfied with malawi_glenn's explanation, but I'm a rabble-rouser! ;-)

I should warn you that mesons are much nastier than positronium. Although on the surface, you might think that they're exactly the same, mesons are bounded by gluons. And the gluons that are involved are NON-perturbative (meaning that the strong nuclear force inside the pion is "infinitely strong" in the sense of perturbation theory).

Positronium is calculable in QED perturbation theory - you really can go ahead and compute the spins, energy states, etc. But you can never do this for the quark-antiquark system (lattice calculations have had some minor success with the heavy mesons, but not the light pions). So there is no known way to compute the spins of the pions.

To see the problem, realize that the gluons are spin-1 particles, and in the nonperturbative limit there are an infinite number of gluons living inside the pion! So where does their spin go? Also, since the strong force is so strong, quark-antiquark pairs can be created and destroyed regularly, and even live a long time inside the pion. Where does their spin go? The same kind of thing happens in positronium, but each time that happens, it is down by a power of the fine structure constant, and thus the effect is controlled in perturbation theory. But again, the strong coupling is infinite now, so all these things are going on unsupressed!

That being said, I should mention that there are models of low-energy strong interaction physics that DO allow you to compute these things like how malawi_glenn said. But they are only models (idealizations), and not the actual physics.

 

[Parlyne]

I would think that any appearance of particle/antiparticle pairs would have to effectively be a loop in a vector boson, with the consequence that the quantum numbers of the pair must add up to those of the vector boson. So, I think that the problem is already fully stated by trying to account for the spins of the vector particles.

 

[blechman]

I would think that any appearance of particle/antiparticle pairs would have to effectively be a loop in a vector boson, with the consequence that the quantum numbers of the pair must add up to those of the vector boson. So, I think that the problem is already fully stated by trying to account for the spins of the vector particles.

I assume that you're referring to the diagram gluon --> q-qbar ---> gluon. This kind of thing only makes sense in perturbation theory. The "sea quarks" that live in the meson are not coming from loops, since the whole point is that perturbation theory breaks down. They really are there: they have nontrivial pdf's (parton distribution functions, for the laypeople). What is true is that their flavor quantum numbers add up to zero, so the only flavor that's left is from the "valence quarks". It's wrong to think of them as "virtual" in the loop-sense; such statements only make sense in perturbation theory, and we're past that here.

 

[malawi_glenn]

Yeah, but I had the feeling that this gut was not at this level yet blechman ;) The sentance "made up of the same two quarks but has spin 1..." got me the feeling that he needed some introduction of spin coupling and comparison with the good ol positrinum, that also has triplet and singlet states. Just the way as you learn these things in school, at least I did hehe =)

thanx for filling in atough;)

 

[blechman]

I agree with you malawi_glenn. genloz: if you found my comments above confusing, don't worry too much about them ;-). All I wanted to say was that in the context of quark models, you can do the positronium thing. But in the real world (QCD) things are actually more complicated, and there is no way to explicitly "compute" the spin of the pion without resorting to quark models. that's all I wanted to say.