The Mystery of 8 Gluons: Uncovering the Secrets of Particle Physics

[tade]

I've only read some popular science books on particle physics so my understanding is pretty basic.



There are three colors and three anticolors. 3 x 3 will give us 9 gluons.

Subtracting the red-antired and the other two leaves us with 6 gluons.



Those science books only explain the first part but they don't explain how we wind up with 8.

 

[Jim Kata]

because there are 8 generators of the lie algebra of su(3)

 

[tade]

because there are 8 generators of the lie algebra of su(3)

So the popular explanation is off-tangent?

 

[yenchin]

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/gluons.html

 

[tom.stoer]

The popular explanation misses the color-neutral gluon, which should behave like a photon (and which must be absent b/c we don't see long-range color-electric forces in nature).

We start with 3 colors for the quarks. We represent these three colors as something like a 3-component vector. Then we act with 3*3 matrices A on these 3-vectors. Now one can represent all 3*3 matrices A as a sum A = Ʃ_aA^aT^a where we use 9 matrices T^a as a vector space basis. The 9 real numbers A^a represent the gluons.

Now we have to make a choice regarding the matrices we are talking about. They could be u(3) = u (1) + su(3) matrices, or they could be su(3) matrices w/o the u(1) part. The u(1) corresponds to the T⁰=1, i.e. to the 3*3 identity matrix. But this would correspond to the photon-like gluon, which we do not see in nature. Therefore we use a=1..8 instand of a=0..8 in the sum Ʃ_a, i.e. we explicitly exclude the photon-like gluon. That's why we can say that QCD has the gauge symmetry SU(3) instead of U(3).

Please note that this difference does not show up on the level of the quarks, so any explanation using quarks only will fail to explain the reason for 8 instead of 9 gluons. And it's not math but nature telling us that there are only 8 gluons.

 

[Bill_K]

The popular explanation misses the color-neutral gluon, which should behave like a photon

Why should it behave like a photon? And what does "behave like a photon" mean?

 

[tade]

Why should it behave like a photon? And what does "behave like a photon" mean?

Both have zero electric charge, zero rest mass and spin 1. A gluon has color charge, a photon has none. (haha that rhymes)So the "white" gluon under SU(3) behaves like a photon and would not play a part in QCD. Hence there are 8 left.But what are the 8 remaining gluons called?

 

[tom.stoer]

Why should it behave like a photon? And what does "behave like a photon" mean?

It decouples from the other gluons a=1..8 b/c [T⁰, T^a]=0, i.e. b/c the structure constants f^0bc=0. That means that all terms in the Yang-Mills eq. with structure constants are 'abelian' (like Maxwell's eq.) if at least one adjoint index is 0. So for the a=0 gluon there is no coupling to other gluons, which means that it behaves like a color-neutral gauge boson. But this is precisley the case for photons.

Looking at the Gauß law constraint (DE)^a-ρ^a = 0 which introduces the self-interaction in the color-Coulomb-potential it becomes evident that this constraint reduces to ∂E⁰-ρ⁰ = 0 for a=0, so I expect a standard Coulomb interaction ~ ρ⁰(x)ρ⁰(y)/|x-y|, neither with any dynamical gluon-dependent integral operator, nor with non-vanishing Fadeev-Popov determinant.

In addition the color charge density ρ⁰ is closely related to the electric charge density of the quark fields. Thefore U(3) ~ U(1) * SU(3) is (up to some algebraic issues) nothing else like QED + QCD.

 

[mfb]

But what are the 8 remaining gluons called?

They don't have a special name (unless you count things like "red antiblue" as name), and their identity depends on your choice of a basis anyway.

 

[tade]

(unless you count things like "red antiblue" as name)

If I do that, how would I list out the 8?

 

[tom.stoer]

You can use the entries of the eight T-matrices to find linear combinations of the color-anticolor pairs.

http://en.wikipedia.org/wiki/Gell-Mann_matrices
http://en.wikipedia.org/wiki/Gluon#Eight_gluon_colors

But listing them simply means writing down a 4-vector field A^a_μ(x) with a spacetime index μ=0..3 and Lie-algebra index a=1..8. This is a complete "list of gluons".

 

[tade]

http://en.wikipedia.org/wiki/Gluon#Eight_gluon_colors

thanks tom. the one in the bottom right is written in a unique way. hmm..

 

[tom.stoer]

the one in the bottom right is written in a unique way

But keep in mind that any other basis T'^a = U T^a U^-1 will work as well.

 

[tade]

But keep in mind that any other basis T'^a = U T^a U^-1 will work as well.

Unfortunately, I'm unable to keep that in mind because I only have a layman understanding

 

[mfb]

You don't have to use the gluons used there. If $\psi$ and $\phi$ are two gluons, you can replace them by $\frac{1}{\sqrt{2}}(\psi+\phi)$ and $\frac{1}{\sqrt{2}}(\psi-\phi)$, for example (keep the other 6 gluons), and get another possible set of 8 gluons.

 

[tom.stoer]

Unfortunately, I'm unable to keep that in mind because I only have a layman understanding

Think about the eight T-matrices as basis vectors in an eight-dim vector space. You can chose any rotated basis you like, physics remains the same, but you have to change the "names" of the basis vectors (the colors)

 

[tade]

Think about the eight T-matrices as basis vectors in an eight-dim vector space. You can chose any rotated basis you like, physics remains the same, but you have to change the "names" of the basis vectors (the colors)

Hmm, thinking about 8-D space? That's a little hard.

But I'm sort of getting the picture.

 

[tom.stoer]

Think about four dimensions; then forget one dimension b/c of the photon-like gluon; now you can rotate the remaining three axes arbitrarily, respecting pairwise perpendicular dictions

 

[tade]

Think about four dimensions; then forget one dimension b/c of the photon-like gluon; now you can rotate the remaining three axes arbitrarily, respecting pairwise perpendicular dictions

So there are nine dimensions but we drop one and it becomes eight?

 

[tom.stoer]

yes; but these eight dimensions are not spacetime dimensions but an abstract mathematical concept; but if QCD symmetry were SU(2) instead of SU(3) you could visualize 2²-1 = 3 gluons using three mutually orthogonal axes in three-space

 

[tade]

yes; but these eight dimensions are not spacetime dimensions but an abstract mathematical concept; but if QCD symmetry were SU(2) instead of SU(3) you could visualize 2²-1 = 3 gluons using three mutually orthogonal axes in three-space

I see. Now my question would be why QCD symmetry uses SU(3), but I think that's for another day.

 

[tom.stoer]

Now my question would be why QCD symmetry uses SU(3), but I think that's for another day.

The answer is simple: nobody knows! There are experimental hints for SU(3) instead of other symmetry groups, but math and theoretical physics would work equally well with other symmetries.

 

[tade]

The answer is simple: nobody knows! There are experimental hints for SU(3) instead of other symmetry groups, but math and theoretical physics would work equally well with other symmetries.

Color charge is often compared to electric charge, but electric charge is so simple, just positive and negative, whereas SU(3) is so complex.

I guess the person who finds out why could find himself richer by 10 million kronor.

 

[tom.stoer]

In physics you never know "why", unfortunately. Why is electric charge - which is related to an U(1) symmetry which is nothing else but a rotation of a pointer in a two dim plane - so simple? Why do we have electric charge at all, not just SU(173) or SU(362524273644)? Why do we live in four spacetime dimensions? Why do the particles behave as they behave? Why is the electron mass 511 keV instead of 798 keV (or any other random number). Why do we see the quark flavours u,d,s,c,b,t, not 12, or 274656, or just u,d,s,c? Knobody knows.

 

[tade]

In physics you never know "why", unfortunately. Why is electric charge - which is related to an U(1) symmetry which is nothing else but a rotation of a pointer in a two dim plane - so simple? Why do we have electric charge at all, not just SU(173) or SU(362524273644)? Why do we live in four spacetime dimensions? Why do the particles behave as they behave? Why is the electron mass 511 keV instead of 798 keV (or any other random number). Why do we see the quark flavours u,d,s,c,b,t, not 12, or 274656, or just u,d,s,c? Knobody knows.

true. We can say "it's just so".

By the way, what did Wittgenstein mean by throwing away the ladder?

 

[tom.stoer]

He means that after a discussion one may find out that the discussion was based on ill-defined terms, therefore the whole discussion may not really make sense. Unfortunately he writes this in a book, so this conclusion may apply to his book as well. Nevtheless I think he is right ;-)

Just google for 'Wittgenstein's ladder'.

I think we observe similar phenomena in science as well. If you look at the history of quantum mechanics it was quite often the case that after having fully understood an explanation or interpretation of a model it became clear that one has to throw it away. Bohr's model is definitely wrong, but w/o this model we would never had a starting point for QM. So after having climbed up the QM ladder using Bohr's model we through away the ladder = Bohr's model.

I doubt that this is what Wittgenstein had in mind but it fits to the history of science in many cases.

 

[tade]

Unfortunately he writes this in a book, so this conclusion may apply to his book as well.

Hmm, you can't apply Wittgenstein's ladder to Wittgenstein's ladder. That would lead to a self-referential paradox. Am I right?

I think we observe similar phenomena in science as well. If you look at the history of quantum mechanics it was quite often the case that after having fully understood an explanation or interpretation of a model it became clear that one has to throw it away.
I doubt that this is what Wittgenstein had in mind but it fits to the history of science in many cases.

I suddenly feel like going into a long discussion about the way in which scientists should conduct their scientific endeavors.

 

[tade]

Hmm, you can't apply Wittgenstein's ladder to Wittgenstein's ladder. That would lead to a self-referential paradox. Am I right?


I suddenly feel like going into a long discussion about the way in which scientists should conduct their scientific endeavors.

They've been several arguments for and against reductionism.