Quark Confinement

I've read conflicting definitions of what happens to the force between quarks as they're pulled apart. Do the gluon tubes form to maintain a constant force between them, or does the force actually increase as the distance between quarks increase?

Is there an easily defined potential between the quarks?

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The force increases, but there is not a well-defined potential. For heavy quarks, there have been several attempts (of varying degrees of success) to create a potential, but for light quarks, a static potential is a poor approximation.

The one I know of is a linear potential, which is pairwise acting between the quarks, like:
$$U_{ij}=k\cdot|r_i-r_j|$$

where the constant should be of the order: $$k\sim m_pc^2/a$$ (mp=mass of proton) (its on Wikipedia I think also) and $$a=10^{-15}$$m is the size of the proton.

But obviously it does not explain why quarks can only exist colorless or you must have three of them in a proton. Only phenomenologically it suggest that a quark will be so energetically enriched if you would try to separate it from the other, that it will generate a swarm of matter-antimatter, because of E=mc^2.

Also if you use this potential it would suggest that a quark in a proton could interact with another quark in another galaxy, and therefore the whole universe would explode because of this large potential energy.

I have an idea that you could derive some kind of potential from the QCD Lagrangian, in terms of the gluonic G-field (compare with electromagnetic field), where L is given by:

Classically you could obtain the potential Uij from an electron if you assume $$\Psi^2=\delta(\vec{r}-\vec{r}_j)$$ for electron j, and then solve for the EM-field, i.e. the electric potential from a point particle, know as the Coulomb interaction. Perhaps you could do a similar trick when solving the equations the gluon-fields (involving the eight Gell-Mann matrices)? Putting back this G into the equations for the dirac-like (Schrödinger) equation for the quark-fields one would then get an effective potential? Stright forward (but messy I guess)! Any have tried? Post the answer here then please...

Doesn't that just lead to the non-Abelian version of Maxwell's equation? In other words, the Yang Mills equation?

$(\partial^{\mu}-igT^aA_a^\mu)^{\mu}F_{\mu\nu} = J_\nu$

Doesn't that just lead to the non-Abelian version of Maxwell's equation? In other words, the Yang Mills equation?

$(\partial^{\mu}-igT^aA_a^\mu)^{\mu}F_{\mu\nu} = J_\nu$
Something like that, but are you sure about this form of the equation really? It seems like the equations you should get from
$G_{\mu\nu,a}G^{\mu\nu}_a$

would be 8 coupled Helmholtz equations (if we set the 4-source J=Psi*Psi*g*... equal to zero). Look at: http://arxiv.org/PS_cache/hep-ph/pdf/0210/0210398v1.pdf" [Broken] at page 27 eq.2.24 (also 2.19).

If we ignore the self-interaction (we only consider interactions between different quarks) and consider time-independent solution for these 8 different A-fields (each A-field contain the 4 usual relativistic components, where A0=the scalar potential, which appears as the potential in the Schrödinger equation), we could maby find Green functions (8x8 matrix inverse Helmholzt-operator?...) to use on a Dirac-delta like source (matter density). Tell me if I'm wrong.

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I suspect that the last few messages are at too high a level to answer the OP's question.

Something like that, but are you sure about this form of the equation really? It seems like the equations you should get from
$G_{\mu\nu,a}G^{\mu\nu}_a$

would be 8 coupled Helmholtz equations (if we set the 4-source J=Psi*Psi*g*... equal to zero). Look at: http://arxiv.org/PS_cache/hep-ph/pdf/0210/0210398v1.pdf" [Broken] at page 27 eq.2.24 (also 2.19).

If we ignore the self-interaction (we only consider interactions between different quarks) and consider time-independent solution for these 8 different A-fields (each A-field contain the 4 usual relativistic components, where A0=the scalar potential, which appears as the potential in the Schrödinger equation), we could maby find Green functions (8x8 matrix inverse Helmholzt-operator?...) to use on a Dirac-delta like source (matter density). Tell me if I'm wrong.
Equation [2.19] is the Lagrangian of Yang-Mills theory, while equation [2.24] is the definition of a non-Abelian curvature tensor. So the formulas you refer to are more or less the definition of the curvature tensor. The equations I wrote down earlier are the (classical) equations of motion you get from a term $\textrm{Trace}[G_{\mu\nu,a}G^{\mu\nu}_a]$ in the action. The "Helmholtz equations" follow by substituting the curvature tensor $G_{\mu\nu}^a$ by its definition in terms of the gauge field. The resulting equation is a mess though, since it's non-linear and the gauge fields do not commute. So the next step of inverting the operator is probably quite tricky.

If you want an introduction to some of these matters, I suggest the book by Baez: Gauge Fields, Knots and Gravity. It's full of errors, but fun to read.

Note that in the quantum case we cannot ignore the contribution coming from self-interaction. QCD has a coupling constant of order 1, meaning that higher order terms do not converge and so perturbation theory breaks down. In the high-energy limit this is less problematic, since the coupling constant is running and due to asymptotic freedom we enter a regime in which perturbation theory is indeed possible. The low-energy limit is still a big mess though.

@ Vanadium 50, sorry if this post is still too technical, but I couldn't refuse answering ;)

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jambaugh
Gold Member
To the OP...
Since we can't observe individual quarks it is difficult to say exactly what is going on inside the nuclei of atoms w.r.t. potentials. What seems to happen is that in trying to separate two quarks the energy quickly rises above the masses of quarks and new quark-anti-quark pairs are produced.

Remember quarks are elements of a Model... a damned good model but still the "reality" of them is not quite fixed in stone. Within the quark model we can test various potentials to see which fit with observations of nuclear collisions but we can't test them directly by measuring forces between individual quarks.

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Remember quarks are elements of a Model... a damned good model but still the "reality" of them is not quite fixed in stone.
Well, yes, but you could say the same thing about atoms.

As Vanadium mentioned in his first post, the problem with potential is mostly that it is unsuitable to describe light quarks, since a potential is a non-relativistic construct. Potential models do a great job for heavy quarks, and non-relativistic QCD is a fairly well advanced path, but it does not address the confinement as it happens in our nucleons.

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Potential models do a great job for heavy quarks,
I'd say "good", rather than "great". The Bc mass predictions were all over the map, It's known that rather different looking potentials give the same energy levels in charmonium and bottomonium. So it does a good job, but IMHO, not anything one can write home about.

Equation [2.19] is the Lagrangian of Yang-Mills theory, while equation [2.24] is the definition of a non-Abelian curvature tensor. So the formulas you refer to are more or less the definition of the curvature tensor. The equations I wrote down earlier are the (classical) equations of motion you get from a term $\textrm{Trace}[G_{\mu\nu,a}G^{\mu\nu}_a]$ in the action. The "Helmholtz equations" follow by substituting the curvature tensor $G_{\mu\nu}^a$ by its definition in terms of the gauge field. The resulting equation is a mess though, since it's non-linear and the gauge fields do not commute. So the next step of inverting the operator is probably quite tricky.

If you want an introduction to some of these matters, I suggest the book by Baez: Gauge Fields, Knots and Gravity. It's full of errors, but fun to read.

Note that in the quantum case we cannot ignore the contribution coming from self-interaction. QCD has a coupling constant of order 1, meaning that higher order terms do not converge and so perturbation theory breaks down. In the high-energy limit this is less problematic, since the coupling constant is running and due to asymptotic freedom we enter a regime in which perturbation theory is indeed possible. The low-energy limit is still a big mess though.

@ Vanadium 50, sorry if this post is still too technical, but I couldn't refuse answering ;)
Ok ,thank you very much for this information, I will look it up some day. Have to understand the non-linearity and also try to get it down more explicit. If you compare self-interaction of electrons vs. electron-electron interaction it gives a very small correction (even if they look like to be of the same order), but you think the situation would be different with the quark-fields? Its not straight forward to understand the difference, but maby I try to find your reference.
/Per

Ok ,thank you very much for this information, I will look it up some day. Have to understand the non-linearity and also try to get it down more explicit. If you compare self-interaction of electrons vs. electron-electron interaction it gives a very small correction (even if they look like to be of the same order), but you think the situation would be different with the quark-fields? Its not straight forward to understand the difference, but maby I try to find your reference.
/Per
The self-interactions are indeed the largest cause of the trouble (note that they arise due to the non-linearity). In the case of quantum electrodynamics the higher order corrections become smaller and smaller. In the case of QCD they become larger and larger. And since there are an infinite amount of them, there is no clue on where the total sum ends up...

jambaugh