And AdmSmith said: "The confinement potential for the quark is often approximated by -4/3*alpha/r + Ar. Thus the force "far away" i.e. longer than 1 fm would be constant. "
Thank you both for referring me to relevant articles. Unfortunately, my ignorance is sufficient to keep me from adequately appreciating them. ( Physicist, your link appears to require a subscription.)
AdmSmith is saying that the potential is proportional to distance for larger distances. This would be equivalent to a constant force, I believe. The other term would dominate for smaller distances and would reflect a force law which is the deriviative of the potential and so it would be an inverse-square law for small distances.
Physicist is saying that the force is proportional to distance. This would be similar to stretching a spring. The potential function would then be a square law.
I will have to let you guys duke it out over this. Without further information it sure seems to be that the two claims are contradictory.
I appreciate Physicist pointing out that I was confusing my ancient and simplified description of inter-hadron force (specifically proton-proton) with inter-quark force.
If I understand what you are saying about inter-hadron force, it is that the resultant force is the sum of all the inter-quark forces in a system which will have perhaps a half-dozen quarks in play. But that this resultant will be determined totally by considerations of the inter-quark behavior. Also, that the very short distance of the inter-hadron force is not necessarily a characteristic of inter-quark forces but rather the inter-quark forces are proportional to distance over some relevant distances.
You're confusing two things. The confinement potential doesn't describe the force between two quarks; it describes (in its region of applicability, out at the periphery of a hadron) the force between a quark and the hadron in which it is bound. As a description of the force between two quarks, the potential you gave is hopelessly wrong, because it is not asymptotically free.
"Asymptotic freedom" is the name for the experimental result that the strong (i.e. color) force between two quarks goes to zero as the distance between them goes to zero. As you can see, the 1/r term in that confinement potential gives an inverse-square force, which becomes infinte at zero distance.
The best layman's description of confinement potentials can be found in chapter 34 ("Quark Confinement") in The Ideas of Particle Physics by G.D. Coughlan and J.E. Dodd. I also recommend chapter 33, "Asymptotic Freedom".