# Quarks and Strong Attraction Calculation

1. Dec 20, 2013

### x BlueRobot

Since Colour Force is Strong Interaction, is there a equation to calculate Strong Interaction or express it mathematically? The closest I came to is the Coupling Constant but it doesn't provide a equation for Colour Force.

I was also looking at the Gluon Field Strength Tensor, is this related to my query?

2. Dec 25, 2013

### Meir Achuz

There are simple models that have been used for calculations with the color force.
The model potentials generally feature a short range attractive interaction something like the Coulomb potential, and a long range confining part which is a linear potential something like that of a spring.
The simplest (but too simple) such potential goes like V=-a/r+kr. It is called the 'Cornell potential' because it was first suggested by people at Cornell.

3. Dec 30, 2013

### tom.stoer

Yes, there are explicit but rather formal and highly complex expressions for the "Coulomb interaction" of the QCD Hamiltonian (plus other interaction terms) depending on the gauge fixing scheme. One brief overview can be found here:

http://physik.uni-graz.at/oberwoelz2012/talks/oberwoelz2012_reinhardt.pdf - slide #9.

$H_\text{C} = - \frac{1}{2} \int J^{-1}\rho J\;(D\partial)^{-1}\,\partial^{2}\,(D\partial)^{-1}\;\rho$

J is the Jacobian determinant (Fadeev-Popov determinant)
ρ is the color charge density carried by quarks and (!) gluons
(D∂)-1 is the Coulomb kernel in the adjoint rep.

Note that in QED the covariant derivative D is replaced by ∂, simply b/c photons do not carry charge and therefore the covariant derivative is trivial in the adjoint rep. (not in the fundamental rep.). Due to the same reason J is trivial. So in QED the integral reduces to the well-known Coulomb force between electric charge densities.

In QCD ρ as well as both operators J and the Coulomb kernel (D∂)-1 depend on the gauge fields A.

Reinhardt (and collaborators) as well as other groups analyzed the IR properties of QCD in great detail. You will find several papers on arxiv.

Last edited: Dec 30, 2013