## The interquark potential

In our initial post about quarkonia we have derived the interquark potential from the gluon propagator. In this post we want to deepen this matter being this central to all hadronic bound states. The gluon propagator is given by

$G(p^2)=\sum_{n=0}^\infty\frac{B_n}{p^2-m_n^2+i\epsilon}$

being

$B_n=(2n+1)\frac{\pi^2}{K^2(i)}\frac{(-1)^{n+1}e^{-(n+\frac{1}{2})\pi}}{1+e^{-(2n+1)\pi}}$

and

$m_n = (2n+1)\frac{\pi}{2K(i)}\sqrt{\sigma}$

with $\sqrt{\sigma}=0.44$ GeV being $\sigma$ the string tension that is just an integration constant of Yang-Mills theory, arising from conformal invariance, to be fixed experimentally. We have obtained this propagator in a series of papers starting from a massless scalar theory. The most relevant of them is here. It is immediate to recognize that this propagator is just an infinite superposition of Yukawa propagators. But the expectations from effective theories are quite different (see Brambilla’s CERN yellow report here). Indeed, a largely used interquark potential is given by

$V(r)=-\frac{a}{r}+\sigma r +b$

but this potential is just phenomenological and not derived from QCD. Rather, as pointed out by Gocharov (see here) this potential is absolutely not a solution of QCD. We note that it would be if the linear term is just neglected as happens at very small distances where

$V_C(r)\approx -\frac{a}{r}+b$.

We can derive this potential from the gluon propagator imposing $p_0=0$ and Fourier transforming in space obtaining

$V(r)=-\frac{\alpha_s}{r}\sum_{n=0}^\infty B_n e^{-m_n r}$

and we can Taylor expand the exponential in r obtaining

$V(r)\approx -\frac{\alpha_s}{r}+Ar+b$

but we see immediately that $A=0$ and so no linear term exists in the potential for heavy quarkonia! This means that we can formulate a relativistic theory of heavy quarkonia by solving the Dirac equation for the corresponding Coulombic-like potential whose solution is well-known and adding the b constant. We will discuss such a spectrum in future posts.

For lighter quarks the situation is more involved as we have to take into account the full potential and in this case no solution is known and one has to use numerical computation. But solving Dirac equation on a computer should be surely easier than treating full QCD.

### 3 Responses to The interquark potential

1. jfa says:

How do you reconcile your result with the lattice results for the interquark potential? They look pretty linear at large distances.

2. mfrasca says:

The question of lattice computations is not that simple. I have taken as an example the paper http://arxiv.org/abs/0806.3554 published on PRD. As for all these kind of papers one take very small volumes (here is $16^4$). All people that worked on the question of gluon and ghost propagators know that for very small volumes you are not able to really understand what is going on. In order to get an understanding a volume of $128^4$ was reached lowering $\beta$ that in that paper above is 6 and there is no need to take it so high. So, what you see as a linear term in such small volumes could be as well a constant at larger volumes reducing the slope (a thing that is seen varying the hopping parameter in the above reference) even if such checks have never been done due to people expectations that such a linear term must be there.

Aside from above considerations, there is something more to say about as e.g. people computing pure Yang-Mills spectrum miss the true ground state of the theory due to a too much large lattice spacing (again a volume effect) that instead is properly hit by the lattice propagator as you can see in my post https://marcofrasca.wordpress.com/2008/06/27/meaning-of-lattice-results-for-the-gluon-propagator/. This is blatantly evident as no one is able to recover $\sigma$ resonance from lattice computations. So, what really matters in the end is agreement of any computation with experiment. And I surely trust reality rather than whatever lattice computation still working at unrealistic volumes.

Finally, my propagator agrees fairly well with large volume lattice computations and numerical solution of Dyson-Schwinger equations and this support is enough for me to trust my conclusions (e.g. see my paper http://arxiv.org/abs/0803.0319 for such a comparison).

Marco

3. […] a comment about my analysis of quarkonia (see here) it was questioned by James Amundson at Fermilab that my potential does not seem to agree with the […]