## The interquark potential

27/08/2008

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.

## Ground state of toponium

27/08/2008

Following the series of posts I started after the beautiful result of BABAR collaboration, now I try to get a prevision for a new resonance, i.e. the ground state of $\bar t t$ quarkonium that is known in literature as toponium. This resonance has a large mass with respect to the others due to t quark being about 37 times more massive than b quark. In this case we have a theoretical reference by Yuri Goncharov (see here) published in Nuclear Physics A (see here). Goncharov assumes a mass for the top quark of 173.25 GeV and gets $\alpha_s(m_t)\approx 0.12$. He has a toponium ground state mass of 347.4 GeV. How does our formula compare with these values?

Let us give again this formula as

$\eta_t(1S)=2m_t-\frac{1}{4}\alpha_s^2m_t+0.876\alpha_s\sqrt{\sigma}$

being $\sqrt{\sigma}=0.44$ GeV. We obtain

$m_{\eta_t}=345.9$ GeV

that is absolutely good being the error of about 0.4% compared to Goncharov’s paper! Now, using the value of PDG $m_t=172.5$ GeV we get our final result

$m_{\eta_t}=344.4$ GeV.

This is the next quarkonium to be seen. Alhtough we can theoretically do this computation, we want just to point out that no toponium could be ever observed due to the large mass and width of the t quark (see here for a computation).