Pion mass

04/09/2008

We have seen in preceding posts how good was the computation of ground states of quarkonia obtaining the interquark potential from the gluon propagator and solving the Dirac equation (see here). Here we try a more ambitious aim: We compute the pion mass from the interquark potential in the limit of very light quarks but assuming them to be not relativistic that is a drastic assumption. So, the interquark potential is given by (see here)

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

but in case of light quarks we can take just the first term, that is

$V(r)\approx -\frac{\alpha_s}{r}B_0e^{-m_0r}$

and so our problem reduces to the one of solving the Schroedinger equation with the Yukawa potential. This is a well-known problem. To get the ground state we have used this paper by A.E.S. Green. Then, our final formula is

$m_\pi=2m_q-\frac{m_0^2}{m_q}\frac{\alpha_s}{2}B_0\left(\frac{\alpha_s}{2}B_0-As^2\right)$

being $m_q=350$ MeV the constituent mass quark, $\alpha_s=1.47$ the strong coupling constant, $m_0=1.19814\sqrt{\sigma}$ with $\sigma=(440MeV)^2$ the string tension, $B_0=1.144231$, $A=1.9875$ and $s=0.03951$ two constants of the energy level computation from Green’s paper. So, finally we get the satisfactory value $m_\pi\approx 140$ MeV in good agreement with experimental value taking into account of how rough was our computation.

We cannot claim this as a full success but rather as a simple exercise showing how knowing the proper gluon propagator can give a serious hint on computation of all the relevant quantities in QCD and this has been the main aim of such analysis.

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.