## 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.