QCD constants from sigma resonance

In our recent paper we were able to compute a relevant property of the $\sigma$ resonance. We have obtained its decay width as (see here):

$\Gamma_\sigma = \frac{6}{\pi}\frac{G^2_{NJL}}{4\pi\alpha_s} m_\sigma f_\pi^4\sqrt{1-\frac{4m_\pi^2}{m_\sigma^2}}$

being $G_{NJL}\approx 3.76\frac{4\pi\alpha_s}{\sigma}$ , $\sigma$ the string tension that is one of the constants to be determined, and $f_\pi\approx 93\ MeV$ the pion decay constant. From asymptotic freedom we know that

$\alpha_s(Q^2)=\frac{12\pi}{(33-2n_f)\ln\left(\frac{Q^2}{\Lambda^2}\right)}$

being $n_f$ the number of flavors and $\Lambda$ the scale where infrared physics sets in and is another constant to be computed. We have computed the mass of the $\sigma$ resonance from the gluon propagator obtaining

$m_\sigma\approx 1.198140235\sqrt{\sigma}$

and we have all the theoretical data to compute $\sqrt{\sigma}$ and $\Lambda$ from experimental data. This can be accomplished using two main references (here and here) that give:

$\sqrt{s_\sigma} = 441^{+16}_{-8}-i279^{+9}_{-14.5}\ MeV$

and

$\sqrt{s_\sigma} = 460^{+18}_{-19}-i255^{+17}_{-18}\ MeV$

respectively. These produce the following values for QCD constants

$\sqrt{\sigma}=368\ MeV\ \Lambda=255\ MeV$

and

$\sqrt{\sigma}=384\ MeV\ \Lambda=266\ MeV.$

We have not evaluated the errors being this a back of envelope computation. A striking result is that the ratio of these constants is the same in both cases giving the pure number 1.44 that I am not able to explain.

These results, besides being truly consistent, are really striking making possible a deep understanding of QCD. I would appreciate any reference about these values and on their determination, both theoretical and experimental, from other processes in QCD. These values are critical indeed for strong interactions.

This entry was posted on Tuesday, December 9th, 2008 at 1:21 pm and is filed under Physics, QCD. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

The Gauge Connection