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.

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