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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