## Mass of the sigma resonance

09/12/2008

One of the most hotly debated properties of the $\sigma$ resonance is the exact determination of its mass. Difficulties arise from its broadness. Indeed, in $\pi\pi$ scattering data this resonance appears with a very large peak that makes difficult a precise determination of the mass and, indeed,  a large body of data is needed to accomplish this. Initially, it was very difficult to accept the existence of this particle and, for some years, disappeared from particle listings of PDG. Recent papers, using Roy equation, proved without doubt the existence of this resonance and gave what appears the most precise determination of the mass and width so far (see here and here). This approach has been recently criticized (see here and, more recently, here) where is claimed that this approach currently underestimates the mass of the particle.

Due to such a situation, we prefer to consider another similar resonance, f0(980), whose mass is better determined giving

$m_{f0(980)}=980\pm 10\ MeV.$

Theoretically, we have built a full computation, starting with the spectrum of Yang-Mills theory, for the mass of all these resonances (see here). This paper does not use properly the mapping theorem but gives the right results. We have identified two kind of spectra (higher order spectra can also be obtained) giving

$m_1(n)=1.198140235\cdot (2n+1)\sqrt{\sigma}$

and

$m_2(n,m)=1.198140235\cdot (2n+2m+2)\sqrt{\sigma}$

being, as usual, $\sigma$ the string tension, a parameter to be computed experimentally. So, one has the spectrum of the $\sigma$ resonance and its excited states by simply taking $m,n=0$ giving

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

and

$m_{\sigma^*}=2\cdot 1.198140235\sqrt{\sigma}.$

So, taking $\sqrt{\sigma}=410\ MeV$ we get easily $m_{\sigma^*}=982\ MeV$ in close agreement with experiments, while $m_\sigma=491\ MeV$ showing that, effectively, one has currently an underestimation of this quantity. With these values we will have from the width of the $\sigma$ resonance the QCD constant $\Lambda=285\ MeV$ (see here).

Finally, a derivation of $\Lambda$ and string tension $\sigma$ from other experimental data would be critical to obtain fixed all the constants of QCD, producing immediately a proper understanding of all physics about $\sigma$ resonance.

## QCD constants from sigma resonance

09/12/2008

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.