## Mass of the sigma resonance

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.