## Narison, Ochs, Mennessier and the width of the sigma

09/01/2009

In order to understand what is going on in the lower part of the meson spectrum of QCD that is currently seen in experiments one would like to have an explicit formula for the width of the sigma. The reason is that we would like to have an idea of its broadness. Being this the infrared limit the only known way to get this would be lattice computations but in this case there is no help. Lattice computations see no sigma resonance anywhere. Narison, Ochs and Mennessier were able to obtain an understanding of this quantity by QCD spectral sum rules here and here. They get the following phenomenological equation

$\Gamma_\sigma=\frac{|g_{\sigma\pi^+\pi^-}|^2}{16\pi m_\sigma}\sqrt{1-\frac{4m_\pi^2}{m_\sigma^2}}$

being the coupling $|g_{\sigma\pi^+\pi^-}|\approx (4\sim 5)\ GeV$ explaining in this way why this resonance is so broad. Their main conclusion, after computing the width of the reaction $\sigma\rightarrow\gamma\gamma$, is that this resonance is a glueball.

In our latest paper (see here) we computed the width of the sigma directly from QCD. We obtained the following equation

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

being $G_{NJL}\approx 3.76\frac{4\pi\alpha_s}{\sigma}$, $\sigma$ the string tension that we take about 410 MeV, and $f_\pi\approx 93\ MeV$ the pion decay constant. The mass was given by

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

This permits us to give the coupling in the Narison, Ochs and Mennessier formula as

$|g_{\sigma\pi^+\pi^-}|\approx 156.47\sqrt{\frac{\alpha_s}{\sigma}}f^2_\pi$

giving in the end

$|g_{\sigma\pi^+\pi^-}|\approx 3.3\sqrt{\alpha_s}\ GeV$

in very nice agreement with their estimation. We can conclude that their understanding of $\sigma$ is quite precise. An interesting conclusion to be drawn here is about how good turn out to be these techniques based on spectral sum rules. The authors call these methods with a single acronym QSSR. They represent surely a valid approach for the understanding of the lower part of QCD spectrum. Indeed, QCD calculations prove that this resonance is a glueball.