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

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.

### 4 Responses to Narison, Ochs, Mennessier and the width of the sigma

1. hermann says:

What do u think about the paper of Colangelo, Di Fazio etc in http://arxiv.org/PS_cache/hep-ph/pdf/0703/0703316v1.pdf ?
Do u believe their approach could be correct ?

2. mfrasca says:

Dear hermann,

I think that this is very illuminating about strengths and weaknesses of such an approach.

Marco

The abstract seems to claim that the sigma is a glueball. I thought the lightest glueball was in the 1.2-1.5GeV range. The sigma never fit standard meson, theory. But these a whole nonet of these 0+ mesons, and a nonet rather want quarks to explain it. Sigma is often claim to be a tetraquark state, with varying degree of success. Its can’t be a ordinary quark anti-quark pair, because quarks have -parity and anti-quarks have +ve, so you get negative party for quark, anti-quark pair.

I seems odd that claim our understanding of the sigma is precise from a decay rate, when we don’t seem to have a good idea of what the sigma actually is, or is made of.

4. mfrasca says:

About the nonet there are several points to fix yet before to claim this is really an explanation. But a tetraquark state should have a larger width for $\gamma\gamma$ decay as shown by Narison et al. One more reason is that the screening potential for lower mass quarks cannot support tetraquark states that instead should be seen for heavier mass quarks. I expect the residual potential to be too weak for these lower states to exist. So, nonet idea may be highly misleading to understand the nature of the sigma.
Finally, my computation uses directly QCD and a gradient expansion on it proving without any doubt that $\sigma$ is indeed a glueball in agreement with the computations of Narison et al. The agreement is really astounding.
As you can see here, there are some explanations around about this particle but none seems to fit the bill properly. The idea that $\sigma$ is a glueball is really innovative and does not appear to contradict any sacred principle of physics. Let us wait and see.