Using an improved "analytic K-matrix model", we reconsider the extraction of
the sigma/f_0(600) and f_0(980) gamma-gamma widths from gamma-gamma to pi-pi
scatterings data of Crystal Ball and Belle. Our main results are summarized in
Tables 3 and 4. The averaged sigma "direct width" to gamma-gamma is 0.16(3) keV
which confirms a previous result of [1] and which does neither favour a large
four-quark (diquark-antidiquark) nor a molecule nor a pure \bar qq component.
The "direct width" of the f_0(980) of 0.28(2) keV is much larger than the
four-quark expectation but can be compatible with a \bar ss or gluonium
component. We also found that the rescattering part of the amplitude is
relatively large indicating an important contribution of the meson loops in the
determination of the gamma-gamma total widths. This is mainly due to the large
couplings of the sigma and f_0(980) to pi-pi and/or \bar KK, which can also be
due to a light scalar gluonium with large OZI violating couplings but not
necessary to a four-quark or molecule state. Our average results for the total
(direct+rescattering) gamma-gamma widths: Gamma_sigma^{tot}= 3.08(82) keV,
Gamma_{f_0}^{tot}= 0.16(1) keV} are comparable with the ones from dispersion
relations and PDG values. Using the parameters from QCD spectral sum rules, we
complete our analysis by showing that the production rates of unmixed scalar
gluonia sigma_B(1) and G (1.5-1.6) agree with the data from J/\psi, phi
radiative and D_s semi-leptonic decays.Comment: 11 pages, 6 figures (1 more figure added in Fig 6), 4 tables :
version appeared on-line in doi:10.1016/j.physletb.2010.12.00
