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Using an improved "analytic K-matrix model", we reconsider the extraction ofthe sigma/f_0(600) and f_0(980) gamma-gamma widths from gamma-gamma to pi-piscatterings data of Crystal Ball and Belle. Our main results are summarized inTables 3 and 4. The averaged sigma "direct width" to gamma-gamma is 0.16(3) keVwhich confirms a previous result of [1] and which does neither favour a largefour-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 thefour-quark expectation but can be compatible with a \bar ss or gluoniumcomponent. We also found that the rescattering part of the amplitude isrelatively large indicating an important contribution of the meson loops in thedetermination of the gamma-gamma total widths. This is mainly due to the largecouplings of the sigma and f_0(980) to pi-pi and/or \bar KK, which can also bedue to a light scalar gluonium with large OZI violating couplings but notnecessary 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 dispersionrelations and PDG values. Using the parameters from QCD spectral sum rules, wecomplete our analysis by showing that the production rates of unmixed scalargluonia sigma_B(1) and G (1.5-1.6) agree with the data from J/\psi, phiradiative and D_s semi-leptonic decays.

sigma and f_0(980) substructures from gamma-gamma to pi-pi, J/psi, phi radiative and D_s semi-leptonic decays