  We systematically reconsider, within an improved "analytic K-matrix model",
the extraction of the sigma = f_0(600) and f_0(980) masses, widths and hadronic
couplings using new Ke4 = K-->pi-pi e nu_e data on pi-pi phase shift below 390
MeV and different sets of pi-pi--> pi-pi / K-K scatterings data from 400 MeV to
1.4 GeV. Our results are summarized in Tables 1, 2 and 5. In units of MeV, the
complex poles are: M_sigma=452(12) - i 260(15) and M_f=981(34) -i 18(11), which
are comparable with some recent high-precision determinations and with PDG
values. Besides some other results, we find: |g_{sigma K+K-}|/|g_{sigma
pi+pi-}|=0.37(6) which confirms a sizeable g_{sigma K+K-} coupling found
earlier, and which disfavours a large pi-pi molecule or four-quark component of
the sigma, while its broad pi-pi width (relative to the one of the rho-meson)
cannot be explained within a \bar qq scenario. The narrow pi-pi width of the
f_0(980) and the large value: |g_{f K+K-}|/|g_{f pi+pi-}|=2.59(1.34), excludes
its pure (\bar uu+\bar dd) content. A significant gluonium component eventually
mixed with \bar qq appears to be necessary for evading the previous
difficulties.
