  A set of well known once subtracted dispersion relations with imposed
crossing symmetry condition is used to modify unitary multichannel $S$
($\pi\pi$, $K \bar K$, and $\eta\eta$) and $P$ ($\pi\pi$, $\rho 2\pi$, and
$\rho\sigma$) wave amplitudes mostly below 1 GeV. Before the modifications,
these amplitudes significantly did not satisfy the crossing symmetry condition
and did not describe the $\pi\pi$ threshold region. Moreover, the pole of the
$S$ wave amplitude related with the $f_0(500)$ meson (former $f_0(600)$ or
$\sigma$) had much smaller imaginary part and bigger real one in comparison
with those in the newest Particle Data Group Tables. Here, these amplitudes are
supplemented by near threshold expansion polynomials and refitted to the
experimental data in the effective two pion mass from the threshold to 1.8 GeV
and to the dispersion relations up to 1.1 GeV. In result the self consistent,
i.e. unitary and fulfilling the crossing symmetry condition, $S$ and $P$ wave
amplitudes are formed and the $\sigma$ pole becomes much narrower and lighter.
To eliminate doubts about the uniqueness of the so obtained sigma pole position
short and purely mathematical proof of the uniqueness of the results is also
presented. This analysis is addressed to a wide group of physicists and aims at
providing a very effective and easy method of modification of, many presently
used, $\pi\pi$ amplitudes with a heavy and broad $\sigma$ meson without
changing of their original mathematical structure.

High Energy Physics - Phenomenology