Modified $\pi\pi$ amplitude with $\sigma$ pole

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.