Global parameterization of $\pi \pi$ scattering up to 2 GeV

We provide global parameterizations of $\pi\pi$ scattering $S0$ and $P$ partial waves up to roughly 2 GeV for phenomenological use. These parameterizations describe the output and uncertainties of previous partial-wave dispersive analyses of $\pi\pi$, both in the real axis up to 1.12 GeV and in the complex plane within their applicability region, while also fulfilling forward dispersion relations up to 1.43 GeV. Above that energy we just describe the available experimental data. Moreover, the analytic continuations of these global parameterizations also describe accurately the dispersive determinations of the $\sigma/f_0(500)$, $f_0(980)$ and $\rho(770)$ pole parameters.