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The measurements of reactions $\pi^- p_\uparrow \to \pi^- \pi^+ n$ at 17.2GeV/c and $\pi^+ n_\uparrow \to \pi^+ \pi^- p$ at 5.98 and 11.85 GeV/c made atCERN with polarized targets provide a model-independent andsolution-independent evidence for a narrow scalar state sigma(750). Theoriginal chi^2 minimization method and the recent Monte Carlo method foramplitude analysis of data at 17.2 GeV/c are in excellent agreement. Bothmethods find that the mass distribution of the measured amplitude $|\overline S|^2\Sigma$ with recoil transversity ``up'' resonates near 750 MeV while theamplitude $|S|^2\Sigma$ with recoil transversity ``down'' is large andnonresonating. The amplitude $|S|^2\Sigma$ contributes as a strong backgroundto S-wave intensity I_S = (|S|^2 + |\overline S |^2)\Sigma$ and distorts thedeterminations of $\sigma$ resonance parameters from $I_S$. To avoid thisproblem we perform a series of Breit-Wigner fits directly to the measureddistribution $|\overline S |^2\Sigma$. The inclusion of various backgroundscauses the width of sigma(750) to become very narrow. Our best fit with$t$-averaged coherent background yields $m_\sigma = 753 \pm 19$ MeV and$\Gamma_\sigma = 108 \pm 53$ MeV. These values are in excellent agreement withEllis-Lanik theorem for the width of scalar gluonium. The gluoniuminterpretation of $\sigma(750)$ is also supported by the absence of$\sigma(750)$ in reactions $\gamma\gamma \to \pi\pi$. We also show how data onpolarized target invalidate essential assumptions of past determinations of$\pi\pi$ phase shifts .Comment: 77 page

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Mass and width of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>(750) scalar meson from measurements of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi><mml:mi>N</mml:mi><mml:mo>→</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mi>−</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo…

Mass and width of theσ(750) scalar meson from measurements ofπN→π−π+