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When calculating the scattering cross-section via the resonance with mass $M$ one can use the Breit-Wigner formula

\begin{equation}\frac{1}{s-M^2+iM\Gamma},\end{equation}

where $\Gamma$ is the total decay width of the resonance. However this is just an approximation, which works for narrow resonances far from thresholds, because the exact formula is

\begin{equation}\frac{1}{s-M_0^2-\Sigma(s)},\end{equation}

where $\Sigma(s)$ is the self-energy. PDG review (http://pdg.lbl.gov/2015/reviews/rpp2015-rev-resonances.pdf) claims one can use

\begin{equation}\frac{1}{s-M^2-i\sqrt{s}\Gamma(s)},\end{equation}

where is the decay width calculated for resonance of mass $\sqrt{s}$instead of $M$. Moreover it is said that if there is a possible decay channel to as state of mass $m$, but $\sqrt{s}$ is below the threshold of $2m$ one needs to calculate the analytical continuation of the decay width. Should also the kinematically forbidden states (2m>M) be included in this calculation?

When threshold is close to the resonance mass $2m\approx M$ using the $\Gamma(s)$ instead of $\Gamma(M^2)$ hugely affects the resonant scattering cross-section, however the inclusion of analytic continuation in $\Gamma(s)$ gives large correction for $\sqrt{s}<2m$ even if $2m\ll M$. Is that right?

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