Deduction of $M(Sigma pi)$ for $Lambda(1405)$ with mixed $T_{ Sigma pi leftarrow K^-p} and T_{ Sigma pi leftarrow Sigmapi}$ from Hemingway's data

We formulated the $Lambda(1405)$ (abbreviated as $Lambda^*$) $rightarrow (Sigmapi)^0$ invariant-mass spectra produced in the $K^- + p rightarrow Sigma^+ (1660) + pi^-$, followed by $Sigma^+ (1660) rightarrow Lambda(1405) + pi^+ rightarrow Sigmapi + pi^+$, processes at $p(K^-) = 4.2$ GeV/$c$, in which both the incident channel for a quasi-bound $K^-p$ state and its decay process to $(Sigma pi)^0$ were taken into account realistically. We calculated $M(Sigma pi)$ spectral shapes using mixed transition matrices, $T_{21} = T_{ Sigmapi leftarrow K^-p}$ and $T_{22} = T_{ Sigma pi leftarrow Sigma pi}$, for various theoretical models involving $Lambda^*$. The asymmetric spectra were compared to old experimental data of Hemingway, and it was found that the mixing of the two channels, written as $(1-f) , T_{21} + f, T_{22}$, gave a better result than considering the individual channels, yielding $f = 0.376_{-0.019}^{+0.021}$, $M(Lambda^*) = 1406.6_{-3.3}^{+3.4}$ MeV/$c^2$ and $Gamma = 70 pm 2$ MeV, nearly consistent with the 2014 PDG values.

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