Study of $K^{+}{\pi }^{-}$ scattering in the reaction $K^{+}p\to K^{+}{\pi }^{-}{\Delta }^{++}$ at $12 \frac{\mathrm{GeV}}{c}$

We have studied $K^{+}{\pi }^{-}$ elastic scattering in the reaction $K^{+}p\to K^{+}{\pi }^{-}{\Delta }^{++}$ at $12 \frac{\mathrm{GeV}}{c}$ and in the $K\pi$ mass interval 800 to 1000 MeV. We have performed a partial-wave analysis in this $K\pi$ mass region, dominated by the $p$-wave resonance $K^{*}\mathrm{}\left(890\right)\mathrm{}$, in order to obtain information about the $s$-wave amplitude. We have extrapolated the $K^{+}{\pi }^{-}$ moments, the total cross section, and $p$-wave cross section to the pion pole. The $p$-wave cross section is close to the unitarity limit and can be described by a Breit-Wigner resonance form, with parameters $M=896\mathrm{}\pm 2$ MeV and $\Gamma =47\mathrm{}\pm 3$ MeV. We then perform an energy-independent phase-shift analysis of the extrapolated moments and total cross section using this Breit-Wigner form for the $p$ wave and a previously determined small negative phase shift for the $I=\frac{3}{2}s$ wave. For the $I=\frac{1}{2}s$-wave phase shift we find the so called "down" solution, which has a phase shift that rises slowly from 20° at $M\left(K\pi \right)=800\mathrm{}$ MeV to 60° at $M\left(K\pi \right)=1000\mathrm{}$ MeV. The energy dependence of this phase shift is well described by an effective range form, with a scattering length $a_0^1=-0.33\mathrm{}\pm 0.05$ F. The so-called "up" solution is eliminated or has large ${\chi }^2$ everywhere except for two overlapping mass intervals at $M\left(K\pi \right)=890\mathrm{} \mathrm{and} 900$ MeV. However, due to limited statistics, we expect two solutions for the $s$ wave very near the mass where the $p$ wave is resonant. We then perform an energy-dependent partial-wave analysis and find again no evidence for an $s$-wave resonance although, due to limited statistics, we could not exclude one at 890 MeV with $\Gamma <7\mathrm{}$ MeV.