$\pi \pi$ Partial-Wave Analysis from Reactions ${\pi }^{+}p\to {\pi }^{+}{\pi }^{-}{\Delta }^{++}$ and ${\pi }^{+}p\to K^{+}{K}^{-}{\Delta }^{++}$ at 7.1 GeV/c

We present results of an energy-dependent phase-shift analysis for $\pi \pi$ energies between 550 and 1150 MeV from reactions ${\pi }^{+}p\to {\pi }^{+}{\pi }^{-}{\Delta }^{++}$ and ${\pi }^{+}p\to K^{+}{K}^{-}{\Delta }^{++}$ at 7.1 GeV/c. The $I=0\mathrm{}$ $s$ wave is parametrized in terms of a 2 × 2 $M$-matrix coupling $\pi \pi$ and $K\overline{K}$ channels. All the obtained solutions rule out the possibility of a narrow $\epsilon$ resonance in the $\rho$ region and are characterized by a very rapid variation of the $I=0\mathrm{}$ $s$-wave amplitude near $K\overline{K}$ threshold. We show that this rapid variation can be explained by a pole in the second Riemann sheet close to the $K\overline{K}$ threshold.