Further search for the decay $K^+ \to \pi^+ \nu \bar \nu$ in the momentum region P < 195 MeV/c

We report the results of a search for the decay $K^+ \to \pi^+ \nu \bar \nu$ in the kinematic region with $\pi^+$ momentum $140 < P < 195$ MeV/c using the data collected by the E787 experiment at BNL. No events were observed. When combined with our previous search in this region, one candidate event with an expected background of $1.22 \pm 0.24$ events results in a 90% C.L. upper limit of $2.2 \times 10^{-9}$ on the branching ratio of $K^+ \to \pi^+ \nu \bar \nu$. We also report improved limits on the rates of $K^+ \to \pi^+ X^0$ and $K^+ \to \pi^+ X^1 X^2$ where $X^0, X^1, X^2$ are hypothetical, massless, long-lived neutral particles.