Search for the decay $K^+\to \pi^+ \nu \bar\nu$ in the momentum region $P_\pi < 195 {\rm ~MeV/c}$

We have searched for the decay $K^+ \to \pi^+ \nu \bar\nu$ in the kinematic region with pion momentum below the $K^+ \to \pi^+ \pi^0$ peak. One event was observed, consistent with the background estimate of $0.73\pm 0.18$. This implies an upper limit on $B(K^+ \to \pi^+ \nu \bar\nu)< 4.2\times 10^{-9}$ (90% C.L.), consistent with the recently measured branching ratio of $(1.57^{+1.75}_{-0.82}) \times 10^{-10}$, obtained using the standard model spectrum and the kinematic region above the $K^+ \to \pi^+ \pi^0$ peak. The same data were used to search for $K^+ \to \pi^+ X^0$, where $X^0$ is a weakly interacting neutral particle or system of particles with $150 < M_{X^0} < 250 {\rm ~MeV/c^2}$.