Optimum capture window in a background-dominated model ($e^x=1+2x,x=W/\tau$)

If genuine pairs arrive at rate $S_0$ before the timing cut, an exposure $T$ gives $S=S_{0}T(1-e^{-W/\tau })$ and expected accidental pairs $B=R_p{R}_{d}WT$. In a background-dominated large-count approximation, maximizing $S/\sqrt{B}$ reduces to maximizing $f(x)=(1-e^{-x})/\sqrt{x}$ for $x=W/\tau >0$. Differentiation gives the positive stationary solution $e^x=1+2x$, or $x\simeq 1.25643$. The endpoint $x=0$ solves the rearranged equation but is outside the domain of this derivative calculation; $f(x)$ tends to zero at both positive-domain extremes.

Thus the chosen $\tau =20\mu \mathrm{s}$ gives $W\simeq 25.13\mu \mathrm{s}$ for this particular criterion, accepting about $0.715$ of genuine delayed signals. This differs from maximizing signal counts alone, which would favor longer windows. Small samples, appreciable signal fluctuations, unknown background, dead time and multiple-candidate vetoes require a fuller likelihood model. The calculation isolates a tradeoff rather than prescribing an experimental cut.