= Optimum capture window in a background-dominated model
{title2=$e^x=1+2x,\quad x=W/\tau$}
If genuine pairs arrive at rate $S_0$ before the timing cut, an exposure $T$ gives $S=S_0T(1-e^{-W/\tau})$ and expected accidental pairs $B=R_pR_dWT$. 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 $x\simeq1.25643$. 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 $W\simeq25.13\,\mu\mathrm s$ 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.
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