How do I calculate the error on the efficiency of a cut, when the numbers of events before and after the cut were obtained as the signal components of two separate fits to signal plus background?
The difficulty of this problem is that background fluctuations introduce correlations between the signal samples before and after the cut. The solution is to produce a histogram of events that fail the cut, and to fit that to signal plus background. We can then use the fact that the number of signal events from a fit to the histogram of events that pass the cuts is uncorrelated with the number of signal events from a fit to the histogram of events that fail the cuts, since no event is present in both histograms. With some further assumptions, we don’t even have to produce this histogram of events that fail the cut, because we can predict the outcome of its fit from the two fits that are given to us. Let N ± dN be the fit result for the number of signal events before the cut, n ± dn the fit result for the number of signal events that pass the cut, and m ± dm the fit result for the number of signal events that fail the cut. The efficiency we are interested in is then p = n/N, but for rea
