Product of sWeights and Efficiency Weightings

Hi experts,

I have a question regarding a method I am trying to implement (as it is not currently working). I am currently trying to write a toy MC that I use to get a value for a lifetime. The steps I take to achieve this are thus.

I extract the shape of the total decay time distribution for the channel from MC, and use this shape to generate candidates (and I also generate combinatorial background using a different shape). I then mass fit the distribution in order to use sPlot to extract an sWeight for each event (as there is a lot of combinatorial background and very little signal). If I was to weight by these sWeights, I’d be left with a distribution which is the true decay time distribution multiplied by the acceptance of the detector (i.e with the background removed).

Because of this aforementioned detector acceptance, I have to correct for the low detector efficiency that occurs at low lifetimes. To do this I generate another decay time distribution using the same shape (extracted from MC) used to generate the candidates, and then reweight this distribution by the inverse of the true lifetime distribution (i.e exp(t/lifetime)) to get just an acceptance distribution. I then sample a binned version of this acceptance distribution to get an efficiency value per binned value of the lifetime.

I then take the reciprocal of these efficiencies and assign these 1/efficiency values to each of my original data points (i.e the ones with an sWeight) depending on their lifetimes values, in order to correct for the acceptance effects (effectively each data has two weights, sWeight and efficiency weight). I then multiply these two weights together to get a product weight and normalise these weights so that their sum is equal to the sum of the sWeights. I then weight the data by this product weight. My hope was this product weight would remove the background and would also leave me with just an exponential distribution of the true lifetime distribution, with the acceptance having been corrected for.

I am finding that this method is not really working, the distribution I am being left with after reweighting by this product weight is not really exponential and the fit does not extract the correct lifetime. I am basically wondering if what I am doing with the weights (i.e multiplying sWeights by another weight) is rigorous, and if anyone has any experience of having done something similar/has tried a similar approach before? I am particularly interested in information about how to account for efficiency after sWeighting.

Any help would be greatly appreciated.