# I Effect of errors due to branching and acceptances

1. Oct 13, 2018

### CAF123

Say I have say 10 measurements of an experimental observable e.g differential cross section with respect to rapidity , one measurement in each bin and corresponding statistical and systematic uncertainties given in % per bin.

Now, suppose I want the value of this dsigma/dy after correcting for e.g acceptance and certain branching fraction.

How are the statistical and systematic errors changed? The acceptance and branching typically come with an error so in what way will they affect the statistical and systematic errors?

2. Oct 13, 2018

### Staff: Mentor

Standard error propagation for the multiplication of two numbers.
It gets more interesting if you have significant migration between the bins or other sources of correlations.

3. Oct 13, 2018

### CAF123

But does the acceptance and branching enter into the statistical or the systematic uncertainty? I think the acceptance is dependent on the rapidity so cannot be systematic (therefore statistical) and the branching would enter in the systematics?

4. Oct 13, 2018

### Staff: Mentor

What exactly is a statistic uncertainty about that? If you repeat the measurement, do you expect a different result?

In general both will come with a statistic and a systematic uncertainty. Treat them separately. Combine all sources of statistical uncertainty, combine all sources of systematic uncertainty.

5. Oct 14, 2018

### CAF123

Thanks, in the article I am studying the acceptance and branching are given each in the form $A \pm \delta A$ only, so how should I interpret the $\delta A$? Added in quadrature of the statistical and systematic errors? If that might be the case, then it seems without further information on the decomposition of this error it is impossible for me to split into statistical and systematic error contributions?

6. Oct 14, 2018

### Staff: Mentor

I don't know, I would have to see the article. If it is both combined then you can't split it of course.
If you have external sources of uncertainty it is common to treat them as separate class. Something like that: $52 \pm 2 \text{(stat)} \pm 3 \text{(syst)} \pm 1 \text{(BF)}$.

7. Oct 15, 2018

### CAF123

Indeed, if I had the errors for the branching and acceptance spelled out as $A \pm \delta A_{\text{stat}} \pm \delta A_{\text{sys}}$ for acceptance and $B\pm \delta B_{\text{stat}} \pm \delta B_{\text{sys}}$ for the branching then assuming independent sources of statistical and systematic errors, my net statistical on d(sigma)/dy would be $$\delta _{\text{stat}} = \sqrt{\delta A_{\text{stat}}^2 + \delta B_{\text{stat}}^2 + \dots}$$ and similarly for the systematic. I think that should be correct.

It's common to see experimental formulae for differential cross sections expressed in terms of parameters that are estimated by experimentalists, e.g the acceptance appears in the denominator. Could you also get the effect of the error on the differential cross section due to the acceptance through partial derivatives? And this would coincide with simply adding in quadrature?

8. Oct 16, 2018

### Staff: Mentor

I guess so - it works if I understand your description correctly.

9. Oct 18, 2018

### CAF123

Can we do this explicitly? Given d(sigma)/dyi = f(Ai,...), where Ai is the acceptance in bin i and dots denote other experimental quantities such as track efficiencies, purities etc..., we have in the notation used in previous posts $$\delta_{\text{stat}}^2 \left( \frac{d \sigma}{d y_i} \right) = \left( \frac{\delta f}{\delta A_i} \delta A_{i,\text{stat}} \right)^2 + \dots$$ so that $$\delta_{\text{stat}} \left( \frac{d \sigma}{d y_i} \right) = \sqrt{\left( \frac{\delta f}{\delta A_i} \delta A_{i,\text{stat}} \right)^2 + \dots}$$ But comparing this to the centered equation in #7, $$\delta_{\text{stat}} = \sqrt{\delta A_{\text{stat}}^2 + \dots}$$ there is an extra multiplicative factor $\left( \frac{\delta f}{\delta A}\right)^2$ in the case where I used partial derivatives. What's the reconciliation?

10. Oct 19, 2018

### Staff: Mentor

Ah, I thought you were talking about relative uncertainties in post 7. If not then you need the additional factor of course. Otherwise not even the units work out.

11. Nov 8, 2018

### CAF123

Indeed, so do you mean to say it only makes sense to add percentage uncertainties in quadrature and if one is given absolute uncertainties then the formula in #9 (2nd equation centered) should be used?

Also, was wondering, it is common to add in quadrature errors due to independent systematic sources. What is the theoretical justification for that? Statistics have the Gaussian description for large number of measurements but the systematics are not described as such.

12. Nov 9, 2018

### Staff: Mentor

That, or convert it to relative uncertainties first.
The variance still behaves just like it does for Gaussian errors. If a source of systematic uncertainty is not even close to a normal distribution (e.g. it is an interval with a minimum and maximum and everything in between is equally likely) and relevant it might make sense to treat it differently.

13. Nov 9, 2018

### dukwon

The statistical uncertainty on your result should only depend on the size of the data sample with which you made the measurement. If you get acceptance from MC and a branching fraction from some other analysis then their uncertainties contribute only to the systematic uncertainties. If you can easily factorise external branching fractions from your result, then it is common to quote its uncertainty separately, so that your result can be improved if better external measurements are made.

14. Nov 9, 2018

### CAF123

I see. So in what sense generally will the acceptance and the branching come with a statistical uncertainty and a systematic uncertainty? (as mfb mentioned in #4) That is, what experimental source would give rise to a statistical uncertainty upon the acceptance and branching?

15. Nov 9, 2018

### dukwon

Say your acceptance is calculated from MC; the statistical uncertainty is just due to the sample size.

If you take branching fractions (please don't just say "branching", it's confusing) from an external measurement, then the statistical uncertainty is due to the size of the dataset used in that analysis.

However, the effect of these uncertainties on your measurement remains systematic: they don't depend on the size of the dataset you make your measurement with.

16. Nov 9, 2018

### CAF123

Thanks. What you say makes sense for me regarding the branching fraction as this is a value found in e.g PDG and found separate from measurement under study.

But the acceptance is determined from the process at hand, so if im studying a decay $A \rightarrow BC$, say, the amount of events selected as a signal enter into the acceptance and this is based on the actual measurement under study, not some prior measurement so would qualify as largely statistical. Or have I misunderstood here? Thanks

17. Nov 10, 2018

### dukwon

I've never seen acceptance be taken from data (how do you count particles that you don't detect?) but if that's what you do, then yeah its statistical uncertainty should enter into the statistical uncertainty of the result.

18. Nov 10, 2018

### CAF123

Ah sorry probably then I’m just misunderstanding how experimentalists determine the acceptance. Is it usually always done through simulations in Monte Carlo event generators and taking the figure from there?

19. Nov 10, 2018

### Staff: Mentor

It is often a mixture of both. You'll never get a number that is completely free of Monte Carlo input but you also don't want to rely exclusively on a good MC description. What is done depends on the individual analysis. Often you can find related datasets that share some features with the main signal sample, but where you can study some of its properties better.

20. Nov 13, 2018

### CAF123

Thank you. Is there also an uncertainty which accounts for the robustness/reliability of the underlying MC method used in determining acceptances? I would say something like a 'model dependent uncertainty' but not sure if that's the correct terminology here.