# Relative Luminosity Systematic

Documentation for a systematic uncertainty introduced in the Run 5 charged pion A_{LL} based on uncertainties in our measurement of the relative luminosity.

### BBC/ZDC Comparison

Murad compared relative luminosities from the BBCs and ZDCs and concluded that we know R3 with an absolute error of (2.45±1.26)x10-4. His analysis is documented here. To translate this uncertainty on R3 into a systematic on A_{LL} we can do

Define the quantity B (B << 1) and use the binomial expansion twice to get an expression for δA_{LL}:

Keep only the leading term to arrive at the end result assuming final Run 5 beam polarizations:

So δA_{LL} from this portion is 4.9e-4 (absolute). (note: in Murad’s link he says 5.6e-4, but in later work he agrees with this value of 4.9e-4)

### Beam-Gas Background

Beam-gas background also contributes to our systematic uncertainty from relative luminosities. Kasia did a study in which she concluded that beam-gas background contributes at most 3e-4 to δA{LL} for the jet result. I’ve already shown above that contributions to δA{LL} from relative luminosity systematics are essentially final-state-independent (that is, the contribution to δA{LL} does not depend on the value of A{LL} in the limit of small asymmetries), so it’s fine to just use Kasia’s number directly for this analysis.

### Summary

If I add these two numbers (4.9, 3.0) in quadrature I get an **absolute uncertainty of 5.75e-4** for this particular systematic. Thanks Murad and Kasia for doing all this analysis!

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