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Systematic Uncertainties in Double Ratio

Systematic Uncertainties in Double Ratio . Manuel Calderon de la Barca Sanchez. Signal PDF Background PDF Resolution. Current Systematic Uncertainties. Analysis Note, Section 9.5 http ://cms.cern.ch/iCMS/jsp/openfile.jsp?tp=draft&files=AN2011_062_v4.pdf. Signal PDF : Current Procedure.

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Systematic Uncertainties in Double Ratio

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  1. Systematic Uncertainties in Double Ratio Manuel Calderon de la Barca Sanchez

  2. Signal PDF • Background PDF • Resolution Current Systematic Uncertainties Analysis Note, Section 9.5 http://cms.cern.ch/iCMS/jsp/openfile.jsp?tp=draft&files=AN2011_062_v4.pdf

  3. Signal PDF : Current Procedure • The mass resolution and CB-tail parameters are fixed in the nominal fit to their MC estimated values a= 1.6 and npow = 2.3. • When calculating the systematic, we change the fixed a and n parameters by a random amount using their covariance matrix. • This is repeated 500 times and the systematic error is taken as three times of the rms/mean.

  4. Background PDF : Current Procedure • The nominal background model is a second order polynomoinalthroughout the 8-14 GeVcc mass-fitting range. • As a variation, a linear model is employed in the restricted mass-fitting range 8-12 GeV/c2. • The difference in the fitted parameters is taken as a systematic. • When performing a simultaneous fit to both samples, the background parameters are allowed to float independently for both samples. • As a fit variation, these parameters are constrained to be the same.

  5. Resolution : Current Procedure • When varying input resolution, the fitter is very stable and robust for the ppsample. • But for the PbPb sample, there is a large fluctuation in the fit result. Since there isn’t enough statistics in data, we studied the HI MC to find out the resolution. • In the nominal case, the resolution is fixed to its MC value(0.092 MeV), which is consistent with the 7 TeVppmeasurement(0.096 MeV). • We vary it with 3 times of the error of the MC uncertainty (+/- 0.004 MeV) to study the systematic. (Note, these should probably be GeV) • For the 500 toy experiments, we fixed the resolution to different value with a gaussian function. • The mean is set to 0.092 and sigma equals to 0.004. • Thesystematic error is taken as the rms/mean of the toy experiments.

  6. Current Results • From: espace web page (Zhen’s page) • https://espace.cern.ch/cms-quarkonia/heavyion/new_systematic.aspx • R_2   = Y(2S) / Y(1S)   :   raw yields ratioR_23 = [Y(2S)+Y(1S)] / Y(1S)   :   raw yields ratio • Chi_2 = R_2(PbPb) / R_2(pp)  :  double ratioChi_23 = R_23(PbPb) / R_23(pp)  :double ratio

  7. Crystal Ball parameter variation • Top: • R23PbPb: 0.001185/0.256 = 0.46% • Bottom: • χ23. : 0.00088/0.3139 = 0.25% • Each has a different set of Crystal Ball parameters • Change in a and npow by random amount using their covariance matrix • Problem: • Size of variation is governed by MC statistics • Both a and npowvary independently for the 1S and for the 2S (and 3S). • This scheme says that the tails will be different for the different states! • We do not have any reason to expect that the crystal ball tails will differ amongst the states.

  8. s variation • Top plot: • R23PbPb: 0.009473/0.2435 = 3.9% • Bottom plot: • χ23. : 0.009698/0.3143 = 3.1% • Vary s = 92 MeV with 3 times of the error of the MC uncertainty (+/- 4MeV) to study the systematic • Problem: • Size of variation is governed by MC statistics • If s varies independently for the 1S, 2S, 3S states, we are saying that resolution is different. • Can the resolution of the 3S be better than the resolution of the 1S? No. But procedure allows it. • Variation of resolution should not be independent for the different states.

  9. Systematic Uncertainty issues • Signal PDF (CB parameters) and Resolution systematic uncertainties • rely on size of MC statistics • allow parameters to vary independently between states (?) • Note: not sure about this, but from looking at Zhen’s procedure, it seems like it does. • Can we do an improved estimate? • Should not depend on the MC statistics • Parameters (either of CB or the s of the mass resolution) should vary according to known physics • Example presented here: • Systematic Uncertainty in Resolution, s. • Variation in Signal PDF (CB parameters) is already a small effect, so will focus only on resolution here.

  10. Do single ratios depend on a fixed s? • Resolution governed by s • Gaussian std. deviation. • Choosing a different (fixed) s will • change yields. • will not change ratio, to first order • R2 = • s cancels in single ratio. • Only if s is different between states will this play a role for single ratios. • There is a variation, but it is not random: • Resolution parameter s increases linearly with pT. Hence it increases with mass. • We need a function s(m) to embody this dependence.

  11. Resolution indeed should not affect ratio • https://espace.cern.ch/cms-quarkonia/heavyion/lowEta.aspx • Left: floating sigma, 79 MeV. (2s+3s)/1s = 0.76, 2s/1s = 0.44 • Right: fixed sigma at 95 MeV. (2s+3s)/1s = 0.75, 2s/1s = 0.44 • (From Zhen’s page): we find that the resolution affects the yields but notaffect the yield ratio • Similar arguments will apply for the variation in the CB shape: it should affect yields but not ratio.

  12. Variation of Resolution with mass • Resolution at J/y mass: • 30 MeV, |y|<1.4 • 47 MeV, 1.4 < |y| < 2.4 • Guesstimate average ~ 36 ± 5 MeV • Resolution at ϒ mass: • 92 MeV (from MC) • 79 MeV • fit to pp data • 114 MeV • fit to PbPb data • Numbers from Zhen’s resolution page.

  13. Proposal: Use s(m) for systematic uncertainty estimate • Default case: • s(J/y) = 36, s(ϒ)=92 • s(ϒ) as in MC • Line Fit to blue points • pp Test case • s(J/y) = 41, s(ϒ)=79 • s(ϒ)as in fit to pp data • Line Fit to red points • PbPbTest case • s(J/y) = 31, s(ϒ)=114 • s(ϒ)as in fit to PbPbdata • Line Fit to green points • Compare ratios obtained from above test cases. • Variation in ratios between cases: systematic uncertainty • Here I do a quick estimate using Gaussians, I’m sure it can be improved.

  14. Single ratios in pp, with default s(m). • Default case, pp • Use 3 Gaussians (one for each state) • Amplitudes: 1s: 2s: 3s = 100 : 44 : 31 • Means : masses of each state • Sigmas (in MeV): 1s : 2s : 3s = 92 : 97.0 : 99.9 • Ratios: • 2s/1s = 0.464 • (2s+3s)/1s = 0.800 • Note: if sigmas were held constant (at any value), ratios would be exactly • 44/100 = 0.44 and • (44+31)/100 = 0.75. • Difference from 0.44 and 0.75 due to change in resolution as a function of mass, s(m) • Next, change slope of s(m)

  15. Systematic uncertainty on single ratio, pp case • Use fit to red points. • Sigmas (in MeV): 1s : 2s : 3s = 79 : 82.4 : 84.3 • Ratios for this case: • 2s/1s = 0.458 • (2s+3s)/1s = 0.790 • Single Ratios, pp: • Systematic Uncertainty • 2s/1s : (0.463-0.458)/0.463 • → 1.1% • (2s+3s) : (0.800-0.790)/0.800 • → 1.3%

  16. Single ratios in PbPb, with default s(m). • Default case, pp • Use 3 Gaussians (one for each state) • Amplitudes: 1s: 2s: 3s = 100 : 16.7 : 9.3 • Means : masses of each state • Sigmas (in MeV): 1s : 2s : 3s = 92 : 97.0 : 99.9 • Ratios: • 2s/1s = 0.1760 • (2s+3s)/1s = 0.2770 • Note: if sigmas were held constant (at any value), ratios would be exactly • 16.7/100 = 0.167 and • (16.7+9.3)/100 = 0.26. • Difference from 0.167 and 0.26 is due to change in resolution as a function of mass, s(m) • Next, change slope of s(m), using PbPb data

  17. Systematic uncertainty on single ratio, PbPb case • Use fit to green points. • Sigmas (in MeV): 1s : 2s : 3s = 114 : 121.3 : 125.7 • Ratios for this case: • 2s/1s = 0.1778 • (2s+3s)/1s = 0.2803 • Single Ratios PbPb : • Systematic Uncertainty • 2s/1s : (0.1778-0.176)/0.176 • → 1.0% • (2s+3s) : (0.2803-0.277)/0.277 • → 1.2%

  18. Systematic Uncertainty on Double Ratio • Default case: • Both pp and PbPbsigmas consistent with MC • Both vary as in fit to blue points. • Double ratio 1: 0.2770/0.8003 = 0.3461 • Systematic change: • pp sigma varies as in fit to pp data. • As in fit to red points. • PbPb sigma varies as in fit to PbPb data. • As in fit to green points. • Double ratio 2 : 0.2803/0.7897 = 0.3549 • Systematic Uncertainty: • (0.3549-0.3461)/0.3549 = 2.5%

  19. Advantages of this methd • Uncertainty estimate • Is data driven • Uses measured resolutions at J/y and ϒ mass. • Does not depend on the MC statistics. • Depends on two reasonable models of the variation of the resolution with mass. • One has identical variation in PbPb and pp • One has a different slope between PbPb and pp

  20. Uncertainty estimate • Does not depend on the MC statistics. • Depends on behavior of detector: variation of the resolution with mass. • Uses two reasonable test cases: • One has identical s(m) variation in PbPb and pp • One has a different slope for s(m) between PbPb and pp • One slope is from MC, the others are data-driven • Slopes are from measured resolutions at J/y and ϒ mass. Conclusion:Systematic Uncertainty on Double Ratiodue to Uncertainty in our knowledge of the mass resolution: 2.5%

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