Systematic Uncertainties in Double Ratio

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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

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
• 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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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)
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%
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
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%
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%
• 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
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%