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# Systematic Uncertainties in Double Ratio - PowerPoint PPT Presentation

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

Current Systematic Uncertainties

Analysis Note, Section 9.5

http://cms.cern.ch/iCMS/jsp/openfile.jsp?tp=draft&files=AN2011_062_v4.pdf

• 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.

• 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.

• 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.

• 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

• 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.

• 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.

• 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.

• 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.

• 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)

• 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

• 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%

• 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%

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

• 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%