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

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Systematic uncertainties in double ratio

Systematic Uncertainties in Double Ratio

Manuel Calderon de la Barca Sanchez


Current systematic uncertainties

  • 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 r atio

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%


Advantages of this methd

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


Systematic uncertainties in double ratio

  • 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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