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### Systematic Uncertainties in Double Ratio

Manuel Calderon de la Barca Sanchez

Signal PDF

- Background PDF
- Resolution

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%

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.

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