Precise measurement of ๐and ๐ช - ias.ust.hkias.ust.hk/program/shared_doc/2020/202001hep/...2...
Transcript of Precise measurement of ๐and ๐ช - ias.ust.hkias.ust.hk/program/shared_doc/2020/202001hep/...2...
Precise measurement of ๐๐พ and ๐ช๐using threshold scan method
Peixun Shen, Paolo Azzurri, Zhijun Liang, Gang Li, Chunxu Yu
NKU, INFN Pisa, IHEP
IAS Program on High Energy Physics 2020
2020/1/18, HKUST
1IAS Program on High Energy Physics 2020 [email protected]
Outline
Motivation
Methodology
Statistical and systematic uncertainties
Data taking schemes
Summary
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Motivation
The mW plays a central role in precision EW
measurements and in constraint on the SM model
through global fit.
๐บ๐น =๐๐ผ
2๐๐2 sin2 ๐๐
1
(1+ฮ๐)
ฮ๐ is the correction, whose leading-order
contributions depend on the ๐๐ก and ๐๐ป
Several ways to measure ๐๐:
The direct method, with kinematically-constrained or mass
reconstructions
Using the lepton end-point energy
๐+๐โ threshold scan method (used in this study )
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W.J. Stirling, arXiv:hep-ph/9503320v1 14 Mar 1995
ILC, arXiv:1603.06016v1 [hep-ex] 18 Mar 2016
FCC-ee, arXiv:1703.01626v1 [hep-ph] 5 Mar 2017
Methodology
Why?
๐๐๐(๐๐, ฮ๐, ๐ )= ๐๐๐๐
๐ฟ๐๐(๐ =
๐๐๐
๐๐๐+๐๐๐๐)
so ๐๐, ฮ๐ can be obtained by fitting the ๐๐๐๐ , with the theoretical formula ๐๐๐
How?
In general, these uncertainties are dependent on ๐ , so it is a optimization problem
when considering the data taking.
If โฆ, then?
With the configurations of ๐ฟ, ฮ๐ฟ, ฮ๐ธ โฆ, we can obtain: ๐๐~?ฮ๐ โผ?
ฮ๐๐, ฮฮ๐๐๐๐๐ ๐ฟ ๐ N๐๐๐ ๐ธ ๐๐ธ โฆโฆ
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Theoretical Tool
The ๐๐๐ is a function of ๐ , ๐๐ and ฮ๐,
which is calculated with the GENTLE
package in this work (CC03)
The ISR correction is also calculated by
convoluting the Born cross sections with
QED structure function, with the radiator
up to NL O(๐ผ2) and O(๐ฝ3)
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Statistical and systematic uncertainties
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Statistical uncertainty
ฮ๐๐๐ = ๐๐๐ รฮ๐๐๐
๐๐๐= ๐๐๐ ร
๐๐๐+๐๐๐๐
๐๐๐
=๐๐๐
๐ฟ๐๐(๐ =
๐๐๐
๐๐๐+๐๐๐๐)
ฮ๐๐ =๐๐๐๐
๐๐๐
โ1ร ฮ๐๐๐ =
๐๐๐๐
๐๐๐
โ1ร
๐๐๐
๐ฟ๐๐
ฮฮ๐ =๐๐๐๐
๐ฮ๐
โ1ร ฮ๐๐๐ =
๐๐๐๐
๐ฮ๐
โ1ร
๐๐๐
๐ฟ๐๐
With ๐ฟ=3.2๐๐โ1, ๐=0.8, ๐=0.9:
ฮ๐๐=0.6 MeV, ฮฮ๐=1.4 MeV (individually)
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Statistical uncertainty
When there are more than one data point, we can measure both ๐๐ and ฮ๐.
With the chisquare defined as:
the error matrix is in the form:
When the number of fit parameter reduce to 1:
ฮ๐๐ =๐๐๐๐
๐๐๐
โ1
ร ฮ๐๐๐ =๐๐๐๐
๐๐๐
โ1
ร๐๐๐
๐ฟ๐๐
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Statistical uncertainty
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Systematic uncertainty
To
tal
Uncorrelated
E
๐๐ธ
๐๐๐๐
Correlated
๐๐๐
๐ฟ
๐
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Energy calibration uncertainty
With ฮ๐ธ, the total energy becomes:
๐ธ = ๐บ ๐ธ๐, ฮ๐ธ + ๐บ(๐ธ๐, ฮ๐ธ)
ฮ๐๐ =๐๐๐
๐๐๐๐
๐๐๐๐
๐๐ธฮ๐ธ
The ฮmW will be large when ฮ๐ธ
increase, and almost independent
with ๐.
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Energy spread uncertainty
With ๐ธ๐ต๐, the ๐๐๐ becomes:
๐๐๐ ๐ธ = 0โ๐๐๐ ๐ธโฒ ร ๐บ ๐ธ, ๐ธโฒ ๐๐ธโฒ
= ๐ ๐ธโฒ ร1
2๐๐ฟ๐ธ๐
โ ๐ธโ๐ธโฒ2
2๐๐ธ2
๐๐ธโฒ
๐๐ธ + ฮ๐๐ธ is used in the simulation, and ๐๐ธ is for
the fit formula.
The ๐๐พ insensitive to ๐น๐ฌwhen taking data
around ๐๐๐. ๐ GeV
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Background uncertainty
The effect of background are in two different ways
1. Stat. part: ฮ๐๐(๐๐ต) =๐๐๐
๐๐๐๐โ
๐ฟ๐๐ต๐๐ต
๐ฟ๐
2. Sys. part: ฮ๐๐(๐๐ต) =๐๐๐
๐๐๐๐โ ๐ฟ๐๐ต๐๐ต
๐ฟ๐โ ฮ๐๐ต
With L=3.2abโ1, ๐๐ต๐๐ต = 0.3pb, ฮ๐๐ต = 10โ3๏ผ
ฮ๐๐(๐๐ต)~0.2 MeV, which has been embodied in the product of ๐ โ ๐,
and ฮ๐๐(๐๐ต) is considerable with the former.
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Correlated sys. uncertainty
The correlated sys. uncertainty includes: ฮ๐ฟ, ฮ๐, ฮ๐๐๐โฆ
Since ๐๐๐๐ = ๐ฟ โ ๐ โ ๐, these uncertainties affect ๐๐๐ in same way.
We use the total correlated sys. uncertainty in data taking optimization:
๐ฟ๐ = ฮ๐ฟ2 + ฮ๐2
ฮ๐๐ =๐๐๐
๐๐๐๐๐๐๐ โ ๐ฟ๐ , ฮฮ๐ =
๐ฮ๐
๐๐๐๐๐๐๐ โ ๐ฟ๐
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Correlated sys. uncertainty
ฮ๐๐ =๐๐๐
๐๐๐๐๐๐๐ โ ๐ฟ๐
Two ways to consider to effect:
Gaussian distribution
๐๐๐ = ๐บ(๐๐๐0 , ๐ฟ๐ โ ๐๐๐
0 )
Non-Gaussian (will cause shift)
๐๐๐ = ๐๐๐0 ร (1 + ๐ฟ๐)
With ๐ฟ๐ = +1.4 โ 10โ4(10โ3) at 161.2GeV
ฮmW~0.24MeV (3MeV)
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Correlated sys. uncertainty
To consider the correlation, the scale factor method is
used,
๐2 = ๐๐ ๐ฆ๐โโโ ๐ฅ๐
2
๐ฟ๐2 +
โโ1 2
๐ฟ๐2 ,
where ๐ฆ๐ , ๐ฅ๐ are the true and fit results, h is a free
parameter, ๐ฟ๐ and ๐ฟ๐ are the independent and
correlated uncertainties.
For the Gaussian consideration, the scale factor can
reduce the effect.
For the non-Gaussian case, the shift of the ๐๐ is
controlled well
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โข Smallest ฮ๐๐, ฮฮ๐ (stat.)
โข Large sys. uncertainties
โข Only for ๐๐ or ฮ๐, without correlation
One point
โข Measure ๐๐ and ฮ๐ simultanously
โข Without the correlation
Two points
โข Measure ๐๐ and ฮ๐ simultaneously, with the correlation
Three points
or more
Da
ta t
ak
ing
sch
emes
Data taking scheme
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Taking data at one point (just for ๐๐พ)
There are two special energy points :
The one which most statistical sensitivity to ๐๐:
ฮ๐๐(stat.) ~0.59 MeV at ๐ธ=161.2 GeV
(with ฮฮ๐ and ฮ๐ธ๐ต๐ effect)
The one ฮ๐๐(stat)~0.65 MeV at ๐ธ โ 162.3 GeV
(with negative ฮฮ๐, ฮ๐ธ๐ต๐ effects)
With ฮ๐ฟ ฮ๐ < 10โ4, ฮ๐๐ต<10โ3, ฮ๐ธ=0.7MeV,
ฮ๐๐ธ=0.1, ฮฮ๐=42MeV)
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โ๐(GeV) 161.2 162.3
๐ธ 0.36 0.37
๐๐ธ 0.20 -
๐๐ต 0.17 0.17
๐ฟ๐ 0.24 0.34
ฮW 7.49 -
Stat. 0.59 0.65
ฮ๐๐(MeV) 7.53 0.84
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Taking data at two energy points
To measure ฮ๐๐ and ฮฮ๐, we scan the energies and the luminosity fraction of
the two data points:
1. ๐ธ1, ๐ธ2 โ [155, 165] GeV, ฮ๐ธ = 0.1 GeV
2. ๐น โก๐ฟ1
๐ฟ2โ 0, 1 , ฮ๐น = 0.05
We define the object function: ๐ = mW + 0.1ฮ๐ to optimize the scan parameters
(assuming ๐๐ is more important than ฮ๐).
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Taking data at two energy points
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The 3D scan is performed, and
2D plots are used to illustrate
the optimization results;
When draw the ฮ๐ change
with one parameter, another
is fixed with scanning of the
third one;
๐ธ1=157.5 GeV, ๐ธ2=162.5
GeV (around ๐๐๐๐
๐ฮ๐=0 ,
๐๐๐๐
๐๐ธ๐ต๐=0)
and F=0.3 are taken as
the result.
(MeV) ๐ ๐๐ฌ ๐๐ฉ ๐น๐ Stat. Total
ฮ๐๐ 0.38 - 0.21 0.33 0.80 0.97
ฮฮ๐ 0.54 0.56 1.38 0.20 2.92 3.32
ฮ๐ฟ(ฮ๐)<10โ4, ฮ๐๐ต<10โ3
๐๐ธ=1 ร 10โ3, ฮ๐ธ=0.7MeVฮ๐๐ธ=0.1%
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Optimization of ๐ธ1
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The procedure of three
points optimization is
similar to two points
๐ธ1 157.5 GeV
๐ธ2 162.5 GeV
๐ธ3 161.5 GeV
๐น1 0.3
๐น2 0.9
Taking data at three energy points
ฮ๐๐~0.98 MeVฮฮ๐ ~3.37 MeV
ฮ๐ฟ(ฮ๐)<10โ4, ฮ๐๐ต<10โ3
๐๐ธ=1 ร 10โ3, ฮ๐ธ=0.7MeVฮ๐๐ธ=0.1
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Summary
The precise measurement of ๐๐ (ฮ๐) is studied (threshold scan method)
Different data taking schemes are investigated, based on the stat. and sys.
uncertainties analysis.
With the configurations :
Thank you๏ผ
ฮ๐ฟ(ฮ๐)<10โ4, ฮ๐๐ต<10โ3
๐๐ธ=1 ร 10โ3, ฮ๐ธ=0.7MeVฮฮ๐=42MeV, ฮ๐๐ธ=0.1
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Backup Slides
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Covariance matrix method
๐ฆ๐ =๐๐
๐, ๐ฃ๐๐ = ๐๐
2 + ๐ฆ๐2๐๐
2
where ๐๐ is the stat. error of ๐๐, ๐๐ is the relative error of ๐
The correlation between data points ๐, j contributes to the
off-diagonal matrix element ๐ฃ๐๐:
Then we minimize: ๐12 = ๐๐๐โ1๐
For this method, The biasness is uncontrollable
(MO Xiao-Hu HEPNP 30 (2006) 140-146)
H. J. Behrend et al. (CELLO Collaboration)
Phys. Lett. B 183 (1987) 400
Dโ Agostini G. Nucl. Instrum. Meth. A346 (1994)
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Scale factor method
This method is used by introducing a free fit parameter to the ๐2:
๐22 = ๐
๐๐ฆ๐โ๐๐2
๐๐2 +
๐โ1 2
๐๐2
๐๐ includes stat. and uncorrelated sys errors, ๐๐ are the correlated errors.
The equivalence of this form and the one from matrix method is
proved in : MO Xiao-Hu HEPNP 30 (2006) 140-146 .
Both the matrix and the factor approach have bias, which may be considerably striking when the data points are quite many or the scale factor is rather large.
According to ref: MO Xiao-Hu HEPNP 31 (2007) 745-749, the unbiased ๐2 is constructed as:
๐32 = ๐
๐ฆ๐โ๐๐๐2
๐๐2 +
๐โ1 2
๐๐2 (used in our previous results)
The central value from ๐22 can be re-scaled, the relative error is still larger than those from ๐3
2 estimation.25