kinematic fitting - desyintroduction: kinematic fitting as a part of data processing > raw data...

Kinematic fittingA powerful tool of event selection and reconstruction

Sergey

Yaschenko, DESY Zeuthen

University of Glasgow, 12.01.2011

Sergey Yaschenko | Kinematic fitting | 12.10.2011 |

Outline

> Introduction

> Techniques

> Applications

> Results of use at HERMES

> Summary

> Literature


Introduction: kinematic fitting as a part of data processing

> Raw data processing

> Detector calibrations

> Track search and reconstruction

> Momentum reconstruction

> Event selection and reconstruction

At this stage 3-momenta of all found tracks are reconstructed

Combine individual tracks to events

Apply certain requirements (cuts) on track correlations to select events of interest and reject the background

Alternatively use kinematic fitting

> Physics analysis


Introduction: kinematic fitting as a tool of event selection

> Kinematic fitting –

adjustment of measured kinematic parameters under certain assumptions (conditions, constraints)

> Aims

Test if assumptions are true

Improve accuracy of measurements

Check if the knowledge of measurement uncertainties is correct

Check for possible systematic uncertainties

> Has been used for more than 50 years in particle physics, considered as a standard tool of event selection and reconstruction

> But also often considered as too complicated and not really necessary


Example 1

> Decay of a particle a

to 3

particles a 1+2+3

> Energy of the primary particle is precisely known and equal to Ea = 30 GeV

> Energies of the secondary particles are equal to E1

= 5 GeV, E2

= 10 GeV

and E3

= 15 GeV

and measured with errors distributed by Gaussian with sigmas

σ1

= 1 GeV, σ2

= 1 GeV

and σ3

= 1 GeV

> Minimization of least-squares functional

> under condition

> If measurement errors are the same the result is trivial:

223

1

2 /)( ifit

iii

EE

0321 afitfitfit EEEE

ifit

i EE3/)( 321 aEEEE


Results of example 1

> Chi square distributed as chi square for one degree of freedom

> Probability distribution

> A measure of the probability that a chi square from the theoretical distribution is greater than chi square obtained from the fit

> Improvement of accuracies of energy measurements by factor of √(2/3)

2χ0 1 2 3 4 5 6 7 8 9 10

10

210

310

410

Probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

Energy [GeV]0 5 10 15 20 25 300

1000

2000

3000

4000

5000

6000

7000

2

);(

dznzf


Background process


to 4

particles a 1+2+3+4

> Energy of the primary particle is precisely known and equal to Ea

=30 GeV


= 3 GeV, E2

= 8 GeV

and E3

= 13 GeV

and measured with errors distributed by Gaussian with sigma σ1

= 1 GeV, σ2

= 1 GeV

and σ3

= 1 GeV

> Particles 4

with energy of 6 GeV

is unmeasured (missed particle)

> Fitting assuming a 1+2+3

hypothesis

2χ0 1 2 3 4 5 6 7 8 9 10

210

310

Energy [GeV]0 5 10 15 20 25 300

1000

2000

3000

4000

5000

6000

7000

Probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

10

210

310

410

510


Missing energy method vs kinematic fitting

> Process of interest: missing energy is peaked near zero, chi square is distributed as chi square for 1 degree of freedom

> Background process: missing energy peak is shifted, chi square distribution is completely different

Missing Energy [GeV]-20 -15 -10 -5 0 5 10 15 200

500

1000

1500

2000

2500

3000

3500

4000

4500


500

1000

1500

2000

2500

3000

3500

4000

4500

2χ0 1 2 3 4 5 6 7 8 9 10

10

210

310

410

2χ0 1 2 3 4 5 6 7 8 9 10

210

310


Missing energy method vs kinematic fitting

> Apply a cut on the missing energy or to chi-square or probability distribution to select the process of interest and reject the background

Probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

Probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

10

210

310

410

510


500

1000

1500

2000

2500

3000

3500

4000

4500


500

1000

1500

2000

2500

3000

3500

4000

4500


Kinematic fitting technique

> Minimization of least squares functional

under constraints

… … …

yi

–

measured kinematic parameters,

ηi

–

fit parameters, σi

–

measurement errors, n –

number of kinematic parameters, m

–

number of constraints

> In case of correlations between kinematic parameters, covariance

matrix Gy

should be used

0),...,,( 212 nf

22

1

2 /)( ii

n

iiy

0),...,,( 211 nf

0),...,,( 21 nmf

n

ijjjiyi

n

ji yGy

1),(

1

2 )()(


Minimization of least-squares functions with constraints

> Method of elements

Using q

equations of constraints eliminate q

of the n kinematic parameters as functions of n-q

parameters

Not always possible, procedure is not automatic

> Method of Lagrange multiplies

Automatic procedure, widely used

Exact solution if constraints depend linearly on parameters

Simple iterative procedure if constraints are non-linear

> Method of orthogonal transformation

Mentioned in one textbook, not widely used

> Method of penalty functions

Automatic procedure

Can be effectively used with constraints of inequality type


Method of Lagrange multipliers

> Measurements give instead of true quantities ηi

values yi

> Measurement errors are normally distributed about zero with standard deviations σi

> m

equations of constraints

> If equations are linear write in the matrix form

> Define

> Minimization of Lagrange function

> Total derivative should vanish

iiiy

22 )(

,0)(

ii

i

E

E

mkfk ,...2,1,0

)(2 BcGL T

yT

00 bB

022 BddGdL T

yT

0byBc


Method of Lagrange multipliers

> Solve system of linear equations

> Obtain

> Can be easily solved

> Estimators of the measurement errors

> Estimators for fit parameters

> Using abbreviation

> Obtain by applying error propagation

01 T

y BBGc

cBBG Ty

11~

~~

y

cBBGBG Ty

Ty

111~

1111~

yBT

yy BGGBGGG

11 TyB BBGG

0

0

BcBG T

yT


Solution of example 1

> In the example 1

> Vector

> Estimation of fit parameters

> Estimation of error matrix of fit parameters

211121112

311

~G

111B

31

111

000000

111

1

23

22

21

BG

)(31~

321 aii EEEEE

aa EEEEEEEE

c

321

3

2

1

111


Example 2


to 3

particles a 1+2+3

> Energy of the primary particle is precisely known and equal to Ea = 30 GeV


= 5 GeV, E2

= 10 GeV

and E3

= 15 GeV


= 1 GeV, σ2

= 1 GeV

and σ3

= 5 GeV

2χ0 1 2 3 4 5 6 7 8 9 10

10

210

310

410

Probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

Energy [GeV]0 5 10 15 20 25 300

1000

2000

3000

4000

5000

6000

iifit

i EE )/()( 2

322

21

2321 iai EEEE



> Pull distributions defined as

> Should be normally distributed around zero with standard deviation equal to unity

> Help to understand if measurement errors are known correctly

> Check for possible systematic effects

Pull 1-10 -8 -6 -4 -2 0 2 4 6 8 100

500

1000

1500

2000

2500

3000

3500

4000

/ ndf 2χ 68.64 / 81Constant 15.5± 3981 Mean 0.00317± -0.00399 Sigma 0.002± 1.001

Pull 2-10 -8 -6 -4 -2 0 2 4 6 8 100

500

1000

1500

2000

2500

3000

3500

4000


Pull 3-10 -8 -6 -4 -2 0 2 4 6 8 100

500

1000

1500

2000

2500

3000

3500

4000

/ ndf 2χ 75.82 / 79

Constant 16.1± 4132

Mean 0.003054± -0.004127 Sigma 0.0022± 0.9649

)()()( 22ii

ii

ii

ii

yypull


Example 3

> Decay of a particle 3

particles a 1+2+3


=30 GeV


= 5 GeV, E2

= 10 GeV

and E3

= 15 GeV


= 1 GeV, σ2

= 1 GeV

and σ3

= 5 GeV

> Measurement of error of E3

is not known and assumed to be 2 GeV

2χ0 1 2 3 4 5 6 7 8 9 10

210

310

410

Probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

22000

Energy [GeV]0 5 10 15 20 25 300

1000

2000

3000

4000

5000

6000



> Pull distributions are wider, indication that something is wrong

with the knowledge of measurement errors

Pull 1-10 -8 -6 -4 -2 0 2 4 6 8 100

200

400

600

800

1000

1200

1400

1600

1800

2000 / ndf 2χ 181.6 / 166

Constant 7.3± 1878 Mean 0.00672± -0.00775 Sigma 0.00± 2.12

Pull 2-10 -8 -6 -4 -2 0 2 4 6 8 100

200

400

600

800

1000

1200

1400

1600

1800

2000 / ndf 2χ 181.6 / 166

Constant 7.3± 1878 Mean 0.00672± -0.00775 Sigma 0.00± 2.12

Pull 3-10 -8 -6 -4 -2 0 2 4 6 8 100

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400 / ndf 2χ 163.3 / 142

Constant 8.6± 2202

Mean 0.005731± -0.006458 Sigma 0.004± 1.809


Example 4

> Decay of a particle 3

particles a 1+2+3


=30 GeV


= 5 GeV, E2

= 10 GeV

and E3

= 15 GeV


= 1 GeV, σ2

= 1 GeV

and σ3

= 5 GeV

> Bias in the measurement of E3

: 17 GeV

instead of 15 GeV

2χ0 1 2 3 4 5 6 7 8 9 10

10

210

310

410

Probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

1400

1600

Energy [GeV]0 5 10 15 20 25 300

1000

2000

3000

4000

5000

6000



> Pull distributions are shifted

Pull 1-10 -8 -6 -4 -2 0 2 4 6 8 100

500

1000

1500

2000

2500

3000

3500

4000

/ ndf 2χ 75.65 / 81Constant 15.5± 3981 Mean 0.0032± 0.3808 Sigma 0.002± 1.001

Pull 2-10 -8 -6 -4 -2 0 2 4 6 8 100

500

1000

1500

2000

2500

3000

3500

4000


Pull 3-10 -8 -6 -4 -2 0 2 4 6 8 100

500

1000

1500

2000

2500

3000

3500

4000



Applications of kinematic fitting

> Event selection

> Background rejection

> Improvement of resolution

> Better understanding of measurement errors, search for possible sources of systematic uncertainties

Analysis of chi square (probability) and pull distributions

> Detector calibration

If number of constraints is enough, some measurements can be considered as unknown parameters, reconstructed by kinematic fitting and then used for calibration


Possible problems

> Measurement errors are not known precisely

> Systematic uncertainties

> Measurement errors are not Gaussian

> If different particles are measured in different coordinate systems –

misalignment between these systems

> Multiple scattering and radiative

effects usually lead to tails in chi-

square and pull distributions


Outline

> Introduction

> Techniques

> Applications

> Results of use at HERMES

> Literature

> Summary


Deeply virtual Compton scattering (DVCS) at HERMES

p p’

e

e’

*γ γ

,t)ξGPDs(x,

ξx+ ξx-

t

p p’

e e’

*γ

γ

p p’

e e’

*γ

γ

> DVCS and Bethe-Heitler: the same initial and final state

> Bethe-Heitler

dominates at HERMES kinematics

> Generalized Parton Distributions (GPDs) accessible through cross section differences and azimuthal

asymmetries via interference term

Bethe-HeitlerDVCS


The HERMES experiment

1

0

2

−1

−2

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LUMINOSITY

CHAMBERSDRIFT

FC 1/2

TARGETCELL

DVC

MC 1−3

HODOSCOPE H0

MONITOR

BC 1/2

BC 3/4 TRD

PROP.CHAMBERS

FIELD CLAMPS

PRESHOWER (H2)

STEEL PLATE CALORIMETER

DRIFT CHAMBERS

TRIGGER HODOSCOPE H1

0 1 2 3 4 5 6 7 8 9 10

RICH270 mrad

270 mrad

MUON HODOSCOPEWIDE ANGLE

FRONTMUONHODO

MAGNET

m

IRON WALL

e+

27.5 GeV

140 mrad

170 mrad

170 mrad

140 mrad

MUON HODOSCOPES

SILICON e'

p'

Gas targets: • Longitudinally polarized H, D• Unpolarized

H, D, 4He, N, Ne, Kr, Xe• Transversely polarized H

Beam:• Longitudinally polarized e+

and e−

with both helicities• Energy 27.6 GeV


DVCS/BH measurement with the Recoil Detector

> Pre-Recoil data

Scattered lepton and photon were detected in the forward spectrometer

Recoil proton was not detected

Exclusivity achieved via missing mass technique

Associated processes (epe∆γ) were not resolved (12%

contribution in the signal)

> Recoil data

Detection of recoil proton

Suppression of the background to <1%

level

1 Tesla superconducting solenoid

Photon Detector (PD)

Scintillating Fiber Tracker

Lepton beam

Silicon Strip Detector

Target cell of unpolarized

target

)2

(GeV2xM

0 10 20 30

DIS

1000

N/N

0

0.1

0.2

0.3

)2

(GeV2xM

0 10 20 30

DIS

1000

N/N

0

0.1

0.2

0.3 data+e data-e

MC sumelastic BHassociated BHsemi-inclusive

e

p


Elastic DVCS/BH and associated background

> Elastic DVCS/BH: ep

epγ

Beam 3-momentum is known precisely (in comparison with 3-momenta of secondary particles)

Target proton is at rest

For e, p,

and γ,

3-momenta are measured with certain precision

> Background from associated process epe∆+γ

with ∆+pπ0

> Not possible to separate by missing mass only

> Low efficiency of selection if one-dimensional cuts (coplanarity, momentum-momentum correlations) are applied


Selection of elastic DVCS/BH using simple cuts

> One-dimensional cuts

Momentum-momentum correlations

Coplanarity

condition

> Efficiency of 70%

at background contamination of 5%

> background contamination of 2%

can be achieved but efficiency drops below 50%

> No improvement of resolutions

Hermes 2007 data

0

500

1000

1500

-2 0 2 4 6 8 10 12 14

Mx2 [ (GeV/c)2 ]

cou

nts Traditional DVCS analysis

(Eγ >5 GeV)

Recoil protonin acceptance

with Coplanarity cut

Coplanarity cutturned around


Kinematic fitting for elastic DVCS/BH

> Nine

kinematic parameters: Scattered electron Photon Proton

> Four

equations of constraints

13

112

111

/1)/tan()/tan(

pyppyppy

zy

zx

)sin/(1 339

38

37

pyyy

00

00

3214

3213

3212

3211

pbeam

beamzzz

yyy

xxx

meeeefppppf

pppfpppf

25

225

224

)/tan()/tan(

eyppyppy

zy

zx


Constraints as functions of parameters

> Constraints are non-linear

> Derivatives of constraints can be calculated analytically or numerically

pbeampe

beam

memyyymyf

pyyyyyyyyf

yyyyyyyyyyf

yyyyyyyyyyf

))(sin/(1/1

))tan(/(11/1//1

/)sin(1/1//

/)cos(1/1//

822

9622

34

8925

246

22

2133

9725

2465

22

21322

9725

2464

22

21311


Fitting procedure

> Minimization of chi square with constraints using penalty term

> If T is large enough constraints are satisfied automatically during the fitting procedure

> In order to equalize penalty contributions from different constraints errors of constraints are formally calculated using error propagation

> Non-linear constraints of inequality type can be included

m

kckk

n

ijjjiyi

n

ji fTyGy

0

22

1),(

1

2 /)()(


Results for elastic DVCS/BH and associated background

> For elastic DVCS/BH (Monte Carlo)

> For associated backgroundProbablility

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

210

310

2χ0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

50

100

150

200

250

2χ0 10 20 30 40 50 60 70 80 90 1000

500

1000

1500

2000

2500

3000

2χ0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

Probablility0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

10

210

310


Efficiency of elastic event selection

cut2χ0 10 20 30 40 50 60 70 80 90 100

Eff

icie

ncy

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

cut2χ0 10 20 30 40 50 60 70 80 90 100

Bac

kgro

un

d c

on

tam

inat

ion

0

0.001

0.002

0.003

0.004

0.005

0.006

> Efficiency of event selection of elastic DVCS/BH events is high at a reasonable cut on chi-square

> Contamination of associated process is well below 1%

> In addition, improvement of resolutions


Improvement of kinematic parameter reconstruction

Momentum [GeV/c]8 10 12 14 16 18 20

p/p

Δ

0

0.005

0.01

0.015

0.02

0.025

Momentum [GeV/c]8 10 12 14 16 18 20

p/p

Δ

0

0.01

0.02

0.03

0.04

0.05

0.06

Momentum [GeV/c]0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

[ra

d]

θ Δ

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Momentum [GeV/c]0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

p/p

Δ

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Momentum of electron Energy of photon

Polar angle of proton Momentum of proton


t resolution

> Mandelstam t=(p-p´)2

, important

physical

observable

> Blue squares

–

reconstruction

using

measured

kinematic

parameters

of electron

and photon

> Magenta triangles

-

reconstruction

using

measured

kinematic

parameters

of recoil

proton

> Light blue

triangles

-

reconstruction

using

measured

kinematic

parameters

of electron

and photon

(excluding

photon

energy) assuming

proton

mass

> Red squares

–

reconstruction

using

kinematic

fitting

]2-t [GeV0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

]2 t

[G

eVΔ

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16


Pull distributions

Pull 1-5 -4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400

/ ndf 2χ 206.9 / 139

Constant 3.4± 370.7

Mean 0.007389± -0.009738 Sigma 0.006± 1.007

Pull 2-5 -4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400

/ ndf 2χ 244.6 / 131Constant 3.5± 372.2 Mean 0.00737± -0.01557 Sigma 0.006± 1.001

Pull 3-5 -4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400


Pull 4-5 -4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400


Pull 5-5 -4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400

/ ndf 2χ 224.6 / 133Constant 3.5± 376.1

Mean 0.007288± -0.006971 Sigma 0.006± 0.992

Pull 6-5 -4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400

/ ndf 2χ 338.9 / 137Constant 3.4± 369.7

Mean 0.007607± 0.008947 Sigma 0.005± 1.003

Pull 7-5 -4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400

450 / ndf 2χ 287.1 / 135

Constant 3.5± 378.7

Mean 0.007220± -0.005882 Sigma 0.0056± 0.9816

Pull 8-5 -4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400

/ ndf 2χ 330.2 / 143Constant 3.5± 369.5

Mean 0.007460± 0.003169 Sigma 0.006± 1.003

Pull 9-5 -4 -3 -2 -1 0 1 2 3 4 50

50

100

150

200

250

300

350

400


Electron Photon Proton


Selection of elastic DVCS/BH events (HERMES data)

> Kinematic fitting is developed and tested on Monte-Carlo

3 particles detected 4 constraints from energy-momentum conservation

Allows to suppress the associated processes and semi-inclusive background to negligible level

> Applied for data for physics analysis

Systematic studies in progress

First physics results expected soon

> Missing mass distribution

No requirement for Recoil

Positively charged Recoil track

Kinematic fit probability > 1%

Kinematic fit probability < 1%]4/c2 [GeVX

2M-4 -2 0 2 4 6 8 10 12 140

500

1000

1500

2000

2500

3000

Data


Summary

> Kinematic fitting is a powerful tool of event selection and reconstruction

> Many applications

Improvement of resolution

Better understanding of measurement errors, search for possible sources of systematic uncertainties

Detector calibration

> Kinematic fitting technique is well developed and described in literature

> Different problems could appear in different experiments, solutions are not always straightforward


Literature (personal choice)

> W.E.Deming, Statistical Adjustment of Data

> S.Brandt, Data Analysis

> R.Frühwirth, M.Regler, R.K.Bock, H. Grote, D. Notz, Data Analysis Techniques for High-Energy Physics

> J.P.Berge, F.T.Solmitz, H.D.Taft, Rev. Scient. Instr. 32 N5 (1961) 538

> R.Boeck

CERN 60-30

> W.G.Davis, Nucl. Instr. Meth A 386 (1997) 498

> S.N.Dymov

et al, Nucl. Instr. Meth A 440 (2000) 431

> M.Williams, C.A.Meyer, CLAS-NOTE 2003-017

> P.Avery, http://www.phys.ufl.edu/~avery

> V.Bloebel, http://www.desy.de/~blobel/

http://www.phys.ufl.edu/~avery

http://www.desy.de/~blobel/

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