Data Analysisin High Energy Physics

Data Analysisin High Energy Physics

Wen-Chen ChangInstitute of Physics, Academia Sinica

章文箴中央研究院 物理研究所

OutlineOutline

• Some Observables of Interest

• Elementary Observables

• More Complex Observables

• Analysis tools.

Some Observables of InterestSome Observables of Interest

• Total Interaction/Reaction Cross Section

• Differential Cross Section

• Particle mass or width

• Branching Ratio

Cross SectionCross Section

• Cross Section defines the strength of a particular interaction between two particles.

Small cross section

Large cross section

Cross SectionInteraction Matrix Element

Cross SectionInteraction Matrix Element

fi

fif

i

f

ifif

fif

i

ia

vv

pM

hv

W

d

d

dpdph

VdN

dEdN

dVUM

Mh

W

v

vn

dcba

22

42

23

*

2

4

1

)2(

/ state final ofdensity Energy

Element Matrix n Interactio

2

Wparticle target pre reactions ofNumber

target the torelative beamincident theofvelocity :

Flux

Scattering Cross SectionScattering Cross Section

• Differential Cross Section

Flux

Target

Unit Area

dsolid angle

– scattering angle

d

dN

FE

d

d s1),(

• Total Cross Section

),()(

Ed

ddE

),(),(

Ed

dxFANEN s

• Average number of scattered into d

Particle life time or widthParticle life time or width

• Most particles studied in particle physics or high energy nuclear physics are unstable and decay within a finite lifetime.

• Particles decay randomly (stochastically) in time. The time of their decay cannot be predicted. Only the the probability of the decay can be determined.

• The probability of decay (in a certain time interval) depends on the life-time of the particle. In traditional nuclear physics, the concept of half-life is commonly used.

Half-LifeHalf-Life

Radioactive Decay of Nuclei with Half Life = 1 Day

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1000000

0 2 4 6 8

Days

Un

dec

ayed

Nu

clei

Half-Life and Mean-LifeHalf-Life and Mean-Life

• The number of particle (nuclei) left after a certain time “t” can be expressed as follows:

• where “” is the mean life time of the particle

• “” can be related to the half-life “t1/2” via the simple relation:

)(1)(

)( 0 tNdt

tdNeNtN

t

0.693)ln( 21

2/1 t

Examples - particlesExamples - particles

Mass (MeV/c2)

or c Type

Proton (p) 938.2723 >1.6x1025 y Very long… Baryon

Neutron (n) 939.5656 887.0 s 2.659x108 km

Baryon

N(1440) 1440 350 MeV Very short! Baryon resonance

(1232) 1232 120 MeV Very short!! Baryon resonance

1115.68 2.632x10-10 s 7.89 cm Strange Baryon resonance

Pion (+-) 139.56995 2.603x10-8 s 7.804 m Meson

Rho - (770)

769.9 151.2 MeV Very short Meson

Kaon (K+-) 493.677 1.2371 x 10-8 s 3.709 m Strange meson

D+- 1869.4 1.057x10-12 s 317 m Charmed meson

Examples - NucleiExamples - Nuclei

Radioactive Decay Reactions Used to data rocks

Parent Nucleus Daughter Nucleus Half-Life (billion year)

Samarium (147Sa) Neodymium (143Nd) 106

Rubidium (87Ru) Strontium (87Sr) 48.8

Thorium (232Th) Lead (208Pb) 14.0

Uranium (238U) Lead (206Pb) 4.47

Potassium (40K) Argon (40Ar) 1.31

Particle WidthsParticle Widths

• By virtue of the fact that a particle decays, its mass or energy (E=mc2), cannot be determined with infinite precision, it has a certain width noted .

• The width of an unstable particle is related to its life time by the simple relation

• h is the Planck constant.h

Decay Widths and Branching FractionsDecay Widths and Branching Fractions

• In general, particles can decay in many ways (modes).

• Each of the decay modes have a certain relative probability, called branching fraction or branching ratio.

• Example (0s) Neutral Kaon (Short)

– Mean life time = (0.89260.0012)x10-10 s– c = 2.676 cm– Decay modes and fractions

mode i+ - (68.61 0.28) %

0 0 (31.39 0.28) %

+ - (1.78 0.05) x10-3

Elementary ObservablesElementary Observables

• Momentum

• Time-of-Flight

• Energy Loss

• Particle Identification

• Invariant Mass Reconstruction

Momentum MeasurementsMomentum Measurements

• Definition– Newtonian Mechanics

– Special Relativity

• But how does one measure “p”?

mvp

mvp

21

1

cv

Momentum Measurements TechniqueMomentum Measurements Technique

• Use a spectrometer with a constant magnetic field B.• Charged particles passing through this field with a

velocity “v” are deflected by the Lorentz force.

• Because the Lorentz force is perpendicular to both the B field and the velocity, it acts as centripetal force “Fc”.

• One finds

BvqFM

R

mvFc

2

qBRmvp

Momentum Measurements TechniqueMomentum Measurements Technique

• Knowledge of B (magnetic field) and R (bending radius) needed to get “p”

• B is determined by the construction/operation of the spectrometer.

• “R” must be measured for each particle.

Bending Magnet in SpectrometerBending Magnet in Spectrometer

12 sinsin

/

/ dzBce

eq

p yxz

Momentum MeasurementMomentum Measurement

+B=0.5 T

Collision Vertex

Trajectory is a helix in 3D; a circle in the transverse plane

qBRmvp

Radius: R

TPC Inside the Solenoid Magnet at SPring-8

TPC Inside the Solenoid Magnet at SPring-8

Pad Plane of TPCPad Plane of TPC

• Inner radius: < 1.25 cm.

• Outer radius: < 30 cm.

• Maximum drift distance : about 70 cm.

• ~1000 pads and ~100 wires for readout,

xy ~350mm and z ~ 500mm,

• B = 1.5 ~ 2.5T.

32-channel SPring-8 FADC cards32-channel SPring-8 FADC cards

• Use TEXONO FADC and IHEP BES version as the starting point.

• 40 MHz clock, maximum 1024 sampling bins for one strobe.

• 10-bit FADC: ADC input 0-2 V range.

• Shift register inside FPGA: max length = 100 time bin.

• On-board threshold suppression performed by FPGA.

• Buffer FIFO: 16 bits x 4096 depth dual port memory, large enough to hold 5 events before issuing IRQ.

• CPLD: controlling VME actions.• Adjustable zero-suppression level

channel by channel and number of events per IRQ.

• VME 9U: 32 channels/module; 8 detachable cards/module; 4 channel/card.

Determination of Particle TrajectoryDetermination of Particle Trajectory

Operation of Time-Projection Chamber

Operation of Time-Projection Chamber

Time-of-Flight (TOF) Measurements Time-of-Flight (TOF) Measurements

• Typically use scintillation detectors to provide a “start” and “stop” time over a fixed distance.

• Electric Signal Produced by scintillation detector

• Use electronic Discriminator• Use time-to-digital-converters (TDC) to measure the

time difference = stop – start.• Given the known distance, and the measured time,

one gets the velocity of the particle 21 cttc

d

t

dv

startstop

Time S (volts)

More Complex ObservablesMore Complex Observables

• Particle Identification

• Invariant Mass Reconstruction

• Identification of decay vertices

Particle IdentificationParticle Identification

• Particle Identification or PID amounts to the determination of the mass of particles. – The purpose is not to measure unknown mass of

particles but to measure the mass of unidentified particles to determine their species e.g. electron, pion, kaon, proton, etc.

• In general, this is accomplished by using to complementary measurements e.g. time-of-flight and momentum, energy-loss and momentum, etc

1m

TOF wall

MWDC 2

MWDC 3

MWDC 1

Dipole Magnet (0.7 T)

Start counter

Silicon VertexDetector

AerogelCherenkov(n=1.03)

LEPS Detector SystemLEPS Detector System

Target, Upstream Spectrometer, Dipole Magnet

Target, Upstream Spectrometer, Dipole Magnet

LH2 Target (50 mm long)

Cherenkov Detector

SSD

Drift Chamber

Start Counter

Dipole Magnet and Drift Chambers

Dipole Magnet and Drift Chambers

Time-of-Flight WallTime-of-Flight Wall

Particle Identification by TOFParticle Identification by TOF

Mass(GeV)

Mo

men

tum

(G

eV

)

K/ separation (positive charge)

K++

Mass/Charge (GeV)

Eve

nts

Reconstructed mass

d

p

K+

K-

+-

TOF

lv

p

vm ,

PID with a TPCPID with a TPC

• The energy loss of charged particles passing through a gas is a known function of their momentum. (Bethe-Bloch Formula)

ln(...)22

222

z

A

ZcmrN

dx

dEeea

STARParticle Identification by dE/dx

dE/dx PID range: ~ 0.7 GeV/c for K/

~ 1.0 GeV/c for K/p

Anti - 3He

How Do We Identifiy Resonances?

How Do We Identifiy Resonances?

Resonance: Broad states with finite widths and lifetimes, which can be formed by collision between the particles into which they decay.

XeeBep

J/

221

221 )()( PPEEM inv

Invariant Mass ReconstructionInvariant Mass Reconstruction

• In special relativity, the energy and momenta of particles are related as follow

• This relation holds for one or many particles. In particular for 2 particles, it can be used to determine the mass of parent particle that decayed into two daughter particles.

42222 cmcpE

Pp 1p

2p

Invariant Mass Reconstruction (cont’d)

Invariant Mass Reconstruction (cont’d)

• Invariant Mass

• Invariant Mass of two particles

• After simple algebra

22242 cpEcm PPP

221

221

42 cppEEcmP

cos2 22121

422

421

42 cppEEcmcmcmP

Example – (1115)ReconstructionExample – (1115)Reconstruction

(1115)p+-

Decay vertex (p-)outside target

Finding V0sFinding V0sproton

pion

Primary vertex

Dalitz PlotDalitz Plot

• The Dalitz plot is a way to represent the entire phase space, viz. all essential kinematical variables, of any three-body final state in a scattered plot or two-dimensional histogram. Dalitz introduced it in 1953.

• Let a reaction be 1+23+4+5. For fixed p1 and p2, i.e. fixed total energy, the physical region of a Dalitz plot is inside a well-defined area, and in the absence of resonances or interferences can be shown to be uniformly populated. Resonant behaviour of two of the final state particles gives rise to a band of higher density, parallel to one of the coordinate axes or along a 45 degree line.

n-Particle Phase space, n=3

2 Observables From four vectors 12 Conservation laws -4 Meson masses -3 Free rotation -3 Σ 2

Usual choice Invariant mass m12

Invariant mass m13

π3

π2pp

π1

Dalitz plot

ppbar000 It’s All a Question of Statistics ...

pp 30

with100 events

ppbar000 It’s All a Question of Statistics ... ...

pp 30

with100 events1000 events

ppbar000 It’s All a Question of Statistics ... ... ...

pp 30

with100 events1000 events10000 events

ppbar000 It’s All a Question of Statistics ... ... ... ...

pp 30

with100 events1000 events10000 events100000 events

ppbar000ppbar000

Offline AnalysisOffline Analysis

Ntuple

RAW DATA

Detector

Electronics

LEPSana

Physics Results

DAQ

Analysis Code

Physics event

Calibration Parameters

Introduction of PAWIntroduction of PAW

Introduction of PAWIntroduction of PAW

Introduction of PAWIntroduction of PAW

PAW:http://wwwasd.web.cern.ch/wwwasd/paw/

PAW:http://wwwasd.web.cern.ch/wwwasd/paw/

ROOT:http://root.cern.ch

ROOT:http://root.cern.ch

ReferencesReferences

• “Analysis Techniques in High Energy Physics”, Claude A Pruneau, Wayne State University. (http://rhic.physics.wayne.edu/REU/talks/Analysis%20Techniques.ppt)

• “Introduction to High Energy Physics”, R.H. Perkins, Cambridge University Press 2000.

• “Data Analysis Techniques for High-Energy Physics”, R. Fruhwirth, M. Regler, R.K. Bock, H. Grote, D. Notz, Cambridge University Press 2000.