data analysis in high energy physics wen-chen chang institute of physics, academia sinica 章文箴...
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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 ...
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ppbar000 It’s All a Question of Statistics ... ...
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ppbar000 It’s All a Question of Statistics ... ... ...
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ppbar000 It’s All a Question of Statistics ... ... ... ...
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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.