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Analysis Techniquesin High Energy Physics
Claude A Pruneau
Wayne State Univeristy
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Overview
• Some Observables of Interest
• Elementary Observables
• More Complex Observables
• Typical Analysis of (Star) data
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Some Observables of Interest
• Total Interaction/Reaction Cross Section
• Differential Cross Section: I.e. cross section vs particle momentum, or production angle, etc.
• Particle life time or width
• Branching Ratio
• Particle Production Fluctuation (HI)
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Cross Section
• Total Cross Section of an object determines how “big” it is and how likely it is to collide with projectile thrown towards it randomly.
Small cross section
Large cross section
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Scattering 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
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Particle life time or width
• Most particles studied in particle physics or high energy nuclear physics are unstable and decay within a finite lifetime. – Some useful exceptions include the electron, and the proton. However
these are typically studied for their own sake but to address some other observable…
• 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.
• In particle physics and high energy nuclear physics, the concept of mean life time or simple life time is usually used. The two are connected by a simple multiplicative constant.
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Half-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
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Half-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:
t
eNtN
0)(
0.693)ln( 21
2/1 t
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Examples - particlesMass (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
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Examples - 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
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Particle 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
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Decay 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
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Elementary Observables
• Momentum
• Energy
• Time-of-Flight
• Energy Loss
• Particle Identification
• Invariant Mass Reconstruction
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Momentum Measurements
• Definition– Newtonian Mechanics
– Special Relativity
• But how does one measure “p”?
mvp
mvp
21
1
cv
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Momentum 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
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Momentum Measurements Technique
• Knowledge of B and R needed to get “p”• B is determined by the construction/operation of
the spectrometer.• “R” must be measured for each particle.• To measure “R”, STAR and CLEO both use a
similar technique.– Find the trajectory of the charged particle through
detectors sensitive to particle energy loss and capable of measuring the location of the energy deposition.
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Star Time-Projection-Chamber (TPC)
• Large Cylindrical Vessel filled with “P10” gas (10% methane, 90% Ar)
• Imbedded in a large solenoidal magnet
• Longitudinal Electrical Field used to supply drive force needed to collect charge produced ionization of p10 gas by the passage of charged particles.
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STAR TPC
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Pad readout
2×12 super-sectors
60 cm
127 cm
190 cm
Outer sector6.2 × 19.5 mm2 pad
3940 pads
Inner sector2.85 × 11.5 mm2 pad
1750 pads
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JT: 20The Berkeley Lab
STARPixel Pad Readout
Readout arranged like the face of a clock - 5,690 pixels per sector
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Momentum Measurement
+B=0.5 T
Collision Vertex
Trajectory is a helix in 3D; a circle in the transverse plane
qBRmvp
Radius: R
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JT: 22The Berkeley Lab
STARAu on Au Event at CM Energy ~ 130 A GeV
Real-time track reconstruction
Pictures from Level 3 online display. ( < 70 ms )
Data taken June 25, 2000.
The first 12 events were captured on tape!
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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)
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Energy Measurement
• Definition– Newtonian
– Special Relativity
221.. mvEK
22 .. mcEKmcE
21
1
cv
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Energy Measurement
• Determination of energy by calorimetry
• Particle energy measured via a sample of its energy loss as it passes through layers of radiator (e.g. lead) and sampling materials (scintillators)
nergyDepositedEkEK ..
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More Complex Observables
• Particle Identification
• Invariant Mass Reconstruction
• Identification of decay vertices
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Particle 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
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PID by TOF
• Since
• The mass can be determined
• In practice, this often amounts to a study of the TOF vs momentum.
mvp p
vm
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PID 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
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STARParticle Identification by dE/dx
dE/dx PID range: ~ 0.7 GeV/c for K/
~ 1.0 GeV/c for K/p
Anti - 3He
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Invariant 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
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Invariant Mass Reconstruction (cont’d)
• Invariant Mass
• Invariant Mass of two particles
• After simple algebra
22242 cpEcm PPP
221
221
42 cppEEcmP
cos2 212122
21
42 ppEEmmcmP
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But – remember the jungle of tracks…
Large likelihood of coupling together tracks that do not actually belong together…
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Example – Lambda Reconstruction
Good pairs have the right invariant mass and accumulate in a peak at the Lambda mass.
Bad pairs produce a more or less continuous background below and around the peak.
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STAR STRANGENESS!
K0s
K+
(Preliminary)
__
_
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Finding V0sproton
pion
Primary vertex
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In case you thought it was easy…
BeforeAfter
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Strange Baryon Ratios
Ratio = 0.73 ± 0.03 (stat)
~0.84 /ev, ~ 0.61/ev_Reconstruct:
Ratio = 0.82 ± 0.08 (stat)
Reconstruct: ~0.006 /ev, ~0.005/ev
_
STAR Preliminary
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Resonances