Analysis Techniquesin High Energy Physics

Claude A Pruneau

Wayne State Univeristy

Overview

• Some Observables of Interest

• Elementary Observables

• More Complex Observables

• Typical Analysis of (Star) data

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)

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

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

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.

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

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

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

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

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

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

Elementary Observables

• Momentum

• Energy

• Time-of-Flight

• Energy Loss

• Particle Identification

• Invariant Mass Reconstruction

Momentum Measurements

• Definition– Newtonian Mechanics

– Special Relativity

• But how does one measure “p”?

mvp

mvp

21

1

cv

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

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.

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.

STAR TPC

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

JT: 20The Berkeley Lab

STARPixel Pad Readout

Readout arranged like the face of a clock - 5,690 pixels per sector

Momentum Measurement

+B=0.5 T

Collision Vertex

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

qBRmvp

Radius: R

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!

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)

Energy Measurement

• Definition– Newtonian

– Special Relativity

221.. mvEK

22 .. mcEKmcE

21

1

cv

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 ..

More Complex Observables

• Particle Identification

• Invariant Mass Reconstruction

• Identification of decay vertices

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

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

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

STARParticle Identification by dE/dx

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

~ 1.0 GeV/c for K/p

Anti - 3He

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

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

But – remember the jungle of tracks…

Large likelihood of coupling together tracks that do not actually belong together…

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.

STAR STRANGENESS!

K0s

K+

(Preliminary)

__

_

Finding V0sproton

pion

Primary vertex

In case you thought it was easy…

BeforeAfter

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

Resonances