Transcript Analysis Techniques in High Energy Physics

Analysis Techniques
in High Energy Physics
G. Bonvicini, C. Pruneau
Wayne State University
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
d
1 dN s
( E , W) 
dW
F dW
Flux
W
dW - solid angle
q – scattering angle
Target
Unit Area
• Total Cross Section
d
 ( E )   dW
( E , W)
dW
• Average number of
scattered into dW
N s ( E , W)  FAN x
d
( E , W)
dW
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
1000000
Undecayed Nuclei
900000
800000
700000
600000
500000
400000
300000
200000
100000
0
0
2
4
6
8
Days
Half-Life and Mean-Life
• The number of particle (nuclei) left after a certain
time “t” can be expressed as follows:
N (t )  N 0 e
-t
t
• where “t” is the mean life time of the particle
• “t” can be related to the half-life “t1/2” via the
simple relation:
t1/ 2  - ln(12)t  0.693t
Examples - particles
Mass
t or G
ct
Type
(MeV/c2)
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
D(1232)
1232
120 MeV
Very short!!
Baryon resonance
L
1115.68
2.632x10-10 s
7.89 cm
Strange Baryon
resonance
Pion (p+-)
139.56995
2.603x10-8 s
7.804 m
Meson
Rho - r(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 mm
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 G.
• The width of an unstable particle is related to its
life time by the simple relation
h
G
t
• h is the Planck constant.
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 (K0s) Neutral Kaon (Short)
– Mean life time = (0.89260.0012)x10-10 s
– ct = 2.676 cm
– Decay modes and fractions
mode
G i/ G
p+ p-
(68.61  0.28) %
p0 p0
(31.39  0.28) %
p+ p- g
(1.78  0.05) x10-3
Elementary Observables
•
•
•
•
•
•
Momentum
Energy
Time-of-Flight
Energy Loss
Particle Identification
Invariant Mass Reconstruction
Momentum Measurements
p  mv
– Special Relativity
p  gmv
• Definition
– Newtonian Mechanics
g 
1
1 - v c 
• But how does one measure “p”?
2
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.
 
FM  qv  B
• Because the Lorentz force is perpendicular to both the B
field and the velocity, it acts as centripetal force “Fc”.
• One finds
m v2
Fc 
R
p  gmv  qBR
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
190 cm
Outer sector
6.2 × 19.5 mm2 pad
3940 pads
Inner sector
2.85 × 11.5 mm2 pad
1750 pads
60 cm
127 cm
Pixel Pad Readout
O
STAR
Readout arranged like the face of a clock - 5,690 pixels per sector
JT: 20
The Berkeley Lab
Momentum Measurement
B=0.5 T
Radius: R
p+
Trajectory is a helix
in 3D; a circle in the
transverse plane
Collision Vertex
p  gmv  qBR
Au on Au Event at CM Energy ~ 130 A GeV
O
STAR
Data taken June 25, 2000.
The first 12 events were captured on tape!
Real-time track reconstruction
Pictures from Level 3 online
display.
( < 70 ms )
JT: 22
The Berkeley Lab
The CLEO detector
A CLEO III hadronic event
A fast charged particle interacts
with matter
• Ionization (atom → ion + electron)
• Atomic excitation (atom → atom + photon
• Molecule breakup (molecule → two
molecules/atoms) (eg, CH4 →CH2+H2)
(molecular breakup can be avoided only if
molecule=atom, that is, noble gases)
Some properties of ionization
• Takes about 10 eV to ionize. Typical yield
1e/40eV.
• Needs an electric field to separate ion and
electron
• Needs “gain” (anywhere from 1000 to 100000),
therefore a very high electric field. We use wires
because the field goes like 1/R →MV/m near the
wire, but moderate away from wire.
Some properties of scintillation
Takes about 3-5 eV. Typical yield 1 photon
per 100 eV
• Light travels far
• Needs a device to turn light signal into
electric signal (photomultiplier)
Neutral particle interacts with
matter
• Neutral particles do not ionize
• When they first interact, they knock out CHARGED
particles which do ionize
• We measure their ENERGY by measuring the
ionization/scintillation of their targets
• They are identified in very dense detectors (calorimeters)
• A 1 GeV neutron will typically interact and be stopped in
1 meter of iron
• A 1 GeV photon will interact and be stopped in 10 cm of
lead
Bowling ball hits golf ball
• Our typical particle is the pion (M=139
MeV)
• It is 270 times heavier than one electron,
and 120 times lighter than a carbon nucleus
• When a pion hits an electron it mostly
loses ENERGY(in gas, 30 keV/cm), but it
is hardly deflected – like a BB hitting a GB
Golf ball hits bowling ball
• But there are nuclei in there too
• When the golf ball hits the bowling ball, it will
lose little energy but the momentum will be
severely affected (multiple scattering). The
bigger the nucleus, the higher the multiple
scattering
• So, most of the energy is released to electrons,
but most of the momentum change is off the
nuclei
Some useful kinematic relations
• Newtonian: p=mv, E=p2/2m, E=pv/2, etc.
• You need two of (p,m,v,E) to find the other
two). We need to know the mass to identify
the particle, and we need to know its
quantity of motion (either p or E)
• Relativistic: p=gmv, E=gm, v=p/E, etc.
• Same deal, different formulas
Energy Measurement
• Definition
– Newtonian
K.E.  mv
2
1
2
– Special Relativity
E  mc2  K.E.  gmc2
g 
1
1 - v c 
2
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)
K .E.  k  DepositedEnergy
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
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
Time
S (volts)
• 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
Dd
gets the velocity of the particle v  Dd 
Dt
c1 t stop - t start  c2 
PID by TOF
p  gmv
gv
• Since
m
p
• The mass can be determined
• In practice, this often amounts to a study of
the TOF vs momentum.
PID with a TPC
• The energy loss of charged particles passing
through a gas is a known function of their
momentum. (Bethe-Bloch Formula)
dE
Z z2
2
2
 2pN a re me c r
ln(...)
2
dx
A
O
Particle Identification by dE/dx
STAR
Anti - 3He
dE/dx PID range:
~ 0.7 GeV/c for K/p
~ 1.0 GeV/c for K/p
Invariant Mass Reconstruction
• In special relativity, the energy and momenta of
particles are related as follow
E  p c m c
2
2 2
2 4
• 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.

pP

p1
 q
p2
Invariant Mass Reconstruction (cont’d)
• Invariant Mass
2 4
2
2 2
mP c  EP - pP c
• Invariant Mass of two particles
  2
2 4
2
mP c  E1  E2  -  p1  p2 c
• After simple algebra
mP c  m  m  2E1E2 - p1 p2 cosq 
2 4
2
1
2
2
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!(Preliminary)
K
_
+
W
_
L
W
f
K 0s
L
X-
_
X
K*
Finding V0s
proton
Primary
vertex
pion
In case you thought it was easy…
After
Before
Strange
Baryon
Ratios
Reconstruct:
Reconstruct:
~0.84 L/ev,
_
~ 0.61 L/ev
~0.006 X-/ev,
_
~0.005 X/ev
STAR Preliminary
Ratio = 0.73 ± 0.03 (stat)
Ratio = 0.82 ± 0.08 (stat)
Resonances
f
K  K-