Monte Carlo Simulation in
Particle Physics
Concezio Bozzi
Istituto Nazionale di Fisica Nucleare
Ferrara (Italy)
IUB, Bremen, Germany, November 28th 2002
Layman’s terms?
The MonteCarlo
method
A word on simulations
• A (computer) simulation applies mathematical
methods to the analysis of complex, real-world
problems and predicts what might happen
depending on various actions/scenarios
• Use simulations when
– Doing the actual experiments is not possible (e.g. the
Greenhouse effect)
– The cost in money, time, or danger of the actual
experiment is prohibitive (e.g. nuclear reactors)
– The system does not exist yet (e.g. an airplane)
– Various alternatives are examined (e.g. hurricane
predictions)
Montecarlo simulation
A numerical simulation
method which uses
sequences of random
numbers to solve
complex problems.
Similarity to games
of chance explains
the name…
Why MonteCarlo?
• Other numerical methods tipically need a mathematical
description of the system (ordinary or partial differential
equations)
• More and more difficult to solve as complexity increases
MC assumes the system is described by probability density
functions which can be modeled with no need to write down
equations.
These PDF are sampled randomly, many simulations are
performed and the result is the average over the number of
observations
A brief history
• Method formally developed by John Von Neumann during
WWII, but already known before
• Fermi used it to simulate neutron diffusion in
the 1930s. He knew the behavior of one neutron,
but he did not have a formula for how a system of
neutrons would behave.
He also used it to demonstrate
the stability of the first manmade nuclear reactor
(Chicago Pile, 1942). His
model had an analogy with heat
diffusion models previously
developed.
Fermi used tables of numbers sorted on a roulette to obtain random
numbers which he then used in his calculations of neutron absorption.
A brief history
• Manhattan Project of WWII (Von
Neumann, Ulam, Metropolis)
– Scientists used it to construct
dampers and shields for the nuclear
bomb, experimentation was too time
consuming and dangerous.
• Extensively used in many disciplines especially
after the advent of high-speed computing:
– Cancer therapy, traffic flow, Dow-Jones forecasting,
oil well exploration, stellar evolution, reactor design,
particle physics, ancient languages deciphering,…
The drunk dart player
• Suppose you are in a
pub and drank a
number of beers…
• …enough to throw
darts randomly
• Did you ever imagine
to be useful to
science?
a
a
r
Target area = p r2, dart board = a2, ratio = Ncircle/Nboard = p r2 / a2
The drunk player gets p !
• From previous page, and if a=2r: p = 4 Ncircle/Nboard
Try this!
• Precision of calculation is 1/sqrt(N)
–
–
–
–
–
100 tries:
10,000 tries:
1,000,000 tries:
100,000,000 tries:
10,000,000,000 tries:
3.1
 0.3
3.14
 0.03
3.142  0.003
3.1416  0.0003
3.14159  0.00003
• Computing power is an issue…how long would
it take to throw 10,000,000,000 darts…that’s why
MC method has becoming popular only quite
recently
Placing rest areas in a motorway
• Define model depending on
– Entry points (will depend e.g. on the population of a
nearby city, time of day, peak- offpeak hours, etc.)
– Car velocities and gasoline consumption
– Journey length
– Exit points
• Throw random numbers to set initial conditions
and evolve
• Repeat experiment several times and look at the
resulting car distribution
• Determine where the majority is located at
lunchtime, or where they run out of gasoline, etc…
Particle physics
The name of the game
• Search for the building
blocks of our world and the
interactions between them
• Carried out with huge
accelerators by studying the
debris from large number of
particle collisions
• The same forces govern the
behaviour of the universe
from its bery beginning (Big
Bang). Strong link between
particle physics and cosmology
Evolution of
the Universe
15 billion years
life on earth,
molecules
form
1 billion years
heavy
elements
stars and
formed
galaxies
in stars
microwave
exist,
1 million years
300,000 years
background
atoms
radiation
form
fills universe
3 minutes
helium
nuclei
formed
1 second
neutrons
and protons
quark "soup"
10
10 s
formed
?
matter
dominates
1015deg
1010deg
109deg
6000o
The Universe began with a “Big
Bang” about 15 billion years ago
4000o
-255o
-270o
The concept of elements
In Aristotle’s philosophy
there were four elements
Today we know that there is
something more fundamental than
earth, water, air, and fire...
By convention there is color,
By convention sweetness,
By convention bitterness,
But in reality there are atoms and space.
-Democritus (c. 400BC)
But is the atom fundamental?
The periodic table
Mendeleev (1869) introduced the
periodic table
This pattern suggests atoms
are made by smaller building blocks!
The structure of atoms
Rutherford (1912)
showed that
atoms
contain a central
nucleus
-10
10 m
Electrons orbit nucleus
with well-defined
energy and ill-defined
positions
Nucleons and quarks
Nuclei are in turn
made of protons
and neutrons
Protons and neutrons
contain quarks
A modern view
of the atom
(not to scale)
A look at the scales
• There is no further
evidence of quark and
electrons
substructures…
The standard model: matter
The standard model: forces
Quantum mechanics
• All particle interactions and decays are
described by quantum mechanics
(relativistic quantum field theory, to be
more precise)
• Particles behave quite differently from
everyday’s experience
– Particle-wave duality: interference
– Pauli exclusion principle (-> chemistry)
– We cannot say what particles will do, but
only what they might do
– QM explains the behaviours of particles
in probabilistic terms
– Mean lifetime, branching fractions, cross
sections, etc.
Electron interference!
Testing the theory
A source-target-detection scheme
That’s how we perceive the world
(bats use sound waves)
Level of detail limited by wavelength
Visible light unfit to analyze anything
smaller than a cell
Going to shorter wavelengths
QM (DeBroglie) says all
particles have wave properties
Use particles as probes
e.g. the electronic miscroscope!
Wavelength is inversely proportional
to particle momentum!
•
Put your probing particle into an accelerator.
•
Give your particle lots of momentum by
speeding it up to very nearly the speed of light.
•
Since the particle now has a lot of momentum,
its wavelength is very short.
•
Slam this probing particle into the target and
record what happens.
The world’s meterstick
Mass and energy
Also, physicists study heavy particles by using light projectiles
E=mc2
Particle accelerators
A linear
accelerator
(cathode tube)
A circular
accelerator
(collider)
Detectors
Fixed target
Collider
LEP at CERN (Geneva)
eElectron (matter)
Annihilation produces energy
mini Big Bang
e+
Positron (antimatter)
Particles and antiparticles are produced
The ALEPH detector
End view
International collaborations
~500-1000 physicists from all the world. Typical costs: 100s M$
The Stanford Linear Accelerator
The Babar detector
The “event”
An event is the result of a collision. We isolate each event, collect
data from it, and check whether the particle processes of that event
agree with the theory we are testing.
Each event is very
complicated since lots of
particles are produced. Most
of these particles have
lifetimes so short that decay
into other particles, leaving
no detectable tracks. So we
look at decay products and
infer from them a particle
existance and its properties
MonteCarlo and
Particle Physics
A typical MC use case
Generate events to simulate detector data. Extremely
useful for
• Detector design and optimization
– complicated, huge and very expensive
– will it work as expected?
– simulation of particle interactions with detectors to
optimize design and cost/benefits ratio
• Geometrical acceptance
• Space resolution
• Energy/momentum resolution
• Physics measurements
– Estimate background, efficiencies, etc.
– Simulate new physics effects or new particles
– Need a lot of simulated events
MC and event simulation
• Particle interactions and decays are
governed by quantum mechanics, so they
are intrinsecally probabilistic
StdHepPrint:StdHep Track info for event 4 :
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StdHepPrint:StdHep
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32 :: 1 :
StdHepPrint:StdHep
StdHepPrint:StdHep
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Optimizing detector acceptance
Study of the process:
e+ e-  p+ p- p0
Angular distribution of p0
decay products for 3 different
energies.
Detector design (Babar detector)
Particle-detector interactions
Electrons and/or photons hit
matter, travel through the
material, interacting with
atoms and their nuclei in
various ways that are easily
predicted by physics.
The path of each particle can
be modeled as a random
walk as collisions with atoms
occur with well-defined
probability.
Easily modeled
by the MC technique!
Let’s simulate an
electromagnetic shower!!!
Incoming
particles
A block of matter
Material validation (Babar detector)
Use known processes
to see if detector
simulation (position in
space, resolution, amount
of material) is reliable
Bremsstrahlung in
Bhabha events
Background
MC
• Use MC simulation to
compute the signal
efficiency and background
contamination.
• Optimize the selection
criteria to get the smallest
error.
Signal
MC
Using MC in physics measurements
• Need to estimate the
reliability of the
simulation, and assign the
correspondent
systematical uncertainty
Discovery of the top quark
(Fermilab, Chicago, 1995)
• Distribution show
invariant mass of decay
products
• Data points clearly above
background, computed
with MC
• Generate several MC
samples corresponding to
different values of the top
quark mass
• Find the mass value which
best fits to data
How much computing power?
• Take e.g. Babar
–
–
–
–
500 million events/year of real data
MC:data at least 3:1, i.e. 1.5 billion events/year
~20 sec/event on a Intel CPU
A single computer will need 1000 years to generate them
To Russia/Japan
To USA
Cern
Use 1000
computers in
parallel
Develop a Grid
Conclusion
• Simulation with random numbers is a quite
general technique
• Can be applied in many different fields
(natural sciences, engineering, finance, etc.)
• Particle physicists use it widely both in
detector design/optimization and
subsequently data analysis
• Needs big computing power
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Monte Carlo simulation in High Energy Physics