Monte Carlo Simulation in Particle Physics

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Monte Carlo Simulation in Particle Physics

• Concezio Bozzi
• Istituto Nazionale di Fisica Nucleare
• Ferrara (Italy)

IUB, Bremen, Germany, November 28th 2002
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Laymans terms?
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The MonteCarlo method
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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)

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Montecarlo simulation

• A numerical simulation method which uses
  sequences of random numbers to solve complex
  problems.

Similarity to games of chance explains the name
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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

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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
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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 man-made 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.
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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,

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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?

Target area p r2, dart board a2, ratio
Ncircle/Nboard p r2 / a2
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The drunk player gets p !

• From previous page, and if a2r p 4
  Ncircle/Nboard
• Try this!
• Precision of calculation is 1/sqrt(N)
• 100 tries 3.1 ? 0.3
• 10,000 tries 3.14 ? 0.03
• 1,000,000 tries 3.142 ? 0.003
• 100,000,000 tries 3.1416 ? 0.0003
• 10,000,000,000 tries 3.14159 ? 0.00003
• Computing power is an issuehow long would it
  take to throw 10,000,000,000 dartsthats why MC
  method has becoming popular only quite recently

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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

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Particle physics
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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

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Evolution of the Universe
The Universe began with a Big Bang about 15
billion years ago
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The concept of elements
In Aristotles 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?
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The periodic table
Mendeleev (1869) introduced the periodic table
This pattern suggests atoms are made by smaller
building blocks!
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The structure of atoms
Rutherford (1912) showed that atoms contain a
central nucleus
Electrons orbit nucleus with well-defined energy
and ill-defined positions
10-10 m
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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)
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A look at the scales

• There is no further evidence of quark and
  electrons substructures

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The standard model matter
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The standard model forces
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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
  everydays experience
• Particle-wave duality interference
• Pauli exclusion principle (-gt 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.

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Testing the theory
A source-target-detection scheme Thats 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
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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.

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The worlds meterstick
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Mass and energy
Also, physicists study heavy particles by using
light projectiles
Emc2
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Particle accelerators
A linear accelerator (cathode tube)
A circular accelerator (collider)
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Detectors
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LEP at CERN (Geneva)
Electron (matter)
Positron (antimatter)
Annihilation produces energy mini Big Bang
Particles and antiparticles are produced
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The ALEPH detector
International collaborations 500-1000 physicists
from all the world. Typical costs 100s M
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The Stanford Linear Accelerator
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The Babar detector
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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
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MonteCarlo and Particle Physics
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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

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MC and event simulation

• Particle interactions and decays are governed by
  quantum mechanics, so they are intrinsecally
  probabilistic

StdHepPrintStdHep Track info for event 4
StdHepPrintTrk Stat Id Dtr1 DtrN
Mom1 MomN Px Py Pz E Vx
Vy Vz StdHepPrint 1 3 e
3 4 0 0 0.05558 0.001356 -3.112
3.113 0.09944 0.33 -1.394 StdHepPrint 2
3 e- 3 4 0 0 -0.165
-0.0004597 8.985 8.986 0.09944 0.33
-1.394 StdHepPrint 3 2 tau 5
6 1 0 1.155 3.309 6.957 7.99
0.09944 0.33 -1.394 StdHepPrint 4 2
tau- 9 10 1 0 -1.264 -3.308
-1.085 4.108 0.09944 0.33
-1.394 StdHepPrint 5 1 anti-nu_tau 0
0 3 0 0.3767 0.1478 1.502 1.555
0.1049 0.3456 -1.361 StdHepPrint 6 2 rho
7 8 3 0 0.7781 3.162
5.455 6.435 0.1049 0.3456 -1.361 StdHepPrint
7 1 pi 0 0 6 0
0.1861 0.08695 0.2127 0.327 0.1049 0.3456
-1.361 StdHepPrint 8 2 pi0 14
15 6 0 0.592 3.075 5.243 6.108
0.1049 0.3456 -1.361 StdHepPrint 9 1
nu_tau 0 0 4 0 -0.3907 -1.938
-0.9116 2.177 0.09512 0.3187
-1.397 StdHepPrint 10 2 a_1- 11
13 4 0 -0.8736 -1.37 -0.1733 1.931
0.09512 0.3187 -1.397 StdHepPrint 11 2 pi0
16 17 10 0 -0.05043 -0.716
-0.1166 0.7396 0.09512 0.3187
-1.397 StdHepPrint 12 2 pi0 18
19 10 0 -0.4634 -0.6317 0.02961 0.7955
0.09512 0.3187 -1.397 StdHepPrint 13 1 pi-
0 0 10 0 -0.3598 -0.02258
-0.08632 0.3961 0.09512 0.3187
-1.397 StdHepPrint 14 1 gamma 0
0 8 0 0.2288 1.168 2.122 2.433
0.1049 0.3456 -1.361 StdHepPrint 15 1
gamma 0 0 8 0 0.3632 1.906
3.121 3.675 0.1049 0.3456
-1.361 StdHepPrint 16 1 gamma 0
0 11 0 -0.03818 -0.4131 -0.0001552 0.4149
0.09512 0.3187 -1.397 StdHepPrint 17 1
gamma 0 0 11 0 -0.01225 -0.3028
-0.1164 0.3247 0.09512 0.3187
-1.397 StdHepPrint 18 1 gamma 0
0 12 0 -0.1236 -0.1387 0.06213 0.1959
0.09512 0.3187 -1.397 StdHepPrint 19 1
gamma 0 0 12 0 -0.3398 -0.493
-0.03252 0.5996 0.09512 0.3187 -1.397
StdHepPrintStdHep Track info for event 3
StdHepPrintTrk Stat Id Dtr1 DtrN
Mom1 MomN Px Py Pz E Vx
Vy Vz StdHepPrint 1 3 e
3 4 0 0 0.05951 -0.0005719 -3.114
3.115 0.09094 0.3304 -0.7858 StdHepPrint 2
3 e- 3 4 0 0 -0.1682
0.002575 8.984 8.985 0.09094 0.3304
-0.7858 StdHepPrint 3 2 tau 5
7 1 0 2.812 3.643 5.011 7.032
0.09094 0.3304 -0.7858 StdHepPrint 4 2
tau- 8 9 1 0 -2.921 -3.641
0.8581 5.068 0.09094 0.3304 -0.7858 StdHepPrint
5 1 anti-nu_tau 0 0 3 0
0.8467 0.8655 2.489 2.768 0.0932 0.3333
-0.7817 StdHepPrint 6 1 mu 0
0 3 0 0.8607 1.613 1.408 2.31
0.0932 0.3333 -0.7817 StdHepPrint 7 1
nu_mu 0 0 3 0 1.105 1.165
1.114 1.954 0.0932 0.3333 -0.7817 StdHepPrint
8 1 nu_tau 0 0 4 0
0.004886 -0.2322 0.2253 0.3236 0.08223 0.3195
-0.7832 StdHepPrint 9 2 a_1- 10
12 4 0 -2.926 -3.409 0.6328 4.745
0.08223 0.3195 -0.7832 StdHepPrint 10 1 pi-
0 0 9 0 -1.17 -1.052
0.1692 1.588 0.08223 0.3195 -0.7832 StdHepPrint
11 1 pi- 0 0 9 0
-1.634 -2.019 0.7127 2.697 0.08223 0.3195
-0.7832 StdHepPrint 12 1 pi 0
0 9 0 -0.1223 -0.3385 -0.2491 0.4594
0.08223 0.3195 -0.7832
StdHepPrintStdHep Track info for event 2
StdHepPrintTrk Stat Id Dtr1 DtrN
Mom1 MomN Px Py Pz E Vx
Vy Vz StdHepPrint 1 3 e
3 5 0 0 0.05871 -0.001051 -3.115
3.115 0.09233 0.33 0.5908 StdHepPrint 2
3 e- 3 5 0 0 -0.1684
-0.002014 8.989 8.99 0.09233 0.33
0.5908 StdHepPrint 3 2 tau 6
8 1 0 2.136 -1.298 -1.221 3.301
0.09233 0.33 0.5908 StdHepPrint 4 2
tau- 9 10 1 0 -2.184 1.238
7.649 8.244 0.09233 0.33
0.5908 StdHepPrint 5 1 gamma 0
0 1 0 -0.0622 0.05684 -0.5539 0.5603
0.09233 0.33 0.5908 StdHepPrint 6 1
anti-nu_tau 0 0 3 0 0.09482 0.006695
0.07114 0.1187 0.09629 0.3276
0.5886 StdHepPrint 7 1 mu 0
0 3 0 1.278 -0.6479 0.1266 1.442
0.09629 0.3276 0.5886 StdHepPrint 8 1
nu_mu 0 0 3 0 0.7632 -0.6571
-1.419 1.74 0.09629 0.3276
0.5886 StdHepPrint 9 1 nu_tau 0
0 4 0 -1.092 -0.09733 1.911 2.203
0.07894 0.3376 0.6378 StdHepPrint 10 2
rho- 11 12 4 0 -1.092 1.336
5.738 6.041 0.07894 0.3376
0.6378 StdHepPrint 11 1 pi- 0
0 10 0 -0.6058 0.2708 1.578 1.718
0.07894 0.3376 0.6378 StdHepPrint 12 2 pi0
13 14 10 0 -0.4862 1.065
4.16 4.324 0.07894 0.3376 0.6378 StdHepPrint
13 1 gamma 0 0 12 0 -0.4173
0.8177 3.101 3.234 0.07894 0.3376
0.6378 StdHepPrint 14 1 gamma 0
0 12 0 -0.06891 0.2471 1.059 1.089
0.07894 0.3376 0.6378
StdHepPrintStdHep Track info for event 1
StdHepPrintTrk Stat Id Dtr1 DtrN Mom1 MomN
Px Py Pz E Vx Vy
Vz StdHepPrint 1 3 e 3 4
0 0 0.0576 -0.0005 -3.1094 3.1099 0.0907
0.3294 -0.8146 StdHepPrint 2 3 e- 3
4 0 0 -0.1676 0.0008 8.9919 8.9934
0.0907 0.3294 -0.8146 StdHepPrint 3 2 tau
5 7 1 0 -4.0722 -2.4796
1.1461 5.2156 0.0907 0.3294 -0.8146 StdHepPrint
4 2 tau- 8 9 1 0
3.9623 2.4799 4.7364 6.8877 0.0907 0.3294
-0.8146 StdHepPrint 5 1 anti-nu_tau 0 0
3 0 -2.8840 -1.3849 0.8917 3.3212 0.0878
0.3277 -0.8138 StdHepPrint 6 1 e
0 0 3 0 -0.5448 -0.5248 0.6025
0.9671 0.0878 0.3277 -0.8138 StdHepPrint 7
1 nu_e 0 0 3 0 -0.6434
-0.5699 -0.3480 0.9273 0.0878 0.3277
-0.8138 StdHepPrint 8 1 nu_tau 0
0 4 0 1.6662 1.8894 2.0134 3.2249
0.0943 0.3317 -0.8103 StdHepPrint 9 2 rho-
10 11 4 0 2.2961 0.5905
2.7230 3.6628 0.0943 0.3317 -0.8103 StdHepPrint
10 1 pi- 0 0 9 0
1.2046 0.0371 1.2295 1.7273 0.0943 0.3317
-0.8103 StdHepPrint 11 2 pi0 12
13 9 0 1.0915 0.5533 1.4935 1.9356
0.0943 0.3317 -0.8103 StdHepPrint 12 1
gamma 0 0 11 0 0.8447 0.4610
1.2430 1.5720 0.0943 0.3317 -0.8103 StdHepPrint
13 1 gamma 0 0 11 0 0.2468
0.0923 0.2505 0.3635 0.0943 0.3317 -0.8103
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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.
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Detector design (Babar detector)
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Particle-detector interactions
Lets simulate an electromagnetic shower!!!
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.
Incoming particles
Easily modeled by the MC technique!
A block of matter
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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
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Using MC in physics measurements

• Use MC simulation to compute the signal
  efficiency and background contamination.
• Optimize the selection criteria to get the
  smallest error.

Signal MC
Background MC

• Need to estimate the reliability of the
  simulation, and assign the correspondent
  systematical uncertainty

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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

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How much computing power?

• Take e.g. Babar
• 500 million events/year of real data
• MCdata at least 31, i.e. 1.5 billion
  events/year
• 20 sec/event on a Intel CPU
• A single computer will need 1000 years to
  generate them

Use 1000 computers in parallel
Develop a Grid
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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