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

• 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.

• Method formally developed by John Von Neumann during WWII, but already known before

Fermi used tables of numbers sorted on a roulette to obtain random numbers which he then used in his calculations of neutron absorption.

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.

• 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,…

A brief history

a

a

r

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 = r2, dart board = a2, ratio = Ncircle/Nboard = r2 / a2

The drunk player gets • From previous page, and if a=2r: = 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 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

The Universe began with a “Big Bang” about 15 billion years ago

-270o

?

heavy elements formed in stars

stars and galaxies exist, atoms form

neutrons

quark "soup"

15 billion years

1 million years

1 second

10-10

1015deg 1010deg109deg

6000o

-255o

3 minuteshelium nuclei formed

microwave background radiation fills universe

300,000 years

4000o

life on earth, molecules form

dominates matter

and protons formed

1 billion years

s

Big Bang

Big Bang

Evolution of the Universe

The concept of elements

In Aristotle’s philosophythere 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 atomscontain a centralnucleus

Electrons orbit nucleuswith well-definedenergy and ill-definedpositions10

-10 m

Nucleons and quarks

Nuclei are in turn made of protons and neutrons

Protons and neutronscontain 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 schemeThat’s how we perceive the world(bats use sound waves)

Level of detail limited by wavelengthVisible light unfit to analyze anythingsmaller 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)

e- e+

Annihilation produces energy mini Big Bang

Electron (matter) Positron (antimatter)

Particles and antiparticles are produced

The ALEPH detector

End viewInternational collaborations

~500-1000 physicists from all the world. Typical costs: 100s M$

The Stanford Linear Accelerator

The Babar detector

The “event”

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

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.

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 2 :StdHepPrint:Trk# Stat Id Dtr1 DtrN Mom1 MomN Px Py Pz E Vx Vy VzStdHepPrint: 1 3 e+ 3 5 0 0 0.05871 -0.001051 -3.115 3.115 0.09233 0.33 0.5908StdHepPrint: 2 3 e- 3 5 0 0 -0.1684 -0.002014 8.989 8.99 0.09233 0.33 0.5908StdHepPrint: 3 2 tau+ 6 8 1 0 2.136 -1.298 -1.221 3.301 0.09233 0.33 0.5908StdHepPrint: 4 2 tau- 9 10 1 0 -2.184 1.238 7.649 8.244 0.09233 0.33 0.5908StdHepPrint: 5 1 gamma 0 0 1 0 -0.0622 0.05684 -0.5539 0.5603 0.09233 0.33 0.5908StdHepPrint: 6 1 anti-nu_tau 0 0 3 0 0.09482 0.006695 0.07114 0.1187 0.09629 0.3276 0.5886StdHepPrint: 7 1 mu+ 0 0 3 0 1.278 -0.6479 0.1266 1.442 0.09629 0.3276 0.5886StdHepPrint: 8 1 nu_mu 0 0 3 0 0.7632 -0.6571 -1.419 1.74 0.09629 0.3276 0.5886StdHepPrint: 9 1 nu_tau 0 0 4 0 -1.092 -0.09733 1.911 2.203 0.07894 0.3376 0.6378StdHepPrint: 10 2 rho- 11 12 4 0 -1.092 1.336 5.738 6.041 0.07894 0.3376 0.6378StdHepPrint: 11 1 pi- 0 0 10 0 -0.6058 0.2708 1.578 1.718 0.07894 0.3376 0.6378StdHepPrint: 12 2 pi0 13 14 10 0 -0.4862 1.065 4.16 4.324 0.07894 0.3376 0.6378StdHepPrint: 13 1 gamma 0 0 12 0 -0.4173 0.8177 3.101 3.234 0.07894 0.3376 0.6378StdHepPrint: 14 1 gamma 0 0 12 0 -0.06891 0.2471 1.059 1.089 0.07894 0.3376 0.6378

StdHepPrint:StdHep Track info for event 3 :StdHepPrint:Trk# Stat Id Dtr1 DtrN Mom1 MomN Px Py Pz E Vx Vy VzStdHepPrint: 1 3 e+ 3 4 0 0 0.05951 -0.0005719 -3.114 3.115 0.09094 0.3304 -0.7858StdHepPrint: 2 3 e- 3 4 0 0 -0.1682 0.002575 8.984 8.985 0.09094 0.3304 -0.7858StdHepPrint: 3 2 tau+ 5 7 1 0 2.812 3.643 5.011 7.032 0.09094 0.3304 -0.7858StdHepPrint: 4 2 tau- 8 9 1 0 -2.921 -3.641 0.8581 5.068 0.09094 0.3304 -0.7858StdHepPrint: 5 1 anti-nu_tau 0 0 3 0 0.8467 0.8655 2.489 2.768 0.0932 0.3333 -0.7817StdHepPrint: 6 1 mu+ 0 0 3 0 0.8607 1.613 1.408 2.31 0.0932 0.3333 -0.7817StdHepPrint: 7 1 nu_mu 0 0 3 0 1.105 1.165 1.114 1.954 0.0932 0.3333 -0.7817StdHepPrint: 8 1 nu_tau 0 0 4 0 0.004886 -0.2322 0.2253 0.3236 0.08223 0.3195 -0.7832StdHepPrint: 9 2 a_1- 10 12 4 0 -2.926 -3.409 0.6328 4.745 0.08223 0.3195 -0.7832StdHepPrint: 10 1 pi- 0 0 9 0 -1.17 -1.052 0.1692 1.588 0.08223 0.3195 -0.7832StdHepPrint: 11 1 pi- 0 0 9 0 -1.634 -2.019 0.7127 2.697 0.08223 0.3195 -0.7832StdHepPrint: 12 1 pi+ 0 0 9 0 -0.1223 -0.3385 -0.2491 0.4594 0.08223 0.3195 -0.7832

StdHepPrint:StdHep Track info for event 4 :StdHepPrint:Trk# Stat Id Dtr1 DtrN Mom1 MomN Px Py Pz E Vx Vy VzStdHepPrint: 1 3 e+ 3 4 0 0 0.05558 0.001356 -3.112 3.113 0.09944 0.33 -1.394StdHepPrint: 2 3 e- 3 4 0 0 -0.165 -0.0004597 8.985 8.986 0.09944 0.33 -1.394StdHepPrint: 3 2 tau+ 5 6 1 0 1.155 3.309 6.957 7.99 0.09944 0.33 -1.394StdHepPrint: 4 2 tau- 9 10 1 0 -1.264 -3.308 -1.085 4.108 0.09944 0.33 -1.394StdHepPrint: 5 1 anti-nu_tau 0 0 3 0 0.3767 0.1478 1.502 1.555 0.1049 0.3456 -1.361StdHepPrint: 6 2 rho+ 7 8 3 0 0.7781 3.162 5.455 6.435 0.1049 0.3456 -1.361StdHepPrint: 7 1 pi+ 0 0 6 0 0.1861 0.08695 0.2127 0.327 0.1049 0.3456 -1.361StdHepPrint: 8 2 pi0 14 15 6 0 0.592 3.075 5.243 6.108 0.1049 0.3456 -1.361StdHepPrint: 9 1 nu_tau 0 0 4 0 -0.3907 -1.938 -0.9116 2.177 0.09512 0.3187 -1.397StdHepPrint: 10 2 a_1- 11 13 4 0 -0.8736 -1.37 -0.1733 1.931 0.09512 0.3187 -1.397StdHepPrint: 11 2 pi0 16 17 10 0 -0.05043 -0.716 -0.1166 0.7396 0.09512 0.3187 -1.397StdHepPrint: 12 2 pi0 18 19 10 0 -0.4634 -0.6317 0.02961 0.7955 0.09512 0.3187 -1.397StdHepPrint: 13 1 pi- 0 0 10 0 -0.3598 -0.02258 -0.08632 0.3961 0.09512 0.3187 -1.397StdHepPrint: 14 1 gamma 0 0 8 0 0.2288 1.168 2.122 2.433 0.1049 0.3456 -1.361StdHepPrint: 15 1 gamma 0 0 8 0 0.3632 1.906 3.121 3.675 0.1049 0.3456 -1.361StdHepPrint: 16 1 gamma 0 0 11 0 -0.03818 -0.4131 -0.0001552 0.4149 0.09512 0.3187 -1.397StdHepPrint: 17 1 gamma 0 0 11 0 -0.01225 -0.3028 -0.1164 0.3247 0.09512 0.3187 -1.397StdHepPrint: 18 1 gamma 0 0 12 0 -0.1236 -0.1387 0.06213 0.1959 0.09512 0.3187 -1.397StdHepPrint: 19 1 gamma 0 0 12 0 -0.3398 -0.493 -0.03252 0.5996 0.09512 0.3187 -1.397

StdHepPrint:StdHep Track info for event 1 : StdHepPrint:Trk# Stat Id Dtr1 DtrN Mom1 MomN Px Py Pz E Vx Vy VzStdHepPrint: 1 3 e+ 3 4 0 0 0.0576 -0.0005 -3.1094 3.1099 0.0907 0.3294 -0.8146StdHepPrint: 2 3 e- 3 4 0 0 -0.1676 0.0008 8.9919 8.9934 0.0907 0.3294 -0.8146StdHepPrint: 3 2 tau+ 5 7 1 0 -4.0722 -2.4796 1.1461 5.2156 0.0907 0.3294 -0.8146StdHepPrint: 4 2 tau- 8 9 1 0 3.9623 2.4799 4.7364 6.8877 0.0907 0.3294 -0.8146StdHepPrint: 5 1 anti-nu_tau 0 0 3 0 -2.8840 -1.3849 0.8917 3.3212 0.0878 0.3277 -0.8138StdHepPrint: 6 1 e+ 0 0 3 0 -0.5448 -0.5248 0.6025 0.9671 0.0878 0.3277 -0.8138StdHepPrint: 7 1 nu_e 0 0 3 0 -0.6434 -0.5699 -0.3480 0.9273 0.0878 0.3277 -0.8138StdHepPrint: 8 1 nu_tau 0 0 4 0 1.6662 1.8894 2.0134 3.2249 0.0943 0.3317 -0.8103StdHepPrint: 9 2 rho- 10 11 4 0 2.2961 0.5905 2.7230 3.6628 0.0943 0.3317 -0.8103StdHepPrint: 10 1 pi- 0 0 9 0 1.2046 0.0371 1.2295 1.7273 0.0943 0.3317 -0.8103StdHepPrint: 11 2 pi0 12 13 9 0 1.0915 0.5533 1.4935 1.9356 0.0943 0.3317 -0.8103StdHepPrint: 12 1 gamma 0 0 11 0 0.8447 0.4610 1.2430 1.5720 0.0943 0.3317 -0.8103StdHepPrint: 13 1 gamma 0 0 11 0 0.2468 0.0923 0.2505 0.3635 0.0943 0.3317 -0.8103

Optimizing detector acceptanceStudy of the process:e+ e- + - 0

Angular distribution of 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.

Let’s simulate an electromagnetic shower!!!

Incoming particles

A block of matter

Easily modeled by the MC technique!

Material validation (Babar detector)

Bremsstrahlung in Bhabha events

Use known processesto see if detector simulation (position in space, resolution, amount of material) is reliable

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.

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

Bac

kgro

und

MC

Sig

nal

MC

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 USA

To Russia/Japan

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