Monte Carlo Simulation in Particle Physics Concezio Bozzi

Monte Carlo Simulation in Particle Physics Concezio Bozzi Istituto Nazionale di Fisica Nucleare Ferrara (Italy) IUB, Bremen, Germany, November 28 th 2002

Layman’s terms?

The Monte. Carlo 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 Monte. Carlo? • 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 1930 s. 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 r 2, dart board = a 2, ratio = Ncircle/Nboard = p r 2 / a 2

The drunk player gets p ! • From previous page, and if a=2 r: p = 4 Ncircle/Nboard Try this! • Precision of calculation is 1/sqrt(N) – – – 100 tries: 10, 000 tries: 1, 000 tries: 100, 000 tries: 10, 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 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 life on earth, molecules form 15 billion years 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 Big ? Ba ng matter dominates 1015 deg 1010 deg 109 deg 6000 o The Universe began with a “Big Bang” about 15 billion years ago 4000 o -255 o -270 o

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 atoms and space. -Democritus (c. 400 BC) 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 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 (De. Broglie) 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=mc 2

Particle accelerators A linear accelerator (cathode tube) A circular accelerator (collider)

Detectors Fixed target Collider

LEP at CERN (Geneva) e. Electron (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: 100 s 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 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

Monte. Carlo 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 Std. Hep. Print: Std. Hep Track info for event 4 : Std. Hep. Print: Trk# Stat Id Dtr 1 Dtr. N Mom 1 Mom. N Px Py Pz E Vx Vy Vz Std. Hep. Print: Std. Hep Track info for event 3 : Std. Hep. Print: Std. Hep Track info 3 info for 4 event for 0 event : 1: Std. Hep. Print: 1 3 e+Track 02 0. 05558 0. 001356 -3. 112 3. 113 0. 09944 0. 33 -1. 394 Std. Hep. Print: Trk# Stat Id Dtr 1 Dtr. N Mom 1 Mom. N Py Pz E Vy Vz Std. Hep. Print: Trk# Dtr. N Mom 1 Mom. N Px Px. Px 8. 985 Py. Py 8. 986 Pz. Pz 0. 09944 E E Vx Vx. Vx Vz. Vz Std. Hep. Print: 2 3 Stat e- Id Id 3 Dtr 1 4 Dtr 1 0 Dtr. N 0 Mom 1 -0. 165 -0. 0004597 0. 33 Vy. Vy -1. 394 Std. Hep. Print: 1 3 e+ 3 4 0 0 0. 05951 -0. 0005719 -3. 114 3. 115 0. 09094 0. 3304 -0. 7858 Std. Hep. Print: e+3 e+ 0 0. 0576 -0. 001051 -0. 0005 -3. 1094 3. 1099 0. 09233 0. 0907 0. 33 0. 3294 0. 5908 -0. 8146 Std. Hep. Print: 31 213 tau+ 53 653 104 0000. 05871 1. 155 3. 309 6. 957 -3. 115 7. 99 3. 115 0. 09944 0. 33 -1. 394 Std. Hep. Print: 2 3 e 3 4 0 0 -0. 1682 0. 002575 8. 984 8. 985 0. 09094 0. 3304 -0. 7858 Std. Hep. Print: e-3 e 53 014 00 -0. 1684 0 -0. 1676 -0. 002014 8. 989 8. 9919 8. 9934 0. 09233 0. 0907 0. 33 0. 3294 0. 5908 -0. 8146 Std. Hep. Print: 42 223 tau 93 10 -1. 264 -3. 3080. 0008 -1. 085 4. 108 0. 09944 0. 33 -1. 394 Std. Hep. Print: 33 232 tau+ 56 785 117 001 2. 812 3. 643 5. 0111. 1461 7. 032 0. 09094 0. 3304 -0. 7858 Std. Hep. Print: tau+ 2. 136 0 -4. 0722 -1. 298 -2. 4796 -1. 221 3. 301 5. 2156 0. 09233 0. 0907 0. 33 0. 3294 0. 5908 -0. 8146 Std. Hep. Print: 5 1 anti-nu_tau 0 0 3 0 0. 3767 0. 1478 1. 502 1. 555 0. 1049 0. 3456 -1. 361 Std. Hep. Print: 4 2 tau 8 9 1 0 -2. 921 -3. 641 0. 8581 5. 068 0. 09094 0. 3304 -0. 7858 Std. Hep. Print: tau 2 tau- 79 10 -2. 184 0 3. 9623 1. 238 2. 4799 7. 649 4. 7364 8. 2446. 8877 0. 09233 0. 330. 3294 0. 5908 Std. Hep. Print: 64 242 rho+ 88 319 001 0. 7781 3. 162 5. 455 6. 435 0. 10490. 0907 0. 3456 -1. 361 -0. 8146 Std. Hep. Print: 5 1 anti-nu_tau 0 0 3 0 0. 8467 0. 8655 2. 489 2. 768 0. 0932 0. 3333 -0. 7817 Std. Hep. Print: 5 5 1 gamma 1 anti-nu_tau 0 0 1 3 0 0 -0. 0622 -2. 8840 0. 05684 -1. 3849 -0. 5539 0. 8917 0. 5603 3. 3212 0. 09233 0. 0878 0. 3277 0. 33 0. 5908 -0. 8138 Std. Hep. Print: 7 1 pi+ 0 6 0 0. 1861 0. 08695 0. 2127 0. 327 0. 1049 0. 3456 -1. 361 Std. Hep. Print: 6 1 mu+ 0 0 3 0 0. 8607 1. 613 1. 408 2. 31 0. 0932 0. 3333 -0. 7817 Std. Hep. Print: anti-nu_tau 1 e+ 14 15 00 060 303 0. 592 000. 09482 -0. 5448 0. 006695 -0. 5248 0. 07114 0. 11870. 0878 0. 096290. 3277 0. 3276 -0. 8138 0. 5886 Std. Hep. Print: 86 261 pi 0 3. 075 5. 2430. 6025 6. 108 0. 9671 0. 1049 0. 3456 -1. 361 Std. Hep. Print: 7 1 nu_mu 0 0 3 0 1. 105 1. 165 1. 114 1. 954 0. 0932 0. 3333 -0. 7817 Std. Hep. Print: mu+ 1 nu_e 00 000 403 030 -0. 3907 1. 278 0 -0. 6434 -0. 6479 -0. 5699 0. 1266 -0. 3480 1. 442 0. 9273 0. 09629 0. 0878 0. 3276 0. 3277 0. 5886 -0. 8138 Std. Hep. Print: 97 171 nu_tau -1. 938 -0. 9116 2. 177 0. 09512 0. 3187 -1. 397 Std. Hep. Print: 8 1 nu_tau 0 0 4 0 0. 004886 -0. 2322 0. 2253 0. 3236 0. 08223 0. 3195 -0. 7832 Std. Hep. Print: 8 8 1 nu_mu 1 nu_tau 0 0 3 4 0 0 0. 7632 1. 6662 -0. 6571 1. 8894 -1. 419 2. 0134 1. 74 3. 2249 0. 09629 0. 0943 0. 3276 0. 3317 0. 5886 -0. 8103 Std. Hep. Print: 10 2 a_111 13 4 0 -0. 8736 -1. 37 -0. 1733 1. 931 0. 09512 0. 3187 -1. 397 Std. Hep. Print: 9 2 a_110 12 4 0 -2. 926 -3. 409 0. 6328 4. 745 0. 08223 0. 3195 -0. 7832 Std. Hep. Print: nu_tau 2 rho- 16 0 17 100 10 114 040 -0. 05043 -1. 092 0 2. 2961 -0. 09733 0. 5905 1. 911 2. 7230 2. 203 3. 6628 0. 07894 0. 0943 0. 3376 0. 3317 0. 6378 -0. 8103 Std. Hep. Print: 119 291 pi 0 -0. 716 -0. 1166 0. 7396 0. 09512 0. 3187 -1. 397 Std. Hep. Print: 10 1 pi 0 0 9 0 -1. 17 -1. 052 0. 1692 1. 588 0. 08223 0. 3195 -0. 7832 Std. Hep. Print: 10 10 2 rho 1 pi 11 0 12 0 4 9 0 0 -1. 092 1. 2046 1. 336 0. 0371 5. 738 1. 2295 6. 041 1. 7273 0. 07894 0. 0943 0. 3376 0. 3317 0. 6378 -0. 8103 Std. Hep. Print: 12 2 pi 0 18 19 10 0 -0. 4634 -0. 6317 0. 02961 0. 7955 0. 09512 0. 3187 -1. 397 Std. Hep. Print: 11 11 pi 00 10 9 0 9 -1. 634 -2. 019 0. 7127 2. 697 0. 08223 0. 3195 -0. 7832 Std. Hep. Print: pi-2 pi 0 000 012 -0. 6058 0 1. 0915 0. 2708 0. 5533 1. 578 1. 4935 1. 718 1. 9356 0. 07894 0. 0943 0. 3376 0. 3317 0. 6378 -0. 8103 Std. Hep. Print: 1311 11 1 pi 1013 00 -0. 3598 -0. 02258 -0. 08632 0. 3961 0. 09512 0. 3187 -1. 397 Std. Hep. Print: 12 1 pi+ 0 0 9 0 -0. 1223 -0. 3385 -0. 2491 0. 4594 0. 08223 0. 3195 -0. 7832 Std. Hep. Print: pi 0 1 gamma 130 14 00 010 0 0. 84471. 168 1. 065 0. 4610 4. 16 1. 2430 4. 324 1. 5720 0. 07894 0. 0943 0. 3376 0. 3317 0. 6378 -0. 8103 Std. Hep. Print: 1412 12 12 gamma 8 110 -0. 4862 0. 2288 2. 122 2. 433 0. 1049 0. 3456 -1. 361 Std. Hep. Print: 13 13 1 gamma 0 0 12 11 0 0 -0. 4173 0. 2468 0. 8177 0. 0923 3. 101 0. 2505 3. 234 0. 3635 0. 07894 0. 0943 0. 3376 0. 3317 0. 6378 -0. 8103 Std. Hep. Print: 15 1 gamma 0 0 8 0 0. 3632 1. 906 3. 121 3. 675 0. 1049 0. 3456 -1. 361 Std. Hep. Print: 1614 11 gamma -0. 06891 -0. 4131 0. 2471 -0. 0001552 1. 059 1. 089 0. 07894 0. 3376 0. 6378 Std. Hep. Print: 00 00 1112 00 -0. 03818 0. 4149 0. 09512 0. 3187 -1. 397 Std. Hep. Print: 17 1 gamma 0 0 11 0 -0. 01225 -0. 3028 -0. 1164 0. 3247 0. 09512 0. 3187 -1. 397 Std. Hep. Print: 18 1 gamma 0 0 12 0 -0. 1236 -0. 1387 0. 06213 0. 1959 0. 09512 0. 3187 -1. 397 Std. Hep. Print: 19 1 gamma 0 0 12 0 -0. 3398 -0. 493 -0. 03252 0. 5996 0. 09512 0. 3187 -1. 397

Optimizing detector acceptance Study of the process: e+ e- ® p+ p- p 0 Angular distribution of p 0 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 Incoming particles with well-defined probability. Easily modeled by the MC technique! Let’s simulate an electromagnetic shower!!! 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