Monte Carlo and Statistical Methods in HEP

Kajari Mazumdar

Course on Particle Physics, TIFR, August, 2009

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

• 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)

What is Monte Carlo simulation?

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

Determination of π in MC technique

Throw darts randomly

• 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

Similarity to games of chance explains the

name name… Other numerical methods typically need a mathematical description

of the system (ordinary or partial differential equations)

• More and more difficult to solve as complexity increases

Why Monte Carlo?

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

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

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.

• 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,…

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

A typical MC use caseA typical MC use case

Detectors

Fixed target

Collider

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.

An 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

Optimization of detector acceptance

Study of the process:e+ e- + - 0

Angular distribution of decay products for 3 different energies.

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.

Particle-Detector interaction

EM shower in a block of matter

Incidentparticle

Use known processes (eg. Bremsstrahlung in Bhabha events)to see if detector simulation (position in space, resolution, amount of material) is reliable

EM shower easily modelled by MC technique.

• Use MC simulation to compute the signal efficiency and background contamination.

• Optimize the selection criteria to get the smallest error.

Using MC in Physics measurements

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

Why bother about simulation

How are MCs used

Detector Simulation using GEANT

Physics processes and cut-off

Simulation of detector response

• 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