QCD and Monte Carlo simulation - desy.de

H. Jung, QCD & MCs, University Antwerp, 2009 1

QCD and Monte Carlo simulation

H. Jung (DESY, University Antwerp)[email protected]

http://www.desy.de/~jung/qcd_and_mc_2009-2010/


Outline of the lectures

24. Nov Intro to Monte Carlo techniques and structure of matter

25. Nov DGLAP: solution with MCs

1. Dec DGLAP/BFKL/CCFM: evolution for small x

2. Dec Cross sections and virtual corrections

15. Dec W/Z production in pp and soft gluon resummation

16. Dec NLO corrections

Exercises in the afternoons: 14:00 – 16:30


Requests to you ...

If things go wrong .. lecture is too easy... too trivial ... too complicated, too chaotic or too boring ...

PLEASE complain immediately !

PLEASE ask questions any time !


Let's start ....


Outline for today

MCs

What is a Monte Carlo simulation ?

Random Numbers

Monte Carlo integration

QCD:

How to learn about the structure of matter ?

partons are free ...

distribution of partons in proton


Literature for Monte Carlo MethodF. James Rep. Prog. Phys., Vol 43, 1145 (1980)

F. James Statistical Methods in Experiemntal Physics World Scientific 2006

S. Weinzierl Introduction to Monte Carlo method hep-ph/0006269

J. Vermaseren, Lectures on Monte Carlo, Madrid 2008 (http://www.nikhef.nl/~form/maindir/courses/course2/course2.html/)

G. Bohm, G. Zech Einfuhrung in Statistik und Messwertanalyse fur Physiker DESY 2007

V. Blobel, E. Lohrmann Statistische und numerische Methoden der Datenanalyse Teubner 1998

Particle Data Book S. Eidelman et al., Physics Letters B592, 1 (2004)

section on: Mathematical Tools (http://pdg.lbl.gov/)Hardware Random Number Generators:

A Fast and Compact Quantum Random Number Generator (

http://arxiv.org/abs/quant-ph/9912118/)

Quantum Random Number Generator (http://www.idquantique.com/products/quantis.htm/)

Hardware random number generator (http://en.wikipedia.org/wiki/)History of Monte Carlo Method

(http://www.geocities.com/CollegePark/Quad/2435/history.html)


What is this ?


What is this ?

Tell the difference between the production of a mini black hole

and a multijet event ?


What is this ?

NO way ...

can only tell

probabilities !


What is this ?

There is nothing

like

THE

BH event !!!


What this is about ...

How to go from text book formula

for DIS

to

ep! e0X

pp! h+X


.... or from gambling


.... to


... via applications in risk management


Why do we need all that ?

because physics and life is more complicated than a simple equation, which can be solved analytically ....

Monte Carlo techniques are

widely used

are of enormous advantages

can be used to simulate any complicated process

are now EVEN used in particle physics theory ...... !!!!! ....


Where is the problem ?

QPM process BGF process

total x-section heavy quarks (charm & bottom)

2-jet 3-jet

O(!s) O(!2s)


Where is the problem ?F2 » !(!

¤p)



2-jet 3-jet

O(!s) O(!2s)




2-jet 3-jet

Where is the problem: hadronic final state

O(!s) O(!2s)

F2 » !(!¤p)

!!jj (rad.)!!jj (rad.)



processes of have not yet been calculated ...

interesting to go closer to outgoing proton remnant

forward jets !!!

O > !3

s





forward jets !!!

O > !3

s





jet veto in Higgs production at LHC

O > !3

s

H. Jung, QCD & MCs, University Antwerp, 2009

How to simulate these processes ?


Monte Carlo methodMonte Carlo method

refers to any procedure that makes use of random numbers

uses probability statistics to solve the problem

Monte Carlo methods are used in:

Simulation of natural phenomena

Simulation of experimental apparatus

Numerical analysis

Random number:








Numerical analysis

Random number:

one of them is 3








Numerical analysis

Random number:

one of them is 3

No such thing as a single random number








Numerical analysis

Random number:

one of them is 3

No such thing as a single random number

A sequence of random numbers is a set of numbers that have

nothing to do with the other numbers in a sequence


Going out to Monte Carlo


Random NumbersIn a uniform distribution of random numbers in [0,1] every number has the same chance of showing up

Note that 0.000000001 is just as likely as 0.5


True Random NumbersRandom numbers from classical physics: coin tossing evolution of such a system can be

predicted, once the initial condition is known... however it is a chaotic process ... extremely sensitive to initial conditions.

Truly Random numbers used for

Cryptography

Confidentiality

AuthenticationScientific Calculation

Lotteries and gambling

do not allow to increase chance of

winning by having a bias ....

too bad

Random numbers from quantum physics: intrinsic random photons on a semi-transparent mirror

Available and tested in MC generator by a summer student

Generator is however very slow...


True Random Numbers

atmospheric noise, which is quite easy to pick up with a normal radio: used by RANDOM.ORG

much more can be found on the web ....


Random NumbersIn a uniform distribution of random numbers in [0,1] every number has the same chance of showing up

Note that 0.000000001 is just as likely as 0.5


Pseudo Random NumbersPseudo Random Numbers

are a sequence of numbers generated by a computer algorithm, usually uniform in the range [0,1]

more precisely: algo's generate integers between 0 and M, and then rn=I

n/M

A very early example: Middle Square (John van Neumann, 1946):

generate a sequence, start with a number of 10 digits, square it, then take the

middle 10 digits from the answer, as the next number etc.:

57721566492 = 33317792380594909291

Hmmmm, sequence is not random, since each number is determined from the

previous, but it appears to be randomthis algorithm has problems .....

BUT a more complex algo does not necessarily lead to better random sequences .... Better us an algo that is well understood


Congruential linear generator

develop our own simple generator

with seed I0

and multiplicative constant a and additive constant c

modulus

!maximal repetition period:

Exercise

m

O(m)

Ij = mod(aIj¡1 + c,m)

Rj =Ij

m


Randomness tests

!Congruential generator is not bad … but it could be better ....

Congruential generator


Randomness tests

!RANLUX much more sophistcated. Developed and used for QCD lattice calcs

RANLUXCongruential generatorM. Lüscher, A portable high-quality random number generator for lattice field theory simulations, Computer Physics Communications 79 (1994) 100 http://luscher.web.cern.ch/luscher/ranlux/index.html

Exercise


Random Number generatorsstudy by B. Domnik (summerstudent 2005)

Compare random number generators with physics process

y spectrum of electron

! observe peaks

! coming from physics ?


Random Number generatorsstudy by B. Domnik (summerstudent 2005)

From now on assume:we have good random number

generator:

we use RANLUX

Compare random number generators with physics process

y spectrum of electron

! observe peaks

! coming from physics ?

BUT coming from bad random number generator


Randomness tests

test uniform distribution

correlation tests

sequence up – sequence down test

gap – test

random walk test

and ...

Blackboard


Expectation values and variance

Expectation value (defined as the average or mean value of function f ):

for uniformly distributed u in [a,b] then

rules for expectation values:

Variance

rules for variance:

if x,y uncorrelated:

if x,y correlated

dG(u) = du/(b¡ a)

E[f ] =

!f(u)dG(u) =

!1

b¡ a

!b

a

f(u)du

!=1

N

N!

i=1

f(ui)

E[cx+ y] = cE[x] + E[y]

V [f ] =

!(f ¡ E[f ])

2dG=

!1

b¡ a

! b

a

(f(u)¡ E[f ])2du

!

V [cx+ y] = c2V [x] + V [y]

V [cx+ y] = c2V [x] + V [y]+2cE [(y ¡E[y]) (x¡E[x])]

Blackboard


Generating distributions

From uniform distributions to distributions for any probability density function

use variable transformation

linear p.d.f:

1/x distribution

f(x) = 2x

u(x) =

!x

0

2tdt = x2

xj =puj

f(x) =1

x

u(x) =

!x

xmin

1

tdt

Fmax ¡ Fmin

xj = xmin

!xmax

xmin

!uj


Generating distributions

Brute Force or Hit & Miss method

use this if there is no easy way to find a analytic integrable function

find

reject if

accept if

Improvements for Hit & Miss method by variable transformation

find

reject if

accept if

c · max f(x)

f(xi) < uj ¢ c

f(xi) > uj ¢ c

c ¢ g(x) > f(x)

f(x) < uj ¢ c ¢ g(x)

f(x) > uj ¢ c ¢ g(x)


Law of large numbersLaw of large numbers

xi independent

random variables, having the same mean and

variances :

for large enough N the sum converges to the correct answer.Convergence

is given with a certain probability …

THIS is a mathematically serious and

precise statement !!!!

1

N

N!

i=1

xi ! µ

!2

i


Central Limit Theorem

!independent on the original sub-distributions

!ixi ¡

!iµi!"

i!2

i

! N(0, 1)

f(x) =1

s

p2!exp

!¡

(x¡ a)2

2s2

!

Central Limit Theorem for large N the sum of independent random variables is always normally (Gaussian) distributed:


Central Limit TheoremCentral Limit Theorem for large N the sum of independent random variables is always normally (Gaussian) distributed:

example: take sum of uniformly distributed random numbers:

f(x) =1

s

p2!exp

!¡

(x¡ a)2

2s2

!

Rn =!n

i=1Ri

E[R1] =!udu = 1/2,

V [R1] =!(u¡ 1/2)2du = 1/12

E[Rn] = n/2V [Rn] = n/12

Blackboard


Central Limit TheoremCentral Limit Theorem for large N the sum of independent random variables is always normally (Gaussian) distributed

!for any starting distribution

!for uniform distribution

!for exponential distribution

Exercise

G. Bohm, G. ZechEinfuhrung in die Statistik und Messwertanalyse fur Physiker


Monte Carlo IntegrationLaw of large numbers

MC estimate converges to true integralCentral limit theorem

MC estimate is asymptotically normally distributed

it approaches a Gaussian density

with effective variance V(f)! to decrease , either reduce V(f) or increase N

advantages for n-dimensional integral ...

i.e. phase space integrals 2 ! n processes

is where other approaches tend to fail

1

N

N!

i=1

f(ui)!1

b¡ a

!b

a

f(u)du

! =

!V [f ]pN

»

1pN

!


Monte Carlo Integration

solve

estimate by

with variance

I =

!b

a

f(x)dx = (b¡ a)E[f(x)]

I » IMC =b¡ a

n

n!

i=1

f(xi)

V [IMC ] = !2

I= V

!b¡ a

n

!

i

f(xi)

!

=(b¡ a)2

nV [f ]

=(b¡ a)2

n

!"if(xi)

2

n¡

!"if(xi)

n

!2!

=1

n

·(b¡ a)2

!if(xi)2

n¡ I

2

MC

!


MC method: advantage of hit & miss

integration ! weighting events

large fluctuations from large weights

weights also to errors applied

difficult to apply further

hadronizationreal events all have weight = 1 !!!

Hit & Miss method:

MC for function f(x): get random number:

R1 in (0,1) and R2 in (0,1) calculate x = R1

reject event if: fx < f

max R2




max R2


MC method: advantage of hit & missintegration ! weighting events





Hit & Miss method:




max R2




max R2


MC method: advantage of hit & missintegration ! weighting events



difficult to apply further hadronizationreal events all have weight = 1 !!!

Hit & Miss method:




max R2




max R2


MC method: advantage of hit & miss

integration ! weighting events





Hit & Miss method:




max R2




max R2


MC method: do even better ...

Importance sampling

MC for function f(x)approximate f(x) ~ g(x)

with g(x) > f(x) simple and integrable generate x according to g(x):

example:

reject event if: f(x) < g(x) R2



example:


!x

xmin

g(x0)dx0 = R1

!xmax

xmin

g(x0)dx0

f(x) = 1/x0.7

g(x) = 1/x

x = xmin ¢!xmax

xmin

!R1



Importance sampling



example:




example:


!x

xmin

g(x0)dx0 = R1

!xmax

xmin

g(x0)dx0

f(x) = 1/x0.7

g(x) = 1/x

x = xmin ¢!xmax

xmin

!R1



Importance sampling



example:




example:


!x

xmin

g(x0)dx0 = R1

!xmax

xmin

g(x0)dx0

f(x) = 1/x0.7

g(x) = 1/x

x = xmin ¢!xmax

xmin

!R1



Importance sampling



example:




example:


!x

xmin

g(x0)dx0 = R1

!xmax

xmin

g(x0)dx0

f(x) = 1/x0.7

g(x) = 1/x

x = xmin ¢!xmax

xmin

!R1



Importance sampling

WOW !!! very efficient even for peaked f(x)



example:




example:


!x

xmin

g(x0)dx0 = R1

!xmax

xmin

g(x0)dx0

f(x) = 1/x0.7

g(x) = 1/x

x = xmin ¢!xmax

xmin

!R1


Importance SamplingMC calculations most efficient for small weight fluctuations:

f(x)dx ! f(x) dG(x)/g(x)

chose point according to g(x) instead of uniformly

f is divided by g(x) = dG(x)/dx

generate x according to:

relevant variance is now V(f/g):

small if g(x) ~ f(x)

how-to get g(x)

(1) g(x) is probability: g(x) > 0 and !dG(x) = 1

(2) integral !dG(x) is known analytically

(3) G(x) can be inverted (solved for x)

(4) f(x)/g(x) is nearly constant, so that V(f/g) is small compared to V(f)

R

!b

a

g(x0)dx0 =

!x

a

g(x0)dx0


Integration: Monte Carlo versus others

One dimensional quadrature

Monte Carlo: Hit & Miss

w =1 and xi chosen randomly

Trapezoidal Rule:

approximate integral in sub-

interval by area of trapezoid

below (above) curveSimpson quadrature

approximate by parabolaGauss quadrature

approximate by higher order

polynomial

I =

!f(x)dx =

n!

i=1

wif(xi)

method err (1d) error

MC n¡1/2 n¡1/2

Trapez n¡2 n¡2/d

Simpson n¡4 n¡4/d

Gauss n¡2m+1 n¡(2m¡1)/d


We have the method,....BUT

HOWTO simulate the physics

????


MC event: hadron and detector level

!s ~ 318 GeV

x ~ 7. 10-5 at Q2 = 4 GeV2


From experiment to measurementtake data • run MC

generator• detector simulation

Compare measurement with

simulation

Upppps ...... all measurements rely on proper MC generators and MC simulation !!!!


From experiment to measurementtake data • run MC

generator• detector simulation

Compare measurement with

simulation

Upppps ...... all measurements rely on proper MC generators and MC simulation !!!!

define visible x - section in kinematic variablescalculate factor C

corr to correct from detector to hadron level

visible x-section on hadron level

d!data

had

dx=d!

data

det

dxCcorr with Ccorr =

d!MC

had

dx

d!MC

det

dx


and now it is our turn ... to write

Monte Carlo programs


.... but first,what about the physics ?

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QCD and Monte Carlo simulation - desy.de