Summary of lessons on MC techniques

1

Summary of lessons on MC techniques

Sara Bolognesi 31 May 2006

Matrix Element Monte Carlo

• amplitude calculation

• phase space integration

• approximations

Parton shower

Hadronization (or Fragmentation)

1

Parton level simulation

Increasing in collider energy

Increasing number of final state particles

Increasing the complexity of the parton level simulation:

More Feynman diagrams and with a complex topology :

• you need to correctly manage the phase space

• more efficient integration techniques have to be developed

• increasing of the difficulty in the amplitude calculation


Amplitude calculation

1

2diagr diagr

terms

N NN

2 2 2 †

1 2 1 2 1 22ReA A A A A A

terms diagrN N

Standard method: TRACE EVALUATION

the number of terms grows with a power 2 of the number of diagrams

EX.: with 2 diagrams -> 3 terms

New technique: HELICITY AMPLITUDE METHODS

you loose spin info by summing on polarization

working with amplitude (not squared!)

you keep the spin info:

smaller cancellations between different terms, the cancellations become more easy to manage

so in reality terms diagrN N spin configurationsyou need to use approximation (ex. m~0)


3

Spinor formalism

A u p Xu p Re-write the amplitude as a product of terms

each of which represents a “piece” of diagram

you can construct the diagrams for

Very simple to computerize the process in an automatic program


EXAMPLE: with these pieces

e e ude v

4

Phase space integration For each intermediate particle you have a propagator so you have a resonance in the amplitude

The dimension of the phase space (i.e. the number of variables on which you have to integrate) grows with 3N-4(N = number of final particles)

You want to have a stable and reliable result and you need it quickly!!

•adaptive iterative algorithm•multi-channel technique

The main strategy is


choose as integration variables those which have the peaks (i.e. align the peaks with the axis)

= PHASE SPACE SPLITTING

make a change of variable to flat the bumps in the amplitude

= IMPORTANCE / STRATIFIED SAMPLING

5


Phase space splitting

21, 1

1

, 12

nj j

j

dMdps n dps j j

Re-write the phase space of N outgoing particles as the product of phase space of N/2 couples of particles

where M(j,j+1) are the variables where you expect the bumps in the amplitude

EXAMPLE:

so you have to choose carefully the splitting accordingly to physics process(eventually you need to use different splitting for the same amplitude -> multi-channel techniques)

ud bbe v

6

Numerical integrationWe can’t computerize an analytical integration but we can apply a numerical integration(better we know our function -> more efficient and precise could be the integration)

MC integration is the best technique:

1E

N E = estimate of the result

N = number of points used to integrate

the error on the estimate converges to the real result not very quickly but it’s independent from the integral dimensions (i.e. the number of variables)

Techniques to improve the convergence are more than welcome!!

importance sampling stratified sampling


Importance sampling

f x f x

I f x dx P x dx dPP x P x

1f x

P x

put more points where the function peaks more

change of variable to pass from x (distributed according with f) to P (rather uniformly distributed)

1P x dx

• choose a probability density function P(x) with those properties

• generate uniform random number z

PRATICALLY:

MATHEMATICS:

THE MAIN IDEA:

• change the variable:

• sample f with x

1x P z

0P x with



Stratified sampling

Put more points where the function varies more quickly

Split your integration interval in N different sub-intervals (thanks to the linearity):

• put the same amount of points in each intervals• use narrower intervals where the function varies more quickly

The error on the total integral is lower if the error on different sub-intervals is of the same order

THE MAIN IDEA:

PRATICALLY:

MATHEMATICS:

9

Adaptive-iterative algorithmBecause you don’t know the shape of the function, you need to learn about that during the integration itself:

split the integration domain in sub-intervals where you put the same amount of points

at the first iteration points equally distributed on all the domain

see where the first integration is less precise

at the second iteration put more points here

and so on…

… but we need to have already the peaks aligned with the axis (thanks to the phase splitting)


Multichannel techniques…but typically you have many peaks in different direction (one peak on the mass of each intermediate resonance)

For each channel use a different mapping i.e. a different set of variables to map your phase space

M(bev) ~ Mtop

M(bb) ~ MZ

define a probability (i) of pick up that channel(the best i are the ones which give roughly the same error for the different channels)


M(ev) ~ MW

M(ev) ~ MW

EXAMPLE:

ud bbe v

11

Approximations in M.E. MC

2

1M

q

22 21 3 1 3 1 3 132 2 2 coseq p p m E E p p

0em

MASSLESS APPROXIMATION

+ to reduce the number of helicity configuration that you have to consider

- could raise some fictitious singularity -> instability in the MC

Anyone MC can’t cover all the phase space

you have to apply clever cuts on the phase space in order to avoid singularities and instability

NOTE: also without massless approximation the mass of an electron can be a trouble!


2 0q 1,3 0E 13 0

se e

22 4~ 10em GeV

12

In general you can’t consider only a subset of diagrams of your process because • you can loose big interference terms or cancellation effects

• you break the Gauge Invariancebut you can save the G.I. by forcing your intermediate particles to be on shell = Narrow Width Approximation

qq->e+e-e+e- ~ qq->ZZ->e+e-e+e-

PRODUCTION TIMES DECAY

Mathematically this corresponds to • neglect spin correlation between production and decay of the intermediate particles• approximate the resonance with a Dirac delta

- you can’t esteem the acceptance of cuts on the invariant mass- you loose spin info (that affects the angular distributions) and all the irreducible backgrounds+ you can use this approx. to introduce NLO corrections

NARROW WIDTH APPROXIMATION AND PRODUCTION TIMES DECAY


~

13


The extension of the NWA from s to t channel (i.e. virtual intermediate particle)

A sort of PDF = probability to find a photon in an electron

scattering of a quasi real photon with a positron

q2 ~ 0

EQUIVALENT VECTOR BOSON APPROXIMATION

WEIZSECKER-WILLIAM APPROXIMATION

The extension of the WWA from the photon to massive vector boson

A sort of PDF =

scattering of real vector boson

q2 ~ MV

A little more tricky because MV doesn’t belong to the allowed values of q2 (q2<0)

probability to find a V in a fermion

14

Parton shower


HARD SCATTERING PARTONIC PROCESS

HADRONIZATION / FRAGMENTATION

high energy (>~100 GeV) “low” energy (~100 MeV)

PARTON SHOWER = radiation from initial and final state (ISR/FSR)

• bulk of the corrections to the LO process

• scale evolution2ˆ hadronss Q M

Approximated with only 1->2 split 1->3 processes

virtual correctionsi.e. only O(s) considered at each split:

15


Bulk of corrections From a perturbative point of view, we have divergences due to

• virtual corrections• soft emission (IR divergences)• collinear emission

equal and opposite, they cancel one with the other

2lnn n s

m

ordinate emission from big angles at larger scale to small angles at lower scale:

Multiple emission:

Physical interpretation:

emitted and intermediate particles as partons (constituent) of the initial particleDGLAP equation = probability of splitting, a sort of PDF

MC algorithm

generate a random number 0<r<1 if r < Prob. of non splitting stop here

else r=Pnosplit(Q2,K2) extract K and simulate splitting (with 4-mom. conservation)

16


HadronizationReconstruct the final stable hadrons from partons radiated during the Parton Shower

Models to describe the reality (i.e. to fit the data) not a coherent theory (no Lagrangian): PS + hadronization

String Fragmentation (Lund model)

color exchange (i.e strong force) between quarks represented with a string

potential proportional to the quarks distance (string length)

when the energy is high, the string crack and give rise to a new couple of quark

MC contains an iterative algorithm to reproduce this process with a probabilistic approach

17

hadron

THE ENDTHE END

Date post:	14-Jan-2016
Category:	Documents
Upload:	opal
View:	19 times
Download:	1 times

Summary of lessons on MC techniques

Documents