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CERN Summer Student LecturePart 2, 20 July 2012

Introduction toMonte Carlo Techniquesin High Energy Physics

Torbjorn Sjostrand

How are complicated multiparticle events created?

How can such events be simulated with a computer?

Lectures Overview

yesterday: Introduction the Standard ModelQuantum Mechanicsthe role of Event Generators

Monte Carlo random numbersintegrationsimulation

today: Physics hard interactionsparton showersmultiparton interactionshadronization

Generators Herwig, Pythia, SherpaMadGraph, AlpGen, . . .common standards

Torbjorn Sjostrand Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2 slide 2/40

The Structure of an Event – 1

Warning: schematic only, everything simplified, nothing to scale, . . .

pp/p

Incoming beams: parton densities



pp/p

ug

W+

d

Hard subprocess: described by matrix elements



pp/p

ug

W+

d

c s

Resonance decays: correlated with hard subprocess



pp/p

ug

W+

d

c s

Initial-state radiation: spacelike parton showers



pp/p

ug

W+

d

c s

Final-state radiation: timelike parton showers



pp/p

ug

W+

d

c s

Multiple parton–parton interactions . . .



pp/p

ug

W+

d

c s

. . . with its initial- and final-state radiation



Beam remnants and other outgoing partons



Everything is connected by colour confinement stringsRecall! Not to scale: strings are of hadronic widths



The strings fragment to produce primary hadrons



Many hadrons are unstable and decay further



These are the particles that hit the detector


Hard Processes

Defines the character of the event:QCD, top, Z0, W±, H0, SUSY, ED, TC, . . .

Obtained by perturbation theory:σZ0 = αemM0 + αemαsM1 + αemα2

sM2 + . . .= LO + NLO + NNLO + . . .

Involves subtle cancellations between real and virtual contributions:

Next-to-leading order (NLO) graphs

Torbjorn Sjostrand PPP 2: Phase sapce and matrix elements slide 42/48

These days • LO easy• NLO tough but doable• NNLO only in very few cases


QED Bremsstrahlung

Accelerated electric charges radiate photons ,

see e.g. J.D. Jackson, Classical Electrodynamics.A charge ze that changes its velocity vector from β to β′ radiates a spectrum ofphotons that depends on its trajectory. In the long-wavelength limit it reduces to

limω→0

d2I

dωdΩ=

z2e2

4π2

˛ε∗

„β′

1− nβ′−

β

1− nβ

«˛2where n is a vector on the unit sphere Ω, ω is the energy of the radiated photon, and

ε its polarization.

1 For fast particles radiation collinear with the initial (β) andfinal (β′) directions is strongly enhanced.

2 dN/dω = (1/ω)dI/dω ∝ 1/ω so infinitely many infinitely softphotons are emitted, but the net energy taken away is finite.

3 For ω → 0 the radiation pattern is independent of the spinof the radiator ⇒ universality.


QCD Bremsstrahlung – 1

QCD ≈ QED with e → eγ ⇒ q → qgαem ⇒ (4/3)αs

More precisely:

dPq→qg ≈αs

2π

dQ2

Q2

4

3

1 + z2

1− zdz

where

mass (or collinear) singularity:

dQ2

Q2≈ dθ2

θ2≈ dm2

m2≈

dp2⊥

p2⊥

soft singularity:

z ≈Eq,after

Eq,before= 1− ω

Eq,beforeso

dz

1− z=

dω

ω


QCD Bremsstrahlung – 2

But QCD is non-Abelian so additionally

g → gg similarly divergent

αs(Q2) diverges for Q2 → 0

DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi)

dPa→bc =αs(Q

2)

2π

dQ2

Q2Pa→bc(z) dz

Pq→qg =4

3

1 + z2

1− z

Pg→gg = 3(1− z(1− z))2

z(1− z)

Pg→qq =nf

2(z2 + (1− z)2) (nf = no. of quark flavours)


The iterative structure

Generalizes to many consecutive emissions if strongly ordered,Q2

1 Q22 Q2

3 . . . (≈ time-ordered).To cover “all” of phase space use DGLAP in whole regionQ2

1 > Q22 > Q2

3 . . ., although only approximately valid.

Must be clever when you write a shower algorithm.

Iteration givesfinal-stateparton showers:

Need soft/collinear cuts to stay away from nonperturbative physics.Details model-dependent, but around 1 GeV scale.


The Parton-Shower Approach

Showers: approximation method for higher-order matrix elements.

Universality: any matrix element reduces to DGLAP in collinear limit.

2 → n ≈ (2 → 2) ⊕ ISR ⊕ FSR

ISR = Initial-State Radiation: Q2i ∼ −m2 > 0 increasing

FSR = Final-State Radiation: Q2i ∼ m2 > 0 decreasing


Reminder: Radioactive Decay and the Veto Algorithm

Recall discussion on radioactive decays from Lecture 1:

Naively P(t) = c =⇒ N(t) = 1− ct.Wrong! Conservation of probabilitydriven by depletion:a given nucleus can only decay once

CorrectlyP(t) = cN(t) =⇒ N(t) = exp(−ct)i.e. exponential dampeningP(t) = c exp(−ct)

For P(t) = f (t)N(t), f (t) ≥ 0, this generalizes to

P(t) = f (t) exp

(−

∫ t

0f (t ′) dt ′

)which can be Monte Carlo-simulated using the veto algorithm.


The Sudakov form factor

Correspondingly, with Q ∼ 1/t (Heisenberg)

dPa→bc =αs

2π

dQ2

Q2Pa→bc(z) dz

× exp

−∑b,c

∫ Q2max

Q2

dQ ′2

Q ′2

∫αs

2πPa→bc(z

′) dz ′

where the exponent is (one definition of) the Sudakov form factor.

A given parton can only branch once, i.e. if it did not already do so

Note that∑

b,c

∫ ∫dPa→bc ≡ 1 ⇒ convenient for Monte Carlo

(≡ 1 if extended over whole phase space, else possibly nothinghappens before you reach Q0 ≈ 1 GeV).


Matching Showers and Matrix Elements

Showers: approximation to matrix elements (MEs).Shower emissions ≈ real part of higher orders.Shower Sudakovs ≈ virtual (loop) part of higher orders.

Real MEs manageable, virtual MEs tough, so want to combine.Key issue is to avoid doublecounting.

Active and rich field of study, with many methods, e.g.:

Merging: correct first shower emission to MEs,using veto algorithm.

CKKW(-L): generate several multiplicities with real MEs,use shower Sudakovs to include virtual corrections.

MLM: real MEs as above, but veto events that showersmigrate to another jet multiplicity.

MC@NLO: split σNLO into real LO+1 event and real+virtualLO ones.

POWHEG: like merging, with overall rate normalized to σNLO.


Parton Distribution Functions (PDFs)

Hadrons are composite: p = uud + gluons + sea qq.Higher virtuality scale Q ⇒ more partons resolved.fi (x ,Q2) = number density of partons i at momentum fraction xand probing scale Q2.

Initial conditions at small Q20 unknown: nonperturbative.

Resolution dependence perturbative, by DGLAP:

DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi)

dfb(x ,Q2)

d(lnQ2)=

∑a

∫ 1

x

dz

zfa(y ,Q2)

αs

2πPa→bc

(z =

x

y

)

σ =

∫ 1

0dx1

∫ 1

0dx2 fi (x1,Q

2) fj(x2,Q2) σ(x1x2E

2cm,Q2)

Continuous ongoing activity to provide best overall fit to data,consistent with theoretical evolution: CTEQ, MSTW, NNPDF, . . .


PDF examples

Convenient plotting interface:http://hepdata.cedar.ac.uk/pdf/pdf3.html


What is Pileup?

Protons in LHC collected in bunches.1 bunch ≈ 1.5× 1011 protons.Two bunches passing through each other ⇒ several pp collisions.Current LHC machine conditions ⇒ µ = 〈n〉 ≈ 20.Pileup introduces no new physics, so is uninteresting here.

But analogy: a proton is a bunch of partons (quarks and gluons)

so several parton–parton collisions when two protons cross .


MultiParton Interactions (MPIs)

Many parton-parton interactionsper pp event: MPI.

Most have small p⊥,∼ 2 GeV⇒ not visible as separate jets,but contribute to event activity.

Solid evidence that MPIs playcentral role for event structure.

Problem:

σint =

∫∫∫dx1 dx2 dp2

⊥ f1(x1, p2⊥) f2(x2, p

2⊥)

dσ

dp2⊥

= ∞

since∫

dx f (x , p2⊥) = ∞ and dσ/dp2

⊥ ≈ 1/p4⊥ →∞ for p⊥ → 0.

Requires empirical dampening at small p⊥,owing to colour screening (proton finite size).

Many aspects beyond pure theory ⇒ model building.


Hadronization

Nonperturbative =⇒ not calculable from first principles!

Model building = ideology + “cookbook”

Two common approaches:

String fragmentation:most ideological

Cluster fragmentation:simplest

Both contain manyparameters to bedetermined from data,preferably LEP e+e− →γ∗/Z0 → qq/qqg/ . . .

DELPHI Interactive AnalysisRun: 39265Evt: 4479

Beam: 45.6 GeV

Proc: 4-May-1994

DAS : 5-Jul-1993 14:16:48

Scan: 3-Jun-1994

TD TE TS TK TV ST PA

Act

Deact

95(145) 0

( 0)

173(204) 0

( 20)

0( 0) 0

( 0)

38( 38) 0

( 42)

0( 0) 0

( 0)

0( 0) 0

( 0)

0( 0) 0

( 0)

X

Y

Z


The Lund String Model

In QCD, for large charge separation, field lines seem to becompressed to tubelike region(s) ⇒ string(s)

by self-interactions among soft gluons in the “vacuum”.

Gives linear confinement with string tension:F (r) ≈ const = κ ≈ 1 GeV/fm ⇐⇒ V (r) ≈ κr

String breaks into hadrons along its length,with roughly uniform probability in rapidity,by formation of new qq pairs that screen endpoint colours.


The Lund Gluon Picture

Gluon = kink on string

Force ratio gluon/ quark = 2,cf. QCD NC/CF = 9/4, → 2 for NC →∞No new parameters introduced for gluon jets!


The HERWIG Cluster Model

1 Introduce forced g → qq branchings2 Form colour singlet clusters3 Clusters decay isotropically to 2 hadrons according to

phase space weightTorbjorn Sjostrand Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2 slide 31/40

Event Generators


The Generator LandscapeGenerator Landscape

Hard Processes

Resonance Decays

Parton Showers

Underlying Event

Hadronization

Ordinary Decays

General-Purpose

HERWIG

PYTHIA

SHERPA

......

Specialized

MadGraph, AlpGen, . . .

HDECAY, . . .

Ariadne/LDC, VINCIA, . . .

PHOJET/DPMJET

none (?)

TAUOLA, EvtGen

specialized often best at given task, but need General-Purpose coreSpecialized often best at given task, but need General-Purpose core


The Workhorses: What are the Differences?

HERWIG, PYTHIA and SHERPA offer convenient frameworksfor LHC physics studies, but with slightly different emphasis:

PYTHIA (successor to JETSET, begun in 1978):• originated in hadronization studies: the Lund string• leading in development of MPI for MB/UE• pragmatic attitude to showers & matching

HERWIG (successor to EARWIG, begun in 1984):• originated in coherent-shower studies (angular ordering)• cluster hadronization & underlying event pragmatic add-on• large process library with spin correlations in decays

SHERPA (APACIC++/AMEGIC++, begun in 2000):• own matrix-element calculator/generator• extensive machinery for CKKW ME/PS matching• hadronization & min-bias physics under development

PYTHIA & HERWIG originally in Fortran, now all C++


A Commercial: MCnet

? “Trade Union” of (QCD) Event Generator developers ?Collects HERWIG, SHERPA and PYTHIA.Also ThePEG, ARIADNE, VINCIA, . . . ,

generator validation (RIVET) and tuning (PROFESSOR)(CERN, Durham, Lund, Karlsruhe, UC London, + associated).

? Funded by EU Marie Curie training network 2007–2010 ?New applications for continued activities: no luck so far.

? Annual Monte Carlo school: ?Next: MCnet-LPCC, 23–27 July 2012, CERN

+ Lectures on QCD & Generators at many other schools +Much relevant material at http://www.montecarlonet.org/

MCPLOTS: repository of comparisons between generators anddata, based on RIVET, see http://mcplots.cern.ch/


The Bigger Picture

Need standardized interfaces: see next!


PDG Particle Codes

A. Fundamental objects

1 d 11 e− 21 g

2 u 12 νe 22 γ 32 Z′0

3 s 13 µ− 23 Z0 33 Z′′0

4 c 14 νµ 24 W+ 34 W′+

5 b 15 τ− 25 h0 35 H0 37 H+

6 t 16 ντ 36 A0 39 Graviton

add − sign for antiparticle, where appropriate

B. Mesons100 |q1|+ 10 |q2|+ (2s + 1) with |q1| ≥ |q2|C. Baryons1000 q1 + 100 q2 + 10 q3 + (2s + 1)with q1 ≥ q2 ≥ q3, or Λ-like q1 ≥ q3 ≥ q2

. . . and many more


Interfaces

LHA: Les Houches Accord,transfers info on processes, cross sections, parton-level events,. . . , via two Fortran commonblocks

LHEF: Les Houches Event Files,same information, but stored as a single plaintext file

HepMC: output of complete generated events(intermediate stages and final state with hundreds of particles)for subsequent detector simulation or analysis

SLHA: SUSY Les Houches Accord,file with info on SUSY (or other BSM) model:parameters, masses, mixing matrices, branching ratios, . . . .

LHAPDF: uniform interface to PDF parametrizations

Binoth LHA: one-loop virtual corrections

FeynRules: nascent standard for input of Lagrangian andoutput of Feynman rules

. . .Torbjorn Sjostrand Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2 slide 38/40

The Road Ahead

Event generators crucial since the start of LHC studies.

Qualitatively predictive already 25 years ago.

Quantitatively steady progress, continuing today:? continuous dialogue with experimental community,? more powerful computational techniques and computers,? new ideas.

As LHC needs to study more rare phenomena and more subtleeffects, generators must keep up by increased precision.

But it often happens that the physics simulations provided bythe Monte Carlo generators carry the authority of data itself.They look like data and feel like data, and if one is not carefulthey are accepted as if they were data.

J.D. Bjorkenfrom a talk given at the 75th anniversary celebration of the Max-Planck Institute of Physics, Munich, Germany,

December 10th, 1992. As quoted in: Beam Line, Winter 1992, Vol. 22, No. 4


Thank You —

— and Good Hunting!

introduction to monte carlo techniques in high …home.thep.lu.se/~torbjorn/talks/cern12b.pdfcern...

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