Monte Carlo simulation for the LHC

Monte Carlo simulation for the LHC 

Johan Alwall 

National Taiwan University and Fermilab 

Lecture 1 

PreSUSY, Beijing, August 8-11, 2012 Monte Carlo simulation for the LHC Johan Alwall

• 

• 

Lecture I: 

Outline of lectures 

➡ New Physics at hadron colliders 

➡ Simulation of collider events 

➡ Parton Showers 

➡ Jet matching between ME and PS 

Lecture II: 

➡ The NLO revolution 

➡ NLO+Parton Showers 

➡ Workflow for BSM phenomenology 

PreSUSY, Beijing, August 8-11, 2012 Monte Carlo simulation for the LHC Johan Alwall 

2

• 

Aims for these lectures 

Get you acquainted with the concepts and tools 

used in event simulation at hadron colliders 

• Answer as many of your questions as I can 

(so please ask questions!) 


3

18 

I%6B12*B1$@ 

Why the LHC? 

• 

) %6;< 9.*(27 ;H %>1?@ 

) 6(*+,-*.@ 

!"##$ 

;#5&23)&4#56.76< 

• 

• 

) 6(11B1B@ 

!"##$ 

) %6;1?@ 

Higgs boson mass “naturally” at 

mass of new physics 

! !"##$%&'$'(%)*$$%(*+,-*../% 

(only known “NP scale”: Planck scale at 

~1018 *+%+01%$2*.1%'3%(14%50/$"2$% 

GeV) 6'(./%7('4(8%9.*(27%$2*.1:% 

;< ;= %>1?@ 

Standard Model only “works” if 

! A+*(B*-B%C'B1.%'(./%4'-7$%"3% 

Higgs mass below ~800 GeV 

+01%!"##$%)*$$%&1.'4%D=

18 

N#C1.$/1.-E 

, G0/&$F #C9: 9H #D.BE 

, DIA #C9: 9J #D.BE#K 


;#5&23)&4#56.76< 

L.=#G82-)$-M 

, C&..1.1E 

")77- 

, #C9::#D.BE 

=./F 

, >(%'%& #C9#D.BE 

• ΔMH contribution must be 

!! #$%&'()*+')%&#,+-'# 

canceled " by bare mass term. For 

*.#$/&$.0.1#*2#*/(.# 

,/--#'.(,3#4%(#5)&.6 

'+&)&7#0.--#'8/&#9:;82-)$-# 

=8)$8#$+'-#%55#'8.# 

?+/1(/')$#0%%>-#/'# 

@9#A.B 

fine-tuning less than 1%, need 

new physics which cuts off the 

quadratic loops at ~1 TeV 

@9: O #D.B P 9: J #D.B P 

!"#$%&'()$((&*&+,%-.%/&0"1&23)&4#56.76&$-&-#3&8+9 : 


5


The Hierarchy problem, together with Dark Matter (and to 

some extent Grand Unification) have been driving New 

Physics model building in past 30 years 

➡ Supersymmetry 

➡ Large Extra Dimensions 

➡ Randall-Sundrum (warped extra dimensions) 

➡ Little Higgs theories 

➡ Composite models/Technicolor/Topcolor/... 

➡ ... (mostly variants/combinations) 

But of course, we might also find something 

completely unexpected! 


6

• 

• 

• 

New Physics at hadron colliders 

The LHC has taken over from the Tevatron! 

Significant luminocities 

➡ Tevatron collected >10 fb -1 in the last 10 years 

➡ Fantastic legacy, including several interesting 

excesses! 

➡ LHC already has a spectacular 10 fb -1 ! 

(perhaps as much as 35 fb -1 by end of this run!) 

➡ Allows ever-more stringent tests of the SM! 

➡ Already found what might be the Higgs boson! 

How interpret excesses? How determine Standard 

Model backgrounds? 

➡ Monte Carlo simulation! (combined with data-driven methods) 


7

Example: CDF excess in W + 2 jets 

WW, WZ 

W + Nobody 

knows what? 

) 

2 

Events/(8 GeV/c 

150 

100 

50 

0 

-50 

CDF collaboration, arXiv:1104.0699 

Bkg Sub Data (4.3 fb 

Gaussian 

WW+WZ 

100 200 

Mjj 

2 

[GeV/c ] 

Background subtracted data (except WW/WZ) 


(a) 

-1 

) 

8


A more complete picture 

) 

2 


700 

600 

500 

400 

300 

200 

100 

0 


100 200 

CDF data (4.3 fb 

Gaussian 2.5% 

WW+WZ 4.8% 

W+Jets 78.0% 

Top 6.3% 

Z+jets 2.8% 

QCD 5.1% 

[GeV/c 


M 

jj 

-1 

) 

(c) 

2 

] 

9


CDF data 


) 

2 


700 

600 

500 

400 

300 

200 

100 

0 


100 200 


Gaussian 2.5% 

WW+WZ 4.8% 

W+Jets 78.0% 

Top 6.3% 

Z+jets 2.8% 

QCD 5.1% 

[GeV/c 


M 

jj 

-1 

) 

(c) 

2 

] 

9


CDF data 

Standard Model 

backgrounds 

(shape from simulation) 


) 

2 


700 

600 

500 

400 

300 

200 

100 

0 


100 200 


Gaussian 2.5% 

WW+WZ 4.8% 

W+Jets 78.0% 

Top 6.3% 

Z+jets 2.8% 

QCD 5.1% 

[GeV/c 


M 

jj 

-1 

) 

(c) 

2 

] 

9


CDF data 

Excess 


backgrounds 



) 

2 


700 

600 

500 

400 

300 

200 

100 

0 


100 200 


Gaussian 2.5% 

WW+WZ 4.8% 

W+Jets 78.0% 

Top 6.3% 

Z+jets 2.8% 

QCD 5.1% 

[GeV/c 


M 

jj 

-1 

) 

(c) 

2 

] 

9


CDF data 

Excess 


backgrounds 



) 

2 


700 

600 

500 

400 

300 

200 

100 

0 

100 200 

We certainly need to know our backgrounds well! 



Gaussian 2.5% 

WW+WZ 4.8% 

W+Jets 78.0% 

Top 6.3% 

Z+jets 2.8% 

QCD 5.1% 

[GeV/c 


M 

jj 

-1 

) 

(c) 

2 

] 

9

Processes at Hadron Colliders 

First: Understand our processes! 

Cross sections at a collider depend on: 

• Coupling strength 

• Coupling to what? 

(light quarks, gluons, heavy quarks, 

EW gauge bosons?) 

• Mass 

• Single production/pair production 


b 

W 

Z 

t 

10

Processes at Hadron Colliders 

First: Understand our processes! 

Cross sections at a collider depend on: 

• Coupling strength 

• Coupling to what? 

(light quarks, gluons, heavy quarks, 

EW gauge bosons?) 

• Mass 

• Single production/pair production 

Expected 

new 

physics 


b 

W 

Z 

t 

10

Master formula 


11

• 

ˆσab→X(ˆs, . . .) 

Parton level 

cross section 


Parton level cross section from matrix element 


11

• 

• 


ˆσab→X(ˆs, . . .) fa(x1)fb(x2) 

Parton level 

cross section 

Parton density 

functions 

Parton level cross section from matrix element 

Parton density (or distribution) functions: 

Process independent, determined by particle type 


11

• 

• 

Tevatron vs. the LHC 

Tevatron: 2 TeV proton-antiproton collider 

⎯⎯ ⎯⎯ 

➡ Most important: q-q annihilation (85% of t t ) 

LHC: 8-14 TeV proton-proton collider 

➡ Most important: g-g annihilation (90% of t t ) 


⎯⎯ 

12

• 

• 

Tevatron vs. the LHC 

Tevatron: 2 TeV proton-antiproton collider 

⎯⎯ ⎯⎯ 

➡ Most important: q-q annihilation (85% of t t ) 

LHC: 8-14 TeV proton-proton collider 

➡ Most important: g-g annihilation (90% of t t ) 


⎯⎯ 

12

inematics 

atically towards low x 

ions for LHC, e.g. 

Parton densities 

Ratio of Luminosity: LHC at 7 TeV vs Tevatron 

pdf’s measured in deep-inelastic scattering! 

!! Power of collider can be 

fully characterized by ratio 

of parton luminosities 

!! Ratio larger for gg than qq 

!! Due to steap rise of gluon 

towards low x 

!! M X =100 GeV 

!! gg: R!10, e.g. Higgs 

!! qq: R!3, e.g. W and Z 

!! M X=800 GeV 

!! gg: R!1000, e.g. SUSY 

At small x (small ŝ), gluon domination. 

At large x valence quarks 

!! qq: R!20, e.g. Z’ 

LHC formidable at large mass – 

For low mass, Tevatron backgrounds smaller 

PreSUSY, Beijing, August 8-11, 2012 Monte Carlo simulation for the LHC Johan Alwall 11! 

13

Back to the processes 

LHC at 7 TeV vs Tevatron 

e 

atio 

qq 

n 

Z 

SY 

PreSUSY, Beijing, August 8-11, 2012 11! Monte Carlo simulation for the LHC Johan Alwall 14



e 

atio 

qq 

n 

Z 

SY 




e 

atio 

qq 

n 

Z 

SY 




e 

atio 

qq 

n 

Z 

SY 


Simulation of collider events 


15

Sherpa artist 


16

2 

1. High-Q Scattering 

2. Parton Shower 

3. Hadronization 4. Underlying Event 


17

2 





18

PreSUSY, Beijing, August 8-11, 2012 Monte Carlo simulation for the LHC Johan Alwall 21

2 





22

List of processes 

implemented 

in Pythia (by hand!) 


23

Automated Matrix Element Generators 


24

• 


High-Q2 scattering processes: In principle infinite number 

of processes for innumerable number of models 


24

• 

• 




Implementation by hand time-consuming, labor intensive 

and error prone 


24

• 

• 

• 






Instead: Automated matrix element generators 

➡ Use Feynman rules to build diagrams 


24

• 

• 

• 

• 






Instead: Automated matrix element generators 

➡ Use Feynman rules to build diagrams 

Given files defining the model content: particles, 

parameters and interactions, allows to generate any 

process for a given model! 


24


• 

• 

Automatic matrix element generators: 

➡ CalcHep / CompHep 

➡ MadGraph 

➡ AMEGIC++ (Sherpa) 

➡ Whizard 

Standard Model only, with fast matrix elements for high 

parton multiplicity final states: 

➡ AlpGen 

➡ HELAC 

➡ COMIX (Sherpa) 


25

2 





26

Parton Shower MC event generators 

• General-purpose tools 

• Complete exclusive description of the events: hard scattering, 

showering, hadronization, underlying event 

• Reliable and well tuned to experimental data. 

most well-known: PYTHIA, HERWIG, SHERPA 

• Significant progress in the development of new showering algorithms 

with the final aim to go to NLO in QCD 

[Nagy, Soper, 2005; Giele, Kosower, Skands, 2007; Krauss, Schumann, 2007] 


27

Parton Shower basics 

The spin averaged (unregulated) splitting functions for the various 

types of branching are: 


29

Parton Shower basics 

The spin averaged (unregulated) splitting functions for the various 

types of branching are: 

Comments: 

* Gluons radiate the most 

* There are soft divergences in z=1 and z=0. 

* Pqg has no soft divergences. 


29

Final-state parton showers 


31


With the Sudakov form factor, we can now implement a finalstate 

parton shower in a Monte Carlo event generator! 


31




1. Start the evolution at the virtual mass scale t0 (e.g. the mass of the 

decaying particle) and momentum fraction z0 = 1 


31






2. Given a virtual mass scale ti and momentum fraction xi at some 

stage in the evolution, generate the scale of the next emission ti+1 

according to the Sudakov probability ∆(ti,ti+1) by solving 

∆(ti+1,ti) = R 

where R is a random number (uniform on [0, 1]). 


31









∆(ti+1,ti) = R 


3. If ti+1 < tcut it means that the shower has finished. 


31









∆(ti+1,ti) = R 



4. Otherwise, generate z = zi/zi+1 with a distribution proportional to 

(αs/2π)P(z), where P(z) is the appropriate splitting function. 


31









∆(ti+1,ti) = R 



4. Otherwise, generate z = zi/zi+1 with a distribution proportional to 

(αs/2π)P(z), where P(z) is the appropriate splitting function. 

5. For each emitted particle, iterate steps 2-4 until branching stops. 


31



32


• The result is a “cascade” or “shower” of partons with ever 

smaller virtualities. 

Matching of Matrix 

Elements and 

Parton Showers 

Lecture 1: QCD 

Johan Alwall 

Plan of the lectures 

Introduction: The 

big picture 

Infrared Behaviour 

of QCD 

Jet Definitions 


Parton branchings 

Evolution 

equations and 

parton densities 

Logarithmic 

resummation 

Sudakov form 

factors 

Angular ordering 

NLL Sudakovs 

Parton showers in 

Monte Carlos 


e - 

e + 

t0 

Due to these successive branchings, the parton cascade or parton show 

develops. Each outgoing line is a source of a new cascade, until all ou 

lines have stopped branching. At this stage, which depends on the cut 

outgoing partons have to be converted into hadrons via a hadronizatio 

32




• The cutoff scale tcut is usually set close to 1 GeV, 

the scale where non-perturbative effects start dominating 


over the perturbative Elements parton and shower. 


Lecture 1: QCD 

Johan Alwall 

Plan of the lectures 

Introduction: The 

big picture 

Infrared Behaviour 

of QCD 

Jet Definitions 


Parton branchings 

Evolution 

equations and 

parton densities 

Logarithmic 

resummation 

Sudakov form 

factors 

Angular ordering 

NLL Sudakovs 

Parton showers in 

Monte Carlos 


e - 

e + 

t0 

tcut 

Due to these successive branchings, the parton cascade or parton show 

develops. Each outgoing line is a source of a new cascade, until all ou 

lines have stopped branching. At this stage, which depends on the cut 

outgoing partons have to be converted into hadrons via a hadronizatio 

32




• The cutoff scale tcut is usually set close to 1 GeV, 

the scale where non-perturbative effects start dominating 

over the perturbative parton shower. 

• At this point, phenomenological 

models are used to simulate 

how the partons turn into 

color-neutral hadrons. 

Hadronization not sensitive to 

the physics at scale t0, but only tcut! 

(can be tuned once and for all!) 


e - 

e + 

t0 

32

• 

• 

• 

• 

Initial-state parton splittings 

So far, we have looked at final-state (time-like) 

splittings 

For initial state, the splitting functions are the same 

However, there is another ingredient - the parton 

density (or distribution) functions (PDFs) 

➡ Probability to find a given parton in a hadron at a 

given momentum fraction x = pz/Pz and scale t 

How do the PDFs evolve with increasing t? 


33

• 

• 

• 

Parton Shower MC event generators 

In both initial-state and final-state showers, the definition of t is 

not unique, as long as it has the dimension of scale: 

Different parton shower generators have made different choices: 

➡ Ariadne: “dipole pT” 

➡ Herwig: E⋅θ 

➡ Pythia (old): virtuality q 2 

➡ Pythia 6.4 and Pythia 8: pT 

➡ Sherpa: v. 1.1.x virtuality q 2 , v. 1.2.x “dipole pT” 

Note that all of the above are complete MC event generators 

with matrix elements, parton showers, hadronization, decay, and 

underlying event simulation. 


37

From Parton Showers to Hadronization 

• The parton shower evolves the hard scattering down to the scale of 

O(1GeV). 

• At this scale, QCD is no longer perturbative. some hadronization model is 

used to describe the transition from the perturbative PS region to the 

non-perturbative hadronization region. 

• Main hadronization models: 

➡ String hadronization (Pythia) 

➡ Cluster hadronization (Herwig) 

• Hadronization only acts locally, not sensitive to high-q 2 scattering. 

e - 

e + 

[Andersson,Gustafson,Ingelman,Sjöstrand (1983)] 

[Webber (1984)] 


38

• 

• 

• 

Detector simulation 

Detector simulation 

➡ Fast general-purpose detector simulators: 

Delphes, PGS (“Pretty good simulations”), AcerDet 

➡ Specify parameters to simulate different experiments 

Experiment-specific fast simulation 

➡ Detector response parameterized 

➡ Run time ms-s/event 

Experiment-specific full simulation 

➡ Full tracking of particles through detector using GEANT 

➡ Run time several minutes/event 

➡ See Michael’s lecture for much more details! 


39

Back to our favorite piece of art! 

2 1. High-Q Scattering 


How do we define the limit between parton shower 

and matrix element? 


40

Matrix Elements vs. Parton Showers 


41


ME 

1. Fixed order calculation 

2. Computationally expensive 

3. Limited number of particles 

4. Valid when partons are hard and 

well separated 

5. Quantum interference correct 

6. Needed for multi-jet description 


41


ME 








Shower MC 

1. Resums logs to all orders 

2. Computationally cheap 

3. No limit on particle multiplicity 

4. Valid when partons are collinear 

and/or soft 

5. Partial interference through 

angular ordering 

6. Needed for hadronization 


41


ME 








Shower MC 





and/or soft 




Approaches are complementary: merge them! 


41


ME 








Shower MC 





and/or soft 




Approaches are complementary: merge them! 

Difficulty: avoid double counting, ensure smooth distributions 


41

(pb/bin) 

T 

/dP 

! 

d 

10 

10 

10 

10 

1 

-1 

-2 

-3 

PS alone vs ME matching 

In a matched sample these differences are irrelevant since the behavior 

at high pt is dominated by the matrix element. 

Q 

Q 

P 

P 

2 

2 

2 

T 

2 

T 

(wimpy) 

(power) 

(wimpy) 

(power) 

tt+0,1,2,3 

partons + Pythia (MMLM) 

of the 2-nd extra jet 

0 50 100 150 200 250 300 350 400 

GeV 


P 

T 

[MadGraph] 

43

Goal for ME-PS merging/matching 

Matrix element 

Parton shower 

2nd QCD radiation jet in 

top pair production at 

the LHC, using 

MadGraph + Pythia 


44

• 


Regularization of matrix element divergence 


Parton shower 






44

• 

• 



Correction of the parton shower for large momenta 


Parton shower 






44

• 

• 

• 




Smooth jet distributions 


Parton shower 






44

• 

• 

• 






Parton shower 

Desired curve 






44

ME 

↓ 

Merging ME with PS 

PS → 

... 


... 

[Mangano] 

[Catani, Krauss, Kuhn, Webber] 

[Lönnblad] 

45

ME 

↓ 


PS → 

DC DC 

DC 

... 


... 

[Mangano] 


[Lönnblad] 

45

ME 

↓ 


kT > Q c 

kT > Q c 

kT > Q c 

PS → 

... 

kT < Q c 

kT < Q c 

kT > Q c 


... 

[Mangano] 


[Lönnblad] 

kT < Q c 

kT < Q c 

45

ME 

↓ 


kT > Q c 

kT > Q c 

kT > Q c 

PS → 

... 

kT < Q c 

kT < Q c 

kT > Q c 

Double counting between ME and PS easily avoided using phase space 

cut between the two: PS below cutoff, ME above cutoff. 


... 

[Mangano] 


[Lönnblad] 

kT < Q c 

kT < Q c 

45

• 

• 

• 


So double counting problem easily solved, but 

what about getting smooth distributions that are 

independent of the precise value of Q c ? 

Below cutoff, distribution is given by PS 

- need to make ME look like PS near cutoff 

Let’s take another look at the PS! 


46


Elements and 


Lecture 2: 

Matching in e + e − 

collisions 

Johan Alwall 

Why Matching? 

Present matching 

approaches 

CKKW matching in 

e + e − collisions 

Overview of the 

CKKW procedure 

Clustering the 

n-jet event 

Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 

Results of CKKW 

matching (Sherpa) 

Difficulties with 

practical 

implementations 

The MLM 

procedure 

Clustering the n-jet event 


1 Find the two partons with smallest jet separation yij 

2 If partons allowed to cluster by QCD splitting rules: combine partons to 

new particle (e.g. q¯q → g, qg → q) 

3 Iterate 1-2 until 2 → 2 process reached (e + e − → q¯q) 

t0 

With the choice of the Durham jet measure, the jet separations di = √ yi Q0 at 

each branching corresponds closely to the kT of that branching, and is therefore 

suitable to use as argument for αs in the branching. 


t1 

tcut tcut 

t2 

tcut 

tcut 

10 / 29 

47


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 






n-jet event 

• Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 



How does the PS generate the configuration above? 





t0 





t1 

tcut tcut 

t2 

tcut 

tcut 

10 / 29 

47


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 






n-jet event 

• Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 





• Probability for the splitting at t1 is given by 




t0 

(∆q(t1,t0)) 





t1 

2 αs(t1) 

tcut tcut 

t2 

tcut 

tcut 

2π Pgq(z) 

10 / 29 

47


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 






n-jet event 

• Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 









t0 

(∆q(t1,t0)) 





t1 

2 αs(t1) 

tcut tcut 

t2 

tcut 

tcut 

2π Pgq(z) 

10 / 29 

47


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 






n-jet event 

• Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 









t0 

(∆q(t1,t0)) 





t1 

2 αs(t1) 

tcut tcut 

t2 

tcut 

tcut 

2π Pgq(z) 

10 / 29 

47


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 






n-jet event 

Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 







|M| 2 (ˆs, p3,p4,...) 





10 / 29 

49


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 





• 


n-jet event 

Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 



To get an equivalent treatment of the corresponding 

matrix element, do as follows: 





|M| 2 (ˆs, p3,p4,...) 





10 / 29 

49


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 





• 


n-jet event 

Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 






1. Cluster the event using some clustering algorithm 

|M| 2 (ˆs, p3,p4,...) 



3 Iterate 1-2 until 2 → 2 process reached (e + e− - this gives us a corresponding “parton → q¯q) shower history” 





10 / 29 

49


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 





• 


n-jet event 

Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 







t0 








t1 

t2 |M| 2 (ˆs, p3,p4,...) 

10 / 29 

49


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 





• 


n-jet event 

Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 







t0 




2. Reweight αs in each clustering vertex with the clustering 


each branching corresponds closely to the kT scale 

of that branching, and is therefore 

suitable to use as argument for αs in the 2 αs(t1) αs(t2) 

branching. 

|M| 2 → |M| 


t1 

αs(t0) 

t2 

αs(t0) 

|M| 2 (ˆs, p3,p4,...) 

10 / 29 

49


Elements and 


Lecture 2: 


collisions 

Johan Alwall 

Why Matching? 


approaches 





• 


n-jet event 

Sudakov 

reweighting 

Vetoed parton 

showers 

Highest 

multiplicity 

treatment 




practical 


The MLM 

procedure 







t0 




2. Reweight αs in each clustering vertex with the clustering 


each branching corresponds closely to the kT scale 

of that branching, and is therefore 

suitable to use as argument for αs in the 2 αs(t1) αs(t2) 

branching. 

3. Use some algorithm to apply the equivalent Sudakov 

suppression (∆q(tcut,t0)) 2 ∆g(t2,t1)(∆q(cut,t2)) 2 

|M| 2 → |M| 

αs(t0) αs(t0) 


t1 

t2 

|M| 2 (ˆs, p3,p4,...) 

10 / 29 

49

le of the procedure 

o simulate pp → W + jets. 

Matching for initial state radiation 

ck (according to the relative cross-section of the processes) a 

Wd ¯d event 

ck momenta according to the pdf-weighted matrix element 

|M u ¯ d→Wd ¯ d (x1, x2, αs(dini))| 2 fu(x1, dini)f¯ d (x2, dini) 

ter the event using the 

variant kT clustering 

, to get nodes d1, d2, d3 as 

ly the αs and Sudakov 

x1 

x2 

PreSUSY, 2 Beijing, August 8-11, 2012 Monte Carlo simulation for the LHC Johan Alwall 

∆g (d2, dini) 2 αs(d2) αs(d1) 

50


• We are of course not interested in e+ e - but p-p(bar) 



- what happens for initial state radiation? 


Wd ¯d event 







x1 

x2 



50





Wd ¯d event 







x1 

x2 



52


• le of the procedure 

Again, use a clustering scheme to get a parton shower 

history 



Wd ¯d event 







x1 

x2 



52


• le of the procedure 

Again, use a clustering scheme to get a parton shower 

history 



Wd ¯d event 







x1 

t1 t2 

x2 

x1’ 

t0 



52

• 

• 

Matching schemes 

We still haven’t specified how to apply the Sudakov 

reweighting to the matrix element 

Three general schemes available in the literature: 

➡ CKKW scheme [Catani,Krauss,Kuhn,Webber 2001; Krauss 2002] 

➡ Lönnblad scheme (or CKKW-L) [Lönnblad 2002] 

➡ MLM scheme [Mangano unpublished 2002; Mangano et al. 2007] 


53

CKKW matching 

[Catani, Krauss, Kuhn, Webber 2001] 

[Krauss 2002] 


54

CKKW matching 

• Apply the required Sudakov suppression 

(∆Iq(tcut,t0)) 2 ∆g(t2,t1)(∆q(tcut,t2)) 2 

analytically, using the best available (NLL) Sudakovs. 


[Krauss 2002] 


54



simulate pp → W + jets. 

k (according to the relative cross-section of the processes) a 

Wd ¯d event 

k momenta according to the pdf-weighted matrix element 






CKKW matching 



• Perform “truncated showering”: Run the parton shower 

starting at t0, but forbid any showers above the cutoff scale tcut. 

t0 


[Krauss 2002] 

, dini)) 


2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) αs(d1) 

dini)) 

αs(dini) αs(dini) 

54





Wd ¯d event 







CKKW matching 





kT1 

kT2 

t0 


[Krauss 2002] 

, dini)) 


2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) αs(d1) 

dini)) 


kT3 

54





Wd ¯d event 







CKKW matching 





kT1 

x 

x 

kT2 

t0 


[Krauss 2002] 

, dini)) 


2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) αs(d1) 

dini)) 


kT3 

54





Wd ¯d event 







CKKW matching 





kT1 

x 

✓ Best theoretical treatment of matrix element 

- Requires dedicated PS implementation 

x 

t0 


[Krauss 2002] 

- Mismatch between analytical kT2 Sudakov and (non-NLL) shower 

• Implemented in Sherpa (v. 1.1) 

, dini)) 


2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) αs(d1) 

dini)) 


kT3 

54

ick (according to the relative cross-section of the processes) a 

Wd ¯d event 

ick momenta according to the pdf-weighted matrix element 

|M u¯d→Wd ¯d (x1, x2, αs(dini))| 2 fu(x1, dini)f¯d (x2, dini) 

ster the event using the 

invariant kT clustering 

e, to get nodes d1, d2, d3 as 

ply the αs and Sudakov 

3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 

αs(dini) 

CKKW-L matching 

αs(d1) 

αs(dini) 

pply initial-state radiation for the incoming u and ¯d starting at 

MW , and final-state radiation for the outgoing d and ¯d starting at d2, 

eto all emissions above dini (in both initial- and final state showers). 

t0 


7 / 23 

[Lönnblad 2002] 

[Hoeche et al. 2009] 

55


Wd ¯d event 







• Cluster back to “parton shower history” 

3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 

αs(dini) 


αs(d1) 

αs(dini) 




t0 


7 / 23 



55


Wd ¯d event 








3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 

αs(dini) 


αs(d1) 

αs(dini) 




t0 


7 / 23 



55


Wd ¯d event 









• Perform showering step-by-step for each step in the parton 

shower history, starting from the clustering scale for that step 

3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 


αs(d1) 




t0 


7 / 23 



55


Wd ¯d event 











3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 


αs(d1) 




kT1 

t0 


7 / 23 



55


Wd ¯d event 











3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 


αs(d1) 




kT1 

t0 


7 / 23 



55


Wd ¯d event 











3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 


αs(d1) 




kT2 

kT1 

t0 


7 / 23 



55


Wd ¯d event 











3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 


αs(d1) 




kT2 

kT1 

t0 


7 / 23 



55


Wd ¯d event 











3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 


αs(d1) 




kT2 

kT1 

t0 


kT4 

7 / 23 



55


Wd ¯d event 











• Veto the event if any shower is harder than the clustering scale 

for the next step (or tcut, if last step) 

3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 


αs(d1) 




kT2 

kT1 

t0 


kT4 

7 / 23 



55


Wd ¯d event 











• Veto the event if any shower is harder than the clustering scale 


• Keep any shower emissions that are softer 7 than / 23 the clustering 

scale for the next step 

3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 


αs(d1) 




kT2 

kT1 

t0 




kT4 

55


Wd ¯d event 











✓• Automatic Veto the event agreement if any shower between is Sudakov harder than and the shower clustering scale 


- Requires dedicated PS implementation 

• Keep any shower emissions that are softer 7 than / 23 the clustering 

➡ Need multiple implementations to compare between showers 

scale for the next step 

• Implemented in Ariadne, Sherpa (v. 1.2), and Pythia 8 

3, dini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 


αs(d1) 




kT2 

kT1 

t0 




kT4 

55

of the procedure 


MLM matching 

(according to the relative cross-section of the processes) a 

d ¯d event 

momenta according to the pdf-weighted matrix element 


r the event using the 

ariant kT clustering 

o get nodes d1, d2, d3 as 

the αs and Sudakov 

ini)) 2 ∆g (d2, dini) 

∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 

αs(dini) 

αs(d1) 

αs(dini) 

ly initial-state radiation for the incoming u and ¯d starting at 

W , and final-state radiation for the outgoing d and ¯d starting at d2, 

all emissions above dini (in both initial- and final state showers). 

t0 

[M.L. Mangano, ~2002, 2007] 

[J.A. et al 2007, 2008] 

PreSUSY, Beijing, August 8-11, 2012 Monte Carlo 7 / 23simulation 

for the LHC Johan Alwall 

56



MLM matching 


d ¯d event 








∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 

αs(dini) 

αs(d1) 

αs(dini) 




t0 

[M.L. Mangano, ~2002, 2007] 

[J.A. et al 2007, 2008] 

• The simplest way to do the Sudakov suppression is to run the 

shower on the event, starting from t0! 



56



MLM matching 


d ¯d event 








∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 

αs(dini) 

αs(d1) 

αs(dini) 




kT1 

kT2 

t0 

kT3 



kT4 

[M.L. Mangano, ~2002, 2007] 

[J.A. et al 2007, 2008] 



56



MLM matching 


d ¯d event 







• Perform jet clustering αs(d1) after PS - if hardest jet kT1 > tcut or 


∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 

there are jets αs(dini) not matched αs(dini) to partons, reject the event 




kT1 

kT2 

t0 

kT3 



kT4 

[M.L. Mangano, ~2002, 2007] 

[J.A. et al 2007, 2008] 



56



MLM matching 


d ¯d event 









∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 





kT1 

kT2 

t0 

kT3 



kT4 

(∆Iq(tcut,t0)) 2 (∆q(tcut,t0)) 2 

[M.L. Mangano, ~2002, 2007] 

[J.A. et al 2007, 2008] 



• The resulting Sudakov suppression from the procedure is 


which turns out to be a good enough approximation of the 

correct expression 

56



MLM matching 


d ¯d event 









∆g (d1, dini) (∆q(d1, 

2 αs(d2) 

dini)) 





kT1 

kT2 

t0 

kT3 



kT4 

(∆Iq(tcut,t0)) 2 (∆q(tcut,t0)) 2 

[M.L. Mangano, ~2002, 2007] 

[J.A. et al 2007, 2008] 



✓ Simplest available scheme 

✓• Allows The resulting matching Sudakov with any suppression shower, without from the modification procedure is 


➡ Sudakov 

which turns 

suppression 

out to be 

not 

a good 

exact, 

enough 

minor 

approximation 

mismatch with 

of 

shower 

the 

• Implemented correct expression in AlpGen, HELAC, MadGraph 

56

• 

• 

• 

Highest multiplicity sample 

In the previous, assumed we can simulate all parton 

multiplicities by the ME 

In practice, we can only do limited number of final-state 

partons with matrix element (up to 4-5 or so) 

For the highest jet multiplicity that we generate with the 

matrix element, we need to allow additional jets above 

the matching scale tcut, since we will otherwise not get a 

jet-inclusive description – but still can’t allow PS radiation 

harder than the ME partons 

➡ Need to replace tcut by the clustering scale for the softest 

ME parton for the highest multiplicity 


57

• 

• 

• 

Back to the “matching goal” 





Parton shower 






58

• 

• 

• 






Parton shower 

Matching scale 






58

• 

• 

• 






Parton shower 







58

• 

• 

• 






Parton shower 







58

• 

• 

• 






Parton shower 







58

• 

• 

• 






Parton shower 


Desired curve 






58

Summary of Matching Procedure 

1. Generate ME events (with different parton multiplicities) 

using parton-level cuts (pT ME /ΔR or kT ME ) 

2. Cluster each event and reweight αs and PDFs based on the 

scales in the clustering vertices 

3. Apply Sudakov factors to account for the required nonradiation 

above clustering cutoff scale and generate parton 

shower emissions below clustering cutoff: 

a. (CKKW) Analytical Sudakovs + truncated showers 

b. (CKKW-L) Sudakovs from truncated showers 

c. (MLM) Sudakovs from reclustered shower emissions 


59

[pb] 

T 

)dE 

T 

(d/dE 

 

2 

10 

min 

ET 

10 

1 

-1 

10 

-2 

10 

Comparing to experiment: W+jets 

(We) 

+ n 

jets 

st 

1 jet 

nd 

2 jet 

rd 

3 jet 

th 

4 jet 

CDF Data 

CDF Run II Preliminary 

 

-1 

dL = 320 pb 

e 

W kin: E 20[GeV]; 

| | 

1.1 

0 50 100 150 200 

min 

Jet Transverse Energy (E ) [GeV] 

T 

W 

T 

e 

 

T 

M 20[GeV/c 

]; E 30[GeV] 

Jets: JetClu R=0.4; | | 30 GeV 

1 2 3 4 

inclusive jet multiplicity, n 

60

MLM matching schemes in MadGraph 

[J.A. et al (2007, 2008)] 

[J.A. et al (2011)] 


61


In MadGraph, there are 3 different MLM-type matching 

schemes differing in how to divide ME vs. PS regions: 

[J.A. et al (2007, 2008)] 

[J.A. et al (2011)] 


61


[J.A. et al (2007, 2008)] 

[J.A. et al (2011)] 



a. Cone jet MLM scheme: 

- Use cuts in pT (pT ME )and ΔR between partons in ME 

- Cluster events after parton shower using a cone jet 

algorithm with the same ΔR and pT match > pT ME 

- Keep event if all jets are matched to ME partons (i.e., all 

ME partons are within ΔR of a jet) 


61


[J.A. et al (2007, 2008)] 

[J.A. et al (2011)] 



a. Cone jet MLM scheme: 

- Use cuts in pT (pT ME )and ΔR between partons in ME 

- Cluster events after parton shower using a cone jet 

algorithm with the same ΔR and pT match > pT ME 

- Keep event if all jets are matched to ME partons (i.e., all 

ME partons are within ΔR of a jet) 

b. kT-jet MLM scheme: 

- Use cut in the Durham kT in ME 

- Cluster events after parton shower using the same kT 

clustering algorithm into kT jets with kT match > kT ME 

- Keep event if all jets are matched to ME partons 

(i.e., all partons are within kT match to a jet) 


61


c. Shower-kT scheme: 

- Use cut in the Durham kT in ME 

- After parton shower, get information from the PS 

generator about the kT PS of the hardest shower 

emission 

- Keep event if kT PS < kT match 


62

How to do matching in MadGraph+Pythia 

Example: Simulation of pp→W with 0, 1, 2 jets 

mg5> generate p p > w+, w+ > l+ vl @0 

mg5> add process p p > w+ j, w+ > l+ vl @1 

mg5> add process p p > w+ j j, w+ > l+ vl @2 

mg5> output 

In run_card.dat: 

… 

… 

… 

1 = ickkw 

0 = ptj 

15 = xqcut 

(comfortable on a laptop) 

Matching on 

kT matching scale 

Matching automatically done when run through 

MadEvent and Pythia! 

No cone matching 


63

How to do matching in MadGraph+Pythia 

• By default, kT-MLM matching is run if xqcut > 0, with the 

matching scale QCUT = max(xqcut*1.4, xqcut+10) 

• For shower-kT, by default QCUT = xqcut 

• If you want to change the Pythia setting for matching 

scale or switch to shower-kT matching: 

In pythia_card.dat: 

… 

! This sets the matching scale, needs to be > xqcut 

QCUT = 30 

! This switches from kT-MLM to shower-kT matching 

! Note that MSTP(81)>=20 needed (pT-ordered shower) 

SHOWERKT = T 


64

How to do validate the matching 

• The matched cross section is found at the end of the 

Pythia log file 

• The matched cross section (for X+0,1,... jets) should be 

close to the unmatched cross section for the 0-jet sample 

• The matching scale (QCUT) should typically be chosen 

around 1/6-1/2 x hard scale (so xqcut correspondingly 

lower) 

• The differential jet rate plots should be smooth 

• When QCUT is varied (within the region of validity), the 

matched cross section or differential jet rates should not 

vary significantly 


65

Matching validation 

W+jets production at the Tevatron for MadGraph+Pythia 

(kT-jet MLM scheme, q 2 -ordered Pythia showers) 

Q match = 10 GeV Q match = 30 GeV 

log(Differential jet rate for 1 → 2 radiated jets ~ pT(2nd jet)) 

Jet distributions smooth, and stable when we vary the matching scale! 


66

Summary of Lecture 1 


67

• 


Despite the apparent enormous complexity of simulation 

of complete collider events, nature has kindly allowed us 

to factorize the simulation into separate steps 


67

• 

• 





The Monte Carlo method allows us to step-by-step 

simulate hard scattering, parton shower, particle decays, 

hadronization, and underlying event 


67

• 

• 

• 








Jet matching between matrix elements and parton 

showers gives crucial improvement of simulation of 

background as well as signal processes 


67

• 

• 

• 

• 








Jet matching between matrix elements and parton 

showers gives crucial improvement of simulation of 

background as well as signal processes 

Next lecture: Next-to-leading order simulations and 

workflow for New Physics simulation! 


67

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