CERN Summer Student Lecture
Part 2, 20 July 2012
Introduction to
Monte Carlo Techniques
in High Energy Physics
Torbjörn Sjöstrand
How are complicated multiparticle events created?
How can such events be simulated with a computer?
Lectures Overview
yesterday:
today:
Torbjörn Sjöstrand
Introduction
the Standard Model
Quantum Mechanics
the role of Event Generators
Monte Carlo
random numbers
integration
simulation
Physics
hard interactions
parton showers
multiparton interactions
hadronization
Generators
Herwig, Pythia, Sherpa
MadGraph, AlpGen, . . .
common standards
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, . . .
p
p/p
Incoming beams: parton densities
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 3/40
The Structure of an Event – 2
W+
u
d
g
p
p/p
Hard subprocess: described by matrix elements
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 4/40
The Structure of an Event – 3
c
s
W+
u
d
g
p
p/p
Resonance decays: correlated with hard subprocess
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 5/40
The Structure of an Event – 4
c
s
W+
u
d
g
p
p/p
Initial-state radiation: spacelike parton showers
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 6/40
The Structure of an Event – 5
c
s
W+
u
d
g
p
p/p
Final-state radiation: timelike parton showers
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 7/40
The Structure of an Event – 6
c
s
W+
u
d
g
p
p/p
Multiple parton–parton interactions . . .
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 8/40
The Structure of an Event – 7
c
s
W+
u
d
g
p
p/p
. . . with its initial- and final-state radiation
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 9/40
The Structure of an Event – 8
Beam remnants and other outgoing partons
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 10/40
The Structure of an Event – 9
Everything is connected by colour confinement strings
Recall! Not to scale: strings are of hadronic widths
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 11/40
The Structure of an Event – 10
The strings fragment to produce primary hadrons
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 12/40
The Structure of an Event – 11
Many hadrons are unstable and decay further
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 13/40
The Structure of an Event – 12



These are the particles that hit the detector
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 14/40
Hard Processes
Defines the character of the event:
QCD, top, Z0 , W± , H0 , SUSY, ED, TC, . . .
Obtained by perturbation theory:
σ̂Z 0 = αem M0 + αem αs M1 + αem αs2 M2
Next-to-leading
graphs
=
LO order
+ (NLO)
NLO
+
NNLO
+
+
...
...
Involves subtle cancellations between real and virtual contributions:
These days
Torbjörn Sjöstrand
• LO easy
• NLO tough but doable
• NNLO only in very few cases
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 15/40
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 0 radiates a spectrum of
photons that depends on its trajectory. In the long-wavelength limit it reduces to
lim
ω→0
˛ „
«˛2
˛
d2 I
z 2 e 2 ˛˛ ∗
β0
β
˛
=
−
˛
dωdΩ
4π 2
1 − nβ 0
1 − nβ ˛
where n is a vector on the unit sphere Ω, ω is the energy of the radiated photon, and
its polarization.
1
For fast particles radiation collinear with the initial (β) and
final (β 0 ) directions is strongly enhanced.
2
dN/dω = (1/ω)dI /dω ∝ 1/ω so infinitely many infinitely soft
photons are emitted, but the net energy taken away is finite.
3
For ω → 0 the radiation pattern is independent of the spin
of the radiator ⇒ universality.
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 16/40
QCD Bremsstrahlung – 1
QCD ≈ QED with
e → eγ ⇒ q → qg
αem ⇒ (4/3)αs
More precisely:
dPq→qg ≈
αs dQ 2 4 1 + z 2
dz
2π Q 2 3 1 − z
where
mass (or collinear) singularity:
2
dp⊥
dQ 2
dθ2
dm2
≈
≈
≈
2
Q2
θ2
m2
p⊥
soft singularity:
Eq,after
ω
dz
dω
z≈
=
=1−
so
Eq,before
Eq,before
1−z
ω
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 17/40
QCD Bremsstrahlung – 2
But QCD is non-Abelian so additionally
g → gg similarly divergent
αs (Q 2 ) diverges for Q 2 → 0
DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi)
dPa→bc
Pq→qg
Pg→gg
Pg→qq
Torbjörn Sjöstrand
αs (Q 2 ) dQ 2
Pa→bc (z) dz
2π
Q2
4 1 + z2
=
3 1−z
(1 − z(1 − z))2
= 3
z(1 − z)
nf 2
=
(z + (1 − z)2 ) (nf = no. of quark flavours)
2
=
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 18/40
The iterative structure
Generalizes to many consecutive emissions if strongly ordered,
Q12 Q22 Q32 . . . (≈ time-ordered).
To cover “all” of phase space use DGLAP in whole region
Q12 > Q22 > Q32 . . ., although only approximately valid.
Must be clever when you write a shower algorithm.
Iteration gives
final-state
parton showers:
Need soft/collinear cuts to stay away from nonperturbative physics.
Details model-dependent, but around 1 GeV scale.
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 19/40
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: Qi2 ∼ −m2 > 0 increasing
FSR = Final-State Radiation: Qi2 ∼ m2 > 0 decreasing
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 20/40
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 probability
driven by depletion:
a given nucleus can only decay once
Correctly
P(t) = cN(t) =⇒ N(t) = exp(−ct)
i.e. exponential dampening
P(t) = c exp(−ct)
For P(t) = f (t)N(t), f (t) ≥ 0, this generalizes to
Z t
0
0
P(t) = f (t) exp −
f (t ) dt
0
which can be Monte Carlo-simulated using the veto algorithm.
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 21/40
The Sudakov form factor
Correspondingly, with Q ∼ 1/t (Heisenberg)
dPa→bc
αs dQ 2
Pa→bc (z) dz
2π Q 2


2
2 Z
0
X Z Qmax
dQ
αs
Pa→bc (z 0 ) dz 0 
× exp −
2
0
2π
2
Q
b,c Q
=
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
P RR
Note that b,c
dPa→bc ≡ 1 ⇒ convenient for Monte Carlo
(≡ 1 if extended over whole phase space, else possibly nothing
happens before you reach Q0 ≈ 1 GeV).
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 22/40
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 showers
migrate to another jet multiplicity.
MC@NLO: split σNLO into real LO+1 event and real+virtual
LO ones.
POWHEG: like merging, with overall rate normalized to σNLO .
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 23/40
Parton Distribution Functions (PDFs)
Hadrons are composite: p = uud + gluons + sea qq.
Higher virtuality scale Q ⇒ more partons resolved.
fi (x, Q 2 ) = number density of partons i at momentum fraction x
and probing scale Q 2 .
Initial conditions at small Q02 unknown: nonperturbative.
Resolution dependence perturbative, by DGLAP:
DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi)
Z
dfb (x, Q 2 ) X 1 dz
x
2 αs
=
fa (y , Q )
Pa→bc z =
d(ln Q 2 )
2π
y
x z
a
Z
σ=
1
Z
dx1
0
1
2
dx2 fi (x1 , Q 2 ) fj (x2 , Q 2 ) σ̂(x1 x2 Ecm
, Q 2)
0
Continuous ongoing activity to provide best overall fit to data,
consistent with theoretical evolution: CTEQ, MSTW, NNPDF, . . .
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 24/40
PDF examples
Convenient plotting interface:
http://hepdata.cedar.ac.uk/pdf/pdf3.html
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 25/40
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 = hni ≈ 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 .
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 26/40
MultiParton Interactions (MPIs)
Many parton-parton interactions
per pp event: MPI.
Most have small p⊥ , ∼ 2 GeV
⇒ not visible as separate jets,
but contribute to event activity.
Solid evidence that MPIs play
central role for event structure.
Problem:
ZZZ
σint =
since
R
2
2
2
dx1 dx2 dp⊥
f1 (x1 , p⊥
) f2 (x2 , p⊥
)
dσ̂
=∞
2
dp⊥
2 ) = ∞ and dσ̂/dp 2 ≈ 1/p 4 → ∞ for p → 0.
dx f (x, p⊥
⊥
⊥
⊥
Requires empirical dampening at small p⊥ ,
owing to colour screening (proton finite size).
Many aspects beyond pure theory ⇒ model building.
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 27/40
Hadronization
Nonperturbative =⇒ not calculable from first principles!
Model building = ideology + “cookbook”
Two common approaches:
String fragmentation:
most ideological
DELPHI Interactive Analysis
Beam: 45.6 GeV
Proc: 4-May-1994
Run: 39265
Evt: 4479
DAS : 5-Jul-1993
14:16:48
Scan: 3-Jun-1994
Act
Deact
TD
TE
TS
TK
TV
ST
PA
95 173
0
38
0
0
0
(145) (204) ( 0) ( 38) ( 0) ( 0) ( 0)
0
0
0
0
0
0
0
( 0) ( 20) ( 0) ( 42) ( 0) ( 0) ( 0)
Cluster fragmentation:
simplest
Both contain many
parameters to be
determined from data,
preferably LEP e+ e− →
γ ∗ /Z0 → qq/qqg/ . . .
Y
Z
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
X
slide 28/40
The Lund String Model
In QCD, for large charge separation, field lines seem to be
compressed 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.
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 29/40
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!
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 30/40
The HERWIG Cluster Model
1
2
3
Introduce forced g → qq branchings
Form colour singlet clusters
Clusters decay isotropically to 2 hadrons according to
phase space weight
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 31/40
Event Generators
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 32/40
Generator Landscape
The Generator Landscape
General-Purpose
Specialized
MadGraph, AlpGen, . . .
Hard Processes
HERWIG
Resonance Decays
Parton Showers
Underlying Event
Hadronization
Ordinary Decays
HDECAY, . . .
PYTHIA
SHERPA
......
Ariadne/LDC, VINCIA, . . .
PHOJET/DPMJET
none (?)
TAUOLA, EvtGen
Specialized often best at given task, but need General-Purpose core
specialized often best at given task, but need General-Purpose core
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 33/40
The Workhorses: What are the Differences?
HERWIG, PYTHIA and SHERPA offer convenient frameworks
for 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++
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 34/40
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 and
data, based on RIVET, see http://mcplots.cern.ch/
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 35/40
The Bigger Picture
Need standardized interfaces: see next!
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 36/40
PDG Particle Codes
A. Fundamental objects
1
2
3
4
5
6
d
u
s
c
b
t
11
12
13
14
15
16
e−
νe
µ−
νµ
τ−
ντ
21
22
23
24
25
g
γ
Z0
W+
h0
32
33
34
35
36
Z0 0
Z00 0
W0 +
H0
A0
37
39
H+
Graviton
add − sign for antiparticle, where appropriate
B. Mesons
100 |q1 | + 10 |q2 | + (2s + 1) with |q1 | ≥ |q2 |
C. Baryons
1000 q1 + 100 q2 + 10 q3 + (2s + 1)
with q1 ≥ q2 ≥ q3 , or Λ-like q1 ≥ q3 ≥ q2
. . . and many more
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 37/40
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 and
output of Feynman rules
...
Torbjörn Sjöstrand
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 subtle
effects, generators must keep up by increased precision.
But it often happens that the physics simulations provided by
the Monte Carlo generators carry the authority of data itself.
They look like data and feel like data, and if one is not careful
they are accepted as if they were data.
J.D. Bjorken
from 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
Torbjörn Sjöstrand
Introduction to Monte Carlo Techniques in High Energy Physics – lecture 2
slide 39/40
Thank You —
— and Good Hunting!