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Monte Carlo Event Generators for the LHCor How to relate Theory with Experiment?
Steffen Schumann
ITP, University of Heidelberg
EMG Annual Retreat 2010
Bingen am Rhein
27. - 29.09. 2010
Introduction & Monte Carlo Techniques
Hard Processes at (Next-to-)Leading Order
Parton Showers & Matching with Fixed Order
Multiple Interactions, Hadronization & Hadron Decays
Steffen Schumann Monte Carlo Event Generators for the LHC
Disclaimer
These lectures will not cover:Heavy-ion collisionsSpecific aspects of heavy-flavour physicsBeyond the Standard Model physicsmost of the details, derivations, ...
Recommended reading:
Ellis, Stirling & Webber
“QCD and Collider Physics”
Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 8, 1 (1996)
Sjostrand, Mrenna & Skands
“PYTHIA 6.4 Physics and Manual”
JHEP 0605 (2006) 026
... papers cited on the way
arXiv:
hep-ph
/0603
175v
2 12
May
2006
Preprint typeset in JHEP style - HYPER VERSION hep-ph/0603175
LU TP 06–13
FERMILAB-PUB-06-052-CD-T
March 2006
PYTHIA 6.4 Physics and Manual
Torbjorn Sjostrand
Department of Theoretical Physics, Lund University,
Solvegatan 14A, S-223 62 Lund, Sweden
E-mail: [email protected]
Stephen Mrenna
Computing Division, Simulations Group, Fermi National Accelerator Laboratory,
MS 234, Batavia, IL 60510, USA
E-mail: [email protected]
Peter Skands
Theoretical Physics Department, Fermi National Accelerator Laboratory,
MS 106, Batavia, IL 60510, USA
E-mail: [email protected]
Abstract: The Pythia program can be used to generate high-energy-physics ‘events’,
i.e. sets of outgoing particles produced in the interactions between two incoming particles.
The objective is to provide as accurate as possible a representation of event properties in
a wide range of reactions, within and beyond the Standard Model, with emphasis on those
where strong interactions play a role, directly or indirectly, and therefore multihadronic
final states are produced. The physics is then not understood well enough to give an exact
description; instead the program has to be based on a combination of analytical results
and various QCD-based models. This physics input is summarized here, for areas such
as hard subprocesses, initial- and final-state parton showers, underlying events and beam
remnants, fragmentation and decays, and much more. Furthermore, extensive information
is provided on all program elements: subroutines and functions, switches and parameters,
and particle and process data. This should allow the user to tailor the generation task to
the topics of interest.
The code and further information may be found on the Pythia web page:
http://www.thep.lu.se/∼torbjorn/Pythia.html.
Keywords: Phenomenological Models, Hadronic Colliders, Standard Model, Beyond
Standard Model.
– 1 –
Steffen Schumann Monte Carlo Event Generators for the LHC
Outline Lecture 1
Motivation/Introduction
LHC challenges: highest energies & highest multiplicitiesA prototypical LHC event: the simulators point of viewThe case for Monte Carlo event generators
Basic Monte Carlo techniquesThe Hit-or-Miss method
importance samplingthe multi-channel approach
The Veto Algorithm
Hard-Process generation
The partonic hadron-hadron cross-section formulaMulti-leg tree-level matrix elementsOne-loop matrix-element calculations
Steffen Schumann Monte Carlo Event Generators for the LHC
Hadron-collision experiments
identify & measure all particles coming from individual pp clashes
⇒ to be confronted with hypotheses for the underlying physics
Steffen Schumann Monte Carlo Event Generators for the LHC
Hadron-collision experiments
identify & measure all particles coming from individual pp clashes
Steffen Schumann Monte Carlo Event Generators for the LHC
The LHC challenge
Objectives for the LHC era
reveal the mechanism of EWSB [discovery of the Higgs?, alternatives?]
search for physics beyond the SM: weak scale SUSY, ED, W’ & Z’, ...
Higgs Huntingobvious production & decay channels buried in QCD backgroundssearch for rare decays like h→ γγlook for associated production modes, e.g. hjj , W /Zh, tth
signals & backgrounds to be understood at high precision
Searching for New Physicslarge cross sections for producing new colored statessubsequent decays into SM final states + Xgeneric signals: #-leptons + #-jets + E/T
SM backgrounds: V+jets, VV+jets, tt+jets, QCD jets new physics encoded in energies, flavours, edges
Typically searches in (rare) multi-particle / multi-jet final states!
Detailed understanding of QCD jet production indispensable!
Steffen Schumann Monte Carlo Event Generators for the LHC
The LHC challenge
Objectives for the LHC era
reveal the mechanism of EWSB [discovery of the Higgs?, alternatives?]
search for physics beyond the SM: weak scale SUSY, ED, W’ & Z’, ...
Higgs Huntingobvious production & decay channels buried in QCD backgroundssearch for rare decays like h→ γγlook for associated production modes, e.g. hjj , W /Zh, tth
signals & backgrounds to be understood at high precision
Searching for New Physicslarge cross sections for producing new colored statessubsequent decays into SM final states + Xgeneric signals: #-leptons + #-jets + E/T
SM backgrounds: V+jets, VV+jets, tt+jets, QCD jets new physics encoded in energies, flavours, edges
Typically searches in (rare) multi-particle / multi-jet final states!
Detailed understanding of QCD jet production indispensable!
Steffen Schumann Monte Carlo Event Generators for the LHC
The LHC challenge
Objectives for the LHC era
reveal the mechanism of EWSB [discovery of the Higgs?, alternatives?]
search for physics beyond the SM: weak scale SUSY, ED, W’ & Z’, ...
Higgs Huntingobvious production & decay channels buried in QCD backgroundssearch for rare decays like h→ γγlook for associated production modes, e.g. hjj , W /Zh, tth
signals & backgrounds to be understood at high precision
Searching for New Physicslarge cross sections for producing new colored statessubsequent decays into SM final states + Xgeneric signals: #-leptons + #-jets + E/T
SM backgrounds: V+jets, VV+jets, tt+jets, QCD jets new physics encoded in energies, flavours, edges
Typically searches in (rare) multi-particle / multi-jet final states!
Detailed understanding of QCD jet production indispensable!
Steffen Schumann Monte Carlo Event Generators for the LHC
The LHC challenge
Objectives for the LHC era
reveal the mechanism of EWSB [discovery of the Higgs?, alternatives?]
search for physics beyond the SM: weak scale SUSY, ED, W’ & Z’, ...
Higgs Huntingobvious production & decay channels buried in QCD backgroundssearch for rare decays like h→ γγlook for associated production modes, e.g. hjj , W /Zh, tth
signals & backgrounds to be understood at high precision
Searching for New Physicslarge cross sections for producing new colored statessubsequent decays into SM final states + Xgeneric signals: #-leptons + #-jets + E/T
SM backgrounds: V+jets, VV+jets, tt+jets, QCD jets new physics encoded in energies, flavours, edges
Typically searches in (rare) multi-particle / multi-jet final states!
Detailed understanding of QCD jet production indispensable!
Steffen Schumann Monte Carlo Event Generators for the LHC
Theoretical Modelling
Monte Carlo Event Generators
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Hard interactionexact matrix elements |M|2
QCD bremsstrahlungparton showers in the initial and final state
Multiple Interactionsbeyond factorization: modelling
Hadronizationnon-perturbative QCD: modelling
Hadron Decaysphase space or effective theories
⇒ stochastic simulation of pseudo data⇒ fully exclusive hadronic final states⇒ direct comparison with experimental data, e.g. ATLAS, CMS, LHCb, DØ, CDF
modulo detector simulation
Herwig, Pythia, Sherpa
Steffen Schumann Monte Carlo Event Generators for the LHC
The case for Monte Carlo generators
theoretical & experimental studies of complex multi-particle physics
large flexibility of physical quantities that can be assessed
disseminate ideas from theorists to experimentalists
Applications I: theorists/experimentalists
predict event rates and topologies ⇒ can estimate feasibility
simulate possible backgrounds ⇒ can devise analysis strategies
Applications II: experimentalists [supplemented with detector simulation]
study detector requirements ⇒ can optimize detector/trigger design
study detector imperfections ⇒ can evaluate acceptance corrections
study detector response ⇒ can unfold data to particle level
Steffen Schumann Monte Carlo Event Generators for the LHC
The case for Monte Carlo generators
new analysis techniques
Searching for the boosted Higgs[Butterworth et al. Phys. Rev. Lett. 100 (2008) 242001]
consider pp → Vh with h→ bb
focus on Higgs with phT > 200 GeV
use subjet info to reduce QCD BG
pp → Vh with h→ bb resurrected
desperate new physics searches
New Physics in all-jet final states[Cilic, S., Son JHEP 0904 (2009) 128]
consider pp → π8π8 → 4jets
fight QCD 4jet background
mass window criterion for jet pairs
discovering color octets seems feasible
Steffen Schumann Monte Carlo Event Generators for the LHC
The case for Monte Carlo generators
The Atlas simulation framework [arXiv:1005.4568]
Steffen Schumann Monte Carlo Event Generators for the LHC
↘
LHC data is coming!
We are here!
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques
“Spatial” problems: without memory
What is the volume of a given body?
Pick a random point, with equal probability in this area.
What is the integrated cross section of a given process?
Pick an event at random, according to the differential cross section.
“Temporal” problems: have memory
Financial systems: What is the probability for a stock to have price X attime t, given the price Y at t0?
Parton shower: What is the probability for a parton to branch at a scale Q,given that it was created at a scale Q0?
In particle physics often combined problems
What is the probability for a parton to branch at Q, with the daughters sharingthe mother momentum in some specific way?
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques: “Spatial” problems
Assume function f (x) in range xmin ≤ x ≤ xmax,where f (x) ≥ 0 everywhere(in practise x is multi-dimensional)
Two standard tasks
1 Calculate (approximatively)
I =
xmax∫xmin
f (x ′)dx ′ = (xmax − xmin)〈f (x)〉
' IN = (xmax − xmin)1
N
N∑i=1
f (xi )± (xmax − xmin)
√〈f 2〉 − 〈f 〉2
N
Example: integrated cross section from the differential one
2 Select x at random according to f (x)
Example: given probability distribution from quantum mechanics
↙random points ∈ [xmin, xmax ]
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques: “Spatial” problems
Select x according to f (x)
as P(x) ∼f (x)∫0
dy = f (x)
equivalent to uniform selection of (x , y) in areaxmin ≤ x ≤ xmax & 0 ≤ y ≤ f (x)
x∫
xmin
f (x ′)dx ′ = #
xmax∫
xmin
f (x ′)dx ′
Analytical solution
known primitive function F (x) and inverse F−1
F (x)− F (xmin) = #(F (xmax)− F (xmin)) = #Atot
x = F−1(F (xmin) + #Atot)
Example: f (x) = e−x for x ≥ 0
F (x) = 1− e−x
1− e−x = # x = − ln(#)
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques: “Spatial” problems
The Hit-or-Miss method: basics
f (x) ≤ fmax in xmin ≤ x ≤ xmax
1 select x = xmin + #1(xmax − xmin)
2 select y = #2fmax
3 if y ≤ f (x) accept point [Nacc++, Ntry++]
if y > f (x) reject point [Nfail++, Ntry++]
As a byproduct the integral is given by:
I =
xmax∫
xmin
f (x)dx ' (xmax − xmin)fmaxNacc
Ntry= Atot
Nacc
Ntry
with its relative variance approaching:
δI
I−→ 1√
Nacc
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques: “Spatial” problems
Example: Battleships
Can we do better?
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques: “Spatial” problemsThe Hit-or-Miss method: importance sampling
f (x) ≤ g(x) in xmin ≤ x ≤ xmax
and G (x) =x∫
xmin
g(x ′)dx ′ & G−1(y) are simple
1 select x = according to distribution g(x)
2 select y = #g(x)
3 if y ≤ f (x) accept point [Nacc++, Ntry++]
if y > f (x) reject point [Nfail++, Ntry++]
Example: f (x) = xe−x and g(x) = Ne−x/2
f (x)
g(x)=
xe−x/2
N
!≤ 1→ find maximum
d
dx
(f (x)
g(x)
)=
1
N
(1− x
2
)e−x/2 !
= 0 x = 2
normalize such that g(2) = f (2) N = 2/e
from G (x) ∼ 1− e−x/2 = #1
x = −2 ln(#1)y = #2g(x) = #22e−(1+x/2)
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques: “Spatial” problems
The Hit-or-Miss method: multi-channel approach
f (x) ≤ g(x) =N∑i=1
αigi (x) in xmin ≤ x ≤ xmax,
where all gi are “nice”, αi ≥ 0 &N∑i=1
αi = 1
1 select i with relative probability αi
xmax∫
xmin
gi (x′)dx ′
2 select x according to gi (x)
3 select y = #g(x) = #N∑i=1
αigi (x)
4 if y ≤ f (x) accept point [Nacc++, Ntry++]
if y > f (x) reject point [Nfail++, Ntry++]
5 adjust apriori weights αi to minimize variance
self-adaptive multi-channel Monte Carlo
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques: “Temporal” problems
The radioactive decay problem
known probability f (t) that something will happen at time t
[nucleus decays, parton branches, transistor fails]
something happens at t only if it didn’t happen at t ′ < t
Define: N(t): probability that nothing happend until t [N(0) = 1]
P(t) = −dN(t)/dt = f (t)N(t): probability for decay at time t
Solution:
N(t) = exp
−
t∫
0
f (t ′)dt ′
P(t) = f (t) exp
−
t∫
0
f (t ′)dt ′
naıve answer P(t) = f (t) modified by exponential suppression
in parton-shower picture, N(t) is called the Sudakov form factor
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques: “Temporal” problems
If F (t) and F−1 exist: Standard solution to find “decay” time tt∫
0
P(t ′)dt ′ = N(0)− N(t) = 1− exp {−(F (t)− F (0))} = 1−#
t = F−1(F (0)− ln(#))
The Veto Algorithm
if f (t) has no simple F (t) or F−1: use “nice” g(t) ≥ f (t)
1 start with i = 0 and t0 = 0
2 increment i and select ti = G−1(G (ti−1)− ln(#1))
3 if f (ti )/g(ti ) ≤ #2 go back to step 2
otherwise, keep ti as “decay” time
next step only depends on the very previous one
Markov chain process
Steffen Schumann Monte Carlo Event Generators for the LHC
Basic Monte Carlo techniques: Summary
use Monte Carlo to perform integrals and sample distributions
only need few points to estimate feach additional point increases accuracy
techniques generalize to many dimensions
typical LHC phase space ∼ d3~p × 100’s particleserror scales as 1/
√N vs. 1/N2/d or 1/N4/d (Trapezoidal or Simpson’s Rule)
suitable for complicated integration regions
kinematic cuts or detector cracks
can sample distributions where exact solutions cannot be found
Veto Algorithm applied to parton shower
Steffen Schumann Monte Carlo Event Generators for the LHC
Hard-Process generation
Steffen Schumann Monte Carlo Event Generators for the LHC
The partonic hadron-hadron cross section
proton bound state of quarks and gluons
universal probability distributionof finding parton a with momentum pa = xaPfa(xa, µ
2F )→ Parton Density Function
↪→ partonic CM energy: s = xaxbspp
Steffen Schumann Monte Carlo Event Generators for the LHC
The partonic hadron-hadron cross section
�'
&
$
%|Mqq→e+e−|2
γ∗/Z0
fq(xq , µ2F ) fq(xq, µ
2F )
proton bound state of quarks and gluons
universal probability distributionof finding parton a with momentum pa = xaPfa(xa, µ
2F )→ Parton Density Function
factorization in hard and soft component (resummed in PDFs)
σpp→Xpart(s;µ2R, µ
2F) ≡
∑ab
∫dxadxb fa(xa, µ
2F)fb(xb, µ
2F) dσab→Xpart(s; {pX}, µ2
R, µ2F)
soft/collinear QCD emissions summed in fa(xa, µ2F) σ is inclusive!
hard production process described through
dσab→Xpart(s; {pX}, µ2R, µ
2F) ≡ |Mab→Xpart(s; {pX}, µ2
R, µ2F)|2 dΦX
↪→ from model Lagrangian, e.g. LSM , LMSSM , ...
allows for probabilistic generation of partonic events {pX} the center piece of each event need to model transition of partons to hadrons: Xpart → Xhad
Steffen Schumann Monte Carlo Event Generators for the LHC
The Hard Process: setting the scene
σpp→Xn =∑
ab
∫dxadxb fa(xa, µ
2F)fb(xb, µ
2F) |Mab→Xn |2 dΦn
generic features
high-dimensional phase space dim[Φn ] = 3n − 4
|Mab→Xn |2 wildly fluctuating over Φn
steep parton densities [parametrization]
state-of-the-art
tradionally based on LO 2→ 2 only
now dedicated 2→ n tree-level codes
extract Feynman rules from Lagrangian Lgenerate compact expressions for |M|2flexible phase-space integrators
↪→ ALPGEN, AMEGIC, COMIX, HELAC, MADGRAPH
automation of one-loop corrections on the way
L(B)SM
↓
↓|Mn|2dΦn
�fZ ′
µ
f
1
Steffen Schumann Monte Carlo Event Generators for the LHC
The Hard Process: setting the scene
σpp→Xn =∑
ab
∫dxadxb fa(xa, µ
2F)fb(xb, µ
2F) |Mab→Xn |2 dΦn
generic features
high-dimensional phase space dim[Φn ] = 3n − 4
dΦn = δ4
(pa + pb −
n∑
i=1
pi
)n∏
i=1
d3~pi(2π)32Ei
↪→ subject to non-trivial cuts �pa
pb
. . .
p1p2
pn−1
pn
|Mab→Xn |2 wildly fluctuating over Φn
steep parton densities [parametrization]
state-of-the-art
tradionally based on LO 2→ 2 only
now dedicated 2→ n tree-level codes
extract Feynman rules from Lagrangian Lgenerate compact expressions for |M|2flexible phase-space integrators
↪→ ALPGEN, AMEGIC, COMIX, HELAC, MADGRAPH
automation of one-loop corrections on the way
L(B)SM
↓
↓|Mn|2dΦn
�fZ ′
µ
f
1
Steffen Schumann Monte Carlo Event Generators for the LHC
The Hard Process: setting the scene
σpp→Xn =∑
ab
∫dxadxb fa(xa, µ
2F)fb(xb, µ
2F) |Mab→Xn |2 dΦn
generic features
high-dimensional phase space dim[Φn ] = 3n − 4
|Mab→Xn |2 wildly fluctuating over Φn
competing resonances [Breit-Wigner]enhanced phase-space regionsmomentum correlationsnumber of diagrams ∼ n! [costly]
steep parton densities [parametrization]
state-of-the-art
tradionally based on LO 2→ 2 only
now dedicated 2→ n tree-level codes
extract Feynman rules from Lagrangian Lgenerate compact expressions for |M|2flexible phase-space integrators
↪→ ALPGEN, AMEGIC, COMIX, HELAC, MADGRAPH
automation of one-loop corrections on the way
L(B)SM
↓
↓|Mn|2dΦn
�fZ ′
µ
f
1
Steffen Schumann Monte Carlo Event Generators for the LHC
The Hard Process: setting the scene
σpp→Xn =∑
ab
∫dxadxb fa(xa, µ
2F)fb(xb, µ
2F) |Mab→Xn |2 dΦn
generic features
high-dimensional phase space dim[Φn ] = 3n − 4
|Mab→Xn |2 wildly fluctuating over Φn
steep parton densities [parametrization]
10-4
10-3
10-2
10-1 1
x
0
0.5
1
1.5
2
2.5
3
3.5
4
xf(
x,Q
2)
u-quark
u-quarkgluon (*0.05)
Q2=(100 GeV)
2
MRST2002NLO
state-of-the-art
tradionally based on LO 2→ 2 only
now dedicated 2→ n tree-level codes
extract Feynman rules from Lagrangian Lgenerate compact expressions for |M|2flexible phase-space integrators
↪→ ALPGEN, AMEGIC, COMIX, HELAC, MADGRAPH
automation of one-loop corrections on the way
L(B)SM
↓
↓|Mn|2dΦn
�fZ ′
µ
f
1
Steffen Schumann Monte Carlo Event Generators for the LHC
The Hard Process: setting the scene
σpp→Xn =∑
ab
∫dxadxb fa(xa, µ
2F)fb(xb, µ
2F) |Mab→Xn |2 dΦn
generic features
high-dimensional phase space dim[Φn ] = 3n − 4
|Mab→Xn |2 wildly fluctuating over Φn
steep parton densities [parametrization]
state-of-the-art
tradionally based on LO 2→ 2 only
now dedicated 2→ n tree-level codes
extract Feynman rules from Lagrangian Lgenerate compact expressions for |M|2flexible phase-space integrators
↪→ ALPGEN, AMEGIC, COMIX, HELAC, MADGRAPH
automation of one-loop corrections on the way
L(B)SM
↓
↓|Mn|2dΦn
�fZ ′
µ
f
1
Steffen Schumann Monte Carlo Event Generators for the LHC
Phase-Space integration: a look into the tool box
build-up suitable channels during amplitude generation
|M|2 ≡∣∣∣∣∣
k∑
i=1
Ai
∣∣∣∣∣
2
=k∑
i=1
|Ai |2 +∑
i 6=j
AiA†j
phase-space map for each topology |Ai |2
�4
1
2
3=⇒
dΦ4 ∝ dφ12,34 d cos θ12,34
×ds12 dφ1,2 d cos θ1,2
×ds34 dφ3,4 d cos θ3,4
with sij = (pi + pj)2
use suitable distributions
use adaptive multi-channel technique to combine channels
Steffen Schumann Monte Carlo Event Generators for the LHC
Phase-Space integration: a look into the tool box
Example: Breit-Wigner propagator – particle with mass M and width Γ
mapping
s = MΓ tan ρ+ M2
ds = MΓ sec2 ρ dρ
yields
I =
smax∫
smin
ds
(s −M2)2 + M2Γ2
=
ρmax∫
ρmin
MΓ sec2 ρ dρ
M2Γ2 tan2 ρ+ M2Γ2
=1
MΓ
ρmax∫
ρmin
dρ δI = 0
60 70 80 90 100√s [GeV]
0.00
0.25
0.50
1
(s-M2)2+M
2Γ
2
Steffen Schumann Monte Carlo Event Generators for the LHC
Matrix-Element generation: a look into the tool box
Helicity amplitudes [e.g. Hagiwara, Zeppenfeld Nucl. Phys. B 274 (1986) 1]
express amplitudes Ai in terms of spinor products u(pi , λi )u(pj , λj)
for assigned helicities & momenta Ai ’s just complex numbers
can perform the sum of amplitudes before squaring
Feynman-diagram based
factor out common pieces
2→ 6 is feasible
Recursive approaches
exponential growth of complexity
2→ 8− 10 doable
n−1
n−2
2
1
Jµ =
n−1∑
i=2
i+2i+1
ii−1
1 2
n−1 n−2
V3
+
n−2∑
i=2j>i
j
j−1
i+2
i+1
1 2 i−1 i
n−1 n−2 j+2 j+1
V4
Steffen Schumann Monte Carlo Event Generators for the LHC
Tree-level matrix elements: state-of-the-art
LHC cross sections [√
s =14 TeV, generic cuts] [Gleisberg, Hoche JHEP 0812 (2008) 039]
σpp→XpartNumber of jets
Process + n QCD jets Order 0 1 2 3 4 5 6
e+νe [pb] α2EMα
nS 5434(5) 1274(2) 465(1) 183.0(6) 77.5(3) 33.8(1) 14.7(1)
e+e− [pb] α2EMα
nS 723.5(4) 187.9(3) 69.7(2) 27.14(7) 11.09(4) 4.68(2) 2.02(2)
e+e− + bb [pb] α2EMα
n+2S
18.86(3) 6.78(2) 3.10(5) 1.536(9) 2.47(2) 1.28(2)
tt [pb] αn+2S
754.8(8) 745(1) 518(1) 309.8(8) 170.4(7) 89.2(4) 44.4(4)
bb [µb] αn+2S
471.2(5) 8.83(2) 1.813(8) 0.459(2) 0.150(1) 0.0531(5) 0.0205(4)
σpp→XpartNumber of jets
Process + n QCD jets Order 0 1 2 3 4 5 6 7 8
pure jets [µb] αnS – – 331.0(4) 22.72(6) 4.95(2) 1.232(4) 0.352(1) 0.1133(5) 0.0369(3)
parton-level event generation
+ fully differential [{pXpart} any distribution can be studied]
+ full momentum/spin correlations
+ theory/pheno analyses
− limited in final-state multiplicity [factorial growth of Feynman diagrams / phase-space maps]
− not directly observable / measurable
− Leading-Order dependence on µ2F & µ2
R
NLO QCD precision desirable
Steffen Schumann Monte Carlo Event Generators for the LHC
Hard Processes at Next-to-Leading Order QCD
The need for higher precision
derive exclusion bounds from a given measurement
non-observation of a given hypothesis, e.g. light Higgs, BSM modelcrucial to know the corresponding production cross sections preciselyderive limits on particle masses, model parameters
counting experiments to extract certain parameters, couplings
stop mass bounds from Tevatron[from the early days, T. Plehn private communication]
10-1
1
100 110 120 130 140 150
∆MLO
∆MNLO
σNLO
σLO
: µ=[m(t)/2 , 2m(t)]
: µ=[m(t)/2 , 2m(t)]
m(t1) [GeV]
σ(pp– → t
1t−
1 + X) · BR(t
1t−
1 → ee + ≥ 2j) [pb]
CDF 95% C.L. upper limit (prel.)
m(χ1o)=m(t
1)/2
Higgs coupling extraction
σ(gg → h→ γγ) ∼ ΓgΓγΓ
σ(gg → h→WW ∗) ∼ ΓgΓW
Γ
σ(qq → qqh, h→ ττ) ∼ ΓW ΓτΓ
...
Γi ∼ giih, Γg ∼ gtth theoretical uncertainties dominate
systematics reduced for σ ratios
Steffen Schumann Monte Carlo Event Generators for the LHC
Hard Processes at Next-to-Leading Order QCDAnatomy of NLO QCD calculations
�γ∗/Z0
�γ∗/Z0
�γ∗/Z0
σNLO2→n =
∫
n+1
d(d)σR +
∫
n
d(d)σV +
∫
n
d(4)σB
real-emission σR IR divergent
(UV renormalized) virtual-corrections σV IR divergent
for IR safe observables sum is finite
Dipole subtraction method [Catani, Seymour Nucl. Phys. B 485 (1997) 291]
σNLO2→n =
∫
n+1
[d(4)σR−d(4)σA
]+
∫
n
[d(4)σB +
∫
loop
d(d)σV+
∫
1
d(d)σA
]
ε=0
subtraction terms yield local approximation for the real emission process
describe the amplitude in the soft & collinear limits [1/ε and 1/ε2 poles]∫
n+1
d(d)σA =∑
dipoles
∫
n
d(d)σB ⊗∫
1
d(d)Vdipole
↪→ universal dipole terms←↩spin- & color correlations
Steffen Schumann Monte Carlo Event Generators for the LHC
Hard Processes at Next-to-Leading Order QCDAnatomy of NLO QCD calculations
�γ∗/Z0
�γ∗/Z0
�γ∗/Z0
σNLO2→n =
∫
n+1
d(d)σR +
∫
n
d(d)σV +
∫
n
d(4)σB
real-emission σR IR divergent
(UV renormalized) virtual-corrections σV IR divergent
for IR safe observables sum is finite
Dipole subtraction method [Catani, Seymour Nucl. Phys. B 485 (1997) 291]
σNLO2→n =
∫
n+1
[d(4)σR−d(4)σA
]+
∫
n
[d(4)σB +
∫
loop
d(d)σV+
∫
1
d(d)σA
]
ε=0
subtraction terms yield local approximation for the real emission process
describe the amplitude in the soft & collinear limits [1/ε and 1/ε2 poles]∫
n+1
d(d)σA =∑
dipoles
∫
n
d(d)σB ⊗∫
1
d(d)Vdipole
↪→ universal dipole terms←↩spin- & color correlations
Steffen Schumann Monte Carlo Event Generators for the LHC
Hard Processes at Next-to-Leading Order QCD
The emerging picture
σNLO2→n =
∫
n+1
[d(4)σR−d(4)σA
]+
∫
n
[d(4)σB +
∫
loop
d(d)σV+
∫
1
d(d)σA
]
ε=0
Monte-Carlo codes
all the tree-level bits
subtraction of singularities∗
efficient phase-space integration
One-Loop codes
Loop amplitudes, i.e. 2<(AVA†B)
Loop integration
1/ε, 1/ε2 coefficients & finite terms
∗ automatedCatani-Seymour subtraction in:
Amegic/Sherpa [Gleisberg, Krauss Eur. Phys. J. C 53 (2008) 501]
MadDipole/MadGraph [Frederix, Gehrmann, Greiner JHEP 1006 (2010) 086]
Helac [Czakon, Papdopoulos, Worek JHEP 0908 (2009) 085]
Frixione-Kunszt-Signer subtraction [Nucl. Phys. B 467 (1996) 399] in:
MadFKS/MadGraph [Frederix et al. JHEP 0910 (2009) 003]
Steffen Schumann Monte Carlo Event Generators for the LHC
Hard Processes at Next-to-Leading Order QCDExample: W + 3jets @ NLO
real emission corrections:
g
q
gg g
qν
g
e
q′ q′
g
ν
Q
Q
e
q ′
q
ν
Q2Q1
Q1
Q2
e
one-loop corrections:
q
q′ν
e
g
Q Qg
q
q′ν
e
g
g g
q′
qν
e
g
g
q
g q′
g
g
ν
e
qν
e
Q Q
q′
g
Steffen Schumann Monte Carlo Event Generators for the LHC
Hard Processes at Next-to-Leading Order QCDExample: W + 3jets @ NLO
real emission corrections:
g
q
gg g
qν
g
e
q′ q′
g
ν
Q
Q
e
q ′
q
ν
Q2Q1
Q1
Q2
e
one-loop corrections:
q
q′ν
e
g
Q Qg
q
q′ν
e
g
g g
q′
qν
e
g
g
q
g q′
g
g
ν
e
qν
e
Q Q
q′
g
recently calculated using BlackHat+Sherpa[Berger et al. Phys. Rev. Lett. 102 (2009) 222001, Phys. Rev. D 80 (2009) 074036 ]
BlackHat: on-shell methods for one-loop amplitudes [arXiv:0808.0941]
Sherpa:
real-emission processesCatani–Seymour subtraction termsphase-space integration
⇒ also completed: Z/γ∗ + 3jets [arXiv:1004.1659] & W + 4jets [arXiv:1009.2338]
Steffen Schumann Monte Carlo Event Generators for the LHC
NLO QCD calculations: W + 3jets
BlackHat+Sherpa: Tevatron results [Phys. Rev. D 80 (2009) 074036]
consider W → eν and SISCone jets with E nth-jetT > 25 GeV & R=0.4
# of jets CDF LO NLO
1 53.5± 5.6 41.40(0.02)+7.59−5.94 57.83(0.12)+4.36
−4.00
2 6.8± 1.1 6.159(0.004)+2.41−1.58 7.62(0.04)+0.62
−0.86
3 0.84± 0.24 0.796(0.001)+0.488−0.276 0.882(0.005)+0.057
−0.138
20 40 60 80 100 120 140 160 180 200
10-4
10-3
10-2
10-1
100
dσ
/ d
ET [
pb /
GeV
]
LONLOCDF data
20 40 60 80 100 120 140 160 180 200
Second Jet ET [ GeV ]
0.5
1
1.5
2 LO / NLOCDF / NLO
NLO scale dependence
W + 2 jets + X
BlackHat+Sherpa
LO scale dependence
ET
jet > 20 GeV, | η
jet | < 2
ET
e > 20 GeV, | η
e | < 1.1
ET
/ > 30 GeV, MT
W > 20 GeV
R = 0.4 [siscone]
√s = 1.96 TeV
µR = µ
F = E
T
W
20 30 40 50 60 70 80 90
10-3
10-2
10-1
dσ
/ d
ET [
pb /
GeV
]
LONLOCDF data
20 30 40 50 60 70 80 90
Third Jet ET [ GeV ]
0.5
1
1.5
2 LO / NLOCDF / NLO
NLO scale dependence
W + 3 jets + X
BlackHat+Sherpa
LO scale dependence
ET
jet > 20 GeV, | η
jet | < 2
ET
e > 20 GeV, | η
e | < 1.1
ET
/ > 30 GeV, MT
W > 20 GeV
R = 0.4 [siscone]
√s = 1.96 TeV
µR = µ
F = E
T
W
Steffen Schumann Monte Carlo Event Generators for the LHC
NLO QCD calculations: W + n-jets
BlackHat+Sherpa: LHC predictions for√s = 7 TeV [arXiv:1009.2338]
Number of jetsσpp→e−νe+n−jets 1 2 3 4
σLO [pb] 264.4(0.2)+22.6−21.4 73.14(0.09)+20.81
−14.92 17.22(0.03)+8.07−4.95 3.81(0.01)+2.44
−1.34
σNLO [pb] 331(1)+15−12 78.1(0.5)+1.5
−4.1 16.9(0.1)+0.2−1.3 3.56(0.03)+0.08
−0.30
50 100 150
10-4
10-3
10-2
10-1
100
dσ
/ d
pT [
pb /
GeV
]
LONLO
50 100 150
First Jet pT [ GeV ]
0.5
1
1.5
2
LO / NLO
50 100 150
50 100 150
Second Jet pT [ GeV ]
50 100 150
50 100 150
Third Jet pT [ GeV ]
50 100 150
10-4
10-3
10-2
10-1
100
50 100 150
Fourth Jet pT [ GeV ]
0.5
1
1.5
2
NLO scale dependence BlackHat+SherpaLO scale dependence
pT
jet > 25 GeV, | η
jet | < 3
ET
e > 20 GeV, | η
e | < 2.5
ET
ν > 20 GeV, M
T
W > 20 GeV
R = 0.5 [anti-kT]
√s = 7 TeV
µR = µ
F = H
T
^ ’ / 2
W- + 4 jets + X
Steffen Schumann Monte Carlo Event Generators for the LHC
Hard Process generation: Summary
Automated tools to generate and evaluate tree-level amplitudes
compact matrix-element expressionsefficient phase-space integrators
Enormous progress in automating NLO calculations
modularity of the problemautomated subtraction terms + improved loop methodsW /Z + 3jets, W + 4jets with BlackHat+Sherpatt + 2jets with Helac-NLO [Bevilacqua et al. Phys. Rev. Lett. 104 (2010) 162002]
ttbb with Helac-NLO [Bevilacqua et al. JHEP 0909 (2009) 109]
e+e− → 5jets with MadFKS/MadGraph [Frederix et al. arXiv:1008.5313 [hep-ph]]
Steffen Schumann Monte Carlo Event Generators for the LHC
↘Wanna reality check???
Steffen Schumann Monte Carlo Event Generators for the LHC