Parton Showers and theirmatching/merging
with fixed-order computations
Rikkert Frederix
Lund University

Iwate Collider School 2024, Feb. 26 - Mar. 2, Appi highland, Iwate, Japan

An LHC collision, factorised

3FLPNNFOEFSBEGÊSHWBSJBOU

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·

·

7JUOFHBUJW··Hard Interaction
Resonance Decays
MECs, Matching & Merging
FSR
ISR*
QED
Weak Showers
Hard Onium
Multiparton Interactions

dˆ0

MPI

··

Figure taken from Bierlich et al., 2022 (Pythia8.3 manual)

MPI

Meson
BaryonAntibaryon· Heavy Flavour

Beam Remnants*
Strings
Ministrings / Clusters
Colour Reconnections
String Interactions
Bose-Einstein & Fermi-Dirac
Primary Hadrons
Secondary Hadrons
Hadronic Reinteractions(*: incoming lines are crossed)2

Hadronisation

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7JUOFHBUJW• The shower stops if all partons are characterised by a scale atthe IR cut-off: Q0 ~ 1 GeV• Physically, we observe hadrons, not (coloured) partons• We need a non-perturbative model in passing from partons tocolourless hadrons• There are two models, based on physical andphenomenological considerations

3

Cluster model

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7JUOFHBUJWThe structure of the perturbative evolution including angular ordering,leads naturally to the clustering in phase-space of colour-singlet parton
pairs (pre-confinement). Long-range correlations are strongly
suppressed. Hadronisation will only act locally, on low-mass colour
singlet clusters.

e-

e+

4

Lund
stringmodel
Quark antiquark color potential and string modelė¿ėΡ

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7JUOFHBUJWI>>Eƒ

From lattice
QCD one sees that the colour confinement potential of a
From lattice QCD one sees that the color confinement potential of a quark-antiquark
2 ∼ kr, with k ~quark-antiquark
linearlywith
theirdistance:
V(r)
Quark
color theirpotential
and string
growsantiquark
linearlygrows
withdistance:
V (r )model⇠ kr , withk ⇠ 0.2 GeV. This is modeled with astringwith(energy
unitpotential
length)ka quark-antiquarkthat getstension
stretched between
the qq̄
0.2 GeV,This
isuniform
modelled
a perstringwithofuniform(energyFromlattice
QCD
one seestensionthat thewith
color confinement
pair.linearly with their distance: V (r ) ⇠ kr , with k ⇠ 0.2 GeV . This is modeled with agrows
per unit
length) k that gets stretched between the qq ̄ pair.
string with uniform tension (energy per unit length) k that gets stretched between the qq̄
f&î  Ρ Ρ °Ρ LŅ  Ρ Ρ Ρ Ρ cΡ  Ρ 4  4Ρ
² ! ²5Ρt! Ρ  Ρ²  ΡΡ² ΡΡΡ Ρ!² Ρ L¸Ņ
ʉ.ΡΡ Ρ Ρ.ΡŅ ! Ρ Ρ.Ρ Ρ Ρ.ΡDĚΡňΡΡ ΡΡ ΡΡ ̴Ρ ÂΡΡΡΡ[ΡƜƭɊʺɕΡ ΡΡΡ ħ Ρ ΡΡ Ρ ΡOΡ[ΡΡ5ΡfΡΡ Ρ  Ρ£v}þΡ¿~þΡ¿«½Ρ Ρ
I>>Eƒė¿ėΡ Ρ ΡΡ² ΡΡ Ρ ΡΡ ΡΡ[ΡΡΡΡf&î  Ρ Ρ °Ρ LŅ  ,ΡΡ Ρ Ρ
Ρ 4 
4Ρ Ρ ²! Ρ !² ΡΡ  !Ρ.Ρ  5Ρ Ρ cΡΡ! Ρ
!3Ρ!
² ! ²5Ρt! Ρ  Ρ²  ΡΡ² ΡΡΡ Ρ!² Ρ L¸Ņʘ  Ρ Ρ  Ρ ΡΡ[Ρ Ρ ΡͯΡ ΡΡ
ʉ.ΡΡ Ρ Ρ.ΡŅ ! Ρ Ρ.Ρ Ρ Ρ.ΡDĚΡňΡΡ ΡΡ ΡΡœΡΡΡ[ΡΡ  Ρ[Ρ ΡΡ ΡΡΡΡ ̴Ρ ÂΡΡΡΡ[ΡƜƭɊʺɕΡΡΡΡΡ ΡΡ Ρ Ρ
2 ħ
H Ρ[ ΡΡ
ΡΡ ŹȟÖ­ Ρ5ΡĆ Ρ Ρ  Ρ Ρ£v}þΡ¿~þΡ¿«½ΡΡOΡ[ΡΡ5ΡfΡΡ Ρ ΡΡ² ΡΡ Ρ
ΡΡ
ΡΡ[ΡΡΡΡ4ΡΡΡ²²²Ρ ! !!Ρ4Ρ>Ρ Ρ Ρu,Ρ   Ρ! Ρ !3Ρ! ΡΡ
²! Ρ !²ΡΡ
 !Ρ.Ρ
 5ΡΡΡ
Ρ[ΡΡ Ρ.3Ρ[ Ρʘ  Ρ Ρ  Ρ ΡΡ[Ρ Ρ ΡͯΡ ΡΡ  -. 7KΡp ΡΡ 4Ρ ̼Ρ Ρ ȆĀ  Ρ Ρ Ρ ý Ρ[Ρ ΡΡ ΡΡΡ ΡœΡΡΡ[ΡΡ Ρ £««JΡΡ «¿þΡΡΡ  Ρ 4Ρ  Ρ   Ρ §=Ρ 4 ΡH Ρ[  Ρ ΡΡ Źȟ[Ö­ Ρ5ΡĆΡ Ρ4Ρ Ρ Ρ²²²Ρ [ΡΡHΡΡΡdΡsΡ! !!Ρ4Ρ>Ρ Ρ Ρu'K

Annu. Rev. Nucl. Part. Sci. 1986.36:253-286. Downloaded from arjournals.annualreviews.org
by Ecole Polytechnique Federal Lusanne on 12/21/09. For personal use only.

pair.

Annu. Rev. Nucl. Part. Sci. 1986.36:253-286. Downloaded from arjournals.annualreviews.org
by Ecole Polytechnique Federal Lusanne on 12/21/09. For personal use only.

Ρ Ρ Ρ Ρ[ΡΡ  Ρ.3Ρ[ Ρ -. 7K p Ρ  Ρ 4 Ρ ̼Ρ Ρ ȆĀ  Ρ Ρ Ρ ý[ Ρ  Ρ £««JΡ «¿þΡ ΡΡ  Ρ 4Ρ  Ρ   Ρ §=Ρ 4 Ρ[ΡΡHΡΡΡdΡsΡ 'K?ŜQaƄOƒ   ƒ !" ū@ĭū@ū "Y|ū@4l;4ū /@)5ū YūĮ)Õ Y45 4gū?ŜQaƄOƒ   ƒ !" ū@ĭū@ū "Y|ū@4l;4ū /@)5ū YūĮ)Õ Y45 4gūAt a certain point it becomes energetically favorable to break the string in two by
creating a new qq̄ pair in the middle of the string.

When quark-antiquarks
are too
far apart,it becomesenergeticallyAt a certain point it becomesenergeticallyfavorable
to break thestring in two by morefavourable
to breakby creatinga new qq ̄ pair in the middle.
creating
a new qq̄thepairstringin the middleof the string.
Paolo Torrielli (EPFL)

Paolo Torrielli (EPFL)

Interfacing NLO with Parton Showers

ThinkTank on Physics @ LHC

Interfacing NLO with Parton Showers

38 / 83

ThinkTank on Physics @ LHC

38 / 83

5

An LHC collision, factorised

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

·

·

7JUOFHBUJW··Hard Interaction
Resonance Decays
MECs, Matching & Merging
FSR
ISR*
QED
Weak Showers
Hard Onium
Multiparton Interactions

dˆ0

MPI

··

Figure taken from Bierlich et al., 2022 (Pythia8.3 manual)

MPI

Meson
BaryonAntibaryon· Heavy Flavour

Beam Remnants*
Strings
Ministrings / Clusters
Colour Reconnections
String Interactions
Bose-Einstein & Fermi-Dirac
Primary Hadrons
Secondary Hadrons
Hadronic Reinteractions(*: incoming lines are crossed)

6

Exclusive observable

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7JUOFHBUJW?

A parton shower program associates one of the possible histories(and pre-histories in case of pp collisions) of an hard event in an
explicit and fully detailed way, such that the sum of the probabilities ofall possible histories is unity.

7

Parton Shower Monte-Carloevent generators

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7JUOFHBUJWA parton shower program associates one of the possible histories(and pre-histories in case of pp) of an hard event in an explicit andfully detailed way, such that the sum of the probabilities of all possible
histories is unity.
• General-purpose tools• Always the first experimental choice• Complete exclusive description of the events: hard scattering,
showering & hadronisation (and underlying event)• Reliable and well-tuned tools• Significant and intense progress in the development of new
showering algorithms with the final aim to go beyond (N)LL in QCD
8

Pythia, Herwig & Sherpa

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7JUOFHBUJW• Significant differences between shower implementations(choice of evolution variable and kernel, momentum mappings, phasespace boundaries, massive quarks, photon emissions, etc.)• All are tuned to data, and describe it reasonably well(typically better than expected from their formal accuracy)• Some are (formally) more correct than others• However, not easy to assess accuracy for a generalobservable• Assessment (and improvement!) of formal accuracy is anactive field of research9Improving Parton Showers
LO merging

Collinear approximation

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7JUOFHBUJW• The parton shower is correct in the collinear limit• But we use it also outside of this limit

dσ/dPT (pb/bin)• Induces great dependence on the details of theimplementationtt (a la Pythia)

10

PT of the 2-nd extra jet
1

10-1

Q2 (wimpy)
Q2 (power)

10-2

P2T (wimpy)
P2T (power)

10-3 0

50

100

150

200

250

300

350

400
GeV

11

Matrix elements vs. Parton showers

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ME

1.
Fixed order calculation
2.
Computationally expensive
3. Limited number of particles
4. Valid when partons are hard andwell separated
5. Quantum interference correct
6. Needed for multi-jet description

7JUOFHBUJWShower 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 throughangular ordering
6.
Needed for hadronisation

Approaches are complementary: merge them!

Avoid double counting, ensure smooth distributions
12

Possible double counting

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7JUOFHBUJWMatrix elements

Parton shower

13

Possible double counting

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7JUOFHBUJWMatrix elements

Parton shower

13

Possible double counting

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7JUOFHBUJWMatrix elements

Parton shower

13

Possible double counting

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7JUOFHBUJWMatrix elements

Parton shower

13

Possible double counting

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7JUOFHBUJWMatrix elements

Parton shower

13

Possible double counting

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7JUOFHBUJWMatrix elements

Parton shower

13

Possible double counting

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJWMatrix elements

Parton shower

13

Possible double counting

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJWMatrix elements

Parton shower

13

Possible double counting

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7JUOFHBUJWMatrix elements

Parton shower
...
...

...
...

13

Possible double counting

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7JUOFHBUJWMatrix elements

Parton shower
...
...

...
...

Possible double counting
between partons frommatrix elements and parton
shower easily avoided byapplying a cut in phase
space
13

Possible double counting

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7JUOFHBUJWMatrix elements

Parton shower
kT < tcut
kT < tcut
kT < tcut
kT > Qc

kT > tcut

kT > tcut
kT > tcut

...

...

kT < tcut

...

...

Possible double counting
between partons frommatrix elements and parton
shower easily avoided byapplying a cut in phase
space
13

Merging ME with PS

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7JUOFHBUJW• So double counting no problem, but what about getting smooth
distributions that are independent of the precise value of tcut?
• Below cutoff, distribution is given by PS- need to make ME look like PS near cutoff• Let’s take another look at the PS!

14

Merging ME with
PS
t
cut

t1

e-

t2

Q2

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7JUOFHBUJW

tcut
tcut

e+
tcut

• How does the PS generate the configuration above (i.e. starting frome+e- -> qqbar events)?
• Probability for the splitting at t1 is given by2
2 ↵s (t1 )( q (Q , t1 ))Pgq (z)2⇡and for the whole tree↵s (t2 )22
2 ↵s (t1 )( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))Pgq (z)Pqg (z 0 )
2⇡
2⇡
15

Merging ME with
PS
t
cut

t1

e-

t2

Q2

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7JUOFHBUJW

tcut
tcut

e+
tcut

• How does the PS generate the configuration above (i.e. starting frome+e- -> qqbar events)?
• Probability for the splitting at t1 is given by2
2 ↵s (t1 )( q (Q , t1 ))Pgq (z)2⇡and for the whole tree↵s (t2 )22
2 ↵s (t1 )( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))Pgq (z)Pqg (z 0 )
2⇡
2⇡
15

Merging ME with
PS
t
cut

t1

e-

t2

Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJW

tcut
tcut

e+
tcut

• How does the PS generate the configuration above (i.e. starting frome+e- -> qqbar events)?
• Probability for the splitting at t1 is given by2
2 ↵s (t1 )( q (Q , t1 ))Pgq (z)2⇡and for the whole tree↵s (t2 )22
2 ↵s (t1 )( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))Pgq (z)Pqg (z 0 )
2⇡
2⇡
15

Merging ME with
PS
t
cut

t1

e-

t2

Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJW

tcut
tcut

e+
tcut

• How does the PS generate the configuration above (i.e. starting frome+e- -> qqbar events)?
• Probability for the splitting at t1 is given by2
2 ↵s (t1 )( q (Q , t1 ))Pgq (z)2⇡and for the whole tree↵s (t2 )22
2 ↵s (t1 )( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))Pgq (z)Pqg (z 0 )
2⇡
2⇡
15

Merging ME with
PS
t
cut

t1

e-

t2

Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW

tcut
tcut

e+
tcut

• How does the PS generate the configuration above (i.e. starting frome+e- -> qqbar events)?
• Probability for the splitting at t1 is given by2
2 ↵s (t1 )( q (Q , t1 ))Pgq (z)2⇡and for the whole tree↵s (t2 )22
2 ↵s (t1 )( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))Pgq (z)Pqg (z 0 )
2⇡
2⇡
15

Merging ME with
PS
t
cut

t1

e-

t2

Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJW

tcut
tcut

e+
tcut

• How does the PS generate the configuration above (i.e. starting frome+e- -> qqbar events)?
• Probability for the splitting at t1 is given by2
2 ↵s (t1 )( q (Q , t1 ))Pgq (z)2⇡and for the whole tree↵s (t2 )22
2 ↵s (t1 )( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))Pgq (z)Pqg (z 0 )
2⇡
2⇡
15

Merging ME with
PS
t
cut

t1

e-

t2

Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJW

tcut
tcut

e+
tcut

• How does the PS generate the configuration above (i.e. starting frome+e- -> qqbar events)?
• Probability for the splitting at t1 is given by2
2 ↵s (t1 )( q (Q , t1 ))Pgq (z)2⇡and for the whole tree↵s (t2 )22
2 ↵s (t1 )( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))Pgq (z)Pqg (z 0 )
2⇡
2⇡
15

Merging ME with
PS
t
cut

t1

e-

t2

Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJW

tcut
tcut

e+
tcut

• How does the PS generate the configuration above (i.e. starting frome+e- -> qqbar events)?
• Probability for the splitting at t1 is given by2
2 ↵s (t1 )( q (Q , t1 ))Pgq (z)2⇡and for the whole tree↵s (t2 )22
2 ↵s (t1 )( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))Pgq (z)Pqg (z 0 )
2⇡
2⇡
15

Merging ME with
PS
t
cut

t1

e-

t2

Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJW

tcut
tcut

e+
tcut

(

2

q (Q , tcut ))2

g (t1 , t2 )(q (t2 , tcut ))2 ↵s (t1 )↵s (t2 )Pgq (z)Pqg (z 0 )
2⇡
2⇡

16

Merging ME with
PS
t
cut

t1

e-

t2

Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJW

tcut
tcut

e+
tcut

(

2

q (Q , tcut ))2

g (t1 , t2 )(q (t2 , tcut ))2 ↵s (t1 )↵s (t2 )Pgq (z)Pqg (z 0 )2⇡
2⇡

Leading Logarithmic approximation of the matrix element
BUT with αs evaluated at the scale of each splitting

16

Merging ME with
PS
t
cut

t1

e-

t2

Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJW

tcut
tcut

e+
tcut

(

2

q (Q , tcut ))2

g (t1 , t2 )(q (t2 , tcut ))2 ↵s (t1 )↵s (t2 )Pgq (z)Pqg (z 0 )2⇡
2⇡

Leading Logarithmic approximation of the matrix element
BUT with αs evaluated at the scale of each splitting
Sudakov suppression due to disallowing additional radiation
above the scale tcut
16

Merging ME with PS

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJWee+|M|2 (ŝ, p3 , p4 , ...)

To get an equivalent treatment of the corresponding matrix element,
do as follows:
1. Cluster the event using some clustering algorithm- this gives us a corresponding “parton shower history”
2. Reweight αs in each clustering vertex with the clustering scale2
2 ↵s (t1 ) ↵s (t2 )|M| ! |M|↵s (Q2 ) ↵s (Q2 )3. Use some algorithm to apply the equivalent Sudakov
2
2
2
suppression( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))17

Merging ME with PS
e

e+

t1
Q2

3FLPNNFOEFSBEGÊSHWBSJBOU

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7JUOFHBUJWt2|M|2 (ŝ, p3 , p4 , ...)

To get an equivalent treatment of the corresponding matrix element,
do as follows:
1. Cluster the event using some clustering algorithm- this gives us a corresponding “parton shower history”
2. Reweight αs in each clustering vertex with the clustering scale2
2 ↵s (t1 ) ↵s (t2 )|M| ! |M|↵s (Q2 ) ↵s (Q2 )3. Use some algorithm to apply the equivalent Sudakov
2
2
2
suppression( q (Q , tcut )) g (t1 , t2 )( q (t2 , tcut ))17

Merging ME with PS

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWCKKW (2004) and MLM (2004)• In summary:• Double counting no problem: we simply throw events away when thematrix-element partons are too soft, or when the parton shower
generates too hard radiation• Applying the matrix-element cut is easy: duringphase-space integration, we only generate eventswith partons above the matching scalekT < tcut

kT > tcut• For the cut on the shower, there are two methods. Throwing events away
after showering is not very efficient, although it is working (“MLMmethod”)• Instead we can also multiply the Born matrix elements by suitable productof Sudakov factors (i.e. the no-emission probabilities) Δ(Qmax, tcut) andstart the shower at the scale tcut (“CKKW method”).
• For a given multiplicity we have<latexit sha1_base64="C/jSNbwVVUWDAHTsvWHyI8fs69U=">AAACT3icdVFLaxsxGNS6jyTuy2mOvYiYgguuWQc3SQ+BkOaQQyEJ2EnA6y5a+VtbWNIu0rclZtE/zKW99W/0kkNLqOxuSvoaEIxmRujTKMmlsBiGX4LavfsPHq6srtUfPX7y9Flj/fmZzQrDYcAzmZmLhFmQQsMABUq4yA0wlUg4T2bvFv75RzBWZLqP8xxGik20SAVn6KW4kUZWTBSLS92OFMOpUSVccunch/J2//7Yub2DWEf9KSBrzeKy39buNca/ErxA515FhyCRxbp12v7LixvNsBMuQe+Q3tabt+E27VZKk1Q4iRufo3HGCwUauWTWDrthjqOSGRRcgqtHhYWc8RmbwNBTzRTYUbnsw9GXXhnTNDN+aaRL9e6Jkilr5yrxycWU9k9vIf7LGxaY7o5KofMCQfOfF6WFpJjRRbl0LAxwlHNPGDfCz0r5lBnG0X9B3Zdw+1L6f3K21elud3qnveb+QVXHKnlBNkmLdMkO2SdH5IQMCCdX5Cv5Rr4Hn4Lr4KZWRWtBRTbIb6it/QCCkbcW</latexit>LOn,excl = Bn ⇥(kT,ntcut )n (Q, tcut )18Collinear approximation

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Dependence on internal parameters reduced
dσ/dPT (pb/bin)• greater predictive power!
tt (a la Pythia)

10

PT of the 2-nd extra jet
1

10-1

Q2 (wimpy)
Q2 (power)

10-2

P2T (wimpy)
P2T (power)

10-3 0

50

100

150

200

250

300

350

400
GeV

19

Collinear approximation

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Dependence on internal parameters reduced
dσ/dPT (pb/bin)• greater predictive power!
tt+0,1,2,3 partons + Pythia (MMLM)

10

PT of the 2-nd extra jet
1

10-1

Q2 (wimpy)
Q2 (power)

10-2

PT2 (wimpy)
PT2 (power)

10-3 0

50

100

150

200

250

300

350

400
GeV

20

NL
dsQC

20

50

Merging results
100

200

500
d01 [GeV]

pp ! ` n̄ + 0,1,2 j @ 13 TeV

10 4

Qcut = 10 GeV
Qcut = 15 GeV
Qcut = 20 GeV

10 3
10 2

Qcut = 30 GeV
Qcut = 40 GeV
Qcut = 60 GeV
Qcut = 100 GeV
Qcut = 200 GeV
inclusive

10 1
1
10 1
10 2

dsLO ( Qcut )/dsLO (20 GeV)

10 3
10 4
1.2
1

CKKW-L
MEPS@LO

0j

1j

2j

0.8
0.6

ds/dd12 [pb/GeV]

ds/dd01 [pb/GeV]

10

0.2
10

20

50

100
3FLPNNFOEFSBEGÊSHWBSJBOU

pp ! ` n̄ + 0,1,2 j @ 13 T

10 4

4WBSU

7JUOFHBUJWQcut = 10 GeV
Qcut = 15 GeV
Qcut = 20 GeV• W+jets production: diff. jet
10 2 0➙1 transition (~ pT
rate forof hardest
jet)
10 1
10 3

1
• Small dependence
on themerging
10 1 scale for smallvalues,~10%210• When10 3 taken too large, thepartonshower cannot fill
10 4
the1.2region
all
the
way
up
0j
MEPS@LOto the1 merging scale
anymore,leading to large
0.8
deficits
0.6
dsLO ( Qcut )/dsLO (20 GeV)

NL
dsQC

0.2

0.4 Lindert,[Kallweit,
Maierhöfer,Pozzorini,
0.2
Schönherr 2016]

0.4
0.2
10

20

50

100

200

500
d01 [GeV]

10

20

50

100

21

ME+PS (at LO)

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Merging matrix elements of various multiplicities with parton
showers improves the predictive power of the parton shower
outside the collinear/soft regions•

These merged samples give good description of the data(except for the total normalisation)• There is a dependence on the parameters responsible for the cut
in phase-space (i.e. the matching scale)• By letting the matrix elements mimic what the parton shower doesin the collinear/soft regions (PDF/alphas reweighting and including
the Sudakov suppression) the dependence is greatly reduced• In practice, one should check explicitly that this is the case byplotting differential jet-rate plots for a couple of values for thematching scale
22

Improving Parton Showers:
NLO matchingAt NLO

3FLPNNFOEFSBEGÊSHWBSJBOU

2
4WBSU

gs

7JUOFHBUJW+ ...

• We have to integrate the real emission over the complete
phase-space of the one particle that can go soft or collinear to
obtain the infra-red poles that will cancel against the virtual
corrections• Hence, we cannot introduce a "cut" that says that:• hard radiation needs to be described by
the matrix elements• and soft radiation by the parton shower• We have to invent a new procedure to match
NLO matrix elements with parton showers

24Naive (wrong) approach

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• In a fixed order calculation we have contributions with m final stateparticles
and with m+1Zfinal stateZ
Z particles
Z

NLO

⇠

d4

m B(

m) +

d4

dd l V (

m

m) +

dd

m+1 R(

m+1 )

loop

• We could try to shower them independently(k)• Let IMC (O) be the parton shower spectrum for an observable O,
showering from a k-body initial condition• We can then try to shower the m and m+1 final states
independently
d NLOwPS= ddO

m (B +Z(m)V ) IMC (O) + d

loop

(m+1)

m+1 R IMC

(O)

25

Naive (wrong) approach

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• In a fixed order calculation we have contributions with m final stateparticles
and with m+1Zfinal stateZ
Z particles
Z

NLO

⇠

d4

m B(

m) +

d4

dd l V (

m

m) +

dd

m+1 R(

m+1 )

loop

• We could try to shower them independently(k)• Let IMC (O) be the parton shower spectrum for an observable O,
showering from a k-body initial condition• We can then try to shower the m and m+1 final states
independently
d NLOwPS= ddO

m (B +Z(m)V ) IMC (O) + d

loop

(m+1)

m+1 R IMC(O)

25

NLO+PS: sources of double counting

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWParton shower
Born+Virtual:
Real emission:

26

NLO+PS: sources of double counting

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWParton shower
Born+Virtual:
Real emission:

26

NLO+PS: sources of double counting

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWParton shower
Born+Virtual:
Real emission:

...
...

26

NLO+PS: sources of double counting

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWParton shower
Born+Virtual:
Real emission:

...
...

26

NLO+PS: sources of double counting

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWParton shower
Born+Virtual:
Real emission:

...
...

• There is double counting between the real emission matrix elements and the parton
shower: the extra radiation can come from the matrix elements or the parton shower• There is also an overlap between the virtual corrections and the Sudakov
suppression in the zero-emission probability• We have to integrate the real emission over the complete phase-space of the one
particle that can go soft or collinear to obtain the infra-red poles that will cancel
against the virtual corrections• We can NOT use the same merging procedure as used at LO (MLM or CKKW):
requiring that all partons should produce separate jets is not infrared safe
26

Double counting in virtual/Sudakov

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• The Sudakov factor Δ (which is responsible for the resummationof all the radiation in the shower) is the no-emission probability• It’s defined to be Δ = 1 - P, where P is the probability for abranching to occur• By using this conservation of probability in this way, Δ contains
contributions from the virtual corrections implicitly• Because at NLO the virtual corrections are alreadyincluded via explicit matrix elements, Δ is double
counting with the virtual corrections• In fact, because the shower is unitary, what we are
double counting in the real emission corrections isexactly equal to what we are double counting in thevirtual corrections (but with opposite sign)!
27

Avoiding double counting

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• There are two widely methods to circumvent this double
counting• MC@NLO (Frixione & Webber)• POWHEG (Nason)• KRKNLO (Cracow group), Vincia (Skands et al.), Geneva (Alioli et al.), …

28

MC@NLO procedure

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWFrixione & Webber (2002)• To remove the double counting, we can add and subtract thesame term to the m and m+1 body configurations
"

Z

Z

+ d

m+1 (Rd MC@NLO= d m (B +
V +
dO
loop
"

d

1 M C)##(m)
IMC (O)(m+1)

M C) )IMC(O)

Where the MC are defined to be the contributionof the parton shower to get from the m body Born
final state to the m+1 body real emission final state

29

MC@NLO procedure

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWParton shower
Born+Virtual:...

Real emission:...

• Double counting is explicitly removed by including the “shower
subtraction terms”

d NLOwPS= ddO

m (B +



+ d

Z

Z
V + d

1 M C)M C)(m+1)
IMC (O)loop

m+1 (R(m)
IMC (O)

30

Negative weights
d NLOwPS= ddO

m (B +



+ d

Z

4WBSU

Z
V + d

1 M C)M C)(m+1)
IMC (O)loop

m+1 (R

3FLPNNFOEFSBEGÊSHWBSJBOU

7JUOFHBUJW(m)
IMC (O)• We generate events for the two terms between the square brackets(S- and H-events) separately• There is no guarantee that these contributions are separately
positive (even though predictions for infra-red safe observables
should always be positive!)• Therefore, when we do event unweighting we can only unweight
the events up to a sign. These signs should be taken into accountwhen doing a physics analysis (i.e. making plots etc.)• The events are only physical when they are showered31

Cost of negative weights

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• The computational costs to generate events negative weights
can be enormous: for physical observables they cancel against
positive weight events, but the overall sample still carries thestatistical uncertainty of the full sample:
Correlation
C± = 0%C± = 50%C± = 100%70
50

Relative cost, c(f )

30
20
15
10
7
5
3
2
1.5
1
0

5

10

15

20

25

30

Fraction of negative weights, f [%]

35

40

45

32

NLO+PS

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Advantages:• Total cross section and differential distributions related tothe hard process are NLO accurate• Reduced renormalisation and factorisation scale
uncertainties• Shower to include multiple emissions; there are models forhadronisation and underlying event• Fully exclusive description of the event• Disadvantages• Other observables (e.g., "multi-jet") are only LO accurate, oronly generated by the shower
33

Merging ME+PS at NLO accuracy

Merging LO ME with PS

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWCKKW (2004) and MLM (2004)• In summary:• Double counting no problem: we simply throw events away when thematrix-element partons are too soft, or when the parton shower
generates too hard radiation
kT < Qc• Applying the matrix-element cut is easy: duringphase-space integration, we only generate eventswith partons above the matching scale

kT > Qc• For the cut on the shower, there are two methods. Throwing events away
after showering is not very efficient, although it is working (“MLMmethod”)• Instead we can also multiply the Born matrix elements by suitableproduct of Sudakov factors (i.e. the no-emission probabilities) Δ(Qmax, Qc)
and start the shower at the scale Qc (“CKKW method”).
• For a given multiplicity we haveLOn,excl = Bn ⇥(kT,n

Qc )

c
(Q
,
Q
)
n
max
35

=

Merging at NLO

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• To make a LO prediction exclusive in the number of jets, we need tomultiply it by a Sudakov damping factor; this is CKKW method:LOn,excl = Bn ⇥(kT,n

Qc )

c
(Q
,
Q
)
n
max

This makes the prediction exclusive at leading logarithmic accuracy• Similarly we can make an NLO prediction exclusive at leading logarithm
⇢
Z
NLO
c
c
=
B
+
V
+
d
R
⇥(k
Q
)
(Q
,
Q
)
n
n
1 n+1
T,n
n
max
n,excl, LL• We can improve here and use the real-emission matrix elements
instead of just the Sudakov:⇢

NLOn,excl, LL =Z Qc

B n + Vn +

d

0

⇢

B n + Vn +1 Rn+1Z Qc

d

1 Rn+1

0

Bn

(1)
c
(Q
,
Q
)
max
n

Bn

(1)
c
(Q
,
Q
)
max
n

⇥(kT,nQc )⇥(kT,n

Qc )

n

c
(Q
,
Q
)
n
max
36

Exclusive MC@NLO:
FxFx merging and MEPS@NLO

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Converting the NLO exclusive predictions in the number of jets
to the MC@NLO event generation is straight-forward:
⇢

Z Qc

c
B
c
S-events:
B n + Vn +d 1 MC Bn (1)
(Q
,
Q
)
⇥(k
Q
) n (Q
max
n
T,n
⇢
Z Qc
0
c
B
c
B
c
nts:
B n + Vn +d 1 MC Bn (1)
(Q
,
Q
)
⇥(k
Q
)
(Q
,
Q
)
max
n
n
T,n
max
0⇢
R
B
R
H-events:
Rn+1 ⇥(kT,nQc ) MC ⇥(kT,nQc ) ⇥(Qc kT,n+1) n (QR
ma
⇢
R
B
R
c
-events:
Rn+1 ⇥(kT,nQc ) MC ⇥(kT,nQc ) ⇥(Qc kT,n+1) n (QR,Q)
max• But the evil is in the details…
37

FxFx merging: Higgs boson production

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW[RF and Frixione]

Figure 3: As in fig. 1, with Sudakov re• Transverse momentum of the Higgs and of the 1stjet.
• Agreement with H+0j at MC@NLO
and H+1j at MC@NLO in their
the lower insets of fig. 2). On the one hand, this overestim
respective regions of phase-space;
Smoothdue
matching
in between;
contributions
to scales close
to the end-points of the mSmall dependence on matching scale(in the eﬀective average performed by the smooth D functi

the other hand, this is not equivalent to assessing the eﬀe38

NLO ( Q )/ds NLO (
dsQCD
cut
QCD

0.8
0.6
0.4

100

200

Qcut = 10 GeV
Qcut = 15 GeV
Qcut = 20 GeV

10 3
10 2

1

10

2

10

3

dsLO ( Qcut )/dsLO (20 GeV)1

MEPS@LO

0j

1j

2j

0.8
0.6
0.4
0.2
10

20

50

100

200

3FLPNNFOEFSBEGÊSHWBSJBOU

0.2

10

10 4
10 3
10 2
10 1

1

4

10
1.2

0.4

20

50

100

200

500
d12 [GeV]
4WBSU

7JUOFHBUJW

Kallweit, 2016]
Lindert,[Kallweit, Lindert, Maierhöfer,ppPozzorini,
Schönherr! ` n̄ + 0,1,2 j @ 13 TeV

Qcut = 30 GeV
Qcut = 40 GeV
Qcut = 60 GeV
Qcut = 100 GeV
Qcut = 200 GeV
inclusive

10 1

10

0.6

500
d01 [GeV]

pp ! ` n̄ + 0,1,2 j @ 13 TeV

10 4

1

0.8

500
d01 [GeV]

10

1

10

2

10

3
4

10
1.2
1

0.8
0.6
0.4

pp ! ` n̄ + 0,1,2 j @ 13 TeV

10 4

Qcut = 10 GeV
Qcut Q= 10=GeV
15 GeVcut
Qcut Q=cut
15=GeV
20 GeV
Qcut = 20 GeV

10 3
10 2

Qcut = 30 GeV
QcutQ= 30=GeV
40 GeVcut
QcutQ= 40=GeV
60 GeVcut
QcutQ=cut60=GeV
100 GeV
QcutQ=cut100
GeVGeV= 200
Qcutinclusive= 200 GeV
inclusive

10 1
1

Sherpa+OpenLoops

50

ds/dd12 [pb/GeV]

20

dsLO ( Qcut )/dsLO (20 GeV)

ds/dd01 [pb/GeV]

10

Electroweak corre

Mergingsystematics: pp ! ` ⌫¯
Merging scaledependence
ds/dd01 [pb/GeV]

0.2

NLO ( Q )/ds NLO (20 GeV)
dsQCD
cut
QCD

NLO ( Q )/ds NLO (
dsQCD
cut
QCD

Next-to-leading order electroweak corrections

10 1
10 2
10 3
10 4
1.2

MEPS@LO
1MEPS@NLO QCD

0j

1j

0j

1j

2j

2j

0.8
0.6
0.4
0.2

0.2

10

10

20

20

50

50

100

100

200

200

500
500 d12 [GeV]

d01 [GeV]

25. Multijet
mergingof the1 ! higher-accurate
0 (left) and 2 ! 1 (right)
k? jet resolutions(R = 0.6)•FigureBesideshavingthesystematics
benefitsfrommatrixelements,) next-to-leading
dead zonesorderin (top)incl.multijetobs. ifmerging
Qcutintoo
in pp ! ` ⌫¯ + jets events in leading order (bottom) andthere is also a smaller merging scale dependence at NLO
the MEPS scheme. Only basic lepton acceptance cuts are applied. The contributions of the individual jet
39

FxFx
• Comparison to data
• Z+jets• Exclusive jet multiplicity
and hardest and 3rd
hardest jet pT spectra• Uncertainty band
contains ren. & fac.
scale, PDF & merging
scale dependence• Rather good agreement
between data andtheory3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWFigure 1: Exclusive jet multiplicity. Data from ref. [28], compared to Herwig++ (leftpanel) 1:andExclusive
Pythia8jet(rightpanel) predictions.
FxFxcompared
uncertainty
envelope (“Var”)
Figuremultiplicity.
Data from The
ref. [28],to Herwig++(left
and thefully-inclusive
centralresult(“inc”) are Theshown
as green
bands andred histograms
panel)
andPythia8 (rightpanel)
predictions.
FxFx
uncertainty
envelope(“Var”)
respectively.
See the endof sect.
2 formorearedetailson as
thegreenlayoutof theandthe fully-inclusive
centralresult(“inc”)
shown
bands
andplots.
red histograms
respectively. See the end of sect. 2 for more details on the layout of the plots.
Figure 1: Exclusive jet multiplicity. Data from ref. [28], compared to Herwig++ (leftpanel) and Pythia8 (right panel) predictions. The FxFx uncertainty envelope (“Var”)
and the fully-inclusive central result (“inc”) are shown as green bands and red histograms
respectively. See the end of sect. 2 for more details on the layout of the plots.

Figure 2: As in fig. 1, for the transverse momentum of the 1st jet.
Figure 2: As in fig. 1, for the transverse momentum of the 1st jet.

Figure 2: As in fig. 1, for the transverse momentum of the 1st jet.

Figure 3: As in fig. 1, for the transverse momentum of the 3rd jet.

40

Conclusions

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• In the last couple of years the accuracy of event generation has
greatly improved, and full automation has been achieved at NLO
accuracy• A lot of freedom in tuning has been replaced by accurate theory
descriptions:• More predictive power• Better control on uncertainties• Greater trust in the measurements• Recent developments for which I have had no time
• NLO EW corrections and the parton shower• Combining NNLO in QCD and the parton shower: MINNLO• MC@NLO-Delta: reducing negative weights in MC@NLO
41