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

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

3FLPNNFOEFSBEGÊSHWBSJBOU

Olivier Mattelaer, last Tuesday
4WBSU

7JUOFHBUJWMonte-Carlo Physics

Mattelaer Olivier

Japan 2023

2

2

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW?
MadGraph5_aMC@NLO
can generate only a coupleof final state particles

3Daniel Jeans, last Monday
3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

e+e- → e+e- h [invisible h decay]@ 250 GeV

7JUOFHBUJW

e+e- → tth [tt → 6q, h → bb]@ 1000 GeV
57⇥ ⇤⌅⇧⌃⌥⌦↵⇥

4

Daniel Jeans, last Monday
3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

e+e- → e+e- h [invisible h decay]@ 250 GeV

7JUOFHBUJW

e+e- → tth [tt → 6q, h → bb]@ 1000 GeV
57⇥ ⇤⌅⇧⌃⌥⌦↵⇥MadGraph5_aMC@NLO generates
the two final state leptons (and theinvisible Higgs boson decay products).
They corresponds directly two thetwo observed particles

4Daniel Jeans, last Monday
3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

e+e- → e+e- h [invisible h decay]@ 250 GeV

7JUOFHBUJW

e+e- → tth [tt → 6q, h → bb]@ 1000 GeV
57⇥ ⇤⌅⇧⌃⌥⌦↵⇥MadGraph5_aMC@NLO generates
the two final state leptons (and theinvisible Higgs boson decay products).
They corresponds directly two thetwo observed particles

MadGraph5_aMC@NLO generates
the 8 final state quarks (of which 4 are(anti-)bottom quarks).
How does this corresponds to theobserved particles?

4Daniel Jeans, last Monday
3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

e+e- → e+e- h [invisible h decay]@ 250 GeV⇥ ⇤⌅⇧⌃⌥⌦↵⇥MadGraph5_aMC@NLO generates
the two final state leptons (and theinvisible Higgs boson decay products).
They corresponds directly two thetwo observed particles

7JUOFHBUJW

e+e- → tth [tt → 6q, h → bb]@ 1000 GeV?

57

MadGraph5_aMC@NLO generates
the 8 final state quarks (of which 4 are(anti-)bottom quarks).
How does this corresponds to theobserved particles?

4

Two possibilities

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Parton showering!

or• Jet clustering!
e+e- → e+e- h [invisible h decay]@ 250 GeV

e+e- → tth [tt → 6q, h → bb]@ 1000 GeV
57⇥ ⇤⌅⇧⌃⌥

⌦↵⇥

5

]

Jet clustering

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Goal:• Cluster particles "that areclose in phase-space" into
single objects: jets
• These jets correspond to
the quarks (or gluons)generated by
MadGraph5_aMC@NLO

e+e- → tth [tt → 6q, h → bb]@ 1000 GeV
57

6

]

Jet clustering

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Goal:• Cluster particles "that areclose in phase-space" into
single objects: jets
• These jets correspond to
the quarks (or gluons)generated by
MadGraph5_aMC@NLO

e+e- → tth [tt → 6q, h → bb]@ 1000 GeV
57

6

Jet clustering

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Sequential algorithm• Define all the distances• dij (between particles i and j) and• diB (between particle i and the beam)• If dij is the smallest, replace particles i and j by a new(pseudo) particle• If diB is the smallest, call particle i a jet and remove it fromthe list• Keep going until no particles are left
7

of the distances and if it is a dij recombining entities i and j, w
removing it from the list of entities. The distances are recalcul
until no entities are left.
The extension relative to the kt and Cambridge/Aachen algor
distance measures:

Jet clustering• Sequential algorithm• Define all the distances

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW

2
∆
ij
2p 2p
dij = min(kti , ktj ) 2 ,R
2p
diB = kti ,particles i and j) and• dij (between
where ∆2ij = (yi − yj )2 + (φi − φj )2 and kti , yi and φi are respec
particle ofthe beam)diB (between
i and• rapidity
and azimuth
particlei. In addition to the usual radiu
parameter p to govern the relative power of the energy versus geo• If dij is the smallest, replace particles i and j by a new
For p = 1 one recovers the inclusive kt algorithm. It can be(pseudo) particle
the behaviour of the jet algorithm with respect to soft radiation
kt smallest,algorithm,
is theitorderingbetwe
diB thei a jetmattersis thecallbecause
particlewhat
and removefrom• If fortheis listmaintained for all positive values of p. The case of p = 0 isinclusive• KeepgoingCambridge/Aachen
until no particles arealgorithm.
left
1This is discussed in the appendix in detail for the iterative cone, and t
iterative cone with split–merge steps (IC-SM) and iterative cone with progressi
7

of the distances and if it is a dij recombining entities i and j, w
removing it from the list of entities. The distances are recalcul
until no entities are left.
The extension relative to the kt and Cambridge/Aachen algor
distance measures:

Jet algorithms• Different clustering algorithms exist• Same event cluster with different
algorithms gives slightly different jet(s)and shapes

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW

2
∆
ij
2p 2p
dij = min(kti , ktj ) 2 ,R
2p
diB = kti ,2 = (y − y )2 + (φ − φ )2 and k , y and φ are respec
where
∆
i
jp = 0ijti p i= − 1ip=1ij
rapidity and azimuth of particle i. In addition to the usual radiu
parameter p to govern the relative power of the energy versus geo
For p = 1 one recovers the inclusive kt algorithm. It can be
the behaviour of the jet algorithm with respect to soft radiation
for the kt algorithm, because what matters is the ordering betwe
is maintained for all positive values of p. The case of p = 0 isinclusive
Cambridge/Aachen
Figure 1: A sample parton-level algorithm.
event (generated with Herwig [8]), together with many random soft
• Anti-k (p = − 1) algorithm most popular at the LHC
T“ghosts”, clustered with four diﬀerent jets algorithms, illustrating the “active” catchment areas of
the resulting hard jets. For kt and Cam/Aachen the detailed shapes are in part determined by the1
This is discussed
the used,
appendixdetailforarethe
iterative cone, and t
specific set inof ghosts
and changein
when
the ghosts
modified.

• All implemented in the fastjet package (https://fastjet.fr)

iterative cone with split–merge steps (IC-SM) and iterative cone with progressi
8

Be careful!

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• The correspondence between quarks/gluons and jets works well
• However:• The jets that come out of the jet algorithm can be arbitrarily soft(i.e., with a very small energy or transverse momentum)• For them to correspond to quarks/gluons computed by
MadGraph5_aMC@NLO, they need to be "hard"• Only consider jets above a threshold• But this is somewhat arbitrary…
• How hard does "hard" need to be to be fine?
• No general rule here… depends on the rest of the event!
• In practice, in your calculation you get a large logarithms
that hamper the convergence of perturbation theory(in the expansion of the strong coupling, each order is larger than the previous)
9

Two possibilities

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Parton showering!

or• Jet clustering!
e+e- → e+e- h [invisible h decay]@ 250 GeV

e+e- → tth [tt → 6q, h → bb]@ 1000 GeV
57⇥ ⇤⌅⇧⌃⌥⌦↵⇥

10

Two possibilities

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Parton showering!

and• Jet clustering!
e+e- → e+e- h [invisible h decay]@ 250 GeV

e+e- → tth [tt → 6q, h → bb]@ 1000 GeV
57⇥ ⇤⌅⇧⌃⌥⌦↵⇥

11

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)

12Olivier Mattelaer, last Tuesday

Master formula for the LHC

µF

`+

`

x1 E

x2 E

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWµF
pplong distancelong distancea,bMattelaer Olivier

dx1 dx2 d
Phase-space
integralˆab
FS fa (x1 , µF )fb (x2 , µF ) ⇥Parton density
functionsJapan 2023

X (ŝ, µF , µR )Parton-level crosssection12

13

Matrix elements in perturbation
theory3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• In matrix element calculations in perturbation theory• "initial state QCD radiation" is included inclusively ("resummed")in the PDFs (and through strong coupling definition) and• "final state QCD radiation" is included through the parton-jet
duality (and through strong coupling definition) LHC master formula• Hence… all is already there!
What to do…?
• "Undo" this resummationand make it explicit

`+

`

x1 E

x2 E

p

p
long distancelong distance

!" 1

214

Why do this?

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Effects are already included (resummed) in fixed-order perturbation theory• for many inclusive observables including shower should not influence the results• but one would miss an extremely rich variety of observables which may play
important roles in experimental analyses.
• When there are large scale differences entering the observables, fixed-order
perturbation theory breaks down!
• this does NOT mean that observable is useless/unimportant: it is just that one is
not using the right tools to describe it.
• It is better to try and find a way to reorganise the computation in order to take
into account emissions close to the singular regions of the phase space, to allorders in perturbation theory.
• "We want to simulate the collisions, hence we want to simulate also the creation of
the hadrons, for which we need parton showering"15

Collinear factorisation
b
θ

Mn+1

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW2c• Consider a process for which two particles are separated by asmall angle θ• In the limit of θ ➞ 0, the contribution is coming from a single parent
particle going on shell: therefore its branching is related to timescales which are very long with respect to the hard subprocess• The inclusion of such a branching cannot change the picture set upby the hard process: the whole emission process must be writablein this limit as the simpler one times a branching probability
16Collinear factorisation
b
θ

Mn+1

c

b

2

a
θ➞0

Mn+1

θ

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW2c• Consider a process for which two particles are separated by asmall angle θ• In the limit of θ ➞ 0, the contribution is coming from a single parent
particle going on shell: therefore its branching is related to timescales which are very long with respect to the hard subprocess• The inclusion of such a branching cannot change the picture set upby the hard process: the whole emission process must be writablein this limit as the simpler one times a branching probability
16Collinear factorisation
b
θ

Mn+1

c

b

2

a
θ➞0

Mn+1

θ

c

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW2θ➞0• Consider a process for which two particles are separated by asmall angle θ• In the limit of θ ➞ 0, the contribution is coming from a single parent
particle going on shell: therefore its branching is related to timescales which are very long with respect to the hard subprocess• The inclusion of such a branching cannot change the picture set upby the hard process: the whole emission process must be writablein this limit as the simpler one times a branching probability
16Collinear factorisation
b
a

Mn+1

θ

c

2

a
θ➞0

Mn

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

2
×a

7JUOFHBUJWb 2c• Consider a process for which two particles are separated by asmall angle θ• In the limit of θ ➞ 0, the contribution is coming from a single parent
particle going on shell: therefore its branching is related to timescales which are very long with respect to the hard subprocess• The inclusion of such a branching cannot change the picture set upby the hard process: the whole emission process must be writablein this limit as the simpler one times a branching probability
16Collinear factorisation
a

Mn+1

b

θ

c

2

a

Mn

θ➞0

2
a
×

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

b

7JUOFHBUJW2c• The process factorises in the collinear limit. This procedure is universal

dt d ↵S|Mn+1 | d n+1 ' |Mn | d n dz
Pa!bc (z)
t
2⇡ 2⇡
2

2

• Notice that what has been roughly called ‘branching fraction’ is actually asingular factor, so one will need to make sense of this definition.
• At the leading contribution to the (n+1)-body cross section the DGLAP
splitting kernels are defined as:


⇥ 2
⇤
z
1 z
2
Pg!qq (z) = TR z + (1 z) ,Pg!gg (z) = CA z(1 z) +
+
,
1 z
z


2
1+z
1 + (1 z)2
Pq!qg (z) = CF,
Pq!gq (z) = CF
.
1 z
z

17

Collinear factorisation
a

Mn+1

b

θ

c

2

a

Mn

θ➞0

2
a
×

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

b

7JUOFHBUJW2c• The process factorises in the collinear limit. This procedure is universal

dt d ↵S|Mn+1 | d n+1 ' |Mn | d n dz
Pa!bc (z)
t
2⇡ 2⇡
2

2

• t can be called the ‘evolution variable’: it can be the virtuality m2 of particle a,or its pT2, or E2θ2 …
22 2
m ' z(1• It represents the hardness of the
branching and tends to 0 in the collinear limit.
• Indeed in the collinear limit one has:so that the factorisation takes place
for all these definitions:
2 2

d✓ /✓

2

= dm /mz)✓ Eap2T ' zm2

2

=

2
2
dpT /pT

18

𝜙Collinear factorisation
a

Mn+1

b

θ

c

2

a

Mn

θ➞0

2
a
×

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

b

7JUOFHBUJW2c• The process factorises in the collinear limit. This procedure is universal

dt d ↵S|Mn+1 | d n+1 ' |Mn | d n dz
Pa!bc (z)
t
2⇡ 2⇡
2

2

• z is the “energy variable”: it is defined to be the energy fraction taken byparton b from parton a• It represents the energy sharing between b and c and tends to 1 in thesoft limit (parton c going soft)•is the azimuthal angle. It can be chosen to be the angle between thepolarisation of a and the plane of the branching
19Collinear factorisation
a

Mn+1

b

θ

c

2

a

Mn

θ➞0

2
a
×

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

b

7JUOFHBUJW2c• The process factorises in the collinear limit. This procedure is universal

dt d ↵S|Mn+1 | d n+1 ' |Mn | d n dz
Pa!bc (z)
t
2⇡ 2⇡
2

2

• This is an amplitude squared: naively one would maybe expect 1/t2
dependence. Why is the square not there?
• It’s due to angular-momentum conservation.
E.g., take the splitting q ⟶ qg: helicity is conserved for the quarks, so thefinal state spin differs by one unity with respect to the initial one. The
scattering happens in a p-wave (orbital angular momentum equal to one),so there is a suppression factor as t ⟶ 0.
• Indeed, a factor 1/t is always cancelled in an explicit computation
20

Multiple
emission
d
a

Mn+2

b

θ

2

θ’

c

e

a
θ, θ’ ➞ 0
θ’ ≪ θMn

3FLPNNFOEFSBEGÊSHWBSJBOU

2
×

4WBSU

a

7JUOFHBUJW

b 2

d 2

c

e

×b

• Now consider Mn+2 as the new core process and use the recipe we used forthe first emission in order to get the dominant contribution to the (n+2)-bodycross section: add a new branching at angle much smaller than theprevious one:dt d ↵S|Mn+2 | d n+2 ' |Mn | d n dz
Pa!bc (z)t2⇡ 2⇡
dt0 0 d 0 ↵S
0
⇥ 0 dzPb!de (z )
t
2⇡ 2⇡
2

2

• This can be done for an arbitrary number of emissions. The recipe to get
the leading collinear singularity is thus cast in the form of an iterative
sequence of emissions whose probability does not depend on the pasthistory of the system: a ‘Markov chain’.
21Multiple
emissions
d
a

Mn+2

b

θ

2

θ’

c

e

a
θ, θ’ ➞ 0
θ’ ≪ θMn

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

2
×

a

7JUOFHBUJW

b 2

d 2

c

e

×b

• The dominant contribution comes from the region where thesubsequently emitted partons satisfy the strong ordering requirement:θ ≫ θ’ ≫ θ’’...
For the rate for multiple emission we getkn+k / ↵S

Z Q2
Q20

dt
t

Z t

0

dt
...
0
Q20 t

Z t(k 2)Q20

dt(k 1)/(k1)
t⇣ ↵ ⌘kSn2⇡klog (Q2 /Q20 )where Q is a typical hard scale and Q0 is a small infrared cutoff that
separates perturbative from non perturbative regimes.
• Each power of αs comes with a logarithm. The logarithm can easily belarge, and therefore we see a breakdown of perturbation theory
22

d
a

Mn+2

b

θ

θ’

c

e

Approximation
2aθ, θ’ ➞ 0
θ’ ≪ θMn

3FLPNNFOEFSBEGÊSHWBSJBOU

2
×

4WBSU

a

7JUOFHBUJW

b 2

d 2

c

e

×b

• We have an approximation of the matrix elements for multiple
emissions• We know that the |Mn|2 is inclusive over all radiation (due to parton/jet
duality and PDF evolution)• However, in our approximation we multiply |Mn|2 by k "branching
fractions" to get |Mn+k|2. These branching fractions are actually
singular factors• How to make sense of this? How to enforce that summing over all
branching fractions adds up to one?
23

d
a

Mn+2

b

θ

θ’

c

e

Approximation
2aθ, θ’ ➞ 0
θ’ ≪ θMn

3FLPNNFOEFSBEGÊSHWBSJBOU

2
×

4WBSU

a

7JUOFHBUJW

b 2

d 2

c

e

×b

• We have an approximation of the matrix elements for multiple
emissions• We know that the |Mn|2 is inclusive over all radiation (due to parton/jet
duality and PDF evolution)• However, in our approximation we multiply |Mn|2 by k "branching
fractions" to get |Mn+k|2. These branching fractions are actually
singular factors• How to make sense of this? How to enforce that summing over all
branching fractions adds up to one?
We are missing the contributions that there was no emission
23

(No-)emission probability

3FLPNNFOEFSBEGÊSHWBSJBOU

a

b 2
4WBSU

7JUOFHBUJWc• The probability for the branching a ⟶ bc between scales t and t+dt is⇥equal todp(t) =

bc

dt
t

d⇤ S
dz
Pa
2⇥ 2⇥

bc (z)• The probability that a parton does NOT split between the scales t andt+dt is given by 1-dp(t)• Probability that particle a does not emit between scales Q⇥2 and t⇤ dtk ⇧d⇤ S
(Q , t) =1
dz
Pa bc (z) =
tk
2⇥ 2⇥
k
bc
⇥
⇥
2
2
⇧
⇧
Q
Q
⇤
dt⇥ d⇤ S
⇥
dz
P
(z)
=
exp
dp(t
)
a bc
⇥
t
2⇥ 2⇥
t
t
2

exp

⌅

bc

Δ(Q2,t) is the Sudakov form factor
24

Sudakov form factor

3FLPNNFOEFSBEGÊSHWBSJBOU• The Sudakov form factor is the heart of the parton shower. It gives theprobability that a parton does not branch between two scales
4WBSU

7JUOFHBUJW*Initial state shower also requires PDF contributions• This no-emission probability needs to be included to interpret thebranchings as probabilities that add up to 1
• Define dPk as the probability for k ordered splittings from leg a at given
scales dP (t ) =(Q2 , t ) dp(t ) (t , Q2 ),11

dP2 (t1 , t2 ) =1

1(Q2 , t1 ) dp(t1 )10(t1 , t2 ) dp(t2 )... = ...
kdPk (t1 , ..., tk ) =(Q2 , Q20 )dp(tl )⇥(tl 1(t2 , Q20 )⇥(t1

t2 ),

tl )

l=1

• Q02 is the hadronisation scale (~1 GeV2). Below this scale we do not trust
the perturbative description for parton splitting anymore• This is what is implemented in a parton shower, taking the scales for the
splitting ti randomly (but weighted according to the no-emission
probability)
25

Unitarity

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWkdPk (t1 , ..., tk )=(Q2 , Q20 )dp(tl )⇥(tl 1

tl )

l=1

• The parton shower has to be unitary (the sum over all
branching trees should be 1). We can explicitly check this byintegrating the probability for k splittingsPk⇤

dPk (t1 , ..., tk ) =2 1(Q , Q0 )
2

k!

⇤ Q2
Q20

⇥k

dp(t)

,

⇥k = 0, 1, ...

• Summing over all number of emissions
⇤

k=0

Pk =

(Q , Q20 )
2

⇤ 1

k=0

k!

⌅ Q2
Q20

⇥k

dp(t)

=

(Q2 , Q20 ) exp

⌅ Q2
Q20

⇥

dp(t) = 1

• Hence, the total probability is conserved

26

Physical interpretation

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• We should not interpret the parton shower to be generating anapproximation of the |Mn+k|2 matrix elements• Rather it is an approximation of NkLO computation of |Mn|2• That is, including the (real-)emission contributions, but also
virtual (no-)emission corrections

27

Initial-state parton showers

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• To simulate parton radiation from the initial state, we start with the hard
scattering, and then “devolve” the DGLAP evolution to get back to theoriginal hadron: backwards evolution!
• i.e. we undo the analytic resummation and replace it with explicit
partons (e.g. in Drell-Yan this gives non-zero pT to the vector boson)• In backwards evolution, the Sudakovs include also the PDFs -- thisfollows from the DGLAP⇤ equation and ensures conservation of⌅⌦ 1
probability:⌥ ⌦ t2⇥Ii (x, t1 , t2 ) = exp

⇧

dt

t1

j

x

dx
x

s (t )

2⇥

Pij

x
x

fi (x , t )fj (x, t ) ⌃This represents the probability that parton i will stay at the same x (nosplittings) when evolving from t1 to t2.
• The shower simulation is now done as in a final state shower

28

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)

29

Hadronisation

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

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

30

Cluster model

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

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+

31

Lund
stringmodel
Quark antiquark color potential and string modelė¿ėΡ

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

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 / 8332

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)

33

Exclusive observable

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

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.

34

Parton Shower Monte-Carloevent generators

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

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
35

Pythia, Herwig & Sherpa

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

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 research

36Extra material

Exclusive observables

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Very exclusive observables are poorly described in perturbation
theory.
• One could take the conservative attitude of considering only
perturbatively well-behaved observables. But one would missan extremely rich variety of observables which may play
important roles in experimental analyses.
• If fixed-order perturbation theory breaks down for an observable,
this does NOT mean that observable is useless/unimportant: it isjust that one is not using the right tools to describe it.
• It is better to try and find a way to reorganise the computation inorder to take into account emissions close to the singularregions of the phase space, to all orders in perturbation theory.
• This can be done in a systematic way: "resummation"!
38

Absence of interference

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• The collinear factorisation picture gives a branching sequence for a given leg
starting from the hard subprocess all the way down to the non-perturbative region.
• Suppose you want to describe two such histories from two different legs:• these two legs are treated in a completely uncorrelated way. And even within
the same history, subsequent emissions are uncorrelated.
• The collinear picture completely misses the possible interference effects between
the various legs
• the extreme simplicity comes with the price of quantum inaccuracy.
• Smart choices improve upon this: soft enhancement (which is purely aninterference contribution) can be included. For this, the evolution variablemust be related to the angle of the emission• Nevertheless, the collinear picture captures the leading contributions: it gives an
excellent description of an arbitrary number of (collinear) emissions:• it is a “resummed computation” and• it bridges the gap between fixed-order perturbation theory and the nonperturbative hadronisation.
39

Cancellation of singularities

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• We have shown that the shower is unitary. However, how are the IR
divergences cancelled explicitly? Let’s show this for the first emission:
Consider the contributions from (exactly) 0 and 1 emissions from leg a:

d⇤
=
⇤n

(Q , Q20 ) +2(Q , Q20 )2• Expanding to first order in αs gives

d⇤
⇥1
⇤n

bc

⇥ Q2
Q20

dt⇥ d⌅ S
dz
Pa
⇥
t
2⇥ 2⇥

bc

dt d⌅ Sdz
Pat 2⇥ 2⇥bc (z) +bc

bc (z)

dt d⌅ Sdz
Pat 2⇥ 2⇥bc (z)• Same structure of the two latter terms, with opposite signs: cancellation ofdivergences between the approximate virtual and approximate real emission
cross sections.
• The probabilistic interpretation of the shower ensures that infrared
divergences will cancel for each emission.

40

Argument of αS

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Each choice of argument for αS is equally acceptable at the leadinglogarithmic accuracy. However, there is a choice that allows one to resum
certain classes of subleading logarithms.
• The higher order corrections to the partons splittings imply that the DGLAP
splitting kernels should be modified: Pa ⟶ bc(z) ⟶ Pa ⟶ bc(z) + αs P’a ⟶ bc(z)• For g ⟶ gg branchings P’a ⟶ bc(z) diverges as -b0 log[z(1-z)] Pa ⟶ bc(z)(just z or 1-z if quark is present)• Recall the one-loop running of the strong couplingS(Q ) =

S

2

1+

(µ )
2

Q2
2
S (µ )b0 log µ2⇥

S(µ ) 12

2Q
S (µ )b0 logµ22⇥• We can therefore include the P’(z) terms by choosing pT2~z(1-z)Q2 asargument of αS:
⇥
⇥
2
⇥
2
2
2
=
z) Pa
S (Q ) Pa
bc (z) + S (Q )Pa bc
S (Q ) 1
S (Q )b log z(1⇥S (z(1z)Q2 )Pabc (z)bc (z)

41

Choice of evolution parameter(Q2 , t) = exp⇤ ⌅ Q2bct

dt⇥ d⇤ S
dz
Pa
⇥
t
2⇥ 2⇥

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW⇥bc (z)• There is a lot of freedom in the choice of evolution parameter t.
It can be the virtuality m2 of particle a or its pT2 or E2θ2 ... For thecollinear limit they are all equivalent
2

d✓ /✓

2

2

= dm /m2=

2
2
dpT /pT• However, in the soft limit (z ⟶ 1) they behave differently• Can we chose it such that we get the correct soft limit?

42

Choice of evolution parameter(Q2 , t) = exp⇤ ⌅ Q2bct

dt⇥ d⇤ S
dz
Pa
⇥
t
2⇥ 2⇥

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW⇥bc (z)• There is a lot of freedom in the choice of evolution parameter t.
It can be the virtuality m2 of particle a or its pT2 or E2θ2 ... For thecollinear limit they are all equivalent
2

d✓ /✓

2

2

= dm /m2=

2
2
dpT /pT• However, in the soft limit (z ⟶ 1) they behave differently• Can we chose it such that we get the correct soft limit?

YES! It should be (proportional to) the angle θ42Angular ordering

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• Radiation inside cones around the original partons is allowed(and described by the eikonal approximation), outside thecones it is zero (after averaging over the azimuthal angle)

photon+photon43

μAngular ordering

Intuitive explanation

3FLPNNFOEFSBEGÊSHWBSJBOU(slide by M. Mangano)

4WBSU

7JUOFHBUJWAn intuitive explanation of angular ordering
Lifetime of the virtual intermediate state:
Lifetime of the virtual2intermediate 2state:
τ < γ/µ = E/µ = 1/(k0θ ) = 1/(k⊥θ)φτ < γ/μ = E/μ! = 1 / (k₀θ!)= 1/(k⊥θ)

kμ!

Distance between q and qbar after τ:
d = φτ = (φ/θ) 1/k⊥

θpDistance between q and qbar after τ:

μ! = (p+k)! = 2E k₀ (1-cosθ)
2 = (p+k)2 = 2E k0 (1-cosθ)∼ E k₀ θ! ∼ E k⊥ θ

d = φτ = (φ/θ) 1/k⊥

∼ E k0 θ2 ∼ E k⊥ θ

If the transverse wavelength of the emitted gluon is longer than
the separationwavelength
between q andof
qbar,
gluon emission
If the transverse
thetheemitted
gluonisis longer than the
suppressed,
becausewill appear
as colouris suppressed,
separation
betweenq theandq qbar
qbar,system
the gluon
emission
neutraldipole-likeemission,
becausethe(=>
q qbarsystemwill suppressed)
appear as colour neutral (i.e. dipole-like emission, suppressed)
Therefore d> 1/k⊥ , which implies

Therefore d>1/k⊥ , which implies
Paolo Torrielli (EPFL)

θ<φ

θ<φ

Interfacing NLO with Parton Showers

12

ThinkTank on Physics @ LHC

25 / 8344

Angular ordering

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• The construction can be iterated to the next
emission, with the result that the emission
angles keep getting smaller and smaller.
• One can generalise it to a generic parton ofcolour charge Qk splitting into two partons iand j, Qk=Qi+Qj. The result is that inside
the cones i and j emit as independent
charges, and outside their angular-orderedcones the emission is coherent and can betreated as if it was directly from colour
charge Qk.
• Angular ordering is automatically satisfied inθ ordered showers! (and straight-forward toaccount for in pT ordered showers)
45

Angular ordering

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJWAngular ordering is:
1. A quantum effect coming from the interference of different
Feynman diagrams.
2. Nevertheless it can be expressed in “a classical fashion”(square of an amplitude is equal to the sum of the squares of two
special “amplitudes”). The classical limit is the dipole-radiation.
3. It is not an exclusive property of QCD (i.e., it is also present inQED) but in QCD produces very non-trivial effects, depending on
how particles are colour connected.

46

Initial-state parton splittings

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW• 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)➡ Naively: Probability to find a given parton in a hadron at agiven momentum fraction x = pz/Pz and scale t• How do the PDFs evolve with increasing t?

47

Initial-state parton splittings

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW

Q2
xn 1 tn 1
x0 t0

x1 t1

xn tn

···

p
Figure 3.5: The struck quark radiating several gluons at successive t and x, such thatt0 ⇤ t1 ⇤ . . . ⇤ tn 1 ⇤ tn ⇤ t = Q2 and x0 > x1 > . . . > xn 1 > xn = x.

• Start with a quark PDF f0(x) at scale t0. After considering a single
parton emission, the probability to find the quark at virtuality t > t0 is
⇤

⇤

dz
dt s
x⇥
f (x, t) = f0 (x) +P (z)f0
⌥
1
⇥
2⇥
z x⇥
z
dz x
t0t dtts
steps, we see that such atradiation would result
1 in

q(x, t) = q0 (x) +⇥t0 t 2⇥xz

P (z) q0z

1
t have⇥⇥⇥• After a second emission,wedts⇤ dz P (z⇥) q
⇤ +

+

x ⇥
= ⇥
0 ⌅⇥
t
1 z⇥
⇥⇥ 2⇥
t
zz
dtt0 s x/z dzx⇤ ⌅ 1f (x, t) = f0 (x) +
P (z)
f0
t
dz
x⇥
s
t
2⇥
z
z
= q0 (x) +t0 ln
P (z)
q0
+
x
z
0
x z⇤
⇤2⇥ t⇤ t⌅⌃
1
⇥⌅
⇧
2
⇥
1
1
⇥
dt
dz
x
1
t
dz
x
sdz
s
⇥
+ + ln
P (z)
P
(z
)
q
⇥
0
⇥P (z )f0
⇥
2! 2⇥
t0
z
z
zz
x
x/z
zz
tt0 ⇥ t ⇥ 2⇥1 x/z z ⇥dt s (t )dzx ⇥⇥ q0 (x) +
P
(z)
q
,t
⇥

f(x/z, t’)

48

The DGLAP equation

3FLPNNFOEFSBEGÊSHWBSJBOU

4WBSU

7JUOFHBUJW

Q2
xn tn

xn 1 tn 1
···

x1 t1

x0 t0
p

Figure 3.5: The struck quark radiating several gluons at successive t and x, such thatt0 ⇤ t1 ⇤ . . . ⇤ tn 1 ⇤ tn ⇤ t = Q2 and x0 > x1 > . . . > xn 1 > xn = x.

• So for multiple parton splittings, we arrive at an integral-differential
Z 1
equation:⇣⌘@ steps, we see that such a radiation
dzwould↵sresult inx
t fi (x, t) =Pij (z)fj,t@tz
x z 2⇡ ⌥
⇥
1

t

dt⇥ s
q(x, t) = q0 (x) +⇥t0 t 2⇥dzx
P (z) q0+zz

• This is the famous DGLAP equation(wherewe havetaken into account
⇥
1
t
⇥
⇥⇥
dz
x
dt s
⇥
P
(z
)
q
=
+
0
the multiple parton species i, j).t tThe
condition
for the⇥⇥ 2⇥ boundary
⇥
⇥
z
zz
x/z
⌅ a1 starting ⇥scale t0 (around 2 GeV).
equation is the initial PDFs fi0s (x)⇤ tat
dz
x
x

0

= q0 (x) +2⇥ln

P (z) q0

t0
x z
⌅⌃2 1
t
dz
P (z)
t0
z
x

+

z

• These starting PDFs are1 fitted
to experimental
data. x ⇥
1
dz ⇥
s
+

⇧

2! 2⇥

⇥ q0 (x) +

ln
t

⇤

dt⇥
⇥

⇥
s (t )

1

dz

x/z z

P (z) q

x

⇥
P
(z
) q0
⇥

,t

⇥

⇥

zz ⇥

⇥
49