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Parton Showers and Matrix Element Merging in Event Generator-
a Mini-Overview
Introduction to ME+PS
Branching and Sudakov factor (no branching)
Matching ME 23 + PS –CKKW and MLM
Mini-Reminder:
NLO calculation and subtraction method
MC@NLO
• Since mid 1980 Pythia and Herwig (and Ariadne) have been widely used
in most particle physics experiments
They provide full topology of final state particles based on:
Born matrix element + Parton shower (leading log approx.) for higher
order for radiation in strong and electromagnetic interaction
+ hadronisation (Lund strong and cluster model)
Late 80/early 90: Procedure to work out merging ME+PS, i.e. ME is used for emission of additional jet besides Born process (Sjostrand, Seymour, Lonnblad)
• Starting from 2001/2002:
A lot of activity started (…triggered by LHC challenge: large phase space!)
1) General Algorithm to merge ME(2n)+PS in leading order
(Mangano, Krauss, Catani, Webber et al.)
included in SHERPA, ALPGEN, HERWIG++
2) Match NLO ME + Parton showers (Frixione, Webber, Nason et al.)
MC@NLO (on top of HERWIG)
ME Generators
Calculation of Hadron-Hadron Cross-Section
ij
ij2
2j2
1i21 t/dσd )μ,(xf )μ,(xf tddx dxσ
2T
2T2
22
2ij /pdpt
us
st/dσd :LO in scattering-qq :e.g.
2
9
4s
p 1p
3p
2p4p
p
241
231
211
)p (p u
)p (p t
)p (p s
:variables mMandelstam
Diverges for low PT0
The calculation of exact matrix elements is difficult (loops, divergences, cancellations between large positive/negative numbers)
The Parton Shower Approximation
FSRISR 22 :in n2 factorize
0Q :likeshowerspace
:Radiation State Inital2i
0Q :likeshowertime
:Radiation State Final2i
Hard 22 process calculation has all (external leg) partons on mass shellHowever, partons can be off-shell for short times (uncertainty principle)close to the hard interaction
Incoming partons radiate harder and harder partons
Outgoing partons radiate softer and softer partons
probablity unit with showers parton afterwards add
ME 22 by n2 :represent
neglected be can Q all :QQ all for if 2i
22i
For more complexreaction often notclear which subdiagramShould be treated asHardest double counting
Final State Parton shower
2121
22
21
113
4
2dxdx
xxxxs
))((
d
gqq
e.g.: Halzen&Martin chap 11
gqqee
2xxx
s/2Ex
321
jj
In the qg-collinear limit x21
)())((
)x-x-(2xxx
z1x
dzdxzx :zEE with
)Q(m s
dQdx
sm
x1
213
22
22
21
3
11iq
2213
2
2
213
2
222
22
112 zzzz
zz
dzz)(1z1
34
QdQ
2π
dx)x(1
xx34
)x(1dx
2πσ
dσ
2
2
2s
11
22
21
2
2s
gqq
collinear :1x or 1x
soft :1x and 1x
21
21
Altarelli-Parisi Splitting Function
dz (z)PQdQ
2π bca2
2s
bca
dP
20
2
ijs
2
log unit per )z(P 2
yprobabilit
with momentum its of z)-(1 and z fractions carrying
k and jpartons into splits i parton a increases Q As
π
α
:evolution of tionInterpreta
z-1
z134
P2
qgq
2z-1z2
n P 2
2f
qqg z)-(1 zz))-(1 z-(1
3P2
ggg
z
z-11
34
P qq
2 g
Iteration over branchingsgives final stateParton shower
The Sudakov Form Factor
))(d
exp(-))(exp(-
)(()()(
T
0
1-n
0i
dtdt
tPTtTP
TtTPTtTPTtP
somethingiisomething
n
iiisomething
n
iiinothingnothing
1
1
01
1
01 10
happens) gP(somethin1happens) P(nothing
)dtdt
(t)dPexp( (T)dP(T)dP
T
0
somethingsomethingfirst
)z)dz(P2πQ
Qdexp( dz (z)P
QdQ
2πdP bca
cb,
Q
Q
s2
2
bca2
2s
bca
2max
2
P to branch first time= P to branch times P that no branch before
Sudakov
Sudakov form factor approximates the virtual loop corrections
valid incollinear&softlimit x,Q2 small
• Details of Parton shower more complex, e.g. coherence, angular ordering
2
2
22
26
Tp
mPythiaIn
E
2
2
2
Q :6.3
Q :.
Q :Herwig In
22
22
2
2
1
1
1
1
3
4
2lnlnlnP
max
min
min
maxqgq ss
s
zz
zz
dzQdQ
Rate for one emission:
Rate for n emissions: nns
n 2ln)(P qgq
Parton shower include all corrections of type(better than analytical leading log)
nns
2ln
• Traditional ME/PS merging, e.g. in Pythia and Herwig
details different in all MC: generate phase space with PS,
correct first or hardest emission with ME probability
If WPS gives (real) parton shower phase space: correction factor WME/ WPS
In this way effectively the splitting functions are replaced by the ME
The reweighting only works, if
in the full phase space of gluon emission
This relation is not valid for higher parton configurations
reweighting procedure has to be satisfied
PS ME Merging
dPS
d ME
1
wME
)(d)(d PSME gqqgqq
Summary - …so far
Parton showers include soft and collinear radiation that is logarithmically enhanced(non-singular contributions are ignored) not enough gluons are emitted that have high energy and large angle from the shower initiator
Matrix elements gives a good description of specific parton topologies wherethe partons are energetic and well separated,They include the interference between amplitudes with same external partonsHowever, in the soft and collinear limit, they neglect interference betweenmultiple gluon emissions, e.g. angular ordering
Jet Rates in NLL Accuracy
2ini1qNLL2 )d,(dΔR
)d(q,)Δd(q,Δ )d(q,P q),(dΔ dq )d,(dΔ 2 R iniginiq
d
d1qgq1qini1q
NLL3
1
ini
No emission from each quark line
No emissionfrom quark line
No emission from internal lines
Branching at d2
Two possible historiesq and qbar can radiate
Cluster partons to jets using KT-algorithmStop at point where 2-jets (d1), 3-jets (d3) are resolved
1d
q2d
inid
inid
inid
)d(q,Δ )d(q,P dq )d,(dΔ 2 inig
d
d1qgq
2ini1q
1
ini
),(
),(),( since
q
ini
ini
dd
dddd
2
121
Sum over all possible branchings!
CKKW-Merging
resolved are jetsadditional
1,2,...n where d..d values resolution
determine to algo-K using partons the Cluster 3)
M to according momenta particle Choose 2)
P with
algo)-K (using n tymultiplici jetthe Select 1)
)(d (using pppp for level tree for ME Calculate 0)
ini1
T
n
(0)n
T
2inisdcba
n0
2
0
0
0
Nk
kk
n
n
reconstruct shower history
“nodal” values for tree diagramSpecifying kT sequence for event
Catani, Krauss, Kuhn, Webber (2001)
General scheme to merge parton showers with ME 2->n
1) Make exclusive ME topologies, exactly 2-jets, 3-jets etc.2) Calculate ME weight for exclusive topologies up to ME cut 3) Make parton shower and veto parton shower above ME-cut
Jet production
Jet evolution
ME exclusive all from onscontributi up sum 8)
created is it where scale is parton each for scale starting
dd with radiation all vetoing
ionconfigurat this for shower parton Generate 7)
1 to return otherwise],,[
of product the if ion,configurat Accept 6)
)(d :line external each for
d node next to d scale at node : line internal each for
)(d/)(d),( :weight Sudokov 5)
)(/)()...( )( :weight 4)
n
cut
s
ini,
kj
ini,ini,
sssss
10
21
R
d
d
dddd
dddd
j
j
kjkj
ninin
CKKW - ME-Weights and PS Veto
One could start at dcut,but this would create adip near dcut, so PS veto approach isbetter
This procedure is included in SHERPAFirst implementation exist for PYTHIA++, HERWIG++
Procedure can be generalised to pp (Krauss 2002)
Had we known the branching tree we should have computed the MEs like that
PS would not emit partons in addition to those in ME (exclusive)
Avoid duble counting, well separated partons already done via ME
CKKW Result
TeV 1.96s at Wpp X
ME+PS Pythia default (dashed)
ME 22
ME 23
ME 24
ME 25
ME 26
of parton i
Note that the CKKWwork up to NLL accuracy(for hadron collisions, noformal proof)
When using CKKW, alwaysmake sure that dini dependenceis small
MLM Mangano (2002)
1) Generate hard parton configuration for given n=Npart with ME, imposing 2) Define tree branching structure using KT-algo allowing only pairing consistent with color flow3) Compute s at the nodal values, but do not apply Sudakov factors4) Shower the hard event without any veto using Herwig/Pythia when done, find Njet jet with cone algorithm with if Npart<Njet reject event 5) Matched jets to hard partons using Only keep events, if each hard parton is uniquely contained in jets Events with Npart<Njet are rejected except for highest multiplicities
6) Define exclusive N-jet sample by requiring Npart =Njet
7) After matching, combine exclusive samples to one inclusive sample
minminmaxi ,, RREE ijTiT
jetT RE ,min
jetsiR ,min
Used in Alpgen
Hard parton
Shower parton
Npart=Njet
Event kept
Npart=Njet=3but Nmatch=2event rejected
Npart<Njet
reject for excl. samplekeep for incl. sample
soft double counting
collinear double counting
This is equivalent totSudakov reweighting
in CKKW (external lines)
Prevent parton shower harder than any emission by ME using cone algo:
Reminder: Next-To-Leading-Order calculations
Born: First-Order: RealFirst Order: Virtual
qqee
:)O( gqqge ee :..
:) O( s
qqees
:)O(
:0x3cancel each other (KNL-theorem), ifinfra-red singularities
One can show that for any observable where the NLO prediction is:
(x) dxd
BB
(x)V 2
dxd
B
sV x
xRs
R
)(dxd
Loop diagram
BR(x) lim 0x
Real and virtual contributions can be regularised by introducing integral in d=4-2dim.
2
)( )log(
1
1
10
1
1
0
21
0
xxdx
xdx
In this case: LO
sVR )( lim
0
BR(x) lim 0x
1
0
2
dxd
dxd
dxd
O(x))-(O limdOd
RVB
xdx
0
where:
(infra-red safeness)
Subtraction Method
1
02
1
02
010
11
)))(())(()())((
dOd
xOOBxOOxR
dxx
dxOOB ssR
))(( ))(()( 0
s 0OOxB
xOOxxR sx
Add and subtract locally a counter-term with same point-wise singular behaviour as R(x):
BR(x) lim 0x
Ellis, Ross, Terrano (1981)
Since
1
021
1
0
2ε )( )(
)(dOd
xxR
dxxx
xRdxxxR
dx sssR
1
0
regularised
Let us look at the real contribution:
By construction this integral is finite
Add and subtract counter-term
The only divergent term has B&V kinematicsand gets cancels against s B/2term of virtual contribution cancellation independent of Observable
1
0
00
xOOBxOOxR
dxOOB
ss
R
)))(())(()())((
2-dOd
22-2
))(()(
))((dxlimdOd 1
0 xBB
VB
BOOxxR
xOO ss
0
0
MC@NLO In previous methods, the IR singularities in ME are cut and bias is correctedMC@NLO includes the virtual diagrams to cancel the IR singularities
Frixione, Webber 2002
1) Total rates are accurate in NLO (normalisation is meaningful, in contrast to LO MC)2) Hard emissions are treated as in NLO computation (up to 23)3) Soft/collinear emissions are treated as in MC, i.e. using PS4) Smooth matching between hard and soft/collinear emissions5) output set of event using standard hadronisation models
Problem: in NLO singularities cancel bin-by-bin, when shower is attached not possible
Event generator including benefits from NLO computations
Objectives:
Basic Scheme:
1) Calculate NLO ME for n-body process using subtraction method (n+1 real, n virtual+Born)2) Calculate analytically how first shower emission off n-body topology populates n+1 phase space3) Subtract the shower expression from the real n+1 ME, consider rest as n-body4) Add shower to n and n+1 events
MC@NLO
)MM(M F)M(M F )(x)f(xfdΦdxdx MCab
ctab
CV,B,ab
22MC
MCab
Rab
32MC2a1a321
absubtdOd
Real MC
Born, virtual ME collinear remainder
Collinear counter-term
Introduce MC countertermsremove spurious NLO termsarising from the evolution of Born ME
Shower generating functionalswhose initial configurationis the 22 and 23 hard partonsreplaces formulafunction insubtraction method
produced are MM Since MCab
Rab weightsnegativesometimes
Introduce MC countertermsremove by hand non branchingprobability of Born term includedin showers
For pp X Y:
Counter terms are constructed by hand to reproduce behaviour ofcollinear singularities, they locally match the singular behaviour of real ME. They are specific to MC implementation So far, only HERWIG
• Standard MC Generators: Pythia, Herwig,…
22 ME (at most) + parton showers+hadronisation
• Matrix Element Generators for specific processes:
AcerMC, Alpgen, Gr@ppa, MadCup, Vecbos• Matrix Element Generator for arbitrary processes:
Amegic++/Sherpa, CompHep, Grace, MadEvent/MadGraph
• ME at NLO precision with event generator
MC@NLO, POWHEG
• Possible future development:
Automatic matching of NLO ME + NLO parton showers
Automatised NLO calculation plus matching• Present theory bottleneck to go to NNLO or NLO for more then 23
(two-loop diagrams)
Overview
LO onlyUnweighted eventsUp to 8 external particles
• F. Krauss, Matrix Elements and Parton Shower in Hadronic Interations, hep-ph/025283• T. Sjostrand, Monte Carlo Generator, hep-ph/0611247• S. Frixione, The Inclusion of Higher-Order QCD corrections into Parton Shower MC
hep-ph/0408316• S. Mrenna and P. Richardson, Matching ME and PS with Herwig and Pythia, hep-ph/0312274
Literature
