QCD & Monte Carlo Event Generators
Frank Krauss
Institute for Particle Physics PhenomenologyDurham University
HiggsTools School, Pre Saint-Didier, 2015
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QCD & Monte Carlo Event Generators
PART I: Introduction
1 QCD Basics: Scales & Kinematics
2 General Monte Carlo
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PART II: Monte Carlo for Perturbative QCD
3 Parton–level Monte Carlo
4 Parton showers – the basics
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PART III: Precision Simulations
5 First improvements
6 Matching
7 Multijet merging
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PART IV: Monte Carlo for Non–Perturbative QCD
8 Hadronisation
9 Underlying Event
10 Summary
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QCD Basics: Scales & Kinematics General Monte Carlo
PART I: INTRODUCTION
F. Krauss IPPP
QCD & Monte Carlo Event Generators
QCD Basics: Scales & Kinematics General Monte Carlo
QCD BASICS
SCALES & KINEMATICS
F. Krauss IPPP
QCD & Monte Carlo Event Generators
QCD Basics: Scales & Kinematics General Monte Carlo
Contents
1.a) Factorisation: an electromagnetic analogy
1.c) QED Initial and Final State Radiation
1.b) DGLAP equations in QED
1.d) Running of αs and bound states
1.e) Hadrons in initial state: DGLAP equations of QCD
1.f) Hadron production: Scales
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Factorisation: an electromagnetic analogy
An electromagnetic analogy
consider a charge Z moving at constant velocity v
v ≅ cv = 0 v ≅ c
at v = 0: radial E field onlyat v = c: B field emerges: ~E ⊥ ~B, ~B ⊥ ~v , ~E ⊥ ~v ,
energy flow ∼ Poynting vector ~S ∼ ~E × ~B, ‖ ~vapproximate classical fields by “equivalent quanta”: photons
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Factorisation: an electromagnetic analogy
spectrum of photons:(in dependence on energy ω and transverse distance b⊥)
dnγ =Z 2α
π· dωω· db
2⊥
b2⊥
electron(Z=1)−→ α
π· dωω· db
2⊥
b2⊥
Fourier transform to transverse momenta k⊥:
dnγ =α
π· dωω· dk
2⊥
k2⊥
note: divergences for k⊥ → 0 (collinear) and ω → 0 (soft)
therefore: Fock state for lepton = superposition (coherent):
|e〉phys = |e〉+ |eγ〉+ |eγγ〉+ |eγγγ〉+ . . .
photon fluctuations will “recombine”
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Factorisation: an electromagnetic analogy
lifetime of electron–photon fluctuations: e(P)→ e(p) + γ(k)
first estimate: use uncertainty relation and Lorentz time dilationP2 = (p + k)2 = M2
virt the virtual mass of the incident electronlife time = life time in rest frame · time dilation
τ ∼ 1
Mvirt· E
Mvirt=
E
(p + k)2∼ E
2Ek(1− cos θ)≈ k
k2 sin2 θ/2≈ ω
k2⊥
second estimate: use uncertainty relation and assume only photonoff-shell
energy balance of photon
P2 = 2p · k + k2 , therefore k2 ≈ −k2⊥ ≈ −2p · k < 0 .
assume photon momentum to be kµ = (ω,~k⊥, k‖),shift in energy for photon going on-shell: δω ∼ k2
⊥/ω, therefore
τγ ∼1
δω≈ ω
k2⊥≈ ω
ω2 sin2 θ≈ 1
ωθ2
lifetime larger with smaller transverse momentum(i.e. with larger transverse distance)
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QED Initial and Final State radiation
QED Initial and Final State Radiation
physical interpretation:equivalent quanta = quantum manifestation of accompanying fields
in absence of interaction: recombination enforced by coherence
but: hard interaction possibly “kicks out” quantum−→ coherence broken−→ equivalent (virtual) quanta become real−→ emission pattern unravels
alternative idea:initial state radiation of photons off incident electron
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QED Initial and Final State radiation
consider final state radiation in γ∗ → `¯
(electron velocities/momenta labelled as v and v ′/p and p′)
classical electromagnetic spectrum from radiation function:(this is from Jackson or any other reasonable book on ED)
d2I
dωdΩ=
e2
4π2
∣∣∣∣~ε ∗ · ( ~v
1− ~v · ~n −~v ′
1− ~v ′ · ~n
)∣∣∣∣2 ,with ε the polarisation vector and ~n(Ω) the direction of the radiation
recast with four–momenta, equivalent photon spectrum:
dN =d3k
(2π)32k0
α
π
∣∣∣∣ε∗µ( pµ
p · k −p′µ
p′ · k
)∣∣∣∣2=
d3k
(2π)32k0
α
π
∣∣∣∣Wpp′;k
∣∣∣∣2with the eikonal Wpp′;k
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QED Initial and Final State radiation
repeat exercise in QFT, Feynman diagrams:
k
p′
p p
p′
k
MX→e+e−γ = eu(p)
[Γ
p/′ − k/
(p′ − k)2γµ − γµ p/+ k/
(p + k)2Γ
]u(p′)ε∗µ(k)
soft−→ eε∗µ(k)
[pµ
p · k −p′µ
p′ · k
]u(p′)Γu(p) = eMX→e+e−γ ·Wpp′;k
manifestation of Low’s theorem:soft radiation independent of spin (→ classical)
(radiation decomposes into soft, classical part with logs – i.e. dominant – and hard collinear part)
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DGLAP equations for QED
DGLAP equations for QED
(Dokshitser–Gribov–Lipatov–Altarelli–Parisi Equations)
define probability to find electron or photon in electron:
at LO in α(noemission) : `(x , k2⊥) = δ(1− x)
and γ(x , k2⊥) = 0
(introduced x = energy fraction w.r.t. physical state)
including emissions:
probabilities changeenergy fraction ξ of lepton parton w.r.t. the physical lepton objectreduced by some fraction z = x/ξreminder: differential of photon number w.r.t. k2
⊥:
dnγ =α
π
dk2⊥
k2⊥
dω
ω↔ dnγ
d log k2⊥
=α
π
dx
x
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DGLAP equations for QED
evolution equations (trivialised)
d`(x , k2⊥)
d log k2⊥
=α(k2⊥)
2π
1∫x
dξ
ξP``(x
ξ, α(k2
⊥)
)`(ξ, k2
⊥)
dγ(x , k2⊥)
d log k2⊥
=α(k2⊥)
2π
1∫x
dξ
ξPγ`
(x
ξ, α(k2
⊥)
)`(ξ, k2
⊥) .
k2⊥ plays the role of “resolution parameter”
the Pab(z) are the splitting functions, encoding quantum mechanicsof the “splitting cross section”, for example (at LO)
P``(z) =
(1 + z2
1− z
)+
+3
2δ(1− z)
if γ → `¯ splittings included, have to add entries/splitting functionsinto evolution equations above
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Running of αs and bound states
Running of αs and bound states
quantum effect due to loops:couplings change with scale
running driven by β–function
β(αs) = µ2R
∂αs(µ2R)
∂µ2R
=β0
4πα2
s +β1
(4π)2α3
s + . . .
with
β0 =11
3CA −
4
3TRnf
β1 =34
3C 2A −
20
3CATRnf − 4CFTRnf
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Running of αs and bound states
Casimir operators in the fundamental and adjoint representation:
CF =N2
c − 1
2Ncand CA = Nc
with Nc = 3 colours and TR = 1/2.
nf = the number of (quark) flavours
the Casimirs correspond to quark and gluon colour charges
explicit expression for strong coupling
αs(µ2R) ≡ g2
s (µ2R)
4π=
1β0
4π logµ2R
ΛQCD2
with ΛQCD the Landau pole of QCD, ΛQCD ≈ 250MeV.
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Hadrons in initial state: DGLAP equations of QCD
Hadrons in initial state: DGLAP equations of QCD
similar to QED case:define probabilities (at LO) to find a parton q – quark or gluon – inhadron h at energy fraction x and resolution parameter/scale Q:
parton distribution function (PDF) fq/h(x , Q2)
scale-evolution of PDFs: DGLAP equations
∂
∂ logQ2
(fq/h(x , Q2)fg/h(x , Q2)
)
=αs(Q
2)
2π
1∫x
dz
z
(Pqq
(xz
)Pqg
(xz
)Pgq
(xz
)Pgg
(xz
) )( fq/h(z , Q2)fg/h(z , Q2)
),
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Hadrons in initial state: DGLAP equations of QCD
QCD splitting functions:
P(1)qq (x) = CF
[1 + x2
(1− x)++
3
2δ(1− x)
]=
[P(1)qq (x)
]+
+ γ(1)q δ(1− x)
P(1)qg (x) = TR
[x2 + (1− x)2
]= P(1)
qg (x)
P(1)gq (x) = CF
[1 + (1− x)2
x
]= P(1)
gq (x)
P(1)gg (x) = 2CA
[x
(1− x)++
1− x
x+ x(1− x)
]+
11CA − 4nf TR
6δ(1− x) =
[P(1)gg (x)
]+
+ γ(1)g δ(1− x) .
remark: IR regularisation by +–prescription &terms ∼ δ(1− x) from physical conditions on splitting functions
(flavour conservation for q → qg and momentum conservation for g → gg , qq)
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Hadrons in initial state: DGLAP equations of QCD
simple idea: proton = |uud〉(valence quarks) only
naively: no interactions−→ fu,d/p ∼ δ
(x − 1
3
)elastic interactions−→ Gaussian smearing
strong interactions:develop “sea” = soft partonswill depend on resolution scaleremember: dn ∝ logω log k2
⊥
in fact, due to g → gg , sea increases much faster,
fsea/p(x , Q2) ∼ x−λ , λ ≈ 1 .
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Hadrons in initial state: DGLAP equations of QCD
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Hadrons in the final state
Hadron production: Scales
consider QCD final state radiation
pattern for q → qg similar to `→ `γ in QED:
dwq→qg =αs(k
2⊥)
2πCF
dk2⊥
k2⊥
dω
ω
[1 +
(1− ω
E
)2]
ω=E(1−z)=
αs(k2⊥)
2πCF
dk2⊥
k2⊥
dz1 + z2
1− z=αs(k
2⊥)
2πCF
dk2⊥
k2⊥
dz P(1)qg (z) .
divergent structures for:
z → 1 (soft divergence) ←→ infrared/soft logarithmsk2⊥ → 0 (collinear/mass divergence) ←→ collinear logarithms
cut regularise with cut-off k⊥,min ∼ 1GeV > ΛQCD
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Hadrons in the final state
find two perturbative regimes:
a regime of jet production, where k⊥ ∼ k‖ ∼ ω k⊥,min andemission probabilities scale like w ∼ αs(k⊥) 1; anda regime of jet evolution, where k⊥,min ≤ k⊥ k‖ ≤ ω and
therefore emission probabilities scale like w ∼ αs(k⊥) log2 k2⊥>∼ 1.
in jet production:standard fixed–order perturbation theory
in jet evolution regime,perturbative parameter not αs any morebut rather towers of exp
[αs log k2
⊥ log k‖]
induces counting of leading logarithms (LL), αsL2n,
next-to leading logarithms (NLL), αsL2n−1, etc.
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Hadrons in the final state
consider spatio-temporal structure in classical QED case
assume a charge comes into existence at t = 0 with v = 0:in its rest frame radial ~E spreads out in sphere r ′ ≤ t′
assume charge moves with v → 1 and Lorentz factor E/m:then in lab frame field at radial distance r⊥ will arrive att = γt′ = Er⊥/m
translate to classical QCD:
light quarks with constituent mass m ≈ ΛQCD ≈ 1/Ror m = mQ for heavy quarks
(assume here typical hadron radius R)
identify r⊥ with typical hadronic size Rthen: hadronization time
t(had) ≈
ER2 for light quarks
ER
mQfor heavy quarks.
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Hadrons in the final state
repeat exercise in quantum mechanics
confining forces associated with gluons withk ≈ k⊥ ≈ k‖ ≈ 1/R ≈ m in hadronic rest frame
demand hadronization time ≥ formation time:
t(form) ≈ k‖k2⊥≤ k‖R
2 ≈ t(had)
therefore k⊥ ≥ 1/R = O (fewΛQCD)
therefore: breakdown of perturbative picture at scales/transversemomenta O (fewΛQCD)
“gluers” replace gluons
transition to bound states (phase transition)
no first–principle understanding: =⇒ models
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Hadrons in the final state
Summary
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GOING MONTE CARLO
GENERAL IDEAS & TECHNIQUES
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Contents
2.a) Prelude: selecting from a distribution
2.b) Monte Carlo integration: basic idea
2.c) Traditional MC simulation
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Selecting from a distribution
Prelude: Selecting from a distribution
a typical Monte Carlo/simulation problem:
wanted: random numbers x ∈ [xmin, xmax], distributed according to(probability) density f (x), i.e.
P(x ∈ [x ′, x ′ + dx ′]) = f (x ′)dx ′
but: only “usual” random numbers # available: “flat” in [0, 1]
exact solution:
must know integral F of density f and its inverse F−1
x given byx∫
xmin
dx ′f (x ′) = #
xmax∫xmin
dx ′f (x ′)
and therefore
x = F−1 [F (xmin) + # (F (xmax)− F (xmin))]
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Selecting from a distribution
case above is very untypical – integral sometimes known, inversepractically never
need a work-around solution: “Hit-or-miss”(solution, if exact case does not work.)
construct good “over-estimator” g(x) (G and G−1 known):
g(x) > f (x) ∀x ∈ [xmin, xmax]
exact algorithm to select a trial-xaccording to over-estimator g
accept x with probability f (x)/g(x)
obvious fall-back choice for g(x):
g(x) = Max[xmin, xmax]f (x).
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Monte Carlo integration: basic idea
Monte Carlo integration
underlying idea: determination of π with random number generator
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Monte Carlo integration: basic idea
MC integration: estimate integral by N probes
I(a,b)f =
b∫a
dxf (x)
−→ 〈I (a,b)f 〉 =
b − a
N
N∑i=1
f (xi ) = 〈f 〉a,b
where xi homogeneously distributed in [a, b]
error estimate from statistical sample =⇒ standard deviation
〈E (a,b)f (N)〉 = σ =
[〈f 2〉a,b − 〈f 〉2a,b
N
]1/2
independent of the number of integration dimensions!=⇒ method of choice for high-dimensional integrals.
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Monte Carlo integration: basic idea
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Monte Carlo integration: refinements
Monte Carlo integration: refinements
want to minimise number of potentially expensive function calls=⇒ need to improve convergence of MC integration.
first basic idea: sample in regions, where f largest( =⇒ corresponds to a Jacobean transformation of integral)
alternative algorithm: minimise error by “smoothing” integrand(”importance sampling”)
assume a function g(x) similar to f (x).f (x)/g(x) smooth =⇒ 〈E(f /g)〉 smallmust sample according to dx g(x) rather than dx :g(x) plays role of probability distribution; we know already how todeal with this!
works, if f (x) is well-known, but hard to generalise.
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Monte Carlo integration: refinements
importance sampling
consider f (x) = cos πx2 and g(x) = 1− x2:
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Monte Carlo integration: refinements
yet another idea: decompose integral in M sub-integrals
〈I (f )〉 =M∑j=1
〈Ij(f )〉
〈E (f )〉2 =M∑j=1
〈Ej(f )〉2
overall variance smallest, if “equally distributed”.(=⇒ sample, where the fluctuations are.)
(”stratified sampling”)
algorithm:
divide interval in bins (variable bin-size or weight);adjust such that variance identical in all bins.
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Monte Carlo integration: refinements
stratified sampling
consider f (x) = cos πx2 and g(x) = 1− x2:
〈I 〉 = 0.637± 0.147/√N
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Monte Carlo integration: refinements
a hybrid of stratified and importance sampling:replace independent bins of stratified sampling with independentfunctions of importance sampling
“bins” – with weight αi – of “eigenfunctions” – gi (x):
=⇒ g(~x) =∑N
i=1 αigi (~x).
in particle physics, this is the method of choice for parton level eventgeneration:
translate each Feynman diagram into one or more channelsoptimise interplay of channels, cuts, etc. through weights αi
optional: add VEGAS to “best” channels
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Traditional MC simulation
Traditional MC simulation
a classical example: two-dimensional Ising model:(spins si fixed on 2-D lattice with nearest neighbour interactions.)
H = −J∑〈ij〉
si sj
evaluation of observable O by summing over all micro states φi,given as spin ensembles (similar to path integral in QFT.)
〈O〉 =
∫Dφi Tr
O(φi) exp
[−H(φi)
kBT
]typical problem in such calculations (integrations!):phase space too large =⇒ need to sample.
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Traditional MC simulation
Metropolis algorithm simulates the canonical ensemble,summing/integrating over micro-states with MC method.
necessary ingredient: interactions among spins in probabilisticlanguage (this will come back to us!)
algorithm:
go over the spins,check whether they flip:
compare Pflip with random numberPflip from energies of the two micro-states (before and after flip) andBoltzmann factors
repeat to equilibrium.evaluate observables directly during run &take thermal average
(average over many steps).
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Traditional MC simulation
why does this work? detailed balance!
consider one spin flip, connecting micro-states 1 and 2.rate of transitions given by the transition probabilities Wif E1 > E2 then W1→2 = 1 and W2→1 = exp
(− E1−E2
kBT
)in thermal equilibrium, both transitions equally often:
P2W2→1 = P1W1→2
takes into account that the respective states are occupied accordingto their Boltzmann factors.
(Pi ∼ exp(−Ei/kBT ))
in principle, all systems in thermal equilibrium can be studied withMetropolis - just need to write transition probabilities in accordancewith detailed balance, as above =⇒ general simulation strategy inthermodynamics.
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Traditional MC simulation
example results on a 10× 10 lattice
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Parton–level Monte Carlo Parton showers – the basics
PART II: MONTE CARLO
FOR PERTURBATIVE QCD
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Parton–level Monte Carlo Parton showers – the basics
MONTE CARLO FOR
PARTON LEVEL
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Parton–level Monte Carlo Parton showers – the basics
Contents
3.a) Calculating matrix elements efficiently
3.b) Phase spacing for professionals
3.c) Including higher order corrections
3.d) Cancellation of IR divergences
3.e) Tools for LHC physics
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Parton–level Monte Carlo Parton showers – the basics
Calculating matrix elements efficiently
Simulating hard processes (signals & backgrounds)
Simple example: t → bW+ → blνl :
|M|2 = 12
(8πα
sin2 θW
)2pt ·pν pb·pl
(p2W−M
2W )2+Γ2
WM2W
Phase space integration (5-dim):
Γ = 12mt
1128π3
∫dp2
Wd2ΩW
4πd2Ω4π
(1− p2
W
m2t
)|M|2
5 random numbers =⇒ four-momenta =⇒ “events”.
Apply smearing and/or arbitrary cuts.
Simply histogram any quantity of interest - no new calculation foreach observable
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Calculating matrix elements efficiently
Calculating matrix elements efficiently
stating the problem(s):
multi-particle final states for signals & backgrounds.need to evaluate dσN :∫
cuts
[N∏i=1
d3qi(2π)32Ei
]δ4
(p1 + p2 −
∑i
qi
)|Mp1p2→N |2 .
problem 1: factorial growth of number of amplitudes.problem 2: complicated phase-space structure.solutions: numerical methods.
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Calculating matrix elements efficiently
example for factorial growth: e+e− → qq + ng
n #diags
0 11 22 83 484 384
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Calculating matrix elements efficiently
obvious: traditional textbook methods (squaring, completenessrelations, traces) fail=⇒ result in proliferation of terms (MiM∗j )
better ideas of efficient ME calculation:=⇒ realise: amplitudes just are complex numbers,=⇒ add them before squaring!
remember: spinors, gamma matrices have explicit formcould be evaluated numerically (brute force)but: Rough method, lack of elegance, CPU-expensive
can do better with smart basis for spinors (see next slide)
this is still on the base of traditional Feynman diagrams!
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Calculating matrix elements efficiently
helicity method:
introduce basic helicity spinors (needs to “gauge”-vectors)write everything as spinor products, e.g.
u(p1, h1)u(p2, h2) = complex numbers.have completeness relations such as
(p/+ m) =⇒ 1
2
∑h
[(1 +
m2
p2
)u(p, h)u(p, h)
+
(1− m2
p2
)v(p, h)v(p, h)
]there are other genuine expressions . . .translate Feynman diagrams into “helicity amplitudes”:complex-valued functions of momenta & helicities.spin-correlations etc. nearly come for free.
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Calculating matrix elements efficiently
taming the factorial growth in the helicity method
by reusing pieces: calculate only once!factoring out: reduce number of multiplications!
can be implemented as a-posteriori manipulations of amplitudes.
; Z
e
+
; Z
e
+
e
e
+
e
+
; Z
e
+
; Z
e
+
e
e
+
e
+
better method: recursion relations (recycling built in).best candidate so far: off-shell recursions
(Dyson-Schwinger, Berends-Giele etc.)
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
Calculating matrix elements efficiently
improvement: off-shell recursion relations
general idea: recursively construct generalised currents Jα(π) for aset π of external particles on their mass shell plus one internal one
Jα(π) = Pα(π)
∑P2(π)
∑Vα1α2α
[S(π1, π2)Vα1α2α Jα1 (π1)Jα2 (π2)]
+∑P3(π)
∑Vα1α2α3α
[S(π1, π2, π3)Vα1α2α3α Jα1 (π1)Jα2 (π2)Jα3 (π3)]
Pα(π) denotes the propagator denominatorS(π1, π2) and S(π1, π2, π3) for symmetry factorsVα for three– and four–particle verticesgo over all permutations of external particles π
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Parton–level Monte Carlo Parton showers – the basics
Calculating matrix elements efficiently
recursion relations particularly powerful due to massive recycling asintegral part of structure −→ bookkeeping problem only
there are sub–classes of particularly simple amplitudes:maximally helicity violating (MHV) amplitudes
all-gluon amplitudes with helicities given(Parke–Taylor/Berends–Giele amplitudes)
A(1+, 2+, . . . , i−, . . . , j−, . . . . . . n+) = ign−2s
〈ij〉4〈12〉〈23〉 . . . 〈(n − 1)n〉〈n1〉
A(1−, 2−, . . . , i+, . . . , j+, . . . . . . n−) = ign−2s
[ij ]4
[12][23] . . . [(n − 1)n][n1].
terms [ij ] etc. are products of two–component left– andright–handed Weyl spinors (particularly simple)
note: all-sign identical amplitudes vanish due to the conservation ofangular momentum
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Parton–level Monte Carlo Parton showers – the basics
Calculating matrix elements efficiently
in principle also factorial growth with number of colours
sampling over colours improves situation.(but still, e.g. naively ' (n − 1)! permutations/colour-ordering for n external gluons).
improved scheme: colour dressing
T ai jT
akl = δi lδkj−
1
Ncδi jδkl ←→
li
j k
− 1
Nc
li
j k
works very well with Berends-Giele recursions
Time [s] for the evaluation of 104 phase space points, sampled over helicities & colour.
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Parton–level Monte Carlo Parton showers – the basics
Phase spacing for professionals
Phase spacing for professionals
(“Amateurs study strategy, professionals study logistics”)
democratic, process-blind integration methods:
Rambo/Mambo: Flat & isotropicR.Kleiss, W.J.Stirling & S.D.Ellis, Comput. Phys. Commun. 40 (1986) 359;
HAAG/Sarge: Follows QCD antenna patternA.van Hameren & C.G.Papadopoulos, Eur. Phys. J. C 25 (2002) 563.
multi-channelling: each Feynman diagram related to a phase spacemapping (= ”channel”), optimise their relative weights
R.Kleiss & R.Pittau, Comput. Phys. Commun. 83 (1994) 141.
main problem: practical only up to O(10k) channels.
some improvement by building phase space mappings recursively:more channels feasible, efficiency drops a bit.
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Parton–level Monte Carlo Parton showers – the basics
Phase spacing for professionals
basic idea of multichannel sampling (again):use a sum of functions gi (~x) as Jacobean g(~x).
=⇒ g(~x) =∑N
i=1 αigi (~x);=⇒ condition on weights like stratified sampling;(“combination” of importance & stratified sampling).
algorithm for one iteration:
select gi with probability αi → ~xj .
calculate total weight g(~xj ) and partial weights gi (~xj )
add f (~xj )/g(~xj ) to total result and f (~xj )/gi (~xj ) to partial
(channel-) results.
after N sampling steps, update a-priori weights.
this is the method of choice for parton level event generation!
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Parton–level Monte Carlo Parton showers – the basics
Phase spacing for professionals
quality measure for integration performance: unweighting efficiency
want to generate events “as in nature”.
basic idea: use hit-or-miss method;
generate ~x with integration method,compare actual f (~x) with maximal value during sampling=⇒ “Unweighted events”.
comments:
unweighting efficiency, weff = 〈f (~xj)/fmax〉 = number of trials foreach event.expect log10 weff ≈ 3− 5 for good integration of multi-particle finalstates at tree-level.maybe acceptable to use fmax,eff = Kfmax with K < 1.problem: what to do with events where f (~xj)/fmax,eff > 1?answer: Add int[f (~xj)/fmax,eff ] = k events and perform hit-or-misson f (~xj)/fmax,eff − k.
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Parton–level Monte Carlo Parton showers – the basics
Including higher order corrections
Including higher order corrections
obtained from adding diagrams with additional:loops (virtual corrections) orlegs (real corrections)
effect: reducing the dependence on µR & µF
NLO allows for meaningful estimate of uncertainties
additional difficulties when going NLO:ultraviolet divergences in virtual correctioninfrared divergences in real and virtual correction
enforceUV regularisation & renormalisationIR regularisation & cancellation
(Kinoshita–Lee–Nauenberg–Theorem)
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Parton–level Monte Carlo Parton showers – the basics
Including higher order corrections
traditional bottleneck of higher–order calculations: virtual parts
algorithm before about 2005:
Passarino–Veltman reduction of tensors in numerator(replace 2p · k = (p + k)2 − p2 − k2)
reduce to scalar master integrals of the form∫dDk
[(p1 + k)2(p2 + k)2 . . . ]
further reduce to integrals with up to four propagators only(but careful: introduces instabilities through “Gram determinants”)
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Parton–level Monte Carlo Parton showers – the basics
Including higher order corrections
about 2005: begin of “NLO revolution”
basic idea: reduce to master integrals numerically by cutting
→ −p123
↓ p3
↑ p1
← p2
ℓ + p12
ℓ + p123
ℓ
ℓ + p1
coefficients of master integrals emerge as solutions of linearequations
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QCD & Monte Carlo Event Generators
Parton–level Monte Carlo Parton showers – the basics
Cancellation of infrared divergences
Cancellation of infrared divergencesneed a mechanism to cancel IR divergences for higher multiplicitiesin final states
toy model in one dimension:
|MRm+1|2 =
1
xR(x) and |MV
m|2 =1
εV ,
where x = gluon energy & regularised in d = 4− 2ε dimensions.Cross section in d dimensions with jet measure F J :
σ =
1∫0
dx
x1+εR(x)F J
1 (x) +1
εVF J
0
infrared safety of jet measure: F J1 (0) = F J
0
=⇒ ”a soft/collinear parton has no effect.”(tricky issue - without it, no reliable NLO calculation!)
KLN theorem: R(0) = V .
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
Cancellation of infrared divergences
rewrite toy-model cross section as
σ =
1∫0
dx
x1+εR(x)F J
1 (x)−1∫
0
dx
x1+εVF J
0 +
1∫0
dx
x1+εVF J
0 +1
εVF J
0
=
1∫0
dx
x1+ε
(R(x)F J
1 (x)− VF J0
)+O(1)VF J
0 .
two separately finite integrals, with no large numbers to beadded/subtracted.
subtraction terms are universal (analytic bit can be calculated onceand for all).
this has been automated in two schemes: Catani-Seymour andFrixione-Kunszt-Signer
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Parton–level Monte Carlo Parton showers – the basics
Cancellation of infrared divergences
general structure of NLO calculation for N-body production
dσ = dΦBBN(ΦB) + dΦBVN(ΦB) + dΦRRN(ΦR)
= dΦB(BN + VN + I(S)
N
)+ dΦR (RN − SN)
phase space factorisation assumed here (ΦR = ΦB ⊗ Φ1)∫dΦ1SN(ΦB ⊗ Φ1) = I(S)
N (ΦB)
process independent subtraction kernels
SN(ΦB ⊗ Φ1) = BN(ΦB) ⊗ S1(ΦB ⊗ Φ1)
I(S)N (ΦB ⊗ Φ1) = BN(ΦB) ⊗ I(S)
1 (ΦB)
with universal S1(ΦB ⊗ Φ1) and I(S)1 (ΦB)
in Catani–Seymour invertible phase space mapping
ΦR ←→ ΦB ⊗ Φ1
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
Cancellation of infrared divergences
Aside: choices . . .
common lore: NLO calculations reduce scale uncertainties
this is, in general, true. however:unphysical scale choices will yield unphysical results
so maybe we have to be a bit smarter than just running NLO code
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
Tools for LHC physics
Availability of exact calculations (hadron colliders)
fixed order matrix elements (“parton level”) are exact to a givenperturbative order. (and often quite a pain!)
important to understand limitations:only tree-level and one-loop level fully automated, beyond: prototyping
done
for some processes
first solutions
n legs1 2 3 4 5 6 7 8 9
1
2
0
m loops
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Parton–level Monte Carlo Parton showers – the basics
Tools for LHC physics
Survey of existing parton-level tools @ tree–level
Models 2 → n Ampl. Integ. public? lang.
ALPGEN SM n = 8 rec. Multi yes FortranAMEGIC++ SM, UFO n = 6 hel. Multi yes C++COMIX SM, UFO n = 8 rec. Multi yes C++COMPHEP SM, LANHEP n = 4 trace 1Channel yes CHELAC SM n = 8 rec. Multi yes FortranMADEVENT SM, UFO n = 6 hel. Multi yes Python/FortranWHIZARD SM, UFO n = 8 rec. Multi yes O’Caml
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Parton–level Monte Carlo Parton showers – the basics
Tools for LHC physics
Survey of existing parton-level tools @ NLO
type technologydependencies on other codes
LOOPTOOLS integrals
ONELOOP integrals
QCDLOOP integrals
COLLIER reduction
CUTTOOLS reduction OPP
FORMCALC reduction PV
NINJA reduction Laurent expansion
SAMURAI reduction
BLACKHAT library (amplitudes) OPP (unitarity)
MCFM library (full calculation) PV & OPP
MJET library (amplitudes) OPP
GOSAM generator (amplitudes) OPPSAMURAI +NINJA +
MADLOOP generator (full calculation) OL+OPPCUTTOOLS +
OPENLOOPS generator (amplitudes) OL+OPPCOLLIER +CUTTOOLS +
RECOLA generator (amplitudes) TRCOLLIER +CUTTOOLS +
HELAC-NLO generator (full calculation) OPPCUTTOOLS +
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QCD & Monte Carlo Event Generators
Parton–level Monte Carlo Parton showers – the basics
GOING MONTE CARLO
PARTON SHOWERS – THE BASICS
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Parton–level Monte Carlo Parton showers – the basics
Contents
4.a) An analogy: radioactive decays
4.b) The pattern of QCD radiation
4.c) Quantum improvements
4.d) Compact notation
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
An analogy: radioactive decays
An analogy: Radioactive decays
consider radioactive decay of an unstable isotope with half-life τ .(and ignore factors of ln 2.)
“survival” probability after time t is given by
S(t) = Pnodec(t) = exp[−t/τ ]
(note “unitarity relation”: Pdec(t) = 1 − Pnodec(t).)
probability for an isotope to decay at time t:
dPdec(t)
dt= −dPnodec(t)
dt=
1
τexp(−t/τ)
now: connect half-life with width Γ = 1/τ .
probability for the isotope to decay at any fixed time t determined by Γ.
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Parton–level Monte Carlo Parton showers – the basics
An analogy: radioactive decays
spice things up now: add time–dependence, Γ = Γ(t ′)
rewrite
Γt −→t∫
0
dt ′Γ
decay-probability at a given time t is given by
dPdec(t)
dt= Γ(t) exp
− t∫0
dt ′Γ(t ′)
= Γ(t)Pnodec(t)
(unitarity strikes again: dPdec(t)/dt = −dPnodec(t)/dt.)
interpretation of l.h.s.:
first term is for the actual decay to happen.
second term is to ensure that no decay before t=⇒ conservation of probabilities.the exponential is - of course - called the Sudakov form factor.
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
The pattern of QCD radiation
The pattern of QCD radiation
a detour: Altarelli-Parisi equation, once more
AP describes the scaling behaviour of the parton distribution function(which depends on Bjorken-parameter and scale Q2)
dq(x , Q2)
d lnQ2=
1∫x
dy
y
[αs(Q2)Pq(x/y)
]q(y ,Q2)
term in square brackets determines the probability that the parton emitsanother parton at scale Q2 and Bjorken-parameter y
(after the splitting, x → yx + (1 − y)x .)
driving term: Splitting function Pq(x)important property: universal, process independent
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Parton–level Monte Carlo Parton showers – the basics
The pattern of QCD radiation
differential cross section for gluon emission in e+e− → jets
dσee→3j
dx1dx2= σee→2j
CFαs
π
x21 + x2
2
(1− x1)(1− x2)
singular for x1,2 → 1.
rewrite with opening angle θqg and gluon energy fraction x3 = 2Eg/Ec.m.:
dσee→3j
d cos θqgdx3= σee→2j
CFαs
π
[2
sin2 θqg
1 + (1− x3)2
x3− x3
]singular for x3 → 0 (“soft”), sin θqg → 0 (“collinear”).
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
The pattern of QCD radiation
re-express collinear singularities
2d cos θqg
sin2 θqg=
d cos θqg1− cos θqg
+d cos θqg
1 + cos θqg
=d cos θqg
1− cos θqg+
d cos θqg1− cos θqg
≈ dθ2qg
θ2qg
+dθ2
qg
θ2qg
independent evolution of two jets (q and q)
dσee→3j ≈ σee→2j
∑j∈q,q
CFαs
2π
dθ2jg
θ2jg
P(z) ,
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Parton–level Monte Carlo Parton showers – the basics
The pattern of QCD radiation
note: same form for any t ∝ θ2:
transverse momentum k2⊥ ≈ z2(1− z)2E 2θ2
invariant mass q2 ≈ z(1− z)E 2θ2
dθ2
θ2≈ dk2
⊥k2⊥≈ dq2
q2
parametrisation-independent observation:(logarithmically) divergent expression for t → 0.
practical solution: cut-off Q20 .
=⇒ divergence will manifest itself as logQ20 .
similar for P(z): divergence for z → 0 cured by cut-off.
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
The pattern of QCD radiation
what is a parton?collinear pair/soft parton recombine!
introduce resolution criterion k⊥ > Q0.
combine virtual contributions with unresolvable emissions:cancels infrared divergences =⇒ finite at O(αs)(Kinoshita-Lee-Nauenberg, Bloch-Nordsieck theorems)
unitarity: probabilities add up to oneP(resolved) + P(unresolved) = 1.
+ =1.
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Parton–level Monte Carlo Parton showers – the basics
The pattern of QCD radiation
the Sudakov form factor, once more
differential probability for emission between q2 and q2 + dq2:
dP =αs
2π
dq2
q2
zmax∫zmin
dzP(z) =: dq2 Γ(q2)
from radioactive example: evolution equation for ∆
−d∆(Q2, q2)
dq2= ∆(Q2, q2)
dPdq2
= ∆(Q2, q2)Γ(q2)
=⇒ ∆(Q2, q2) = exp
− Q2∫q2
dk2Γ(k2)
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
The pattern of QCD radiation
maximal logs if emissions ordered
impacts on radiation pattern: in each emission t becomes smaller
q21 > q2
2 > q23 , q2
1 > q′22
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
Quantum improvements
Quantum improvements
improvement: inclusion of various quantum effects
trivial: effect of summing up higher orders (loops) αs → αs(k2⊥)
much faster parton proliferation, especially for small k2⊥.
avoid Landau pole: k2⊥ > Q2
0 Λ2QCD =⇒ Q2
0 = physical parameter.
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
Quantum improvements
soft limit for single emission also universal
problem: soft gluons come from all over (not collinear!)quantum interference? still independent evolution?
answer: not quite independent.
consider case in QED
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Parton–level Monte Carlo Parton showers – the basics
Quantum improvements
assume photon into e+e− at θee and photon off electron at θphoton momentum denoted as k
energy imbalance at vertex: kγ⊥ ∼ k‖θ, hence ∆E ∼ k2⊥/k‖ ∼ k‖θ
2.
formation time for photon emission: ∆t ∼ 1/∆E ∼ k‖/k2⊥ ∼ 1/(k‖θ
2).
ee-separation: ∆b ∼ θee∆t
must be larger than transverse wavelength of photon:θee/(k‖θ
2) > 1/k⊥ = 1/(k‖θ)
thus: θee > θ must be satisfied for photon to form
angular ordering as manifestation of quantum coherence
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
Quantum improvements
pictorially:
=⇒gluons at large angle from combined colour charge!
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Parton–level Monte Carlo Parton showers – the basics
Quantum improvements
experimental manifestation:∆R of 2nd & 3rdjetinmulti − jeteventsinpp − collisions
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Parton–level Monte Carlo Parton showers – the basics
Parton showers, compact notation
Parton showers, compact notation
Sudakov form factor (no-decay probability)
∆(K)ij,k (t, t0) = exp
[−
t∫t0
dt
t
αs
2π
∫dz
dφ
2πKij,k(t, z , φ)︸ ︷︷ ︸splitting kernel for
(ij)→ ij (spectator k)
]
evolution parameter t defined by kinematicsgeneralised angle (HERWIG ++) or transverse momentum (PYTHIA, SHERPA)
will replacedt
tdz
dφ
2π−→ dΦ1
scale choice for strong coupling: αs(k2⊥) resums classes of higher logarithms
regularisation through cut-off t0
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
Parton showers, compact notation
“compound” splitting kernels Kn and Sudakov form factors ∆(K)n
for emission off n-particle final state:
Kn(Φ1) =αs
2π
∑all ij,k
Kij,k(Φij,k) , ∆(K)n (t, t0) = exp
[−
t∫t0
dΦ1Kn(Φ1)
]
consider first emission only off Born configuration
dσB = dΦN BN(ΦN)
·
∆(K)N (µ2
N , t0) +
µ2N∫
t0
dΦ1
[KN(Φ1)∆
(K)N (µ2
N , t(Φ1))
]︸ ︷︷ ︸
integrates to unity −→ “unitarity” of parton shower
further emissions by recursion with Q2 = t of previous emission
F. Krauss IPPP
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Parton–level Monte Carlo Parton showers – the basics
Parton showers, compact notation
analyse connection to QT resummation formalism
consider standard Collins-Soper-Sterman formalism (CSS):
dσAB→X
dydQ2⊥
= dΦX Bij(ΦX ) ·∫
d2b⊥(2π)2
exp(i~b⊥ · ~Q⊥)︸ ︷︷ ︸guarantee 4-mom conservation
Wij(b; ΦX )︸ ︷︷ ︸higher orders
with
Wij(b; ΦX ) =
collinear bits︷ ︸︸ ︷Ci (b; ΦX , αs)Cj(b; ΦX , αs)
loops︷ ︸︸ ︷Hij(αs)
exp
− Q2X∫
1/b2⊥
dk2⊥
k2⊥
(A(αs(k
2⊥)) log
Q2X
k2⊥
+ B(αs(k2⊥))
)︸ ︷︷ ︸
Sudakov form factor, A, B expanded in powers of αs
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Parton–level Monte Carlo Parton showers – the basics
Parton showers, compact notation
analyse structure of emissions above
logarithmic accuracy in logµN
k⊥(a la CSS)
possibly up to next-to leading log,
if evolution parameter ∼ transverse momentum,if argument in αs is ∝ k⊥ of splitting,if Kij,k → terms A1,2 and B1 upon integration
(OK, if soft gluon correction is included, and if Kij,k → AP splitting kernels)
resummed
in PS
L
αn
m
NLL
in CSS k⊥ typically is the transverse momentum of produced system, inparton shower of course related to the cumulative effect of explicitmultiple emissions
resummation scale µN ≈ µF given by (Born) kinematics –simple for cases like qq′ → V , gg → H, . . .tricky for more complicated cases
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First improvements Matching Multijet merging
ROUND III: PRECISION MONTE CARLO
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
FIRST IMPROVEMENTS:
ME CORRECTIONS
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Contents
5.a) Improving event generators
5.b) Matrix-element corrections
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Improving event generators
Improving event generatorsThe inner working of event generators. . . simulation: divide et impera
hard process:fixed order perturbation theory
traditionally: Born-approximation
bremsstrahlung:resummed perturbation theory
hadronisation:phenomenological models
hadron decays:effective theories, data
”underlying event”:phenomenological models
F. Krauss IPPP
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First improvements Matching Multijet merging
Improving event generators
. . . and possible improvementspossible strategies:
improving the phenomenological models:
“tuning” (fitting parameters to data)replacing by better models, based on more physics
(my hot candidate: “minimum bias” and “underlying event” simulation)
improving the perturbative description:
inclusion of higher order exact matrix elements and correctconnection to resummation in the parton shower:
“NLO-Matching” & “Multijet-Merging”
systematic improvement of the parton shower:next-to leading (or higher) logs & colours
F. Krauss IPPP
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First improvements Matching Multijet merging
Improving event generators
remember structure of NLO calculation for N-body production
dσ = dΦBBN(ΦB) + dΦBVN(ΦB) + dΦRRN(ΦR)
= dΦB(BN + VN + I(S)
N
)+ dΦR (RN − SN)
phase space factorisation assumed here (ΦR = ΦB ⊗ Φ1)∫dΦ1SN(ΦB ⊗ Φ1) = I(S)
N (ΦB)
process independent subtraction kernels
SN(ΦB ⊗ Φ1) = BN(ΦB) ⊗ S1(ΦB ⊗ Φ1)
I(S)N (ΦB ⊗ Φ1) = BN(ΦB) ⊗ I(S)
1 (ΦB)
with universal S1(ΦB ⊗ Φ1) and I(S)1 (ΦB)
in Catani–Seymour invertible phase space mapping
ΦR ←→ ΦB ⊗ Φ1
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First improvements Matching Multijet merging
Matrix element corrections
Matrix element corrections
parton shower ignores interferencestypically present in matrix elements
pictorially
ME :
∣∣∣∣ +∣∣∣∣2
PS :
∣∣∣∣ ∣∣∣∣2+∣∣∣∣ ∣∣∣∣2 0
0.2
0.4
0.6
0.8
1
1x
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ME over PS
form many processes RN < BN ×KN
typical processes: qq′ → V , e−e+ → qq, t → bW
practical implementation: shower with usual algorithm, but rejectfirst/hardest emissions with probability P = RN/(BN ×KN)
F. Krauss IPPP
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First improvements Matching Multijet merging
Matrix element corrections
analyse first emission, given by
dσB = dΦN BN(ΦN)
·
∆(R/B)N (µ2
N , t0) +
µ2N∫
t0
dΦ1
[RN(ΦN × Φ1)
BN(ΦN)∆
(R/B)N (µ2
N , t(Φ1))
]︸ ︷︷ ︸
once more: integrates to unity −→ “unitarity” of parton shower
radiation given by RN (correct at O(αs))(but modified by logs of higher order in αs from ∆
(R/B)N
)
emission phase space constrained by µN
also known as “soft ME correction”hard ME correction fills missing phase space
used for “power shower”:µN → Epp and apply ME correction
resummed
in PS
L
αn
m
NLL
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
PRECISION MONTE CARLO
NLO MATCHING
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Contents
6.a) Basic idea
6.b) Powheg
6.c) MC@NLO
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Basic idea
NLO matching: Basic idea
parton shower resums logarithmsfair description of collinear/soft emissionsjet evolution (where the logs are large)
matrix elements exact at given orderfair description of hard/large-angle emissionsjet production (where the logs are small)
adjust (“match”) terms:
cross section at NLO accuracy &correct hardest emission in PS to exactlyreproduce ME at order αs
(R-part of the NLO calculation)(this is relatively trivial)
maintain (N)LL-accuracy of parton shower(this is not so simple to see)
resummed
in PS
L
αn
m
NLL
F. Krauss IPPP
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First improvements Matching Multijet merging
POWHEG
POWHEG
reminder: Kij,k reproduces process-independent behaviour of RN/BN insoft/collinear regions of phase space
dΦ1RN(ΦN+1)
BN(ΦN)IR−→ dΦ1
αs
2πKij,k(Φ1)
define modified Sudakov form factor (as in ME correction)
∆(R/B)N (µ2
N , t0) = exp
− µ2N∫
t0
dΦ1RN(ΦN+1)
BN(ΦN)
,assumes factorisation of phase space: ΦN+1 = ΦN ⊗ Φ1
typically will adjust scale of αs to parton shower scale
F. Krauss IPPP
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First improvements Matching Multijet merging
POWHEG
define local K -factors
start from Born configuration ΦN with NLO weight:(“local K -factor”)
dσ(NLO)N = dΦN B(ΦN)
= dΦN
BN(ΦN) + VN(ΦN) + BN(ΦN)⊗ S︸ ︷︷ ︸
VN (ΦN )
+
∫dΦ1 [RN(ΦN ⊗ Φ1)− BN(ΦN)⊗ dS(Φ1)]
by construction: exactly reproduce cross section at NLO accuracy
note: second term vanishes if RN ≡ BN ⊗ dS(relevant for MC@NLO)
F. Krauss IPPP
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First improvements Matching Multijet merging
POWHEG
analyse accuracy of radiation pattern
generate emissions with ∆(R/B)N (µ2
N , t0):
dσ(NLO)N = dΦN B(ΦN)
×
∆(R/B)N (µ2
N , t0) +
µ2N∫
t0
dΦ1RN(ΦN ⊗ Φ1)
BN(ΦN)∆
(R/B)N (µ2
N , k2⊥(Φ1))
︸ ︷︷ ︸integrating to yield 1 - “unitarity of parton shower”
radiation pattern like in ME correction
pitfall, again: choice of upper scale µ2N (this is vanilla POWHEG!)
apart from logs: which configurations enhanced by local K -factor( K -factor for inclusive production of X adequate for X+ jet at large p⊥?)
F. Krauss IPPP
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First improvements Matching Multijet merging
POWHEG
large enhancement at high pT ,h
can be traced back to large NLO correction
fortunately, NNLO correction is also large → ∼ agreement
F. Krauss IPPP
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First improvements Matching Multijet merging
POWHEG
improving POWHEG
split real-emission ME as
R = R(
h2
p2⊥ + h2︸ ︷︷ ︸R(S)
+p2⊥
p2⊥ + h2︸ ︷︷ ︸R(F )
)
can “tune” h to mimick NNLO - or other(resummation) result
differential event rate up to first emission
dσ = dΦB B(R(S))
[∆(R(S)/B)(s, t0) +
s∫t0
dΦ1R(S)
B ∆(R(S)/B)(s, k2⊥)
]
+dΦR R(F )(ΦR)
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MC@NLO
MC@NLO
MC@NLO paradigm: divide RN in soft (“S”) and hard (“H”) part:
RN = R(S)N +R(H)
N = BN ⊗ dS1 +HN
identify subtraction terms and shower kernels dS1 ≡∑ij,k
Kij,k
(modify K in 1st emission to account for colour)
dσN = dΦN BN(ΦN)︸ ︷︷ ︸B+V
[∆
(K)N (µ2
N , t0) +
µ2N∫
t0
dΦ1Kij,k(Φ1) ∆(K)N (µ2
N , k2⊥)
]
+dΦN+1HN
effect: only resummed parts modified with local K -factor
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MC@NLO
phase space effects: shower vs. fixed order
Sherpa
NLO
α = 1
α = 0.3
α = 0.1
α = 0.03
α = 0.01
α = 0.003
α = 0.001
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1Rapidity separation betw. Higgs and leading jet in pp → h+ X
dσ/d
∆y(h,
1stjet)
[pb]
-4 -3 -2 -1 0 1 2 3 40.5
1
1.5
2
∆y(h, 1st jet)
Ratio
problem: impact of subtraction terms on local K -factor(filling of phase space by parton shower)
studied in case of gg → H above
proper filling of available phase space by parton shower paramount
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MC@NLO
MC@NLO for light jets: jet-p⊥
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bSherpa+BlackHat
b ATLAS data
Phys. Rev. D86 (2012) 014022
Sherpa MC@NLO
µR = µF = 14 HT , µQ = 1
2 p⊥Sherpa MC@NLO
µR = µF = 14 H
(y)T , µQ = 1
2 p⊥
µR, µF variation
µQ variation
MPI variation
10 2 10 310−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
1
10 1
10 2
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10 4
10 5
10 6
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10 8Inclusive jet transverse momenta in different rapidity ranges
p⊥ [GeV]
dσ/dp⊥[pb/GeV
]
b b b b b b b b b b b b b b b b
|y| < 0.3
2 3
0.5
1
1.5
MC/data
b b b b b b b b b b b b b b b b
0.3 < |y| < 0.8
2 3
0.5
1
1.5
MC/data
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2.8 < |y| < 3.6
2 3
0.5
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1.5
MC/data
b b b b b b
3.6 < |y| < 4.4
10 2 10 3
0.5
1
1.5
p⊥ [GeV]
MC/data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MC@NLO
MC@NLO for light jets: dijet mass
b
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b
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bb
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bbb
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b
bSherpa+BlackHat
b ATLAS data
Phys. Rev. D86 (2012) 014022
Sherpa MC@NLO
µR = µF = 14 HT , µQ = 1
2 p⊥Sherpa MC@NLO
µR = µF = 14 H
(y)T , µQ = 1
2 p⊥
µR, µF variation
µQ variation
MPI variation
10−1 110−1
1
10 1
10 2
10 3
10 4
10 5
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10 8
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10 10
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10 12
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10 17
10 18
10 19Dijet invariant mass spectra in different rapidity ranges
m12 [TeV]
dσ/dm
12[pb/TeV
]
b b
4.0 < y∗ < 4.4
1
0.51
1.5
MC/data
b b b b b b b b b
3.5 < y∗ < 4.0
1
0.51
1.5
MC/data
b b b b b b b b b b b b
3.0 < y∗ < 3.5
1
0.51
1.5
MC/data
b b b b b b b b b b b b b b b b
2.5 < y∗ < 3.0
1
0.51
1.5
MC/data
b b b b b b b b b b b b b b b b b b b
2.0 < y∗ < 2.5
1
0.5
1
1.5
MC/data
b b b b b b b b b b b b b b b b b b b b
1.5 < y∗ < 2.0
1
0.5
1
1.5
MC/data
b b b b b b b b b b b b b b b b b b b b b
1.0 < y∗ < 1.5
1
0.5
1
1.5MC/data
b b b b b b b b b b b b b b b b b b b b
0.5 < y∗ < 1.0
1
0.5
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1.5
MC/data
b b b b b b b b b b b b b b b b b b b b
y∗ < 0.5
10−1 1
0.5
1
1.5
m12 [GeV]
MC/data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MC@NLO
MC@NLO for light jets: azimuthal decorrelations
bb
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b
Sherpa+BlackHat
b CMS data
Phys. Rev. Lett. 106 (2011) 122003
Sherpa MC@NLO
µR = µF = 14 HT , µQ = 1
2 p⊥
1.6 1.8 2 2.2 2.4 2.6 2.8 3
10−2
10−1
1
10 1
10 2
10 3
10 4
10 5
Dijet azimuthal decorrelation in various plead⊥ bins
∆φ
1/σ
dσ/d
∆φ[pb]
300 GeV< plead⊥
b b b b b b b b b b
0.5
1
1.5
MC/data
200 GeV< plead⊥ < 300 GeV
b b b b b b b b b b b b b b b
0.5
1
1.5
MC/data
140 GeV< plead⊥ < 200 GeV
b b b b b b b b b b b b b b b b b b b b
0.5
1
1.5
MC/data
110 GeV< plead⊥ < 140 GeV
b b b b b b b b b b b b b b b b b b b b
0.5
1
1.5
MC/data
80 GeV< plead⊥ < 110 GeV
b b b b b b b b b b b b b b b b b b b b
1.6 1.8 2 2.2 2.4 2.6 2.8 3
0.5
1
1.5
∆φ
MC/data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MC@NLO
MC@NLO for light jets: R32 & forward energy flow
Sherpa+BlackHat
b
b
b
b
b
bbbbbbb b
b b b b bb b b
bb b
bb
bb b
b
b CMS data
Phys. Lett. B 702 (2011) 336Sherpa MC@NLO
µR = µF = 14 HT , µQ = 1
2 p⊥
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
3 jets over 2 jets ratio (anti-kt R=0.5)
R32
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
0.5 1 1.5 2 2.5
0.80.91.01.11.2
HT [TeV]
MC/data
Sherpa+BlackHat
b
b
b
b
b
b CMS data
JHEP 1111 (2011) 148Sherpa MC@NLO
µR = µF = 14 HT , µQ = 1
2 p⊥
0
100
200
300
400
500
600
700Forward energy flow in dijet events, p
jets⊥ > 20 GeV
dE/d
η[G
eV]
b b b b b
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0.6
0.8
1
1.2
1.4
η
MC/data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MC@NLO
MC@NLO for light jets: jet vetoes
bb
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bb b
b b
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70 GeV < p⊥ < 90 GeV
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150 GeV < p⊥ < 180 GeV+3
180 GeV < p⊥ < 210 GeV+4
210 GeV < p⊥ < 240 GeV+5
240 GeV < p⊥ < 270 GeV+6
Forward-backward
selection
Sherpa+BlackHat
b ATLAS data
JHEP 09 (2011) 053
Sherpa MC@NLO
µR = µF = 14 HT , µQ = 1
2 p⊥
µF, µR variation
µQ variation
MPI variation
0 1 2 3 4 5 60
2
4
6
8
10Inclusive jet transverse momenta in different rapidity ranges
∆y
GapFraction
b b b b b b b b b b
240 GeV < p⊥ < 270 GeV0.5
1
1.5
MC/data
b b b b b b b b b b b
210 GeV < p⊥ < 240 GeV0.5
1
1.5
MC/data
b b b b b b b b b b b b
180 GeV < p⊥ < 210 GeV0.5
1
1.5
MC/data
b b b b b b b b b b b b
150 GeV < p⊥ < 180 GeV0.5
1
1.5
MC/data
b b b b b b b b b b b b
120 GeV < p⊥ < 150 GeV0.5
1
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1
1.5
MC/data
b b b b b b b b b b b b
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0 1 2 3 4 5 60.5
1
1.5
∆y
MC/data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
PRECISION MONTE CARLO
MULTIJET MERGING
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Contents
7.a) Basic idea
7.b) Multijet merging at LO
7.c) Multijet merging at NLO
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Basic idea
Multijet merging: basic idea
parton shower resums logarithmsfair description of collinear/soft emissionsjet evolution (where the logs are large)
matrix elements exact at given orderfair description of hard/large-angle emissionsjet production (where the logs are small)
combine (“merge”) both:result: “towers” of MEs with increasingnumber of jets evolved with PS
multijet cross sections at Born accuracymaintain (N)LL accuracy of parton shower
resummed in PS
exact ME
LO 5jet, but also
NLO 4jet
L
αn
m
NLLexact ME
LO 4jet
4
4
4
4
4
5
5
5
5
5
5
5
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Basic idea
separate regions of jet production and jetevolution with jet measure QJ
(“truncated showering” if not identical with evolution parameter)
matrix elements populate hard regime
parton showers populate soft domain
/ GeV W
p
0 20 40 60 80 100 120 140 160 180 200
[ p
b/G
eV
] W
/d
pσ
d
-210
-110
1
10
210
SHERPA
Wp
W + 0jet
W + 1jet
W + 2jet
W + 3jet
W + 4jet
D0 Data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Basic idea
Why it works: jet rates with the parton shower
consider jet production in e+e− → hadronsDurham jet definition: relative transverse momentum k⊥ > QJ
fixed order: one factor αS and up to log2 Ec.m.
QJper jet
use Sudakov form factor for resummation &replace approximate fixed order by exact expression:
R2(QJ) =[∆q(E 2
c.m., Q2J )]2
R3(QJ) = 2∆q(E 2c.m., Q
2J )
E 2c.m.∫
Q2J
dk2⊥
k2⊥
[αs(k
2⊥)
2πdzKq(k2
⊥, z)
×∆q(E 2c.m., k
2⊥)∆q(k2
⊥, Q2J )∆g (k2
⊥, Q2J )
]F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at LO
Multijet merging at LO
expression for first emission
dσ = dΦN BN[
∆(K)N (µ2
N , t0) +
µ2N∫
t0
dΦ1KN∆(K)N (µ2
N , tN+1)Θ(QJ − QN+1)
]
+dΦN+1 BN+1 ∆(K)N (µ2
N+1, tN+1)Θ(QN+1 − QJ)
note: N + 1-contribution includes also N + 2, N + 3, . . .(no Sudakov suppression below tn+1, see further slides for iterated expression)
potential occurence of different shower start scales: µN,N+1,...
“unitarity violation” in square bracket: BNKN −→ BN+1
(cured with UMEPS formalism, L. Lonnblad & S. Prestel, JHEP 1302 (2013) 094 &
S. Platzer, arXiv:1211.5467 [hep-ph] & arXiv:1307.0774 [hep-ph])
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at LO
dσ =nmax−1∑n=N
dΦn Bn
(n − N) extra jets︷ ︸︸ ︷[n−1∏j=N
Θ(Qj+1 − QJ)
] no emissions off internal lines︷ ︸︸ ︷[n−1∏j=N
∆(K)j (tj , tj+1)
]
×[
∆(K)n (tn, t0)
︸ ︷︷ ︸no emission
+
tn∫t0
dΦ1Kn∆(K)n (tn, tn+1)Θ(QJ − Qn+1)
︸ ︷︷ ︸next emission no jet & below last ME emission
]
+dΦnmax Bnmax
[nmax−1∏j=N
Θ(Qj+1 − QJ)
][nmax−1∏j=N
∆(K)j (tj , tj+1)
]
×[
∆(K)nmax
(tnmax , t0) +
tnmax∫t0
dΦ1Knmax ∆(K)nmax
(tnmax , tnmax+1)
]
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at LO
Di-photons @ ATLAS: mγγ, p⊥,γγ, and ∆φγγ in showers
(arXiv:1211.1913 [hep-ex])
[p
b/G
eV
]γγ
/dm
σd
410
310
210
110
1
1Ldt = 4.9 fb ∫ Data 2011,
1.2 (MRST2007)×PYTHIA MC11c
1.2 (CTEQ6L1)×SHERPA MC11c
ATLAS
= 7 TeVs
data
/SH
ER
PA
00.5
11.5
22.5
3
[GeV]γγm
0 100 200 300 400 500 600 700 800
data
/PY
TH
IA
00.5
11.5
22.5
3
[p
b/G
eV
]γγ
T,
/dp
σd
510
410
310
210
110
1
10
1Ldt = 4.9 fb ∫ Data 2011,
1.3 (MRST2007)×PYTHIA MC11c
1.3 (CTEQ6L1)×SHERPA MC11c
ATLAS
= 7 TeVs
data
/SH
ER
PA
00.5
11.5
22.5
3
[GeV]γγT,
p
0 50 100 150 200 250 300 350 400 450 500
data
/PY
TH
IA
00.5
11.5
22.5
3
[p
b/r
ad
]γγφ
∆/d
σd
1
10
2101
Ldt = 4.9 fb ∫ Data 2011,
1.2 (MRST2007)×PYTHIA MC11c
1.2 (CTEQ6L1)×SHERPA MC11c
ATLAS
= 7 TeVs
data
/SH
ER
PA
00.5
11.5
22.5
3
[rad]γγ
φ∆
0 0.5 1 1.5 2 2.5 3
data
/PY
TH
IA
00.5
11.5
22.5
3
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at LO
Aside: Comparison with higher order calculations [
pb
/Ge
V]
γγ/d
mσ
d
410
310
210
110
1
1Ldt = 4.9 fb ∫ Data 2011,
DIPHOX+GAMMA2MC (CT10)
NNLO (MSTW2008)γ2
ATLAS
= 7 TeVs
data
/DIP
HO
X
00.5
11.5
22.5
3
[GeV]γγm
0 100 200 300 400 500 600 700 800
NN
LO
γdata
/2
00.5
11.5
22.5
3
[p
b/G
eV
]γγ
T,
/dp
σd
510
410
310
210
110
1
10
1Ldt = 4.9 fb ∫ Data 2011,
DIPHOX+GAMMA2MC (CT10)
NNLO (MSTW2008)γ2
ATLAS
= 7 TeVs
data
/DIP
HO
X
00.5
11.5
22.5
3
[GeV]γγT,
p
0 50 100 150 200 250 300 350 400 450 500
NN
LO
γdata
/2
00.5
11.5
22.5
3
[p
b/r
ad
]γγφ
∆/d
σd
1
10
2101
Ldt = 4.9 fb ∫ Data 2011,
DIPHOX+GAMMA2MC (CT10)
NNLO (MSTW2008)γ2
ATLAS
= 7 TeVs
data
/DIP
HO
X
00.5
11.5
22.5
3
[rad]γγ
φ∆
0 0.5 1 1.5 2 2.5 3
NN
LO
γdata
/2
00.5
11.5
22.5
3
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MENLOPS
A step towards multijet-merging at NLO: MENLOPS
combine matching for lowest multiplicity with multijet merging
interpolating local K -factor for reweighting hard emissions
kN(ΦN+1) =BNBN
(1− HN
BN+1
)+HN
BN+1−→
BN/BN for soft emission
1 for hard emission
dσ = dΦN BN[
∆(K)N (µ2
N , t0) +
µ2N∫
t0
dΦ1KN∆(K)N (µ2
N , tN+1)Θ(QJ − QN+1)
]
+dΦN+1HN∆(K)N (µ2
N , tN+1)Θ(QJ − QN+1)
+dΦN+1 kN BN+1∆(K)N (µ2
N , tN+1)Θ(QN+1 − QJ)
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MENLOPS
Transverse momentum of W & Z boson
ATLAS, arXiv:1108.6308, arXiv:1107.2381
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MENLOPS
Z+jets: inclusive quantitiesATLAS, arXiv:1111.2690
[1/G
eV]
T/d
Hσ
) d
- l+ l
→* γZ
/σ
(1/
-610
-510
-410
-310
-210
-110
= 7 TeV)sData 2011 (
ALPGEN
SHERPA
+ SHERPAATHLACKB
ATLAS )µ 1 jet (l=e,≥)+ -l+ l→*(γZ/-1
L dt = 4.6 fb∫ jets, R = 0.4tanti-k
| < 4.4jet
> 30 GeV, |yjet
Tp
100 200 300 400 500 600 700 800 900 1000
NLO
/ D
ata
0.5
1
1.5 + SHERPAATHLACKB
100 200 300 400 500 600 700 800 900 1000
MC
/ D
ata
0.60.8
11.21.4 ALPGEN
[GeV]TH100 200 300 400 500 600 700 800 900 1000
MC
/ D
ata
0.60.8
11.21.4 SHERPA
) [p
b]je
t N≥
) +
- l
+ l→
*(γ(Z
/σ
-310
-210
-110
1
10
210
310
410
510
610
= 7 TeV)sData 2011 (ALPGENSHERPAMC@NLO
+ SHERPAATHLACKB
ATLAS )µ)+jets (l=e,-l+ l→*(γZ/-1
L dt = 4.6 fb∫ jets, R = 0.4tanti-k
| < 4.4jet
> 30 GeV, |yjet
Tp
0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥
NLO
/ D
ata
0.60.8
11.21.4 + SHERPAATHLACKB
0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥
MC
/ D
ata
0.60.8
11.21.4 ALPGEN
jetN
0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥
MC
/ D
ata
0.60.8
11.21.4 SHERPA
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MENLOPS
Z+jets: jet transverse momentaATLAS, arXiv:1111.2690
[1/G
eV]
jet
T/d
pσ
) d
- l+ l
→* γZ
/σ
(1/
-710
-610
-510
-410
-310
-210
-110
= 7 TeV)sData 2011 (ALPGENSHERPAMC@NLO
+ SHERPAATHLACKB
ATLAS )µ 1 jet (l=e,≥)+ -l+ l→*(γZ/-1
L dt = 4.6 fb∫ jets, R = 0.4tanti-k
| < 4.4jet
> 30 GeV, |yjet
Tp
100 200 300 400 500 600 700
NLO
/ D
ata
0.60.8
11.21.4 + SHERPAATHLACKB
100 200 300 400 500 600 700
MC
/ D
ata
0.60.8
11.21.4 ALPGEN
(leading jet) [GeV]jetT
p100 200 300 400 500 600 700
MC
/ D
ata
0.60.8
11.21.4 SHERPA
[1/G
eV]
jet
T/d
pσ
) d
- l+ l
→* γZ
/σ
(1/
-710
-610
-510
-410
-310
-210
= 7 TeV)sData 2011 (
ALPGEN
SHERPA
+ SHERPAATHLACKB
ATLAS )µ 2 jets (l=e,≥)+ -l+ l→*(γZ/-1
L dt = 4.6 fb∫ jets, R = 0.4tanti-k
| < 4.4jet
> 30 GeV, |yjet
Tp
100 200 300 400 500
NLO
/ D
ata
0.60.8
11.21.4 + SHERPAATHLACKB
100 200 300 400 500
MC
/ D
ata
0.60.8
11.21.4 ALPGEN
(2nd leading jet) [GeV]jetT
p100 200 300 400 500
MC
/ D
ata
0.60.8
11.21.4 SHERPA
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MENLOPS
Z+jets: jet transverse momentaATLAS, arXiv:1111.2690
[1/G
eV]
jet
T/d
pσ
) d
- l+ l
→* γZ
/σ
(1/
-610
-510
-410
-310 = 7 TeV)sData 2011 (
ALPGEN
SHERPA
+ SHERPAATHLACKB
ATLAS )µ 3 jets (l=e,≥)+ -l+ l→*(γZ/-1
L dt = 4.6 fb∫ jets, R = 0.4tanti-k
| < 4.4jet
> 30 GeV, |yjet
Tp
40 60 80 100 120 140 160 180 200
NLO
/ D
ata
0.60.8
11.21.4 + SHERPAATHLACKB
40 60 80 100 120 140 160 180 200
MC
/ D
ata
0.60.8
11.21.4 ALPGEN
(3rd leading jet) [GeV]jetT
p40 60 80 100 120 140 160 180 200
MC
/ D
ata
0.60.8
11.21.4 SHERPA
[1/G
eV]
jet
T/d
pσ
) d
- l+ l
→* γZ
/σ
(1/
-610
-510
-410
-310
= 7 TeV)sData 2011 (
ALPGEN
SHERPA
+ SHERPAATHLACKB
ATLAS )µ 4 jets (l=e,≥)+ -l+ l→*(γZ/-1
L dt = 4.6 fb∫ jets, R = 0.4tanti-k
| < 4.4jet
> 30 GeV, |yjet
Tp
30 40 50 60 70 80 90 100
NLO
/ D
ata
0.60.8
11.21.4 + SHERPAATHLACKB
30 40 50 60 70 80 90 100
MC
/ D
ata
0.60.8
11.21.4 ALPGEN
(4th leading jet) [GeV]jetT
p30 40 50 60 70 80 90 100
MC
/ D
ata
0.60.8
11.21.4 SHERPA
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MENLOPS
Z+jets: correlation of leading jetsATLAS, arXiv:1111.2690
|jj y∆
/d|
σ)
d- l
+ l→* γ
Z/
σ(1
/
-410
-310
-210
-110 = 7 TeV)sData 2011 (
ALPGEN
SHERPA
+ SHERPAATHLACKB
ATLAS )µ 2 jets (l=e,≥)+ -l+ l→*(γZ/-1
L dt = 4.6 fb∫ jets, R = 0.4tanti-k
| < 4.4jet
> 30 GeV, |yjet
Tp
0 1 2 3 4 5 6
NLO
/ D
ata
0.60.8
11.21.4 + SHERPAATHLACKB
0 1 2 3 4 5 6
MC
/ D
ata
0.60.8
11.21.4 ALPGEN
| (leading jet, 2nd leading jet)jj y∆|0 1 2 3 4 5 6
MC
/ D
ata
0.60.8
11.21.4 SHERPA
[1/G
eV]
jj/d
mσ
) d
- l+ l
→* γZ
/σ
(1/
-610
-510
-410
-310
= 7 TeV)sData 2011 (
ALPGEN
SHERPA
+ SHERPAATHLACKB
ATLAS )µ 2 jets (l=e,≥)+ -l+ l→*(γZ/-1
L dt = 4.6 fb∫ jets, R = 0.4tanti-k
| < 4.4jet
> 30 GeV, |yjet
Tp
0 100 200 300 400 500 600 700 800 900 1000
NLO
/ D
ata
0.60.8
11.21.4 + SHERPAATHLACKB
0 100 200 300 400 500 600 700 800 900 1000
MC
/ D
ata
0.60.8
11.21.4 ALPGEN
(leading jet, 2nd leading jet) [GeV]jjm
0 100 200 300 400 500 600 700 800 900 1000
MC
/ D
ata
0.60.8
11.21.4 SHERPA
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MENLOPS
Z+jets: ∆φZj in unboosted sample
CMS, arXiv:1301.1646
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
MENLOPS
Z+jets: ∆φZj in boosted sample
CMS, arXiv:1301.1646
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Multijet-merging at NLO: MEPS@NLO
basic idea like at LO: towers of MEs with increasing jet multi(but this time at NLO)
combine them into one sample, remove overlap/double-counting
maintain NLO and (N)LL accuracy of ME and PS
this effectively translates into a merging of MC@NLO simulations andcan be further supplemented with LO simulations for even higher finalstate multiplicities
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
First emission(s), once more
dσ = dΦN BN[
∆(K)N (µ2
N , t0) +
µ2N∫
t0
dΦ1KN∆(K)N (µ2
N , tN+1)Θ(QJ − QN+1)
]
+dΦN+1HN∆(K)N (µ2
N , tN+1)Θ(QJ − QN+1)
+dΦN+1 BN+1
(1 +BN+1
BN+1
µ2N∫
tN+1
dΦ1KN
)Θ(QN+1 − QJ)
·[
∆(K)N+1(tN+1, t0) +
tN+1∫t0
dΦ1KN+1∆(K)N+1(tN+1, tN+2)
]
+dΦN+2HN+1∆(K)N (µ2
N , tN+1)∆(K)N+1(tN+1, tN+2)Θ(QN+1 − QJ) + . . .
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
pH⊥ in MEPS@NLO
pp → h + jetsSherpa S-MC@NLO
0 50 100 150 200 250 30010−4
10−3
10−2
10−1
Transverse momentum of the Higgs boson
p⊥(h) [GeV]
dσ
/dp ⊥
[pb/
GeV
] first emission byMC@NLO
MC@NLO pp → h + jetfor Qn+1 > Qcut
restrict emission offpp → h + jet toQn+2 < Qcut
MC@NLO
pp → h + 2jets forQn+2 > Qcut
iterate
sum all contributions
eg. p⊥(h)>200 GeVhas contributions fr.multiple topologies
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
pH⊥ in MEPS@NLO
pp → h + jetspp → h + 0j @ NLO
0 50 100 150 200 250 30010−4
10−3
10−2
10−1
Transverse momentum of the Higgs boson
p⊥(h) [GeV]
dσ
/dp ⊥
[pb/
GeV
]
first emission byMC@NLO , restrict toQn+1 < Qcut
MC@NLO pp → h + jetfor Qn+1 > Qcut
restrict emission offpp → h + jet toQn+2 < Qcut
MC@NLO
pp → h + 2jets forQn+2 > Qcut
iterate
sum all contributions
eg. p⊥(h)>200 GeVhas contributions fr.multiple topologies
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
pH⊥ in MEPS@NLO
pp → h + jetspp → h + 0j @ NLOpp → h + 1j @ NLO
0 50 100 150 200 250 30010−4
10−3
10−2
10−1
Transverse momentum of the Higgs boson
p⊥(h) [GeV]
dσ
/dp ⊥
[pb/
GeV
]
first emission byMC@NLO , restrict toQn+1 < Qcut
MC@NLO pp → h + jetfor Qn+1 > Qcut
restrict emission offpp → h + jet toQn+2 < Qcut
MC@NLO
pp → h + 2jets forQn+2 > Qcut
iterate
sum all contributions
eg. p⊥(h)>200 GeVhas contributions fr.multiple topologies
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
pH⊥ in MEPS@NLO
pp → h + jetspp → h + 0j @ NLOpp → h + 1j @ NLO
0 50 100 150 200 250 30010−4
10−3
10−2
10−1
Transverse momentum of the Higgs boson
p⊥(h) [GeV]
dσ
/dp ⊥
[pb/
GeV
]
first emission byMC@NLO , restrict toQn+1 < Qcut
MC@NLO pp → h + jetfor Qn+1 > Qcut
restrict emission offpp → h + jet toQn+2 < Qcut
MC@NLO
pp → h + 2jets forQn+2 > Qcut
iterate
sum all contributions
eg. p⊥(h)>200 GeVhas contributions fr.multiple topologies
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
pH⊥ in MEPS@NLO
pp → h + jetspp → h + 0j @ NLOpp → h + 1j @ NLOpp → h + 2j @ NLO
0 50 100 150 200 250 30010−4
10−3
10−2
10−1
Transverse momentum of the Higgs boson
p⊥(h) [GeV]
dσ
/dp ⊥
[pb/
GeV
]
first emission byMC@NLO , restrict toQn+1 < Qcut
MC@NLO pp → h + jetfor Qn+1 > Qcut
restrict emission offpp → h + jet toQn+2 < Qcut
MC@NLO
pp → h + 2jets forQn+2 > Qcut
iterate
sum all contributions
eg. p⊥(h)>200 GeVhas contributions fr.multiple topologies
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
pH⊥ in MEPS@NLO
pp → h + jetspp → h + 0j @ NLOpp → h + 1j @ NLOpp → h + 2j @ NLO
0 50 100 150 200 250 30010−4
10−3
10−2
10−1
Transverse momentum of the Higgs boson
p⊥(h) [GeV]
dσ
/dp ⊥
[pb/
GeV
]
first emission byMC@NLO , restrict toQn+1 < Qcut
MC@NLO pp → h + jetfor Qn+1 > Qcut
restrict emission offpp → h + jet toQn+2 < Qcut
MC@NLO
pp → h + 2jets forQn+2 > Qcut
iterate
sum all contributions
eg. p⊥(h)>200 GeVhas contributions fr.multiple topologies
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
pH⊥ in MEPS@NLO
pp → h + jetspp → h + 0j @ NLOpp → h + 1j @ NLOpp → h + 2j @ NLOpp → h + 3j @ LO
0 50 100 150 200 250 30010−4
10−3
10−2
10−1
Transverse momentum of the Higgs boson
p⊥(h) [GeV]
dσ
/dp ⊥
[pb/
GeV
]
first emission byMC@NLO , restrict toQn+1 < Qcut
MC@NLO pp → h + jetfor Qn+1 > Qcut
restrict emission offpp → h + jet toQn+2 < Qcut
MC@NLO
pp → h + 2jets forQn+2 > Qcut
iterate
sum all contributions
eg. p⊥(h)>200 GeVhas contributions fr.multiple topologies
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
pH⊥ in MEPS@NLO
pp → h + jetspp → h + 0j @ NLOpp → h + 1j @ NLOpp → h + 2j @ NLOpp → h + 3j @ LO
0 50 100 150 200 250 30010−4
10−3
10−2
10−1
Transverse momentum of the Higgs boson
p⊥(h) [GeV]
dσ
/dp ⊥
[pb/
GeV
]
first emission byMC@NLO , restrict toQn+1 < Qcut
MC@NLO pp → h + jetfor Qn+1 > Qcut
restrict emission offpp → h + jet toQn+2 < Qcut
MC@NLO
pp → h + 2jets forQn+2 > Qcut
iterate
sum all contributions
eg. p⊥(h)>200 GeVhas contributions fr.multiple topologies
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
pH⊥ in MEPS@NLO
pp → h + jetspp → h + 0j @ NLOpp → h + 1j @ NLOpp → h + 2j @ NLOpp → h + 3j @ LO
0 50 100 150 200 250 30010−4
10−3
10−2
10−1
Transverse momentum of the Higgs boson
p⊥(h) [GeV]
dσ
/dp ⊥
[pb/
GeV
]
first emission byMC@NLO , restrict toQn+1 < Qcut
MC@NLO pp → h + jetfor Qn+1 > Qcut
restrict emission offpp → h + jet toQn+2 < Qcut
MC@NLO
pp → h + 2jets forQn+2 > Qcut
iterate
sum all contributions
eg. p⊥(h)>200 GeVhas contributions fr.multiple topologies
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
MEPS@NLO: example results for e−e+ → hadrons
b
b
bbb b b b b b b b b b b b b b b b b b b b b b b b b b b
bbbbb
b
b
b
bb
b
b b
b
b b
b
b
b ALEPH data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
10−6
10−5
10−4
10−3
10−2
10−1
Durham jet resolution 3 → 2 (ECMS = 91.2 GeV)
1/
σd
σ/dln(y
23)
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
1 2 3 4 5 6 7 8 9 10
0.6
0.8
1
1.2
1.4
− ln(y23)
MC/data
b
b
b
b
b
bbbbbbb b
bbb b
bb b b b b b b b b b b
bbbb
b
b
b
b
b
b
b
bb b
b b b
b
b ALEPH data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo10−5
10−4
10−3
10−2
10−1
Durham jet resolution 4 → 3 (ECMS = 91.2 GeV)
1/
σd
σ/dln(y
34)
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
2 3 4 5 6 7 8 9 10 11
0.6
0.8
1
1.2
1.4
− ln(y34)
MC/data
b
b
b
bb
bbbbbbbbbbbb b b b b b b b b b
bbbb
b
b
b
b
b
bbb b b b
b
b
b
b ALEPH data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo10−5
10−4
10−3
10−2
10−1
Durham jet resolution 5 → 4 (ECMS = 91.2 GeV)
1/
σd
σ/dln(y
45)
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
4 5 6 7 8 9 10 11 12
0.6
0.8
1
1.2
1.4
− ln(y45)
MC/data
b
b
b
b
b
b
b
b
b
b
b
bbbbbb b b b b b b b
bbb
b
b
b
b
b
b
b
b bb b
bb b
b
b
b
b ALEPH data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
10−5
10−4
10−3
10−2
10−1
Durham jet resolution 6 → 5 (ECMS = 91.2 GeV)
1/
σd
σ/dln(y
56)
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
4 5 6 7 8 9 10 11 12 13
0.6
0.8
1
1.2
1.4
− ln(y56)
MC/data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
MEPS@NLO: example results for e−e+ → hadrons
b
bb
bb
b
bbbbb b b b b b b b b b b b b b b b b b b b b b b
bbbbbbbb
b
b ALEPH data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo10−3
10−2
10−1
1
10 1
Thrust (ECMS = 91.2 GeV)
1/
σd
σ/dT
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
0.6
0.8
1
1.2
1.4
T
MC/data
b
b
b
b
b
b OPAL data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
10−4
10−3
10−2
10−1Moments of 1− T at 91 GeV
1/
σd
σ/d〈(1−
T)n〉
b b b b b
1 2 3 4 5
0.6
0.8
1
1.2
1.4
n
MC/data
b
b
b
b
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bbbbbb
b
b
bb
b
b
b ALEPH data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo10−3
10−2
10−1
1
C-Parameter (ECMS = 91.2 GeV)
1/
σd
σ/dC
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
0 0.2 0.4 0.6 0.8 1
0.6
0.8
1
1.2
1.4
C
MC/data
b
b
b
b
b
b OPAL data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
10−2
10−1
Moments of C at 91 GeV
1/
σd
σ/d〈C
n〉
b b b b b
1 2 3 4 5
0.6
0.8
1
1.2
1.4
n
MC/data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Example: MEPS@NLO for W+jets(up to two jets @ NLO, from BlackHat, see arXiv: 1207.5031 [hep-ex])
pjet⊥ > 30GeV
pjet⊥ > 20GeV(×10)
b
b
b
b
b
b
b
b
b
b
b
b ATLAS data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
0 1 2 3 4 5
10 1
10 2
10 3
10 4
Inclusive Jet Multiplicity
Njet
σ(W
+≥
Njetjets)[pb]
pjet⊥ > 20GeV
b
b
b b
b
0.15
0.2
0.25
0.3
σ(≥
Njetjets)/
σ(≥
Njet−
1jets)
pjet⊥ > 30GeV
b
b
bb
b
0 1 2 3 4 5
0.1
0.15
0.2
0.25
Njet
σ(≥
Njetjets)/
σ(≥
Njet−
1jets)
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
W+ ≥ 1 jet (×1)
W+ ≥ 2 jets (×0.1)
W+ ≥ 3 jets (×0.01)
b
b
b
b
b
b
b
b
b
b
bb b
b
b
b
b
b
b
bb
bb b
bb
b
b
b b
b ATLAS data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
10−4
10−3
10−2
10−1
1
10 1
10 2
10 3
First Jet p⊥
dσ/dp⊥[pb/GeV
]
b b b b b b b b b b
0.20.40.60.8
11.21.41.61.8
MC/data
b b b b b b b b b b
0.20.40.60.8
11.21.41.61.8
MC/data
b b b b b b b b b b
50 100 150 200 250 300
0.20.40.60.8
11.21.41.61.8
p⊥ [GeV]
MC/data
W+ ≥ 2 jets (×1)
W+ ≥ 3 jets (×0.1)
b
b
b
b
b
b
b
bb
b
b
bb
b
b ATLAS data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
10−3
10−2
10−1
1
10 1
10 2
Second Jet p⊥
dσ/dp⊥[pb/GeV
]
b b b b b b b
0.20.40.60.8
11.21.41.61.8
MC/data
b b b b b b b
40 60 80 100 120 140 160 180
0.20.40.60.8
11.21.41.61.8
p⊥ [GeV]
MC/data
W+ ≥ 3 jets
b
b
b
b
b
b
b ATLAS data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
10−3
10−2
10−1
1
Third Jet p⊥
dσ/dp⊥[pb/GeV
]
b b b b b b
40 60 80 100 120 140
0.20.40.60.8
11.21.41.61.8
p⊥ [GeV]
MC/data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
W+ ≥ 1 jet (×1)
W+ ≥ 2 jets (×0.1)
W+ ≥ 3 jets (×0.01)
b
bb
b
b
bb
bb
bb
b
b
b
bb
bb
bb
b
bb
b
b
b
b
b bb
bb
b bb
b
b ATLAS data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
100 200 300 400 500 600 700
10−4
10−3
10−2
10−1
1
10 1
10 2
10 3
HT [GeV]
dσ/dH
T[pb/GeV
]
b b b b b b b b b b b b b
0.20.40.60.8
11.21.41.61.8
MC/data
b b b b b b b b b b b b
0.20.40.60.8
11.21.41.61.8
MC/data
b b b b b b b b b b b
100 200 300 400 500 600 700
0.20.40.60.8
11.21.41.61.8
HT [GeV]
MC/data
bb
b bb
bb
b b
b
b
b
b
b
b
b
b
b
bb
bb
bb b b b b
b ATLAS data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
20
40
60
80
100
∆R Distance of Leading Jets
dσ/d
∆R[pb]
b b b b b b b b b b b b b b b b b b b b b b b b b b b b
1 2 3 4 5 6 7 80
0.5
1
1.5
2
∆R(First Jet, Second Jet)
MC/data
b bb b
b b b bb
bb
b
b
b
b
b
b ATLAS data
MePs@NloMePs@Nlo µ/2 . . . 2µ
MEnloPSMEnloPS µ/2 . . . 2µ
Mc@Nlo
20
40
60
80
100
120
140
Azimuthal Distance of Leading Jets
dσ/d
∆φ[pb]
b b b b b b b b b b b b b b b b
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
∆φ(First Jet, Second Jet)
MC/data
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Results for Higgs boson production through gluon fusion
parton-shower level, Higgs boson does not decay
setup & cuts:jets: anti-kt, p⊥ ≥ 20 GeV, R = 0.4, |η| ≤ 4.5dijet cuts: at least 2 jets with p⊥ ≥ 25 GeVWBF cuts: mjj ≥ 400 GeV, ∆yjj ≥ 2.8
jet multiplicity plots:0-jet excl.: no jet with p⊥ ≥ 20, 25, 30 GeV2-jet incl.: at least two jets with p⊥ ≥ 20, 25, 30 GeV
SHERPA with H + 0, 1, 2(NLO) + 3(LO) jets, Qcut = 20GeV
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Inclusive observables for gg → H
Sherpa MePs@NloµR = µCKKW
µR = mh
µR = H′T
10−2
10−1
1Higgs boson transverse momentum
dσ/dp⊥[pb/GeV
]
0 20 40 60 80 100
0.6
0.8
1
1.2
1.4
p⊥(h) [GeV]
Ratioto
µR=
mh
Sherpa MePs@NloµR = µCKKW
µR = mh
µR = H′T
0
1
2
3
4
5Higgs boson rapidity
dσ/dy[pb]
-4 -3 -2 -1 0 1 2 3 4
0.6
0.8
1
1.2
1.4
y(h)
Ratioto
µR=
mh
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Exclusive observables for gg → H
Sherpa MePs@NloµR = µCKKW
µR = mh
µR = H′T
10−3
10−2
10−1
1Higgs boson transverse momentum (njet = 0)
dσ/dp⊥[pb/GeV
]
0 20 40 60 80 100
0.60.8
11.21.41.61.8
p⊥(h) [GeV]
Ratioto
µR=
mh
Sherpa MePs@NloµR = µCKKW
µR = mh
µR = H′T
10−5
10−4
10−3
10−2
10−1
Transverse momentum of the H j1 system
dσ/dp⊥[pb/GeV
]
0 50 100 150 200 250 300
0.60.8
11.21.41.61.8
p⊥(hj1)
Ratioto
µR=
mh
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
gg → H after WBF cuts
Sherpa MePs@NloµR = µCKKW
µR = mh
µR = H′T
VBF cuts
0
0.0005
0.001
0.0015
0.002Transverse momentum of the Higgs boson
dσ/dp⊥[pb/GeV
]
0 50 100 150 200 250 300
0.60.8
11.21.41.61.8
p⊥(h) [GeV]
Ratioto
µR=
mh
Sherpa MePs@NloµR = µCKKW
µR = mh
µR = H′T
VBF cuts
10−6
10−5
10−4
10−3
10−2Transverse momentum of the H j1 j2 system
dσ/dp⊥[pb/GeV
]
0 50 100 150 200 250 300
0.60.8
11.21.41.61.8
p⊥(hj1 j2)
Ratioto
µR=
mh
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
gg → H after WBF cuts
Sherpa MePs@NloµR = µCKKW
µR = mh
µR = H′T
VBF cuts
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Azumuthal separation of the two leading jets
dσ/d
∆φ[pb]
0 0.5 1 1.5 2 2.5 3
0.60.8
11.21.41.61.8
∆φ(j1, j2)
Ratioto
µR=
mh
Sherpa MePs@NloµR = µCKKW
µR = mh
µR = H′T
VBF cuts
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Azumuthal separation of the Higgs and the two leading jets
dσ/d(π
−∆
φ)[pb]
0 0.2 0.4 0.6 0.8 1
0.60.8
11.21.41.61.8
π − ∆φ(h, j1 j2)
Ratioto
µR=
mh
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Quark mass effects
include effects of quark masses
reweight NLO HEFT with LO ratio:
dσ(NLO)mass ≈ dσ
(NLO)HEFT ×
dσ(LO)mass
dσ(LO)HEFT
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Quark mass effects – results
top mass effect in MEPS@NLO
(on Higgs–p⊥)
0 100 200 300 400 500 600
-410
-310
-210
-110
1=173 GeVtm
H+0j NLO
=173 GeVtmH+1j NLO
=173 GeVtmH+2j LO
∞ →tmH+jets NLO
=173 GeVtmH+jets NLO
GeVfb
T,Hdp
σd
[GeV]T,H
p0 100 200 300 400 500 600
0.5
1
∞→tm(N)LO
=173 GeVtm(N)LO
comparison S-MC@NLO– HRES(top–loop only)
p p → H√s = 8TeV
Finite top mass
HRes NLO+NLL
Sherpa S-MC@NLO
10−7
10−6
10−5
10−4
10−3
Higgs boson transverse momentum
dσ/dp⊥[pb/GeV
]
0 100 200 300 400 5000
0.5
1
1.5
2
p⊥ [GeV]
Ratio
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
b–mass effects
b–mass effects more tricky
relevant only for (negative) interference of top– and bottom–loops(bottom2 double Yukawa - supressed)
but: cannot start shower at mH
radiation “sees” bottom at all scales above mb
=⇒ must use full theory there
pT spectrum naively “squeezed” – funny shapes
LO multijet merging improves situation
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Higgs backgrounds: inclusive observables in W+W−+jets
Sherpa+OpenLoops
MEPS@NLO 4ℓ+ 0, 1jMC@NLO 4ℓNLO 4ℓNLO 4ℓ+ 1j
10−4
10−3
10−2
Transverse momentum of leading jet
dσ/dpT[pb/GeV]
10 1 10 2
0.6
0.8
1
1.2
1.4
pT [GeV]
Ratio
Sherpa+OpenLoops
MEPS@NLO 4ℓ+ 0, 1jMC@NLO 4ℓ
NLO 4ℓ
10−8
10−7
10−6
10−5
10−4
10−3
Total transverse energy
dσ/dHT[pb/GeV]
10 2 10 3
0.6
0.8
1
1.2
1.4
HT [GeV]
Ratio
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Higgs backgrounds: jet vetoes in W+W−+jets
Sherpa+OpenLoops
MEPS@NLO 4ℓ+ 0, 1jMC@NLO 4ℓ
NLO 4ℓ
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Integrated cross section in the exclusive 0-jet bin
σ(pTjet<pmax
T)[pb]
10 20 30 40 50 60 70 80 90 100
0.850.9
0.951.0
1.051.1
1.15
pmaxT [GeV]
Ratio
Sherpa+OpenLoops
MEPS@NLO 4ℓ+ 0, 1jMC@NLO 4ℓNLO 4ℓ+ 1j
0
0.05
0.1
0.15
0.2
0.25
Integrated cross section in the inclusive 1-jet bin
σ(pTjet>pminT
)[pb]
10 20 30 40 50 60 70 80 90 100
0.70.80.91.01.11.21.3
pminT [GeV]
Ratio
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Higgs backgrounds: gluon-induced processes W+W−+jets
include (LO-) merged loop2 contributions of gg → VV (+1 jet)
Sherpa+OpenLoops
MEPS@NLO 4ℓ+ 0, 1jMEPS@LOOP2 4ℓ+ 0, 1j
10−6
10−5
10−4
10−3
10−2
Transverse momentum of leading jet
dσ/dpT[pb/GeV]
10 1 10 2
0.02
0.04
0.06
0.08
pT [GeV]
dσ/d
σMEPS@NLO
Sherpa+OpenLoops
MEPS@NLO 4ℓ+ 0, 1jMEPS@LOOP2 4ℓ+ 0, 1j
10−6
10−5
10−4
10−3
10−2
Invariant mass of oppositely charged leptons
dσ/dm
ℓℓ[pb/GeV]
0 50 100 150 200 250 300
0.02
0.04
0.06
0.08
mℓℓ [GeV]
dσ/d
σMEPS@NLO
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Higgs backgrounds: jet vetoes in W+W−+jets
Sherpa+OpenLoops
MEPS@LOOP2 4ℓ+ 0, 1j4ℓ+ 0j4ℓ+ 1j
LOOP2+PS 4ℓ
10−9
10−8
10−7
10−6
10−5
10−4
Transverse momentum of leading jet
dσ/dpT[pb/GeV]
10 1 10 20
0.20.40.60.8
11.21.4
pT [GeV]
Ratio
Sherpa+OpenLoops
MEPS@LOOP2 pp→ 4ℓ+ 0, 1jMEPS@LOOP2 gg→ 4ℓ+ 0, 1g
10−6
10−5
10−4
10−3Transverse momentum of leading jet
dσ/dpT[pb/GeV]
10 1 10 20
0.5
1
1.5
2
pT [GeV]
Ratio
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Relevant observables for VH → 3`: E/T
Signal
WH(WW)
WH(ττ)
ZH(WW)
ZH(ττ)
WH(ZZ)
ZH(ZZ)
Background
WZ
WWW
WWZ
ZZ
WZZ
ZZZ
0
0.05
0.1
0.15
0.2
0.25
0.3Missing transverse energy after Z and top veto
dσ/d/Emiss
⊥[fb/GeV
]
Signal
WH(WW)
WH(ττ)
ZH(WW)
ZH(ττ)
WH(ZZ)
ZH(ZZ)
Background
WZ
WWW
WWZ
ZZ
WZZ
ZZZ
0
0.01
0.02
0.03
0.04
0.05Missing transverse energy in Z-depleted sample
dσ/dEmiss
⊥[fb/GeV
]
0 50 100 150 200 250
0.9
1.0
1.1Accumulated background uncertainty
Accumulated signal uncertainty
Emiss⊥ [GeV]
Relativeuncertainty
0 50 100 150 200
0.9
1.0
1.1Accumulated background uncertainty
Accumulated signal uncertainty
Emiss⊥ [GeV]
Relativeuncertainty
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Multijet merging at NLO
Relevant observables for VH → 3`: m123 & ∆R01
Signal
WH(WW)
WH(ττ)
ZH(WW)
ZH(ττ)
WH(ZZ)
ZH(ZZ)
Background
WZ
WWW
WWZ
ZZ
WZZ
ZZZ
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1Trilepton invariant mass after Z and top veto
dσ/dm
3ℓ[fb/GeV
]
Signal
WH(WW)
WH(ττ)
ZH(WW)
ZH(ττ)
WH(ZZ)
ZH(ZZ)
Background
WZ
WWW
WWZ
ZZ
WZZ
ZZZ
0
0.5
1
1.5
2
2.5Angular separation of closest opposite-sign lepton pair in Z-depleted sample
dσ/d
∆R01[fb]
50 100 150 200 250 300
0.9
1.0
1.1
Accumulated signal uncertainty
Accumulated background uncertainty
m3ℓ [GeV]
Relativeuncertainty
0 1 2 3 4 5
0.9
1.0
1.1Accumulated background uncertainty
Accumulated signal uncertainty
∆R01
Ratio
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Other merging approaches: FxFx & friends
Differences between MEPS@NLO, UNLOPS & FxFx
FxFx MEPS@NLO UNLOPS
ME all internal V external all externalCOMIX or AMEGIC++
V from OPENLOOPS, , MJET, . . .
shower external intrinsic intrinsicHERWIG or PYTHIA PYTHIA
∆N analytical from PS from PSΘ(QJ) a-posteriori per emission per emissionQJ -range relatively high > Sudakov regime ≈ Sudakov regime
(but changed)
≈ 10% ≈ 10%
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Other merging approaches: FxFx & friends
FxFx: validation in Z+jets
(Data from ATLAS, 1304.7098, with HERWIG++)
(green: 0, 1, 2 jets + uncertainty band from scale and PDF variations, red: MC@NLO)
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Other merging approaches: FxFx & friends
FxFx: validation in Z+jets
(Data from ATLAS, 1304.7098, with HERWIG++)
(green: 0, 1, 2 jets + uncertainty band from scale and PDF variations, red: MC@NLO)
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Other merging approaches: FxFx & friends
FxFx: QJ dependence in tt
(R.Frederix & S.Frixione, JHEP 1212 (2012) 061)
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Other merging approaches: FxFx & friends
Aside: merging without QJ - the MINLO approach
(K.Hamilton, P.Nason, C.Oleari & G.Zanderighi, JHEP 1305 (2013) 082)
based on POWHEG + shower from PYTHIA or HERWIG
up to today only for singlet S production, gives NNLO + PS
basic idea:
use S+jet in POWHEG
push jet cut to parton shower IR cutoff
apply analytical NNLL Sudakov rejection weight for intrinsic line in Bornconfiguration
(kills divergent behaviour at order αs)
don’t forget double-counted terms
reweight to NNLO fixed order
F. Krauss IPPP
QCD & Monte Carlo Event Generators
First improvements Matching Multijet merging
Other merging approaches: FxFx & friends
for H production
(K.Hamilton, P.Nason, E.Re & G.Zanderighi, JHEP 1310 (2013) 222)
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
ROUND IV: SIMULATING SOFT QCD
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
SIMULATING SOFT QCD
HADRONISATION
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
Contents
8.a) QCD radiation, once more
8.b) Hadronisation: General thoughts
8.c) The string model
8.d) The cluster model
8.e) Practicalities
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
QCD radiation, once more
QCD radiation, once more
remember QCD emission pattern
dwq→qg =αs(k
2⊥)
2πCF
dk2⊥
k2⊥
dω
ω
[1 +
(1− ω
E
)].
spectrum cut-off at small transverse momenta and energies by onset ofhadronization, at scales R ≈ 1 fm/ΛQCD
two (extreme) classes of emissions: gluons and gluersdetermined by relation of formation and hadronization times
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
QCD radiation, once more
gluers formed at times R, with momenta k‖ ∼ k⊥ ∼ ω>∼ 1/R
assuming that hadrons follow partons,
dN(hadrons) ∼Q∫
k⊥>1/R
dk2⊥
k2⊥
CF αs(k2⊥)
2π
[1 +
(1− ω
E
)] dω
ω
∼ CFαs(1/R2)
πlog(Q2R2)
dω
ω
or - relating their energyn with that of the gluers -
dN(hadrons)/d log ε = const. ,
a plateau in log of energy (or in rapidity)
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
QCD radiation, once more
impact of additional radiation
new partons must separate before they can hadronize independently
therefore, one more time
tform ∼ k‖k2⊥
tsep ∼ Rθ ∼ tform (Rk⊥)
thad ∼ k‖R2 ∼ tform (Rk⊥)2 .
for gluers Rk⊥ ≈ 1: all times the same
naively; new & more hadrons following new partons
but: colour coherenceprimary and secondary partons not separated enough in
1/R<∼ ω(hadron)
<∼ 1/(Rθ)
and therefore no independent radiation
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
Hadronisation: General thoughts
Hadronisation: General thoughts
confinement the striking feature of low–scale sotrng interactions
transition from partons to their bound states, the hadrons
the Meissner effect in QCD
QED:
QCD:
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
Hadronisation: General thoughts
linear QCD potential in Quarkonia – like a string
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
Hadronisation: General thoughts
combine some experimental facts into a naive parameterisation
in e+e− → hadrons: exponentially decreasing p⊥, flat plateau in y forhadrons
try “smearing”: ρ(p2⊥) ∼ exp(−p2
⊥/σ2)
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
Hadronisation: General thoughts
use parameterisation to “guesstimate” hadronisation effects:
E =
∫ Y
0
dydp2⊥ρ(p2
⊥)p⊥ cosh y = λ sinhY
P =
∫ Y
0
dydp2⊥ρ(p2
⊥)p⊥ sinh y = λ(coshY − 1) ≈ E − λ
λ =
∫dp2⊥ρ(p2
⊥)p⊥ = 〈p⊥〉 .
estimate λ ∼ 1/Rhad ≈ mhad, with mhad 0.1-1 GeV.
effect: jet acquire non-perturbative mass ∼ 2λE(O(10GeV) for jets with energy O(100GeV)).
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Hadronisation: General thoughts
similar parametrization underlying Feynman-Field model for independentfragmentation
recursively fragment q → q′+ had, where
transverse momentum from (fitted) Gaussian;
longitudinal momentum arbitrary (hence from measurements);
flavour from symmetry arguments + measurements.
problems: frame dependent, “last quark”, infrared safety, no direct linkto perturbation theory, . . . .
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The string model
The string model
a simple model of mesons: yoyo strings
light quarks (mq = 0) connected by string, form a meson
area law: m2had ∝ area of string motion
L=0 mesons only have ’yo-yo’ modes:
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The string model
turn this into hadronisation model e+e− → qq as test case
ignore gluon radiation: qq move away from each other, act as point-likesource of string
intense chromomagnetic field within string:more qq pairs created by tunnelling and string break-up
analogy with QED (Schwinger mechanism):dP ∼ dxdt exp
(−πm2
q/κ), κ = “string tension”.
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The string model
string model = well motivated model, constraints on fragmentation(Lorentz-invariance, left-right symmetry, . . . )
how to deal with gluons?
interpret them as kinks on the string =⇒ the string effect
infrared-safe, advantage: smooth matching with PS.
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The cluster model
The cluster model
underlying idea: preconfinement/LPHD
typically, neighbouring colours will end in same hadron
hadron flows follow parton flows −→ don’t produce any hadrons at placeswhere you don’t have partons
works well in large–Nc limit with planar graphs
follow evolution of colour in parton showers
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The cluster model
paradigm of cluster model: clusters as continuum of hadron resonances
trace colour through shower in Nc →∞ limit
force decay of gluons into qq or dd pairs, form colour singlets fromneighbouring colours, usually close in phase space
mass of singlets: peaked at low scales ≈ Q20
decay heavy clusters into lighter ones or into hadrons(here, many improvements to ensure leading hadron spectrum hardenough, overall effect: cluster model becomes more string-like)
if light enough, clusters will decay into hadrons
naively: spin information washed out, decay determined through phasespace only → heavy hadrons suppressed (baryon/strangeness suppression)
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Hadronisation Underlying Event Summary
The cluster model
self–similarity of parton showerwill end with roughly the same localdistribution of partons, with roughly thesame invairant mass for colour singlets
adjacent pairs colour connected,form colourless (white) clusters.
clusters (“≈ excited hadrons)decay into hadrons
F. Krauss IPPP
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Observables
Observables
in the following a selection of data from the LEP collaboration relevantfor the tuning of hadronisation models
all compared with an actual tune of SHERPA
typically, PYTHIA does as good (or sometimes even slightly better)
so, this is the level we talk about these days, agreement of 5% or betterover large ranges of observables and scales
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Observables
b
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F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
Observables
b
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F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
Observables
b
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10−5
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F. Krauss IPPP
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Observables
b
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F. Krauss IPPP
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Hadronisation Underlying Event Summary
Observables
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F. Krauss IPPP
QCD & Monte Carlo Event Generators
Hadronisation Underlying Event Summary
Observables
b
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F. Krauss IPPP
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Hadronisation Underlying Event Summary
Practicalities
practicalities of hadronisation models: parameters
kinematics of string or cluster decay: 2-5 parameters
must “pop” quark or diquark flavours in string or cluster decay — cannotbe completely democratic or driven by masses alone−→ suppression factors for strangeness, diquarks 2-10 parameters
transition to hadrons, cannot be democratic over multiplets−→ adjustment factors for vectors/tensors etc. 2-6 parameters
tuned to LEP data, overall agreement satisfying
validity for hadron data not quite clear(beam remnant fragmentation not in LEP.)
there are some issues with inclusive strangeness/baryon production
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SIMULATING SOFT QCD
UNDERLYING EVENT
F. Krauss IPPP
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Contents
8.a) Multiple parton scattering
8.b) Modelling the underlying event
8.c) Some results
8.d) Practicalities
F. Krauss IPPP
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Multiple parton scattering
Multiple parton scattering
hadrons = extended objects!
no guarantee for one scatteringonly.
running of αS
=⇒ preference for soft scattering.
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Multiple parton scattering
first experimental evidencefor double–parton scattering:events with γ + 3 jets:
cone jets, R = 0.7,ET > 5 GeV; |ηj | <1.3;“clean sample”: twosoftest jets with ET < 7GeV;
cross section for DPS
σDPS =σγjσjjσeff
σeff ≈ 14± 4 mb.
F. Krauss IPPP
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Multiple parton scattering
more measurements, also atLHC
ATLAS results from W + 2jets
W+
ν
ℓ
qg
q
g
νq
ℓ
W+
⊗q
njets∆
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Eve
nts
/ 0.0
3
0
0.02
0.04
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0.12
=7 TeVs unfolded data, νWlFit distributionA+H+J particle-level template A PYTHIA particle-level template B
ATLAS
-1 Ldt=36 pb∫
[GeV]s
210 310 410
[mb]
eff
σ
02
4
68
101214
16
182022
AFS (4 jets - no errors given)UA2 (4 jets - lower limit)CDF (4 jets)
+ 3 jets)γCDF ( + 3 jets)γD0 (
ATLAS (W + 2 jets)
ATLAS
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Multiple parton scattering
Proton AntiProton
Multiple Parton Interactions
PT(hard)
Outgoing Parton
Outgoing Parton
Underlying EventUnderlying Event
but: how to define the underlying event?
1 everything apart from the hard interaction, but including IS showers, FSshowers, remnant hadronisation.
2 remnant-remnant interactions, soft and/or hard.
3 lesson: hard to define
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Multiple parton scattering
origin of MPS: parton–parton scattering cross section exceedshadron–hadron total cross section
σhard(p⊥,min) =
s/4∫p2⊥,min
dp2⊥dσ(p2
⊥)
dp2⊥
> σpp,total
for low p⊥,min
remember
dσ(p2⊥)
dp2⊥
=
1∫0
dx1dx2f (x1, q2)f (x2, q
2)dσ2→2
dp2⊥
〈σhard(p⊥,min)/σpp,total〉 ≥ 1
depends strongly on cut-off p⊥,min (energy-dependent)!
F. Krauss IPPP
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Modelling the underlying event
Modelling the underlying event
take the old PYTHIA model as example:
start with hard interaction, at scale Q2hard.
select a new scale p2⊥ from
exp
− 1
σnorm
Q2hard∫
p2⊥
dp′⊥2 dσ(p2
⊥)
dp′⊥2
with constraint p2
⊥ > p2⊥,min
rescale proton momentum (“proton-parton = proton with reduced energy”).
repeat until no more allowed 2→ 2 scatter
F. Krauss IPPP
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Modelling the underlying event
Modelling the underlying event
possible refinements:
may add impact-parameter dependence −→ more fluctuations
add parton showers to UE
“regularisation” to dampen sharp dependence on p⊥,min: replace 1/t inMEs by 1/(t + t0), also in αs.
treat intrinsic k⊥ of partons (→ parameter)
model proton remnants (→ parameter)
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Some results in Z production
Some results for MPS in Z production
observables sensitive to MPS
classical analysis: transverse regions inQCD/jet events
idea: find the hardest system, orientevent into regions:
toward region along systemaway region back-to-backtransverse regions
typically each in 120
F. Krauss IPPP
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Some results in Z production
F. Krauss IPPP
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Hadronisation Underlying Event Summary
Some results in Z production
F. Krauss IPPP
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Hadronisation Underlying Event Summary
Some results in Z production
F. Krauss IPPP
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Hadronisation Underlying Event Summary
Practicalities
see some data comparison in Minimum Bias
practicalities of underlying event models: parameters
profile in impact parameter space 2-3 parameters
IR cut-off at reference energy, its energy evolution, dampening paramterand normalisation cross section
4 parameters
treating colour connections to rest of event 2-5 parameters
tuned to LHC data, overall agreement satisfying
energy extrapolation not exactly perfect, plus other process categoriessuch as diffraction etc..
F. Krauss IPPP
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Practicalities
SUMMARY
F. Krauss IPPP
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Summary
Systematic improvement of event generatorsby including higher orders has been at the coreof QCD theory and developments in the pastdecade:
multijet merging (“CKKW”, “MLM”)NLO matching (“MC@NLO”, “POWHEG”)MENLOPS NLO matching & mergingMEPS@NLO (“SHERPA”, “UNLOPS”,“MINLO”, “FxFx”)
/home/krauss/TeX/Private/limitationsdemotivator.jpg
multijet merging an important tool for many relevant signals andbackgrounds - pioneering phase at LO & NLO over
complete automation of NLO calculations done−→ must benefit from it!
(it’s the precision and trustworthy & systematic uncertainty estimates!)
F. Krauss IPPP
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Famous last screams
in Run-II we’ll be in for a ride:
more statisticsmore energy
more channelsmore precision
more fun
. . . and all with QCD . . .
oh, and btw.: the first NNLO+PS are out!
F. Krauss IPPP
QCD & Monte Carlo Event Generators
Top Related