qcd & monte carlo event generators

QCD & Monte Carlo Event Generators

Frank Krauss

Institute for Particle Physics PhenomenologyDurham University

HiggsTools School, Pre Saint-Didier, 2015

F. Krauss IPPP


PART I: Introduction

1 QCD Basics: Scales & Kinematics

2 General Monte Carlo

F. Krauss IPPP


PART II: Monte Carlo for Perturbative QCD

3 Parton–level Monte Carlo

4 Parton showers – the basics

F. Krauss IPPP


PART III: Precision Simulations

5 First improvements

6 Matching

7 Multijet merging

F. Krauss IPPP


PART IV: Monte Carlo for Non–Perturbative QCD

8 Hadronisation

9 Underlying Event

10 Summary

F. Krauss IPPP


QCD Basics: Scales & Kinematics General Monte Carlo

PART I: INTRODUCTION

F. Krauss IPPP



QCD BASICS

SCALES & KINEMATICS

F. Krauss IPPP



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

F. Krauss IPPP



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

F. Krauss IPPP




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”

F. Krauss IPPP




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)

F. Krauss IPPP



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

F. Krauss IPPP




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

F. Krauss IPPP




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)

F. Krauss IPPP



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

F. Krauss IPPP




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

F. Krauss IPPP



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

F. Krauss IPPP




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.

F. Krauss IPPP



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)

),

F. Krauss IPPP




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)

F. Krauss IPPP




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 .

F. Krauss IPPP




F. Krauss IPPP



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

F. Krauss IPPP




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.

F. Krauss IPPP




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.

F. Krauss IPPP




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

F. Krauss IPPP




Summary

F. Krauss IPPP



GOING MONTE CARLO

GENERAL IDEAS & TECHNIQUES

F. Krauss IPPP



Contents

2.a) Prelude: selecting from a distribution

2.b) Monte Carlo integration: basic idea

2.c) Traditional MC simulation

F. Krauss IPPP



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))]

F. Krauss IPPP



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).

F. Krauss IPPP



Monte Carlo integration: basic idea

Monte Carlo integration

underlying idea: determination of π with random number generator

F. Krauss IPPP




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.

F. Krauss IPPP




F. Krauss IPPP



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.

F. Krauss IPPP




importance sampling

consider f (x) = cos πx2 and g(x) = 1− x2:

F. Krauss IPPP




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.

F. Krauss IPPP




stratified sampling

consider f (x) = cos πx2 and g(x) = 1− x2:

〈I 〉 = 0.637± 0.147/√N

F. Krauss IPPP




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

F. Krauss IPPP



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.

F. Krauss IPPP




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).

F. Krauss IPPP




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.

F. Krauss IPPP




example results on a 10× 10 lattice

F. Krauss IPPP


Parton–level Monte Carlo Parton showers – the basics

PART II: MONTE CARLO

FOR PERTURBATIVE QCD

F. Krauss IPPP



MONTE CARLO FOR

PARTON LEVEL

F. Krauss IPPP



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

F. Krauss IPPP



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

F. Krauss IPPP





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.

F. Krauss IPPP




example for factorial growth: e+e− → qq + ng

n #diags

0 11 22 83 484 384

F. Krauss IPPP




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!

F. Krauss IPPP




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.

F. Krauss IPPP




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




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

F. Krauss IPPP




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

F. Krauss IPPP




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.

F. Krauss IPPP



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.

F. Krauss IPPP




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!

F. Krauss IPPP




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.

F. Krauss IPPP



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)

F. Krauss IPPP




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”)

F. Krauss IPPP




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

F. Krauss IPPP



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




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

F. Krauss IPPP




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




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



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

F. Krauss IPPP




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

F. Krauss IPPP




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 +

F. Krauss IPPP



GOING MONTE CARLO

PARTON SHOWERS – THE BASICS

F. Krauss IPPP



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



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.

F. Krauss IPPP



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



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

F. Krauss IPPP




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




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) ,

F. Krauss IPPP




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




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




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




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



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




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

F. Krauss IPPP




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




pictorially:

=⇒gluons at large angle from combined colour charge!

F. Krauss IPPP




experimental manifestation:∆R of 2nd & 3rdjetinmulti − jeteventsinpp − collisions

F. Krauss IPPP



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




“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




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

F. Krauss IPPP




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

F. Krauss IPPP


First improvements Matching Multijet merging

ROUND III: PRECISION MONTE CARLO

F. Krauss IPPP



FIRST IMPROVEMENTS:

ME CORRECTIONS

F. Krauss IPPP



Contents

5.a) Improving event generators

5.b) Matrix-element corrections

F. Krauss IPPP



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




. . . 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




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

F. Krauss IPPP



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




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



PRECISION MONTE CARLO

NLO MATCHING

F. Krauss IPPP



Contents

6.a) Basic idea

6.b) Powheg

6.c) MC@NLO

F. Krauss IPPP



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



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



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



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



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



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



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



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



MC@NLO

MC@NLO for light jets: jet-p⊥

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

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

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

10 3

10 4

10 5

10 6

10 7

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


0.3 < |y| < 0.8

2 3

0.5

1

1.5

MC/data


0.8 < |y| < 1.2

2 3

0.5

1

1.5

MC/data

b b b b b b b b b b b b b b b

1.2 < |y| < 2.1

2 3

0.5

1

1.5

MC/data

b b b b b b b b b b b b

2.1 < |y| < 2.8

2 3

0.5

1

1.5

MC/data

b b b b b b b b b

2.8 < |y| < 3.6

2 3

0.5

1

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



MC@NLO

MC@NLO for light jets: dijet mass

b

b

b

b

b

bb

b

b

b

b

b

bbb

b

b

b

b

b

b

b

b

bb

b

b

bb

bbb

b

b

b

b

b

b

b

b

b

b

b

bb

b

bb

b

b

bb

b

bb

b

b

b

b

b

b

b

b

b

bb

bb

b

b

bb

b

b

bb

b

b

b

b

b

b

b

b

b

b

bb

bb

bb

b

bb

bb

b

b

b

b

b

b

b

b

b

b

b

b

bb

bb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

bb

bbb

bb

b

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

10 6

10 7

10 8

10 9

10 10

10 11

10 12

10 13

10 14

10 15

10 16

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


3.0 < y∗ < 3.5

1

0.51

1.5

MC/data


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


0.5 < y∗ < 1.0

1

0.5

1

1.5

MC/data


y∗ < 0.5

10−1 1

0.5

1

1.5

m12 [GeV]

MC/data

F. Krauss IPPP



MC@NLO

MC@NLO for light jets: azimuthal decorrelations

bb

bb

b

b

bb

b

b

b

b

b

b

b

b

b

b

bb

b

bb

b

bb

b

b

b

b

b

b

b

b

b

b

b

b

bb

bb

b

bb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

bb

b

bb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

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



0.5

1

1.5

MC/data



0.5

1

1.5

MC/data



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



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



MC@NLO

MC@NLO for light jets: jet vetoes

bb

bb

bb b

bb b

b b

bb

bb

bb

bb b b b b

bb

bb

bb

bb

b b bb

bb

bb

bb

b b bb b

b

bb

bb

bb b

bb b b b

b

bb

bb

b bb b

b b b

b

bb

bb

b bb b b

70 GeV < p⊥ < 90 GeV

90 GeV < p⊥ < 120 GeV+1

120 GeV < p⊥ < 150 GeV+2

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


180 GeV < p⊥ < 210 GeV0.5

1

1.5

MC/data


150 GeV < p⊥ < 180 GeV0.5

1

1.5

MC/data


120 GeV < p⊥ < 150 GeV0.5

1

1.5

MC/data


90 GeV < p⊥ < 120 GeV0.5

1

1.5

MC/data


70 GeV < p⊥ < 90 GeV

0 1 2 3 4 5 60.5

1

1.5

∆y

MC/data

F. Krauss IPPP



PRECISION MONTE CARLO

MULTIJET MERGING

F. Krauss IPPP



Contents

7.a) Basic idea

7.b) Multijet merging at LO

7.c) Multijet merging at NLO

F. Krauss IPPP



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



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



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



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




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




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,



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,



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




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,


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,


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



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



MENLOPS

Transverse momentum of W & Z boson

ATLAS, arXiv:1108.6308, arXiv:1107.2381

F. Krauss IPPP



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


| < 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



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


| < 4.4jet

> 30 GeV, |yjet

Tp

100 200 300 400 500 600 700

NLO

/ D

ata

0.60.8


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


ALPGEN

SHERPA

+ SHERPAATHLACKB

ATLAS )µ 2 jets (l=e,≥)+ -l+ l→*(γZ/-1


| < 4.4jet

> 30 GeV, |yjet

Tp

100 200 300 400 500

NLO

/ D

ata

0.60.8


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



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



| < 4.4jet

> 30 GeV, |yjet

Tp

40 60 80 100 120 140 160 180 200

NLO

/ D

ata

0.60.8


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


ALPGEN

SHERPA

+ SHERPAATHLACKB



| < 4.4jet

> 30 GeV, |yjet

Tp

30 40 50 60 70 80 90 100

NLO

/ D

ata

0.60.8


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



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



| < 4.4jet

> 30 GeV, |yjet

Tp

0 1 2 3 4 5 6

NLO

/ D

ata

0.60.8


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


ALPGEN

SHERPA

+ SHERPAATHLACKB



| < 4.4jet

> 30 GeV, |yjet

Tp

0 100 200 300 400 500 600 700 800 900 1000

NLO

/ D

ata

0.60.8


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



MENLOPS

Z+jets: ∆φZj in unboosted sample

CMS, arXiv:1301.1646

F. Krauss IPPP



MENLOPS

Z+jets: ∆φZj in boosted sample

CMS, arXiv:1301.1646

F. Krauss IPPP



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




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




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




pH⊥ in MEPS@NLO

pp → h + jetspp → h + 0j @ NLO

0 50 100 150 200 250 30010−4

10−3

10−2

10−1


p⊥(h) [GeV]

dσ

/dp ⊥

[pb/

GeV

]

first emission byMC@NLO , restrict toQn+1 < Qcut



MC@NLO


iterate



F. Krauss IPPP




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


p⊥(h) [GeV]

dσ

/dp ⊥

[pb/

GeV

]




MC@NLO


iterate



F. Krauss IPPP




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


p⊥(h) [GeV]

dσ

/dp ⊥

[pb/

GeV

]




MC@NLO


iterate



F. Krauss IPPP




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


p⊥(h) [GeV]

dσ

/dp ⊥

[pb/

GeV

]




MC@NLO


iterate



F. Krauss IPPP




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



Mc@Nlo10−5

10−4

10−3

10−2

10−1


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



Mc@Nlo10−5

10−4

10−3

10−2

10−1


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



Mc@Nlo

10−5

10−4

10−3

10−2

10−1


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




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



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



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



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



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




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



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




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



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



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



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




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



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


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



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



Mc@Nlo

20

40

60

80

100

120

140

Azimuthal Distance of Leading Jets

dσ/d

∆φ[pb]


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




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




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


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




Exclusive observables for gg → H


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


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




gg → H after WBF cuts


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


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




gg → H after WBF cuts


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


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




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




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




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




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




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




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


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




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


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




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



Emiss⊥ [GeV]

Relativeuncertainty

F. Krauss IPPP




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 background uncertainty

m3ℓ [GeV]

Relativeuncertainty

0 1 2 3 4 5

0.9

1.0



∆R01

Ratio

F. Krauss IPPP



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




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




FxFx: QJ dependence in tt

(R.Frederix & S.Frixione, JHEP 1212 (2012) 061)

F. Krauss IPPP




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




for H production

(K.Hamilton, P.Nason, E.Re & G.Zanderighi, JHEP 1310 (2013) 222)

F. Krauss IPPP


Hadronisation Underlying Event Summary

ROUND IV: SIMULATING SOFT QCD

F. Krauss IPPP



SIMULATING SOFT QCD

HADRONISATION

F. Krauss IPPP



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 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




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




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



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




linear QCD potential in Quarkonia – like a string

F. Krauss IPPP




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




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)).

F. Krauss IPPP




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, . . . .

F. Krauss IPPP



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:

F. Krauss IPPP



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”.

F. Krauss IPPP



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.

F. Krauss IPPP



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

F. Krauss IPPP



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)

F. Krauss IPPP



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



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

F. Krauss IPPP



Observables

b

b

b

b

b

b

b

b

b

b

b

b DataAnalysis.LEP91.2.5

1

10 1

Thrust, 1 − T, at 91 GeV

1/σ

dσ

/d(1

−T)

0 0.05 0.1 0.15 0.2 0.25 0.3

0.6

0.8

1

1.2

1.4

1 − T

MC

/Dat

a

b

b

b

b

b

b

b

b

b


10−1

1

10 1Thrust major, Tmaj, at 91 GeV

1/σ

dσ

/d

T maj

0 0.1 0.2 0.3 0.4 0.5 0.6

0.6

0.8

1

1.2

1.4

Tmaj

MC

/Dat

ab

b

b

bb

b

b

b

b

b

b


10−1

1

10 1

Thrust minor, Tmin, at 91 GeV

1/σ

dσ

/d

T min

0 0.05 0.1 0.15 0.2 0.25 0.3

0.6

0.8

1

1.2

1.4

Tmin

MC

/Dat

a

b

b

b

b

b

b

b

b


10−1

1

10 1

Oblateness, O, at 91 GeV

1/σ

dσ

/dO

0 0.1 0.2 0.3 0.4 0.5

0.6

0.8

1

1.2

1.4

O

MC

/Dat

a

F. Krauss IPPP



Observables

b

b

b

b

b

b

b

b

bb

bb

b b b b b b b b b b b b b b b b b


10−2

10−1

1

10 1

10 2

Integrated 2-jet rate with Durham algorithm (91.2 GeV)

R2

10−4 10−3 10−2 10−1

0.6

0.8

1

1.2

1.4

yDurhamcut

MC

/Dat

a

b

b

b

b

b

b

b

b

b

bb

bb b b b b b b b b

bb

bb

bb

b

b

b

b


10−1

1

10 1


R3

10−4 10−3 10−2 10−1

0.6

0.8

1

1.2

1.4

yDurhamcut

MC

/Dat

ab

bb

b

b

b

b

b

b

b

bb

b b b bb

bb

bb

bb

b

b

b

b

b

b

b


10−2

10−1

1

10 1


R4

10−4 10−3 10−2

0.6

0.8

1

1.2

1.4

yDurhamcut

MC

/Dat

a

b

bb

b

b

b

b

b

b

b

bb

b b b bb

b

b

b

b

b

b

b

b

b

b

b


10−2

10−1

1

10 1


R5

10−5 10−4 10−3 10−2

0.6

0.8

1

1.2

1.4

yDurhamcut

MC

/Dat

a

F. Krauss IPPP



Observables

b

b

b

b

bb

b

b

b

b

b

b

b


10−5

10−4

10−3

10−2

10−1

1

10 1Durham jet resolution 2 → 1 (ECMS = 91.2 GeV)

1/σ

dσ

/d

ln(y

12)

0 0.2 0.4 0.6 0.8 1 1.2

0.6

0.8

1

1.2

1.4

− ln(y12)

MC

/Dat

ab

b

bbbb 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

b

bb

b

b b

b

bb

b

b


10−5

10−4

10−3

10−2

10−1


1/σ

dσ

/d

ln(y

23)

1 2 3 4 5 6 7 8 9 10

0.6

0.8

1

1.2

1.4

− ln(y23)

MC

/Dat

ab

b

b

b

b

b

bbbbbbbbbbbbbbb b b b b b b b

bbbb

b

b

b

b

b

b

b

b

b

b bbb b

b


10−4

10−3

10−2

10−1


1/σ

dσ

/d

ln(y

34)

2 3 4 5 6 7 8 9 10 11

0.6

0.8

1

1.2

1.4

− ln(y34)

MC

/Dat

a

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


10−5

10−4

10−3

10−2

10−1


1/σ

dσ

/d

ln(y

45)

4 5 6 7 8 9 10 11 12

0.6

0.8

1

1.2

1.4

− ln(y45)

MC

/Dat

a

F. Krauss IPPP



Observables

b

b

b b b

b b bb


18

20

22

24

26

28

30

Charged multiplicity at a function of energy

Nch

80 100 120 140 160 180 200

0.6

0.8

1

1.2

1.4

ECMS/GeV

MC

/Dat

ab b

b

b

bb

b b b bb

bb

bb

b

bb

b

b

bb

b

b

b

b


10−5

10−4

10−3

10−2

10−1

Charged multiplicity distribution

1/N

dN

/d

Nch

10 20 30 40 50

0.6

0.8

1

1.2

1.4

Nch

MC

/Dat

a

b bbbbbbb b b

bbbbb

bb

b

b

b

b

b


10−2

10−1

1

10 1

10 2

All events scaled momentum

1/σ

dσ

/d

x p

0 0.2 0.4 0.6 0.8 1

0.6

0.8

1

1.2

1.4

xp

MC

/Dat

a

b

b

b

b

b

b

bb

bb

b b b b b b b b b b b b b bb

bb


10−2

10−1

1

10 1Log of scaled momentum, log(1/xp)

Nd

σ/

dlog

(1/

x p)

0 1 2 3 4 5

0.6

0.8

1

1.2

1.4

log(1/xp)

MC

/Dat

a

F. Krauss IPPP



Observables

b b b bbbbbbb bb b b b b b b

bbb

bb

bb

bb

bb

bb

bb

bb

b

b

b


10−1

1

10 1

10 2

π+ scaled momentum

dN

/d

x p

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.6

0.8

1

1.2

1.4

xp

MC

/Dat

a

b bb bb bb b b b b b b b b

bb b

b bb b

bb

bb

bb

b

bb

b


10−3

10−2

10−1

1

10 1

p+ scaled momentum

dN

/d

x p

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.6

0.8

1

1.2

1.4

xp

MC

/Dat

a

b b

b

b

b

b

b

b


10−1

1

10 1

ρ spectrum

1/σ

dσ

/d

x p

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6

0.8

1

1.2

1.4

xp

MC

/Dat

a

b

b

b


10−1

Σ±(1385) scaled momentum

1/σ

dσ

/d

x p

0.05 0.1 0.15 0.2 0.25 0.3

0.6

0.8

1

1.2

1.4

xp

MC

/Dat

a

F. Krauss IPPP



Observables

b

b

bb

b

b

b

bb

b

b

bb

b

b

bb

b


0

0.002

0.004

0.006

0.008

0.01

0.012Scaled energy of D∗± in e+e− → Z → hadronic at

√s = 91.2 GeV

1/N

Zha

dd

N(D

∗ )/

dX

E

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6

0.8

1

1.2

1.4

XE

MC

/Dat

a

bb

b

bb

b

b

b

b


10−3

10−2

10−1

J/ψ scaled momentum

1/σ

dσ

/d

x p/

0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6

0.8

1

1.2

1.4

xp

MC

/Dat

ab

b b b bb b

bb

bb

b

b

b

b

b

b

b

b

b

b

b


0

0.5

1

1.5

2

2.5

3

3.5

b quark fragmentation function f (xweakB )

1/N

dN

/d

x B

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6

0.8

1

1.2

1.4

xB

MC

/Dat

a

b

bb

b

b

b

bb

b

b


1

b quark fragmentation function f (xweakB )

1/N

dN

/d

x B

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.6

0.8

1

1.2

1.4

xB

MC

/Dat

a

F. Krauss IPPP



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

F. Krauss IPPP



SIMULATING SOFT QCD

UNDERLYING EVENT

F. Krauss IPPP



Contents

8.a) Multiple parton scattering

8.b) Modelling the underlying event

8.c) Some results

8.d) Practicalities

F. Krauss IPPP



Multiple parton scattering


hadrons = extended objects!

no guarantee for one scatteringonly.

running of αS

=⇒ preference for soft scattering.

F. Krauss IPPP




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




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

0.06

0.08

0.1

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

F. Krauss IPPP




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

F. Krauss IPPP




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



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





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)

F. Krauss IPPP



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



Some results in Z production

F. Krauss IPPP



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



Practicalities

SUMMARY

F. Krauss IPPP



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



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

Documents