[PPT] - Simulating NNLO QCD corrections for processes with giant K factors PowerPoint Presentation

SLIDE 1

Simulating NNLO QCD corrections for processes with giant K factors

Sebastian Sapeta

LPTHE, UPMC, CNRS, Paris

in collaboration with Gavin Salam and Mathieu Rubin 1

HP 2 .3rd, Florence, 14-17 September 2010

1M.Rubin, G.P.Salam and SS, arXiv:1006.2144 [hep-ph]

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 1 / 13

SLIDE 2

The problem of giant K factors

◮ Z+j at the LHC

pt,Z pt,hardest jet HT,jets =

all jets pt,j

10-1 1 10 102 103 104 250 500 750 1000 dσ/dV [fb / 100 GeV] V = pt,Z [GeV]

pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV, Z → e+e-

LO 250 500 750 1000 V = pt,j1 [GeV] LO 250 500 750 1000 V = HT,jets [GeV]

MCFM 5.7, CTEQ6M

LO

LO:

g Z q

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 2 / 13

SLIDE 3

The problem of giant K factors

◮ Z+j at the LHC

pt,Z pt,hardest jet HT,jets =

all jets pt,j

10-1 1 10 102 103 104 250 500 750 1000 dσ/dV [fb / 100 GeV] V = pt,Z [GeV]

pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV, Z → e+e-

LO NLO 250 500 750 1000 V = pt,j1 [GeV] LO NLO 250 500 750 1000 V = HT,jets [GeV]

MCFM 5.7, CTEQ6M

LO NLO

NLO:

O ` αewα2

s

´

g Z q g

O

` αewα2

s ln2 pt,j1/MZ

´

Z g g q

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 2 / 13

SLIDE 4

What do we have and what is missing?

◮ The large K factor for the Z+jet comes from the new “dijet type” topologies

that appear at NLO

Z g g q Z g g q

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 3 / 13

SLIDE 5

What do we have and what is missing?

◮ The large K factor for the Z+jet comes from the new “dijet type” topologies

that appear at NLO

Z g g q Z g g q ◮ though formally NLO diagrams for Z+jet, these are in fact leading

contributions to pt,j1 and HT spectra

◮ this raises doubts about the accuracy of these predictions ◮ need for subleading contributions for Z+jet, in this case NNLO

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 3 / 13

SLIDE 6

What do we have and what is missing?

◮ The large K factor for the Z+jet comes from the new “dijet type” topologies

that appear at NLO

Z g g q Z g g q ◮ though formally NLO diagrams for Z+jet, these are in fact leading

contributions to pt,j1 and HT spectra

◮ this raises doubts about the accuracy of these predictions ◮ need for subleading contributions for Z+jet, in this case NNLO

Z+j at NNLO = Z+3j tree + Z+2j 1-loop + Z+j 2-loop

Z+2j at NLO

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 3 / 13

SLIDE 7

What do we have and what is missing?

◮ The large K factor for the Z+jet comes from the new “dijet type” topologies

that appear at NLO

Z g g q Z g g q ◮ though formally NLO diagrams for Z+jet, these are in fact leading

contributions to pt,j1 and HT spectra

◮ this raises doubts about the accuracy of these predictions ◮ need for subleading contributions for Z+jet, in this case NNLO

Z+j at NNLO = Z+3j tree + Z+2j 1-loop + Z+j 2-loop

Z+2j at NLO

◮ 2-loop part

◮ we need it to cancel IR and collinear divergences from Z+2j at NLO result ◮ it will have the topology of Z+j at LO so it will not contribute much to the

cross sections with giant K-factor

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 3 / 13

SLIDE 8

The basic idea

How to cancel the infrared and collinear singularities?

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13

SLIDE 9

The basic idea

How to cancel the infrared and collinear singularities?

◮ use unitarity to simulate the divergent part of 2-loop diagrams

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13

SLIDE 10

The basic idea

How to cancel the infrared and collinear singularities?

◮ use unitarity to simulate the divergent part of 2-loop diagrams

LoopSim procedure

input: event with n final state particles

utput:

all n − k final state particle events (equivalently all k-loop events) LoopSim

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13

SLIDE 11

The basic idea

How to cancel the infrared and collinear singularities?

◮ use unitarity to simulate the divergent part of 2-loop diagrams

LoopSim procedure

input: event with n final state particles

utput:

all n − k final state particle events (equivalently all k-loop events) LoopSim

◮ notation:

¯ nLO – simulated 1-loop ¯ n¯ nLO – simulated 2-loop and simulated 1-loop ¯ nNLO – simulated 2-loop and exact 1-loop

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13

SLIDE 12

The basic idea

How to cancel the infrared and collinear singularities?

◮ use unitarity to simulate the divergent part of 2-loop diagrams

LoopSim procedure

input: event with n final state particles

utput:

all n − k final state particle events (equivalently all k-loop events) LoopSim

◮ notation:

¯ nLO – simulated 1-loop ¯ n¯ nLO – simulated 2-loop and simulated 1-loop ¯ nNLO – simulated 2-loop and exact 1-loop

◮ this will work very well for the processes with large K factors e.g.

σ¯

nNLO = σNNLO

1 + O
α2

s

KNNLO

,

KNNLO KNLO ≫ 1

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13

SLIDE 13

The LoopSim method: ¯ nLO, ¯ n¯ nLO etc.

4 2 1 3

beam Input event

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 5 / 13

En,l – input event with n final state particles and l loops Ub

l

–

perator producing event with b Born particles and l loops

Ub

∀

–

perator generating all necessary loop diagrams at given order

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 6 / 13

SLIDE 20

Including exact loops

En,l – input event with n final state particles and l loops Ub

l

–

perator producing event with b Born particles and l loops

Ub

∀

–

perator generating all necessary loop diagrams at given order

How to introduce exact loop contributions?

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 6 / 13

SLIDE 21

Including exact loops

En,l – input event with n final state particles and l loops Ub

l

–

perator producing event with b Born particles and l loops

Ub

∀

–

perator generating all necessary loop diagrams at given order

How to introduce exact loop contributions?

Ub

∀(En,0) ◮ generate all diagrams from the tree level event

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 6 / 13

SLIDE 22

Including exact loops

En,l – input event with n final state particles and l loops Ub

l

–

perator producing event with b Born particles and l loops

Ub

∀

–

perator generating all necessary loop diagrams at given order

How to introduce exact loop contributions?

Ub

∀(En,0)

+ Ub

∀(En−1,1) ◮ generate all diagrams from the tree level event ◮ generate all diagrams from the 1-loop event

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 6 / 13

SLIDE 23

Including exact loops

En,l – input event with n final state particles and l loops Ub

l

–

perator producing event with b Born particles and l loops

Ub

∀

–

perator generating all necessary loop diagrams at given order

How to introduce exact loop contributions?

Ub

∀(En,0)

+ Ub

∀(En−1,1)

− Ub

∀(Ub 1 (En,0)) ◮ generate all diagrams from the tree level event ◮ generate all diagrams from the 1-loop event ◮ remove all approximate diagrams from Ub ∀(En,0) that have exact

counterparts provided by Ub

∀(En−1,1)

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 6 / 13

SLIDE 24

Including exact loops

En,l – input event with n final state particles and l loops Ub

l

–

perator producing event with b Born particles and l loops

Ub

∀

–

perator generating all necessary loop diagrams at given order

How to introduce exact loop contributions?

Ub

∀(En,0)

+ Ub

∀(En−1,1)

− Ub

∀(Ub 1 (En,0)) ◮ generate all diagrams from the tree level event ◮ generate all diagrams from the 1-loop event ◮ remove all approximate diagrams from Ub ∀(En,0) that have exact

counterparts provided by Ub

∀(En−1,1) ◮ inclusion of exact loops helps reducing scale uncertainties ◮ straightforward generalization to arbitrary number of exact loops

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 6 / 13

SLIDE 25

Validation

2MZ, 58 GeV]

> 58 GeV ¯ nLO vs NLO very good excellent perfect (not guaranteed) (expected) (expected)

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 8 / 13

SLIDE 30

Drell-Yan at NNLO: spectrum of harder lepton

102 103 104 105 10 20 30 40 50 60 70 80 90 100 dσ/dpt,max [fb/GeV] pt,max [GeV] pp, 14 TeV mZ/2 < µ < 2mZ MCFM 5.3 66 < me+e- < 116 GeV LO NLO

–

nLO 102 103 104 105 10 20 30 40 50 60 70 80 90 100 dσ/dpt,max [fb/GeV] pt,max [GeV] pp, 14 TeV, mZ/2 < µ < 2mZ MCFM 5.3, DYNNLO 1.0 66 < me+e- < 116 GeV NLO

–

nNLO NNLO

◮ giant K factor due to a boost caused by initial state radiation ◮ the agreement between NLO and ¯

nLO may serve as a indication whether the method works for a given observable, Z@¯

nLO = Z@LO+LoopSim ◦ (Z+j@LO)

◮

three regions of pt,max : 1

2MZ

[ 1

2MZ, 58 GeV]

> 58 GeV ¯ nLO vs NLO very good excellent perfect and ¯ nNLO vs NNLO (not guaranteed) (expected) (expected)

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 8 / 13

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¯ nNLO predictions for LHC

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 9 / 13

SLIDE 32

Z+jet at ¯ nNLO = Z+j@NLO + LoopSim◦(Z+2j@NLOonly)

pt,Z

0.5 1 1.5 2 2.5 3 250 500 750 1000 1250 K-factor wrt LO pt,Z (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

(A)

g Z q

(B)

Z g g q

(C)

Z g g q

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 10 / 13

SLIDE 33

Z+jet at ¯ nNLO = Z+j@NLO + LoopSim◦(Z+2j@NLOonly)

pt,Z

0.5 1 1.5 2 2.5 3 250 500 750 1000 1250 K-factor wrt LO pt,Z (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep)

◮ pt,Z: no correction; topology (A) dominant at high pt,Z

(extra loops w.r.t. NLO do not change much)

(A)

g Z q

(B)

Z g g q

(C)

Z g g q

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 10 / 13

SLIDE 34

Z+jet at ¯ nNLO = Z+j@NLO + LoopSim◦(Z+2j@NLOonly)

pt,Z pt,hardest jet

0.5 1 1.5 2 2.5 3 250 500 750 1000 1250 K-factor wrt LO pt,Z (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) 2 4 6 8 10 250 500 750 1000 1250 K-factor wrt LO pt,j1 (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

◮ pt,Z: no correction; topology (A) dominant at high pt,Z

(extra loops w.r.t. NLO do not change much)

(A)

g Z q

(B)

Z g g q

(C)

Z g g q

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 10 / 13

SLIDE 35

Z+jet at ¯ nNLO = Z+j@NLO + LoopSim◦(Z+2j@NLOonly)

pt,Z pt,hardest jet

0.5 1 1.5 2 2.5 3 250 500 750 1000 1250 K-factor wrt LO pt,Z (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) 2 4 6 8 10 250 500 750 1000 1250 K-factor wrt LO pt,j1 (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep)

◮ pt,Z: no correction; topology (A) dominant at high pt,Z

(extra loops w.r.t. NLO do not change much)

◮ pt,j: small correction; ¯

nNLO is like NLO for the dominant (B) and (C) configurations and it behaves like healthy NLO

(A)

g Z q

(B)

Z g g q

(C)

Z g g q

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 10 / 13

SLIDE 36

Z+jet at ¯ nNLO = Z+j@NLO + LoopSim◦(Z+2j@NLOonly)

pt,Z pt,hardest jet HT,jets

0.5 1 1.5 2 2.5 3 250 500 750 1000 1250 K-factor wrt LO pt,Z (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) 2 4 6 8 10 250 500 750 1000 1250 K-factor wrt LO pt,j1 (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) 1 10 100 1000 500 1000 1500 2000 2500 K-factor wrt LO HT,jets (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

◮ pt,Z: no correction; topology (A) dominant at high pt,Z

(extra loops w.r.t. NLO do not change much)

◮ pt,j: small correction; ¯

nNLO is like NLO for the dominant (B) and (C) configurations and it behaves like healthy NLO

(A)

g Z q

(B)

Z g g q

(C)

Z g g q

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 10 / 13

SLIDE 37

Z+jet at ¯ nNLO = Z+j@NLO + LoopSim◦(Z+2j@NLOonly)

pt,Z pt,hardest jet HT,jets

0.5 1 1.5 2 2.5 3 250 500 750 1000 1250 K-factor wrt LO pt,Z (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) 2 4 6 8 10 250 500 750 1000 1250 K-factor wrt LO pt,j1 (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) 1 10 100 1000 500 1000 1500 2000 2500 K-factor wrt LO HT,jets (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep)

◮ pt,Z: no correction; topology (A) dominant at high pt,Z

(extra loops w.r.t. NLO do not change much)

◮ pt,j: small correction; ¯

nNLO is like NLO for the dominant (B) and (C) configurations and it behaves like healthy NLO

◮ HT, jets: significant correction; K factor ∼ 2; given that it is

more like going from LO to NLO this may happen sometimes, especially for nontrivial observables like HT; can we understand it here?

(A)

g Z q

(B)

Z g g q

(C)

Z g g q

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 10 / 13

SLIDE 38

HT type observables at ¯ nNLO for Z+jet and for dijets

◮ Z+jet at NNLO like dijets at NLO

(same topology, Z only provides the enhancement O

αEW ln2 pt,j1/mZ
)

Z q g g g q g g g Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 11 / 13

SLIDE 39

HT type observables at ¯ nNLO for Z+jet and for dijets

◮ Z+jet at NNLO like dijets at NLO

(same topology, Z only provides the enhancement O

αEW ln2 pt,j1/mZ
)

Z q g g g q g g g 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 500 1000 1500 2000 2500 K-factor wrt NLO HT,jets (GeV) MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

Z+j

–

nNLO/NLO Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 11 / 13

SLIDE 40

HT type observables at ¯ nNLO for Z+jet and for dijets

◮ Z+jet at NNLO like dijets at NLO

(same topology, Z only provides the enhancement O

αEW ln2 pt,j1/mZ
)

Z q g g g q g g g 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 500 1000 1500 2000 2500 K-factor wrt NLO HT,jets (GeV) MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

Z+j

–

nNLO/NLO 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 500 1000 1500 2000 K factor wrt LO HT [GeV] NLOjet++, CTEQ6M anti-kt, R=0.7 pp, 7 TeV

dijets

NLO/LO

◮ HT for dijets receives large contributions at NLO!

◮ caused by appearance of the third jet from initial state radiation Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 11 / 13

SLIDE 41

Dijets at ¯ nNLO HT,n =

n hardest jets pt,jet

0.5 1 1.5 2 2.5 3 100 200 300 400 500 600 700 800 900

ratio to LO (xµ=1)

1 2

− HT,2 [GeV] pp, 7 TeV, anti-kt R=0.7 NLOJet++, CTEQ6M

LO NLO

–

nNLO (µ)

–

nNLO (RLS)

◮ HT,2: central value and scale uncertainties stay the same: adding

NNLO corrections without proper finite part cannot improve the result

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 12 / 13

SLIDE 42

Dijets at ¯ nNLO HT,n =

n hardest jets pt,jet

0.5 1 1.5 2 2.5 3 100 200 300 400 500 600 700 800 900

ratio to LO (xµ=1)

1 2

− HT,2 [GeV] pp, 7 TeV, anti-kt R=0.7 NLOJet++, CTEQ6M

LO NLO

–

nNLO (µ)

–

nNLO (RLS) 0.5 1 1.5 2 2.5 3 100 200 300 400 500 600 700 800 900

ratio to LO (xµ=1)

1 2

− HT,3 [GeV] pp, 7 TeV, anti-kt R=0.7 NLOJet++, CTEQ6M

LO NLO

–

nNLO (µ)

–

nNLO (RLS)

◮ HT,2: central value and scale uncertainties stay the same: adding

NNLO corrections without proper finite part cannot improve the result

◮ HT,3 converges, significant reduction of scale uncertainty: the

bservable comes under control at ¯

nNLO

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 12 / 13

SLIDE 43

Dijets at ¯ nNLO HT,n =

n hardest jets pt,jet

0.5 1 1.5 2 2.5 3 100 200 300 400 500 600 700 800 900

ratio to LO (xµ=1)

1 2

− HT,2 [GeV] pp, 7 TeV, anti-kt R=0.7 NLOJet++, CTEQ6M

LO NLO

–

nNLO (µ)

–

nNLO (RLS) 0.5 1 1.5 2 2.5 3 100 200 300 400 500 600 700 800 900

ratio to LO (xµ=1)

1 2

− HT,3 [GeV] pp, 7 TeV, anti-kt R=0.7 NLOJet++, CTEQ6M

LO NLO

–

nNLO (µ)

–

nNLO (RLS) 0.5 1 1.5 2 2.5 3 100 200 300 400 500 600 700 800 900

ratio to LO (xµ=1)

1 2

− HT [GeV] pp, 7 TeV, anti-kt R=0.7 NLOJet++, CTEQ6M

LO NLO

–

nNLO (µ)

–

nNLO (RLS)

◮ HT,2: central value and scale uncertainties stay the same: adding

NNLO corrections without proper finite part cannot improve the result

◮ HT,3 converges, significant reduction of scale uncertainty: the

bservable comes under control at ¯

nNLO

◮ HT does not converge: again caused by the initial state radiation, this

time a second emission which shifts the distribution of HT to higher values and causes no effect for the HT,3 distribution

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 12 / 13

SLIDE 44

Summary

◮ several cases of observables with giant NLO K factor exist ◮ those large corrections arise due to appearance of new topologies at NLO ◮ we developed a method, called LoopSim, which allows one to obtain

approximate NNLO corrections for such processes

◮ the method is based on unitarity and makes use of combining NLO results

for different multiplicities

◮ we gave arguments why the method should produce meaningful results and

we validated it against NNLO Drell-Yan and also NLO Z+j and NLO dijets

◮ we computed approximated NNLO corrections to Z+j and dijets at the LHC

finding, depending on observable, either indication of convergence of the perturbative series or further corrections

◮ the latter has been understood and attributed to the initial state radiation

Outlook

◮ processes with W , multibosons, heavy quarks, . . .

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 13 / 13

SLIDE 45

BACKUP SLIDES

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 14 / 13

SLIDE 46

The LoopSim method: some more details

For a given input En event with n final state particles the weights of all diagrams generated by LoopSim sum up to zero (unitarity)

all diagrams

wn =

υ

ℓ=0

(−1)ℓ υ ℓ

= 0 ,

ℓ − number of loops, υ − maximal ℓ

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 15 / 13

SLIDE 47

The LoopSim method: some more details

For a given input En event with n final state particles the weights of all diagrams generated by LoopSim sum up to zero (unitarity)

all diagrams

wn =

υ

ℓ=0

(−1)ℓ υ ℓ

= 0 ,

ℓ − number of loops, υ − maximal ℓ The principle of the method is simple. There is, however, a number of issues that need to be addressed to fully specify the procedure and make it usable:

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 15 / 13

SLIDE 48

The LoopSim method: some more details

For a given input En event with n final state particles the weights of all diagrams generated by LoopSim sum up to zero (unitarity)

all diagrams

wn =

υ

ℓ=0

(−1)ℓ υ ℓ

= 0 ,

ℓ − number of loops, υ − maximal ℓ The principle of the method is simple. There is, however, a number of issues that need to be addressed to fully specify the procedure and make it usable:

◮ infrared and collinear safety ◮ conservation of four-momentum ◮ choice of jet definition (algorithm, value of R) ◮ treatment of flavour (e.g. for processes with vector bosons)

◮ Z boson can be emitted only from quarks and never itself emits

◮ extension to input events with exact loops

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 15 / 13

SLIDE 49

Scale dependence: Z + jet

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.25 0.5 1 2 4 µ / (p2

t,j1 + M2 Z)1/2

900 < pt_jet < 1200 GeV LO(mu) NLO(mu) nNLO(mu) 1 2 3 4 5 6 7 0.25 0.5 1 2 4 µ / (p2

t,j1 + M2 Z)1/2

900 < pt_jet < 1200 GeV NLO(mu)/LO(mu) nNLO(mu)/LO(mu) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.25 0.5 1 2 4 µ / (p2

t,j1 + M2 Z)1/2

900 < HT < 1200 GeV LO(mu) NLO(mu) nNLO(mu) 10 20 30 40 50 60 70 80 90 100 0.25 0.5 1 2 4 µ / (p2

t,j1 + M2 Z)1/2

900 < HT < 1200 GeV NLO(mu)/LO(mu) nNLO(mu)/LO(mu) Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 16 / 13

SLIDE 50

Reference-observable method

Take a reference observable identical at LO to the observable A σ(A)

Z+j@NNLO = σ(ref) Z+j@NNLO + (σ(A) − σ(ref))Z+j@NNLO

= σ(ref)

Z+j@NNLO + (σ(A) − σ(ref))Z+2j@NLO

If the reference observable converges well σ(A)

Z+j@NNLO ≃ σ(ref) Z+j@NLO + (σ(A) − σ(ref))Z+2j@NLO

2 3 4 5 6 7 8 250 500 750 1000 1250 K-factor wrt LO pt,j1 (GeV)

pp, 14 TeV MCFM 5.7 CTEQ6M anti-kt, R=0.7 pt,j1 > 200 GeV

LS – nNLO ref – nNLO 1 10 100 250 500 750 1000 1250 K-factor wrt LO HT,jets (GeV)

pp, 14 TeV MCFM 5.7, CTEQ6M anti-kt, R=0.7 pt,j1 > 200 GeV

LS – nNLO ref – nNLO

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 17 / 13

SLIDE 51

Z+jet at NLO

◮ Z + j@¯

nLO = Z + j@LO + LoopSim ◦ (Z + 2j@LO)

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 18 / 13

SLIDE 52

Z+jet at NLO

◮ Z + j@¯

nLO = Z + j@LO + LoopSim ◦ (Z + 2j@LO)

0.5 1 1.5 2 2.5 3 250 500 750 1000 1250 K-factor wrt LO pt,Z (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nLO (µ dep)

–

nLO (RLS dep)

◮ pt,Z (lack of large K-factor):

◮ finite loop contributions matter ◮ correctly reproduced dip towards pt = 200 GeV Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 18 / 13

SLIDE 53

Z+jet at NLO

◮ Z + j@¯

nLO = Z + j@LO + LoopSim ◦ (Z + 2j@LO)

0.5 1 1.5 2 2.5 3 250 500 750 1000 1250 K-factor wrt LO pt,Z (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nLO (µ dep)

–

nLO (RLS dep) 2 4 6 8 10 250 500 750 1000 1250 K-factor wrt LO pt,j1 (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nLO (µ dep)

–

nLO (RLS dep) 1 10 100 1000 500 1000 1500 2000 2500 K-factor wrt LO HT,jets (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nLO (µ dep)

–

nLO (RLS dep)

◮ pt,Z (lack of large K-factor):

◮ finite loop contributions matter ◮ correctly reproduced dip towards pt = 200 GeV

◮ pt,j, HT,jets (giant K-factor):

◮ very good agreement between ¯

nLO and NLO large

Z g g q

small

g Z q g Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 18 / 13

SLIDE 54

Z+jet at NLO

◮ Z + j@¯

nLO = Z + j@LO + LoopSim ◦ (Z + 2j@LO)

0.5 1 1.5 2 2.5 3 250 500 750 1000 1250 K-factor wrt LO pt,Z (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nLO (µ dep)

–

nLO (RLS dep) 2 4 6 8 10 250 500 750 1000 1250 K-factor wrt LO pt,j1 (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nLO (µ dep)

–

nLO (RLS dep) 1 10 100 1000 500 1000 1500 2000 2500 K-factor wrt LO HT,jets (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nLO (µ dep)

–

nLO (RLS dep)

◮ pt,Z (lack of large K-factor):

◮ finite loop contributions matter ◮ correctly reproduced dip towards pt = 200 GeV

◮ pt,j, HT,jets (giant K-factor):

◮ very good agreement between ¯

nLO and NLO large

Z g g q

small

g Z q g

◮ small R uncertainties – driven only by subleading diagrams

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 18 / 13

SLIDE 55

HT type observables at ¯ nNLO for Z+jet and for dijets

◮ Z+jet at NNLO like dijets at NLO

(same topology, Z only provides the enhancement O

αs ln2 pt,j1/mZ
)

Z q g g g q g g g Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 19 / 13

SLIDE 56

HT type observables at ¯ nNLO for Z+jet and for dijets

◮ Z+jet at NNLO like dijets at NLO

(same topology, Z only provides the enhancement O

αs ln2 pt,j1/mZ
)

Z+j

1 10 100 1000 500 1000 1500 2000 2500 K-factor wrt LO HT,jets (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) Z q g g g q g g g Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 19 / 13

SLIDE 57

HT type observables at ¯ nNLO for Z+jet and for dijets

◮ Z+jet at NNLO like dijets at NLO

(same topology, Z only provides the enhancement O

αs ln2 pt,j1/mZ
)

Z+j

1 10 100 1000 500 1000 1500 2000 2500 K-factor wrt LO HT,jets (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) Z q g g g q g g g

dijets

10-6 10-5 10-4 10-3 10-2 10-1 1 10 102 200 400 600 800 1000 dσ/dV [nb/GeV]

1 2 −HT [GeV]

NLOjet++, CTEQ6M anti-kt, R=0.7 pp, 7 TeV LO pt,j2 & HT/2 NLO HT/2

◮ HT for dijets receives large contributions at NLO!

◮ caused by appearance of the third jet from

initial state radiation

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 19 / 13

SLIDE 58

HT type observables at ¯ nNLO for Z+jet and for dijets

◮ Z+jet at NNLO like dijets at NLO

(same topology, Z only provides the enhancement O

αs ln2 pt,j1/mZ
)

Z+j

1 10 100 1000 500 1000 1500 2000 2500 K-factor wrt LO HT,jets (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) Z q g g g q g g g

dijets

10-6 10-5 10-4 10-3 10-2 10-1 1 10 102 200 400 600 800 1000 dσ/dV [nb/GeV]

1 2 −HT [GeV]

NLOjet++, CTEQ6M anti-kt, R=0.7 pp, 7 TeV LO pt,j2 & HT/2 NLO HT/2

◮ HT for dijets receives large contributions at NLO!

◮ caused by appearance of the third jet from

initial state radiation

◮ if the same is valid for Z + j we should see only

small correction for HT,j2 = 2

i=1 pt,ji

Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 19 / 13

SLIDE 59

HT type observables at ¯ nNLO for Z+jet and for dijets

◮ Z+jet at NNLO like dijets at NLO

(same topology, Z only provides the enhancement O

αs ln2 pt,j1/mZ
)

Z+j

1 10 100 1000 500 1000 1500 2000 2500 K-factor wrt LO HT,jets (GeV)

MCFM 5.7, CTEQ6M pp, 14 TeV anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (RLS dep) Z q g g g q g g g

dijets

10-6 10-5 10-4 10-3 10-2 10-1 1 10 102 200 400 600 800 1000 dσ/dV [nb/GeV]

1 2 −HT [GeV]

NLOjet++, CTEQ6M anti-kt, R=0.7 pp, 7 TeV LO pt,j2 & HT/2 NLO HT/2

◮ HT for dijets receives large contributions at NLO!

◮ caused by appearance of the third jet from

initial state radiation

◮ if the same is valid for Z + j we should see only

small correction for HT,j2 = 2

i=1 pt,ji

◮ and indeed it is small!

Z+j

1 10 100 1000 500 1000 1500 2000 2500 K-factor wrt LO HT,j2 (GeV)

pp, 14 TeV MCFM 5.7, CTEQ6M anti-kt, R=0.7 pt,j1 > 200 GeV

LO NLO

–

nNLO (µ dep)

–

nNLO (R LS dep) Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 19 / 13