Stefan Prestel Radcor/Loopfest 2015 UCLA, June 19, 2015...Th. l in exp. Ds l l k VH l x 7 TeV CMS measurement (L £s-1) 8 TeV CMS measurement (L £ 19.6 fb-1) 7 TeV Theory prediction

NNLO matching and parton shower developments

Stefan Prestel

Radcor/Loopfest 2015

UCLA, June 19, 2015

(in collaboration with Stefan Höche and Ye Li)

1 / 38

Outline

• Motivation / introduction:• A pessimist’s view on NLO matching, LO merging, NLO merging

and NNLO matching (e.g. everything that I usually turn to)• New parton showers for PYTHIA and SHERPA• Summary

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Event generators should describe multijet observables

..

jetsN0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

) [p

b]je

ts(W

+N

σ

-210

-110

1

10

210

310

410

510

610 ATLAS

jets, R=0.4,tanti-k

| < 4.4j

> 30 GeV, |yj

Tp

) + jetsν l→W(Data,

-1 = 7 TeV, 4.6 fbs+SHERPAATHLACKB

HEJALPGENSHERPAMEPS@NLO jetsN

0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4 +SHERPAATHLACKB

ATLAS

jetsN

0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4 HEJ

jetsN

0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

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d. /

Dat

a

0.6

0.8

1

1.2

1.4 ALPGEN

jetsN0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4

SHERPA

MEPS@NLO

(Figures taken from EPJC 75 (2015) 2 82 and CMS summary ofhttps://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsCombined) 3 / 38


..

jetsN0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

) [p

b]je

ts(W

+N

σ

-210

-110

1

10

210

310

410

510

610 ATLAS

jets, R=0.4,tanti-k

| < 4.4j

> 30 GeV, |yj

Tp




0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4 +SHERPAATHLACKB

ATLAS

jetsN

0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4 HEJ

jetsN

0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4 ALPGEN

jetsN0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4

SHERPA

MEPS@NLO

(Figures taken from EPJC 75 (2015) 2 82 and CMS summary ofhttps://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsCombined)

.

[pb]

σP

rodu

ctio

n C

ross

Sec

tion,

-110

1

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310

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510

CMS PreliminaryMar 2015

All results at: http://cern.ch/go/pNj7W 1j≥ 2j≥ 3j≥ 4j≥ Z 1j≥ 2j≥ 3j≥ 4j≥ γW γZ WW WZ ZZ WW

→γγqqllEW γWV tt 1j 2j 3j t-cht tW s-cht γtt ttZ

σ∆ in exp. Hσ∆Th.

ggH qqHVBF VH ttH

CMS 95%CL limit

)-1 5.0 fb≤7 TeV CMS measurement (L


7 TeV Theory prediction


3 / 38


..

jetsN0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

) [p

b]je

ts(W

+N

σ

-210

-110

1

10

210

310

410

510

610 ATLAS

jets, R=0.4,tanti-k

| < 4.4j

> 30 GeV, |yj

Tp




0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4 +SHERPAATHLACKB

ATLAS

jetsN

0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4 HEJ

jetsN

0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4 ALPGEN

jetsN0≥ 1≥ 2≥ 3≥ 4≥ 5≥ 6≥ 7≥

Pre

d. /

Dat

a

0.6

0.8

1

1.2

1.4

SHERPA

MEPS@NLO

(Figures taken from EPJC 75 (2015) 2 82 and CMS summary ofhttps://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsCombined)

.

[pb]

σP

rodu

ctio

n C

ross

Sec

tion,

-110

1

10

210

310

410

510

CMS PreliminaryMar 2015

All results at: http://cern.ch/go/pNj7W 1j≥ 2j≥ 3j≥ 4j≥ Z 1j≥ 2j≥ 3j≥ 4j≥ γW γZ WW WZ ZZ WW

→γγqqllEW γWV tt 1j 2j 3j t-cht tW s-cht γtt ttZ

σ∆ in exp. Hσ∆Th.

ggH qqHVBF VH ttH

CMS 95%CL limit





.Combine multiple accurate fixed-order calculations with eachother, and with shower resummation, into a single flexibleprediction. =⇒ Matching / merging

3 / 38

NLO+PS matched predictions

..

..

NLO+PS methods mostly behave well, but may sometimes showlarge differences

⋄ in the real-emission dominated region, since (smearing of NLO”seed” cross section into) real emission treated differently;⇒ Merging with further accurate multi-jet calculations helps.⋄ in the resummation region, since exponentiation different.⇒ Compare to inclusive resummation, improve shower accuracy.

4 / 38


..

..

NLO+PS methods mostly behave well, but may sometimes showlarge differences⋄ in the real-emission dominated region, since (smearing of NLO”seed” cross section into) real emission treated differently;⇒ Merging with further accurate multi-jet calculations helps.

⋄ in the resummation region, since exponentiation different.⇒ Compare to inclusive resummation, improve shower accuracy.

4 / 38


..

..

NLO+PS methods mostly behave well, but may sometimes showlarge differences⋄ in the real-emission dominated region, since (smearing of NLO”seed” cross section into) real emission treated differently;⇒ Merging with further accurate multi-jet calculations helps.⋄ in the resummation region, since exponentiation different.⇒ Compare to inclusive resummation, improve shower accuracy.

4 / 38

Leading-order merging

..

Start with seed cross sections and no-emission probabilities

BnFn(Φn) , ∆(t0, t) = exp

− t0∫t

dΦradαs(ΦR)P(ΦR)

∆u(tn, tMS) = 1−

tn∫tMS

Bn+1Fn+1(Φn+1)∆(tn, t) (”unitarised Sudakov”)

where Fm(Φm) = Θ(t(Φm)− tMS), t0 ∼ Q. Two multiplicities merged by

Bn∆u(t0, tMS)∆(tMS, tcut)O0(S+0)

+

t0∫dΦR

[Bnαs(ΦR)P(ΦR)(1− Fn+1)∆u(t0, tMS)∆(tMS, tcut) + Bn+1Fn+1∆(t0, t1)∆(t1, t2)

]O1(S+1)

+

t0∫dΦRR Bn+1Fn+1∆(t0, t1)∆(t1, t2)αs(ΦR)P(ΦR)O2(S+2) + further emissions in (t2, tcut]

5 / 38


..



− t0∫t


∆u(tn, tMS) = 1−

tn∫tMS

Bn+1Fn+1(Φn+1)∆(tn, t) (”unitarised Sudakov”)

where Fm(Φm) = Θ(t(Φm)− tMS), t0 ∼ Q. Three multiplicities merged by


+

t0∫dΦR


]O1(S+1)

+

t0∫dΦRR

[Bn+1Fn+1∆(t0, t1)(1− Fn+2)∆u(t1, tMS)∆(tMS, tcut)αs(ΦR)P(ΦR)

+ Bn+2Fn+2∆(t0, t1)∆(t1, t2)∆(t2, t3)]O2(S+2) + further emissions in (t3, tcut]

6 / 38


..



− t0∫t


∆u(tn, tMS) = 1−

tn∫tMS

Bn+1Fn+1(Φn+1)∆(tn, t)

where Fm(Φm) = Θ(t(Φm)− tMS), t0 ∼ Q. Three multiplicities merged by


+

t0∫dΦR


]O1(S+1)

+

t0∫dΦRR

[Bn+1Fn+1∆(t0, t1)(1− Fn+2)∆u(t1, tMS)∆(tMS, tcut)αs(ΦR)P(ΦR)

+ Bn+2Fn+2∆(t0, t1)∆(t1, t2)∆(t2, t3)]O2(S+2) + further emissions in (t3, tcut]

.

Merging methods valid for any multiplicity.Merging methods use shower Sudakov factors ⇒ Improve whenshowers improve.NB: Usual CKKW approximation ∆u(tn, tMS)→ ∆(tn, tMS) calls fortMS ≫ tcut or BnP(Φ)→ Bn+1. We use tMS = O(1GeV) later.

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NLO merging

Any NLO matching method contains only approximate multi-jetkinematics (given by PS).Any LO merging method X only ever contains approximate virtualcorrections.

Use the full NLO results for any multi-jet state!

NLO multi-jet merging from LO scheme X:

⋄ Subtract approximate X O(αs)-terms, add multiple NLO calculations.

⋄ Make sure fixed-order calculations do not overlap careful by cutting,vetoing events and/or vetoing emissions (e.g. remove real contribution ton-jet in favour of n + 1-jet NLO).

⋄ Adjust higher orders to suit other needs (e.g. to preserve the inclusivecross section).

⇒ X@NLO

7 / 38

NLO-merged predictions for Higgs + jets

..

Figure: p⊥,H and ∆ϕ12 for gg→H after merging (H+0)@NLO, (H+1)@NLO,(H+2)@NLO, (H+3)@LO, compared to other generators.

⇒ LH study: The generators come closer together if NLO merging is employed.

Still O(50%) differences in resummation region – importantuncertainty for analyses that use jet binning!

8 / 38

NLO-merged predictions for Higgs + jets

..

Figure: p⊥,H and ∆ϕ12 for gg→H after merging (H+0)@NLO, (H+1)@NLO,(H+2)@NLO, (H+3)@LO, compared to other generators.

⇒ LH study: The generators come closer together if NLO merging is employed.

Still O(50%) differences in resummation region – importantuncertainty for analyses that use jet binning!

8 / 38

The next step(s): Matching @ NNLO

Aim: For important processes – lumi monitors like Drell-Yan, precisionstudies (ggH, ZH, WBF,…) – reduce uncertainties and remove personalbias. But make sure all other improvements stay intact!

Observation: If an NLO merged calculation leads to a well-defined zero-jetinclusive cross section, it is easy to upgrade this cross section to NNLO.

=⇒ Fulfilled by two NLO merging schemes: MiNLO and UNLOPS

9 / 38

Ways to matching @ NNLO

B0

[O0ΠS+0 +

∫1

B1

B0O1ΠS+0

]∼ B0O0 −

∫1

B1O0ΠS+0 +∫1

B1O1ΠS+0

NNLOPS:Start from MiNLO, upgrade analyticCKKW Sudakov to match integralof q⊥ onto zero-jet NLO cross sec-tion (LO+NLL calculation of q⊥),then include differential NNLO K-factor.aaBenefits: Analytic control over 1stemission, improved resummation.aaLimitations: Process-dependent,uses pre-tabulated K-factors.aa

a

UN2LOPSStart from UNLOPS, refine uni-tarised Sudakov factor so that onlyone-jet NLO cross section survivesat O(α2

s), then include NNLO jet-vetoed cross section for NNLO ac-curacy.aaBenefits: Easy, process- andshower-independent.aaLimitations: Does does not showerα2sδ(p⊥) terms, or only a subset –

i.e. has bin edges. 10 / 38

UN2LOPS (Drell-Yan)

..

b

b bb b

bbb

b

b

bb

b

b

b

b

b

b

CMS PRD85(2012)032002b

UN2LOPS

mll/2 < µR/F < 2 mll

mll/2 < µQ < 2 mll10−6

10−5

10−4

10−3

10−2

10−1Z pT reconstructed from dressed electrons

1/

σd

σ/

dp

T,Z

[1/

GeV

]

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

1 10 1 10 2

0.6

0.8

1

1.2

1.4

pT,Z [GeV]

MC

/D

ata

b b bb b b b

bb b b

b bb

bb

b

b

b

b

b

b

b

b

b bbbb

bb

b

b

b

b

b

b

b

b

b bbb b

b bb b b b b b b b

b ATLAS (arXiv:1201.1276)

UN2LOPS

mlν/2 < µR/F < 2 mlν

mlν/2 < µQ < 2 mlν

0

20

40

60

80

100

120

140

160

Lepton-Jet Rapidity Difference

dσ

/d

∆y

[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 b b 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 -2 0 2 40

0.20.40.60.8

11.21.41.61.8

y(Lepton)− y(First Jet)M

C/

Da

ta

CMS data for Z-boson p⊥. UN2LOPS does quite well. Large band at lowp⊥ reflects log scale and shower modelling. ATLAS data for chargedcurrent well described.

11 / 38

UN2LOPS (Drell-Yan)

..

b

b bb b

bbb

b

b

bb

b

b

b

b

b

b

CMS PRD85(2012)032002b

UN2LOPS

mll/2 < µR/F < 2 mll

mll/2 < µQ < 2 mll10−6

10−5

10−4

10−3

10−2

10−1Z pT reconstructed from dressed electrons

1/

σd

σ/

dp

T,Z

[1/

GeV

]

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

1 10 1 10 2

0.6

0.8

1

1.2

1.4

pT,Z [GeV]

MC

/D

ata

b b bb b b b

bb b b

b bb

bb

b

b

b

b

b

b

b

b

b bbbb

bb

b

b

b

b

b

b

b

b

b bbb b

b bb b b b b b b b

b ATLAS (arXiv:1201.1276)

UN2LOPS

mlν/2 < µR/F < 2 mlν

mlν/2 < µQ < 2 mlν

0

20

40

60

80

100

120

140

160

Lepton-Jet Rapidity Difference

dσ

/d

∆y

[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 b b 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 -2 0 2 40

0.20.40.60.8

11.21.41.61.8

y(Lepton)− y(First Jet)M

C/

Da

ta

CMS data for Z-boson p⊥. UN2LOPS does quite well. Large band at lowp⊥ reflects log scale and shower modelling. ATLAS data for chargedcurrent well described.

.UN2LOPS code and plots available fromhttp://www.slac.stanford.edu/∼shoeche/pub/nnlo/

11 / 38

UN2LOPS (Higgs production)

She

rpa

MC

LOPS2UNHqTNNLOMC@NLO

= 14 TeVs

H<2mR/F

µ/2< H m

H<2mQ

µ/2< H m HqTH<2m

Qµ/2< H m

[pb/

GeV

]T,

H/d

pσd

-210

-110

1

Rat

io to

HqT

0.81

1.2

[GeV]T,H

p0 20 40 60 80 100 120 140 160 180 200

..

She

rpa

MC


= 14 TeVs

H<2mR/F

µ/2< H m

H<mQ

µ/4< H m HqTH<m

Qµ/4< H m

[pb/

GeV

]T,

H/d

pσd

-210

-110

1

Rat

io to

HqT

0.81

1.2

[GeV]T,H

p0 20 40 60 80 100 120 140 160 180 200

..

p⊥ of the Higgs-boson for two different matching schemes inUN2LOPS – mimicing the philosophical differences between commonNLO matching schemes.

We have NLO matching, NLO merging and NNLO matching. Still,predictions differ appreciably because of limited shower accuracy.aa

12 / 38


She

rpa

MC


= 14 TeVs

H<2mR/F

µ/2< H m

H<2mQ

µ/2< H m HqTH<2m

Qµ/2< H m

[pb/

GeV

]T,

H/d

pσd

-210

-110

1

Rat

io to

HqT

0.81

1.2

[GeV]T,H

p0 20 40 60 80 100 120 140 160 180 200

..

She

rpa

MC


= 14 TeVs

H<2mR/F

µ/2< H m

H<mQ

µ/4< H m HqTH<m

Qµ/4< H m

[pb/

GeV

]T,

H/d

pσd

-210

-110

1

Rat

io to

HqT

0.81

1.2

[GeV]T,H

p0 20 40 60 80 100 120 140 160 180 200

..

p⊥ of the Higgs-boson for two different matching schemes inUN2LOPS – mimicing the philosophical differences between commonNLO matching schemes.

We have NLO matching, NLO merging and NNLO matching. Still,predictions differ appreciably because of limited shower accuracy.aa

12 / 38

=⇒ Back to Parton Shower Resummation

The PS prediction for an observable O is

Fa⃗(Φn, tc, t0; O) = Fa⃗(Φn, tc, t0) O(Φn) +∫ t0

tc

d̄tt̄

d lnFa⃗(Φn, t̄, t0)d ln t̄

Fa⃗′(Φ′n+1, tc, t̄; O)

with no-emission probabilities and Sudakov factors defined by

Fa(x, t, µ2) = fa(x, t)∆a(t, µ2) = fa(x, µ2) Πa(x, t, µ2) .

Now if tc → 0, then we find

Fa⃗(Φn, tc, t0; O)→∫ t0 d̄t

t̄d lnFa⃗(Φn, t̄, t0)

d ln t̄Fa⃗′(Φ′

n+1, t, t̄; O)

which allows a comparison to analytic resummation formulae.

13 / 38

=⇒ Back to Parton Shower Resummation

The PS prediction for an observable O is

Fa⃗(Φn, tc, t0; O) = Fa⃗(Φn, tc, t0) O(Φn) +∫ t0

tc

d̄tt̄

d lnFa⃗(Φn, t̄, t0)d ln t̄

Fa⃗′(Φ′n+1, tc, t̄; O)

with no-emission probabilities and Sudakov factors defined by

Fa(x, t, µ2) = fa(x, t)∆a(t, µ2) = fa(x, µ2) Πa(x, t, µ2) .

Now if tc → 0, then we find

Fa⃗(Φn, tc, t0; O)→∫ t0 d̄t

t̄d lnFa⃗(Φn, t̄, t0)

d ln t̄Fa⃗′(Φ′

n+1, t, t̄; O)

which allows a comparison to analytic resummation formulae.

13 / 38

Parton Shower Sudakovs

The main object on which showers are built are Sudakov form factors. Toimprove showers, we should improve the accuracy of the Sudakov factorsby comparing

∆AN(ρ0, ρ1) = exp

(−∫ ρ1

ρ0

dρ

ρ

[ln(Q2

ρ

)∑i

(αs

2π

)iAi −

∑i

(αs

2π

)iBi

])

∆PS(ρ0, ρ1) = exp(−∫ ρ0

ρ1

dρ

ρ

∫ zmax

zmin

dzαs(ρ)2πK(z, ρ)

)where A and B are related to γcusp and γi (e.g. A1 = Ca × 1, A2 = Ca × γ

(2)cusp)

”Good” parton shower should yield a sensible result for the perturbativeexponent (A′s, B′s). Higher-order normalisation (C′s, H′s) may beincluded through matching.

14 / 38

Reliable Event Generator predictions?

..

We have hints that current parton showers (e.g. PYTHIA, SHERPA) arenot ideal in the Sudakov region. This can mean

• Problems in the (matching of shower and) fixed-order parts• Problems in the parton showers• Problems in the non-perturbative modelling• Bugs

⇒ We want a simple1, theoretically clean2, extendable parton shower, toso that comparison to observable-based resummation becomes possible.a1 Simply splitting functions, simple phase space boundaries.2 Eikonal in soft limit, AP kernels in collinear limit, collinear anomalousdimensions unchanged, flavour/momentum sum rules, no choices introducingiffy sub-leading logs.

15 / 38

Reliable Event Generator predictions?

..

We have hints that current parton showers (e.g. PYTHIA, SHERPA) arenot ideal in the Sudakov region. This can mean

• Problems in the (matching of shower and) fixed-order parts• Problems in the parton showers• Problems in the non-perturbative modelling• Bugs

⇒ We want a simple1, theoretically clean2, extendable parton shower, sothat comparison to observable-based resummation becomes possible.a1 Simple splitting functions, simple phase space boundaries.2 Eikonal in soft limit, AP kernels in collinear limit, collinear anomalousdimensions unchanged, flavour/momentum sum rules, no choices introducingiffy sub-leading logs.

.

We have defined a new dipole-parton shower, and implementedindependently in the PYTHIA and SHERPA to minimise bug contamination(arXiv:1506.05057). Codes will become PYTHIA and SHERPA plugins.

15 / 38

Deriving a DIRE shower

Catani-Seymour dipoles are a nice starting point becauseA capture the correct divergence structure (including coherence)B correct single log for q⊥ (even with trivial phase space boundaries).C come with exact phase space factorisation

16 / 38


Catani-Seymour dipoles are a bad starting point because1. different for initial-initial, initial-final, final-initial, final-final← unified by good phase space parametrization + combining withJacobian

2 partial-fractioned eikonal can make analytic comparisonscumbersome ← remedied by symmetric evolution variable, andby demoting dipoles to ”damping functions” used to recover thecollinear anomalous dimensions.

3 not immediately applicable for evolution, because not normalized bymomentum / flavour sum rules ← demand sum rules exactly.

Addressing these points completely fixes all (II, IF, FI, FF) dipoles to

Pqq(z, κ2) = 2 CF[( 1− z

(1− z)2 + κ2

)+−

1 + z2

]+

32CF δ(1− z)

Pgg(z, κ2) = 2 CA[( 1− z

(1− z)2 + κ2

)+

+z

z2 + κ2 − 2 + z(1− z)]

+ δ(1− z)( 11

6CA −

23nfTR)

Pgq(z, κ2) = 2 CF[ zz2 + κ2 −

2− z2

]Pqg(z, κ2) = TR

[z2 + (1− z)2

]

17 / 38






Pqq(z, κ2) = 2 CF[( 1− z

(1− z)2 + κ2

)+−

1 + z2

]+

32CF δ(1− z)

Pgg(z, κ2) = 2 CA[( 1− z

(1− z)2 + κ2

)+

+z

z2 + κ2 − 2 + z(1− z)]

+ δ(1− z)( 11

6CA −

23nfTR)

Pgq(z, κ2) = 2 CF[ zz2 + κ2 −

2− z2

]Pqg(z, κ2) = TR

[z2 + (1− z)2

]

17 / 38






Pqq(z, κ2) = 2 CF[( 1− z

(1− z)2 + κ2

)+−

1 + z2

]+

32CF δ(1− z)

Pgg(z, κ2) = 2 CA[( 1− z

(1− z)2 + κ2

)+

+z

z2 + κ2 − 2 + z(1− z)]

+ δ(1− z)( 11

6CA −

23nfTR)

Pgq(z, κ2) = 2 CF[ zz2 + κ2 −

2− z2

]Pqg(z, κ2) = TR

[z2 + (1− z)2

]

17 / 38






Pqq(z, κ2) = 2 CF[( 1− z

(1− z)2 + κ2

)+−

1 + z2

]+

32CF δ(1− z)

Pgg(z, κ2) = 2 CA[( 1− z

(1− z)2 + κ2

)+

+z

z2 + κ2 − 2 + z(1− z)]

+ δ(1− z)( 11

6CA −

23nfTR)

Pgq(z, κ2) = 2 CF[ zz2 + κ2 −

2− z2

]Pqg(z, κ2) = TR

[z2 + (1− z)2

]17 / 38

DIRE ordering variables

a

b

i

k

j

k

i i

j

ia

II IF FF FF

a

More concretely, we use the phase space variables

ρii = saisbisab

saibsab

zii = 1− sbisab

ρif = saisiksai + sak

zif = 1− siksai + sak

ρfi =sajsij

sai + sajzfi = sai

sai + saj

ρff =sijsjk

sij + sik + sjkzff =

sij + siksij + sik + sjk

Kinematical p⊥

Ariandne p⊥

18 / 38

DIRE ordering variables

a

b

i

k

j

k

i i

j

ia

II IF FF FF

a

More concretely, we use the phase space variables

ρii = saisbisab

saibsab

zii = 1− sbisab

ρif = saisiksai + sak

zif = 1− siksai + sak

ρfi =sajsij

sai + sajzfi = sai

sai + saj

ρff =sijsjk

sij + sik + sjkzff =

sij + siksij + sik + sjk

Kinematical p⊥ for small p⊥, but covers full p⊥ range even for small ρstart

Ariandne p⊥ 19 / 38

Improvements

Ordering variables were chosen to obtain very simple phase spaceboundaries and splitting functions, so that a comparison withanalytic results is possible.

=⇒ The soft terms can directly be rescaled with higher-order cuspanomalous dimensions to get A2 and A3:

2(1− z)(1− z)2 + κ2 −→

2(1− z)(1− z)2 + κ2

(1 + αs

2πγ

(1)cusp + α2

s4π2γ

(2)cusp

)

Note: αs not in the CMW scheme (two-loop cusp is absorbed via redefinition of ΛQCD),as this can introduce problematic collinear anomalous dimensions beyond NLL(depending on shower details).

20 / 38

Cross-validation at the sub-permille level: Durham jet rates

y23 × 2

y34 × 10−1

y45 × 10−2

y56 × 10−3

Dir

eP

S

e+e−→ qq̄ @ 91.2 GeV

Sherpa

Pythia

10−2

10−1

1

101

102

103

104

Differential jet resolution at parton level (Durham algorithm)

dσ

/d

log

10

yn

n+

1[p

b]

y23

-2 σ

0 σ

2 σ

Dev

iati

on

y34

-2 σ

0 σ

2 σ

Dev

iati

on

y45

-2 σ

0 σ

2 σ

Dev

iati

on

y56

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5

-2 σ

0 σ

2 σ

log10 yn n+1

Dev

iati

on

21 / 38

Cross-validation at the sub-permille level: Durham jet rates

y23 × 2

y34 × 10−1

y45 × 10−2

y56 × 10−3

Dir

eP

S

τ+

τ−→ [h → gg] @ 125 GeV

Sherpa

Pythia

10−3

10−2

10−1

1

101

102

103

Differential jet resolution at parton level (Durham algorithm)

dσ

/d

log

10

yn

n+

1[p

b]

y23

-2 σ

0 σ

2 σ

Dev

iati

on

y34

-2 σ

0 σ

2 σ

Dev

iati

on

y45

-2 σ

0 σ

2 σ

Dev

iati

on

y56

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5

-2 σ

0 σ

2 σ

log10 yn n+1

Dev

iati

on

Lepton colliders .

22 / 38

Cross-validation at the sub-permille level: k⊥ clustering scales

d01

d12 × 10−1

d23 × 10−2

d34 × 10−3

Dir

eP

S

pp → e+e− @ 8 TeV

66 < mll < 116 GeV2

Sherpa

Pythia

10−3

10−2

10−1

1

10 1

10 2

10 3

Differential kT-jet resolution at parton level

dσ

/d

log

10(d

nm

/G

eV)

[pb

]

d01

-2 σ

0 σ

2 σ

Dev

iati

on

d12

-2 σ

0 σ

2 σ

Dev

iati

on

d23

-2 σ

0 σ

2 σ

Dev

iati

on

d34

0.5 1 1.5 2 2.5

-2 σ

0 σ

2 σ

log10(dnm/GeV)

Dev

iati

on

23 / 38

Cross-validation at the sub-permille level: k⊥ clustering scales

d01

d12 × 10−1

d23 × 10−2

d34 × 10−3

Dir

eP

S

pp → τ+

τ− @ 8 TeV

Sherpa

Pythia10−5

10−4

10−3

10−2

10−1

1Differential kT-jet resolution at parton level

dσ

/d

log

10(d

nm

/G

eV)

[pb

]

d01

-2 σ

0 σ

2 σ

Dev

iati

on

d12

-2 σ

0 σ

2 σ

Dev

iati

on

d23

-2 σ

0 σ

2 σ

Dev

iati

on

d34

0.5 1 1.5 2 2.5

-2 σ

0 σ

2 σ

log10(dnm/GeV)

Dev

iati

on

Hadron colliders .

24 / 38

Cross-validation at the sub-permille level: Jet scales

d02 × 2

d23 × 10−1

d34 × 10−2

d45 × 10−3

Dir

eP

S

e−q→ e−q @ 300 GeV

Q2> 100 GeV2

Sherpa

Pythia

10−6

10−5

10−4

10−3

10−2

10−1

1

10 1Differential kT-jet resolution at parton level (Breit frame)

dσ

/d

log

10(d

nm

/G

eV)

[pb

]

d02

-2 σ

0 σ

2 σ

Dev

iati

on

d23

-2 σ

0 σ

2 σ

Dev

iati

on

d34

-2 σ

0 σ

2 σ

Dev

iati

on

d45

0.5 1 1.5 2

-2 σ

0 σ

2 σ

log10(dnm/GeV)

Dev

iati

on

25 / 38

Cross-validation at the sub-permille level: Jet scales

d02 × 2

d23 × 10−1

d34 × 10−2

d45 × 10−3

Dir

eP

S

τ−g→ τ

−g @ 300 GeV

Sherpa

Pythia10−12

10−11

10−10

10−9

10−8

10−7Differential kT-jet resolution at parton level (Breit frame)

dσ

/d

log

10(d

nm

/G

eV)

[pb

]

d02

-2 σ

0 σ

2 σ

Dev

iati

on

d23

-2 σ

0 σ

2 σ

Dev

iati

on

d34

-2 σ

0 σ

2 σ

Dev

iati

on

d45

0.5 1 1.5 2

-2 σ

0 σ

2 σ

log10(dnm/GeV)

Dev

iati

on

Lepton-hadron colliders .First ever DIS prediction with PYTHIA8!

26 / 38

LEP data comparisons (plain showering)

..

b bb

b

b

b

b

b

bb

b b bb

bb

b

bb

bb

bb

bb

b

b

b

b

b

b

b

Dir

eP

S

b JADE/OPAL data

Eur.Phys.J. C17 (2000) 19

Dire

10−1

1

10 1

10 2

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

dσ

/d

y2

3

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

10−4 10−3 10−2 10−1

0.6

0.8

1

1.2

1.4

yDurham23

MC

/D

ata

bb

b

b

b

b

bb

b b b b bb

bb

b

b

b

b

b

b

b

b

b

b

b

b

b

Dir

eP

S

b JADE/OPAL data

Eur.Phys.J. C17 (2000) 19

Dire

10−2

10−1

1

10 1

10 2

10 3Differential 3-jet rate with Durham algorithm (91.2 GeV)

dσ

/d

y3

4

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

10−4 10−3 10−2 10−1

0.6

0.8

1

1.2

1.4

yDurham34

MC

/D

ata

b b bb

b

b

b

bb

b b b b bb

bb

b

b

b

b

b

b

b

b

b

b

b

b

Dir

eP

S

b JADE/OPAL data

Eur.Phys.J. C17 (2000) 19

Dire

10−2

10−1

1

10 1

10 2

10 3


dσ

/d

y4

5

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

10−5 10−4 10−3 10−2

0.6

0.8

1

1.2

1.4

yDurham45

MC

/D

ata

b bb

b

b

bb

b b b b bb

bb

b

b

b

b

b

b

b

b

b

b

b

Dir

eP

S

b JADE/OPAL data

Eur.Phys.J. C17 (2000) 19

Dire

10−2

10−1

1

10 1

10 2

10 3


dσ

/d

y5

6

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

10−5 10−4 10−3 10−2

0.6

0.8

1

1.2

1.4

yDurham56

MC

/D

ata

27 / 38


..

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

Dir

eP

S

b ALEPH data

Eur.Phys.J. C35 (2004) 457

Dire

10−3

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

/D

ata

b

b

b

b

bb 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

b

b

Dir

eP

S

b ALEPH data

Eur.Phys.J. C35 (2004) 457

Dire

10−4

10−3

10−2

10−1

1

10 1

Total jet broadening (ECMS = 91.2 GeV)

1/

σd

σ/

dB

T

b b b b b b b b b b b b b b b b 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 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0.6

0.8

1

1.2

1.4

BT

MC

/D

ata

b

b

b

b

b b b b b b b b b b 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

Dir

eP

S

b ALEPH data

Eur.Phys.J. C35 (2004) 457

Dire

10−3

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

/D

ata

b

b

b

b

bb

bb

b bb

bb

bb b

b b bb b b

bb b

b b b

b bb

b

b b

b

Dir

eP

S

b ALEPH data

Eur.Phys.J. C35 (2004) 457

Dire

10−4

10−3

10−2

10−1

1

10 1

10 2

Aplanarity (ECMS = 91.2 GeV)

1/

σd

σ/

dA

b b b b b b b b b b b 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 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.6

0.8

1

1.2

1.4

A

MC

/D

ata

28 / 38


..

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

Dir

eP

S

b ALEPH data

Eur.Phys.J. C35 (2004) 457

Dire

10−3

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

/D

ata

b

b

b

b

bb 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

b

b

Dir

eP

S

b ALEPH data

Eur.Phys.J. C35 (2004) 457

Dire

10−4

10−3

10−2

10−1

1

10 1

Total jet broadening (ECMS = 91.2 GeV)

1/

σd

σ/

dB

T

b b b b b b b b b b b b b b b b 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 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0.6

0.8

1

1.2

1.4

BT

MC

/D

ata

b

b

b

b

b b b b b b b b b b 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

Dir

eP

S

b ALEPH data

Eur.Phys.J. C35 (2004) 457

Dire

10−3

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

/D

ata

b

b

b

b

bb

bb

b bb

bb

bb b

b b bb b b

bb b

b b b

b bb

b

b b

b

Dir

eP

S

b ALEPH data

Eur.Phys.J. C35 (2004) 457

Dire

10−4

10−3

10−2

10−1

1

10 1

10 2

Aplanarity (ECMS = 91.2 GeV)

1/

σd

σ/

dA

b b b b b b b b b b b 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 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0.6

0.8

1

1.2

1.4

A

MC

/D

ata

.Nice description of LEP data.

28 / 38

LHC data comparisons (1-jet CKKW-L merged)

..

b

b b b b b b b bb

bbbbb b b b

bbb

b

b

b

b

b

b

b b b b b b b bb

bbbbbb b b b

bb

b

b

b

b

b

b

b b b b b b b b bbb b

bb b b b b

bb

b

b

b

b

b

Dir

eP

S

0 ≤ |yZ| ≤ 1

1 ≤ |yZ| ≤ 2 (×0.1)

2 < |yZ| ≤ 2.4 (×0.01)

b ATLAS data

JHEP 09 (2014) 145

ME+PS (1-jet)

5 ≤ Qcut ≤ 20 GeV

1 210−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1pT spectrum, Z→ ee (dressed)

1σ

fid

dσ

fid

dp

T[G

eV−

1]


0 ≤ |yZ| ≤ 1

1 2

0.8

0.9

1.0

1.1

1.2

| |

MC

/D

ata


1 ≤ |yZ| ≤ 2

1 2

0.8

0.9

1.0

1.1

1.2

| |

MC

/D

ata


2 < |yZ| ≤ 2.4

1 10 1 10 2

0.8

0.9

1.0

1.1

1.2

| |

pT,ll [GeV]

MC

/D

ata

29 / 38


..

b b b b b b b b b b b b b b b b b b b bbbbbb

b

b

b

b

b

b

b

b

b

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

bbbb

b

b

b

b

b

b

b

bb

Dir

eP

S

|yZ| < 0.8

0.8 ≤ |yZ| ≤ 1.6 (×0.1)

1.6 < |yZ| (×0.01)

b ATLAS data

Phys.Lett. B720 (2013) 32

ME+PS (1-jet)


3 2 1

10−4

10−3

10−2

10−1

1

101

φ∗η spectrum, Z→ ee (dressed)

1σ

fid

.

dσ

fid

.d

φ∗ η

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

|yZ| < 0.8

3 2 1

0.8

0.9

1.0

1.1

1.2

→ | |

MC

/D

ata


0.8 ≤ |yZ| ≤ 1.6

3 2 1

0.8

0.9

1.0

1.1

1.2

→ ≤ | |

MC

/D

ata


1.6 < |yZ|

10−3 10−2 10−1 1

0.8

0.9

1.0

1.1

1.2

→ | | ≥

φ∗η

MC

/D

ata

30 / 38


..

b b b b b b b b b b b b b b b b b b b bbbbbb

b

b

b

b

b

b

b

b

b

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

bbbb

b

b

b

b

b

b

b

bb

Dir

eP

S

|yZ| < 0.8

0.8 ≤ |yZ| ≤ 1.6 (×0.1)

1.6 < |yZ| (×0.01)

b ATLAS data

Phys.Lett. B720 (2013) 32

ME+PS (1-jet)


3 2 1

10−4

10−3

10−2

10−1

1

101

φ∗η spectrum, Z→ ee (dressed)

1σ

fid

.

dσ

fid

.d

φ∗ η


|yZ| < 0.8

3 2 1

0.8

0.9

1.0

1.1

1.2

→ | |

MC

/D

ata


0.8 ≤ |yZ| ≤ 1.6

3 2 1

0.8

0.9

1.0

1.1

1.2

→ ≤ | |

MC

/D

ata


1.6 < |yZ|

10−3 10−2 10−1 1

0.8

0.9

1.0

1.1

1.2

→ | | ≥

φ∗η

MC

/D

ata

.Not too terrible description of Drell-Yan data.

30 / 38

LHC data comparisons (plain showering)

..

��

��

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� ��π ��π ��π ��π� ��π� π

��

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31 / 38


..

b

b

b

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b

b

b

bb

bb

b

b

b

b

b

b

b

b

b

b

b

bb

b

bb

b

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b

b

b

b

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b

b

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b

b

b

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b

b

b

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b

b

b

b

b

b

b

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b

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b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

Dir

eP

S

×10−1

×10−2

×10−3

×10−4

×10−5

×10−6

×10−7

×10−8

b

ATLAS data

Phys.Rev.Lett. 106 (2011) 172002

Dire

0.4 0.5 0.6 0.7 0.8 0.9 1.0

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

1

10 1Dijet azimuthal decorrelations

∆φ [rad/π]

1/

σd

σ/

d∆

φ[π

/ra

d]

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

110 < pmax⊥

/GeV < 160

0.60.8

11.21.4

⊥

MC

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ata


160 < pmax⊥

/GeV < 210

0.60.8

11.21.4

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ATLAS data

Phys.Rev.Lett. 106 (2011) 172002

Dire

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.Need multi-jet prediction…so is a shower maybe already enough?

31 / 38

Summary

• Fixed-order + parton shower continues to be an active field,but we need to also improve showers.We can combine multiple LO calculations, or multiple NLOcalculations, or match pp→ colour singlett at NNLO +PS.

• Now the main issue is the accuracy of the shower.• For simple processes, we can in principle match terms in the

shower with analytic results.…needed a new, cleanly defined shower for that though.

• We introduced a new shower combining the simplicity ofparton showers and the symmetry of dipole showers.

• Results of the new DIRE shower in PYTHIA and SHERPA. lookpromising. We hope to improve this shower in the future.

32 / 38

Back-up supplement

32 / 38

References

POWHEGJHEP 0411 (2004) 040 JHEP 0711 (2007) 070 POWHEG-BOX (JHEP 1006 (2010) 043)MC@NLOOriginal (JHEP 0206 (2002) 029) Herwig++ (Eur.Phys.J. C72 (2012) 2187)Sherpa (JHEP 1209 (2012) 049) aMC@NLO (arXiv:1405.0301)NLO matching results and comparisonsPlots taken from Ann.Rev.Nucl.Part.Sci. 62 (2012) 187 Plots taken from JHEP 0904 (2009) 002Tree-level merging MLM (Mangano, http://www-cpd.fnal.gov/personal/mrenna/tuning/nov2002/mlm.pdf. Talkpresented at the Fermilab ME/MC Tuning Workshop, Oct 4, 2002, Mangano et al. JHEP 0701 (2007) 013)Pseudoshower (JHEP 0405 (2004) 040)CKKW (JHEP 0111 (2001) 063, JHEP 0208 (2002) 015)CKKW-L (JHEP 0205 (2002) 046, JHEP 0507 (2005) 054, JHEP 1203 (2012) 019)METS (JHEP 0911 (2009) 038, JHEP 0905 (2009) 053)Unitarised mergingPythia (JHEP 1302 (2013) 094) Herwig (JHEP 1308 (2013) 114) Sherpa (arXiv:1405.3607)FxFx: Jet matching @ NLO: JHEP 1212 (2012) 061MEPS@NLOJHEP 1304 (2013) 027 JHEP 1301 (2013) 144 Plots taken from arXiv:1401.7971UNLOPS JHEP 1303 (2013) 166 arXiv:1405.1067MiNLO: Original (JHEP 1210 (2012) 155) Improved (JHEP 1305 (2013) 082)MiNLO-NNLOPS: JHEP 1310 (2013) 222 arXiv:1407.2940arXiv:1501.04637UN2LOPS: arXiv:1405.3607 arXiv:1407.3773

Parton shower basics

Parton showers are unitary all-order operators:

PS[σME

+0

]

= σPS+0 + σPS

+1 + σ+ ≥ 2

= σME+0 ΠS+0 (ρ0, ρmin) ←− ..0 emissions in [ρ0, ρmin]

+ σME+0 ΠS+0 (ρ0, ρ1) αsw0

f P0ΠS+1 (ρ1, ρmin) ←− ..1 emission in [ρ0, ρmin]

+ σME+0 ΠS+0 (ρ0, ρ1) αsw0

f P0ΠS+1 (ρ1, ρ2) αsw1f P1[ΠS+2 (ρ2, ρmin) + . . .

]↑ ↑

..2 or more emissions in [ρ0, ρmin]!= σME

+0

The no-emission probabilities

ΠS+i (ρ1, ρ2) = exp

{−∫ ρ1

ρ2

dραswifPi

}define exclusive cross sections and remove the overlap between samples!

34 / 38



PS[σME

+0

]= σPS

+0 +

σPS+1 + σ+ ≥ 2


+

σME+0 ΠS+0 (ρ0, ρ1) αsw0


+ σME+0 ΠS+0 (ρ0, ρ1) αsw0


]↑ ↑


+0



{−∫ ρ1

ρ2

dραswifPi


34 / 38



PS[σME

+0

]= σPS

+0 + σPS+1 +

σ+ ≥ 2


+ σME+0 ΠS+0 (ρ0, ρ1) αsw0


+

σME+0 ΠS+0 (ρ0, ρ1) αsw0


]↑ ↑


+0



{−∫ ρ1

ρ2

dραswifPi


34 / 38



PS[σME

+0

]= σPS

+0 + σPS+1 + σ+ ≥ 2


+ σME+0 ΠS+0 (ρ0, ρ1) αsw0


+ σME+0 ΠS+0 (ρ0, ρ1) αsw0


]↑ ↑


+0



{−∫ ρ1

ρ2

dραswifPi


34 / 38



PS[σME

+0

]= σPS

+0 + σPS+1 + σ+ ≥ 2


+ σME+0 ΠS+0 (ρ0, ρ1) αsw0


+ σME+0 ΠS+0 (ρ0, ρ1) αsw0


]↑ ↑


+0



{−∫ ρ1

ρ2

dραswifPi


34 / 38

Next-to-leading order calculations

Pen-and-paper: Add Born + Virtual + Real.

⟨O⟩NLO =∫

BnO(Φn)dΦn +∫

VnOn(Φn)dΦn +∫

Bn+1O(Φn)dΦn+1

Reality: Phase space integral separately divergent ⇒ ..Add zero!

⟨O⟩NLO =∫ [

Bn + Vn +∫

Dn+1

]O(Φn)dΦn +

∫ [Bn+1O(Φn+1)−Dn+1O(Φ′

n)]dΦn+1

Real reality: States Φn+1 and Φ′n are correlated. ⇒ Problematic, since

further manipulations (e.g. hadronisation) can spoil the cancellations⇒ ..Add more zeros!

⟨O⟩NLO =∫ [

Bn + Vn + In +∫

dΦrad(

B′n+1 −Dn+1

)]O(Φn)dΦn

+∫ (

Bn+1 − B′n+1)O(Φn+1)

+∫ (

B′n+1O(Φn+1)− B′

n+1O(Φn))←− ..That’s the O(αs) of a PS step!

35 / 38



⟨O⟩NLO =∫

BnO(Φn)dΦn +∫

VnOn(Φn)dΦn +∫

Bn+1O(Φn)dΦn+1


⟨O⟩NLO =∫ [

Bn + Vn +∫

Dn+1

]O(Φn)dΦn +

∫ [Bn+1O(Φn+1)−Dn+1O(Φ′

n)]dΦn+1



⟨O⟩NLO =∫ [

Bn + Vn + In +∫

dΦrad(

B′n+1 −Dn+1

)]O(Φn)dΦn

+∫ (

Bn+1 − B′n+1)O(Φn+1)

+∫ (

B′n+1O(Φn+1)− B′

n+1O(Φn))←− ..That’s the O(αs) of a PS step!

35 / 38



⟨O⟩NLO =∫

BnO(Φn)dΦn +∫

VnOn(Φn)dΦn +∫

Bn+1O(Φn)dΦn+1


⟨O⟩NLO =∫ [

Bn + Vn +∫

Dn+1

]O(Φn)dΦn +

∫ [Bn+1O(Φn+1)−Dn+1O(Φ′

n)]dΦn+1



⟨O⟩NLO =∫ [

Bn + Vn + In +∫

dΦrad(

B′n+1 −Dn+1

)]O(Φn)dΦn

+∫ (

Bn+1 − B′n+1)O(Φn+1)

+∫ (

B′n+1O(Φn+1)− B′

n+1O(Φn))←− ..That’s the O(αs) of a PS step! 35 / 38

CKKW(-L)

Aim: Combine multiple tree-level calculations with each other and (PS)resummation. Fill in soft and collinear regions with parton shower.

⟨O⟩ = B0O(S+0j)

−∫

dρ B0P0(ρ)Θ(1)> wfwαsΠS+0(ρ0, ρ)O(S+0j)

+∫

B1Θ(1)> wfwαsΠS+0(ρ0, ρ)O(S+1j)

−∫

dρ B1P1(ρ)Θ(2)> wfwαsΠS+0(ρ0, ρ1)ΠS+1(ρ1, ρ)O(S+1j)

+∫

B2Θ(2)> wfwαsΠS+0(ρ0, ρ1)ΠS+1(ρ1, ρ)O(S+2j)

Changes inclusive cross sections=⇒ Can contain numerically large (sub-leading) logs.=⇒ Needs fixing!

35 / 38

Bug vs. Feature in CKKW(-L)The ME includes terms that are not compensated by the PS approximatevirtual corrections (i.e. Sudakov factors).These are the improvements that we need to describe multiple hard jets!

If we simply add samples, the “improvements” will degrade the inclusivecross section: σinc will contain ln(tMS) terms.

The inclusive cross section does not contain logs relatedto cuts on higher multiplicities.

Traditional approach: Don’t use a too small merging scale.→ Uncancelled terms numerically not important.

Unitary approach1:Use a (PS) unitarity inspired approach exactly cancel the dependenceof the inclusive cross section on tMS.

1 JHEP1302(2013)094 (Leif Lönnblad, SP), JHEP1308(2013)114 (Simon Plätzer) 35 / 38

Unitarised ME+PS

Aim: If you add too much, then subtract what you add!

⟨O⟩ = B0O(S+0j)

−∫

dρ B1Θ(1)> wfwαsΠS+0(ρ0, ρ)O(S+0j) −

∫dρ B2Θ(2)

>Θ(1)

<wfwαsΠS+0 (ρ0,ρ)O(S+0j)

+∫

B1Θ(1)> wfwαsΠS+0(ρ0, ρ)O(S+1j)

−∫

dρ B2Θ(2)> wfwαsΠS+0(ρ0, ρ1)ΠS+1(ρ1, ρ)O(S+1j)

+∫

B2Θ(2)> wfwαsΠS+0(ρ0, ρ1)ΠS+1(ρ1, ρ)O(S+2j) +

∫B2Θ(2)

>Θ(1)

<wfwαsΠS+0 (ρ0,ρ1)O(S+2j)

Inclusive cross sections preserved by construction.Cancellation between different ”jet bins”.⇒ Statistics needs fixing.

35 / 38

NLO matching with MC@NLO

Aim: Achieve NLO for inclusive +0-jet, and LO for inclusive +1-jetobservables and attach PS resummation.To get there, remember that the (regularised) NLO cross section is

BNLO = [Bn + Vn + In]O0 +∫

dΦrad (Bn+1O1 −Dn+1O0)

= [Bn + Vn + In]O0 +∫

dΦrad (Sn+1O0 −Dn+1O0)

+∫

dΦrad (Sn+1O1 − Sn+1O0) +∫

dΦrad (Bn+1O1 − Sn+1O1)

where Sn+1 are some additional “transfer functions”, e.g. the PS kernels.Red term is the O(αs) part of a shower from Bn. ⇒ Discard from BNLO.Thus, we have the seed cross section⌢

BNLO =[

Bn + Vn + In +∫

dΦrad (Sn+1 −Dn+1)]O0 +

∫dΦrad (Bn+1 − Sn+1)O1

This is not the NLO result…but showering the O0-part will restore this!=⇒ NLO +PS accuracy!

35 / 38

UMEPS, MC@NLO-style (Plätzer JHEP 1308 (2013) 114)

Aim: Combine multiple tree-level calculations with each other and (PS)resummation. Fill in soft and collinear regions with parton shower.

⟨O⟩ = B0ΠS+0(ρ0, ρMS)O(S+0j)

−∫

dρ [B1 − B0P0(ρ)] Θ(1)> wfwαsΠS+0(ρ0, ρ)O(S+0j)

+∫

B1Θ(1)> wfwαsΠS+0(ρ0, ρ)ΠS+1(ρ, ρMS)O(S+1j)

−∫

dρ [B2 − B1P1(ρ)] Θ(2)> wfwαsΠS+0(ρ0, ρ1)ΠS+1(ρ1, ρ)O(S+1j)

+∫

B2Θ(2)> wfwαsΠS+0(ρ0, ρ1)ΠS+1(ρ1, ρ)O(S+2j) +

∫B2Θ(2)

>Θ(1)

<wfwαsΠS+0 (ρ0,ρ1)O(S+2j)

Inclusive cross sections preserved by construction.Less cancellation between different ”jet bins” fixed.=⇒ Statistics okay.

35 / 38

Comparison of NLO merging schemes

FxFx: Restricts the range of merging scales. Cross section changesFxFx: thus numerically small.FxFx: Probably fewest counter events.

MEPS@NLO: Improved, colour-correct Sudakov of MC@NLO for theMEPS@NLO: first emission. Larger tMS range.MEPS@NLO: Smaller cross section changes.MEPS@NLO: Improved resummation in process-independent way.

UNLOPS: Inclusive observables strictly NLO correct. Further showerUNLOPS: improvements also directly improve the results.UNLOPS: Many counter events if done naively.

MiNLO: applies analytical (N)NLL Sudakov factors, which cancelMiNLO: problematic logs, only merging two multiplicities.MiNLO: Was moulded into an NNLO matching.

36 / 38

The UNLOPS method

Start with UMEPS:

⟨O⟩ =∫

dϕ0

{O(S+0j)

(B0+ B̃0 −

∫sB̃1→0 +

∫sB1→0 −

[∫B̂1→0

]−1,2

−∫sB↑2→0 −

∫B̂2→0

)

+∫O(S+1j)

(B̃1 +

[B̂1

]−1,2

−[∫

B̂2→1

]−2

)+∫ ∫

O(S+2j)B̂2

}

Or for the case of M different NLO calculations, and N tree-level calculations:

⟨O⟩ =M−1∑m=0

∫dϕ0

∫· · ·∫O(S+mj)

{B̃m +

[B̂m

]−m,m+1

+∫sBm+1→m

−M∑

i=m+1

∫sB̃i→m −

M∑i=m+1

[∫B̂i→m

]−i,i+1

−M∑

i=m+1

∫sB↑

i+1→m −N∑

i=M+1

∫B̂i→m

}

+∫

dϕ0

∫· · ·∫O(S+Mj)

{B̃M +

[B̂M

]−M,M+1

−[∫

B̂M+1→M

]−M

−N∑

i=M+1

∫B̂i+1→M

}

+N∑

n=M+1

∫dϕ0

∫· · ·∫O(S+nj)

{B̂n −

N∑i=n+1

∫B̂i→n

}

36 / 38

The UNLOPS method

Remove all unwanted O(αns )- and O(αn+1

s )-terms:

⟨O⟩ =∫

dϕ0

{O(S+0j)

(B0+ B̃0 −

∫sB̃1→0 +

∫sB1→0 −

[∫B̂1→0

]−1,2

−∫sB↑2→0 −

∫B̂2→0

)

+∫O(S+1j)

(B̃1 +

[B̂1

]−1,2

−[∫

B̂2→1

]−2

)+∫ ∫

O(S+2j)B̂2

}


⟨O⟩ =M−1∑m=0

∫dϕ0

∫· · ·∫O(S+mj)

{B̃m +

[B̂m

]−m,m+1

+∫sBm+1→m

−M∑

i=m+1

∫sB̃i→m −

M∑i=m+1

[∫B̂i→m

]−i,i+1

−M∑

i=m+1

∫sB↑

i+1→m −N∑

i=M+1

∫B̂i→m

}

+∫

dϕ0

∫· · ·∫O(S+Mj)

{B̃M +

[B̂M

]−M,M+1

−[∫

B̂M+1→M

]−M

−N∑

i=M+1

∫B̂i+1→M

}

+N∑

n=M+1

∫dϕ0

∫· · ·∫O(S+nj)

{B̂n −

N∑i=n+1

∫B̂i→n

}

36 / 38

The UNLOPS method

Add full NLO results:

⟨O⟩ =∫

dϕ0

{O(S+0j)

(B0+ B̃0 −

∫sB̃1→0 +

∫sB1→0 −

[∫B̂1→0

]−1,2

−∫sB↑2→0 −

∫B̂2→0

)

+∫O(S+1j)

(B̃1 +

[B̂1

]−1,2

−[∫

B̂2→1

]−2

)+∫ ∫

O(S+2j)B̂2

}


⟨O⟩ =M−1∑m=0

∫dϕ0

∫· · ·∫O(S+mj)

{B̃m +

[B̂m

]−m,m+1

+∫sBm+1→m

−M∑

i=m+1

∫sB̃i→m −

M∑i=m+1

[∫B̂i→m

]−i,i+1

−M∑

i=m+1

∫sB↑

i+1→m −N∑

i=M+1

∫B̂i→m

}

+∫

dϕ0

∫· · ·∫O(S+Mj)

{B̃M +

[B̂M

]−M,M+1

−[∫

B̂M+1→M

]−M

−N∑

i=M+1

∫B̂i+1→M

}

+N∑

n=M+1

∫dϕ0

∫· · ·∫O(S+nj)

{B̂n −

N∑i=n+1

∫B̂i→n

}

36 / 38

The UNLOPS method

Unitarise:

⟨O⟩ =∫

dϕ0

{O(S+0j)

(B0+ B̃0 −

∫sB̃1→0 +

∫sB1→0 −

[∫B̂1→0

]−1,2

−∫sB↑2→0 −

∫B̂2→0

)

+∫O(S+1j)

(B̃1 +

[B̂1

]−1,2

−[∫

B̂2→1

]−2

)+∫ ∫

O(S+2j)B̂2

}


⟨O⟩ =M−1∑m=0

∫dϕ0

∫· · ·∫O(S+mj)

{B̃m +

[B̂m

]−m,m+1

+∫sBm+1→m

−M∑

i=m+1

∫sB̃i→m −

M∑i=m+1

[∫B̂i→m

]−i,i+1

−M∑

i=m+1

∫sB↑

i+1→m −N∑

i=M+1

∫B̂i→m

}

+∫

dϕ0

∫· · ·∫O(S+Mj)

{B̃M +

[B̂M

]−M,M+1

−[∫

B̂M+1→M

]−M

−N∑

i=M+1

∫B̂i+1→M

}

+N∑

n=M+1

∫dϕ0

∫· · ·∫O(S+nj)

{B̂n −

N∑i=n+1

∫B̂i→n

}

36 / 38

Deriving an UN2LOPS matchingWe basically follow a “merging strategy”:

• Pick calculations to combine (two MC@NLOs) with each otherand with the PS resummation.

• Remove kinematic overlaps between the two MC@NLOs bydividing the one-jet phase space.

• Reweight one-jet MC@NLO (to make it exclusive ↔ want todescribe hardest jet with this),remove all undesired terms at O(α1+1

s )and make sure that the whole thing is numerically stable.Reweight subtractions with ΠS+0 to be able to group themwith virtuals.

• Add and subtract reweighted one-jet MC@NLO, (→ unitarise)to ensure inclusive zero-jet cross section is unchanged w.r.t.NLO.

• Remove all terms up to O(α2s) in the zero-jet contribution,

replace by NNLO jet-vetoed cross section.

Work with Stefan Höche and Ye Li. 36 / 38

UN2LOPS matching

Aim: Combine just two NLO calculations, then upgrade to NNLO directly.

Start over again, now combining MC@NLO’s because those are resonablystable. Thus:⋄ Use 0-jet matched (MC@NLO 0) and 1-jet matched calculation (MC@NLO 1).⋄ Remove hard (qT > ρMS) reals in MC@NLO 0.⋄ Reweight B1 of MC@NLO 1 with “zero-jet Sudakov” factor ΠS+0/αs running.⋄ Reweight NLO part B̃

R1 of MC@NLO 1 with “zero-jet Sudakov” factor.

⋄ Subtract erroneous O(α+1s ) terms multiplying B1.

⋄ Reweight subtractions with ΠS+0 to be able to group them with B̃R1 .

⋄ Put ρMS → ρc < 1GeV. (→ MC@NLO 0 becomes exclusive NLO)⋄ Unitarise by subtracting the processed MC@NLO ′

1 from the “zero-qT bin”.⋄ Remove all terms up to α2

s from the “zero-qT bin” and add the qT-vetoedNNLO cross section.

⇒ σinclusive @ NNLO, resummation as accurate as Sudakov, stats fine.NNLO logarithmic parts from qT-vetoed TMDs (EFT calculation),

hard coefficients from qT-subtraction (i.e. DYNNLO, HNNLO),power corrections from MC@NLO 1.

Work with Stefan Höche and Ye Li.36 / 38

UN2LOPS matching

Aim: Combine just two NLO calculations, then upgrade to NNLO directly.

Start over again, now combining MC@NLO’s because those are resonablystable. Thus:⋄ Use 0-jet matched (MC@NLO 0) and 1-jet matched calculation (MC@NLO 1).⋄ Remove hard (qT > ρMS) reals in MC@NLO 0.⋄ Reweight B1 of MC@NLO 1 with “zero-jet Sudakov” factor ΠS+0/αs running.⋄ Reweight NLO part B̃

R1 of MC@NLO 1 with “zero-jet Sudakov” factor.

⋄ Subtract erroneous O(α+1s ) terms multiplying B1.

⋄ Reweight subtractions with ΠS+0 to be able to group them with B̃R1 .

⋄ Put ρMS → ρc < 1GeV. (→ MC@NLO 0 becomes exclusive NLO)⋄ Unitarise by subtracting the processed MC@NLO ′

1 from the “zero-qT bin”.⋄ Remove all terms up to α2

s from the “zero-qT bin” and add the qT-vetoedNNLO cross section.

⇒ σinclusive @ NNLO, resummation as accurate as Sudakov, stats fine.NNLO logarithmic parts from qT-vetoed TMDs (EFT calculation),

hard coefficients from qT-subtraction (i.e. DYNNLO, HNNLO),power corrections from MC@NLO 1.

Work with Stefan Höche and Ye Li.35 / 38

UN2LOPS matching

O(UN2LOPS) =∫dΦ0

¯̄BqT,cut0 (Φ0) O(Φ0)

+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))]

B1(Φ1) O(Φ0)

+∫qT,cut

dΦ1 Π0(t1, µ2Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))

B1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)]

B̃R1(Φ1) O(Φ0) +

∫qT,cut

dΦ1Π0(t1, µ2Q) B̃R

1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ2

[1−Π0(t1, µ2

Q)]

HR1 (Φ2) O(Φ0) +

∫qT,cut

dΦ2 Π0(t1, µ2Q) HR

1 (Φ2)F2(t2, O)

+∫qT,cut

dΦ2 HE1 (Φ2)F2(t2, O)

Notice: This is just an extention of the old Sudakov veto algorithm:Run trial shower on the reconstructed zero-jet state,If trial shower produces an emission, keep zero-jet kinematics and stop;else start PS off one-jet state. 35 / 38

UN2LOPS matching

O(UN2LOPS) =∫dΦ0


+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))]

B1(Φ1) O(Φ0)

+∫qT,cut

dΦ1 Π0(t1, µ2Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))

B1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)]

B̃R1(Φ1) O(Φ0) +

∫qT,cut


1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ2

[1−Π0(t1, µ2

Q)]

HR1 (Φ2) O(Φ0) +

∫qT,cut


1 (Φ2)F2(t2, O)

+∫qT,cut

dΦ2 HE1 (Φ2)F2(t2, O)

Note that this is just an extention of the old Sudakov veto algorithm:Run trial shower on the reconstructed zero-jet state,If trial shower produces an emission, keep zero-jet kinematics and stop;else start PS off one-jet state. 35 / 38

UN2LOPS matching

O(UN2LOPS) =∫dΦ0


+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))]

B1(Φ1) O(Φ0)

+∫qT,cut

dΦ1 Π0(t1, µ2Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))

B1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)]

B̃R1(Φ1) O(Φ0) +

∫qT,cut


1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ2

[1−Π0(t1, µ2

Q)]

HR1 (Φ2) O(Φ0) +

∫qT,cut


1 (Φ2)F2(t2, O)

+∫qT,cut

dΦ2 HE1 (Φ2)F2(t2, O)

Note:[1−Π0(t1, µ2

Q)]

B̃R1 comes from using qT-vetoed cross sections.

Note:[1−Π0(t1, µ2

Q)]

B̃R1 etc. comes from using qT-vetoed cross sections.

If trial shower produces an emission, keep zero-jet kinematics and stop;else start PS off one-jet state. 35 / 38

UN2LOPS matching

O(UN2LOPS) =∫dΦ0


+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))]

B1(Φ1) O(Φ0)

+∫qT,cut

dΦ1 Π0(t1, µ2Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))

B1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)]

B̃R1(Φ1) O(Φ0) +

∫qT,cut


1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ2

[1−Π0(t1, µ2

Q)]

HR1 (Φ2) O(Φ0) +

∫qT,cut


1 (Φ2)F2(t2, O)

+∫qT,cut

dΦ2 HE1 (Φ2)F2(t2, O)[

1−Π0(t1, µ2Q)]


¯̄BqT,cut0 + B̃R

1 + HR1 + HE

1 = BNNLO

Other terms drop out in inclusive observables.else start PS off one-jet state. 35 / 38

UN2LOPS matching

O(UN2LOPS) =∫dΦ0


+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))]

B1(Φ1) O(Φ0)

+∫qT,cut

dΦ1 Π0(t1, µ2Q)(w1(Φ1) + w(1)

1 (Φ1) + Π(1)0 (t1, µ2

Q))

B1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ1

[1−Π0(t1, µ2

Q)]

B̃R1(Φ1) O(Φ0) +

∫qT,cut


1(Φ1) F̄1(t1, O)

+∫qT,cut

dΦ2

[1−Π0(t1, µ2

Q)]

HR1 (Φ2) O(Φ0) +

∫qT,cut


1 (Φ2)F2(t2, O)

+∫qT,cut

dΦ2 HE1 (Φ2)F2(t2, O)[

1−Π0(t1, µ2Q)]


Orange terms do not contain any universal αs corrections present in the PS.H1 do not contribute in the soft/collinear limit.=⇒ PS accuracy is preserved. 35 / 38


She

rpa

MC

NNLONLOHNNLO

= 14 TeVs

NLOH<2mR/F

µ/2< H m NNLOH<2m

R/Fµ/2< H m

[pb]

H/d

yσd

2

4

6

8

10

12

Rat

io to

NN

LO

0.960.98

11.021.04

Hy

-4 -3 -2 -1 0 1 2 3 4

She

rpa

MC

NNLONLOHNNLO

= 14 TeVs

NLOH<2mR/F

µ/2< H m NNLOH<2m

R/Fµ/2< H m

[pb/

GeV

]T,

H/d

pσd

-210

-110

1

10

Rat

io to

NN

LO0.960.98

11.021.04

[GeV]T,H

p0 20 40 60 80 100 120 140 160 180 200

Rapidity and p⊥ of the Higgs-boson, comparing SHERPA-NNLO andHNNLO. Our independent NNLO calculation works nicely.

36 / 38

Shower tricks: Weighted shower (JHEP 1209 (2012) 049)For frequently negative contributions, or if we do not want to findan ideal overestimate, we can use a weighted shower instead:

• For each phase space point, store the full splitting probabilityf, the differential overestimate g, and an auxiliary weighth = sgn (f) · g.

• Pick/discard splitting as usual according to abs(

fg

). If the

emission……was picked, multiply wpicked = h

gg−fh−f to event weight

…was discarded, multiply wdiscarded = hg to event weight

If the differential overestimate is not ideal, i.e. fg = a > 1, then use

fg→ f

(ka) · gand h→ (ka) · h

with an additional factor k ≳ 1, and continue as before.37 / 38

DIRE Anomalous DimensionsThe anomalous dimensions

γab(N, κ2) =∫ 1

0zNPab(z, κ2) are given by (1)

γqq(N, κ2) = 2CF Γ(N, κ2)− CF (2N + 3)(N + 1)(N + 2)

+ γq

γgq(N, κ2) = 2CF K(N, κ2)− CF (N + 3)(N + 1)(N + 2)

γgg(N, κ2) = 2CA Γ(N, κ2) + 2CA K(N, κ2)− 2CA (N + 3)(N + 1)(N + 2)

− 2CAN + 3

+ γg

γqg(N, κ2) = − TR N(N + 1)(N + 2)

+ 2TRN + 3

(2)where

2Γ(N, κ2) =2 3F2

(1, 1, 3

2 ; N+32 , N+4

2 ;−κ−2)(N + 1)(N + 2) κ2 − ln 1 + κ2

κ2 ,

2K(N, κ2) =2 2F1

(1, N+2

2 ; N+42 ;−κ−2)

(N + 2) κ2 .

(3)

38 / 38

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