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

2 / 38 Event generators should describe multijet observables

106 W(→ lν) + jets 1.4 BLACKHAT+SHERPA

) [pb] ATLAS Data, 1.2 jets anti-kt jets, R=0.4, -1 j j s = 7 TeV, 4.6 fb 1 5 p > 30 GeV, |y | < 4.4 10 T BLACKHAT+SHERPA

(W+N 0.8 Pred. / Data

σ HEJ ATLAS 0.6 ALPGEN ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7 104 SHERPA 1.4 HEJ Njets MEPS@NLO 1.2

103 1 0.8 Pred. / Data 0.6 2 10 ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7 1.4 ALPGEN Njets 1.2 . 10 1 0.8 Pred. / Data 1 0.6 ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7

1.4 SHERPA Njets 10-1 1.2 1 0.8 Pred. / Data -2 MEPS@NLO 10 0.6 ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7 ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7

Njets Njets

(Figures taken from EPJC 75 (2015) 2 82 and CMS summary of

https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsCombined) 3 / 38 Event generators should describe multijet observables

106 W(→ lν) + jets 1.4 BLACKHAT+SHERPA

) [pb] ATLAS Data, 1.2 jets anti-kt jets, R=0.4, -1 j j s =Mar 7 TeV, 2015 4.6 fb 1 5 p > 30 GeV, |y | < 4.4 CMS Preliminary 10 T BLACKHAT+SHERPA

(W+N 0.8 Pred. / Data -1 σ HEJ ATLAS ≤ 0.6 7 TeV CMS measurement (L 5.0 fb ) -1 ALPGEN ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7 ≤ [pb] 5 8 TeV CMS measurement (L 19.6 fb ) 104 10

σ SHERPA 1.4 HEJ 7 TeV Theory prediction Njets MEPS@NLO 1.2 8 TeV Theory prediction 3 4 1 10 10 CMS 95%CL limit 0.8 Pred. / Data 0.6 2 10 3 ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7 10 1.4 ALPGEN Njets 1.2 . 10 102 1 0.8 Pred. / Data 1 0.6 ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7

10 1.4 SHERPA Njets 10-1 1.2 1 1

Production Cross Section, 0.8 Pred. / Data -2 MEPS@NLO 10 0.6 -1 ≥0 ≥1 ≥2 ≥3 10≥4 ≥5 ≥6 ≥7 ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7

Njets Njets γγ→ W ≥1j ≥2j ≥3j ≥4j Z ≥1j ≥2j ≥3j ≥4j Wγ Zγ WW WZ ZZ EW WVγ tt 1j 2j 3j t tW t ttγ ttZ ggH VBF VH ttH WW qqll t-ch s-ch qqH ∆σ ∆σ All results at: http://cern.ch/go/pNj7 Th. H in exp.

(Figures taken from EPJC 75 (2015) 2 82 and CMS summary of

https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsCombined) 3 / 38 Event generators should describe multijet observables

106 W(→ lν) + jets 1.4 BLACKHAT+SHERPA

) [pb] ATLAS Data, 1.2 jets anti-kt jets, R=0.4, -1 j j s =Mar 7 TeV, 2015 4.6 fb 1 5 p > 30 GeV, |y | < 4.4 CMS Preliminary 10 T BLACKHAT+SHERPA

(W+N 0.8 Pred. / Data -1 σ HEJ ATLAS ≤ 0.6 7 TeV CMS measurement (L 5.0 fb ) -1 ALPGEN ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7 ≤ [pb] 5 8 TeV CMS measurement (L 19.6 fb ) 104 10

σ SHERPA 1.4 HEJ 7 TeV Theory prediction Njets MEPS@NLO 1.2 8 TeV Theory prediction 3 4 1 10 10 CMS 95%CL limit 0.8 Pred. / Data 0.6 2 Combine10 multiple3 accurate fixed-order≥0 ≥1 calculations≥2 ≥3 ≥4 ≥5 with≥6 ≥7 each 10 1.4 ALPGEN Njets other, and with shower resummation,1.2 . into a single flexible 10 102 1 ⇒ 0.8 prediction. = Matching /Pred. / Data merging 1 0.6 ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7

10 1.4 SHERPA Njets 10-1 1.2 1 1

Production Cross Section, 0.8 Pred. / Data -2 MEPS@NLO 10 0.6 -1 ≥0 ≥1 ≥2 ≥3 10≥4 ≥5 ≥6 ≥7 ≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7

Njets Njets γγ→ W ≥1j ≥2j ≥3j ≥4j Z ≥1j ≥2j ≥3j ≥4j Wγ Zγ WW WZ ZZ EW WVγ tt 1j 2j 3j t tW t ttγ ttZ ggH VBF VH ttH WW qqll t-ch s-ch qqH ∆σ ∆σ All results at: http://cern.ch/go/pNj7 Th. H in exp.

(Figures taken from EPJC 75 (2015) 2 82 and CMS summary of

https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsCombined) 3 / 38 .

.

⋄ 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.

NLO+PS matched predictions

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

4 / 38 .

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

NLO+PS matched predictions

.

NLO+PS methods mostly behave well, but may sometimes show large 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.

4 / 38 NLO+PS matched predictions

.

.

NLO+PS methods mostly behave well, but may sometimes show large 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   ∫t0   BnFn(Φn) , ∆(t0, t) = exp − dΦradαs(ΦR)P(ΦR)

t

∫tn u − ∆ (tn, tMS) = 1 Bn+1Fn+1(Φn+1)∆(tn, t) (”unitarised Sudakov”)

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

u O Bn∆ (t0, tMS)∆(tMS, tcut) 0(S+0) ∫t0 [ ] − u O + dΦR Bnαs(ΦR)P(ΦR)(1 Fn+1)∆ (t0, tMS)∆(tMS, tcut) + Bn+1Fn+1∆(t0, t1)∆(t1, t2) 1(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 Leading-order merging

Start with seed cross sections and no-emission probabilities   ∫t0   BnFn(Φn) , ∆(t0, t) = exp − dΦradαs(ΦR)P(ΦR)

t

∫tn u − ∆ (tn, tMS) = 1 Bn+1Fn+1(Φn+1)∆(tn, t) (”unitarised Sudakov”)

tMS

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

u O Bn∆ (t0, tMS)∆(tMS, tcut) 0(S+0) ∫t0 [ ] − u O + dΦR Bnαs(ΦR)P(ΦR)(1 Fn+1)∆ (t0, tMS)∆(tMS, tcut) + Bn+1Fn+1∆(t0, t1)∆(t1, t2) 1(S+1)

∫t0 [ − u + dΦRR Bn+1Fn+1∆(t0, t1)(1 Fn+2)∆ (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 Leading-order merging

Start with seed cross sections and no-emission probabilities   ∫t0   BnFn(Φn) , ∆(t0, t) = exp − dΦradαs(ΦR)P(ΦR)

t

∫tn ∆u( , ) = − B (Φ )∆( , ) Merging methods validtn fortMS any multiplicity.1 n+1Fn+1 n+1 tn t Merging methods use shower SudakovtMS factors ⇒ Improve when − ∼ . whereshowersFm(Φ improve.m) = Θ(t(Φm) tMS), t0 Q. Three multiplicities merged by u → NB: Usual CKKW approximation ∆ (tn, tMS) ∆(tn, tMS) calls for B ∆u( , )∆( , )O ( ) ≫n t0 tMS tMS tcut→0 S+0 O tMS tcut or BnP(Φ) Bn+1. We use tMS = (1GeV) later. ∫t0 [ ] − u O + dΦR Bnαs(ΦR)P(ΦR)(1 Fn+1)∆ (t0, tMS)∆(tMS, tcut) + Bn+1Fn+1∆(t0, t1)∆(t1, t2) 1(S+1)

∫t0 [ − u + dΦRR Bn+1Fn+1∆(t0, t1)(1 Fn+2)∆ (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 NLO merging

Any NLO matching method contains only approximate multi-jet kinematics (given by PS). Any LO merging method X only ever contains approximate virtual corrections.

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 to n-jet in favour of n + 1-jet NLO). ⋄ Adjust higher orders to suit other needs (e.g. to preserve the inclusive cross section).

⇒ X@NLO

7 / 38 .

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

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. 8 / 38 NLO-merged predictions for Higgs + jets

.

Still O(50%) differences in resummation region – important Figure: p⊥,H and ∆ϕ12 for gg→H after merging (H+0)@NLO, (H+1)@NLO, (H+2)@NLO,uncertainty (H+3)@LO, for analyses compared that touse other jet generators. binning!

⇒ LH study: The generators come closer together if NLO merging is employed. 8 / 38 The next step(s): Matching @ NNLO

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

Observation: If an NLO merged calculation leads to a well-defined zero-jet inclusive 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 [ ∫ ] ∫ ∫ B O 1 O ∼ O − O O B0 0ΠS+0 + 1ΠS+0 B0 0 B1 0ΠS+0 + B1 1ΠS+0 B0 1 1 1 NNLOPS: UN2LOPS Start from MiNLO, upgrade analytic Start from UNLOPS, refine uni- CKKW Sudakov to match integral tarised Sudakov factor so that only of q⊥ onto zero-jet NLO cross sec- one-jet NLO cross section survives O 2 tion (LO+NLL calculation of q⊥), at (αs ), then include NNLO jet- then include differential NNLO K- vetoed cross section for NNLO ac- curacy. factor. a aa aa Benefits: Analytic control over 1st Benefits: Easy, process- and emission, improved resummation. shower-independent. aa aa Limitations: Process-dependent, Limitations: Does does not shower 2 uses pre-tabulated K-factors. αs δ(p⊥) terms, or only a subset – aa i.e. has bin edges. 10 / 38 UN2LOPS (Drell-Yan)

Z pT reconstructed from dressed electrons Lepton-Jet Rapidity Difference − 10 1 b b b b b b 160 b b [pb] 1201 1276 b ATLAS (arXiv: . ) y b /GeV] − b b b 2 2 b ∆ 1 10 UN LOPS b [ b 140 b b b

Z b /d < < , mlν/2 µR/F 2 mlν b

b σ T 120 b p −3 d m /2 < µ < 2 m b lν Q lν b 10 b b /d b b

σ 100 b

d b −4 b σ 10 80 b b b 1/ b b CMS PRD85(2012)032002 b b −5 60 10 UN2LOPS b b b b < < 40 b m /2 µ / 2 m b ll R F ll b b −6 b b b m /2 < µ < 2 m b 10 ll Q ll 20 b b b b b b b b b b b b b b b b b . 0 b b b b b b 1.4 1.8 1.6 1.2 1.4 1.2 1 b b bbbbbbbbbbbbbbb b 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb b b b 0.8 . 0 6 MC/Data 0 8 MC/Data . 0.4 0.6 0.2 0 1 10 1 10 2 -4 -2 0 2 4 − pT,Z [GeV] y(Lepton) y(First Jet) CMS data for Z-boson p⊥. UN2LOPS does quite well. Large band at low p⊥ reflects log scale and shower modelling. ATLAS data for charged current well described.

11 / 38 UN2LOPS (Drell-Yan)

Z pT reconstructed from dressed electrons Lepton-Jet Rapidity Difference − 10 1 b b b b b b 160 b b [pb] 1201 1276 b ATLAS (arXiv: . ) y b /GeV] − b b b 2 2 b ∆ 1 10 UN LOPS b [ b 140 b b b

Z b /d < < , mlν/2 µR/F 2 mlν b

b σ T 120 b p −3 d m /2 < µ < 2 m b lν Q lν b 10 b b /d b b

σ 100 b

d b −4 b σ 10 80 b b b 1/ b b CMS PRD85(2012)032002 b b −5 60 10 UN2LOPS b b b b < < 40 b m /2 µ / 2 m b ll R F ll b b −6 < < b b b 10 m /2 µ 2 m b 2 ll Q ll 20 b b b b b b b UN LOPS b b b code and plots available from b b b b b b b . 0 b b b b b b 1.4 1.8 1∼.6 http://www.slac.stanford.edu/1.2 1.4 shoeche/pub/nnlo/ 1.2 1 b b bbbbbbbbbbbbbbb b 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb b b b 0.8 . 0 6 MC/Data 0 8 MC/Data . 0.4 0.6 0.2 0 1 10 1 10 2 -4 -2 0 2 4 − pT,Z [GeV] y(Lepton) y(First Jet) CMS data for Z-boson p⊥. UN2LOPS does quite well. Large band at low p⊥ reflects log scale and shower modelling. ATLAS data for charged current well described.

11 / 38 . .

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

UN2LOPS (Higgs production)

s = 14 TeV s = 14 TeV UN2LOPS UN2LOPS HqT HqT Sherpa MC 1 1 Sherpa MC [pb/GeV] NNLO [pb/GeV] NNLO MC@NLO MC@NLO T,H T,H /dp /dp σ σ d d 10-1 10-1

m /2< µ <2m m /2< µ <2m H R/F H H R/F H m /2< µ <2m m /4< µ

1.2 1.2 1 1 0.8 0.8 Ratio to HqT 0 20 40 60 80 100 120 140 160 180 200 Ratio to HqT 0 20 40 60 80 100 120 140 160 180 200 p [GeV] p [GeV] T,H T,H

p⊥ of the Higgs-boson for two different matching schemes in UN2LOPS – mimicing the philosophical differences between common NLO matching schemes.

12 / 38 UN2LOPS (Higgs production)

s = 14 TeV s = 14 TeV UN2LOPS UN2LOPS HqT HqT Sherpa MC 1 1 Sherpa MC [pb/GeV] NNLO [pb/GeV] NNLO MC@NLO MC@NLO T,H T,H /dp /dp σ σ d d 10-1 10-1

m /2< µ <2m m /2< µ <2m H R/F H H R/F H m /2< µ <2m m /4< µ

1.2 . 1.2 . 1 1 0.8 0.8 Ratio to HqT 0 20 40 60 80 100 120 140 160 180 200 Ratio to HqT 0 20 40 60 80 100 120 140 160 180 200 p [GeV] p [GeV] T,H T,H

Wep⊥ haveof theNLO Higgs-bosonmatching, forNLO twomerging different and matchingNNLO matching. schemes in Still, predictionsUN2LOPS – differ mimicing appreciably the philosophical because of differences limited shower between accuracy. common aaNLO matching schemes.

12 / 38 The PS prediction for an observable O is ∫ t0 d¯t d ln F⃗a(Φn,¯t, t0) ′ F (Φ , , ; ) = F (Φ , , ) (Φ ) + F ′ (Φ , ,¯; ) ⃗a n tc t0 O ⃗a n tc t0 O n ¯ ¯ ⃗a n+1 tc t O tc t d ln t with no-emission probabilities and Sudakov factors defined by

2 2 2 2 Fa(x, t, µ ) = fa(x, t)∆a(t, µ ) = fa(x, µ )Πa(x, t, µ ) .

Now if tc → 0, then we find ∫ t0 d¯t d ln F⃗a(Φn,¯t, t0) ′ F (Φ , t , t ; O) → F ′ (Φ , t,¯t; O) ⃗a n c 0 ¯t d ln¯t ⃗a n+1 which allows a comparison to analytic resummation formulae.

=⇒ Back to Parton Shower Resummation

13 / 38 =⇒ Back to Parton Shower Resummation

The PS prediction for an observable O is ∫ t0 d¯t d ln F⃗a(Φn,¯t, t0) ′ F (Φ , , ; ) = F (Φ , , ) (Φ ) + F ′ (Φ , ,¯; ) ⃗a n tc t0 O ⃗a n tc t0 O n ¯ ¯ ⃗a n+1 tc t O tc t d ln t with no-emission probabilities and Sudakov factors defined by

2 2 2 2 Fa(x, t, µ ) = fa(x, t)∆a(t, µ ) = fa(x, µ )Πa(x, t, µ ) .

Now if tc → 0, then we find ∫ t0 d¯t d ln F⃗a(Φn,¯t, t0) ′ F (Φ , t , t ; O) → F ′ (Φ , t,¯t; O) ⃗a n c 0 ¯t d ln¯t ⃗a n+1 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. To improve showers, we should improve the accuracy of the Sudakov factors by comparing ( [ ]) ∫ ( ) ( ) ( ) ρ1 2 ∑ i ∑ i AN dρ Q αs αs ∆ (ρ0, ρ1) = exp − ln Ai − Bi ρ ρ ρ 2π 2π ( ∫ 0 ∫ i ) i ρ0 dρ zmax α (ρ) ∆PS(ρ , ρ ) = exp − dz s K(z, ρ) 0 1 ρ π ρ1 zmin 2

× × (2) where A and B are related to γcusp and γi (e.g. A1 = Ca 1, A2 = Ca γcusp)

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

14 / 38 Reliable Event Generator predictions?

We have hints that current parton showers (e.g. PYTHIA, SHERPA) are not 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, to so that comparison to observable-based resummation becomes possible. a 1 Simply splitting functions, simple phase space boundaries. 2 Eikonal in soft limit, AP kernels in collinear limit, collinear anomalous dimensions unchanged, flavour/momentum sum rules, no choices introducing iffy sub-leading logs.

15 / 38 Reliable Event Generator predictions?

We have hints that current parton showers (e.g. PYTHIA, SHERPA) are not 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, so that comparison to observable-based resummation becomes possible. a 1 Simple splitting functions, simple phase space boundaries. 2 Eikonal in soft limit, AP kernels in collinear limit, collinear anomalous Wedimensions have defined unchanged, a new flavour/momentum dipole-parton shower, sum rules, and implementedno choices introducing independentlyiffy sub-leading inlogs. 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 because A 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 2 partial-fractioned eikonal can make analytic comparisons cumbersome ← remedied by symmetric evolution variable, and by demoting dipoles to ”damping functions” used to recover the collinear anomalous dimensions. 3 not immediately applicable for evolution, because not normalized by momentum / flavour sum rules ← demand sum rules exactly.

Addressing these points completely fixes all (II, IF, FI, FF) dipoles to [( ) ] − 2 1 z 1 + z 3 Pqq(z, κ ) = 2 CF − + CF δ(1 − z) (1 − z)2 + κ2 2 2 [( )+ ] ( ) − 2 1 z z 11 2 Pgg(z, κ ) = 2 CA + − 2 + z(1 − z) + δ(1 − z) CA − n TR (1 − z)2 + κ2 z2 + κ2 6 3 f [ ]+ [ ] − 2 z 2 z 2 2 2 Pgq(z, κ ) = 2 CF − Pqg(z, κ ) = TR z + (1 − z) z2 + κ2 2

Deriving a DIRE shower

Catani-Seymour dipoles are a bad starting point because 1. different for initial-initial, initial-final, final-initial, final-final ← unified by good phase space parametrization + combining with Jacobian

17 / 38 3 not immediately applicable for evolution, because not normalized by momentum / flavour sum rules ← demand sum rules exactly.

Addressing these points completely fixes all (II, IF, FI, FF) dipoles to [( ) ] − 2 1 z 1 + z 3 Pqq(z, κ ) = 2 CF − + CF δ(1 − z) (1 − z)2 + κ2 2 2 [( )+ ] ( ) − 2 1 z z 11 2 Pgg(z, κ ) = 2 CA + − 2 + z(1 − z) + δ(1 − z) CA − n TR (1 − z)2 + κ2 z2 + κ2 6 3 f [ ]+ [ ] − 2 z 2 z 2 2 2 Pgq(z, κ ) = 2 CF − Pqg(z, κ ) = TR z + (1 − z) z2 + κ2 2

Deriving a DIRE shower

Catani-Seymour dipoles are a bad starting point because 1. different for initial-initial, initial-final, final-initial, final-final ← unified by good phase space parametrization + combining with Jacobian 2 partial-fractioned eikonal can make analytic comparisons cumbersome ← remedied by symmetric evolution variable, and by demoting dipoles to ”damping functions” used to recover the collinear anomalous dimensions.

17 / 38 Addressing these points completely fixes all (II, IF, FI, FF) dipoles to [( ) ] − 2 1 z 1 + z 3 Pqq(z, κ ) = 2 CF − + CF δ(1 − z) (1 − z)2 + κ2 2 2 [( )+ ] ( ) − 2 1 z z 11 2 Pgg(z, κ ) = 2 CA + − 2 + z(1 − z) + δ(1 − z) CA − n TR (1 − z)2 + κ2 z2 + κ2 6 3 f [ ]+ [ ] − 2 z 2 z 2 2 2 Pgq(z, κ ) = 2 CF − Pqg(z, κ ) = TR z + (1 − z) z2 + κ2 2

Deriving a DIRE shower

Catani-Seymour dipoles are a bad starting point because 1. different for initial-initial, initial-final, final-initial, final-final ← unified by good phase space parametrization + combining with Jacobian 2 partial-fractioned eikonal can make analytic comparisons cumbersome ← remedied by symmetric evolution variable, and by demoting dipoles to ”damping functions” used to recover the collinear anomalous dimensions. 3 not immediately applicable for evolution, because not normalized by momentum / flavour sum rules ← demand sum rules exactly.

17 / 38 Deriving a DIRE shower

Catani-Seymour dipoles are a bad starting point because 1. different for initial-initial, initial-final, final-initial, final-final ← unified by good phase space parametrization + combining with Jacobian 2 partial-fractioned eikonal can make analytic comparisons cumbersome ← remedied by symmetric evolution variable, and by demoting dipoles to ”damping functions” used to recover the collinear anomalous dimensions. 3 not immediately applicable for evolution, because not normalized by momentum / flavour sum rules ← demand sum rules exactly.

Addressing these points completely fixes all (II, IF, FI, FF) dipoles to [( ) ] − 2 1 z 1 + z 3 Pqq(z, κ ) = 2 CF − + CF δ(1 − z) (1 − z)2 + κ2 2 2 [( )+ ] ( ) − 2 1 z z 11 2 Pgg(z, κ ) = 2 CA + − 2 + z(1 − z) + δ(1 − z) CA − n TR (1 − z)2 + κ2 z2 + κ2 6 3 f [ ]+ [ ] z 2 − z P (z, κ2) = 2 C − P (z, κ2) = T z2 + (1 − z)2 gq F 2 2 qg R z + κ 2 17 / 38 DIRE ordering variables II IF FF FF

i i i i a a a j j

b k k More concretely, we use the phase space variables

saisbi saib sbi ρii = Kinematical p⊥ zii = 1 − sab sab sab saisik − sik ρif = zif = 1 sai + sak sai + sak sajsij sai ρfi = zfi = sai + saj sai + saj sijsjk sij + sik ρff = Ariandne p⊥ zff = sij + sik + sjk sij + sik + sjk

18 / 38 DIRE ordering variables II IF FF FF

i i i i a a a j j

b k k More concretely, we use the phase space variables Kinematical p⊥ for small p⊥, but covers full p⊥ range even for small ρstart saisbi saib sbi ρii = zii = 1 − sab sab sab saisik − sik ρif = zif = 1 sai + sak sai + sak sajsij sai ρfi = zfi = sai + saj sai + saj sijsjk sij + sik ρff = zff = sij + sik + sjk sij + sik + sjk Ariandne p⊥ 19 / 38 Improvements

Ordering variables were chosen to obtain very simple phase space boundaries and splitting functions, so that a comparison with analytic results is possible.

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

( ) 2(1 − z) 2(1 − z) α α2 −→ 1 + s γ(1) + s γ(2) (1 − z)2 + κ2 (1 − z)2 + κ2 2π cusp 4π2 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

Differential jet resolution at parton level (Durham algorithm) PS [pb]

1 ×

4 y23 2 ire + 10 D n n y

10 3 10 × −1 y34 10 /d log

σ 2 d 10 × −2 y45 10

10 1 × −3 y56 10

1 Sherpa 10−1 Pythia − e+e → qq¯ @ 91.2 GeV 10−2

2 σ y23 0 σ

Deviation -2 σ

2 σ y34 0 σ

Deviation -2 σ

2 σ y45 0 σ

Deviation -2 σ

σ 2 y56 0 σ 21 / 38 Deviation -2 σ

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 log10 yn n+1 Cross-validation at the sub-permille level: Durham jet rates

Differential jet resolution at parton level (Durham algorithm) PS

[pb] 10 3 1 ire

+ × y23 2 D n n y 10 2 10 × −1 y34 10

/d log 10 1 σ d × −2 y45 10 1 × −3 y56 10 10−1

− Sherpa 10 2 Pythia Lepton colliders . τ+τ− → → 10−3 [h gg] @ 125 GeV

2 σ y23 0 σ

Deviation -2 σ

2 σ y34 0 σ

Deviation -2 σ

2 σ y45 0 σ

Deviation -2 σ

σ 2 y56 0 σ 22 / 38 Deviation -2 σ

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 log10 yn n+1 Cross-validation at the sub-permille level: k⊥ clustering scales

Differential kT-jet resolution at parton level 3 10 PS [pb] )

d01 ire D 10 2 /GeV − nm d × 10 1

d 12 ( 1

10 10 × −2 d23 10 /d log 1 σ d × −3 d34 10 10−1

Sherpa 10−2 Pythia → + − −3 pp e e @ 8 TeV 10 < < 2 66 mll 116 GeV

2 σ d01 0 σ

Deviation -2 σ

2 σ d12 0 σ

Deviation -2 σ

2 σ d23 0 σ

Deviation -2 σ

2 σ d 34 0 σ 23 / 38 Deviation -2 σ

0.5 1 1.5 2 2.5 dnm log10( /GeV) Cross-validation at the sub-permille level: k⊥ clustering scales

Differential kT-jet resolution at parton level 1 PS [pb] ) d01 ire 10−1 D /GeV

nm − d × 1

( d12 10 −2

10 10

× −2 /d log d23 10

σ −3

d 10

× −3 d34 10 10−4

Sherpa 10−5 Pythia Hadron colliders . − pp → τ+τ @ 8 TeV

2 σ d01 0 σ

Deviation -2 σ

2 σ d12 0 σ

Deviation -2 σ

2 σ d23 0 σ

Deviation -2 σ

2 σ d 34 0 σ 24 / 38 Deviation -2 σ

0.5 1 1.5 2 2.5 dnm log10( /GeV) Cross-validation at the sub-permille level: Jet scales

Differential kT-jet resolution at parton level (Breit frame) 10 1 PS [pb] )

× ire 1 d02 2 D

/GeV −1

nm 10

d −1 ( × d23 10 10 10−2 − /d log d × 10 2 σ 10−3 34 d

−4 × −3 10 d45 10 Sherpa 10−5 Pythia − − 10−6 e q → e q @ 300 GeV Q2 > 100 GeV2

2 σ d02 0 σ

Deviation -2 σ

2 σ d23 0 σ

Deviation -2 σ

2 σ d34 0 σ

Deviation -2 σ

2 σ d 45 0 σ 25 / 38 Deviation -2 σ

0.5 1 1.5 2 dnm log10( /GeV) Cross-validation at the sub-permille level: Jet scales

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

[pb] ×

) d02 2 ire

10−8 D /GeV

nm − d × 1

( d23 10 −9

10 10

/d log × −2 σ −10 d34 10

d 10

−11 × −3 10 d45 10

Sherpa 10−12 Pythia Lepton-hadron colliders . − − τ g → τ g @ 300 GeV

First ever DIS prediction2 σ with PYTHIA8! d02 0 σ

Deviation -2 σ

2 σ d23 0 σ

Deviation -2 σ

2 σ d34 0 σ

Deviation -2 σ

2 σ d 45 0 σ 26 / 38 Deviation -2 σ

0.5 1 1.5 2 dnm log10( /GeV) LEP data comparisons (plain showering)

Differential 2-jet rate with Durham algorithm (91.2 GeV) Differential 3-jet rate with Durham algorithm (91.2 GeV) 10 3 b 23 34 b b b b b b PS b PS y b b y b b 2 b b b b

/d b /d b b b 10 ire 2 ire σ b σ b

D 10 b D d b b d b b b b b b b b b 1 b 1 b b 10 b b 10 b b b b b b b b b b b b b JADE/OPAL data 1 JADE/OPAL data b b b 1 Eur.Phys.J. C17 (2000) 19 b Eur.Phys.J. C17 (2000) 19 b Dire −1 Dire b b 10 b b −1 10 −2 b b 10

1.4 1.4 1.2 1.2 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbb

MC/Data 0.8 MC/Data 0.8 0.6 0.6 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 yDurham yDurham 23 . 34 Differential 4-jet rate with Durham algorithm (91.2 GeV) Differential 5-jet rate with Durham algorithm (91.2 GeV)

b b b b 45 b 56 b 3 b b b b PS PS

y b y b 10 b 3 b b b b 10 b b b /d b /d ire b b ire σ b σ b 2 D b D d 10 b d 2 b b b b 10 b b b b b b b b 1 b 10 b 1 10 b b b b 1 b JADE/OPAL data b JADE/OPAL data b 1 b 17 2000 19 17 2000 19 Eur.Phys.J. C ( ) Eur.Phys.J. C ( ) b − b 10 1 Dire −1 Dire b 10 b

− b 2 − b 10 10 2 b b

1.4 1.4 1.2 1.2 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbb 1 bbbbbbbbbbbbbbbbbbbbbbbbbb

MC/Data 0.8 MC/Data 0.8 0.6 0.6 10−5 10−4 10−3 10−2 10−5 10−4 10−3 10−2 27 / 38 Durham Durham y45 y56 LEP data comparisons (plain showering)

E = E = Thrust ( CMS 91.2 GeV) Total jet broadening ( CMS 91.2 GeV) T T b 1 b b b b PS PS b b B 10 b b b 1 b b

/d b b b 10 b /d

σ b b b b b ire b ire σ b d b b b b D D

b d b b b σ b b b b b

b σ 1 b b b b b b 1/ b b 1 b b b b 1/ b b b b b b b b b b b b − b 1 b b b b 10 ALEPH data − b b 10 1 b b b Eur.Phys.J. C35 (2004) 457 b b −2 Dire b b b 10 b ALEPH data −2 b 10 b Eur.Phys.J. C35 (2004) 457 b b −3 b Dire 10 −3 b 10 b − 10 4 1.4 1.4 1.2 1.2 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

MC/Data 0.8 MC/Data 0.8 0.6 0.6 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 T . BT E = E = C-Parameter ( CMS 91.2 GeV) Aplanarity ( CMS 91.2 GeV) 2 10 b C A b b b b PS PS b b b /d

b /d b b b

σ b b σ b ire 1 ire b b d b d 10 b b b b b b D ALEPH data D 1 b b σ b b σ b b b b b b b b 35 2004 457 b b b Eur.Phys.J. C ( ) 1/ b b b b 1/ b b 1 b b b Dire b b b −1 b 10 b b b b ALEPH data − b b 1 b 10 b b b Eur.Phys.J. C35 (2004) 457 b b b b b b b Dire b − b −2 2 b b 10 10 b b b b b

b b b b − b 10 3 b − b 3 b 10 b − 10 4 1.4 1.4 1.2 1.2 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

MC/Data 0.8 MC/Data 0.8 0.6 0.6 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 28 / 38 C A LEP data comparisons (plain showering)

E = E = Thrust ( CMS 91.2 GeV) Total jet broadening ( CMS 91.2 GeV) T T b 1 b b b b PS PS b b B 10 b b b 1 b b

/d b b b 10 b /d

σ b b b b b ire b ire σ b d b b b b D D

b d b b b σ b b b b b

b σ 1 b b b b b b 1/ b b 1 b b b b 1/ b b b b b b b b b b b b − b 1 b b b b 10 ALEPH data − b b 10 1 b b b Eur.Phys.J. C35 (2004) 457 b b −2 Dire b b b 10 b ALEPH data −2 b 10 b Eur.Phys.J. C35 (2004) 457 b b −3 b Dire 10 −3 b 10 b − 10 4 1.4 1.4 1.2 1.2 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

MC/Data 0.8 MC/Data 0.8 0.6 0.6 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 T . BT Nice descriptionE = of LEP data. E = C-Parameter ( CMS 91.2 GeV) Aplanarity ( CMS 91.2 GeV) 2 10 b C A b b b b PS PS b b b /d

b /d b b b

σ b b σ b ire 1 ire b b d b d 10 b b b b b b D ALEPH data D 1 b b σ b b σ b b b b b b b b 35 2004 457 b b b Eur.Phys.J. C ( ) 1/ b b b b 1/ b b 1 b b b Dire b b b −1 b 10 b b b b ALEPH data − b b 1 b 10 b b b Eur.Phys.J. C35 (2004) 457 b b b b b b b Dire b − b −2 2 b b 10 10 b b b b b

b b b b − b 10 3 b − b 3 b 10 b − 10 4 1.4 1.4 1.2 1.2 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb 1 bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

MC/Data 0.8 MC/Data 0.8 0.6 0.6 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 28 / 38 C A LHC data comparisons (1-jet CKKW-L merged)

pT spectrum, Z→ ee (dressed) −1

] 10 b b 1 b b b PS − b b b b b 0 ≤|yZ| ≤ 1 b −2 b ire b

GeV 10 b b b D [ b b b b b b b b b b T b b fid

p b b σ b d 3 1 ≤|y | ≤ 2 (×0.1) d − Z b b 10 b b b b b b b fid 1 b b b b σ b b b b b b b b b b −4 b b 10 2 < |y | ≤ 2.4 (×0.01) b Z b b b b b b b b b b −5 b b 10 b b

b ATLAS data b 10−6 JHEP 09 (2014) 145 b b b ME+PS (1-jet) −7 10 5 ≤ Qcut ≤ 20 GeV b

b 10−8 | | . b 10−9 1.2 1 2 1.1 0 ≤|yZ| ≤ 1 1.0 b b b bbbbbbbbbbbbbbbbbbb b b b b 0.9 MC/Data 0.8 | | 1.2 1 2 1.1 1 ≤|yZ| ≤ 2 1.0 b b b bbbbbbbbbbbbbbbbbbb b b b b 0.9 MC/Data 0.8 | | 1.2 1 2 < 1.1 2 |yZ| ≤ 2.4 1.0 b b b bbbbbbbbbbbbbbbbbbb b b b b 0.9 MC/Data 0.8 29 / 38 1 10 1 10 2 pT,ll [GeV] LHC data comparisons (1-jet CKKW-L merged)

∗ φη spectrum, Z→ ee (dressed) ∗ η b fid. 1 b b b b PS φ b b

σ 10 b b b d b

d b b b b < b |y | 0.8 b ire 1 b

fid. Z b

σ b b D b b b b b b b 1 b b b b b b b b b b b b b b b 0.8 ≤|yZ| ≤ 1.6 (×0.1) b b b b b b b b −1 b b b b b b b 10 b b b b b b b b b b b b b < b 1.6 |yZ| (×0.01) b b b b b b b b −2 b b b 10 b b b b b b b b ATLAS data b b −3 b 10 Phys.Lett. B720 (2013) 32 b b

1 b ME+PS ( -jet) b

5 ≤ Qcut ≤ 20 GeV b b −4 10 b

b →. || b 1.2 3 2 1 < 1.1 |yZ| 0.8 1.0 b b b bbbbbbbbbbbbbbbbbbbbbbbb b bbbbbb 0.9 MC/Data 0.8 → ≤|| 1.2 3 2 1 1.1 0.8 ≤|yZ| ≤ 1.6 1.0 b b b bbbbbbbbbbbbbbbbbbbbbbbb b bbbbbb 0.9 MC/Data 0.8 → ||≥ 1.2 3 2 1 < 1.1 1.6 |yZ| 1.0 b b b bbbbbbbbbbbbbbbbbbbbbbbb b bbbbbb 0.9 MC/Data 0.8 30 / 38 10−3 10−2 10−1 1 ∗ φη LHC data comparisons (1-jet CKKW-L merged)

∗ φη spectrum, Z→ ee (dressed) ∗ η b fid. 1 b b b b PS φ b b

σ 10 b b b d b

d b b b b < b |y | 0.8 b ire 1 b

fid. Z b

σ b b D b b b b b b b 1 b b b b b b b b b b b b b b b 0.8 ≤|yZ| ≤ 1.6 (×0.1) b b b b b b b b −1 b b b b b b b 10 b b b b b b b b b b b b b < b 1.6 |yZ| (×0.01) b b b b b b b b −2 b b b 10 b b b b b b b b ATLAS data b b −3 b 10 Phys.Lett. B720 (2013) 32 b b

1 b ME+PS ( -jet) b

5 ≤ Qcut ≤ 20 GeV b b −4 10 b

b Not too terrible description→ of Drell-Yan. || data. b 1.2 3 2 1 < 1.1 |yZ| 0.8 1.0 b b b bbbbbbbbbbbbbbbbbbbbbbbb b bbbbbb 0.9 MC/Data 0.8 → ≤|| 1.2 3 2 1 1.1 0.8 ≤|yZ| ≤ 1.6 1.0 b b b bbbbbbbbbbbbbbbbbbbbbbbb b bbbbbb 0.9 MC/Data 0.8 → ||≥ 1.2 3 2 1 < 1.1 1.6 |yZ| 1.0 b b b bbbbbbbbbbbbbbbbbbbbbbbb b bbbbbb 0.9 MC/Data 0.8 30 / 38 10−3 10−2 10−1 1 ∗ φη LHC data comparisons (plain showering) Slide taken from Monday’s CMS summary by Matthias Weber ���������������������������������������������������

����������������� �� � �� �� �� ����������� ��� ����������� ������� �������� ���� ��� �������� ������ ���������������� � ����������� ����������� ���������� �����

����� �� �������� ���������������������� � φΔ �� σ � ��� ���������������������� ���������� �������������������� �������������������� � ���������������

� �� ����� �� ��������������������� ��� ��� σ ��� � � � ����������� � ����������� ������������������������� � � � � π �� π �� π �� � π �� � π �� π ��� � � �� �� �� �� �� π �� ����� �������������� � �� � ������������������������ � � ������������������������ � ��� � ������������������������ �� � ������������������������ � � ��� ��������������������� ��������������������� ��� � � ����� � �� �� �� � �� � �� π ����� � � φΔ ��� ��� σ � ������� ���������� ������� ���������� � � � ��� . ��� � �����

σ � ��� � ��� �

�� ��� ��� �� ���� � ������� ���������� ������� ���������� � � � ��� ���� � ��� ���� � π �� π �� π �� � π �� � π �� π � �� �� �� � �� � �� π �������������������� �������������������� � � � ���� ��� � �� �� �� �� �π �� � π π π π π � φΔ ����� ��� �����

� �� �� �� � �� � �� π � � π �������������������� �������������������� � � � ��� ���������������������������������� � ���

�� �� �� �� �� π �������������������� �������������������� ����������������������������������� � � � ��� ������������������������������������� � ��� ������������ � ��π ��π ��π ��π�π � ����ππ� ��π ��π ��π�π ��π� φΔ ����� φΔ ����� ����� ����� ���������������� ��������������� �������������� ��� ������������������� ����� 31 / 38 LHC data comparisons (plain showering) Dijet azimuthal decorrelations 10 1 ⊥ b PS 1.4 b max /rad] b 110 < p /GeV < 160

ire ⊥ b 1.2 π [ ATLAS data D b 1 b b b bbbbbbbbbbbb φ Phys.Rev.Lett. 106 (2011) 172002 b

∆ 1 b 0.8

b MC/Data ⊥ Dire b . /d b 0 6 b σ b 1.4 d b max b 160 < /GeV < 210 b p⊥ σ 1.2 b −1 b b b b bbbbbbbbbbbb

1/ b 1 10 b b b b 0.8 b

MC/Data ⊥ b b 0.6 b b b 1.4 b max b 210 < /GeV < 260 −2 b 1.2 p⊥ 10 b b b b b b b bbbbbbbbbbbb b 1 b b b b 0.8

MC/Data ⊥ b b 0.6 b − b b . 3 b 1 4 max 10 b b 260 < /GeV < 310 − 1.2 p⊥ × 1 b b b 10 b b b bbbbbbbbbbbb b b b 1 b b 0.8 b

b MC/Data ⊥ − b 0.6 10 4 b b b 1.4 b max b b 310 < /GeV < 400 b p⊥ −2 b 1.2 ×10 b b b 1 b b b bbbbbbbbbbbb b b b b .0.8 − b 5 b 10 MC/Data ⊥ b 0.6

b b b . − 1 4 max × 3 b < < 10 b 400 p /GeV 500 b 1.2 ⊥ b 1 b b b b b b b b b b −6 b b b . 10 b b 0 8

MC/Data ⊥ b 0.6 × −4 . 10 b b 1 4 b max b . 500 < p⊥ /GeV < 600 b 1 2 −7 b 1 b b b b b b b b b 10 b b b b 0.8 − ×10 5 MC/Data 0.6 ⊥ b b . 1 4 max −6 b 600 < /GeV < 800 − ×10 1.2 p⊥ 10 8 b b b b b b b b b b 1 0.8

MC/Data ⊥ × −7 b 0.6 10 1.4 −9 b max > 10 1.2 p⊥ /GeV 800 − 1 b b b b ×10 8 0.8 MC/Data 0.6 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 31 / 38 ∆φ [rad/π] ∆φ [rad/π] LHC data comparisons (plain showering) Dijet azimuthal decorrelations 10 1 ⊥ b PS 1.4 b max /rad] b 110 < p /GeV < 160

ire ⊥ b 1.2 π [ ATLAS data D b 1 b b b bbbbbbbbbbbb φ Phys.Rev.Lett. 106 (2011) 172002 b

∆ 1 b 0.8

b MC/Data ⊥ Dire b . /d b 0 6 b σ b 1.4 d b max b 160 < /GeV < 210 b p⊥ σ 1.2 b −1 b b b b bbbbbbbbbbbb

1/ b 1 10 b b b b 0.8 b

MC/Data ⊥ b b 0.6 b b b 1.4 b max b 210 < /GeV < 260 −2 b 1.2 p⊥ 10 b b b b b b b bbbbbbbbbbbb b 1 b b b b 0.8

MC/Data ⊥ b b 0.6 b − b b . 3 b 1 4 max 10 b b 260 < /GeV < 310 − 1.2 p⊥ × 1 b b b 10 b b b bbbbbbbbbbbb b b b 1 b b 0.8 b

b MC/Data ⊥ − b 0.6 10 4 b b b 1.4 b max b b 310 < /GeV < 400 b p⊥ −2 b 1.2 ×10 b b b 1 b b b bbbbbbbbbbbb b b b b .0.8 − b 5 b 10Need multi-jet prediction…so is aMC/Data shower maybe already⊥ enough? b 0.6

b b b . − 1 4 max × 3 b < < 10 b 400 p /GeV 500 b 1.2 ⊥ b 1 b b b b b b b b b b −6 b b b . 10 b b 0 8

MC/Data ⊥ b 0.6 × −4 . 10 b b 1 4 b max b . 500 < p⊥ /GeV < 600 b 1 2 −7 b 1 b b b b b b b b b 10 b b b b 0.8 − ×10 5 MC/Data 0.6 ⊥ b b . 1 4 max −6 b 600 < /GeV < 800 − ×10 1.2 p⊥ 10 8 b b b b b b b b b b 1 0.8

MC/Data ⊥ × −7 b 0.6 10 1.4 −9 b max > 10 1.2 p⊥ /GeV 800 − 1 b b b b ×10 8 0.8 MC/Data 0.6 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 31 / 38 ∆φ [rad/π] ∆φ [rad/π] 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 NLO calculations, 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 of parton showers and the symmetry of dipole showers. • Results of the new DIRE shower in PYTHIA and SHERPA. look promising. We hope to improve this shower in the future.

32 / 38 Back-up supplement

32 / 38 References

POWHEG JHEP 0411 (2004) 040 JHEP 0711 (2007) 070 POWHEG-BOX (JHEP 1006 (2010) 043) MC@NLO Original (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 comparisons Plots taken from Ann.Rev.Nucl.Part.Sci. 62 (2012) 187 Plots taken from JHEP 0904 (2009) 002 Tree-level merging MLM (Mangano, http://www-cpd.fnal.gov/personal/mrenna/tuning/nov2002/mlm.pdf. Talk presented 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 merging Pythia (JHEP 1302 (2013) 094) Herwig (JHEP 1308 (2013) 114) Sherpa (arXiv:1405.3607) FxFx: Jet matching @ NLO: JHEP 1212 (2012) 061 MEPS@NLO JHEP 1304 (2013) 027 JHEP 1301 (2013) 144 Plots taken from arXiv:1401.7971 UNLOPS JHEP 1303 (2013) 166 arXiv:1405.1067 MiNLO: Original (JHEP 1210 (2012) 155) Improved (JHEP 1305 (2013) 082) MiNLO-NNLOPS: JHEP 1310 (2013) 222 arXiv:1407.2940 arXiv:1501.04637 UN2LOPS: arXiv:1405.3607 arXiv:1407.3773 PS PS = σ+0 + σ+1 + σ+ ≥ 2 ME ←− = σ+0 ΠS+0 (ρ0, ρmin) 0 emissions. in [ρ0, ρmin]

ME 0 + σ+ ΠS (ρ0, ρ1) αsw P0ΠS (ρ1, ρmin) ←− 1 emission. in [ρ0, ρmin] 0 +0 f +1 [ ] ME 0 1 + σ+0 ΠS+0 (ρ0, ρ1) αswf P0ΠS+1 (ρ1, ρ2) αswf P1 ΠS+2 (ρ2, ρmin) + ... ↑ ↑

2 or more emissions. in [ρ0, ρmin]

! ME = σ+0 The no-emission probabilities { ∫ } ρ1 − i ΠS+i (ρ1, ρ2) = exp dραswfPi ρ2 define exclusive cross sections and remove the overlap between samples!

Parton shower basics Parton showers are unitary all-order operators: [ ] ME PS σ+0

34 / 38 PS σ+1 + σ+ ≥ 2

ME 0 σ+ ΠS (ρ0, ρ1) αsw P0ΠS (ρ1, ρmin) ←− 1 emission. in [ρ0, ρmin] 0 +0 f +1 [ ] ME 0 1 + σ+0 ΠS+0 (ρ0, ρ1) αswf P0ΠS+1 (ρ1, ρ2) αswf P1 ΠS+2 (ρ2, ρmin) + ... ↑ ↑

2 or more emissions. in [ρ0, ρmin]

! ME = σ+0 The no-emission probabilities { ∫ } ρ1 − i ΠS+i (ρ1, ρ2) = exp dραswfPi ρ2 define exclusive cross sections and remove the overlap between samples!

Parton shower basics Parton showers are unitary all-order operators: [ ] ME PS PS σ+0 = σ+0 + ME ←− = σ+0 ΠS+0 (ρ0, ρmin) 0 emissions. in [ρ0, ρmin] +

34 / 38 σ+ ≥ 2

[ ] ME 0 1 σ+0 ΠS+0 (ρ0, ρ1) αswf P0ΠS+1 (ρ1, ρ2) αswf P1 ΠS+2 (ρ2, ρmin) + ... ↑ ↑

2 or more emissions. in [ρ0, ρmin]

! ME = σ+0 The no-emission probabilities { ∫ } ρ1 − i ΠS+i (ρ1, ρ2) = exp dραswfPi ρ2 define exclusive cross sections and remove the overlap between samples!

Parton shower basics Parton showers are unitary all-order operators: [ ] ME PS PS PS σ+0 = σ+0 + σ+1 + ME ←− = σ+0 ΠS+0 (ρ0, ρmin) 0 emissions. in [ρ0, ρmin] ME 0 ←− + σ+0 ΠS+0 (ρ0, ρ1) αswf P0ΠS+1 (ρ1, ρmin) 1 emission. in [ρ0, ρmin] +

34 / 38 The no-emission probabilities { ∫ } ρ1 − i ΠS+i (ρ1, ρ2) = exp dραswfPi ρ2 define exclusive cross sections and remove the overlap between samples!

Parton shower basics Parton showers are unitary all-order operators: [ ] ME PS PS PS σ+0 = σ+0 + σ+1 + σ+ ≥ 2 ME ←− = σ+0 ΠS+0 (ρ0, ρmin) 0 emissions. in [ρ0, ρmin]

ME 0 + σ+ ΠS (ρ0, ρ1) αsw P0ΠS (ρ1, ρmin) ←− 1 emission. in [ρ0, ρmin] 0 +0 f +1 [ ] ME 0 1 + σ+0 ΠS+0 (ρ0, ρ1) αswf P0ΠS+1 (ρ1, ρ2) αswf P1 ΠS+2 (ρ2, ρmin) + ... ↑ ↑

2 or more emissions. in [ρ0, ρmin]

! ME = σ+0

34 / 38 Parton shower basics Parton showers are unitary all-order operators: [ ] ME PS PS PS σ+0 = σ+0 + σ+1 + σ+ ≥ 2 ME ←− = σ+0 ΠS+0 (ρ0, ρmin) 0 emissions. in [ρ0, ρmin]

ME 0 + σ+ ΠS (ρ0, ρ1) αsw P0ΠS (ρ1, ρmin) ←− 1 emission. in [ρ0, ρmin] 0 +0 f +1 [ ] ME 0 1 + σ+0 ΠS+0 (ρ0, ρ1) αswf P0ΠS+1 (ρ1, ρ2) αswf P1 ΠS+2 (ρ2, ρmin) + ... ↑ ↑

2 or more emissions. in [ρ0, ρmin]

! ME = σ+0 The no-emission probabilities { ∫ } ρ1 − i ΠS+i (ρ1, ρ2) = exp dραswfPi ρ2 define exclusive cross sections and remove the overlap between samples! 34 / 38 Reality: Phase space integral separately divergent ⇒ Add zero!. ∫ [ ∫ ] ∫ [ ] ⟨O⟩NLO O O − O ′ = Bn + Vn + Dn+1 (Φn)dΦn + Bn+1 (Φn+1) Dn+1 (Φn) dΦn+1

′ ⇒ Real reality: States Φn+1 and Φn are correlated. Problematic, since further manipulations (e.g. hadronisation) can spoil the cancellations ⇒ Add more. zeros! ∫ [ ∫ ] ( ) ⟨O⟩NLO ′ − O = Bn + Vn + In + dΦrad Bn+1 Dn+1 (Φn)dΦn ∫ ( ) − ′ O + Bn+1 Bn+1 (Φn+1) ∫ ( ) ′ O − ′ O ←− O + Bn+1 (Φn+1) Bn+1 (Φn) That’s the (αs.) of a PS step!

Next-to-leading order calculations Pen-and-paper: Add Born + Virtual + Real. ∫ ∫ ∫

NLO ⟨O⟩ = BnO(Φn)dΦn + VnOn(Φn)dΦn + Bn+1O(Φn)dΦn+1

35 / 38 ′ ⇒ Real reality: States Φn+1 and Φn are correlated. Problematic, since further manipulations (e.g. hadronisation) can spoil the cancellations ⇒ Add more. zeros! ∫ [ ∫ ] ( ) ⟨O⟩NLO ′ − O = Bn + Vn + In + dΦrad Bn+1 Dn+1 (Φn)dΦn ∫ ( ) − ′ O + Bn+1 Bn+1 (Φn+1) ∫ ( ) ′ O − ′ O ←− O + Bn+1 (Φn+1) Bn+1 (Φn) That’s the (αs.) of a PS step!

Next-to-leading order calculations Pen-and-paper: Add Born + Virtual + Real. ∫ ∫ ∫

NLO ⟨O⟩ = BnO(Φn)dΦn + VnOn(Φn)dΦn + Bn+1O(Φn)dΦn+1

Reality: Phase space integral separately divergent ⇒ Add zero!. ∫ [ ∫ ] ∫ [ ] ⟨O⟩NLO O O − O ′ = Bn + Vn + Dn+1 (Φn)dΦn + Bn+1 (Φn+1) Dn+1 (Φn) dΦn+1

35 / 38 Next-to-leading order calculations Pen-and-paper: Add Born + Virtual + Real. ∫ ∫ ∫

NLO ⟨O⟩ = BnO(Φn)dΦn + VnOn(Φn)dΦn + Bn+1O(Φn)dΦn+1

Reality: Phase space integral separately divergent ⇒ Add zero!. ∫ [ ∫ ] ∫ [ ] ⟨O⟩NLO O O − O ′ = Bn + Vn + Dn+1 (Φn)dΦn + Bn+1 (Φn+1) Dn+1 (Φn) dΦn+1

′ ⇒ Real reality: States Φn+1 and Φn are correlated. Problematic, since further manipulations (e.g. hadronisation) can spoil the cancellations ⇒ Add more. zeros! ∫ [ ∫ ] ( ) ⟨O⟩NLO ′ − O = Bn + Vn + In + dΦrad Bn+1 Dn+1 (Φn)dΦn ∫ ( ) − ′ O + Bn+1 Bn+1 (Φn+1) ∫ ( ) ′ O − ′ O ←− O + Bn+1 (Φn+1) Bn+1 (Φn) That’s the (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⟩ O = B0 (S+∫0j) − (1) O dρ B0P0(ρ)Θ> wfwαs ΠS+0 (ρ0, ρ) (S+0j) ∫ (1) O + B1Θ> wfwαs ΠS+0 (ρ0, ρ) (S+1j) ∫ − (2) O dρ B1P1(ρ)Θ> wfwαs ΠS+0 (ρ0, ρ1)ΠS+1 (ρ1, ρ) (S+1j) ∫ (2) O + B2Θ> wfwαs ΠS+0 (ρ0, ρ1)ΠS+1 (ρ1, ρ) (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 approximate virtual 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 inclusive cross section: σinc will contain ln(tMS) terms.

The inclusive cross section does not contain logs related to 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 dependence

of 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⟩ O = B0 ∫(S+0j) ∫ (1) ( ) ( ) − O − ρ B Θ 2 Θ 1 Π (ρ ,ρ)O( ) dρ B1Θ> wfwαs ΠS+0 (ρ0, ρ) (S+0j) d 2 > < wfwαs S+0 0 S+0j ∫ (1) O + B1Θ> wfwαs ΠS+0 (ρ0, ρ) (S+1j) ∫ − (2) O dρ B2Θ> wfwαs ΠS+0 (ρ0, ρ1)ΠS+1 (ρ1, ρ) (S+1j) ∫ ∫ (2) ( ) ( ) O + B Θ 2 Θ 1 Π (ρ ,ρ )O( ) + B2Θ> wfwαs ΠS+0 (ρ0, ρ1)ΠS+1 (ρ1, ρ) (S+2j) 2 > < wfwαs S+0 0 1 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-jet observables 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! 35 / 38 =⇒ NLO +PS accuracy! 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⟩ = B Π (ρ , ρMS)O(S+ ) 0 ∫S+0 0 0j − − (1) O dρ [B1 B0P0(ρ)] Θ> wfwαs ΠS+0 (ρ0, ρ) (S+0j) ∫ (1) O + B1Θ> wfwαs ΠS+0 (ρ0, ρ)ΠS+1 (ρ, ρMS) (S+1j) ∫ − − (2) O dρ [B2 B1P1(ρ)] Θ> wfwαs ΠS+0 (ρ0, ρ1)ΠS+1 (ρ1, ρ) (S+1j) ∫ ∫ (2) ( ) ( ) O + B Θ 2 Θ 1 Π (ρ ,ρ )O( ) + B2Θ> wfwαs ΠS+0 (ρ0, ρ1)ΠS+1 (ρ1, ρ) (S+2j) 2 > < wfwαs S+0 0 1 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 changes FxFx: thus numerically small. FxFx: Probably fewest counter events.

MEPS@NLO: Improved, colour-correct Sudakov of MC@NLO for the

MEPS@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 shower UNLOPS: improvements also directly improve the results. UNLOPS: Many counter events if done naively.

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

36 / 38 Or for the case of M different NLO calculations, and N tree-level calculations: ∫ ∫ ∫ { ∫ ∑M−1 [ ] e b ⟨O⟩ = dϕ0 ··· O(S+mj) Bm + Bm + Bm+1→m −m,m+1 m=0 s ∫ [ ∫ ] ∫ ∫ } ∑M ∑M ∑M ∑N ↑ − e → − b → − − b → Bi m Bi m Bi+1→m Bi m i=m+1 s i=m+1 −i,i+1 i=m+1 s i=M+1 ∫ ∫ ∫ { [ ∫ ] ∫ } [ ] ∑N e b b b + dϕ0 ··· O(S+Mj) BM + BM − BM+1→M − Bi+1→M −M,M+1 −M i=M+1 ∫ ∫ ∫ { ∫ } ∑N ∑N b b + dϕ0 ··· O(S+nj) Bn − Bi→n n=M+1 i=n+1

The UNLOPS method

Start with UMEPS: ∫ { ( ∫ ∫ [∫ ] ∫ ∫ ) ⟨O⟩ O e − e − b − ↑ − b = dϕ0 (S+0j) B0+ B0 B1→0 + B1→0 B1→0 B2→0 B2→0 ( s s ) −1,2 s ∫ [ ] [∫ ] ∫ ∫ } e b b b + O(S+1j) B1 + B1 − B2→1 + O(S+2j)B2 −1,2 −2

36 / 38 Or for the case of M different NLO calculations, and N tree-level calculations: ∫ ∫ ∫ { ∫ ∑M−1 [ ] e b ⟨O⟩ = dϕ0 ··· O(S+mj) Bm + Bm + Bm+1→m −m,m+1 m=0 s ∫ [ ∫ ] ∫ ∫ } ∑M ∑M ∑M ∑N ↑ − e → − b → − − b → Bi m Bi m Bi+1→m Bi m i=m+1 s i=m+1 −i,i+1 i=m+1 s i=M+1 ∫ ∫ ∫ { [ ∫ ] ∫ } [ ] ∑N e b b b + dϕ0 ··· O(S+Mj) BM + BM − BM+1→M − Bi+1→M −M,M+1 −M i=M+1 ∫ ∫ ∫ { ∫ } ∑N ∑N b b + dϕ0 ··· O(S+nj) Bn − Bi→n n=M+1 i=n+1

The UNLOPS method

O n O n+1 Remove all unwanted (αs )- and (αs )-terms: ∫ { ( ∫ ∫ [ ∫ ] ∫ ∫ ) ⟨O⟩ O e − e − b − ↑ − b = dϕ0 (S+0j) B0+ B0 B1→0 + B1→0 B1→0 B2→0 B2→0 ( s s ) −1,2 s ∫ [ ] [ ∫ ] ∫ ∫ } e b b b + O(S+1j) B1 + B1 − B2→1 + O(S+2j)B2 −1,2 −2

36 / 38 Or for the case of M different NLO calculations, and N tree-level calculations: ∫ ∫ ∫ { ∫ ∑M−1 [ ] e b ⟨O⟩ = dϕ0 ··· O(S+mj) Bm + Bm + Bm+1→m −m,m+1 m=0 s ∫ [ ∫ ] ∫ ∫ } ∑M ∑M ∑M ∑N ↑ − e → − b → − − b → Bi m Bi m Bi+1→m Bi m i=m+1 s i=m+1 −i,i+1 i=m+1 s i=M+1 ∫ ∫ ∫ { [ ∫ ] ∫ } [ ] ∑N e b b b + dϕ0 ··· O(S+Mj) BM + BM − BM+1→M − Bi+1→M −M,M+1 −M i=M+1 ∫ ∫ ∫ { ∫ } ∑N ∑N b b + dϕ0 ··· O(S+nj) Bn − Bi→n n=M+1 i=n+1

The UNLOPS method

Add full NLO results: ∫ { ( ∫ ∫ [ ∫ ] ∫ ∫ ) ⟨O⟩ O e − e − b − ↑ − b = dϕ0 (S+0j) B0+ B0 B1→0 + B1→0 B1→0 B2→0 B2→0 ( s s ) −1,2 s ∫ [ ] [ ∫ ] ∫ ∫ } e b b b + O(S+1j) B1 + B1 − B2→1 + O(S+2j)B2 −1,2 −2

36 / 38 Or for the case of M different NLO calculations, and N tree-level calculations: ∫ ∫ ∫ { ∫ ∑M−1 [ ] e b ⟨O⟩ = dϕ0 ··· O(S+mj) Bm + Bm + Bm+1→m −m,m+1 m=0 s ∫ [ ∫ ] ∫ ∫ } ∑M ∑M ∑M ∑N ↑ − e → − b → − − b → Bi m Bi m Bi+1→m Bi m i=m+1 s i=m+1 −i,i+1 i=m+1 s i=M+1 ∫ ∫ ∫ { [ ∫ ] ∫ } [ ] ∑N e b b b + dϕ0 ··· O(S+Mj) BM + BM − BM+1→M − Bi+1→M −M,M+1 −M i=M+1 ∫ ∫ ∫ { ∫ } ∑N ∑N b b + dϕ0 ··· O(S+nj) Bn − Bi→n n=M+1 i=n+1

The UNLOPS method

Unitarise: ∫ { ( ∫ ∫ [ ∫ ] ∫ ∫ ) ⟨O⟩ O e − e − b − ↑ − b = dϕ0 (S+0j) B0+ B0 B1→0 + B1→0 B1→0 B2→0 B2→0 ( s s ) −1,2 s ∫ [ ] [ ∫ ] ∫ ∫ } e b b b + O(S+1j) B1 + B1 − B2→1 + O(S+2j)B2 −1,2 −2

36 / 38 Deriving an UN2LOPS matching We basically follow a “merging strategy”: • Pick calculations to combine (two MC@NLOs) with each other and with the PS resummation. • Remove kinematic overlaps between the two MC@NLOs by dividing the one-jet phase space. • Reweight one-jet MC@NLO (to make it exclusive ↔ want to describe hardest jet with this), O 1+1 remove all undesired terms at (αs ) and make sure that the whole thing is numerically stable.

Reweight subtractions with ΠS+0 to be able to group them with virtuals. • Add and subtract reweighted one-jet MC@NLO,(→ unitarise) to ensure inclusive zero-jet cross section is unchanged w.r.t. NLO. O 2 • Remove all terms up to (αs ) in the zero-jet contribution, replace by NNLO jet-vetoed cross section.

Work with Stefan Höche and Ye Li. 36 / 38 Start over again, now combining MC@NLO’s because those are resonably stable. 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. ⋄ eR Reweight NLO part B1 of MC@NLO 1 with “zero-jet Sudakov” factor. ⋄ O +1 Subtract erroneous (αs ) terms multiplying B1. ⋄ eR Reweight subtractions with ΠS+0 to be able to group them with B1 . ⋄ Put ρMS → ρc < 1GeV. (→ MC@NLO 0 becomes exclusive NLO) ⋄ ′ Unitarise by subtracting the processed MC@NLO 1 from the “zero-qT bin”. ⋄ 2 Remove all terms up to αs from the “zero-qT bin” and add the qT-vetoed NNLO 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.

UN2LOPS matching

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

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 resonably stable. 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. ⋄ eR Reweight NLO part B1 of MC@NLO 1 with “zero-jet Sudakov” factor. ⋄ O +1 Subtract erroneous (αs ) terms multiplying B1. ⋄ eR Reweight subtractions with ΠS+0 to be able to group them with B1 . ⋄ Put ρMS → ρc < 1GeV. (→ MC@NLO 0 becomes exclusive NLO) ⋄ ′ Unitarise by subtracting the processed MC@NLO 1 from the “zero-qT bin”. ⋄ 2 Remove all terms up to αs from the “zero-qT bin” and add the qT-vetoed NNLO 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 ∫ 2 O(UN LOPS) ¯ qT,cut = dΦ0 B0 (Φ0) O(Φ0) ∫ [ ( )] − 2 (1) (1) 2 + dΦ1 1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) O(Φ0) q ∫ T,cut ( ) 2 (1) (1) 2 F¯ + dΦ1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) 1(t1, O) q ∫ T,cut [ ] ∫ − 2 ˜ R 2 ˜ R F¯ + dΦ1 1 Π0(t1, µQ) B1 (Φ1) O(Φ0) + dΦ1Π0(t1, µQ) B1 (Φ1) 1(t1, O) q q ∫ T,cut [ ] ∫ T,cut − 2 R 2 R F + dΦ2 1 Π0(t1, µQ) H1 (Φ2) O(Φ0) + dΦ2 Π0(t1, µQ) H1 (Φ2) 2(t2, O) ∫qT,cut qT,cut E F + dΦ2 H1 (Φ2) 2(t2, O) qT,cut 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 ∫ 2 O(UN LOPS) ¯ qT,cut = dΦ0 B0 (Φ0) O(Φ0) ∫ [ ( )] − 2 (1) (1) 2 + dΦ1 1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) O(Φ0) q ∫ T,cut ( ) 2 (1) (1) 2 F¯ + dΦ1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) 1(t1, O) q ∫ T,cut [ ] ∫ − 2 ˜ R 2 ˜ R F¯ + dΦ1 1 Π0(t1, µQ) B1 (Φ1) O(Φ0) + dΦ1Π0(t1, µQ) B1 (Φ1) 1(t1, O) q q ∫ T,cut [ ] ∫ T,cut − 2 R 2 R F + dΦ2 1 Π0(t1, µQ) H1 (Φ2) O(Φ0) + dΦ2 Π0(t1, µQ) H1 (Φ2) 2(t2, O) ∫qT,cut qT,cut E F + dΦ2 H1 (Φ2) 2(t2, O) qT,cut 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 ∫ 2 O(UN LOPS) ¯ qT,cut = dΦ0 B0 (Φ0) O(Φ0) ∫ [ ( )] − 2 (1) (1) 2 + dΦ1 1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) O(Φ0) q ∫ T,cut ( ) 2 (1) (1) 2 F¯ + dΦ1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) 1(t1, O) q ∫ T,cut [ ] ∫ − 2 ˜ R 2 ˜ R F¯ + dΦ1 1 Π0(t1, µQ) B1 (Φ1) O(Φ0) + dΦ1Π0(t1, µQ) B1 (Φ1) 1(t1, O) q q ∫ T,cut [ ] ∫ T,cut − 2 R 2 R F + dΦ2 1 Π0(t1, µQ) H1 (Φ2) O(Φ0) + dΦ2 Π0(t1, µQ) H1 (Φ2) 2(t2, O) ∫qT,cut qT,cut E F + dΦ2 H1 (Φ2) 2(t2, O) qT,cut [ ] − 2 ˜ R Note: [1 Π0(t1, µQ)] B1 comes from using qT-vetoed cross sections. − 2 ˜ R Note: 1 Π0(t1, µQ) B1 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 ∫ 2 O(UN LOPS) ¯ qT,cut = dΦ0 B0 (Φ0) O(Φ0) ∫ [ ( )] − 2 (1) (1) 2 + dΦ1 1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) O(Φ0) q ∫ T,cut ( ) 2 (1) (1) 2 F¯ + dΦ1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) 1(t1, O) q ∫ T,cut [ ] ∫ − 2 ˜ R 2 ˜ R F¯ + dΦ1 1 Π0(t1, µQ) B1 (Φ1) O(Φ0) + dΦ1Π0(t1, µQ) B1 (Φ1) 1(t1, O) q q ∫ T,cut [ ] ∫ T,cut − 2 R 2 R F + dΦ2 1 Π0(t1, µQ) H1 (Φ2) O(Φ0) + dΦ2 Π0(t1, µQ) H1 (Φ2) 2(t2, O) ∫qT,cut qT,cut E F + dΦ2 H1 (Φ2) 2(t2, O) q[T,cut ] − 2 ˜ R 1 Π0(t1, µQ) B1 comes from using qT-vetoed cross sections. R ¯ qT,cut ˜ R E B0 + B1 + H1 + H1 = BNNLO Other terms drop out in inclusive observables.

else start PS off one-jet state. 35 / 38 UN2LOPS matching ∫ 2 O(UN LOPS) ¯ qT,cut = dΦ0 B0 (Φ0) O(Φ0) ∫ [ ( )] − 2 (1) (1) 2 + dΦ1 1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) O(Φ0) q ∫ T,cut ( ) 2 (1) (1) 2 F¯ + dΦ1 Π0(t1, µQ) w1(Φ1) + w1 (Φ1) + Π0 (t1, µQ) B1(Φ1) 1(t1, O) q ∫ T,cut [ ] ∫ − 2 ˜ R 2 ˜ R F¯ + dΦ1 1 Π0(t1, µQ) B1 (Φ1) O(Φ0) + dΦ1Π0(t1, µQ) B1 (Φ1) 1(t1, O) q q ∫ T,cut [ ] ∫ T,cut − 2 R 2 R F + dΦ2 1 Π0(t1, µQ) H1 (Φ2) O(Φ0) + dΦ2 Π0(t1, µQ) H1 (Φ2) 2(t2, O) ∫qT,cut qT,cut E F + dΦ2 H1 (Φ2) 2(t2, O) q[T,cut ] − 2 ˜ R 1 Π0(t1, µQ) B1 comes from using qT-vetoed cross sections. 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 UN2LOPS (Higgs production)

12 10

[pb] s = 14 TeV s = 14 TeV H NNLO 10 NLO Sherpa MC /dy Sherpa MC [pb/GeV] σ HNNLO d

T,H 1 µ 8 mH /2< <2mH NLO µR/F /dp m /2< <2m NNLO H R/F H σ 6 d 10-1 4 NNLO NLO HNNLO m /2< µ <2m NLO 2 H R/F H -2 m /2< µ <2m NNLO 10 H R/F H 1.04 1.04 1.02 1.02 1 1 0.98 0.98 0.96 0.96 -4 -3 -2 -1 0 1 2 3 4 0 20 40 60 80 100 120 140 160 180 200 Ratio to NNLO Ratio to NNLO y p [GeV] H T,H

Rapidity and p⊥ of the Higgs-boson, comparing SHERPA-NNLO and HNNLO. 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 find an ideal overestimate, we can use a weighted shower instead: • For each phase space point, store the full splitting probability f, the differential overestimate g, and an auxiliary weight = sgn ( ) · . h f g ( ) • f Pick/discard splitting as usual according to abs g . If the emission… h g−f …was picked, multiply wpicked = g h−f to event weight h …was discarded, multiply wdiscarded = g to event weight

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

f f → and h → (ka) · h g (ka) · g with an additional factor k ≳ 1, and continue as before. 37 / 38 DIRE Anomalous Dimensions The anomalous dimensions ∫ 1 2 N 2 γab(N, κ ) = z Pab(z, κ ) are given by (1) 0 C (2N + 3) γ (N, κ2) = 2C Γ(N, κ2) − F + γ qq F (N + 1)(N + 2) q C (N + 3) γ (N, κ2) = 2C K(N, κ2) − F gq F (N + 1)(N + 2) 2C (N + 3) 2C γ (N, κ2) = 2C Γ(N, κ2) + 2C K(N, κ2) − A − A + γ gg A A (N + 1)(N + 2) N + 3 g T N 2T γ (N, κ2) = − R + R qg (N + 1)(N + 2) N + 3 (2) where ( ) 3 N+3 N+4 −2 2 2 3F2 1, 1, ; , ; −κ 1 + κ 2Γ(N, κ2) = 2 2 2 − ln , (N + 1)(N + 2) κ2 κ2 ( ) (3) 2 F 1, N+2 ; N+4 ; −κ−2 ( , κ2) = 2 1 2 2 . 2K N 2 (N + 2) κ 38 / 38