NNLO QCD predictions for the LHC with antenna subtraction

João Pires (MPI Munich)

Les Houches Workshop, Physics at the TeV colliders June 7, 2017

Anatomy of an NNLO calculation

Assume all matrix elements are available

• Tree level matrix elements (RR) 2➝ n+2

• One-loop matrix elements (RV) 2➝ n+1

• Two loop matrix elements (VV) 2➝ n

• double-unresolved

• single-unresolved

• single-unresolved

• 1/𝜀2 ; 1/𝜀

• 1/𝜀4 ; 1/𝜀3 ; 1/𝜀2 ; 1/𝜀 ;

} Form NLO correction to 2->n+1

Infrared singularities: real radiationNLO

• single collinear: pa // pb

• single soft: Ea→0

NNLO

• Triple collinear: pa // pb // pc

• Double single collinear: pa // pb ; pc // pd

• Soft/collinear: Ea→0 , pb // pc

• Double soft: Ea→0 , Eb→0

• One-loop virtual correction to NLO singularities

NNLO antenna subtraction

• mimic RR,RV in unresolved limits

• analytically cancel the poles in RV and VV matrix elements

• NNLO cross section with each line finite and integrable in d=4 dimensions

Implementation in parton-level event generator

• Generate particle momenta for (n), (n+1), (n+2)

• Reconstruct observable

• Weight with squared matrix elements

• Subtract/add real radiation singularities

Colour ordering• QCD amplitudes in colour basis

All gluon

Quark pair plus gluons

• Real radiation infrared singularities only between colour adjacent partons

• Well defined patterns from colour-connections

Antenna subtraction at NNLO (RR)

• (a) one unresolved parton → three parton antenna function Xijk

• (b) two colour-connected unresolved partons → four parton antenna function Xijkl

• (c) two almost colour connected unresolved partons → strongly ordered product of non-independent three parton antenna functions XijkXKlm, XklmXijK

• (d) two colour unconnected unresolved partons → product of independent three parton antenna functions XijkXnop

• (e) subtracts large angle soft radiation → soft factor Sajc

d�SNNLO = d�S,a

NNLO + d�S,bNNLO + d�S,c

NNLO + d�S,dNNLO + d�S,e

NNLO

i j k Il L

i j k l Im MK

j k I Ki n o p N P

Two colour connected partons

A04(iq, jg, kg, lq̄)

Sijkl

Pqgg!Q(w, x, y)

Sq;ggq̄Pqg!Q(z)

Pqg!Q(z)Pq̄g!Q̄(y)

pi//pj//pk

Ek ! 0, pi//pj

pi //p

j ; pk //p

l

smoothly interpolates colour connected unresolved limits

Ej,Ek

! 0

• Phase space mapping (i,j,k,l)→(I,L)

pI = xpi + r1pj + r2pk + zpl

pL = (1� x)pi + (1� r1)pj + (1� r2)pk + (1� z)pl

d�m+2(p1, . . . , pm+2) = d�m(p1, . . . , pI , pL, . . . , pm+2) · d�Xijkl(pi, pj , pk, pl; pI , pL)

pi//pj//pk

Ek ! 0, pi//pj

pi //p

j ; pk //p

l

smoothly interpolates colour connected unresolved limits

pI = pipL = pl

pI = pi + pj + pkpL = pl

Ej,Ek

! 0

{pi, pj , pk, pl} ! {pI , pL}

pI = pi + pjpL = pl

pI = pi + pjpL = pk + pl

Antenna functions and types• all antennae can be derived from physical matrix elements in QCD

• colour ordered pair of hard partons (radiators) with radiation in between

• hard quark-antiquark pair

• hard quark-gluon pair

• hard gluon-gluon pair

• three parton antennae ➝ one unresolved parton

• four-parton antennae ➝ two unresolved partons

• can be at tree level or one loop

• can be massless or massive

• all have three antenna types

• final-final antenna

• initial-final antenna

• initial-initial antenna

X03 (i, j, k)

X13 (i, j, k)

X04 (i, j, k, l)

Angular averaging• Antenna functions are scalar objects → do not subtract angular correlations in gluon

splitting

• Angular correlations vanish after integration over the azimuthal angle

• Make fully local subtraction by combining phase space points related to each other by a 90 degree rotation of the system of unresolved partons

Antenna subtraction at work

Double unresolved emission

• Generate phase space trajectories that approach singular region of the phase space

• Infrared behaviour of subtraction term mimics the behaviour of the matrix element

0

1000

2000

3000

4000

5000

6000

0.99997 0.99998 0.99999 1 1.00001 1.00002 1.00003

# ev

ents

R

double soft limit for gg→gggg

#PS points=10000x=(s-sij)/s

x=10-4

x=10-5

x=10-6

1487 outside the plot 317 outside the plot 59 outside the plot

0

500

1000

1500

2000

2500

3000

3500

0.99996 0.99998 1 1.00002 1.00004

# ev

ents

R

Triple collinear limit for gg→gggg

#PS points=10000x=sijk/s

x=10-7

x=10-8

x=10-9

1419 outside the plot77 outside the plot17 outside the plot

1 2

ijk

l

1 2

i

j

l

k

Double virtual antenna contribution

• Integrated double unresolved emission of RR process tree level double soft function

• Integrated iterated NLO emissions of RR process

• Integrated single unresolved emission from RV process ∝ tree level single soft function

• Integrated single unresolved emission of RV process one loop single soft function

Double virtual antenna contribution

• integrated operators in analytic one-to-one correspondence with operator of Catani

J(2,1)2

(I1)2

Pros and cons Antenna subtraction

• local method with phase space averaging → good control on the numerical accuracy of the final result, RR, RV, VV separately finite

• analytic IR pole cancellation at NNLO → good control on the correctness of the pole cancellation

• double precision

• universal method works for general jet multiplicity → no additional building blocks needed

• pp→jj,Hj,Zj @ NNLO

• subtraction terms for a fixed colour structure reusable

• involves many mappings/subtraction terms as expected for a local method → needs caching system to store mappings

NNLOJET parton level generator

• parton level generator based on antenna subtraction to compute fully differential cross sections at NNLO in QCD

• all antenna functions implemented with a common syntax and structure

• allows testing of matrix element/subtraction term cancellations with singular phase space trajectories

• allows explicit 1/e pole cancelation with one and two loop matrix elements with FORM

• interface to phase space generation, Monte Carlo integration and fully flexible histograming

• interface to applfast nnlo tables in development

[X. Chen, J. Cruz-Martinez, J.Currie, A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, A. Huss, M.Jaquier, T.Morgan, J. Niehues, JP]

M. Sutton and K. Rabbertz tomorrow afternoon

NNLOJET parton level generator

• list of processes available in NNLOJET

• pp->H,W,Z

• pp -> H+jet

• pp -> Z+jet

• pp -> 2 jets

• ep -> 2jets

• …

[X. Chen, J. Cruz-Martinez, J.Currie, A. Gehrmann-De Ridder, T. Gehrmann, N. Glover, A. Huss, M.Jaquier, T.Morgan, J. Niehues, JP]

Single jet inclusive cross section

ATLAS jets

Theory setup

• NNPDF3.0_nnlo

• anti-kT jet algorithm

• µR=µF={pT1, pT}

• vary scales by factors of 2 and 1/2

Comparison to data

• ATLAS 7 TeV 4.5 fb-1

• R=0.4

0.6 0.8

1 1.2 1.4

NNLOJET

NLO

Rat

io to

dat

a

|yj| < 0.5

ATLAS, 7 TeV, anti-kt jets, R=0.4, NNPDF3.0 µ=pT1µ=pT

0.6 0.8

1 1.2 1.4

0.5 < |yj| < 1.0

0.6 0.8

1 1.2 1.4

1.0 < |yj| < 1.5

0.6 0.8

1 1.2 1.4

1.5 < |yj| < 2.0

0.6 0.8

1 1.2 1.4

2.0 < |yj| < 2.5

0.6 0.8

1 1.2 1.4

100 1000

2.5 < |yj| < 3.0

pT (GeV)

Ratio to NLO

• asymmetric scale band variation

• underestimated at small pT due to turn over of the NLO coefficient

• 20% uncertainty for central high pT jets rising to 40% for forward jets

Comparison to data

• non perturbative effects < 2% effect [JHEP 1509, 141 (2015)]

• data favours the pT1 scale choice at NLO

0.6 0.8

1 1.2 1.4

NNLOJET

NN

LO R

atio

to d

ata

|yj| < 0.5

ATLAS, 7 TeV, anti-kt jets, R=0.4, NNPDF3.0 µ=pT1µ=pT

0.6 0.8

1 1.2 1.4

0.5 < |yj| < 1.0

0.6 0.8

1 1.2 1.4

1.0 < |yj| < 1.5

0.6 0.8

1 1.2 1.4

1.5 < |yj| < 2.0

0.6 0.8

1 1.2 1.4

2.0 < |yj| < 2.5

0.6 0.8

1 1.2 1.4

100 1000

2.5 < |yj| < 3.0

pT (GeV)

Ratio to NNLO

• symmetric scale band variation

• pT1!=pT effects enlarged at NNLO

• 10% scale uncertainty at low pT and percent level scale uncertainty at high pT

Comparison to data

• data favours the pT scale choice at NNLO

• NNLO effects around +10% at low pT and small at high pT

• Shape of NNLO/NLO k-factor is getting steeper going to the forward rapidity slices

µR = µF = pT1µR = µF = pT

0.8 0.9

1 1.1 1.2

NNLOJET

K fa

ctor

|yj| < 0.5

ATLAS, 7 TeV, anti-kt jets, R=0.4

NLO/LONNLO/LONNLO/NLO

0.8 0.9

1 1.1 1.2

0.5 < |yj| < 1.0

0.8 0.9

1 1.1 1.2

1.0 < |yj| < 1.5

0.8 0.9

1 1.1 1.2

1.5 < |yj| < 2.0

0.8 0.9

1 1.1 1.2

2.0 < |yj| < 2.5

0.8 0.9

1 1.1 1.2

100 200 500 1000

2.5 < |yj| < 3.0

NNPDF3.0

pT (GeV)

0.8 0.9

1 1.1 1.2

NNLOJET

K fa

ctor

|yj| < 0.5

ATLAS, 7 TeV, anti-kt jets, R=0.4

NLO.pt/LONNLO.pt/LONNLO.pt/NLO

0.8 0.9

1 1.1 1.2

0.5 < |yj| < 1.0

0.8 0.9

1 1.1 1.2

1.0 < |yj| < 1.5

0.8 0.9

1 1.1 1.2

1.5 < |yj| < 2.0

0.8 0.9

1 1.1 1.2

2.0 < |yj| < 2.5

0.8 0.9

1 1.1 1.2

100 200 500 1000

2.5 < |yj| < 3.0

NNPDF3.0

pT (GeV)

• NNLO effects around -10% at low pT and small at high pT

• Shape of NNLO/NLO k-factor is getting flatter going to the forward rapidity slices

• Scale choice has a potential interplay with consistent fit of jet data in PDF’s for all rapidity slices

µR = µF = pT1µR = µF = pT

• Different behaviour in the NNLO scale variation

• Scale uncertainty much smaller than the difference between the two scale choices

• Difference in the prediction with either scale choice is beyond the scale variation uncertainty

• Lack of a theoretically well motivated preference motivates further study of this issue

Scale variation

Dijet inclusive cross section

ATLAS jetsTheory setup

• MMHT2014 nnlo

• anti-kT jet algorithm

• pT1>100 GeV; pT2>50 GeV;

• |yj1| , |yj2| < 3.0

• µR=µF={mjj, <pT>}

• vary scales by factors of 2 and 1/2

Comparison to data

• ATLAS 7 TeV 4.5 fb-1

• R=0.4

(GeV)jjm

310

ratio

to d

ata

0.5

1

1.5

210×5 310×2 310×3 310×5

⟩T

p⟨=µ (GeV)jjm

310

ratio

to d

ata

0.5

1

1.5

jets, R=0.4, 0.0 < |y*| < 0.5TATLAS 7 TeV, anti-kNNLOJET

jj=mµ

LO NLO NNLO

(GeV)jjm

310

ratio

to d

ata

0.5

1

1.5

210×5 310×2 310×3 310×5

⟩T

p⟨=µ (GeV)jjm

310

ratio

to d

ata

0.5

1

1.5

jets, R=0.4, 1.5 < |y*| < 2.0TATLAS 7 TeV, anti-kNNLOJET

jj=mµ

LO NLO NNLO

0.0 < |y⇤| < 0.5 1.5 < |y⇤| < 2.0

y⇤ =1

2(yj1 � yj2)m2

jj = (pj1 + pj2)2

• Largely overlapping scale bands at small y* with either scale choice

• At large y* we observe with 𝜇=<PT> large negative NLO corrections, non-overlapping scale bands and residual NLO,NNLO scale uncertainty of ~100%,~20%

• Good theoretical motivation to use 𝜇=mjj as central scale choice

(GeV)jjm310

ratio

1

1.5

2

210×5 310×2 310×3 310×5

2.5 < |y*| < 3.0 (GeV)jjm310

ratio

1

1.5

22.0 < |y*| < 2.5 (GeV)jjm

310

ratio

1

1.5

21.5 < |y*| < 2.0 (GeV)jjm

310

ratio

1

1.5

21.0 < |y*| < 1.5 (GeV)jjm

310

ratio

1

1.5

20.5 < |y*| < 1.0 (GeV)jjm

310

ratio

1

1.5

2

)jj

=mµ jets, R=0.4 (TATLAS 7 TeV, anti-kNNLOJET

0.0 < |y*| < 0.5

NLO/LONNLO/LONNLO/NLO

(GeV)jjm310

ratio

to N

LO

0.6

0.8

1

1.2

1.4

210×5 310×2 310×3 310×5

2.5 < |y*| < 3.0 (GeV)jjm310

ratio

to N

LO

0.6

0.8

1

1.2

1.4 2.0 < |y*| < 2.5 (GeV)jjm310

ratio

to N

LO

0.6

0.8

1

1.2

1.4 1.5 < |y*| < 2.0 (GeV)jjm310

ratio

to N

LO

0.6

0.8

1

1.2

1.4 1.0 < |y*| < 1.5 (GeV)jjm310

ratio

to N

LO

0.6

0.8

1

1.2

1.4 0.5 < |y*| < 1.0 (GeV)jjm310

ratio

to N

LO

0.6

0.8

1

1.2

1.4

)jj

=mµ jets, R=0.4 (TATLAS 7 TeV, anti-kNNLOJET

0.0 < |y*| < 0.5

NLONNLONNLOxEW

• Excellent convergence of the perturbative expansion; NNLO/NLO < 10% and flat

• Improved description of the dijet data at NNLO

jj/mRµ1

/d|y

*| (fb

/GeV

)jj

/dm

σ2 d

4

6

8

10

12

14

310×

0.2 0.5 2.0 4.0

jets, R=0.4T

ATLAS 7 TeV, anti-kNNLOJET

< 950 GeV 1.5 < |y*| < 2.0jj850 GeV < m

LO

NLO

NNLO

=0.5jj/mF

µ=1.0jj/m

Fµ

=2.0jj/mF

µ

jj/mRµ1

/d|y

*| (fb

/GeV

)jj

/dm

σ2 d

10

15

20

25

30

35

40

45

50

0.2 0.5 2.0 4.0

jets, R=0.4T

ATLAS 7 TeV, anti-kNNLOJET

< 2550 GeV 2.5 < |y*| < 3.0jj2120 GeV < m

LO

NLO

NNLO

=0.5jj/mF

µ=1.0jj/m

Fµ

=2.0jj/mF

µ

• Overlapping NLO and NNLO scale bands

• Significant reduction in scale dependence of the prediction at NNLO

• Residual scale uncertainty <5% smaller than experimental uncertainty on the observable

Scale variation

ConclusionsSubstancial progress in NNLO calculations in past couple of years

• several different approaches for isolating IR singularities

• several new calculations available

Antenna subtraction

• local IR phase space subtraction scheme with analytic pole cancellation

• RR, RV, VV contributions separately finite and integrable in d=4

• formalism implemented in a fully flexible parton level generator

• new processes can be added using the existing common syntax structures

• distribution of results via applfast interface (to APPLGRID and fastNLO)

M. Sutton and K. Rabbertz tomorrow afternoon

BACKUP

Single jet inclusive scale choicetwo widely used scale choices:

• µR=µF={pT1, pT}

• leading jet pT in the event pT1

• individual jet pT

• high pT jets are back to back ⇒ pT —> pT1

Single jet inclusive scale choicetwo widely used scale choices:

• µR=µF={pT1, pT}

• leading jet pT in the event pT1

• individual jet pT

• high pT jets are back to back ⇒ pT —> pT1

• pT!=pT1 for:

• 3jet events

• 3rd jet outside fiducial jet cuts

⇒ with pT choice the real emission event with different R gives rise to a different scale ⇒ larger R ⇒ harder scale ⇒ pT —> pT1

• at NLO the pT1 scale choice generates the same hard scale for the event independent of the value of R