NNLO Corrections Using Antenna Subtraction & Applications

NNLO Corrections Using Antenna Subtraction& Applications

Alexander Huss

Subtracting Infrared SingularitiesBeyond NLO

Higgs Centre for Theoretical PhysicsEdinburgh — April 10th 2018

This talk

Part 1. Antenna Subtraction Formalism

Part 2. Example: Drell–Yan

Part 3. Going beyond...

↪→ Transverse Momentum Spectrum

↪→ Projection-to-Born Method

Anatomy of an NNLO calculation

σNNLO =

∫ΦZ+3

(

dσRRNNLO

− dσSNNLO

)

I single-unresolvedI double-unresolved

+

∫ΦZ+2

(

dσRVNNLO

− dσTNNLO

)

I single-unresolvedI 1/ε

2 , 1/ε

+

∫ΦZ+1

(

dσVVNNLO

− dσUNNLO

)

I 1/ε4 , 1/ε3 , 1/ε2 , 1/ε

∑�nite (Kinoshita–Lee–Nauenberg & factorization)

−0

Non-trivial cancellation of infrared singularities

Di�erent methods:I Antenna subtraction . . . . . . . . . . . . . . . . . . . . . . .

[Gehrmann–De Ridder, Gehrmann, Glover ’05]I CoLorFul subtraction . . . . . . . . . . . . . . . . . . . . . .

[Del Duca, Somogyi, Trocsanyi ’05]I qT subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[Catani, Grazzini ’07]I Sector-improved residue subtraction . . . . . .

[Czakon ’10], [Boughezal, Melnikov, Petriello ’11]I N-jettiness subtraction . . . . . . . . . . . . . . . . . . .

[Gaunt, Stahlhofen, Tackmann, Walsh ’15][Boughezal, Focke, Liu, Petriello ’15]

I Projection-to-Born . . . . . . . . . . . . . . . . . . . . . . . .[Cacciari, et al. ’15]

I Nested so�-collinear subtraction . . . . . . . . .[Caola, Melnikov, Rontsch ’17]

. . .

Anatomy of an NNLO calculation

σNNLO =

∫ΦZ+3

(

dσRRNNLO

− dσSNNLO

)

+

∫ΦZ+2

(

dσRVNNLO

− dσTNNLO

)

+

∫ΦZ+1

(

dσVVNNLO

− dσUNNLO

)

∑�nite

−0

Approaches: subtraction, slicing









. . .

NNLO using Antenna

σNNLO =

∫ΦZ+3

(dσRR

NNLO − dσSNNLO

)

+

∫ΦZ+2

(dσRV

NNLO − dσTNNLO

)I dσS

NNLO, dσTNNLO:

mimic dσRRNNLO, dσRV

NNLO

in unresolved limits

I dσTNNLO, dσU

NNLO:analytic cancellation ofpoles in dσRV

NNLO, dσVVNNLO+

∫ΦZ+1

(dσVV

NNLO − dσUNNLO

)

∑�nite −0

⇒ each line suitable for numerical evaluation in D = 4









. . .

Antenna factorisation

I antenna formalism operates on colour-ordered amplitudesI exploit universal factorisation properties in IR limits

|A0m+1(. . . , i, j, k, . . .)|2 j unresolved−−−−−−−→ X0

3 (i, j, k) |A0m(. . . , I, K, . . .)|2︸︷︷︸

partial amplitude︸︷︷︸

antenna function+ mapping

{pi, pj , pk} → {pI , pK}

︸︷︷︸reduced ME

I captures multiple limits∗ and smoothly interpolates between them

limit X03 (i, j, k) mapping

pj → 02sik

sijsjkpI → pi, pK → pk

pj ‖ pi1

sijPij(z) pI → (pi + pj), pK → pk

pj ‖ pk1

sjkPkj(z) pI → pi, pK → (pj + pk)

∗ c.f. dipoles: X03 (i, j, k) ∼ Dij,k +Dkj,i

Antenna subtraction — building blocks

I X(. . .) based on physical matrix elements X =

qq︷︸︸︷A, B, C ,

qg︷︸︸︷D, E,

gg︷︸︸︷F, G, H

X03 (i, j, k) =

|A03(i, j, k)|2

|A02(I , K)|2

, X04 (i, j, k, l) =

|A04(i, j, k, l)|2

|A02(I , L)|2

,

X13 (i, j, k) =

|A13(i, j, k)|2

|A02(I , K)|2

−X03 (i, j, k)

|A12(I , K)|2

|A02(I , K)|2

,

A03(iq, jg, kq) =

∣∣∣∣∣ γ∗ iq

kq

jg

∣∣∣∣∣2 / ∣∣∣∣∣ γ∗ Iq

Kq

∣∣∣∣∣2

I integrating the antennae ←→ phase-space factorization

dΦm+1(. . . , pi, pj , pk, . . .)

= dΦm(. . . , pI , pK , . . .)dΦXijk (pi, pj , pk; pI + pK)

X 0,13 (i, j, k) =

∫dΦXijkX

0,13 (i, j, k), X 0

4 (i, j, k, l) =

∫dΦXijklX

04 (i, j, k, l)

Antenna subtraction — building blocks

I X(. . .) based on physical matrix elements X =

qq︷︸︸︷A, B, C ,

qg︷︸︸︷D, E,

gg︷︸︸︷F, G, H

X03 (i, j, k) =

|A03(i, j, k)|2

|A02(I , K)|2

, X04 (i, j, k, l) =

|A04(i, j, k, l)|2

|A02(I , L)|2

,

X13 (i, j, k) =

|A13(i, j, k)|2

|A02(I , K)|2

−X03 (i, j, k)

|A12(I , K)|2

|A02(I , K)|2

,

A03(iq, jg, kq) =

∣∣∣∣∣ γ∗ iq

kq

jg

∣∣∣∣∣2 / ∣∣∣∣∣ γ∗ Iq

Kq

∣∣∣∣∣2

I integrating the antennae ←→ phase-space factorization

dΦm+1(. . . , pi, pj , pk, . . .)

= dΦm(. . . , pI , pK , . . .)dΦXijk (pi, pj , pk; pI + pK)

X 0,13 (i, j, k) =

∫dΦXijkX

0,13 (i, j, k), X 0

4 (i, j, k, l) =

∫dΦXijklX

04 (i, j, k, l)

All building blocks known!

X03 , X0

4 , X13 and integrated

counterparts X 03 , X 0

4 , X 13

∀ con�gurations relevant at hadron colliders↪→ �nal–�nal, initial–�nal, initial–initial

Antenna subtraction

NLOI real:

dσS,NLO

∼ dΦm+1(. . . , pi, pj , pk, . . .) X03 (i, j, k) |A0

m(. . . , I, K, . . .)|2 J (pi)

∼ dΦm(. . . , pI , pK , . . .) dΦXijk X03 (i, j, k)︸︷︷︸

integrate

|A0m(. . . , I, K, . . .)|2 J (pi)

I virtual:

dσT,NLO ∼ −dΦm X 03 (sij) |A0

m(. . . , i, j, . . .)|2 J (pi)

NNLOI double real: dσS ∼ X0

3 |A0m+1|2, X0

4 |A0m|2, X0

3 X03 |A0

m|2I real–virtual: dσT ∼ X 0

3 |A0m+1|2, X0

3 |A1m|2, X1

3 |A0m|2

I double virtual: dσU = (collect rest) ∼ X |A0,1m |2

What about those angular terms?I Antenna subtraction: Xl

n |Am|2 ↔ spin averaged!I angular terms in gluon splittings:

Pg→qq =2

sij

[−gµν + 4z(1− z) k

µ⊥k

ν⊥

k2⊥

]↪→ subtraction non-local in these limits!↪→ vanish upon azimuthal-angle (ϕ) average (⇒ do not enter X )

sol. 1: supplement angular terms in the subtractionsol. 2: exploit ϕ dependence & average in the phase space

A∗µkµ⊥k

ν⊥

k2⊥Aν ∼ cos(2ϕ+ ϕ0)

⇒ add ϕ & (ϕ+π/2)!

He.¥#÷g#|.~r −→ PSgen. −→

[{pi, pj , . . .}{p′i, p′j , . . .}

](i‖j)−−−→

[ {pϕi , pϕj , . . .}{pϕ+π/2i , p

ϕ+π/2j , . . .}

]

Checks of the calculation — unresolved limits

Double-real levelI dσS → dσRR

(single- & double-unresolved)

bin the ratio: dσS/dσRR unresolved−−−−−−→ 1

q q → Z + g3 g4 g5 @ tree(g3 so� & g4 ‖ q)

0

200

400

600

800

1000

0.9999 0.99992 0.99994 0.99996 0.99998 1 1.00002 1.00004 1.00006 1.00008 1.0001

2 outside the plot ( 0, 0)



#phase space points = 1000

Soft collinear - 3, 2/4

x=10-7

x=10-8

x=10-9

(approach limit: xi = 10−7, 10

−8, 10

−9)

Real–virtual levelI dσT → dσRV

(single-unresolved)

bin the ratio: dσT/dσRV unresolved−−−−−−→ 1

q q → Z + g3 g4 @ 1-loop(g4 ‖ q)

0

200

400

600

800

1000

0.999 0.9992 0.9994 0.9996 0.9998 1 1.0002 1.0004 1.0006 1.0008 1.001

43 outside the plot

0 outside the plot

0 outside the plot


1/4 collineare0

x=10-8

x=10-9

x=10-10

(approach limit: xi = 10−8, 10

−9, 10

−10)

Checks of the calculation — pole cancellation

DimReg: D = 4− 2ε

Double-virtual levelI Poles

(dσVV − dσU

)= 0

2-loop, (1-loop)2 ; 1/ε4, . . ., 1/ε

Real–virtual levelI Poles

(dσRV − dσT

)= 0

1-loop ; 1/ε2, 1/ε

pole coe�cient: dσT/dσRV ≡ 1

q q → Z + g g @ 1-loop(1/ε coe�cient)

0

200

400

600

800

1000

0.99999 1.00000 1.00000 1.00001 1.00001

0 outside the plot

0 outside the plot

0 outside the plot


normale1

x=10-7

x=10-8

x=10-9

NNLOjet

X. Chen, J. Cruz-Martinez, J. Currie, R. Gauld, A. Gehrmann-De Ridder,T. Gehrmann, E.W.N. Glover, AH, I. Majer, T. Morgan, J. Niehues, J. Pires,

D. Walker

Common framework for NNLO corrections using Antenna Subtraction

I parton-level event generatorI based on antenna subtractionI test & validation frameworkI APPLfast-NNLO interface

(Work in progress)[Britzger, Gwenlan, AH, Morgan, Sutton, Rabbertz]

I . . .

Processes:I pp→ V → ¯ + 0, 1 jetsI pp→ H + 0, 1 jetsI pp→ H + 2 jets (VBF)I pp→ dijetsI ep→ 1, 2 jetsI e+e− → 3 jetsI . . .

Colour Decomposition (Example : qoi→gg£ )

1 • 3 n •3 1 3

(1-931-94)(a) I

;mugn;Hamden.If:Tug

,as

z • 2 • 4 2 •

^Kellyn-•peue41welllbs¥52.3sentIke; (T94T")2 2 • 3 2 a Cncz

a

'

TIMEY'

'

I.E..gg T;afaa39~fF3T4-T4Tay2 2

3- Cncz

4

.

⇒ µgq→ggz = (T "T9)aaAf(1%3g,4g,2g,Z) ←"castes

"

+ ( Ta' -193¥.

A ,( 1g,4g,3g,2g,Z ) ← >

" (b)- (c)"

Dtell - Yan : 99 → gluons channel

drRRnrtq@il1q3g.4gRgTI2tIAiAq4g3gi2gH2-tnelATillq3gi4gRgMdrRVrr1Ajl1qi3gi2gH2-Nte1AIHq3gRgjpdrWn1AIHqRoiH2-tNe1AI2C1q2A12xt2RefAIl4.3g.4g.2aFAIHa.4g.3a2aiH-IfIHg.3g.4a2gtt-1Aillq3g.4g.2DT-IAitlq4gi3gi2aiH2AIHgi3g.4a2gt-AIllq.3g.4g.L

, ) + AIH , ,4g,3g,2 , )

Subtraction Term for 1A; Hq,3g ,4g ,2h12 IRI

td8Hq3g

, 4g) little , F4)g , # 12

+ dikixg , } ) |qu , ,, #

, a ,p}±

Ns¥t¥¥ertm ⇒ single:}

Subtraction Term for IAIHQ,3g ,4g ,2gH' IRI

tdJHq3g

, 4g) little ,F4)g , # 12

+ dikixg , } ) |qq ,#

, a ,p}±%£÷¥ertm

⇒ single:3 ,

+ Aida , 3g,4g,2⇒ tdicpqzgyz }double :3&4

Subtraction Term for

1AIH%3g,4g, #

TIRItdztlq,3g,4g) last ,F4)g,2gtt

+

diaiy.gs/a;a..T#.a,pljNs?itfIEetm=siinn8Yei

:!-

singularitiesdouble : 324

double : 384

+ Aida,3±4±⇒HiMai⇒t} IndemnitiesHindle"

?u )double : 324

- dillq ,3g,4g ) ttsltq ,1544,25 ) IAIHTTIAI

'

single :3

- d3°l4i4q3g) Azohg ,#g. 2.) hang

,Egyp} single '. 4

DTNE

Subtraction Term for 1944, 3g, 2h12 IRI

- [ £ Dils . ) + 12 Dils " ) ]1A;(n , ,3g,⇒p}±Nstftr?¥eertm⇒ Poles

Subtraction Term for 1AlHq,3g, 2h12 IRI

- [ £ Dils . ) + ED;ls. } ) ]1A;(n , ,3g,2⇒p}±Nstftr?¥eetm⇒ Poles

+ Asha ,3gR .) IAIHT ,I⇒12

+ Ailing ,2⇒ hang ,Iap}YeIpeYxYt;D single :3

Subtraction Term for

1AlH%3g,2gH

' K

. l ;D :c %) + £4.419143

.si#Mjsgp!g&gEasnEYetmIsPn0gY:3

+ Asha ,3gR .) IAIHTRIH"

single :3

+ AIH,,3a2⇒1A;ai,Iat}t%¥x¥¥d

ssipngtuiiaasities Poles

+ I Edits . )+£Ds% " ) ] Aina ,3g, ⇒ Hickey )(Ying!+ MF

-

DONE

Subtraction Term for htitlg, 2h12 #

} from AI @ RR- Ails . ) 1A :( 4,4712

g- Asian Hitler 't

laeyapgxxaggnne

,@ rv } " " s

- Ask . ) /Az°aaR⇒t )+ MF

-

DONE

Transverse MomentumSpectrum

p

p

g

qZ/γ

`−

`+

recoil

pZT 6= 0

Inclusive pT spectrum from X + jet

p

p

g

qZ/γ

`−

`+

recoil

pZT 6= 0

p p → X + recoil

I fully inclusive in QCD emissionsI require recoil: pXT > pXT, cut

⇒ can use X + jet calculation

X + jet@ NNLOI H + jet . . . . . . . . . . . . . . . . . . . . . . . (Antenna, N-jettiness, Sector-improved R.S.)I W + jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Antenna, N-jettiness)I Z + jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Antenna, N-jettiness)I γ + jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (N-jettiness)

; validation & opportunity for benchmarks

The pT spectrum

I �xed-order prediction diverges for pT → 0

I large logarithms: lnk(pT/M)/p2T ; all-order resummation needed!

⇒ matching: dσmatched = dσf.o. + dσres. − dσres.|exp.

Compare the logs — �xed-order vs. resummation

-6

-4

-2

0

2

4

100

101

102

Non-Singular [pb]

pTH[GeV]

-120

-80

-40

0

40

80

120

pTH

> 0.7 GeV

PDF4LHC15 nnlo mc

µR=µF=1/2⋅mH

NNLOJET Preliminary p p → H + ≥ 0 jet mH=125 GeV √s = 13 TeV

pTHdσ/dpTH

[pb]

LO FONLO only FO

NNLO only FOLO Singular

NLO only SingularNNLO only Singular

PRELIMINARY

[Chen,Gehrmann,Glover,AH,Li,Neill,Schulze,Stewart,Zhu]

I excellent agreement within stat. errors ∼ 1%

I predictions down to pHT = 0.7 GeV

I important cross check ; matching of NNLO and N3LL

Matched pT spectrum

0.5

1

1.5

0 20 40 60 80 100 120

Ratio

to

each

central

pTH

[GeV]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

NNLOJET⊕SCET p p → H + ≥ 0 jet mH=125 GeV √s = 13 TeV

dσ/dpTH

[pb]

NNLOLO⊕NLL

NLO⊕NNLLNNLO⊕N3LL

PRELIMINARY

[Chen,Gehrmann,Glover,AH,Li,Neill,Schulze,Stewart,Zhu]

I NNLO & NNLO ⊕ N3LL start to deviate @ pT . 30 GeV

I reduction of uncertainties by more than a factor of twoI NLO ⊕ NLL −→ NNLO ⊕ N3LL: large impact in peak region

Projection-to-Born Method

∫ (−

)

Deep Inelastic Scattering

`

p

qV

precise probe to resolve the inner structure of the protonI PDF constraintsI αs extraction (+ running)

[Eur.Phys.J.C77(2017)no.11,791]

I DIS 2 jet @ NNLO[Currie, Gehrmann, Niehues ’16]

[Currie, Gehrmann, AH, Niehues ’17]

⇐ precise αs determination

I DIS 1 jet @ N3LO[Currie, Gehrmann, Glover, AH, Niehues, Vogt. ’18]

Fully Di�erential N3LO

DIS 2 jet@NNLO

[Currie, Gehrmann, Niehues ’16][Currie, Gehrmann, AH, Niehues ’17]

Projection-to-Born

⊕[Cacciari, et al. ’15]

DIS structurefunction

@N3LO[Moch, Vermaseren, Vogt ’05]

= DIS fullydi�erential @N3LO

Projection-to-Born

inclusive DIS (structure function)

σincl.NLO = αsαsαsαsαsαsαsαsαsαsαsαsαsαsαsαsαs +

∫

1, incl.

+

∫

1, diff.

(−

)

only “special” processes but not restricted to any orderinclusive X @ NnLO

+ X + jet @ Nn-1LO

}; X @ NnLO

Born kinematics: Q2 = −q2 > 0, x =−Q2

2P · q

Projection-to-Born

di�erential DIS (w/o IR treatment)

σdiff.NLO = αsαsαsαsαsαsαsαsαsαsαsαsαsαsαsαsαs

+

∫

1, incl.

+

∫

1, diff.

(−

)


+ X + jet @ Nn-1LO

}; X @ NnLO


2P · q

Projection-to-Born

di�erential DIS (w/ IR treatment)

σdiff.NLO = αsαsαsαsαsαsαsαsαsαsαsαsαsαsαsαsαs +

∫

1, incl.

+

∫

1, diff.

(−

)


+ X + jet @ Nn-1LO

}; X @ NnLO


2P · q

Projection-to-Born

di�erential DIS (w/ IR treatment)

σdiff.NLO = αsαsαsαsαsαsαsαsαsαsαsαsαsαsαsαsαs +

∫

1, incl.

+

∫

1, diff.

(−

)

︸︷︷︸DIS structure function @ NLO

︸︷︷︸DIS 2 jet @ LO


+ X + jet @ Nn-1LO

}; X @ NnLO


2P · q

Validation up to NNLO — Antenna vs. P2B

0.91

1.1

-1 -0.5 0 0.5 1 1.5 2 2.5 3

-5%

+5%

NNLOJET √s‾ = 300 GeV

P2B

/ An

tenn

a

ηjet

NNLO coefficient

0.981

1.02

-1%

+1%


NLO coefficient

0

1000

2000

3000

4000

5000

6000

7000

8000NNLOJET √s‾ = 300 GeV

dσ/d

ηjet [p

b]

e p → e + jet + X

LO NLO NNLO


LO NLO NNLO

0.91

1.1

10 100

-5%

+5%


P2B

/ An

tenn

a

ETjet

[GeV]

NNLO coefficient

0.981

1.02

-1%

+1%


NLO coefficient

10-2

10-1

100

101

102

103

104


dσ/d

E Tjet [p

b/Ge

V]

e p → e + jet + X

LO NLO NNLO


LO NLO NNLO

NLO coe�cient: . 5‰NNLO coe�cient: . 2%

}; agreement @ full NNLO: . 1‰

Di�erential distributions at N3LO

00.20.40.60.8

11.21.41.6

-1 -0.5 0 0.5 1 1.5 2 2.5 3


Rati

o to

NN

LO

ηjet

0

1000

2000

3000

4000

5000

6000

7000


dσ/d

ηjet [p

b]

e p → e + jet + X

ZEUS

LONLO

NNLON3LO


ZEUS

LONLO

NNLON3LO

0.60.70.80.9

11.11.21.3

10 100


Rati

o to

NN

LO

ETjet

[GeV]

10-2

10-1

100

101

102

103

104


dσ/d

E Tjet [p

b/Ge

V]

e p → e + jet + X

ZEUS

LONLO

NNLON3LO


ZEUS

LONLO

NNLON3LO

I for the �rst time: overlapping scale bands agreement with dataI reduction of scale uncertainties

Jet Rates

10-5

10-4

10-3

0.01

0.1

1

0 0.02 0.04 0.06 0.08


NLO O(αs)

ycut

R(0+1)R(1+1)R(2+1)


NLO O(αs)

R(0+1)R(1+1)R(2+1)

0 0.02 0.04 0.06 0.08


NNLO O(α2s)

ycut

e p → e + jet + X

R(0+1)R(1+1)R(2+1)R(3+1)


NNLO O(α2s)

R(0+1)R(1+1)R(2+1)R(3+1)

0 0.02 0.04 0.06 0.08


N3LO O(α3s)

ycut

R(0+1)R(1+1)R(2+1)R(3+1)R(4+1)


N3LO O(α3s)

R(0+1)R(1+1)R(2+1)R(3+1)R(4+1)

Jet rates:

R(n+1) = N(n+1)/Ntot

Jade algorithm↪→ cluster partons if:

2EiEj(1− cos θij)

W 2< ycut

Projection-to-Born — an “antenna” view

Consider the real-emission subtraction in the antenna subtraction formalismfor H + 0jet (@ LC):∫ {

dσRH+0jet − dσSNLO

H+0jet

}=

∫dΦH+1

{A3g0H(1g, 2g, 3g,H) J (ΦH+1)

− F 03 (1g, 2g, 3g) A2g0H(1g, 2g,H) J (ΦH+0)

}

=

∫dΦH+1 A3g0H(1g, 2g, 3g,H)

{J (ΦH+1)− J (ΦH+0)

}




H+0jet

}=

∫dΦH+1

{A3g0H(1g, 2g, 3g,H) J (ΦH+1)

− F 03 (1g, 2g, 3g) A2g0H(1g, 2g,H) J (ΦH+0)

}

=

∫dΦH+1 A3g0H(1g, 2g, 3g,H)

{J (ΦH+1)− J (ΦH+0)

}

Antennae = ratios of physical Matrix Elements:

F 03 (ig, jg, kg) ≡ A3g0H(ig, jg, kg,H)

A2g0H(ig, kg,H)




H+0jet

}=

∫dΦH+1

{A3g0H(1g, 2g, 3g,H) J (ΦH+1)

− F 03 (1g, 2g, 3g) A2g0H(1g, 2g,H) J (ΦH+0)

}=

∫dΦH+1 A3g0H(1g, 2g, 3g,H)

{J (ΦH+1)− J (ΦH+0)

}




H+0jet

}=

∫dΦH+1

{A3g0H(1g, 2g, 3g,H) J (ΦH+1)

− F 03 (1g, 2g, 3g) A2g0H(1g, 2g,H) J (ΦH+0)

}=

∫dΦH+1 A3g0H(1g, 2g, 3g,H)

{J (ΦH+1)− J (ΦH+0)

}

⇒ Simple processes where antenna ' real-emission Matrix Element! Projection-to-Born

Similarly at NNLO:X04 &X

03 ×X

03 are “projections” of RR ME & NLO(+jet) subtraction term.

Summary & Outlook — tl;dr

I Antenna subtraction successfully applied to many important processes:↪→ pp→ X + 0, 1 jets (X = H, Z, W)

↪→ pp→ H + 2 jets (VBF)

↪→ pp→ dijets (N2c , NcNF , N

2F )

↪→ ep→ 1, 2 jets↪→ e+e− → 3 jets

⇒ subtraction set up for: pp→ “colour neutral” + 0, 1, 2 jets

I inclusive pHT spectrum: NNLO prediction matched to N3LL

↪→ �xed order: stable predictions down to pHT = 0.7 GeV

; is it good enough for qT-subtraction @ N3LO?!

I Projection-to-Born method ⊕ Antenna subtraction↪→ �rst fully di�erential N3LO prediction: inclusive jets in DIS↪→ method also applicable for colour-neutral �nal states pp in collisions

Thank you

Backup Slides

Antenna subtraction @ NLO

[J. Currie , E.W.N. Glover, S. Wells ’13]

dσT ∼ J(1)n M0

n

dσS ∼ X03M0

n

dσT :

dσS :

1

Antenna subtraction @ NNLO

[J. Currie , E.W.N. Glover, S. Wells ’13]

dσU,A ∼ −β0

� J (1)n M1

n + J (1)n M1

n dσU,B ∼ 12J (1)

n ⊗ J (1)n M0

n dσU,C ∼ J (2)n M0

n

dσT,b1 ∼ X03M1

n + X03J (1)

n M0n

dσT,b3 ∼ −β0

� X03M0

n + β0

� X03

�|s|µ2

�−�

M0n

dσT,c ∼ −�

1

dσS,c + dσT,c1 + dσT,c2

dσT,a ∼ J(1)n+1M

0n+1 dσT,b2 ∼ X1

3M0n + J

(1)X X0

3M0n − MXX0

3J(1)2 M0

n

dσS,a dσS,b2dσS,c dσS,b1dσS,d

dσU :

dσT :

dσS :

NNLO Corrections Using Antenna Subtraction & Applications

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