colorful nnlo – completely local subtractions for fully ... · the problem consider the nnlo...

Colorful NNLO – Completely local subtractions for fullydifferential predictions at NNLO

Gabor Somogyi

MTA-DE Particle Physics Research Group,Debrecen

with V. Del Duca, C. Duhr, A. Kardos,

F. Tramontano, Z. Trocsanyi

Radcor-Loopfest 2015, UCLA

June 18, 2015

Gabor Somogyi | Colorful NNLO | page 1

Message

Message: can compute NNLO cross sections like you always thought you would

1. Compute relevant IR factorization formulae

2. Use them to construct general, explicit, local subtractions

3. Integrate subtractions once and for all, verify pole cancellation

4. Apply the generic scheme to specific process


Outline

1. The problem

2. The recipe

3. Integrating the subtractions

4. Cancellation of poles

5. Application: H → bb

6. Conclusions and outlook


The problem


The problem

Consider the NNLO correction to a generic m-jet observable

σNNLO = σRRm+2 + σRV

m+1 + σVVm =

∫

m+2dσRR

m+2Jm+2 +

∫

m+1dσRV

m+1Jm+1 +

∫

m

dσVVm Jm .

◮ matrix elements for σRRm+2 (tree) and σRV

m+1 (1-loop) known for many processes

◮ σVVm (2-loop) know for 4 parton, V+3 parton processes, higher multiplicities are on

the horizon

◮ the three contributions are separately infrared divergent in d = 4 dimensions

Double real

◮ kin. singularities asone or two partonsunresolved: up toO(ǫ−4) poles from PSintegration

◮ no explicit ǫ poles

Real-virtual

◮ kin. singularities asone parton unresolved:up to O(ǫ−2) polesfrom PS integration

◮ explicit ǫ poles up toO(ǫ−2)

Double virtual

◮ kin. singularitiesscreened by jetfunction: PSintegration finite

◮ explicit ǫ poles up toO(ǫ−4)


The problem

Consider the NNLO correction to a generic m-jet observable


m+1 + σVVm =

∫

m+2dσRR

m+2Jm+2 +

∫

m+1dσRV

m+1Jm+1 +

∫

m

dσVVm Jm .

KLN theorem

Infrared singularities cancel between real and virtual quantum corrections at the sameorder in perturbation theory, for sufficiently inclusive (i.e. IR safe) observables.

However

How to make this cancellation explicit, so that the various contributions can becomputed numerically? Need a method to deal with implicit poles.


Several approaches – why this one?

Colorful NNLO: Completely Local subtRactions for Fully differential predictions at NNLO

◮ general and explicit expressions, including color and flavor(automation, color space notation is used)

◮ fully local counterterms, taking account of all color and spin correlations(mathematical rigor, efficiency)

◮ analytic cancellation of explicit ǫ poles in loop amplitudes(mathematical rigor)

◮ option to constrain subtractions to near singular regions(efficiency, important check)

◮ very algorithmic construction(valid at any order in perturbation theory)


The recipe


Structure of the NNLO correction

Rewrite the NNLO correction as a sum of three terms


m+1 + σVVm = σNNLO

m+2 + σNNLOm+1 + σNNLO

m

each integrable in four dimensions

σNNLOm+2 =

∫

m+2

{

dσRRm+2Jm+2 − dσ

RR,A2m+2 Jm −

[

dσRR,A1m+2 Jm+1 − dσ

RR,A12m+2 Jm

]}

σNNLOm+1 =

∫

m+1

{[

dσRVm+1 +

∫

1dσ

RR,A1m+2

]

Jm+1 −[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1

]

Jm

}

σNNLOm =

∫

m

{

dσVVm +

∫

2

[

dσRR,A2m+2 − dσ

RR,A12m+2

]

+

∫

1

[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1]}

Jm







m


σNNLOm+2 =

∫

m+2

{


RR,A2m+2 Jm −

[


RR,A12m+2 Jm

]}

σNNLOm+1 =

∫

m+1

{[

dσRVm+1 +

∫

1dσ

RR,A1m+2

]

Jm+1 −[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1

]

Jm

}

σNNLOm =

∫

m

{

dσVVm +

∫

2

[

dσRR,A2m+2 − dσ

RR,A12m+2

]

+

∫

1

[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1]}

Jm

1. dσRR,A2m+2 regularizes the doubly-unresolved limits of dσRR

m+2







m


σNNLOm+2 =

∫

m+2

{


RR,A2m+2 Jm −

[


RR,A12m+2 Jm

]}

σNNLOm+1 =

∫

m+1

{[

dσRVm+1 +

∫

1dσ

RR,A1m+2

]

Jm+1 −[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1

]

Jm

}

σNNLOm =

∫

m

{

dσVVm +

∫

2

[

dσRR,A2m+2 − dσ

RR,A12m+2

]

+

∫

1

[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1]}

Jm


m+2

2. dσRR,A1m+2 regularizes the singly-unresolved limits of dσRR

m+2







m


σNNLOm+2 =

∫

m+2

{


RR,A2m+2 Jm −

[


RR,A12m+2 Jm

]}

σNNLOm+1 =

∫

m+1

{[

dσRVm+1 +

∫

1dσ

RR,A1m+2

]

Jm+1 −[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1

]

Jm

}

σNNLOm =

∫

m

{

dσVVm +

∫

2

[

dσRR,A2m+2 − dσ

RR,A12m+2

]

+

∫

1

[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1]}

Jm


m+2


m+2

3. dσRR,A12m+2 accounts for the overlap of dσRR,A1

m+2 and dσRR,A2m+2







m


σNNLOm+2 =

∫

m+2

{


RR,A2m+2 Jm −

[


RR,A12m+2 Jm

]}

σNNLOm+1 =

∫

m+1

{[

dσRVm+1 +

∫

1dσ

RR,A1m+2

]

Jm+1 −[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1

]

Jm

}

σNNLOm =

∫

m

{

dσVVm +

∫

2

[

dσRR,A2m+2 − dσ

RR,A12m+2

]

+

∫

1

[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1]}

Jm


m+2


m+2


m+2 and dσRR,A2m+2

4. dσRV,A1m+1 regularizes the singly-unresolved limits of dσRV

m+1







m


σNNLOm+2 =

∫

m+2

{


RR,A2m+2 Jm −

[


RR,A12m+2 Jm

]}

σNNLOm+1 =

∫

m+1

{[

dσRVm+1 +

∫

1dσ

RR,A1m+2

]

Jm+1 −[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1

]

Jm

}

σNNLOm =

∫

m

{

dσVVm +

∫

2

[

dσRR,A2m+2 − dσ

RR,A12m+2

]

+

∫

1

[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1]}

Jm


m+2


m+2


m+2 and dσRR,A2m+2

4. dσRV,A1m+1 regularizes the singly-unresolved limits of dσRV

m+1

5. (∫

1 dσRR,A1m+2 )

A1 regularizes the singly-unresolved limit of∫

1 dσRR,A1m+2


Use known ingredients

Collinear and soft factorization of QCD matrix elements at NNLO known

◮ Tree level 3-parton splitting functions and double soft gg and qq currents

(Campbell, Glover 1997; Catani, Grazzini 1998;Del Duca, Frizzo, Maltoni 1999; Kosower 2002)

◮ One-loop 2-parton splitting functions and soft gluon current

(Bern, Dixon, Dunbar, Kosower 1994; Bern, Del Duca, Kilgore,Schmidt 1998-9; Kosower, Uwer 1999; Catani, Grazzini 2000;

Kosower 2003)


Use known ingredients

Collinear and soft factorization of QCD matrix elements at NNLO known

◮ Tree level 3-parton splitting functions and double soft gg and qq currents

(Campbell, Glover 1997; Catani, Grazzini 1998;Del Duca, Frizzo, Maltoni 1999; Kosower 2002)

◮ One-loop 2-parton splitting functions and soft gluon current

(Bern, Dixon, Dunbar, Kosower 1994; Bern, Del Duca, Kilgore,Schmidt 1998-9; Kosower, Uwer 1999; Catani, Grazzini 2000;

Kosower 2003)

But note

◮ unresolved regions in phase space overlap

◮ quantities appearing in factorization formulae are only well-defined in the strict limit


Colorful NNLO - general features

Construction based on universal IR limit formulae

◮ Altarelli-Parisi splitting functions, soft currents (tree and one-loop, triple APfunctions)

◮ simple and general procedure for matching of limits using physical gauge

◮ extension based on momentum mappings that can be generalized to any number ofunresolved partons

Fully local in color ⊗ spin space

◮ no need to consider the color decomposition of real emission ME’s

◮ azimuthal correlations correctly taken into account in gluon splitting

◮ can check explicitly that the ratio of the sum of counterterms to the real emissioncross section tends to unity in any IR limit

Straightforward to constrain subtractions to near singular regions

◮ gain in efficiency

◮ independence of physical results on phase space cut is strong check

Given completely explicitly for any process with non colored initial stateGabor Somogyi | Colorful NNLO | page 10

Kinematic singularities cancel in RR

Can check the ratio of the double real emission matrix element and the sum of allsubtractions for all IR limits tends to one.

ratio = subtractions/RR


Kinematic singularities cancel in RV

Can check the ratio of the real-virtual matrix element and the sum of all subtractions forall IR limits tends to one.

ratio = subtractions/(RV + RR,A1)


RR and RV contributions finite

Regularized RR and RV contributions finite, can be computed by standard MCtechniques. Implementation for general m in progress ⇒ see Adam Kardos’ talk

NNLO∗ = B+ R+ V+ RR+ RV


Integrating the subtractions


Integrating the subtractions

Momentum mappings used to define the counterterms

{p}n+pR

−→ {p}n ⇒ dφn+p({p};Q) = dφn({p}(R)n ;Q)[dp

(R)p,n ]

◮ implement exact momentum conservation, recoil distributed democratically (can begeneralized to any p)

◮ different collinear and soft mappings (R labels precise limit)

◮ exact factorization of phase space

Counterterms are products (in color and spin space) of

◮ factorized ME’s independent of variables in [dp(R)p,n ]

◮ singular factors (AP functions, soft currents), to be integrated over [dp(R)p,n ]

XR ({p}n+p) =(

8παsµ2ǫ)p

SingR (p(R)p )⊗ |M

(0)n ({p}

(R)n )|2

Can compute once and for all the integral over unresolved partons

∫

p

XR ({p}n+p) =(

8παsµ2ǫ)p

[ ∫

p

SingR (p(R)p )

]

⊗ |M(0)n ({p}

(R)n )|2


List of basic integrals

Int status

I(k)1C ,0

✔

I(k)1C ,1

✔

I(k)1C ,2

✔

I(k)1C ,3

✔

I(k)1C ,4

✔

I(k,l)1C ,5

✔

I(k,l)1C ,6

✔

I(k)1C ,7

✔

I1C ,8 ✔

Int status

I(k)12S ,1

✔

I(k)12S ,2

✔

I(k)12S ,3

✔

I(k)12S ,4

✔

I(k)12S ,5

✔

I12S ,6 ✔

I12S ,7 ✔

I12S ,8 ✔

I12S ,9 ✔

I12S ,10 ✔

I12S ,11 ✔

I12S ,12 ✔

I12S ,13 ✔

Int status

I1S ,0 ✔

I1S ,1 ✔

I1S ,2 ✔

I(k)1S ,3

✔

I1S ,4 ✔

I1S ,5 ✔

I1S ,6 ✔

I1S ,7 ✔

Int status

I(k)12CS ,1

✔

I12CS ,2 ✔

I12CS ,3 ✔

Int status

I1CS ,0 ✔

I1CS ,1 ✔

I(k)1CS ,2

✔

I1CS ,3 ✔

I1CS ,4 ✔

Int status

I(j,k,l,m)2C ,1

✔

I(j,k,l,m)2C ,2

✔

I(j,k,l,m)2C ,3

✔

I(j,k,l,m)2C ,4

✔

I(−1,−1,−1,−1)2C ,5

✔

I(k,l)2C ,6

✔

Int status

I(k,l)12C ,1

✔

I(k,l)12C ,2

✔

I(k)12C ,3

✔

I(k,l)12C ,4

✔

I(k)12C ,5

✔

I(k)12C ,6

✔

I(k)12C ,7

✔

I(k)12C ,8

✔

I(k)12C ,9

✔

I(k)12C ,10

✔

Int status

I(k)2CS ,1

✔

I(k)2CS ,2

✔

I(2)2CS ,2

✔

I(k)2CS ,3

✔

I(k)2CS ,4

✔

I(k)2CS ,5

✔

Int status

I2S ,1 ✔

I2S ,2 ✔

I2S ,3 ✔

I2S ,4 ✔

I2S ,5 ✔

I2S ,6 ✔

I2S ,7 ✔

I2S ,8 ✔

I2S ,9 ✔

I2S ,10 ✔

I2S ,11 ✔

I2S ,12 ✔

I2S ,13 ✔

I2S ,14 ✔

I2S ,15 ✔

I2S ,16 ✔

I2S ,17 ✔

I2S ,18 ✔

I2S ,19 ✔

I2S ,20 ✔

I2S ,21 ✔

I2S ,22 ✔

I2S ,23 ✔

✔: pole coefficients known analytically, finite numerically


An example

The double soft subtraction term leads to the following integral, among others:

I2S ,2(Yik,Q ; ǫ, y0, d′0) = −

4Γ4(1− ǫ)

πΓ2(1 − ǫ)

By0 (−2ǫ, d ′0)

ǫYik,Q

∫ y0

0dy y−1−2ǫ(1− y)d

′0−1+ǫ

×

∫ 1

−1d(cos ϑ) (sin ϑ)−2ǫ

∫ 1

−1d(cosϕ) (sinϕ)−1−2ǫ

[

f (ϑ, ϕ; 0)]−1[

f (ϑ,ϕ;Yik,Q)]−1

×[

Y (y , ϑ, ϕ;Yik,Q)]−ǫ

2F1

(

− ǫ,−ǫ, 1− ǫ,1− Y (y , ϑ, ϕ;Yik,Q))

where

f (ϑ, ϕ;Yik,Q) = 1− 2√

Yik,Q(1 − Yik,Q) sinϑ cosϕ− (1 − 2Yik,Q)χ cos ϑ

Y (y , ϑ, ϕ;χ) =4(1 − y)Yik,Q

[2(1 − y) + y f (ϑ, ϕ; 0)][2(1− y) + y f (ϑ, ϕ;Yik,Q)]


An example

This integral is equal to (y0 = 1, d ′0 = 3− 3ǫ)

I2S ,2(Y ; ǫ, 1, 3− 3ǫ) =

=1

2ǫ4−

1

ǫ3

[

ln(Y ) − 3

]

+1

ǫ2

[

2Li2(1− Y ) + ln2(Y )− π2 −

(

2

1− Y

−1

2(1− Y )2+

9

2

)

ln(Y ) +1

2(1− Y )+ 16

]

+1

ǫ

[

5

3

(

18Li3(1 − Y )

5+

6Li3(Y )

5

−6Li2(1− Y ) ln(Y )

5−

2

5ln3(Y ) +

3

5ln(1− Y ) ln2(Y ) + π2 ln(Y )−

78ζ3

5

)

+

(

3

1− Y−

3

4(1 − Y )2+

15

4

)

(

2Li2(1− Y ) + ln2(Y ))

− 6π2 −

(

27

2(1 − Y )

−13

4(1− Y )2+

91

4

)

ln(Y ) +19

4(1 − Y )+

163

2

]

+O(ǫ0)

◮ Note the Y → 1 limit is finite

limY→1

I2S ,2(Y ; ǫ, 1, 3− 3ǫ) =1

2ǫ4+

3

ǫ3+

1

ǫ2

(

71

4− π2

)

+1

ǫ

(

393

4− 6π2 − 24ζ3

)

+O(ǫ0)

◮ Finite term is computed numerically


Solving the integrals

Strategy for computing the master integrals

1. write phase space in terms ofangles and energies

2. angular integrals in terms ofMellin-Barnes representations

3. resolve the ǫ poles by analyticcontinuation

4. MB integrals to Euler-typeintegrals, pole coefficients are finiteparametric integrals

1. choose explicit parametrization ofphase space

2. write the parametric integralrepresentation in chosen variables

3. resolve the ǫ poles by sectordecomposition

4. pole coefficients are finiteparametric integrals

5. evaluate the parametric integrals in terms of multiple polylogs

6. simplify result (optional)


Cancellation of poles


Integrated approximate cross sections

Recall the NNLO correction is a sum of three terms




m


σNNLOm+2 =

∫

m+2

{


RR,A2m+2 Jm −

[


RR,A12m+2 Jm

]}

σNNLOm+1 =

∫

m+1

{[

dσRVm+1 +

∫

1dσ

RR,A1m+2

]

Jm+1 −[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1]

Jm

}

σNNLOm =

∫

m

{

dσVVm +

∫

2

[

dσRR,A2m+2 − dσ

RR,A12m+2

]

+

∫

1

[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1]}

Jm

Integrated approximate cross sections

◮ After summing over unobserved flavors, all integrated approximate cross sections canbe written as products (in color space) of various insertion operators with lowerpoint cross sections.

◮ Can be computed once and for all (though admittedly lots of tedious work).

◮ Poles are computed analytically, finite part numerically.


Poles cancel: VV contribution finite

After adding all integrated approximate cross sections the double virtual contributionmust be finite in ǫ.

σNNLOm =

∫

m

{

dσVVm +

∫

2

[

dσRR,A2m+2 − dσ

RR,A12m+2

]

+

∫

1

[

dσRV,A1m+1 +

(

∫

1dσ

RR,A1m+2

)

A1

]}

Jm

◮ Have checked the cancellation of the 1ǫ4

and 1ǫ3

poles analytically for any number ofjets (i.e., with m symbolic).

◮ Have checked m = 2 (e+e− → qq, H → bb) explicitly and we find that all polescancel.

◮ Have checked m = 3 (e+e− → qqg) explicitly and we find that all poles cancel.


Example: H → bb

The double virtual contribution has the following pole structure (µ2 = m2H)

dσVV

H→bb=

(

αs(µ2)

2π

)2

dσB

H→bb

{

2C2F

ǫ4+

(

11CACF

4+ 6C2

F −CFnf

2

)

1

ǫ3

+

[(

8

9+

π2

12

)

CACF +

(

17

2− 2π2

)

C2F −

2CFnf

9

]

1

ǫ2

+

[(

−961

216+

13ζ3

2

)

CACF +

(

109

8− 2π2 − 14ζ3

)

C2F+

65CFnf

108

]

1

ǫ

}

(Anastasiou, Herzog, Lazopoulos, arXiv:0111.2368)


Example: H → bb

The double virtual contribution has the following pole structure (µ2 = m2H)

dσVV

H→bb=

(

αs(µ2)

2π

)2

dσB

H→bb

{

2C2F

ǫ4+

(

11CACF

4+ 6C2

F −CFnf

2

)

1

ǫ3

+

[(

8

9+

π2

12

)

CACF +

(

17

2− 2π2

)

C2F −

2CFnf

9

]

1

ǫ2

+

[(

−961

216+

13ζ3

2

)

CACF +

(

109

8− 2π2 − 14ζ3

)

C2F+

65CFnf

108

]

1

ǫ

}

(Anastasiou, Herzog, Lazopoulos, arXiv:0111.2368)

The sum of the integrated approximate cross sections gives (µ2 = m2H)

∑

∫

dσA =

(

αs(µ2)

2π

)2

dσB

H→bb

{

−2C2F

ǫ4+

(

−11CACF

4− 6C2

F +CFnf

2

)

1

ǫ3

+

[(

−8

9−

π2

12

)

CACF +

(

−17

2+ 2π2

)

C2F+

2CFnf

9

]

1

ǫ2

+

[(

961

216−

13ζ3

2

)

CACF +

(

−109

8+ 2π2 + 14ζ3

)

C2F −

65CFnf

108

]

1

ǫ

}

(Del Duca, Duhr, GS, Tramontano, Trocsanyi,arXiv:1501.07226)


Example: e+e− → 3 jets

The double virtual contribution has the following pole structure (µ2 = s)

dσVV3 = Poles

(

A(2×0)3 + A

(1×1)3

)

+ F inite(

A(2×0)3 + A

(1×1)3

)

where

Poles(

A(2×0)3 + A

(1×1)3

)

= 2

[

−(

I(1)qqg (ǫ)

)2−

β0

ǫI(1)qqg (ǫ)

+ e−ǫγΓ(1 − 2ǫ)

Γ(1 − ǫ)

(

β0

ǫ+ K

)

I(1)qqg (2ǫ) + H

(2)qqg

]

A03(1q , 3g , 2q)

+ 2I(1)qqg (ǫ)A

1×03 (1q , 3g , 2q)

with

H(2)qqg =

eǫγ

4ǫΓ(1 − ǫ)

[(

4ζ3 +589

432−

11π2

72

)

Nc +

(

−1

2ζ3 −

41

54−

π2

48

)

+

(

− 3ζ3 −3

16+

π2

4

)

1

Nc

+

(

−19

18+

π2

36

)

Ncnf +

(

−1

54−

π2

24

)

nf

Nc

+5

27n2f

]

(Gehrmann-De Ridder, Gehrmann, Glover, Heinrich,arXiv:0710.0346)




dσVV3 = Poles

(

A(2×0)3 + A

(1×1)3

)

+ F inite(

A(2×0)3 + A

(1×1)3

)

Adding the sum of the integrated approximate cross sections gives

Poles(

A(2×0)3 + A

(1×1)3

)

+ Poles∑

∫

dσA = 117k terms




dσVV3 = Poles

(

A(2×0)3 + A

(1×1)3

)

+ F inite(

A(2×0)3 + A

(1×1)3

)

Adding the sum of the integrated approximate cross sections gives

Poles(

A(2×0)3 + A

(1×1)3

)

+ Poles∑

∫

dσA = 117k terms

◮ zero numerically in any phase space point

◮ zero analytically after simplification using symbol technology (C. Duhr)


Application: H → bb


Higgs decay to b-quarks

Consider H → bb decay at NNLO:

◮ admittedly the simplest case

◮ but this just amounts to having to sum less terms in general formulae

Inclusive decay rate


Higgs decay to b-quarks

Consider H → bb decay at NNLO:

◮ admittedly the simplest case

◮ but this just amounts to having to sum less terms in general formulae

Differential distributions

◮ pseudorapidity of highest energy jet (right) and leading jet energy (left)


Constrained subtractions

We can constrain subtractions to near singular regions: α0 ∈ (0, 1]

◮ poles cancel numerically (α0 = 0.1)

dσVV

H→bb+

∑

∫

dσA =5.4× 10−8

ǫ4+

3.9× 10−5

ǫ3+

3.3× 10−3

ǫ2+

6.7× 10−3

ǫ+O(1)

Err

(

∑

∫

dσA

)

=3.1× 10−5

ǫ4+

5.0× 10−4

ǫ3+

8.1× 10−3

ǫ2+

7.7× 10−2

ǫ+O(1)

◮ results unchanged

0.0

0.5

1.0

1.5

2.0

2.5

3.0

dΓ

d|η

1|[M

eV]

0.0 0.5 1.0 1.5 2.0 2.5 3.0

|η1|

Durham clustering at ycut = 0.05, µ = mH

Γ(α0 = 1)Γ2(α0 = 1)Γ3(α0 = 1)Γ4(α0 = 1)

Γ(α0 = 0.1)Γ3(α0 = 0.1)Γ2(α0 = 0.1)Γ4(α0 = 0.1)


Constrained subtractions

We can constrain subtractions to near singular regions: α0 ∈ (0, 1]

◮ improved efficiency

α0 1 0.1

timing (rel.) 1 0.40

〈Nsub〉 52 14.5

〈Nsub〉 is the average number of subtraction terms computed


Conclusions and outlook


Conclusions and outlook

Colorful NNLO framework

◮ Completely Local subtRactions for Fully differential predictions at NNLO

◮ construction of subtraction terms based on IR limit formulae

◮ analytic integration of subtraction terms is feasible with modern integrationtechniques

◮ demonstrated cancellation of ǫ poles for m = 2 and m = 3

◮ worked out in full detail for processes with no colored particles in the initial state

First application: Higgs boson decay into a b and anti-b quark

Next steps

◮ e+e− → 3 jets is almost finished

◮ extension to hadronic initial states conceptually understood and on the way


colorful nnlo – completely local subtractions for fully ... · the problem consider the nnlo...

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