jet production in the colorful nnlo frameworkadam kardos ! university of debrecen, mta-de particle...

Adam Kardos !

University of Debrecen, MTA-DE Particle Physics Research Group

and NCSR “Demokritos", Athens

!

Jet production in the colorful NNLO framework

QCD matters

•Relatively large coupling ⇒ final state is QCD dominated. •Key processes have irreducible QCD background and/or QCD corrections.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

d�2j

d|⌘1|[M

eV]

0.0 0.5 1.0 1.5 2.0 2.5 3.0

|⌘1|

Durham clustering at ycut = 0.05

µ 2 [0.5, 2]mHLONLONNLO

Del Duca et al. arXiv:1501.07226H bb decay @ NNLO-

•High precision experiments demand high precision predictions.

•Large scale uncertaint(y)(ies).

Road to high precision predictions

How to increase precision?

Go one step further in the perturbative expansion:

� = �LO + �NLO + �NNLO + . . .

The aim: numerical calculation kinematical singularities due to soft/collinear parton emissions are treated with local subtractions.@NLO:

�NLO =

Z

m+1

hd�RmJm+1 � d�

R,Am+1Jm

i+

Z

m

d�Vm +

Z

1d�R,Am+1

�Jm

Singly unresolved

The Colorful NNLO framework


@NNLO:�NNLO =

=

Z

m+2

n

d�RRm+2Jm+2 � d�RR,A2m+2 Jm �

h

d�RR,A1m+2 Jm+1 � d�RR,A12m+2 Jm

io

"=0

+

Z

m+1

(

✓

d�RVm+1 +

Z

1d�RR,A1m+2

◆

Jm+1 �"

d�RV,A1m+1 +

✓

Z

1d�RR,A1m+2

◆A1#

Jm

)

"=0

+

Z

m

(

d�VVm +

Z

2

h

d�RR,A2m+2 � d�RR,A12m+2

i

+

Z

1

"

d�RV,A1m+1 +

✓

Z

1d�RR,A1m+2

◆A1#)

"=0

Jm

The Colorful NNLO framework was worked out for colorless initial states by Del Duca, Somogyi, Trocsanyi et al., for detailed information see Gabor’s talk…


@NNLO:

Z

m

(d�VVm +

Z

2

hd�RR,A2m+2 � d�

RR,A12m+2

i+

Z

1

"d�RV,A1m+1 +

✓Z

1d�RR,A1m+2

◆A1#)

"=0

Jm

m-parton line:

•Free of kinematical singularities (jet-function!). •Contains the integrals of the subtraction terms ( see the talk of Gabor on thursday afternoon, too).

•State of the art: Two-loop amplitude ( & integration of subtraction terms).


@NNLO:m+1-parton line:

•Only singly unresolved regions present. •But RV has different factorization properties (one-loop amplitude is involved).

•State of the art: fast and precise evaluation of the involved one-loop amplitude (even in the singly unresolved region!).

Z

m+1

(✓d�RVm+1 +

Z

1d�RR,A1m+2

◆Jm+1 �

"d�RV,A1m+1 +

✓Z

1d�RR,A1m+2

◆A1#Jm

)

"=0


@NNLO:

m+2-parton line:

•Both singly and doubly unresolved regions present. •Also spurious ones! e.g.: kinematical singularities of A2 in singly and kinematical singularities of A1 in doubly unresolved regions.

Z

m+2

n

d�RRm+2Jm+2 � d�RR,A2m+2 Jm �

h

d�RR,A1m+2 Jm+1 � d�RR,A12m+2 Jm

io

"=0


m+2-parton line:

•RR SME still tree-level, but contains two partons more compared to the Born one.

•Several subtraction terms. For O(100). •Generation of UB PS points (momentum mappings) and the corresponding UB SMEs.

•Application of physical cuts. •Filling histograms.

Bottleneck(s):

e+ e� ! u ū g g g


•Several subprocesses contribute. •Various subtraction terms have to be calculated. •Subtraction terms present a twofold problem: heavy bookkeeping due to the immense number of terms, a clever grouping is needed to ease up (a bit) the computational burden.

Bottom line:

⇒More profitable to automatize subtractions!

A numerical implementation of the Colorful NNLO framework

A numerical implementation

For demonstration we take e+ e- to 3-jet production.

The code is organized directory-wise, for a new process a new folder has to be created:


The generation of all subprocesses is automatic:


Investigating for possible numerical relations between SMES:


Automatic detection of all singular regions:


An NNLO calculation is extremely complex. Due to this complexity it is good practice to make as much checks as possible.

In our code the following ones are built in:

•Check upon individual subtraction terms, e.g.:

limpi||pr||ps

Cirs|MRR|2

= 1

•Checking bookkeeping and overall consistency by checking complete lines, e.g.:

limpi||pr, ps!0

A1 +A2 �A12|MRR|2

= 1


Testing the subtraction terms in all limits (even in quad precision):

A numerical implementationTesting the whole m+2 parton line:

Doubly unresolved

Singly unresolved:


Behavior of the m+2 parton line in a triple and double collinear limit.

2

5

10

2

5

10

2

2

5

10

3

2

5

10

4

#ofevents

0.999 0.9995 1.0 1.0005 1.001

ratio

y567 = 10�6

y567 = 10�8

y567 = 10�10

e+ e� ! q q̄ g g gA1+A2�A12

|MRR|2, p5||p6||p7

2

5

10

2

5

10

2

2

5

10

3

2

5

10

4

#ofevents

0.999 0.9995 1.0 1.0005 1.001

ratio

y56 = y47 = 10�4

y56 = y47 = 10�5

y56 = y47 = 10�6

e+ e� ! q q̄ g g gA1+A2�A12

|MRR|2

p5||p6, p4||p7


Behavior of the m+2 parton line in a collinear and a soft limit.

2

5

10

2

5

10

2

2

5

10

3

2

5

10

4

#ofevents

0.999 0.9995 1.0 1.0005 1.001

ratio

y56 = 10�6

y56 = 10�8

y56 = 10�10

e+ e� ! q q̄ g g gA1+A2�A12

|MRR|2, p5||p6

2

5

10

2

5

10

2

2

5

10

3

2

5

10

4

#ofevents

0.999 0.9995 1.0 1.0005 1.001

ratio

y6 = 10�4

y6 = 10�5

y6 = 10�6

e+ e� ! q q̄ g g gA1+A2�A12

|MRR|2

p6 ! 0

Towards phenomenology

e+ e- ➞3 jets

Considering only e+ + e�!�⇤!3 jets

NNLO* : no VV and I operators, hence non-physical, though still informative.ps = 90GeV

µR = mZmZ = 91.1876GeV , mW = 80.385GeV↵s(mZ) = 0.118, 1/↵EM = 132.23

Calculating key event shape variables. For a precise definition, see: Gehrmann-De Ridder et al., arXiv:0711.4711 and also Kunszt et al. QCD at LEP, ETH-PT-89-39.

The whole calculation took 4+4 hours on 300 cores.

e+ e- ➞3 jets

0

5

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15

20

25

BT

d�

dBT[pb]

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

BT

NNLO⇤

e+ e� ! �⇤ ! 3jetps = 90GeV

µR = mZ

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BW

d�

dBW[pb]

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

BW

NNLO⇤

e+ e� ! �⇤ ! 3jetps = 90GeV

µR = mZ

Total and wide jet broadening:

Preliminary

e+ e- ➞3 jets

The jet transition variable and large hemisphere invariant mass:

Preliminary

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12

⇢d� d⇢[pb]

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4⇢

NNLO⇤

e+ e� ! �⇤ ! 3jetps = 90GeV

µR = mZ

0.0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Y3d�

dY3[pb]

0 1 2 3 4 5 6 7 8 9 10

� log Y3

NNLO⇤

e+ e� ! �⇤ ! 3jetps = 90GeV

µR = mZ

e+ e- ➞3 jets

The jet transition variable and large hemisphere invariant mass:

Preliminary

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16

(1�

T)d

�dT[pb]

0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

1-T

NNLO⇤

e+ e� ! �⇤ ! 3jetps = 90GeV

µR = mZ

0

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14

16

Cd�

dC[pb]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

C

NNLO⇤

e+ e� ! �⇤ ! 3jetps = 90GeV

µR = mZ

e+ e- ➞3 jets

… and even a completely new one, never appeared in the literature before:

Prel

imin

ary

0

1

2

3

4

5

6

7

sin

2�d⌃EEC

dcos�[pb]

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

cos�

NNLO⇤

e+ e� ! �⇤ ! 3jetps = 90GeV

µR = mZ

Energy-energy correlation:

Calculation cost: 7 hours @ 300 cores.

Conclusions

Conclusions

•A general numerical NNLO framework is introduced to manage subtractions for the case of e+ e- annihilation.

•It contains all subtraction terms, the user only has to provide his/her matrix elements (the I operators will also be included).

•The framework was applied to 3-jet production to obtain some key distributions.

•The inclusion of the I operators and the VV part for 3-jet production is in progress.

Thank you for your attention!

jet production in the colorful nnlo frameworkadam kardos ! university of debrecen, mta-de particle...

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