nnlo qcd predictions for the lhc with antenna subtractionpiresleshouches17.pdfdouble soft limit for...
TRANSCRIPT
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NNLO QCD predictions for the LHC with antenna subtraction
João Pires (MPI Munich)
Les Houches Workshop, Physics at the TeV colliders June 7, 2017
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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
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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
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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
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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
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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
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Two colour connected partons
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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
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• 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)
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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
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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)
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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
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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
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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
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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
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Double virtual antenna contribution
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• integrated operators in analytic one-to-one correspondence with operator of Catani
J(2,1)2
(I1)2
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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
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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
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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]
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Single jet inclusive cross section
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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
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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
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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
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• 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
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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
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Dijet inclusive cross section
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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
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(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
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(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
![Page 30: NNLO QCD predictions for the LHC with antenna subtractionpiresleshouches17.pdfdouble soft limit for gg →gggg #PS points=10000 x=(s-sij)/s x=10-4 x=10-5 x=10-6 1487 outside the plot](https://reader036.vdocuments.mx/reader036/viewer/2022070204/60f3bd19933cef2b4b4bdd16/html5/thumbnails/30.jpg)
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
![Page 31: NNLO QCD predictions for the LHC with antenna subtractionpiresleshouches17.pdfdouble soft limit for gg →gggg #PS points=10000 x=(s-sij)/s x=10-4 x=10-5 x=10-6 1487 outside the plot](https://reader036.vdocuments.mx/reader036/viewer/2022070204/60f3bd19933cef2b4b4bdd16/html5/thumbnails/31.jpg)
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
![Page 32: NNLO QCD predictions for the LHC with antenna subtractionpiresleshouches17.pdfdouble soft limit for gg →gggg #PS points=10000 x=(s-sij)/s x=10-4 x=10-5 x=10-6 1487 outside the plot](https://reader036.vdocuments.mx/reader036/viewer/2022070204/60f3bd19933cef2b4b4bdd16/html5/thumbnails/32.jpg)
BACKUP
![Page 33: NNLO QCD predictions for the LHC with antenna subtractionpiresleshouches17.pdfdouble soft limit for gg →gggg #PS points=10000 x=(s-sij)/s x=10-4 x=10-5 x=10-6 1487 outside the plot](https://reader036.vdocuments.mx/reader036/viewer/2022070204/60f3bd19933cef2b4b4bdd16/html5/thumbnails/33.jpg)
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
![Page 34: NNLO QCD predictions for the LHC with antenna subtractionpiresleshouches17.pdfdouble soft limit for gg →gggg #PS points=10000 x=(s-sij)/s x=10-4 x=10-5 x=10-6 1487 outside the plot](https://reader036.vdocuments.mx/reader036/viewer/2022070204/60f3bd19933cef2b4b4bdd16/html5/thumbnails/34.jpg)
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