Towards Jet Cross Sections at NNLO for hadron colliders Aude - - PowerPoint PPT Presentation

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Towards Jet Cross Sections at NNLO for hadron colliders

Aude Gehrmann-De Ridder

HP2.3 Firenze 15.09.2010

SLIDE 2

Large production rates for

Standard Model processes

jets top quark pairs vector bosons

Allow precision measurements

masses couplings parton distributions

Require precise theory: NNLO

HP2.3 Firenze Aude Gehrmann-De Ridder

Expectations at LHC

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WJS2009

jet(ET

jet > 100 GeV)

jet(ET

jet > s/20)

jet(ET

jet > s/4)

Higgs(MH=120 GeV)

200 GeV

LHC Tevatron

events / sec for L = 10

33 cm

• 2s
• 1

b tot

proton - (anti)proton cross sections

W Z t

500 GeV

nb s (TeV)

• J. Stirling

SLIDE 3

Processes measured to few per cent accuracy

e+e- 3 jets, 2+1 jet production in DIS hadron collider processes:

jet production vector boson (+jet) production top quark pair production

Processes with potentially large perturbative corrections

Higgs or vector boson pair production

prediction stable only at NNLO

HP2.3 Firenze Aude Gehrmann-De Ridder

Where are NNLO corrections needed?

SLIDE 4

Require three principal ingredients (here: pp 2j)

two-loop matrix elements

explicit infrared poles from loop integral

known for all massless 2 2 processes

• ne-loop matrix elements

explicit infrared poles from loop integral and implicit poles from soft/collinear emission

usually known from NLO calculations

tree-level matrix elements

implicit poles from two partons unresolved

known from LO calculations

Challenge: combine contributions into parton-level generator need method to extract implicit infrared poles

HP2.3 Firenze Aude Gehrmann-De Ridder

NNLO calculations

SLIDE 5

Solutions

sector decomposition: expansion in distributions, numerical

integration (T. Binoth, G. Heinrich; C. Anastasiou, K. Melnikov, F. Petriello; M. Czakon)

applied to Higgs and vector boson production

(C. Anastasiou, K. Melnikov, F. Petriello)

subtraction: add and subtract counter-terms: process-

independent approximations in all unresolved limits, analytical integration

several well-established methods at NLO qT subtraction applied to Higgs and vector boson production

(S. Catani, M. Grazzini; with L. Cieri, G. Ferrera, D. de Florian)

antenna subtraction for jet observables in e+e- processes

(T. Gehrmann, E.W.N. Glover, AG)

HP2.3 Firenze Aude Gehrmann-De Ridder

NNLO calculations

SLIDE 6

NNLO corrections small

(T. Gehrmann, E.W.N. Glover, G. Heinrich, AG; S. Weinzierl)

stable perturbative prediction resummation not needed theory error below 2%

hadronization corrections

much smaller than for event shapes

data with different jet resolution

correlated

fit at ycut = 0.02 consistent results with other resolution

s = 0.1175±0.0020(exp)±0.0015(th)

(G. Dissertori, T. Gehrmann, E.W.N. Glover, G. Heinrich, H. Stenzel, AG) HP2.3 Firenze Aude Gehrmann-De Ridder

s from three-jet rate at NNLO

log10(ycut) 3 jet / had

Q = MZ s (MZ) = 0.1189 ALEPH data NNLO NLO LO 0.25 0.5 0.75

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(%)

2 4 6

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• 1

s(MZ) ln(ycut) Q=MZ central result with stat. uncertainty total uncertainty

total error perturbative hadronisation experimental statistical

ln(ycut) s(MZ) 0.1 0.1025 0.105 0.1075 0.11 0.1125 0.115 0.1175 0.12 0.1225 0.125

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0.001 0.002 0.003 0.004 0.005

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SLIDE 7

Structure of NNLO m-jet cross section at hadron colliders

with:

Partonic contributions: Subtraction terms: Mass factorization terms:

Challenge: construction and integration of subtraction terms

HP2.3 Firenze Aude Gehrmann-De Ridder

NNLO Subtraction

dˆ σNNLO =

• dΦm+2
• dˆ

σR

NNLO − dˆ

σS

NNLO

• +
• dΦm+1
• dˆ

σV,1

NNLO + dˆ

σMF,1

NNLO − dˆ

σV S,1

NNLO

• +
• dΦm
• dˆ

σV,2

NNLO + dˆ

σMF,2

NNLO

• +
• dΦm+2

dˆ σS

NNLO +

• dΦm+1

dˆ σV S,1

NNLO

dˆ σR

NNLO

dˆ σV,1

NNLO

dˆ σV,2

NNLO

dˆ σS

NNLO

dˆ σV S,1

NNLO

dˆ σMF,1

NNLO

dˆ σMF,2

NNLO

SLIDE 8

real radiation contribution to m-jet cross section antenna subtraction term:

antenna describes soft and collinear radiation off a hard parton pair

HP2.3 Firenze Aude Gehrmann-De Ridder

Antenna subtraction at NLO

dσR = N

• dΦm+1|Mm+1|2J(m+1)

m

(p1, . . . , pm+1)

dσS

NLO = N

• dΦm+1(p1, . . . , pm+1; q)
• j

X0

ijk |Mm|2 J(m) m (p1, . . . , ˜

pI, ˜ pK, . . . , pm+1)

1 1 i j k I i j k I m+1 m+1 K K

X0

ijk

SLIDE 9

colour-ordered pair of hard partons (radiators)

quark-antiquark pair quark-gluon pair gluon-gluon pair

three-parton antenna one unresolved parton four-parton antenna two unresolved partons at tree-level or at one loop radiators in initial or final state:

three types of antennae: final-final, initial-final, initial-initial

all antenna functions derived from physical

HP2.3 Firenze Aude Gehrmann-De Ridder

Colour-ordered antenna functions

|M|2

SLIDE 10

Colour-connected double unresolved case

final-final initial-final initial-initial

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Subtraction for hadronic processes at NNLO

I I

L L i j k l i j k l

i j k I i j k I l l JKL JKL

i k j I K i k j I K l l

SLIDE 11

Double real radiation at NNLO for

Contributions from all tree-level 2 4 processes Test case: (E.W.N. Glover, J. Pires)

three topologies according to initial state gluon positions

antenna subtraction terms constructed, implemented and

tested in all unresolved limits

HP2.3 Firenze Aude Gehrmann-De Ridder

Hadron collider processes at NNLO

pp → 2j gg → gggg

dσR

NNLO

= N 2 Nborn αs 2π 2 dΦ4(p3, . . . , p6; p1, p2)

• 2

4!

• P (i,j,k,l)∈(3,4,5,6)

A0

6(ˆ

1g, ˆ 2g, ig, jg, kg, lg)J(4)

2 (pi, . . . , pl)

+ 2 4!

• P (i,j,k,l)∈(3,4,5,6)

A0

6(ˆ

1g, ig, ˆ 2g, jg, kg, lg)J(4)

2 (pi, . . . , pl)

+ 2 4!

• PC(i,j,k,l)∈(3,4,5,6)

A0

6(ˆ

1g, ig, jg, ˆ 2g, kg, lg)J(4)

2 (pi, . . . , pl)

SLIDE 12

Subtraction terms involve only gluon-gluon antennae

in final-final, initial-final, initial-initial,

for colour-connected double unresolved limits

in all configurations,

for oversubtracted single unresolved limits and colour unconnected double unresolved limits

for single unresolved limits

Need to identify hard radiators for phase space mapping

HP2.3 Firenze Aude Gehrmann-De Ridder

Antenna subtraction for gg gggg

• F 0

3

F 0

3 ⊗ F 0 3

F 0

4

SLIDE 13

Identification of hard radiators Problem: each parton is radiator or unresolved

due to colour cyclicity of quark-gluon and gluon-gluon antennae

• nly in final-final case (initial state fixes radiators)

decompose into sub-antennae, e.g. each sub-antenna has

different phase space mapping, fixed hard and unresolved partons

was done for four-parton quark-gluon antenna functions previously

based on N=1 SUSY relations among splitting functions

achieved now for gluon-gluon antenna (E.W.N. Glover, J. Pires) eight sub-antennae contained in

HP2.3 Firenze Aude Gehrmann-De Ridder

Antenna subtraction for gg gggg

F 0

3 (1, 2, 3) = f 0 3 (1, 3, 2) + f 0 3 (3, 2, 1) + f 0 3 (2, 1, 3)

F 0

4

F 0

4

SLIDE 14

Check of the subtraction terms (E.W.N. Glover, J. Pires)

choose scaling parameter x for each limit generate phase space trajectories into each limit require reconstruction of two hard jets compute ratio (matrix element)/(subtraction term): Example: double soft limit :

HP2.3 Firenze Aude Gehrmann-De Ridder

Antenna subtraction for gg gggg

1 2 i j l k

1000 2000 3000 4000 5000 6000 0.99997 0.99998 0.99999 1 1.00001 1.00002 1.00003 double soft limit for gggggg ratio |MRR|2/Sterm #PS points=10000 x=(s-sij)/s x=10-4 x=10-5 x=10-6 1487 outside the plot 317 outside the plot 59 outside the plot 0.9 1 1.1 1.2 10-6 10-5 10-4 10-3 10-2 10-1 x 102 104 106 108 1010 1012 1014 1016 1018 1020 ratio |MRR|2/Sterm double soft limit for gggggg matrix element |MRR|2 (in GeV-2) subtr.term Sterm (in GeV-2) x=(s-sij)/s pt=100 GeV pt=200 GeV pt=300 GeV

|MRR|2/Sterm

sij s

SLIDE 15

Check of the subtraction terms (E.W.N. Glover, J. Pires)

Example: triple collinear final state limit Example: triple collinear initial state limit

HP2.3 Firenze Aude Gehrmann-De Ridder

Antenna subtraction for gg gggg

1 2 ijk l

1 2 l i j k

500 1000 1500 2000 2500 3000 3500 4000 4500 0.995 0.996 0.997 0.998 0.999 1 1.001 1.002 1.003 1.004 1.005 triple collinear limit for gggggg ratio |MRR|2/Sterm #PS points=10000 x=s1jk/s x=-10-7 x=-10-8 x=-10-9 2039 outside the plot 1010 outside the plot 177 outside the plot 500 1000 1500 2000 2500 3000 3500 0.99996 0.99998 1 1.00002 1.00004 triple collinear limit for gggggg ratio |MRR|2/Sterm #PS points=10000 x=sijk/s x=10-7 x=10-8 x=10-9 1419 outside the plot 77 outside the plot 17 outside the plot

sijk → 0

s1jk → 0

SLIDE 16

Antenna subtraction for

successful proof-of-principle of antenna subtraction starting point for implementation of all 2 4 processes

Next steps

implementation of virtual single unresolved 2 3 processes integration of antenna functions

Final-final known (T. Gehrmann, E.W.N. Glover, AG) Initial-final known (A. Daleo, T. Gehrmann, G. Luisoni, AG) Initial-initial in progress (R. Boughezal, M. Ritzmann, AG)

HP2.3 Firenze Aude Gehrmann-De Ridder

Jet production at hadron colliders

gg → gggg

SLIDE 17

Analytical integration over unresolved part of phase space only

phase space integrals reduced to masters (C. Anastasiou, K. Melnikov) Final-final: , one scale: q2

1 4 tree level (4 master integrals) 1 3 one loop (3 master integrals)

Initial-final: , two scales: q2, x

2 3 tree level (9 master integrals) 2 2 one loop (6 master integrals)

Initial-initial: , three scales: q2, x1, x2

2 3 tree level (32 master integrals) 2 2 one loop (5 master integrals)

HP2.3 Firenze Aude Gehrmann-De Ridder

Integrated NNLO antenna functions

q → k1 + k2 + k3(+k4)

q + p1 → k1 + k2(+k3) p1 + p2 → q + k1(+k2)

(See talk G. Luisoni)

SLIDE 18

are crossings of final-final antennae: four-parton case

HP2.3 Firenze Aude Gehrmann-De Ridder

Initial-initial antenna functions

quark-antiquark antennae A0

4

A0

4

• q,

g, g, q

• , A0

4

• q, g,

g, q

• , A0

4

• q, g, g,

q

• , A0

4

• q,

g, g, q

• A0

4

• A0

4

• q,

g, g, q

• ,

A0

4

• q, g, g,

q

• ,

A0

4

• q,

g, g, q

• B0

4

B0

4

• q,

q′, q′, q

• , B0

4

• q, q′, q′,

q

• , B0

4

• q,

q′, q′, q ∗ C0

4

C0

4

• q,

q, q, q

• , C0

4

• q, q,

q, q

• , C0

4

• q,

q, q, q ∗, C0

4

• q, q,

q, q ∗ quark-gluon antennae D0

4

D0

4

• q,

g, g, g

• , D0

4

• q, g,

g, g

• , D0

4

• q,

g, g, g

• , D0

4

• q,

g, g, g

• E0

4

E0

4

• q,

q′, q′, g

• , E0

4

• q, q′, q′,

g

• , E0

4

• q,

q′, q′, g

• , E0

4

• q,

q′, q′, g

• ,
• E0

4

• E0

4

• q,

q′, q′, g

• ,

E0

4

• q, q′, q′,

g

• ,

E0

4

• q,

q′, q′, g

• ,

E0

4

• q,

q′, q′, g

• gluon-gluon antennae

F 0

4

F 0

4

• g,

g, g, g

• , F 0

4

• g, g,

g, g

• G0

4

G0

4

• g,

q, q, g

• , G0

4

• g, q,

q, g

• , G0

4

• g, q, q,

g

• , G0

4

• g,

q, q, g

• G0

4

• G0

4

• g,

q, q, g

• ,

G0

4

• g, q, q,

g

• ,

G0

4

• g,

q, q, g

• H0

4

H0

4

• q,

q, q′, q′ , H0

4

• q, q,

q′, q′

SLIDE 19

Double real radiation

phase space factorization (A. Daleo, T. Gehrmann, D. Maitre)

require collinear rescaling: use Lorentz boost: constrained with right behaviour in all unresolved limits

HP2.3 Firenze Aude Gehrmann-De Ridder

Integrated initial-initial antenna functions

p1 → x1p1 p2 → x2p2

q → ˜ q = x1p1 + x2p2

ˆ x1 = s12 − sj2 − sk2 s12 s12 − s1j − s1k − sj2 − sk2 + sjk s12 − s1j − s1k 1

2

ˆ x2 = s12 − s1j − s1k s12 s12 − s1j − s1k − sj2 − sk2 + sjk s12 − sj2 − sk2 1

2

dΦm+2(k1, . . . , km+2; p1, p2) = dΦm(˜ k1, . . . , ˜ ki, ˜ kl, . . . , ˜ km+2; x1p1, x2p2) ×δ(x1 − ˆ x1) δ(x2 − ˆ x2) [dkj] [dkk]dx1 dx2

SLIDE 20

(R. Boughezal, M. Ritzmann, AG)

are linear combinations of 32 master integrals

coefficients contain poles in and rational factors in x1,x2 endpoint behaviour: (1-x1)-1-2(1-x2)-1-2 R(x1,x2) expansion in distributions around endpoints x1,x2 = 1 pole structure up to -4 need to know the masters a priori up to transcendentality 4

HP2.3 Firenze Aude Gehrmann-De Ridder

Integrated initial-initial antennae

X 0

il(xi, xl) =

1 C2()

• [dkj] [dkk] δ(xi − ˆ

xi)δ(xl − ˆ xl)X0

il,jk(pi, pj, pk, pl)

SLIDE 21

Masters are calculated in different regions

Hard region (x1 1, x2 1)

up to transcendentality 2, yielding GHPL of weight 2 in x1,x2

Collinear regions (x1 = 1, x2 1 or x2 =1, x1 1)

up to transcendentality 3, yielding HPL of weight 3 in x1 or x2

Soft region (x1 = 1 and x2 = 1)

up to transcendentality 4, yielding constants

use differential equations in x1,x2 to compute masters in

hard and collinear regions

HP2.3 Firenze Aude Gehrmann-De Ridder

Initial-initial integrated antenna

SLIDE 22

First step: integrated antennae with two quark flavours

crossings of:

quark-antiquark antenna: quark-gluon antenna: gluon-gluon antenna:

contain 12 (out of 32) master integrals

Full set in progress

HP2.3 Firenze Aude Gehrmann-De Ridder

Initial-initial integrated antenna

SLIDE 23

One-loop single unresolved real radiation:

phase-space overconstrained no integrals, just expansions analytically continue master integrals from final-final kinematics

• ne-loop boxes and bubbles

expand in distributions

in progress

HP2.3 Firenze Aude Gehrmann-De Ridder

Initial-initial antenna functions

dΦm+1(k1, . . . , km+1; p1, p2) = dΦm(˜ k1, . . . , ˜ kj−1, ˜ kj+1, . . . , ˜ km+1; x1p1, x2p2) δ(x1 − ˆ x1) δ(x2 − ˆ x2) [dkj] dx1 dx2

ˆ x1 = s12 − sj2 s12 s12 − s1j − sj2 s12 − s1j 1

2

ˆ x2 = s12 − s1j s12 s12 − s1j − sj2 s12 − sj2 1

2

SLIDE 24

Towards subtraction for top quark pair production at NNLO

First step: NLO antenna subtraction for and

previous NLO results in dipole subtraction (A. Bredenstein, A. Denner,

• S. Dittmaier, S. Pozzorini; G. Bevilacqua et al., K. Melnikov, M. Schulze)

require massive phase space mappings require massless antennae: final-final, initial-final, final-final require massive antennae: final-final (M. Ritzmann, AG), initial-final (new)

need flavour-violating quark-antiquark antennae (new)

constructed colour-ordered antenna subtraction terms for

HP2.3 Firenze Aude Gehrmann-De Ridder

Antenna subtraction with massive particles

t¯ t t¯ t + j

q¯ q → t¯ tg qg → t¯ tq gg → t¯ tg q¯ q → t¯ tgg q¯ q → t¯ tq¯ q gg → t¯ tgg

(G. Abelof, AG)

SLIDE 25

Towards jet cross sections at NNLO for hadron colliders:

Progress on the antenna subtraction formalism

Implementation of antenna subtraction for double real

radiation corrections to

Subtraction terms constructed and tested in all unresolved limits

Status of integrated NNLO antennae

Remaining NNLO initial-initial antennae under way

Massive antenna formalism under development

Subtraction terms for top pair production at NNLO under

construction

HP2.3 Firenze Aude Gehrmann-De Ridder

Conclusions

SLIDE 26

HP2.3 Firenze Aude Gehrmann-De Ridder

Backup slides

SLIDE 27

HP2.3 Firenze Aude Gehrmann-De Ridder

e+e- 3 jets and event shapes at NNLO

NNLO results triggered

Progress on resummation

N3LL for 1-T (T. Becher, M. Schwartz, R. Abbate et al.) and

MH (Y. Chien, M. Schwartz)

Progress on hadronization

Shape function approach for 1-T

(R. Abbate et al.)

Dispersive model to NNLO

(T. Gehrmann, G. Luisoni, M. Jaquier)

Reanalysis of data from

LEP/PETRA/Tristan

0.11 0.115 0.12 0.125 0.13

NNLO event shape moments, analytic power corr. (JADE/OPAL: Gehrmann, Jaquier, Luisoni) NNLO+N3LLA thrust, shape function (LEP/PETRA/SLD/AMY: Abbate et al.) NNLO+N3LLA heavy jet mass (ALEPH/OPAL: Chien, Schwartz) NNLO+N3LLA thrust (ALEPH/OPAL: Becher, Schwartz) NNLO three-jet rate (ALEPH: Dissertori et al.) NNLO+NLLA event shapes (JADE: Bethke et al.) NNLO+NLLA event shapes (ALEPH: Dissertori et al.) NNLO event shapes (ALEPH: Dissertori et al.) exp. th. PDG 2010: 0.1184 ± 0.0007