e + e 3 jets and event shapes Classical QCD observable testing - - PowerPoint PPT Presentation

▶
e e 3 jets and event shapes
SMART_READER_LITE
LIVE PREVIEW

e + e 3 jets and event shapes Classical QCD observable testing - - PowerPoint PPT Presentation

Nov 03, 2022 306 likes •526 views

e + e 3 jets at NNLO Thomas Gehrmann in collaboration with: A. Gehrmann-De Ridder, E.W.N. Glover, G. Heinrich Universit at Z urich S T A U T R I S I C R E E N V I S N I S U MDCCC XXXIII RADCOR 2007 e + e

slide-1
SLIDE 1

e+e− → 3 jets at NNLO

Thomas Gehrmann in collaboration with: A. Gehrmann-De Ridder, E.W.N. Glover, G. Heinrich Universit¨ at Z¨ urich

T U R I C E N S I S U N I V E R S I T A S

XXXIII MDCCC

RADCOR 2007

e+e− → 3 jets at NNLO – p.1

slide-2
SLIDE 2

e+e− → 3 jets and event shapes

Classical QCD observable

testing ground for QCD: perturbation theory, power corrections and logarithmic resummation precision measurement of strong coupling constant αs current error on αs from jet observables dominated by theoretical uncertainty:

  • S. Bethke, 2006

αs(MZ) = 0.121 ± 0.001(experiment)±0.005(theory) theoretical uncertainty largely from missing higher orders current status: NLO plus NLL resummation

Theoretical description

easier than at hadron colliders, since coloured partons only in final state: no initial state emission, no parton distributions new calculational methods first developed for e+e−, then extended to hadronic processes

e+e− → 3 jets at NNLO – p.2

slide-3
SLIDE 3

e+e− → 3 jets and event shapes

Event shape variables

assign a number x to a set of final state momenta: {p}i → x e.g. Thrust in e+e− T = max

n

Pn

i=1 |

pi · n| Pn

i=1 |

pi| limiting values: back-to-back (two-jet) limit: T = 1 spherical limit: T = 1/2

Ecm=91.2 GeV Ecm=133 GeV Ecm=161 GeV Ecm=172 GeV Ecm=183 GeV Ecm=189 GeV Ecm=200 GeV Ecm=206 GeV

T ALEPH

O(s

2) + NLLA

1/ d/dT 10

  • 2

10

  • 1

1 10 10 2 10 3 10 4 10 5 10 6 10 7 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

e+e− → 3 jets at NNLO – p.3

slide-4
SLIDE 4

e+e− → 3 jets and event shapes

Standard Set of LEP Thrust (E. Farhi) T = max

n

n X

i=1

| pi · n| ! / n X

i=1

| pi| ! Heavy jet mass (L. Clavelli, D. Wyler) M2

i /s =

1 E2

vis

„ X

k∈Hi

| pk| «2 C-parameter: eigenvalues of the tensor (G. Parisi) Θαβ = 1 P

k |

pk| P

k pα k pβ k

P

k |

pk| Jet broadenings (S. Catani, G. Turnock, B. Webber) Bi = „ X

k∈Hi

| pk × nT | « / „ 2 X

k

| pk| « BW = max(B1, B2) BT = B1 + B2 3j → 2j transition parameter in Durham algorithm yD

23

S.Catani, Y.L.Dokshitzer, M.Olsson, G.Turnock, B.Webber

0.06 0.08 0.1 0.12 0.14 0.16 0.18

αS (MZ2)

EEC AEEC JCEF 1-Thr O C BMax BSum ρH ρS ρD D2E0 D2P0 D2P D2Jade D2Durham D2Geneva D2Cambridge

  • w. average :

αS(MZ2) = 0.1232 ± 0.0116 χ2/ndf = 71 / 17 ρeff = 0.635 ferr = 3.38

DELPHI

xµ = 1 e+e− → 3 jets at NNLO – p.4

slide-5
SLIDE 5

e+e− → 3 jets and event shapes

Current status: NLO and NLL

NLO calculations of event shapes and 3j R.K. Ellis, D.A. Ross, A.E. Terrano; Z. Kunszt

  • J. Vermaseren, K.F

. Gaemers, S.J. Oldham; L. Clavelli, D. Wyler

  • K. Fabricius, I. Schmitt, G. Kramer, G. Schierholz

NLO parton level event generators for 3j EVENT: Z. Kunszt, P . Nason EERAD: W. Giele, E.W.N. Glover EVENT2: S. Catani, M. Seymour NLO parton level event generators for 4j MenloParc: L.D. Dixon, A. Signer EERAD2: J. Campbell, M. Cullen, E.W.N. Glover Debrecen: Z. Nagy, Z. Trocsanyi Mercurito: D. Kosower, S. Weinzierl NLL resummation

  • S. Catani, L. Trentadue, G. Turnock, B. Webber

Power corrections

  • G. Korchemsky, G. Sterman; Y. Dokshitzer, B.R. Webber

e+e− → 3 jets at NNLO – p.5

slide-6
SLIDE 6

Ingredients to NNLO e+e− → 3-jet

Two-loop matrix elements |M|2 2-loop,3 partons explicit infrared poles from loop integrals

  • L. Garland, N. Glover, A. Koukoutsakis, E. Remiddi, TG

(RADCOR 00/02);

  • S. Moch, P

. Uwer, S. Weinzierl One-loop matrix elements |M|2 1-loop,4 partons explicit infrared poles from loop integral and implicit infrared poles due to single unresolved radiation

  • Z. Bern, L. Dixon, D. Kosower, S. Weinzierl;
  • J. Campbell, D.J. Miller, E.W.N. Glover

Tree level matrix elements |M|2 tree,5 partons implicit infrared poles due to double unresolved radiation

  • K. Hagiwara, D. Zeppenfeld;

F .A. Berends, W.T. Giele, H. Kuijf;

  • N. Falck, D. Graudenz, G. Kramer

Infrared Poles cancel in the sum

e+e− → 3 jets at NNLO – p.6

slide-7
SLIDE 7

NNLO Infrared Subtraction

Structure of NNLO m-jet cross section: dσNNLO = Z

dΦm+2

“ dσR

NNLO − dσS NNLO

” + Z

dΦm+1

“ dσV,1

NNLO − dσV S,1 NNLO

” + Z

dΦm

dσV,2

NNLO +

Z

dΦm+2

dσS

NNLO +

Z

dΦm+1

dσV S,1

NNLO ,

dσS

NNLO: real radiation subtraction term for dσR NNLO

dσV S,1

NNLO: one-loop virtual subtraction term for dσV,1 NNLO

dσV,2

NNLO: two-loop virtual corrections

Each line above is finite numerically and free of infrared ǫ-poles − → numerical programme

e+e− → 3 jets at NNLO – p.7

slide-8
SLIDE 8

Numerical Implementation

Structure of e+e− → 3 jets program:

5 parton channel 4 parton channel 3 parton channel dΦq¯

qggg

dΦq¯

qgg

dΦq¯

qg

Monte Carlo Phase Space dσR

NNLO − dσS NNLO

dσV,2

NNLO

+

  • dσV S,1

NNLO dΦX3

+

  • dσS

NNLO dΦX4

dσV,1

NNLO − dσV S,1 NNLO

✲ {pi}5 ✲ {pi}4 ✲ {pi}3 Cross section ✲ {pi}5, w ✲ {pi}4, w ✲ {pi}3, w Definition of Observables 5 parton → 3 jet 4 parton → 3 jet 3 parton → 3 jet w, {C, S, T} w, {C, S, T} w, {C, S, T} ✲ ✲ ✲ ✲ ⊕ Histograms

σ3j

dσ/dT dσ/dS dσ/dC

e+e− → 3 jets at NNLO – p.8

slide-9
SLIDE 9

Numerical Implementation

Antenna subtraction

NLO: M. Cullen, J. Campbell, E.W.N. Glover; D. Kosower; A. Daleo, D. Maitre, TG NNLO: A. Gehrmann-De Ridder, E.W.N. Glover, TG (RADCOR 05) construct subtraction terms from physical 1 → 3 and 1 → 4 matrix elements each antenna function interpolates between all limits associated to one or two unresolved partons integrated subtraction terms cancel infrared pole structure of two-loop matrix element

  • S. Catani; G. Sterman, M.E. Yeomans-Tejeda

Checks

cancellation of infrared poles in 3-parton and 4-parton channel convergence of subtraction terms towards matrix elements along phase space trajectories distributions in raw phase space variables independence on phase space cut y0

e+e− → 3 jets at NNLO – p.9

slide-10
SLIDE 10

Colour structure at NNLO

Decomposition into leading and subleading colour terms

dσNNLO = (N2 − 1) " N2 ANNLO + BNNLO + 1 N2 CNNLO + NNF DNNLO + NF N ENNLO + N2

F FNNLO + NF,γ

„ 4 N − N « GNNLO # last term: closed quark loop coupling to vector boson NF,γ = “P

q eq

”2 P

q e2 q

was found to be O(10−4) in NLO 4j final states L.D. Dixon, A. Signer will be negelected here

e+e− → 3 jets at NNLO – p.10

slide-11
SLIDE 11

Event shapes at NNLO

NNLO expression for Thrust

(1 − T) 1 σhad dσ dT = “αs 2π ” A(T) + “αs 2π ”2 (B(T) − 2A(T)) + “αs 2π ”3 (C(T) − 2 B(T) − 1.64 A(T)) with LO contribution A(T), NLO contribution B(T)

10 20 30 0.1 0.2 0.3 0.4

(1-T) d A d T

1-T

100 200 300 0.1 0.2 0.3 0.4

(1-T) d B d T

1-T

e+e− → 3 jets at NNLO – p.11

slide-12
SLIDE 12

Event shapes at NNLO

Individual colour structures

10000 20000 0.1 0.2 0.3 0.4

N2

  • 2500

2500 5000 7500 0.1 0.2 0.3 0.4

N0

  • 200
  • 100

100 0.1 0.2 0.3 0.4

1/N2

  • 20000
  • 10000

0.1 0.2 0.3 0.4

NF N (1-T)d C d T 1-T

  • 500
  • 250

250 500 0.1 0.2 0.3 0.4

NF/N 1-T

  • 2000

2000 4000 6000 0.1 0.2 0.3 0.4

N2

F

1-T

dominated by leading colour N2 and NF N sizable contributions from N0, NF /N and N2

F

negligible contribution from 1/N2

e+e− → 3 jets at NNLO – p.12

slide-13
SLIDE 13

Results

NNLO thrust distribution

2000 4000 6000 8000 0.1 0.2 0.3 0.4

(1-T) d C d T

1-T

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4

1-T (1-T) 1/σhad dσ/d T

Q = MZ αs (MZ) = 0.1189 NNLO NLO LO

NNLO corrections sizable theory error reduced by 30–40 % large 1 − T: need hadronization corrections small 1 − T: two-jet region, need matching onto NLL resummation Work in progress: G. Luisoni, TG mean value 1 − T: A = 2.101 B = 44.98 C = 1095 ± 130 1 − T(αs = 0.1189) = 0.0398 + 0.0146 + 0.0068

e+e− → 3 jets at NNLO – p.13

slide-14
SLIDE 14

Results

NNLO heavy jet mass and C-parameter

heavy jet mass M 2

H/s

C-parameter

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4

NNLO NLO LO Q = MZ αs (MZ) = 0.1189

MH MH 1/σhad dσ/d MH

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1

NNLO NLO LO Q = MZ αs (MZ) = 0.1189

C C 1/σhad dσ/d C

heavy jet mass (closely related to thurst) has very small NNLO corrections NNLO corrections for C large again require matching onto NLL resummation and hadronization corrections Sudakov shoulder in C = 0.75 also requires resummation

  • S. Catani, B. Webber

e+e− → 3 jets at NNLO – p.14

slide-15
SLIDE 15

Results

NNLO jet broadenings

wide jet boadening BW total jet boadening BT

0.2 0.4 0.6 0.1 0.2 0.3

NNLO NLO LO Q = MZ αs (MZ) = 0.1189

BW BW 1/σhad dσ/d BW

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4

NNLO NLO LO Q = MZ αs (MZ) = 0.1189

BT BT 1/σhad dσ/d BT

relative magnitude of NNLO corrections smaller than for thurst NNLO corrections for BW smaller than for BT again require matching onto NLL resummation and hadronization corrections

e+e− → 3 jets at NNLO – p.15

slide-16
SLIDE 16

Summary and Outlook

completed calculation of NNLO corrections to event shapes related to e+e− → 3j constructed parton-level event generator, based on antenna subtraction method size of NNLO corrections not uniform: small: BW , MH/s, moderate: BT , substantial: C, T still running: yD

23, RD 3j

comparison with data just started next steps: matching onto resummation, hadronization corrections

e+e− → 3 jets at NNLO – p.16

slide-17
SLIDE 17

Backup

e+e− → 3 jets at NNLO – p.17

slide-18
SLIDE 18

Thrust at NNLO

Checks

  • 10000
  • 6000
  • 2000

2000 6000 10000 x 10 2 4 6 8 10

N2

  • 80000
  • 60000
  • 40000
  • 20000

20000 2 4 6 8 10

N0

  • 2000

2000 4000 6000 8000 10000 2 4 6 8 10

1/N2

  • 1000

1000 2000 3000 x 10 2 2 4 6 8 10

NF N d C d T

  • Ln(1-T)
  • 40000
  • 30000
  • 20000
  • 10000

10000 2 4 6 8 10

NF/N

  • Ln(1-T)

10000 20000 30000 2 4 6 8 10

N2

F

y0 = 10-7 y0 = 10-6 y0 = 10-5

  • Ln(1-T)

e+e− → 3 jets at NNLO – p.18

slide-19
SLIDE 19

Errors at NNLO

5 10 15 20 25 0.1 0.2 0.3 0.4

NNLO NLO LO

MH δ (%)

5 10 15 20 25 0.2 0.4 0.6 0.8 1

NNLO NLO LO

C δ (%)

5 10 15 20 25 0.1 0.2 0.3

NNLO NLO LO

BW δ (%)

5 10 15 20 25 0.1 0.2 0.3 0.4

NNLO NLO LO

BT δ (%)

e+e− → 3 jets at NNLO – p.19

slide-20
SLIDE 20

Numerical computation

zBox1 and zBox2 supercomputers

zBox1 288 processors, 2.2 GHz AMD Athlon 0.57 TFlops built in-house from

  • ff-the-shelf components
  • J. Stadel, B. Moore

zBox2 500 processors, 2.6 GHz Opteron 852 5.2 TFlops built by Sun microsystems used mostly by our computational astrophysics group

e+e− → 3 jets at NNLO – p.20