Jets in e Annihilation - SMU PhysicsJets in e+e Annihilation Œ p.16. Simple hadronisation model Y y...
Transcript of Jets in e Annihilation - SMU PhysicsJets in e+e Annihilation Œ p.16. Simple hadronisation model Y y...
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Jets in e+e− AnnihilationNigel Glover
IPPP, University of Durham
CTEQ, Rhodes, July 2006
Jets in e+e− Annihilation – p.1
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Structure of hadronic events
A two jet event A three jet event
Y
XZ
200 . cm.
Cen t r e o f sc r een i s ( 0 . 0000 , 0 . 0000 , 0 . 0000 ) 50 GeV2010 5
Run : even t 4093 : 1000 Da t e 930527 T ime 20716 Ebeam 45 . 658 Ev i s 99 . 9 Emi ss - 8 . 6 V t x ( - 0 . 07 , 0 . 06 , - 0 . 80 ) Bz=4 . 350 Th r us t =0 . 9873 Ap l an=0 . 0017 Ob l a t =0 . 0248 Sphe r =0 . 0073
C t r k (N= 39 Sump= 73 . 3 ) Eca l (N= 25 SumE= 32 . 6 ) Hca l (N=22 SumE= 22 . 6 ) Muon (N= 0 ) Sec V t x (N= 3 ) Fde t (N= 0 SumE= 0 . 0 )
Y
XZ
200 . cm.
Cen t r e o f sc r een i s ( 0 . 0000 , 0 . 0000 , 0 . 0000 ) 50 GeV2010 5
Run : even t 2542 : 63750 Da t e 911014 T ime 35925 Ebeam 45 . 609 Ev i s 86 . 2 Emi ss 5 . 0 V t x ( - 0 . 05 , 0 . 12 , - 0 . 90 ) Bz=4 . 350 Th r us t =0 . 8223 Ap l an=0 . 0120 Ob l a t =0 . 3338 Sphe r =0 . 2463
C t r k (N= 28 Sump= 42 . 1 ) Eca l (N= 42 SumE= 59 . 8 ) Hca l (N= 8 SumE= 12 . 7 ) Muon (N= 1 ) Sec V t x (N= 0 ) Fde t (N= 2 SumE= 0 . 0 )
Jets in e+e− Annihilation – p.2
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Structure of hadronic events
e+
e-
γ
(Z0/γ)*
q
q̄
dū
and many moreuds
dc̄
}
}
}
π-
Λ0 → π-p+
D+ → K0π+
√s’ ~ 1 GeVParton Level Hadron Level
Electro-weak Production Parton Shower Hadronisation
Here we will mostly discuss the hard scattering parton level phaseJets in e+
e− Annihilation – p.3
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O(αs) corrections to e+e− → hadrons
Real Gluon emission
Mqq̄g ∝ gs ⇒ |Mqq̄g|2 ∝ αs
Virtual Gluon emission
Mqq̄ ∼ 1 + αs
Jets in e+e− Annihilation – p.4
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O(αs) corrections to e+e− → hadrons
Note that
|Mqq|^2 =
+
+
x
x
x
O(1)
O(a_s)
O(a_s^2)
At NLO, we are only interested in the interference of the one-loopamplitude with the tree-graph
Jets in e+e− Annihilation – p.5
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Phase space for real emission
Because of momentum con-servation, q, q̄ and g lie in aplane.
q
g_
=
q
Useful variables are the en-ergy fractions and invariantmasses x1
x2
singularities
xi =2Ei√
s, x1 + x2 + x3 = 2
yij =2pi.pj√
s= 1 − xk
Jets in e+e− Annihilation – p.6
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Event shape variables
global observable characterising structure of hadronic event
e.g. Thrust in e+e−
T = max~n
∑ni=1 |~pi · ~n|
∑ni=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( s2) + 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
Jets in e+e− Annihilation – p.7
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O(αs) Thrust distribution
In terms of x1 and x2, the NLO cross section is given by
1
σ0qq̄
d2σ
dx1dx2=
αsCF2π
x21 + x22
(1 − x1)(1 − x2)
For a three particle event,
T = max xi
Therefore, we can divide thephase space into three re-gions corresponding to theThrust value.
x2
x1
x3=T
x2
x1
=T
=T
Jets in e+e− Annihilation – p.8
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O(αs) Thrust distribution
For the region where T = x1,the boundaries are
2(1 − T ) < x2 < T
x1
x1=x2=T
x1=x3=T
1
σ0qq̄
dσ
dT|1 =
αsCF2π
∫ T
2(1−T )
T 2 + x22(1 − T )(1 − x2)
=αsCF
2π
[
1 + T 2
(1 − T ) log(
2T − 11 − T
)
+8 − 14T + 3T 2
2(1 − T )
]
Exercise: do the same for the other two regions.
Jets in e+e− Annihilation – p.9
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O(αs) Thrust distribution
Adding up the three contributions together
1
σ0qq̄
dσ
dT=
αsCF2π
[
2(3T 2 − 3T + 2)T (1 − T ) log
(
2T − 11 − T
)
− 3(3T − 2)(2 − T )1 − T
]
T > 2/3 when x1 = x2 = x3 = TAs T → 1,
1
σ0qq̄
dσ
dT∼ αsCF
2π
[
4
(1 − T ) log(
1
1 − T
)
− 31 − T
]
Virtual contribution at T = 1Expect large hadronisation corrections as T → 1
Jets in e+e− Annihilation – p.10
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O(αs) Thrust distribution
0.6 0.7 0.8 0.9 1T
10-2
10-1
100
101
102
1/σ o
dσ/
dT
OPALO(αs) with αs = 0.2434
deficiency at small T due to kinematic boundshape good 0.75 < T < 0.95
Jets in e+e− Annihilation – p.11
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Spin of the gluon
If the gluon is a scalar, itwould be evident in the eventshape.
Lint ∼ ḡsΨ̄iT aijΦaΨj
leads to
1
σ0qq̄
d2σ
dx1dx2∼ x
23
2(1 − x1)(1 − x2)
1
σ0qq̄
dσ
dT=
ᾱsCF2π
1
2
[
2 log
(
2T − 11 − T
)
+(3T − 2)(4 − 3T )
1 − T
]
Jets in e+e− Annihilation – p.12
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NLO corrections to thrust distribution
At NLO, get contributions from double radiation
and virtual graphs
1
σ0
dσ
dT=
αs(µ)
2πA(T ) +
(
αs(µ)
2π
)2 [
b0A(T ) ln
(
µ2
s
)
+ B(T )
]
renormalisation term and genuine NLO contributionJets in e+e− Annihilation – p.13
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NLO corrections to thrust distribution
0.5 0.6 0.7 0.8 0.9 1T
10-1
100
101
102
103
104
105
1/σ o
dσ/
dT
AB
0.6 0.7 0.8 0.9 1T
10-2
10-1
100
101
102
1/σ o
dσ/
dT
OPALO(αs) with αs = 0.2434O(αs2) with αs = 0.14
As T → 1, A divergespositive, B divergesnegativeT > 1/
√3 at NLO
Better agreement overwider range of TMore sensible value of αsStill problems as T → 1
Jets in e+e− Annihilation – p.14
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Large logarithms in thrust distribution
As T → 1,
1
σ0qq̄
dσ
dT∼ αsCF
2π
[
4
(1 − T ) log(
1
1 − T
)
− 31 − T
]
define cross section for T > τ which is fraction of events with T > τ ,
R(τ) =
∫ 1
τ
dT1
σ0qq̄
dσ
dT
∼ 1 − αsCFπ
ln2(1 − τ)
Singularity at T = 1 cancelled by one-loop two-parton contribution
When αs ln2(1 − τ) large, i.e. τ ∼ 0.95 cannot trust perturbationtheory. In fact,
R(τ) ∼ 1 − αsCFπ
ln2(1 − τ) + 12
(
αsCFπ
)2
ln4(1 − τ)Jets in e+e− Annihilation – p.15
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Large logarithms in thrust distribution
R(τ) ∼ exp(
−αsCFπ
ln2(1 − τ))
so that as τ → 1, R(τ) → 0
This is the SUDAKOV form factor effectfor event to have very high thrust, must have radiated very fewgluons
⇒ very improbablec.f data, very improbable to have only 2 hadron event
can also resum next-to-leading logs for many event shapes sothat calculations believable when αs ln(1 − τ) small
Jets in e+e− Annihilation – p.16
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Simple hadronisation model
yY
pT
dy
Parton produces a tube in (y, pT ) space of light hadrons w.r.t. initialparton direction with transverse mass density µ
Ejet = µ
Z Y
0
cosh y dy = µ sinh Y
Pjet = µ
Z Y
0
sinh y dy = µ(cosh Y − 1)
⇒ m2jet = E2jet − P 2jet = 2µPjet or Ejet ∼ Pjet + µ
where µ ∼ 0.5 − 1 GeV from experiment Jets in e+e− Annihilation – p.17
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Hadronisation and Thrust
2 jet event:
Tparton = 1
Thadron =2Pjet
Q=
2(Ejet − µ)Q
= 1 − 2µQ
i.e.δT = −2µ
Q
Mean value of thrust
〈1 − T 〉 = 0.33αs + 1.0α2s
+1 GeV
Q
Mark JTASSODELCOMark II
JADE
AMYDELPHIL3ALEPHOPALSLD
√−−s [GeV]
〈1-T
〉
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 20 40 60 80 100 120 140 160 180
Jets in e+e− Annihilation – p.18
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The triple gluon vertex
Four jet matrix elements aresensitive to the triple gluonvertex - (makes event moreplanar)
To quantify this effect studye.g.Bengtsson-Zerwas angle
cos θBZ =( ~p1 × ~p2).( ~p3 × ~p4)
| ~p1|| ~p2|| ~p3|| ~p4|
for four jets (Ei, ~pi) with E1 >E2 > E3 > E4
Jets in e+e− Annihilation – p.19
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Probing non-abelian structure of QCD
Probabilities for parton splitting
2
2
2 q → qg ∝ CF αs
g → gg ∝ CAαs
g → qq̄ ∝ TRαs
In QCD,
CF =N2 − 1
2N, CA = N, TR =
1
2
All splittings present in O(α2s) event shapes, i.e.
B(T ) = CF α2s (CF BCF (T ) + CABCA(T ) + TRBTR(T ))
which can be fit to data Jets in e+e− Annihilation – p.20
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Probing non-abelian structure of QCD
Fixing the quadratic casimirsof QCD in 4 jet events
OPALCA 3.02 ± 0.25 ± 0.49
CF 1.34 ± 0.13 ± 0.22
αs(MZ) 0.120 ± 0.011 ± 0.020
ALEPHCA 2.93 ± 0.14 ± 0.49
CF 1.35 ± 0.07 ± 0.22
αs(MZ) 0.119 ± 0.006 ± 0.022
ExpectCA/CF = 9/4 TR/CF = 3/8
ALEPH
0
0.2
0.4
0.6
0.8
1
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25CA/CF
T R/C
F
SO(2)
SU(4)
68% CL contour
ALEPH-1997 OPAL-2001
Jets in e+e− Annihilation – p.21
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Probing non-abelian structure of QCD
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6
U(1)3
SU(1)
SU(2)
SU(4)
SU(5)Combined resultSU(3) QCD
ALEPH 4-jet
OPAL 4-jet
Event Shape
OPAL NggDELPHI FF
CF
CA
86% CL error ellipses
CA = 2.89 ± 0.01 ± 0.21(syst)CF = 1, 30 ± 0.01 ± 0.09(syst)
Jets in e+e− Annihilation – p.22
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Differences between quark and gluon jets
QCD predicts that quarks andgluons fragment differentlybecause of their differentcolour charges
Fundamental prediction:the number of soft gluonsemitted within a gluon jetshould be ∼twice that forquark jet
rg/q ≡〈n〉gluon〈n〉quark
∼ CACF
= 2.25
Valid for soft particles and un-biased jets.For hard gluon emission,quarks and gluons are similarrg/q ∼ 1
5
10
15
20
25
10 102
Ecm
, pT
[GeV]
Nch
Nch
from e+e
- to qq
Fit
Ngg
=2(Nqqg
- Nqq
)
Ngg
(CLEO)
Ngg
(OPAL)
Eden (a)
DELPHI
Jets in e+e− Annihilation – p.23
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The running coupling in perturbative QCD
dαs/d lnµ2 = −β0 α2s − β1 α3s − β2 α4s − β3 α5s − . . .
Four-loop coeff.:van Ritbergen, Vermaseren, Larin; Czakon
-0.2
-0.1
0
0 0.1 0.2 0.3 0.4 0.5αS
β(αS)
1−loop2−loop3−loop4−loop
nf = 4, MS
µ2 (GeV2)
αS( µ2)
αS(M 2 ) = 0.115
3...5 flavours
Z
↓↓
↓
Zϒ
J/ψ
0
0.1
0.2
0.3
0.4
0.5
0.6
1 10 10 2 10 3 10 4
Jets in e+e− Annihilation – p.24
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The running coupling from LEP
Combined results for αs(MZ)
τ decays : = 0.1180 ± 0.0030RZ : = 0.1226
+0.0058−0.0038
shapes : = 0.1202 ± 0.0050
Final combined result from LEP
αs(MZ) = 0.1195 ± 0.0034
Errors dominated by theoretical uncertainties.Bethke, hep-ex/0406058
Jets in e+e− Annihilation – p.25
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e+e− event shapes
Current state of the art is3 NLO perturbation theory,3 more sophisticated NLL
resummation3 better modelling of
hadronisation correc-tions
PSfrag replacements
0.110
0.110
0.115
0.115
0.120
0.120
0.125
0.125
0.130
0.130
αS(MZ)
αS(MZ)
T
T
MH
MH
C
C
BT
BT
BW
BW
y23
y23
All
All
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( s2) + 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
Jets in e+e− Annihilation – p.26
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Why go beyond NLO?
In many cases, the uncertainty from the pdf’s and from the choice ofrenormalisation scale give uncertainties that are as big or biggerthan the experimental errors.e.g. theoretical uncertainties in αs extraction from pp̄ → jet are due torenormalisation scale and pdf’s
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
Stro
ng C
oupl
ing
Cons
tant
αs(E
T)
0
10
0 50 100 150 200 250 300 350 400 450
%
Systematic uncertainties
Transverse Energy (GeV)
0.110.120.130.140.15
100 200 300 400 (GeV)
α s(M
Z)
0.10
0.12
0.14
0.16
0 50 100 150 200 250 300 350 400 450
αs(MZ) as function of ET for µ=ETUncertainties due to the µ scale
(a)
0.10
0.12
0.14
0.16
0 50 100 150 200 250 300 350 400 450
αs(MZ) as function of ET for CTEQ4MUncertainties due to the PDF choice
(b)
Transverse Energy (GeV)
Stro
ng C
oupl
ing
Cons
tant
αs(M
Z)
αs(MZ) = 0.1178+6%−4%(scale)
+5%−5%(pdf) Jets in e+e− Annihilation – p.27
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Why do we vary renormalisation scale?
• The theoretical prediction should be independent of µR• The change due to varying the scale is formally higher order. If
an observable Obs is known to order αNs then,
∂
∂ ln(µ2R)
N∑
0
An(µR)αns (µR) = O
(
αN+1s)
.
• So the uncertainty due to varying the renormalisation scale isway of guessing the uncalculated higher order contribution.
Jets in e+e− Annihilation – p.28
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Why do we vary renormalisation scale?
• . . . but the variation only produces copies of the lower orderterms
Obs = A0αs(µR) +(
A1 + b0A0 ln
(
µ2Rµ20
))
αs(µR)2
A1 will contain logarithms and constants that are not present inA0 and therefore cannot be predicted by varying µR.For example, A0 may contain infrared logarithms L up to L2,while A1 would contain these logarithms up to L4.
• µR variation is only an estimate of higher order terms• A large variation probably means that predictable higher order
terms are large - but doesnt say anything about A1.
Jets in e+e− Annihilation – p.29
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Renormalisation scale dependence
For example, pp̄ → jet, scale dependence
dσ
dET= α2s(µR)A
+ α3s(µR) (B + 2b0LA)
+ α4s(µR)(
C + 3b0LB + (3b20L
2 + 2b1L)A)
with L = log(µR/ET ). The NNLO coefficient C is unknown.
The curves show guessesC = 0 (solid) and C = ±B2/A(dashed).Scale dependence is signifi-cantly reduced.
1 2 3 4 5
µ_R / E_T
0
0.2
0.4
0.6
0.8
1
dσ /
dE_
T at
E_T
= 10
0 G
eV
LONLO"NNLO""NNLO"+"NNLO"-
Jets in e+e− Annihilation – p.30
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Jet algorithms
Also there is a mismatch between the number of hadrons and thenumber of partons in the event. At NLO at most two partons make ajet - while at NNLO three partons can combine to form the jet
LO NLO NNLO
Perturbation theory starts to reconstruct the shower⇒ better matching of jet algorithm between theory and experiment⇒ need for better jet algorithms
Jets in e+e− Annihilation – p.31
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Description of the initial state
LO At lowest order final state has no transverse momentum
NLO Single hard radiation gives final state transverse momentum,even if no additional jet observed
Jets in e+e− Annihilation – p.32
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Description of the initial state
NNLO Double radiation on one side or single radiation off eachincoming particle gives more complicated transversemomentum to final state
Jets in e+e− Annihilation – p.33
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Higher orders and power corrections
NLO Phenomenological power corrections match data withcoefficient of 1/Q extracted from data.
〈1 − T 〉 ∼ 0.33αs + 1.0α2s +λ
Q
At NLO, λ ∼ 1 GeV gives a good description of the data.
〈1−T 〉 with NLO and no powercorrection and NLO withpower correction λ = 1 GeV.
The power correction param-eterises the unknown higherorders as well as the genuinenon-perturbative correction 0
0.05
0.1
0.15
0.2
20 40 60 80 100 120 140 160 180 200Q (GeV)
Jets in e+e− Annihilation – p.34
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Higher orders and power corrections
NNLO Higher orders partially remove need for power correction
〈1 − T 〉 ∼ 0.33αs + 1.0α2s + Aα3s +λ GeV
Q
If we guess A = 3, then λ = 0.5 GeV is good fit.
〈1 − T 〉 with NLO andλ = 1 GeV, "NNLO" withλ = 0.5 GeV and "All orders"with no power correction.
At present data not goodenough to tell difference be-tween 1/Q and 1/ log(Q/Λ)3. 0
0.05
0.1
0.15
0.2
20 40 60 80 100 120 140 160 180 200Q (GeV)
Jets in e+e− Annihilation – p.35
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Gauge boson production at the LHC
Jets in e+e− Annihilation – p.36
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Gauge boson production at the LHC
Gold-plated processAnastasiou, Dixon, Melnikov, Petriello
At LHC NNLO perturbative accuracy better than 1%⇒ use to determine parton-parton luminosities at the LHC
Jets in e+e− Annihilation – p.37
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Event shapes at NNLO
Two-loop matrix elements
|M|22-loop,3 partons explicit infrared poles from loop integralsGarland, Gehrmann, Glover, Koukoutsakis,
Remiddi:Moch, Uwer, Weinzierl
One-loop matrix elements
|M|21-loop,4 partonsexplicit infrared poles from loop integral andimplicit infrared poles due to single unresolved ra-diation
Bern, Dixon, Kosower, Weinzierl;Campbell, Miller, Glover
Tree level matrix elements|M|2tree,5 partons implicit infrared poles due to double unresolved ra-
diationHagiwara, Zeppenfeld;Berends, Giele, Kuijf
Infrared Poles cancel in the sumJets in e+e− Annihilation – p.38
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QED-type contributions to e+e− → 3 jets
Gehrmann-De Ridder, Gehrmann, Glover, Heinrich
-30
-20
-10
0
10
20
30
40
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51-T
1/N2y0 = 10
-5
y0 = 10-6
y0 = 10-7
-800-700-600-500-400-300-200-100
0 100 200 300
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51-T
NF/N
y0 = 10-5
y0 = 10-6
y0 = 10-7
-1000
0
1000
2000
3000
4000
5000
6000
7000
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51-T
NF2
y0 = 10-5
y0 = 10-6
y0 = 10-7
independent of phasespace cut y0CPU time about 1 day on2.8 GHz Athlon
Jets in e+e− Annihilation – p.39
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Summary
QCD studies at LEP have led to a significant increase of knowledgeabout hadron production and the dynamics of quarks and gluons athigh energies.
These studies demonstrate QCD as a consistent theory whichaccurately describes the phenomenology of the Strong Interaction
Future developments in this field are within reach: NNLO QCDcalculations and predictions for jet and event shape observables willsoon be available; they will initiate further analyses of the LEP datawhich will provide even more accurate and more detaileddeterminations of αs
Bethke, hep-ex/0406058
Jets in e+e− Annihilation – p.40
Structure of hadronic eventsStructure of hadronic events${cal O}(alpha _s)$corrections to $e^+e^- o $ hadrons${cal O}(alpha _s)$corrections to $e^+e^- o $ hadronsPhase space for real emissionEvent shape variables${cal O}(alpha _s)$Thrust distribution${cal O}(alpha _s)$Thrust distribution${cal O}(alpha _s)$Thrust distribution${cal O}(alpha _s)$Thrust distributionSpin of the gluonNLO corrections to thrust distributionNLO corrections to thrust distributionLarge logarithms in thrust distributionLarge logarithms in thrust distributionSimple hadronisation modelHadronisation and ThrustThe triple gluon vertexProbing non-abelian structure of QCDProbing non-abelian structure of QCDProbing non-abelian structure of QCDDifferences between quark and gluon jetsThe running coupling in perturbative QCDThe running coupling from LEP$e^+e^-$ event shapesWhy go beyond NLO?Why do we vary renormalisation scale?Why do we vary renormalisation scale?Renormalisation scale dependenceJet algorithmsDescription of the initial stateDescription of the initial stateHigher orders and power correctionsHigher orders and power correctionsGauge boson production at the LHCGauge boson production at the LHCEvent shapes at NNLOQED-type contributions to $e^+e^- o 3$~jetsSummary