Post on 22-Feb-2022
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
RGI / Effective Charges in Practise
Klaus Hamacher
Fachbereich Mathematik & NaturwissenschaftenFachgruppe Physik
Bergische Universität Wuppertal& DELPHI Collaboration
Workshop on 1. Principles Non–Perturbative QCDof Hadron Jets
Paris, 12 Janvier 2006
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Outline
1 Introductory CommentsPapersMass Effects & Corrections
2 O(α2s) Results from Shape Distributions
Effect of Changing the Renormalisation Scale
3 O(α2s) Results from Mean Values
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
4 Summary
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
PapersMass Effects & Corrections
Talk is mainly based on:
DELPHI Collab., Consistent measurements of αs from preciseoriented event shape distributions ..., (Eur. Phys. J. C14, 557)
Standard event shapes (depending on polar angle)
DELPHI Collab., A study of the energy evolution of event shapedistributions and their means ..., (Eur. Phys. J. C29, 285)
Data
– radiative DELPHI Z events at ECM =45, 66 and 76 GeV
– DELPHI data from LEP1 and LEP2 (ECM =89 to 202 GeV)
– low energy data from various experiments
Observables
– usual event shape means:T , C, Major , BT , BW , (EEC, JCEF )
– M2h/s/E2
vis in alternative definitions “E/p-scheme”!
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
PapersMass Effects & Corrections
Mass Effects for Event Shapes / Means
Hadron masses influence shape observables.Two types of observables:
• e.g. Thrust : mass–dependence via E–conservation .
• e.g. Jet Masses: direct mass–dependence;avoid by choosing:
(p-scheme) (E-scheme)
(~p, E) −→ (~p,∣∣~p∣∣) (~p, E) −→ (pE , E)
Furthermore:
• Transverse momentum from B–decays .
Estimate effects using Monte Carlo models;b-correction: calculate b+udsc/udsc;
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
PapersMass Effects & Corrections
Mass Effects for Event Shape Means
Ms2/Evis
2
Mh2/Evis
2
Bsum
Bmax
1-Thrust
Major
C
correction to E-scheme
b mass correction
Cor
rect
ion
Fac
tor
ECM[GeV]
0.90.95
11.05
0.90.95
11.05
0.90.95
11.05
0.90.95
11.05
0.90.95
11.05
0.90.95
11.05
20 40 60 80 100 120 140 160 180 200
0.90.95
11.05
full linecorrection to E–schemedashed lineB–correction
=⇒ For Jet-masseschange to E-scheme
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
PapersMass Effects & Corrections
Mass Effects for Event Shapes
udsc
/uds
cb-1
Bmax
-0.04-0.03-0.02-0.01
00.010.020.030.04
0 0.05 0.1 0.15 0.2 0.25 0.3
udsc
/uds
cb-1
Bsum
-0.04-0.03-0.02-0.01
00.010.020.030.04
0 0.050.1 0.150.20.250.3 0.35
Correction slightly changes slope of distributions!
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Effect of Changing the Renormalisation Scale
Description of Event Shape Spectra
O(α2s)–prediction for events shape distributions
1σtot
dσ
df= Af · αs(µ) + (Af · (β0 log xµ − 2) + Bf ) αs(µ)2
• Change of xµ = µ/ECM“turns ” the prediction
• O(α2s) prediction for xµ = 1
in general inconsistent with data
→ αs results depend on fit range
→ Good χ2 only if xµ is optimised !
• Bad features are partly inheritedto the matched predictions.
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Effect of Changing the Renormalisation Scale
αs = αs(T ) – the Worst CaseO(α2
s) from Siggi Hahns thesis Matched
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Effect of Changing the Renormalisation Scale
Compare T , BT , C, MH , y , BW
1- T
Thrust
0.5M
H/E
Heavy
JetM
ass
0.4
0.4
0.6
0.6
0.81.0C
C-param
eter
BT
Tota
lJetB
roadening
0.0
0.00.0
0.00.0
0.2
0.20.2
0.20.2
0.3
0.3
0.30.3
BW
Wide
JetB
roadening
0.0010.01
0.10.1
0.1
0.10.1
0.50.5
0.50.5
0.50.5
0.60.6
0.60.6
0.60.6
0.70.7
0.70.7
0.70.7
0.80.8
0.80.8
0.80.8
0.90.9
0.90.9
0.90.9
1.01.0
1.01.0
1.01.0
2.02.0
2.02.0
2.02.0
y23
Durham
y23
data/fitOPAL (Matth. Ford´s Thesis) my eye fit note:B/A ratios
T , BT , C & 15MH , y small(er)BW negative
non-optimalscales =⇒
biases of αs
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Effect of Changing the Renormalisation Scale
αs Results - ln R Matched Theory
(from M.Fords thesis & LEPQCDWg info)
OPAL ADLO
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
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
Observe the expected pattern:T , BT , (C?) high ; MH , y central ; BW small
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Effect of Changing the Renormalisation Scale
O(α2s) & Experimental Optimisation
lg ( x )
0
20
40
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
0.11
0.12
0.13
0.14
0.151-T
EEC
CJCEF
O D
DELPHI
D2Geneva
D2Jade
D2Durham
xµ quadratic!0.1 0.12 0.14 αS (MZ
2)
EECAEECJCEF1-ThrOCBMaxBSum Hρ S DD2
E0
D2P0
D2P
D2Jade
D2Durham
D2Geneva
D2Cambridge
w. average
DELPHI xµ = 1
α S(MZ2 )
χ 2/ndf
0.1232(116)
71 / 17
0.1 0.12 0.14
xµ exp. opt.
6.2 / 17
0.1168(26)
ρ
ρ
Experimental optimisation strongly improves consistency!
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Effect of Changing the Renormalisation Scale
Experimental vs. ECH Scales
-5
-4
-3
-2
-1
0
1
2
-5 -4 -3 -2 -1 0 1 2
DELPHI
lg (xµECH)
lg (
x µEX
P)
EXP vs. ECHρ = 0.75 ± 0.11
T
Geneva
Bsum
ρD
JCEF
ρH
Durh.
ρs
Cam.
AEEC
D2p0JADE
D2E0
O
EECBmax
C
D2p
In plot xµ quadratic!
Experimental scalescorrelate ρ = 0.75with ECH or PMS scales
Errors:ECH: scale variation in fit rangeEXP: fit error
Influence of mass effectsnot considered !
10σ(P(lg(xECHµ /xEXP
µ ))) ∼ 2.5We “understand” the largerange of opt. scales in MS!
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
Naive Power Corrections
Assess power terms of means using:
〈f 〉 = 〈f 〉O(α2s)
+ 〈f 〉pow
〈fpow 〉 = C1/ECM
Find expected strongly differingpower terms.
Surprising:“Non–perturbative” p.t.’s correlatewith perturbative ratio B/A =⇒.
Power terms predominantly account for higher order pert. effects?
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
Renormalisation Group Invariant (RGI) P.T.
Demand observable R = 〈f 〉/A to fulfil RGE:
QdRdQ
= −bR2(1 + ρ1R + ρ2R2 + . . .) = bρ(R) .
b, ρi are scheme invariant . =⇒
b lnQΛR
=1R− ρ1 ln
(1 +
1ρ1R
)+
vanishes in second order︷ ︸︸ ︷∫ R
0dx
(1
ρ(x)+
1x2(1 + ρ1x)
)Simple RGI applies to observables depending on a single E scale.Numerically RGI=ECH.Exact conversion to MS.ΛR
ΛMS= eB/2Ab
(2c1
b
)−c1/bInclude power terms by:
ρ(x) → ρ(x)− K0
bx−c1/be−1/bx
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
RGI – Description of Shape Observable Means
arbi
trar
y sc
ale
√s–[GeV]
DELPHI
TOPAZAMYMKIIHRSPLUTOTASSOL3JADEDELPHI
<C-Parameter>
<1-Thrust>
<Major>
0.9
1
2
3
4
5
6
7
8
102
arbi
trar
y sc
ale
√s–[GeV]
JADEL3DELPHI
<Bmax>
<Bsum>
<Mh2/Evis
2 E def.>
<Ms2/Evis
2 E def.>
0.9
1
2
3
4
5
6
7
8
102
full lineRGI
dashedMSsame Λ
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
RGI – for 〈y〉
αs comes out somewhat lower ! ⇒ String effect?
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
RGI & Means Need no Significant Power Terms
Fitting RGI with power–terms to manyobservable means yields:
K0 compatible with 0 ⇔ No P.C.!
Virtue of both:RGI and inclusiveness of mean values.
Presence of genuinepower terms for means unclear !
Possible contribution:O(∼ 2%) (rel.) at the Z.
K0
1-Thrust
Major
C-Parameter
Mh2/Evis
2 E def.
Ms2/Evis
2 E def.
Bsum
Bmax
EEC 30o-150o
JCEF 110o-160o
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
Cmp. αs from Means Obtained with Various Methods
RGI yields smallest scatter; 〈αs(MZ )〉 = 0.121± 0.002
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
“Predict” Power Terms using RGI
Set RGI and Power Modelpredictions equal:
〈R〉RGI · A=〈f 〉pert + 〈f 〉pow
Solve for α0.
“Predicted” α0 follows trend of the data.
Agrees better than any presumeduniversal value ∼ 0.5.
α00 0.2 0.4 0.6 0.8 1
C-Parameter
1-Thrust
Mh2/Evis
2 E def.
Ms2/Evis
2 E def.
Bsum
Bmax
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
Measurement of the β–Function
RGI bases on the RGE ,β–function can be measured without prior αs determination!
∂R−1
∂ ln Q2 = βR =β0
4π
(1 +
β1
2πβ0R + ρ2R2 + . . .
)=⇒ measure βR ≈ β directly from a mean event shape.Measurement is valid even “outside” QCD.
Got consistent results from several observables,now use 〈1− T 〉 (most data).
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Naive Power CorrectionRGIRGI vs. Power TermsMeasuring the β–Function
Measurement of the β–Function
Using DELPHI and low energy data:
∂R−1
∂ ln Q= 1.38± 0.05
Uncertainty due to power terms small!
β0 = 7.86± 0.32 nf = 4.75± 0.44
Compare “cross checks” via αs:
LEP evt. shapes & world dataRτ , F2, · · ·
β0 = 7.67± 1.63 β0 = 7.76± 0.44
6
8
10
12
14
16
18
20
22
8 910 20 30 40 50 60 80 100 200Ecm (GeV)
β0 QCD
QCD+Gluinos
β0 fit
DELPHIL3AMYTOPAZJADETASSOPEP5PLUTOHRS
DELPHI
1/ <
1−T
>
light gluinos excluded
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise
Introductory CommentsO(α2
s ) Results from Shape DistributionsO(α2
s ) Results from Mean ValuesSummary
Summary
• O(α2s) predictions with PMS/ECH scales
better describe event shape distributions than MS!χ2/ndf of αs average from 18 observabes:
MS :7117
ECH :1917
exp. opt. :6.217
• Inadequate MS scales bias matched results !
• RGI can describe E dependence of means without power terms.⇒ Leads to a direct measurement of the β–function ...
Klaus Hamacher, Bergische Universität Wuppertal RGI / Effective Charges in Practise