RGI / Effective Charges in Practise

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




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





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”!






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;






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






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!






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.






αs = αs(T ) – the Worst CaseO(α2

s) from Siggi Hahns thesis Matched






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






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






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!






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!






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?






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






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






RGI – for 〈y〉

αs comes out somewhat lower ! ⇒ String effect?






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






Cmp. αs from Means Obtained with Various Methods

RGI yields smallest scatter; 〈αs(MZ )〉 = 0.121± 0.002






“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






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).






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





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 ...





Summary

RGI measurement of theβ–functionnicely confirmsthe QCD gauge group!


RGI / Effective Charges in Practise

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