Measurements of S and the
QCD Colour Factors using
Four-Jet Observables from
Hadronic Z Decays
Sílvia Bravo i Gallart
IFAE, Barcelona
Moriond-QCD, 18 March 2002
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Outline
 Motivation for the Studies
 Theoretical Framework
 Analysis description
 Results on measurements of S from R4
 Results on the combined measurement of S
and the Colour Factors
 Conclusions
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Motivation
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QCD =(?) theory for the description of the strong
interaction of quarks and gluons
LEP Experiments, as ALEPH, provided good conditions for
QCD studies
High statistics
High energy
Clean environment
New NLO calculations for 4-jet observables allowed us to perform
Measurements of the strong
coupling constant alone and
together with the QCD colour
factors from 4-jet observables
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1st S measurement
from a 4-jet observable
Stringent test of QCD
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Theoretical Framework
Predictions for Four-Jet Observables
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QCD shows its full gauge
structure only in order S2
R4 + 4-jet angular
correlations a good set of
observables to perform a
stringent test of QCD
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The 4-jet Rate
(resummation of large logarithms exists)


4
2
3
R4 
    B4  ycut  1  2b0   ln x       C 4  ycut   3 B4  ycut 
2
 tot
 ln ycut
m
S
 s  C F
   
2
x  
MZ
n
m ,n
with
Where the Born and NLO functions
B 4  B 04  B 4x x  B 4y y
C4  C04  C4x x  C4y y  C4xx x 2  C4xy xy  C4yy y 2
depend linearly and quadratically on the
Colour Factor Ratios
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x
CA
CF
y
TR
CF
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The 4-jet angular correlations
Bengtsson  Zerwas :
E1  E2  E3  E4
 
 







Cos  BZ  Cos  p1  p2 , p3  p4  
Korner  Schierholt z  Willrodt :
 
 
 
 
1

Cos  KSW   Cos    p1  p4 ,  p2  p3     p1  p3 ,  p2  p4  
2

Modified Nachtmann  Reiter :
 


Cos *NR  Cos  p1  p2 ,  p3  p4  


Two lowest energy jets :
 
Cos α34   Cos   p3 , p4  
1 d
2
3
    B4  A 1  2 b0   lnx       C 4  A
 dA
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The Analysis Description
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Two different analyses but same method
S from R4
Simultaneously S and
the colour factors.
Hadronic Selection on 1994-95 ALEPH Data
Efficiency: 90.2%
Non-hadronic bckg ~0.3%
Durham-E clustering algorithm, ycut=0.008 for 4-jet events
Binned distributions are constructed from data (with
correlations)
• R4 is measured at 60 points -12.0  ln(ycut)  -0.2
• Each angular observable has 20 bins
• NLO prediction from DEBRECEN (resummation for R4,
normalized
distributions for
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i Gallart
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The theoretical distributions are then corrected to HL
After correction for detector effects...
the following 2 is minimized
   i  j
2
1
ij
 j  Dat j  Th jDL
Where the Nentries in the DL vectors (i.e. the fit range) is determined
by a cut in the total correction to each bin – 10% –.
Finally the sources of systematic uncertainty are studied:
Corrections, fit range, selection cuts, etc.
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Angular correlations
CosX i
i bin  
CosX PL i
HL
C
had

bin 
Correction for hadronization effects
& missing higher orders
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Four-jet rate
bin
PL before the showering and
HL & PL dist. obtained from 4parton ME option in PYTHIA
C det  ycut
Hadronization
Corrections
X 4DL  ycut
ibin   HL
X 4  ycut
HL
R
had
4  ycut 
C  ycut   PL
R4  ycut 
PL after showering and HL & PL
dist. obtained from std PYTHIA
Detector
Corrections
ibin 
For all observables
ibin 
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Total Corrections
C
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tot
C
det
C
had
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Measurements of the Strong
Coupling Constant from the
Four-Jet Rate
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Method I : x fixed to 1
Standard in Aleph:
•fit and all systematics with x=1
• scale uncertainty 0.5 < x < 2.
Method II : x also fitted
The strong coupling constant and the scale
are fitted together for all systematics
Method III : x fixed to xopt
Experimentally optimized scale Method. The strong coupling
constant and the scale are fitted. The fitted scale is called
the exp.opt.scale and is fixed to this value for all
systematics. The scale uncertainty 0.5xopt < x <2 xopt
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Method I : x fixed to 1
 S M Z   0.1170  0.0001
Method II : x also fitted
 S  M Z   0.1175  0.0002
x   0.73  0.05
Method III : x fixed to xopt
 S  M Z   0.1175  0.0001
x   0.73
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2
N dof
 27.6
5
2
N dof
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 4.8
4
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Systematic Studies
Fit Range
Selection Criteria
Hadronization Corrections
Detector Corrections
Theoretical Predictions





Other Checks Performed
Hadronization Corrections
Scale dependence
Fits over different ranges



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Method I :
x fixed to 1
S = 0.1170  0.0001(stat)  0.0014(sys)
If systematic error added quadratically then 0.0022
for S
S = 0.1175  0.0002(stat)  0.0013(sys)
Method II :
x also fitted
If systematic error added quadratically then 0.0014
for S
Method III :
x fixed to xopt
S = 0.1175  0.0001(stat)  0.0007(sys)
If systematic error added quadratically then 0.0033
for S
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Results in good agreement with previous ALEPH results,
as well as with a similar analysis by DELPHI
S = 0.1170  0.0001(stat)  0.0014(sys)
Theoretical uncertainty dominates the total
error
0.0022
quad.
ALEPH Preliminary Resummed 3-jet Event Shapes
S = 0.1203  0.0001(stat)  0.0068(sys)
DELPHI Preliminary NLO 4-jet rate, EOS Method
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Durham: S = 0.1178  0.0005(stat) 
0.0029(sys)
Cambridge: S = 0.1203  0.0005(stat) 
0.0021(sys)
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Simultaneous Measurement
of the Strong Coupling
Constant and the QCD
Colour Factors
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The simultaneous measurement of the Strong Coupling
Constant and the QCD Colour Factors has been performed
using resummed predictions for R4 and normalized NLO for
the 4-jet angular observables.
Method as described previously. x was fixed to 1.
  M Z   0.0255  0.0003
 S  M Z   0.119  0.006
x  2.17  0.06
C A  2.93  0.14
y  0.37  0.02
C F  1.35  0.07
y  0.450
 xy  0.845
QCD expected: CA=3 & CF=4/3
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Systematic Studies
Fit Range
Selection Criteria & Detector Corrections = Experimental
Hadronization Corrections
Theoretical Predictions




Other Checks Performed
Hadronization Corrections
Two- and Three-parton Backgrounds
Sensitivity to each observable
Dependence on the ycut




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Final results on the combined
measurement
  M Z   0.0255  0.0003( stat )  0.0014( sys )
x  2.17  0.06( stat )  0.09( sys )
y  0.37  0.02( stat )  0.07( sys )
 S  M Z   0.119  0.006( stat )  0.026( sys )
C A  2.93  0.14( stat )  0.58( sys )
C F  1.35  0.07( stat )  0.26( sys )
OPAL Published Results: S=0.120±0.011(stat)±0.020(sys)
CA=3.02±0.25(stat)±0.49(sys)
CF=1.34±0.13(stat)±0.22(sys)
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Conclusions
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Three different measurements of the strong
coupling constant from the four-jet rate have
been presented, all of them in agreement with
previous results and representing one of the
most precise measurements at present. This is
the first measurement of the strong coupling
constant from a four-jet observable.
A simultaneous measurement of the strong
coupling constant and the colour factors has been
presented, showing a good agreement with
previous results.
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