ÉCD Analysis of ventshape Moments in e·e

QCD Analysis of Eventshape Moments
in e+e -Annihilation
Christoph Pahl
Max-Plank-Institut
fu
 r Physik, Mu
 nhen
Ringberg workshop on non-perturbative QCD of jets
8-10 January 2007
Shloss Ringberg
(last):
nnvgl.
nn
alte Vortr
age, insb. OB's Korr.
.
Measurement
Theoretial onepts
Perturbative tests of moments and variane
Nonperturbative tests:
{ Dispersive model
{ Shape funtion
{ Single dressed gluon approximation
Conlusion
Thrust
1 T.
1
0P
j
p
~
n
j
i
i
A;
T = max P
~n
Thrust-axis
i
~nT , two hemispheres H1;2 .
jpij
Like T , however ~nT ?~nT .
Thrust minor T : Like T , however ~
nT ?~nT ?~nT .
Oblateness O = T : T : .
Thrust major
Tmaj:
maj:
min
maj
Spheriity
maj:
min:
min
S.
S =
P ip p
P i 2i ;
i pi
Eigenvalues < < ; S = (Q + Q ) .
1
2
3
2
3
C -Parameter.
1
2
P
(
pipi )=jpij
= P jp j ;
i i
Eigenvalues j , C = 3( + + ) .
i
1
Jet Broadening
2
2
3
3
1
B T , BN , BW .
1
0P
j
p
~
n
j
H
i
T
i
2
A
Bk = P
2 i jpij
B = B + B , B = min(B ; B ) , B = max(B ; B ) .
Normed jet mass M , M . Invariant mass M ; in hemispheres H ; .
p
p
M = max(M ; M )= s , M = min(M ; M )= s .
Durham twojet - transition parameter y23 . Durham jet sheme with resolution variable
2min(Ei ; Ej ) (1 os );
y=
k
T
1
2
N
H
H
1
2
1
2
W
1
L
2
12
L
1
2
y23 = yut , where 2 7! 3 jets.
2
Evis
12
2
2
ij
.
Moments of eventshape distributions
statistial error
hyni = 1
Z
:
tot
d dy ; n = 1:::5 :
dy
h
y ni hyn i
(hyni) =
N
2
1/σtot. (1-T) dσ/d(1-T)
2
1/σtot. dσ/d(1-T)
yn
10
7.5
5
2.5
0
2
:
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
(1-T)
1/σtot. (1-T) dσ/d(1-T)
0.03
0.006
0.02
3
2
1/σtot. (1-T) dσ/d(1-T)
(1-T)
0.01
0.004
0.002
0
0
0.1
0.2
0.3
0.4
0
0.5
0
(1-T)
-2
0.15
0.1
0.2
0.3
0.4
0.5
(1-T)
-3
0.6
0.4
5
4
1/σtot. (1-T) dσ/d(1-T)
1/σtot. (1-T) dσ/d(1-T)
x 10
x 10
0.5
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.2
0
0
(1-T)
1 (1
tot
:
T )n d(1dT ) , n=1...5 .
0.1
0.2
0.3
0.4
0.5
(1-T)
Measurement
.
10 5
LEP
σ [pb]
10
4
_
e+ e → Hadrons
CESR DORIS
10 3
PEP PETRA


√s’/ s > 0.10
TRISTAN
10
2
+ _
_
e+e →γ γ
+ _
e e →µ µ
10
0
20
40

√s’/s > 0.85
60
80
100 120 140 160_ 180
√s [GeV]
p
s
Integrated
number of
(GeV)
luminosity
(pb 1 )
seleted
14
1.46
1722
22
2.41
1383
35
154.0
34860
38
8.28
1584
44
28.8
3896
91
14.7
395695
133
11.26
630
177
78.16
1576
197
629.23
9193
JADE: 9% bb
OPAL: 2..6% 4f
events
Results at Hadronlevel, experimental
.
orretions applied:
1
10
10
〈Cn〉
〈MHn 〉
〈(1-T)n〉
PYTHIA
HERWIG
ARIADNE
1
1
-1
10
-2
-1
n=1
10
10
10
-3
10
-2
n=3
n=4
n=5
200
100
200
100
〈BTn 〉
n
n
〉
〈BW
1
1
10
-1
-1
〈y23 〉
PYTHIA
HERWIG
ARIADNE
10
10
10
-2
10
10
n=2
-4
n=4
10
200
-3
-4
-4
√
s (GeV)
n=1
n=3
10
100
-2
-3
-3
10
-1
-2
10
10
200
√
s (GeV)
√
s (GeV)
√
s (GeV)
10
n=2
-4
100
10
-1
100
200
√
s (GeV)
-5
n=5
100
200
√
s (GeV)
Monte arlo models reproduing data well.
.
Variane of eventshape distributions:
-2
x 10
Var (MH)
Var (1-T)
0.525
x 10
0.012
-2
2
)
Var (MH
Var (C)
PYTHIA
0.35
0.5
HERWIG
0.011
0.045
ARIADNE
0.325
0.475
0.01
0.45
0.04
0.3
0.009
0.425
0.275
0.4
0.035
0.008
0.25
0.375
0.007
0.225
0.35
0.03
0.006
0.2
0.325
0.005
0.3
0.025
0.175
0.004
0
100
200
0
√
s (GeV)
x 10
200
0
100
x 10
-2
0.5
0.3
0.45
200
√
s (GeV)
x 10
-2
0.22
0.2
0.18
0.25
0.225
0.35
100
Var y23
0.275
0.4
0
Var BW
PYTHIA
HERWIG
ARIADNE
0.325
200
√
s (GeV)
√
s (GeV)
Var BT
-2
100
0.16
0.2
0.14
0.3
0.175
0.15
0.12
0.125
0.1
0.25
0
p
100
200
√
s (GeV)
0
100
200
√
s (GeV)
0
100
200
√
s (GeV)
High s: Peaks more narrow { but tail important.
Good desription by MC LLA.
Partonlevel tests
.
Preditions in NLO.
Hadronisation orretion: Monte Carlo Models
PYTHIA, HERWIG, ARIADNE.
ni
h
y
C had = n Parton
hy iHadron
2
1
0
n
〈BW
〉
2
1
1
2
3
4
0
5
1
2
3
4
〈C 〉
n
2
3
4
0
5
1
2
3
4
〈BTn 〉
0
〈MHn 〉
2
3
4
0
5
Chad
PYTHIA
HERWIG
ARIADNE
〈C 〉
n
2
n
3
4
3
4
5
PYTHIA
HERWIG
ARIADNE
n
〈y23 〉
1
1
2
3
4
0
5
1
2
3
4
5
n
0
n
3
PYTHIA
HERWIG
ARIADNE
2
5
2
n
2
〈BTn 〉
PYTHIA
HERWIG
ARIADNE
2
1
1
1
3
Chad
PYTHIA
HERWIG
ARIADNE
1
1
0
5
3
2
1
4
n
Chad
Chad
PYTHIA
HERWIG
ARIADNE
3
n
0
3
2
2
1
5
n
3
1
2
PYTHIA
HERWIG
ARIADNE
n
〈BW
〉
1
Chad
n
〈y23 〉
2
PYTHIA
HERWIG
ARIADNE
2
3
1
1
n
n
Chad
Chad
PYTHIA
HERWIG
ARIADNE
1
0
0
5
3
2
〈(1-T) 〉
1
n
3
PYTHIA
HERWIG
ARIADNE
Chad
n
PYTHIA
HERWIG
ARIADNE
Chad
〈(1-T) 〉
3
Chad
PYTHIA
HERWIG
ARIADNE
2
3
Chad
3
Chad
3
1
1
2
3
4
n
14 GeV
0
5
1
2
n
91 GeV
Experimental systematis ombination
JADE/OPAL: Minimum Overlap
fi2; j2g :
p
Theory variation x R= s = 0:5 ::: 2:0 .
ovij = Min
〈MHn 〉
3
4
5
n
Perturbative Fits:
hyni, y =1
T , C , BT , BW , y23, MH ; n = 1:::5 .
0.1
〈(1-T) 〉
〈C 〉
1
1
0.075
0.3
0.05
0.015
0.2
〈(1-T) 〉
2
〈C 〉
2
0.15
0.01
0.1
0.005
0.1
〈(1-T) 〉
〈C 〉
3
0.002
0.001
x 10
3
0.075
0.05
-3
0.06
〈(1-T) 〉
〈C 〉
4
0.5
4
0.04
0.25
0.02
x 10
-3
0.04
〈(1-T) 〉
5
0.1
〈C 〉
5
0.02
0
0
25
50
75
100
125
150
175
200
√
s (GeV)
0
25
50
75
100
125
150
175
200
√
s (GeV)
150
175
200
√
s (GeV)
1
〈y23 〉
0.04
0.02
0.1
2
〈y23 〉
0.004
0.002-2
x 10
0.05
0.1
3
〈y23 〉
〈MH3 〉
0.03
0.05
x 10
〈MH2 〉
0.075
0.02
0.01
-3
4
〈y23 〉
0.1
〈MH4 〉
0.01
0.005
0-4
x 10
5
〈y23 〉
0.2
〈MH5 〉
0.004
0.002
0
0
25
50
75
100
125
150
175
All ts 2=d:o:f :=O(1) .
200
√
s (GeV)
0
25
50
75
100
125
〈(1-T) 〉
αs(MZ)
n
〈C 〉
n
〈BTn 〉
n
〈BW
〉
n
〈y23 〉
〈MHn 〉
JADE
0.18
and
OPAL
0.16
0.14
0.12
0.1
0.08
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
2 3 4 5 1 2 3 4 5
2 3 4 5
n
errors: stat.+exp. / had.+x .
〈(1-T) 〉
K = Bn/An
n
〈C 〉
n
〈BTn 〉
n
〈BW
〉
n
〈y23 〉
〈MHn 〉
80
60
40
20
0
-20
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
n
ombining 0 < K < 25 :
(M ) = 0:1273 0:0007(stat:) 0:0010(exp:)
= 0:1273 0:0073 (tot:)
s
Z0
:
:
+0 0009
0 0023
(had:)
:
:
+0 0069
0 0053
(theo:)
Non-perturbative QCD: Dispersive
Model (Dokshitzer et al.)
Parametrized by
1 Z
0(I ) = 0 dQ spt:+npt:(Q2) ; IR mathing sale I
I
I
' 2 GeV :
Shift of the dierential distribution
d dpt:
=
(y
dy
dy
Universal
4 CF
P = 2
M
Naively:
h i
yn
=
Z1
0
ay P ) ; observable dependent ay :
8
>
<
>
:
0
dy yn
2
66
4
39
>
2 )775=
R >;
R 2
2
s(R) + f ( ) s (
I
dpt:
d Z1
n
dy dy (y + ayP ) dy ;
0
I
Q
gives:
hy1i = hyipt: + ay P + O(1 =Q 2 !?nn OHab ::herleiten )
hy2i = hy2ipt: + 2hyipt: ay P + (ay P )2+:::
hy3i = hy3ipt: + 3hy2ipt: ay P + 3hyipt: (ay P )2
+(ay P )3
hy4i = hy4ipt: + 4hy3ipt: ay P + 6hy2ipt: (ay P )2
+4hy ipt: (ay P )3 + (ay P )4
hy5i = hy5ipt: + 5hy4ipt: ay P + 10hy3ipt: (ay P )2
+10hy 2ipt: (ay P )3 + 5hy ipt: (ay P )4 + (ay P )5 +O(1 =Q 6 !
P alulated more ompletely for hBT1 i and hBW1 i .
Dispersive Model Fits:
hyni, y =1
T , C , BT , BW , y23, MH ; n = 1:::5 .
0.15
JADE
OPAL
〈(1-T) 〉
1
0.1
〈(1-T) 〉
2
0.02
1
0.2
0.3
0.05
JADE
OPAL
〈C 〉
0.4
〈C 〉
2
0.2
0.01
0.1
0.2
〈(1-T) 〉
〈C 〉
3
0.004
3
0.1
0.002-2
x 10
0.1
〈(1-T) 〉
4
0.05
x 10
4
0.05
-3
0.0906
〈(1-T) 〉
5
0.2
〈C 〉
5
0.0633
0.1
0
〈C 〉
0.1
0.036
0
25
50
75
100
125
150
175
200
√
s (GeV)
0.0087
0
25
50
75
100
125
150
175
200
√
s (GeV)
0.125
0.2
JADE
OPAL
〈BT1 〉
0.15
JADE
OPAL
1
〈BW
〉
0.1
0.075
0.1
0.04
0.015
〈BT2 〉
2
〈BW
〉
0.01
0.02
0.0094
〈BT3 〉
0.0068
3
〈BW
〉
0.002
0.0042
0.001-3
x 10
0.0016
0.002
〈BT4 〉
0.2
0.001
x 10
4
〈BW
〉
0.3
-3
0.1-4
x 10
〈BT5 〉
0.4
5
〈BW
〉
0.4
0.2
0.2
0
25
50
75
100
125
150
175
200
√
s (GeV)
0
25
50
75
100
125
150
175
200
√
s (GeV)
Good qualitative desription.
Theory variations: x = 0:5 ... 2.0 / M 20% / I 1 GeV .
Fit values s(MZ0 ) , 0(I ) :
〈Cn〉
〈BTn 〉
n
n
〈BW
〉
〈y23 〉
〈MHn 〉
αs
〈(1-T)n〉
0.17
stat. uncertainty
+ exp. systematics
+ xµ-variation
+ Μ-, µI-var.
0.16
0.15
0.14
0.13
0.12
0.11
α0
0.1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 2 4
n
Simple average from h(1 T ) i, hC i, hB i, hB i, hy :: i and hM ; i gives
(M ) = 0:1174 0:0002(stat:) 0:0018(exp:) 0:0047(x)
0:0003(I ) 0:0001(M)
= 0:1174 0:0050(tot:) ;
(I ) = 0:484 0:003(stat:) 0:006(exp:) 0:026(x) 0:046(M)
= 0:484 0:053(tot:) :
1
Z0
s
0
1
1
T
1
W
1 5
23
24
H
Shape
Funtion
(Korhemsky
et
al.)
.
Dierential eventshape distribution by onvolution of perturbative distribution with universal shape funtion. By Integration:
1
h1 T i = h1 T iNLO + Q
3
2
3 1 66
s(Q2) 77
hC i = hC iNLO + 2 Q 41 5:73 2 5
1
2
2
hMH i = hMH iNLO + 2Q
1
2
h(1
+ 2 h1 T iNLO + 2
NLO
Q
Q 3
2
3 1 66
s(Q2) 77
2
2
hC i = hC iNLO + 2 Q 42 hC iNLO 4:30 2 5
2
3
2
2
s(Q ) 77
9 2 66
1
11
:
46
+
4
5
4 Q2
2
2(Q)
hMH4 i = hMH4 iNLO + Q1 hMH2 iNLO + 2 +4Æ
Q2
1 , 2 rst/seond moment of the shape funtion: universal. 2(Q) also (non inlusive ontribution).
T )2i = h(1
T )2 i
Shape Funtion Fits hyni
y =1 T , C , MH2 : n = 1:::2
0.15
〈(1-T) 〉
1
0.1
0.05
0
〈C 〉
1
0.4
0.2
0
0.1
〈MH2 〉
0.075
0.05
0.025
0
0
25
50
75
0.02
100
125
150
175
200
√
s (GeV)
125
150
175
200
√
s (GeV)
〈(1-T) 〉
2
0.01
0
0.3
〈C 〉
2
0.2
0.1
0
0.015
〈MH4 〉
0.01
0.005
0
0
25
50
75
100
Good qualitative desription, but moments not sensitive
to higher power orretions / 1=Q2 .
Theory variation: x = 0:5 ... 2.0 .
Fit values s(MZ0 ) , 1 (GeV) :
〈MHn 〉
〈Cn〉
αs
〈(1-T)n〉
stat. uncertainty
+ exp. systematics
+ xµ-variation
0.18
0.17
0.16
0.15
0.14
0.13
0.12
λ1
0.11
1
0.8
0.6
0.4
0.2
1
2
1
2
2
4
n
MH2 dierential distribution, NLLA + shape funtion:
1 = 1.22 GeV .
Eventshape distribution variane
d
dBT
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0
BT
0.2
0.3
BT
14 GeV: Var(B )=0.0025
91 GeV: Var(B )=0.0036
T
d
d(1 T )
0.1
T
20
17.5
15
12.5
10
7.5
5
2.5
0
0
0.1
0.2
0.3
1 T
14 GeV: Var(1 T )=0.0038
0
0.1
0.2
0.3
1 T
91 GeV: Var(1 T )=0.0037 (hadron level)
Webber, dispersive model: Leading power orretions anel,
Var(y) = hy 2ipt: hy i2pt: :
.
Perturbative variane ts:
0.005
Var (1-T)
0.004
JADE
OPAL
0.003
Var (C)
0.04
0.03
0.005
Var (BT)
0.004
0.003
0
25
50
75
0.003
100
125
150
175
200
√
s (GeV)
Var (BW)
0.002
x 10
JADE
OPAL
-2
0.25
Var (y23)
0.2
0.15
0.1
0.004
2
Var (MH
)
0.003
0.002
0
25
50
75
100
125
=d:o:f : 1 .
2
s(MZ0 )
Var(1 T ) Var(C)
0:105
0:086
OB:0.087
150
175
200
√
s (GeV)
Var(B ) Var(B ) Var(y ) Var(M )
0:070 0:103
0:107 0:084
T
W
23
2
H
Eventshape variane from shape
funtion
Var(y) = hy 2ipt: hy i2pt: ; for y = 1 T and y = MH2 .
.
{ already studied.
hC i2
Var(C) = hC 2i
pt:
pt:
1
3:23 s(Q2)
Q
Var (C)
0.045
0.04
0.035
0.03
JADE
OPAL
0.025
0
50
100
150
200
250
√
s (GeV)
s(MZ0 )=0.095
Seems not appropriate at high
ps , f. Var(B
T) above.
Single
Dressed
Gluon
approximation
.
(Gardi et al.)
First ontribution to Skeleton Expansion; beta funtion resummation.
h1 T i = h1 T ipt: + 1
h(1
T )2i = h(1
h(1
T )3i = h(1
(h(1
T )4i = h(1
h(1
T )n
ipt:
Q
2 2
2
T ) ipt: + 2 + 3
Q Q
T )3ipt: + 32 + 33
Q Q
T )4ipt: + 42 + 45
Q Q
1
X
=
n=1
SDG-graphs in O(sn)
Approximation: \inlusive" thrust. Coupling in \abar"-sheme
a(R ) 2
sMS (2R)=
5
sMS (2R )
3 0
1
Perturbative predition
h(1 T )ni : = d a + d ! a
+ ( 31 d + d ) + d a
!
5
+ ( d + d ) + 2 d + d a
+ (d + 51 d 2 d ) + ( 133 d d ) + 3 d !
3
+ 2 d + d a
9 d ) + ( 103 d + d + d ) + ( 77
d
12
2
+(6 d d ) !
7
7
35
+( 6 d + 2 d ) + d + 2 d a
0
pt
2
0
2
1
4
2
2
2
3
2
1
3
2
1
1
1
2
0
2
3
2
3
1
1
3
4
1
2
2
0
1
5
5
5
0
2
4
2
1
2
0
2
3
2
4
0
2
4
1
4
3
1
1
4
3
1
3
0
3
0
2
1
2
0
2
0
2
0
4
1
1
2
2
5
6
3
0
2
2
0
Perturbative Predition
.
Is an asymptoti series
Best approximation by trunating near the minimal term
Analysis for arbitrary trunation order O(a2) ... O(a6) .
LO omplete
NLO approximating well only for h(1 T )1i
10
-1
10
-
A1⋅ a1
10
-1
-2
-
A2⋅ a1
B2⋅ a2
10
SDG
B1⋅ a2
-2
10
-3
SDG
-
C2⋅ a3
-
C1⋅ a3
D1⋅ a
〈(1-T) 〉
1
1
10
10
10
-4
-4
-
〈(1-T) 〉
2
-
E1⋅ a5
F1⋅ a6
2
10
10
√
s (GeV)
1
√
s (GeV)
-2
10
-3
-1
A3⋅ a
10
-3
-
A4⋅ a1
B4⋅ a2
-4
10
B3⋅ a2
-4
SDG
10
10
-5
SDG
-
-C4⋅ a3
-5
-4
10
-
E2⋅ a5
F2⋅ a6
10
-
10
D2⋅ a4
-6
-D3⋅ a
-C3⋅ a3
-5
-E3⋅ a
-F3⋅ a6
〈(1-T) 〉
3
-7
1
10
10
2
10
√
s (GeV)
10
10
3
-
-D4⋅ a4
-6
〈(1-T) 〉
4
-7
1
sMS (MZ0 )=0.12
10
-
-E4⋅ a5
-
-F4⋅ a6
2
10
√
s (GeV)
.
Fits SDG + power orretion:
_4
SDG in Ο(a )
0.15
1
〈(1-T) 〉
SDG+λ1/Q
SDG
NLO
LO
〈(1-T) 〉
SDG+λ2/Q
SDG
NLO
LO
〈(1-T) 〉
SDG+λ3/Q
SDG
NLO
LO
〈(1-T) 〉
SDG+λ4/Q
SDG
NLO
LO
1
0.1
0.05
JADE
OPAL
0
2
0.02
2
0.01
0
2
0.004
3
0.002
0-2
x 10
2
4
0.1
0.05
0
0
25
50
75
100
125
150
175
200
√
s (GeV)
_5
SDG in Ο(a )
0.15
1
〈(1-T) 〉
SDG+λ1/Q
SDG
NLO
LO
〈(1-T) 〉
SDG+λ2/Q
SDG
NLO
LO
〈(1-T) 〉
SDG+λ3/Q
SDG
NLO
LO
〈(1-T) 〉
SDG+λ4/Q
SDG
NLO
LO
1
0.1
0.05
JADE
OPAL
0
2
0.02
2
0.01
0
2
0.004
3
0.002
0-2
x 10
2
4
0.1
0.05
0
0
25
50
75
100
2:::4 0 .
125
150
175
200
√
s (GeV)
. sMS(MZ0 ) and leading power orretion
oeÆient i as a funtion of trunation
order
λi
αs
|λι| (GeVmi)
αs(MZ0)
〈(1-T)4〉
〈(1-T)3〉
0.18
〈(1-T)2〉
〈(1-T)1〉
0.17
1
0.16
0.15
0.14
10
-1
0.13
0.12
0.11
2
3
4
5
6
nmax
2
3
4
5
6
nmax
Measuring sMS(MZ0) from h(1 T )1i
Theoretial unertainty:
4 = 0 vs. 4 = in O(s6) : tiny eet.
2 7! 2 2 , 2 7! 0:5 2 : very small eet in O(s5) .
NLO
NLO = 134% : vary NNLO 34% .
32
2
SDG
s(MZ0 ) = 0:1186 0:0007(stat:) 0:0014(exp:)+00::0033
0028(theo:)
= 0:1186 0:0037(tot:) :
Conlusion
Eventshape moments and variane measured
Perturbative NLO predition adequate for some moments
Not for variane
Unomplete perturbative desription aets all tested nonperturbative models
Universality of nonperturbative parameters an be stated
with some moments
s(MZ ) measured most preisely from SDG approximation
Going
0
from rst to higher moments: Perturbative and
nonperturbative problems