QCD at high energyALEPH-1997 OPAL-2001 LP01 Paolo Nason - QCD at high energy 9 Heavy Quark mass e...
Transcript of QCD at high energyALEPH-1997 OPAL-2001 LP01 Paolo Nason - QCD at high energy 9 Heavy Quark mass e...
QCD at high energy
Paolo Nason
INFN, sez. di Milano,and Universita di Milano-Bicocca
LP01 Rome 23-28 July 2001
QCD at high energy
• very lively and active field: 60 abstracts submitted to
the conference
• QCD at end of LEP hera: emphasis shifts from testing
the theory to pushing its precision
• Theory Standard: NLO.
• Tackle the study of power suppressed effects
Status of phenomenology
• Large body of tests in many different areas
• Overall agreement with theory
• Some minor discrepancies (need for better calculations)
• Some major discrepancies (interesting puzzles)
LP01 Paolo Nason - QCD at high energy 2
PLAN OF THE TALK
• Few examples of current results
in e+e−, ep and pp physics
• Power suppressed effects
• Towards NNLO
• Some puzzling results
LP01 Paolo Nason - QCD at high energy 3
QCD in e+e−→hadrons:
• QCD studies at highest available energies
• 4-jets NLO studies
• Fits to QCD color factors and αS
• Charge multiplicity studies
• Bose-Einstein correlations, color reconnection
• Power corrections, energy evolution of shape variables
distributions
• Heavy quark mass effects
• Fragmentation function studies
• γγ collision studies
LP01 Paolo Nason - QCD at high energy 4
Theoretical basis
NLO formulae for 3−jets
αS(µ2)A+ α2S(µ2)B(µ2/Q2) + . . . K.Ellis, Ross, Terrano
and 4−jets, infrared safe quantities
α2S(µ2)C + α3
S(µ2)D(µ2/Q2) + . . .Signer, Dixon;Nagy, Trocsanyi;Campbell, Cullen, Glover;Weinzierl, Kosower
Heavy quark mass effects
included in the 3−jets resultBernreuther, Brandenburg, Uwer;Rodrigo, Santamaria, Bilenky;P.N., Oleari
LP01 Paolo Nason - QCD at high energy 5
Resummation of Sudakov effects at NLL level:
needed in the limit of thin jets
R(y) = F (αS) eLg1(αSL)+g2(αSL) Catani, Dokshitzer,Olsson, Turnock, Webber
where R is a jet rate, y the jet thickness, L = − log y.
Hadronization and power corrections:
suppressed as 1/Q, but still important at LEP energies.
Estimated using Montecarlo hadronization models.
Renormalon inspired models
provide an alternativeDokshitzer, Marchesini,Webber
LP01 Paolo Nason - QCD at high energy 6
Shape variables at highest energy
L3: thrust, heavy jet
mass, total jet broadening,
wide jet broadening and C
parameter measured at
different energies.0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
20 40 60 80 100 120 140 160 180 200 220
Reduced √�
s� ′
√�s =91.2 GeV
LEP 1.5LEP 2QCD EvolutionConstant α� s�
√s (GeV)α s
From fit:
αS(MZ) = 0.1220± 0.0011(exp.)± 0.0061(th.)
LP01 Paolo Nason - QCD at high energy 7
Fits to QCD color factors (OPAL, ALEPH)
• Jet resolution yij = 2min(E2i , E
2j )(1− cos θij)/E
2vis
• Differential 2-jet rate D2(y23) = 1σtot
dσdy23
(OPAL)
• 4-jet rate R4(ycut)
• 4-jet, ycut = 0.008, E1 > E2 > E3 > E4
– χBZ = 6 [(~p1 ∧ ~p2), (~p3 ∧ ~p4)]
– ΘNR = 6 [((~p1 − ~p2), (~p3 − ~p4)]
– ΦKSW = 〈 6 [(~p1 ∧ ~p4), (~p2 ∧ ~p3)]〉3↔4
– cos(α34) = cos(6 [(~p3, ~p4]) (ALEPH)• Color dependence
D2 = αSCF . . .+ α2SCF(CF . . .+ CA . . .+ TFnf . . .)
R4 = α2SCF(CF . . .+ CA . . .+ TFnf . . .) + α3
S . . .
LP01 Paolo Nason - QCD at high energy 8
• Z peak data (1991/1995)
• hadronization corrections:
Montecarlo models
(fixed color factors?)
OPAL:
CA = 3.02± 0.25± 0.49
CF = 1.34± 0.13± 0.22
αS(MZ) = 0.120±0.011±0.020
ALEPH:
CA = 2.93± 0.14± 0.49
CF = 1.35± 0.07± 0.22
αS(MZ) = 0.119±0.006±0.022
ALEPH PRELIMINARY
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 3.25
CA/CF
TR
/CF
SO(2)
SU(4)
This analysis, 68% CL contour
ALEPH-1997 OPAL-2001
LP01 Paolo Nason - QCD at high energy 9
Heavy Quark mass effects
comparison of light and heavy quark hadronic events
exposes the effects of the heavy quark mass.
Two kind of analysis:
• Fit the b quark mass from the ratio of 3-jet rates in
b-quark jets and light quark jets
B3 =Rb3Rdusc3
(which can be computed at NLO including mass effects)
• Compare the αS determination in light quark events and
b, c quark events (tests of flavour independence of αS)
LP01 Paolo Nason - QCD at high energy 10
Results from b-mass fits to
B3, called (improperly?)
a test of the running of the
b-quark mass, from various
experiments.
DELPHI has new results:
mb(MZ) (in GeV) =
Duhram: 2.67± 0.25(stat.)
±0.34(had.)± 0.27(th)
Cambridge: 2.61± 0.18(stat.)
±0.47(had.)± 0.18(tag)
±0.12(th)1
1.5
2
2.5
3
3.5
4
4.5
5
10 102
Q=√s [GeV]
mb(
Q2 )
[GeV
]
PDG (Production threshold)this analysisALEPHBrandenburg et al (SLD)DELPHI
OPAL
Measurement of the b massin high energy processes
LP01 Paolo Nason - QCD at high energy 11
Flavour independence of
the strong coupling:
α(b/c)S /α
(uds)S is much more
consistent with 1 if heavy
flavour mass effects are
included
(a)
c massless
b massive
1-TMH
BW
C
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
(b)
c massive
b massless
αs�Q�
/ α suds
OPAL
y23
LP01 Paolo Nason - QCD at high energy 12
HERA
• Structure function studies
• Jet studies in DIS
• Jet studies in photoproduction
• Shape variable and power correction studies
• Production of Heavy flavours
LP01 Paolo Nason - QCD at high energy 13
Example: αS from jets in DIS
ZEUS0.1
0.12
0.14
0.16
0.18
20 40 60 80 100
α s
αs�PDG
ZEUS 96-97
αs� from R2+1�
a)
-0.01
0
0.01
20 40 60 80 100
∆αs(E
S)
b)
-0.01
0
0.01
20 40 60 80 100
∆αs(T
h)
Q (GeV)
c)
Zeus: αS(MZ) = 0.1166
±0.0019(stat.)+0.0024−0.0033 (exp.)
+0.0057−0.0044 (th.+pdf)
0�
.1
0�
.12
0�
.14
0�
.16
0�
.18
0�
.2
0�
.22
0�
.24
0�
.26
10 10 2
E T / GeV
H1
α s� α� s from inclusive jet cross section
for CTEQ5M1 parton densities
150 < Q 2�
< 5000 GeV 2�
α s (E T )�
α s (M Z )�
µ� r = E T
inclusive k ⊥� algo.
H1: αS(MZ) = 0.1186
±0.0007(stat.)±0.0030(exp.)+0039−0.0045(th.) +0.0033
−0.0023 (pdf)
LP01 Paolo Nason - QCD at high energy 14
3 jet study from H1
• Recent NLO Calculation by
Nagy and Trocsanyi
• Jets defined with kT cluster
algorithm
• ET > 5 GeV, M3jet > 25 GeV
0.2�
0.4�
0.6�
10 102
103
Q2 (GeV2)
R3/
2 =
σ3j
et /
σ 2jet
H1 data
QCD: LO (1+δ�
hadr� )
QCD: NLO (1+δ�
hadr� )
�
α� s� (MZ� ) = 0.118 ±0.006
gluon: CTEQ5M1 ±15% 0.5 < (µ� r� /
ET
) < 2�
Mn-jet > 25 GeV
10-3
10-2
10-1
1
10
10 102
103
dσ3j
et/d
Q
�
2 (pb
/GeV
2 )
H1 dataQCD:
�
NLO (1+δ�
hadr� )
�
NLOLO
M3jet� > 25 GeV
0.5�
1
1.5
2
10 102
103
Q2 (GeV2)�
data
/ th
eory
H1 data / NLO (1+δhadr)α� s (M
Z
� ) = 0.118 �
±0.006�
0.5 < (µ r� / ET
� ) < 2gluon: CTEQ5M1 ±15%
LP01 Paolo Nason - QCD at high energy 15
0�
0.5�
1
1.5
2
-0.8 -0.4 0�
0.4�
0.8�
cos θ3�
1/σ 3j
et d
σ 3jet
/d c
os
�
θ 3
H1 dataQCD NLOQCD LOPhase Space
150 < Q2 < 5000 GeV2
0�
0.2�
0.4�
0.6�
0.8�
0�
1 2 3�
ψ3�
1/σ 3j
et d
σ 3jet
/d
�
ψ3
H1 dataQCD NLOQCD LOPhase Space
150 < Q2 < 5000 GeV2
LP01 Paolo Nason - QCD at high energy 16
TEVATRON
• Inclusive jet versus Et and pseudorapidities
• Jets with cluster (kT) algorithms
• Jet structure
• W/Z and γ production
LP01 Paolo Nason - QCD at high energy 17
10-1
1
10
10 2
10 3�
10 4
10 5�
10 6�
10-6
10-5
10-4
10-3
10-2
10-1
1
DØ Inclusive Jets |η| < 3, present measurement
CDF/DØ Inclusive Jets |η| < 0.7ZEUS 95 BPC+BPT+SVTX &�
H1 95 SVTX + H1 96 ISRZEUS 96-97 & H1 94-97
E665
CCFR
BCDMS
NMC
SLAC�� �
��
� �
�
1
10
10 2�
10 3
10 4
10 5�
10 6�
10 7�
50 100 150 200 250 300 350 400 450 500
0.0 �
≤ |η| < 0.50.5 �
≤ |η| < 1.01.0 ≤ |η| < 1.51.5 ≤ |η| < 2.02.0 ≤ |η| < 3.0
����� ���� � �� ����
�����������!
"$#&%('*)*+-,-.0/0%
D0 jet analysis at large rapidity extends sensitivity to
PDF’s at small x and large Q2. Impressive agreement
with NLO theoretical calculations
LP01 Paolo Nason - QCD at high energy 18
D0 data, JETRAD (O(α3S))
-0.40
0.40.81.21.6
50 100 150 200 250 300 350 400 450 500
-0.40
0.40.81.21.6
50 100 150 200 250 300 350 400 450 500
-0.40
0.40.81.21.6
50 100 150 200 250 300 350 400 450 500
-0.40
0.40.81.21.6
50 100 150 200 250 300 350 400 450 500
-0.8-0.4
00.40.81.21.6
50 100 150 200 250 300 350 400 450 500
��������
��� � ����� �
�������������
��� �"! #�$%#'& ���)(
���)(*! #�$%#'& +,� �
+,� �"! #�$%#'& +,�)(
+,�)(*! #�$%#'& -'� �
-'� �"! #�$%#'& .'� �
CTEQ4HJ (•) and CTEQ4M (◦)
-0.40
0.40.81.21.6
50 100 150 200 250 300 350 400 450 500
-0.40
0.40.81.21.6
50 100 150 200 250 300 350 400 450 500
-0.40
0.40.81.21.6
50 100 150 200 250 300 350 400 450 500
-0.40
0.40.81.21.6
50 100 150 200 250 300 350 400 450 500
-0.8-0.4
00.40.81.21.6
50 100 150 200 250 300 350 400 450 500
��������
��� � ����� �
�������������
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+,�)(*! #�$%#'& -'� �
-'� �"! #�$%#'& .'� �
MRSTg↑ (•) and MRST (◦)
LP01 Paolo Nason - QCD at high energy 19
CDF dijet study: look at ET of one
central jet (0.1 < η1 < 0.7) where
the other jet lies in several different
pseudorapidity intervals.
Sensitivity to PDF at large x
enhanced;
Qualitatively good description of
data,
quantitative problems
LP01 Paolo Nason - QCD at high energy 20
No PDF set is found that
fits the data satisfactorily.
Other problematic areas:
• Direct photon at low
transverse momenta
• b production
In all these cases
discrepances could be
attributed to unknown
higher order effects.
ET (GeV)
(dat
a -
QC
D)/
QC
D a)�
-0.2
0�
0.2�
0.4�
0.6�
0.8�
1
50�
100 150 200 250 300�
350�
400ET (GeV)
(dat
a -
QC
D)/
QC
D b)�
-0.25
0�
0.25�
0.5�
0.75�
1
1.25
1.5
1.75
2
2.25
2.5�
50�
100 150 200 250 300�
350�
400
ET (GeV)
(dat
a -
QC
D)/
QC
D c)�
-0.4
-0.2
0�
0.2�0.4
�0.6
�0.8
�1
1.2
1.4
1.6
50�
100 150 200 250ET (GeV)
(dat
a -
QC
D)/
QC
D d)�
-0.5-0.25
0�
0.25�
0.5�
0.75�
11.251.5
1.752
�2.252.5
�
50�
100 150 200
LP01 Paolo Nason - QCD at high energy 21
Power Corrections
(Movilla Fernandez, Bethke,
Biebel, Kluth)
•√S = 14 to 189 GeV
• several shape variables
considered
• fit to αS and to the
parameter α0 in the power
correction model of
Dockshitzer, Marchesini
and Webber1
10 2
10 4
10 6�
10 8�
10 10
10 12
10 14
10 16
10 18
0 0.1 0.2 0.3
TASSO 14 GeV
TASSO 22 GeV
HRS 29 GeV
MARK2 29 GeV
TASSO 35 GeV
JADE 35 GeV�TASSO 44 GeV
JADE 44 GeV�AMY 55 GeV
SLD 91 GeV
L3 91 GeV
DELPHI 91 GeV
ALEPH 91 GeV
OPAL 91 GeV
L3 133 GeV
DELPHI 133 GeV
ALEPH 133 GeV
OPAL 133 GeV
L3 161 GeV
DELPHI 161 GeV
OPAL 161 GeV
L3 172 GeV
DELPHI 172 GeV
OPAL 172 GeV
L3 183 GeV
DELPHI 183 GeV
OPAL 183 GeV
L3 189 GeV
OPAL 189 GeV
1-T
1/σ dσ/d(1-T)
LP01 Paolo Nason - QCD at high energy 22
0.3
0.4
0.5
0.6
0.7
0.8
0.1 0.11 0.12 0.13
average<1-T><MH
2><BT><BW><C>
αS(MZ)
α0
b)
Mean Values
αS(MZ) = 0.1187±0.0014±0.0001+0.0028−0.0015
α0(2 GeV) = 0.485±0.013±0.001+0.065−0.043
(fit + syst. + theo. errors)
0.4
0.5
0.6
0.7
0.8
0.9
1
0.08 0.09 0.1 0.11 0.12 0.13
average1-TMH, MH
2
BTBWC
αS(MZ)
α0
a)
Distributions
αS(MZ) = 0.1111±0.0004±0.0020+0.0044−0.0031
α0(2 GeV) = 0.579±0.005±0.011+0.099−0.071
(fit + syst. + theo. errors)
αS(MZ0) = 0.1171+0.0032−0.0020 , α0(2 GeV) = 0.513+0.066
−0.045
LP01 Paolo Nason - QCD at high energy 23
Power corrections
studies at HERA
0.3�
0.4�
0.5�
0.6�
0.7�
0.1�
0.12�
0.14�
αs(MZ)�
α_0(
µ Ι = 2
GeV
)
τ�
B
ρ
τ� C�
C
ρ0�
Power Correction Fits(stat. and exp. syst. uncertainties)
�
H1
LP01 Paolo Nason - QCD at high energy 24
What we know about power corrections
• In Re+e−, 1/Q4 (from O.P.E.)
(why τ hadronic physics works so well)
• In DIS, 1/Q2 (from O.P.E.)
(why scaling sets in early in Q2)
• In Jets, 1/Q (from many points of view...)
(why Jet physics is so hard)
Can we understand Jets better?
LP01 Paolo Nason - QCD at high energy 25
Power Correction Model
Basic idea: from QED analogy, fluctuations
of gluons into virtual and real states cause αS
running. Physical quantities written as:
F (Q2) =∫dk2F(Q2, k2)αS(k2) .
(Unknown) low energy behaviour of αS
determines power corrections. Power correction
law determined by low k2 behaviour of F(Q2, k2)
F (Q2) ∝1
k→ 1/Q correction
F(Q2, k2): physical quantity computed with a gluon of mass k2.
Scheme by Dockshitzer, Marchesini, Webber.
Several similar suggestions: Akhoury and Zakharov, Sterman and
Korchemsky, etc.
LP01 Paolo Nason - QCD at high energy 26
Cannot work as is for shape variables (Seymour, P.N.)
Shape variables “know” what is
inside the blob.
Thrust: decreases for a generic
blob, does not change in some
kinematic configurations
1− t ≈ |k|/Q
1− t ≈ 0
Dokshitzer, Lucenti, Marchesini and Salam have examined these
effects at the two-loop level.
The effect amounts to a factor of order unity (Milan factor) in front
of the 1-loop (naive) formula.
This factor has the same value for several common shape variables
If 2-loop same as 1-loop, what about higher loops?
LP01 Paolo Nason - QCD at high energy 27
On the bright side:
• One parameter model of power corrections
• Also power suppressed effects depend upon color factors:
Kluth etal have used it to fit QCD color factors without relying
on Montecarlo hadronization models.
• Some support from data
• Alternative model to estimate power corrections:
low αS values from distributions.
Study using DELPHI data: PMS type of data
analysis (RGI) gives very good fit to data without
power corrections
Reinhardt,Flagmeyer,Hamaker, Passonand Wicke
So: Power corrections or NNLO?
LP01 Paolo Nason - QCD at high energy 28
αs(MZ)
RGI+K0�
0.1�
0.11�
0.12�
0.13�
0.14�
0.15�
αs(MZ)
Had. Correction
αs(MZ)
pure RGI
0.1�
0.11�
0.12�
0.13�
0.14�
0.15�
0.1�
0.11�
0.12�
0.13�
0.14�
0.15�
αs(MZ)
1/q Powercorrection
0.1�
0.11�
0.12�
0.13�
0.14�
0.15�
1-Thrust
Major
C-Parameter
Mh�2 /Evis
2 E def.
Ms2/Evis
2 E def.
Bsum
Bmax�
EEC 30o-150o
JCEF 110o-160o
LP01 Paolo Nason - QCD at high energy 29
Remarkable progress in NNLO calculations
Current goal: jet production in hadronic collisions.
• Double soft limit (Berends and Giele, 1989)
• Double collinear and soft-collinear (Campbell and Glover, 1998;
Catani and Grazzini, 1999; Del Duca, Frizzo and Maltoni, 2000)
• 2→3 amplitude at one loop level (Bern, Dixon and Kosower, 1993;
Kunszt, Signer and Trocsanyi, 1994).
• Collinear limit of one loop amplitudes (Bern, Dixon, Dunbar and
Kosower, 1994; Kosower, 1999; Kosower and Uwer, 1999)
• Soft limit of one loop amplitudes (Bern, Del Duca, Schmidt, 1998;
Bern, Del Duca, Kilgore and Schmidt 1999; Catani and Grazzini,
2000)
• Some analytic phase space integrals (Gehrman, Glover; Catani, de
Florian, Grazzini)
LP01 Paolo Nason - QCD at high energy 30
The two loop 2→2 contribution recentlycomputed (Anastasiou, Glover, Oleari, Tejeda-Yeomans; 2001).
Recursion relations among feynman integrals reduce the problem to
the computation of a small number of master integrals.
The hardest of those (double box, planar and crossed) only recently
solved (Smirnov, 1999; Tausk, 1999).
All ingredients needed for a full NLO computation of jetcross sections in hadronic collisions are in place now.
The implementation into a useful result is still a formidable task.
But it does not look far...
LP01 Paolo Nason - QCD at high energy 31
Puzzle in b production
LP01 Paolo Nason - QCD at high energy 32
e+e−→e+e−bb
• At LEP, direct and single resolved
comparable
• L3 uses the electrons or muons from
semileptonic b decays.
• the Pt of the lepton (relative to closest
jet) fitted to extract the c, b and light
quark fraction:
σ(e+e−→e+e−bbX)µ = 14.9±2.8±2.6pb ,
σ(e+e−→e+e−bbX)e = 10.9± 2.9± 2.0pb
• OPAL (µ only) gets comparable results
• c cross section in agreement with
calculations, excess of b production
LP01 Paolo Nason - QCD at high energy 33
√s (GeV)
σ(e+ e– →
e+ e– cc
–,b
b– X
) (p
b) mc� =1.3 GeV
mc� =1.7 GeV
mb� =4.5 GeV
mb� =5.0 GeV
NLO QCDDirect
●�
e+e– → e+e–cc–X
■ e+e– → e+e–bb–X
L3
1
10
10 2
10 3
0�
50�
100 150 200
OPAL preliminary
5�
10
15
20
25
30�
60�
80�
100 120 140 160 180 200 220 240√
�s�¬
ee� [GeV�
]
σ(e+ e
- → e
+ e- b
b- ) [p
b]
5�
10
15
20
25
30�
OPAL (µ,√s¬=189-202 GeV, prel.)
L3 (µ,e,√s¬=189-202 GeV, prel.)
NLO (direct+resolved, GRV)
NLO (direct only)
mb� =4.5 GeV, µ=mb
� /2
bands:
mb� =5.2 GeV, µ=2mb
�
LP01 Paolo Nason - QCD at high energy 34
b excess at HERA
New H1 results confirm previous findings:
• σ(b) from impact parameter fit in
photoproduction
• σ(b) in DIS
both in excess over QCD calculations.
Q2 (GeV2)
Dat
a / T
heor
y
H1 µ pT� rel
H1 µ impact param. (prel.)
ZEUS e- pTrel
NLO QCD
σvis (ep → b X)
∫∫0
1
2
3
4
5
6
1 10 102� ��� �
������ ������
Theoretical uncertainties are small. In DIS production, no resolved
contribution. Similarities with γγ result.
Very difficult to justify!
LP01 Paolo Nason - QCD at high energy 35
Conclusions and outlook
• Convincing tests of PQCD
• Few problematic areas, some serious puzzles
• NLO calculations are the standard now
• True progress can only come going from NLO to NNLO
• Hadron colliders are the near future: we need a quality
jump in QCD.
LP01 Paolo Nason - QCD at high energy 36
EXTRA SLIDES
LP01 Paolo Nason - QCD at high energy 37
D2
fit range
OPAL dataR-matchedlnR-matchedfitted lnR-matched
-ln(y23)
Cto
t
0
0.025
0.05
0.075
0.1
0.125
0.15
0.175
0.2
0.225
0.25
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
0.60.8
11.21.4
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
LP01 Paolo Nason - QCD at high energy 38
R4
fit range
OPAL dataNLOR-matchedfitted R-matched
log(ycut)
Cto
t
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1
0.60.8
11.21.4
-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1
LP01 Paolo Nason - QCD at high energy 39
D0, jet cross sections with
kT cluster algorithm.
Resolution parameter:
dij = min(pt2i , pt
2j )
∆R2ij
D2,
∆R2ij = ∆Φ2
ij + ∆η2ij.
Merge ij if dij is the smallest of
all pt2k and dkl.
Choosing D = 1 roughly
corresponds to R = 0.7 in
standard jet definition
LP01 Paolo Nason - QCD at high energy 40
LP01 Paolo Nason - QCD at high energy 41
Running of b mass
Running of αS: a 3-jet rate goes like
R3 = C × αS(µ2)
[1 +
αS
π
(A+ b0 log
µ2
Q2
)]αS goes like
αS(Q2) = αS(µ2)
[1 +
αS
π
(b0 log
µ2
Q2
)]You measure R3, you prove the running.
For the running mass, the commonly used quantity goes like
Rbd3 − 1 = −18.62m2b
Q2
[1 +
αS
π
(−6.1955− 0.483 log
m2b
Q2
)]while the running mass goes like
m2(Q2) = m2[1 +
αS
π
(−
8
3+ 2 log
m2b
Q2
)]Thus: find a quantity that runs like a mass to prove the running.
LP01 Paolo Nason - QCD at high energy 42