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