Introduction to QCD slides - P.Hoyer.pdf - High Energy Physics Group

Paul Hoyer Mugla 2010 

Introduction to QCD 

Paul Hoyer 

University of Helsinki 

Akyaka - Mugla, Turkey 

27.8 - 4.9 2010 

1

As simple as 1-2-3? 


The Standard Model 

SU(3) x SU(2)L x U(1) 

QCD 

Electroweak 

Not exactly: The strong, electromagnetic and weak interactions 

manifest themselves very differently in Nature 

SU(2)L x U(1): Where does the lagrangian come from? 

SU(3): What does the lagrangian do? 

2


The Discovery of the Strong Interaction 

Rutherford’s experiment 1911: 

The positive charges in matter are 

located in a tiny nucleus, whose radius 

is ~ 10 –5 of the atomic size. 

⇒ There must be a strong, short-ranged 

force to counteract the Coulomb repulsion 

Thomson’s atom: Rutherford’s atom: 

F = α 

r 2 

Rnucleus 

Ratom 

� 10 −5 

3

Molecules 


Matter 

- 

e 

- 

e 

- 

e 

Atom 

The Standard Model A. Pich - CERN Summer Lectures 2008 

- 

e 

Electron 

Nucleus 

p 

n n 

n 

pp 

p 

n 

p 

n p n 

p 

n n 

n p 

p 

Proton Neutron 

Quarks 

4


The Pion as a carrier of the strong force 

In 1935 Hideki Yukawa suggested the existence of a new, strongly interacting 

particle “U” (later named the pion). The strength and range of the strong 

interaction could be understood as arising from pion exchange. 

• • • 

π 

• • • 

π 

π 

N 

• • • 

R < ∼ 1/Mπ 

Postulating a new particle was considered very bold in those days, when 

only a handful of elementary particles were known: photon, proton, neutron, 

electron, (neutrino). 

It has been suggested that “social pressure” may have kept physicists in the 

West from a similar proposal. 

Even today in QCD, the pion is special: It is a Goldstone boson. 

5


Birth of Yang-Mills Theory (1954) 

Quantum ElectroDynamics (QED) is invariant under local (space and time - 

dependent) gauge transformations: 

Electron field: 

Photon field: 

ψ(x) → e ieΛ(x) ψ(x) 

Aµ(x) → Aµ(x) − ∂µΛ(x) 

Λ(x) may be any 

regular function 

In 1954, Yang and Mills generalized this local U(1) gauge symmetry to the 

SU(2) group of isospin, with the proton and neutron forming an SU(2) 

� � 

p 

isospin doublet just as in Yukawa’s theory: 

n 

This established the structure of non-abelian gauge symmetry. 

Nature found, however, different uses of YM theories. 

⇐ As in classical ED! 

6

Matter field: 


ψ(x) → U(x)ψ(x) 

Physical Review 96 (1954) 191 

For a local gauge transformation defined by an SU(2) matrix U(x), Yang and 

Mills found that the theory is symmetric provided the fields transform as: 

Gauge field: Aµ(x) → U(x)Aµ(x)U † (x) − i 

g U(x)∂µU † (x) 

The same rule holds for any group, such as SU(3). In QED, 

g is the same 

for all ψ! 

U(x) =e ieΛ(x) Then U1U2 = U2U1, i.e., all group elements commute, 

hence the U(1) gauge symmetry is said to be abelian. 

7

† † 

2 * 2 matrices: " " " " " " # ; det " " 1 

" 

" exp ! 

a 1 a 

Fundamental Representation: ! F " ! 

2 Pauli 



+ i - , 

a a 

Commutation Relation: 

a a† 

a 

; ! " ! ; Tr ( ! ) " 0 ; a = 1 , ! , 3 

a b abc c 

/ , 0 

1 

! ! 

2 

" i . ! 

1 $ 0 1% 2 $ 0 # i % 3 $ 1 0 % 

! " & ' ; ! " & ' ; ! " & ' 

( 1 0) ( i 0 ) ( 0 # 1) 

8

a a 

Fundamental Representation: ! F ! # Gell-Mann 

$ 0 1 0% $ 0 " i 0% $ 1 0 0% $ 0 0 1 % 

1 & ' 2 & ' 3 & ' 4 & ' 

# ! 

& 

1 0 0 

' 

; # ! 

& 

i 0 0 

' 

; # ! 

& 

0 " 1 0 

' 

; # ! 

& 

0 0 0 

' 

& 0 0 0' & 0 0 0' & 0 0 0' & 1 0 0' 

( ) ( ) ( ) ( ) 

$ 0 0 " i % $ 0 0 0% $ 0 0 0 % $ 1 0 0 % 

5 & ' 6 & ' 7 & ' 8 1 & ' 

# ! 

& 

0 0 0 

' 

; # ! 

& 

0 0 1 

' 

; # ! 

& 

0 0 " i 

' 

; # ! 0 1 0 

3 

& ' 

& i 0 0 ' &0 1 0' &0 i 0 ' &0 0 " 2' 

( ) ( ) ( ) ( ) 

1 123 147 156 246 257 345 367 1 458 1 678 1 

f ! f ! " f ! f ! f ! f ! " f ! f ! f ! 

2 3 3 2 



a b abc c 

[ ! , ! ] ! if ! 

1 

2 

9

Yang-Mills 


Particles found in Experiments 

Pauli ν 

using particle accelerators 

Dirac e + 

Einstein γ 

Yukawa π ± 

using cosmic rays 

10 

http://fafnir.phyast.pitt.edu/particles/

Mesons 


Quark Model classification of Hadrons 

q¯q 

Hadrons may be formed 

with any combination of 

q = u,d,s,c,b,t 

π + 

Free quarks have never been seen: 

p 

Baryons 

qqq 

Were quarks mere mathematical rules, or true particles? (Gell-Mann) 

12


Quarks have Color 

Three quark colors were introduced by O.W. Greenberg in 1964, to make the 

quark model of the proton compatible with the Pauli exclusion principle: 

Baryon wave functions must be antisymmetric under the interchange of 

quarks. In the Quark Model, the space – spin wf is symmetric. The color wf is 

antisymmetric, rescuing the Pauli principle. 

The antisymmetric color wave function 

(A,B,C = red, blue, yellow) means that 

the proton is a color singlet (does not 

change under gauge transformations). 

Mesons are also color singlets: 

|π〉 = � 

A 

q A ¯q A 

|p〉 = � 

A,B,C 

U|p〉 = |p〉 

U|π〉 = |π〉 

εABCq A q B q C 

13

Quarks are for real: Pointlike scattering of electrons 

High energy electrons scatter from pointlike quarks inside the proton: 

e +q → e +q (in analogy to Rutherford’s experiment) 

The struck quark flies out of the proton and “hadronizes” into a spray (jet) of 

hadrons (mostly pions). 

At relativistic energies quark-antiquark pairs are created to ensure that all 

quarks end up as constituents of mesons or baryons. 

Deep Inelastic Scattering (DIS): 

e + p → e + anything 

SLAC 1969: Ee = 20 GeV 


14


The search for asymptotic freedom 

The SLAC data (1969) showed that the proton 

charge is located in apparently free, pointlike 

constituents, presumably the quarks proposed 

earlier based on the hadron spectrum. 

This raised the question: 

“How can the quarks be nearly free inside the proton, 

yet bind so strongly together that they do not exist as free particles?” 

It was known that the coupling 

strength of QED “runs”, ie., 

α = α(Q 2 ) increases as the 

distance scale r = 1/Q decreases. 

If there were theories where α = α(Q 2 ) 

decreases with r this might explain 

the SLAC data. 

α(r) 

e 

p 

QED 

γ * 

e 

q 

π 

... 

p 

15 

log(r)


Quantum Chromodynamics (1972) 

The QCD Lagrangian defines the interactions of quarks and gluons: 

LQCD = � 

f 

Looks like QED! But the gauge symmetry is 

SU(3) of color (Greenberg, 1964) 

ψ → Uψ U ⊂ SU(3) 

All quarks u,d,s,c,b,t have 

the same strong coupling g 

¯ψf (i/∂ − g /A − mf )ψf − 1 

4 F µν Fµν 

αs = g2 

4π 

Note: In QED the abelian U(1) symmetry allows 

different electric charges e for different particles 

It remains a mystery why: 

ψ = 

⎛ 

⎝ q 

q 

q 

⎞ 

⎠ 

eu = − 2 

3 ee 

|ep + ee|/e < 1.0 · 10 −21 

20



# 

! 

" 

S. Bethke 

2006 

! $% & ' ()**+, - ()((*( 

! 

" 

! $% & ' ()**+! - ()((!. 

# 

2004 

21


Asymptotic Freedom 

Asymptotic Freedom 

“What this year's Laureates 

discovered was something that, at 

first sight, seemed completely 

contradictory. The interpretation of 

their mathematical result was that the 

closer the quarks are to each other, 

the weaker is the 'colour charge'. 

When the quarks are really close to 

each other, the force is so weak that 

they behave almost as free particles. 

This phenomenon is called 

‘asymptotic freedom’. The converse 

is true when the quarks move apart: 

the force becomes stronger when the 

distance increases.” 

! S(r) 

1/r 

22 

From P. Skands


Data creates Theory 

Data 

Discriminating 

Observables 

Fits 

Phenomenolog 

ical Models 

Individual 

Essential 

Features 

Quark Model, 

Eightfold 

Way, … 

(Solvable) 

Theory 

Complete 

description 

Quantum 

Chromo- 

Dynamics 

23 

Peter Skands, ESHEP-10


Puzzles of QCD 

QCD must have Color Confinement: Only color singlet mesons and baryons 

can propagate over long distances. 

Color confinement is verified numerically in numerical lattice simulations 

and modelled phenomenologically, but the mechanism is still poorly understood. 

Baryons and mesons are bound states of quarks. What are the wave functions? 

The quarks and gluons bound in hadrons are highly relativistic. 

Data and models exist, but no calculations comparable to QED atoms. 

How can we compare data on hadron final states with perturbative QCD? 

Factorization theorems allow to express physical measurements in terms of hard, 

perturbative subprocesses and universal, measurable quark and gluon distributions. 

24


Perspective: The divisibility of matter 

Since ancient times we have wondered whether matter can be divided into 

smaller parts ad infinitum, or whether there is a smallest constituent. 

Democritus, ~ 400 BC 

Vaisheshika school 

Common sense suggest that these are the two possible alternatives. 

However, physics requires us to refine our intuition. 

Quantum mechanics shows that atoms (or molecules) are the identical 

smallest constituents of a given substance 

– yet they can be taken apart into electrons, protons and neutrons. 

Hadron physics gives a new twist to this age-old puzzle: Quarks can be 

removed from the proton, but cannot be isolated. Relativity – the creation of 

matter from energy – is the new feature which makes this possible. 

We are fortunate to be here to address – and hopefully develop an 

understanding of – this essentially novel phenomenon! 

25

Instantaneous Coulomb interaction 


Understanding charge screening in QCD 

What went wrong with the nice physical argument of QED? 

= N _1 

* n 2_ 

c * 

3 f 3 

Transverse gluons (and quarks) 

Instantaneous Coulomb interaction 

= +N 4 

* 

c 

Vacuum fluctuations of transverse fields 

Yu. L. Dokshitzer hep-ph/0306287 

26 

Physical, transverse 

gluons screen as QED 

Coulomb gluons give 

the opposite sign!

In QCD, mass terms can be directly introduced in the lagrangian. 

In the SM all masses result from “Yukawa interactions” involving the Higgs 

field, but they are unconstrained by the theory. 

The quark masses inferred 

from experiment are: 

mu = 1.5 ... 3.3 MeV 

md = 3.5 ... 6.0 MeV 

ms = 104 ± 30 MeV 

mc = 1.27 ± .10 GeV 

mb = 4.20 ± .15 GeV 

mt = 171.2 ± 2.1 GeV 


Quark masses: Nature’s gift 

} m > ΛQCD , mass effects large 

Non-relativistic bound states 

Scale of strong 

interactions 

27

Binding energy 

[meV] 

0 

-1000 

-3000 

-5000 

-7000 

3 1 S 0 

2 1 S 0 

1 1 S 0 


Charmonium – the Positronium of QCD 

Since mc >> ΛQCD, the c¯c mesons are nearly non-relativistic 

3 3 S 1 

2 3 S 1 

1 3 S 1 

2 1 P 1 

8·10 -4 eV 

Ionisationsenergie 

2 3 P2 

2 3 P1 

2 3 P0 

3 1 D 2 

10 -4 eV 

e + e − 

3 3 D2 

3 3 D1 

3 3 D2 

~ 600 meV 

e + e- 0.1 nm 

Mass [MeV] 

4100 

3900 

3700 

3500 

3300 

3100 

2900 

c(3590) 

c(2980) 

(4040) 

(3770) 

(3686) 

(3097) 

3 D1 

hc(3525) 

3 P2 (~ 3940) 

3 P1 (~ 3880) 

3 P0 (~ 3800) 

2(3556) 

1(3510) 

0(3415) 

C 

1 D 2 

DD 

3 D 3 

(~ 3800) 

3 D2 

1 fm 

c¯c 

C 

28

e 

p 

γ * 

e 


Quarks move relativistically inside the proton 

q 

DIS measures the 

fraction x of the 

proton energy 

which is carried 

by the quarks, 

anti-quarks and 

gluons. 

For non-relativistic 

internal motion the 

x-distribution would 

be sharply peaked 

Non-relativistic uud state 

29


Origin of the proton mass 

The u, d quarks in the proton have small masses 

2mu + md 

mp 

� 

10 MeV 

938 MeV 

� 1% 

99% of the proton mass 

is due to interactions! 

1% is due to Higgs. 

⇒ Ultra-relativistic state 

Compare this with positronium (e + e – ), the lightest QED atom: 

2me 

mpos 

� 100.00067% 

Binding energy is tiny wrt mc 2 

⇒ Nonrelativistic state 

The compatibility of the non-relativistic⎥p〉 = ⎥uud〉 quark model description 

of the proton with its ultra-relativistic parton model picture remains a mystery 

– but a mystery that we can address within QCD. 

Both are supported by data: = ?? 

⎥p〉 

uud 

30 

gluon 

q¯q

Unlike atoms, hadrons have 

no ionization threshold, 

where the quark constituents 

would be liberated. 

Hadron masses are 

generated by the 

potential and kinetic 

energies of the 

constituents 

Not yet explained 

by QCD 


Hadron mass spectrum is relativistic 

R e ! (m ) 

2 

8.00 

7.00 

6.00 

5.00 

4.00 

3.00 

2.00 

1.00 

P.Desgrolard, M.Giffon, E.Martynov, E.Predazzi, hep-ph/0006244 

Spin 

" (770) 

# (782) 

Regge trajectory 

f 2(1270) 

a 2 (1318) 

" 3(1700) 

# (1670) 

3 

f 4 (2050) 

a (2020) 

4 

0.00 

0 .00 2.00 4.0 0 6.00 8.00 10.00 

m 2 (G eV ) 

2 

(2350) 

" 5 

f (2510) 

6 

a (2 450) 

6 

For unknown 

reasons, hadron 

spins are 

proportional to 

their mass 2 

31

√+%&'( 

!"#$$%$&'()#*%+,-. 

!" # 

!" 

!" $! 

π- 

π, 

!" $! 


! "#$ %&'()* 

! !" !" # 

./0 0 1 2 3 4 5 6 7 .8 08 18 28 

⇓ 

! !" !" # 

⇓ 

elastic 

Total and Elastic Hadron Cross Sections 

0/0 1 2 3 4 5 6 7 .8 08 18 28 38 48 

π ∓ d total 

π − p elastic 

π − p total 

Not yet explained 

by QCD 

! "#$ %&'()* 

Figure 40.13: Total and elastic cross sections for π ± p and π ± d (total only) collisions as a function of laboratory beam momentum and total 

center-of-mass energy. Corresponding computer-readable data files may be found at http://pdg.lbl.gov/current/xsect/. (Courtesy of the 

COMPAS Group, IHEP, Protvino, August 2005) 

32


Heavy Ion Collisions 

The quest for a new phase of matter at RHIC, LHC,... 

33


34 

P. Skands


Hadron masses from Lattice calculations 

35 

P. Skands


The accuracy of QED perturbation theory 

Many of our most accurate predictions come from QED atoms. 

For example, the 2S1/2 – 8S1/2 splitting in Hydrogen: 

Δ(2S1/2 – 8S1/2)H = 770 649 350 012.0(8.6) kHz EXP 

= 770 649 350 016.1(2.8) kHz QED 

The QED result is based on perturbation theory: 

– an expansion in α = e 2 /4π ≈ 1/137.035 999 11(46) 

U.D. Jentschura et al, 

PRL 95 (2005) 163003 

However, the series must diverge since for any α = e2/4π < 0 the electron 

charge e is imaginary: The Hamiltonian is not hermitian and probability not 

conserved. F. Dyson 

The perturbative expansion is believed to be divergent (asymptotic). 

The good agreement of data with QED is fortuituous, from a 

theoretical point of view. For a discussion of the truncation effects in asymptotic 

expansions see Y. Meurice, hep-th/0608097 

QCD perturbation theory has many features in common with that of QED. 

36


Applications of perturbative QCD to data 

The QCD perturbative expansion successfully describes short distance 

processes, which involve high virtualities Q 2 and small αs(Q 2 ). 

Long distance dynamics is “universal”, i.e., independent of the hard process. 

Measuring the soft quark distribution and fragmentation functions in one 

process one can then predict measurable hadron cross sections. 

The applications of PQCD depends on Factorization Theorems, which hold to 

all orders in αs at “Leading Twist”, for sufficiently inclusive processes. 

The “Higher Twist” corrections are power-suppressed in the hard scale, 

∝ 1/Q 2 and can usually not be predicted. 

37

dσ 

dX 

Factorization Theorem 

Factorization: expresses the independence of long-wavelength (soft) 

Factorization emission on the nature of the hard (short-distance) process. 

� � 

� 

= 

a,b 

f 

ˆX f 

f!x,Q i " 

fa(xa, Q 2 i )fb(xb, Q 2 i ) dˆσab→f(xa, xb, f, Q 2 i , Q2 f ) 

d ˆ Xf 

! 

ˆ 

" 

! sum over long!wavelength histories 

leading to a with xa at the scale Qi 2 

� 

� 

! 

Paul Hoyer Mugla 2010 + (At H.O. each of these defined in a specific scheme, usually MS) P. Skands 

ˆ 

X 

D 

! 

p = x j 

! 

! 

P proton 

Parton distribution 

fa(xa, Q 

functions (PDF) 

2 i )fb(xb, Q 2 i ) dˆσab→f(xa, xb, f, Q2 i , Q2 f ) 

b, Q 2 i ) dˆσab→f(xa, xb, f, Q2 i , Q2 f ) 

rization 

a,b 

f 

ˆX f 

d ˆ Xf 

fa(xa, Q (ISR) 

2 i )fb(xb, Q 2 i ) dˆσab→f(xa, xb, f, Q2 i , Q2 f ) 

d ˆ Xf 

d ˆ Xf 

D( ˆ Xf → X, Q 2 i , Q 2 f) 

Illustration by M. Mangano 

X 

D( ˆ Xf → X, Q 2 i , Q 2 f) 

38 

D( ˆ Fragmentation 

Xf → X 

Function (FF) 

! Sum over long!wavelength histories 

from at Qf 2 D( to X 

ˆ Xf → X, Q 2 i , Q 2 f) (FSR and Hadronization)


e ! 

# q 

e " q 

" # 

" # 

$ ( ee & hadrons) $ ( ee & qq) 

q 

R ' 

! ! 

NC 

Q 

" # " # " # " # - 

$ ( ee & % % ) $ ( e e & % % ) 

, 

( 

) 

) 

* 

) 

) 

) 

+ 

- 

, ( u, d, s) 

, ( u, d, s, c) 

, ( u, d, s, c, 

b) 


2 

3 

10 

9 

11 

9 

N 

N 

N 

C 

C 

C 

q 

2 

q 

40

ξ = 

1 + 

= log E + p � 

� m 2 + p 2 ⊥ 


Q + M X 

2x 

� 

1 + 4m2 px2 /Q2 c¯c b¯b s¯s 

� − log tan(θ/2) 

R = σ(e+ e − → hadrons) 

σ(e + e − → µ + µ − ) 

=3 � 

q 

e 2 q 

� 

1+ αs 

π 

� 

41


LEP determination of neutrino number 

e + e − → hadrons 

at Z peak 

42

:02 PM 

e + 

e – 


Traces of Quarks and Gluons in the final state?! 

e + e – → 2 jets e + e – → 3 jets 

Z 

q 

_ 

q 

h’s 

h’s 

Q 2 = m 2 Z 

e + 

e – 

Z 

q 

_ 

q 

g 

h’s 

h’s 

43 

h’s

No exclusive amplitudes for charged particles (I) 

The identification of the data on e + e – → 2 jets with the QCD process 

e + e − → q¯q 

Gauge invariance dictates that amplitudes 

with external charged particles vanish: 


requires care, since the latter does not exist! 

This is a general feature of gauge theories, and as such is best illustrated by 

the QED process e + e − → µ + µ − 

A(e + e − → µ + µ − ) = 0 

This is because the amplitude must be invariant under local U(1) gauge 

transformations. Multiplying one of the external fermions by 

U = e iπ = – 1 we get A → – A. 

44

No exclusive amplitudes for charged particles (II) 

In the perturbation expansion the Born 

term is well-defined and ≠ 0: 

This problem shows up at order α 2 as an 

infrared singularity in the loop integral for k → 0: 

This may be seen without calculation: 

The two fermion propagators ∝ k, e.g.: 

(p1 − k) 2 − m 2 µ = −2p1 · k + k 2 ∝ k 

The photon propagator ∝ k 2 , giving a log singularity at k = 0 


e + 

e – 

e + 

e – 

γ* 

γ* 

q 

� 

0 

p1 

p2 

d 4 k 

k 4 

⇒ The exclusive process e + e – → µ + µ – is ill defined. 

µ + 

µ – 

µ + 

k 

µ – 

45

No exclusive amplitudes for charged particles (III) 

Two charged particles at different positions x, y must be connected by a 

photon string to be gauge invariant: 


¯ψ(y)exp 

� 

ie 

� y 

x 

dzνA ν � 

(z) ψ(x) 

The photon (gauge) field serves as a connection, which “informs” about the 

choice of gauge at each point in space. 

In perturbation theory, the missing string causes an infrared singularity at the 

one photon correction level. 

But we previously saw that there is no problem with the total e + e – cross 

section, which includes the µ + µ – final state?! 

To see how the IR singularity cancels in the total cross section we may use 

the optical theorem. 

46

Unitarity (white): 

σtot(s) = � 


Some Theorems 

X 

� 

Before QCD 

E.g., Lorentz inv., unitarity and the optical theorem 

SS † = 1 

“something will happen” 

note: includes “no” scattering 

The Lagrangian of QCD 

Optical Theorem 

As a consequence of the unitarity of the scattering matrix: 

the total cross section may be expressed in terms of the 

Unitarity (white) 

imaginary part of the forward elastic amplitude: 

SS † =1 

dΦX |MX| 2 = 8π 

Optical Theorem (white) √ Im [Mel(θ = 0)] 

s 

σtot(s) = 

µ 

)( D µ)ij ψ j q− mq ¯ ψ i qψqi− 1 

4 F a µνF aµν 

The Lagrangian of QCD in white 

µ 

)( D µ)ij ψ j q− mq ¯ ψ i qψqi− 1 

X ~ XX 

2 

X 

10 


SS † = 1 

Total = The Lagrangian that can of happen QCD 

dΦX |M X | 2 = 8π 

√ Im [M 

s 

el(θ = 0) ] 

Sum over everything 

“Square Root” of 

nothing happening 

L = ¯ ψ i q(iγ µ )( D µ)ij ψ j q− mq ¯ ψ i qψqi− 1 


The Lagrangian of QCD in white 

L = ¯ ψ i q(iγ µ )( D µ)ij ψ j q− mq ¯ ψ i qψqi− 1 


The sum over all states X becomes a completeness sum on the rhs. 

QED satisfies unitarity at each order of α. 

= 

47 

P. Skands

µ + 


µ + 

γ* γ* 

µ – 

Optical Theorem for σtot(e in QED (I) 

+ e − ) 

O (α) O � α 2� 

µ – 

γ ∗ → µ + µ − 

µ + 

µ – 

γ 

µ + 

µ – 

γ ∗ → µ + µ − γ 

µ + 

µ – 

µ + 

µ – 

γ ∗ → µ + µ − 

At O(α 2 ) there are two contributions to the imaginary part (dashed line). 

The IR singularity cancels between them, i.e., between different final states! 

Since the γ*→ γ* amplitude does not have external charges, it “has to be” regular. 

It means that even in QED we must define cross sections such that they include 

(arbitrarily soft) photons. There are no free, “bare electrons”. 

48


Optical Theorem for σtot(e in QED (II) 

+ e − ) 

γ* γ* 

k k 

Since the IR singularity is only at k = 0, it suffices to include only photons with 

k < k0, for arbitrarily small k0. 

The criterion is to sum over all states that are degenerate in energy, 

since these have an asymptotically long formation time Δt ∼ 1/ΔE 

In QED, collinear IR singularities are regulated by the electron mass: 

49 

Kinoshita- 

Lee-Nauenberg 

(KLN) theorem 

p 

Ee = 

xp 

(1–x)p 

+ 

� p2 + m2 e Ee+γ = � (xp) 2 + m2 e + (1 − x)|p| 

For |p| >> me the 

collinear e + γ state has 

nearly the same energy 

as the single electron e. 

⇒ In QED, collinear bremsstrahlung is enhanced by a factor log(|p|/me).


Removing IR sensitivity in QCD 

In QCD even soft gluons can change the color of the quark 

⇒ we can never measure the color of a quark! 

Also: Want to sum over all soft and collinear divergences up to a 

“Factorization scale” µ, large enough to apply PQCD for Q > µ. 

Define IR safe cross sections, which can be calculated in PQCD 

using the factorization theorem, and measured in experiments. 

50

e�nition 


Infrared Safe observables 

An obser vable is infrared safe if it is insensitive to 

SOFT radiation: 

Adding any number of infinitely soft particles should not 

change the value of the obser vable 

COLLINEAR radiation: 

Splitting an existing particle up into two comoving particles 

each with half the original momentum should not change 

the value of the obser vable 

51 

P. Skands

LO: 

σ NLO 

� 

1 (R) = 

� LO, NLO, etc 

|M (0) 

2Re[M (1) 

52 

σ Z → 2 1-loop: 

qk 

LO 

(R) = |M X+1 (0) 

X+1 |2 

Cross sections 

� 

at NLO 

σBorn = |M 

Z → 3 jets: 

(0) 

X |2 

Z → 2 1-loop: 

qk 

g 

qi � 

ik 

σ 

qk 

qi 

a 

qk 

LO 

X+1 (R) = |M 

R 

(0) 

X+1 |2 

� 

X = σBorn + |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

Z decay: 

q q 

q q 

� 

|M| 

colours 

2 Z → 3 jets: 

qk 

qi 

g 

qi 

= 

jk 

qk 

qi 

a g 

qi 

ik 

Z → 2 1-loop: 

qk 

a 

g 

qi � 

ik 

|M 

R 

qk 

qi 

a 

(0) 

X+1 |2 

Z → 3 jets: 

qk � 

qi 

g 

qi 

jk 

qk 

qi 

a g 

qi 

ik 

Removing IR sensitivity in QCD at NLO 

a 

g 

qi 

qk 

orem (Kinoshita-Lee-Nauenberg) 

ities cancel at complete order (only finite te 

ik 

� 

qk 

qi 

a 

σBorn + |M qk 

(note: Not the 1-loop diagram sq 

(0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

M X X ] 

|M (0) 

X |2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

M X X ] 

�� 

rn+Finite |M (0) 

X+1 |2 

� �� 

+Finite 2Re[M 

σ NLO 

O, etc 

� 

σBorn = |M 

= X σBorn(1 + K) 

(0) 

X |2 

σ LO 

� 

(R) = |M X+1 

R 

(0) 

X+1 |2 

� 

σBorn + |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

M X X ] 

Z → 2 1-loop: 

LO: 

g 

NL Theorem (Kinoshita-Lee-Nauenberg) 

Singularities cancel at complete order (only finite terms left 

Lemma: only � after some hard �work 

ik 

σ 

(note: Not the 1-loop diagram squared) 

LO 

X+1 (R) = |M 

R 

(0) 

X+1 |2 

σ NLO 

� 

X = σBorn + |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ NLO 

� 

X = |M (0) 

X |2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

�� 

NLO 

X = σBorn+Finite |M (0) 

X+1 |2 

� �� 

+Finite 2Re[M (1) (0 

X M X 

σ NLO 

LO, NLO, etc 

� 

σBorn = |M 

X = σBorn(1 + K) 

� � 

� � 

(0) 

X |2 

σ LO 

� 

X+1 (R) = |M 

R 

(0) 

X+1 |2 

σ NLO 

� 

X = σBorn + |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ NLO 

� 

X = |M (0) 

X |2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ 


NL Theorem (Kinoshita-Lee-Nauenberg) 

Singularities cancel at complete order (only finite terms left over) 

Lemma: only after some hard work 

�� 

NLO 

� 

X = |M (0) 

X |2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

�� 

NLO 


X+1 |2 

� �� 

+Finite 2Re[M (1) (0)∗ 

X M X ] 

� 

σ NLO 


NNLO 

X = σ NLO 

X + 

� � 

|M (1) 

X |2 + 2Re[M (2) (0)∗ 

X M X ] 

� � 

+ 2Re[M (1) 

σBorn = |M 

(0) 

X+1M X+ 

(0) 

X |2 

σ LO 

� 

X+1 (R) = |M 

R 

(0) 

X+1 |2 

σ NLO 

� 

X = σBorn + |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ NLO 

� 

X = |M (0) 

X |2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ NLO 

�� 


X+1 |2 

� �� 

+Finite 2Re[M (1) (0)∗ 

X M X ] 

� 

σ NLO 


!1(e + e − → q ¯q(g)) = !0(e + e − � 

→ q ¯q) 1 + "s(ECM) 

+ O(" 

# 

2 X 

� 

+ |M 


(Kinoshita-Lee-Nauenberg) 

ncel at complete order (only finite terms left over) 

after some hard work 

� 

s) 

29 

(0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

�� 

te |M (0) 

X+1 |2 

� �� 

+Finite 2Re[M (1) (0)∗ 

X M X ] 

� 

LO 

= σBorn(1 + K) 

� 

|M (1) 

X |2 + 2Re[M (2) (0)∗ 

X M X ] 

� � 

+ 2Re[M (1) (0)∗ 

X+1M X+1 ]+ 

� 

|M (0) 

X+2 |2 

σBorn = |M (0) 

X |2 

� 

LO 

X+1 (R) = |M 

R 

(0) 

X+1 |2 

� 

|M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

| 2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

�� 

nite |M (0) 

X+1 |2 

� �� 

+Finite 2Re[M (1) (0)∗ 

X M X ] 

� 

NLO 


)) = !0(e 

� 

� � 

� 

+ e − � 

→ q ¯q) 1 + "s(ECM) 

+ O(" 

# 

2 ... 

� 

s) 

Z decay: 

q 

X+1 |2 + 

q q 


q 

� 

|M| 2 = 

colours 

� 

∝ δijδ ∗ ji 

= Tr[δij] 

= NC 

Z decay: 

NNLO = σ NLO + 

8 

q 


= Tr[δij] 

= NC 

q 

q q 

� 

|M| 2 = 

colours 


= Tr[δij] 

= NC 

qi 

qi 

qk qk 

g jk 

a 

� 

(1) 2 (2) (0)∗ 

M (0)∗ 

R 

X qk 

qk 

] 

g ik 

Z → 3 jets: a 

qi 

qi 

qk 

g jk 

8 

a 

qi 

16 

8 

qi 

qk 

g ik 

a 

8 

qi 

qi 

� � � � 

|M (1) | 2 + 2Re[M (2) M (0)∗ (1) (0)∗ ] 

29 

qk 

qi 

qi 

qk 

qk 

a 

16 

+ 

qk 

qk 

16 

2Re[M (1) 

P. Skands 

(0) 2

Cross 

σσ Sections at NNLO 

X+1 (R) 

Sections 

= |MX+1 | 

at NNLO 

X+1 (R) = |MX+1 | 

R 

NLO 

X = σBorn + |M 

NLO 

(0) 

X+1 |2 + 2Re[M (1) (0)∗ 

X M X ] 

� 

O 

= |M (0) 

X |2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

�� 

O 

= σBorn+Finite |M (0) 

X+1 |2 

� �� 

+Finite 2Re[M (1) (0)∗ 

X M X ] 

σ 

Z → 2 1-loop 1-Loop squared: ! 1-Loop 

Z → 2 1-loop 

1-Loop 

squared: 

! Real (X+1) 

� 

qk qk 

qj qj 

NNLO 

qi 

qi 

qi 

qk 

NLO 

� 

X = σBorn + |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ NLO 

� 

X = |M (0) 

X |2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ 


Z → 2 1-loop 

1-Loop 

squared: 

! Real (X+1) 

NNLO �� 

� 

qk qk 

qj qj 

LO 

X+1 (R) = X+1 

R 

|2 

σ NLO 

� 

X = σBorn + |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ NLO 

� 

X = |M (0) 

X |2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ 


Z → 2 1-loop 

1-Loop 

squared: 

! Real (X+1) 

NNLO �� 

� 

qk qk 

qj qj 

NLO 

� 

X = σBorn + |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

σ NLO 

� 

X = |M (0) 

X |2 � 

+ |M (0) 

X+1 |2 � 

+ 2Re[M (1) (0)∗ 

X M X ] 

Removing IR sensitivity in QCD at NNLO 


Z → 2 1-loop 

1-Loop 

squared: 

! Real (X+1) 

�� 

� 

qk qk 

qj qj 

σ NLO 

σ = σBorn+Finite 

NLO σ = σBorn+Finite 

NLO 

= σBorn+Finite 

X X 

LO = σ NLO 


g ik 

σ NLOg 

qi 

ik 

a 

σX NLO qi qi 

qi 

X 

qk qk qk 

X + 

� � 

X + 

� � 

X + X + 

σ NNLO 

= σ NLO 

σ NNLO 

= σ NLO 

σ NNLO 

= σ NLO 

X X 

Cross Sections at NNLO 

qi qi 

qi 

a 

gik a 

gik |M (0) 

X+1 |2 |M qi 

(0) 

X+1 |2 |M qi 

(0) 

X+1 |2 qi 

a 

g 

= σBorn(1 qi + K) 

ik 

ga ik g 

a 

ik 

a 

= σBorn(1 qi 

σ + K) 

NLO 

a 

σ = σBorn(1 qi + K) 

NLO 

= σBorn(1 qi + K) 

qk 

�� 

g ik 

qk qk qk 

qk 

|M (1) 

X |2 + 2Re[M (2) (0)∗ 

|M 

X M (1) 

X |2 + 2Re[M (2) (0)∗ 

X M X ] 

|M + 

(1) 

X |2 + 2Re[M (2) (0)∗ 

X M X ] 

|M + 

(1) 

X |2 + 2Re[M (2) (0)∗ 

X M X ] + 

Z → 2 2-loop: 

15 

15 

15 

Z → 2 2-loop: 

qk 

qk 

qj 

qj qk 

qi 

qi qj 

gij g jk gij a g jk 

g b 

ij 

a g jk 

+Finite 

g 

qi 

qj 

ij 

a g 

qk 

jk 

qi a b b 

qi 

qi 

qi 

qj 

qj 

qj 

qk 

qk 

qk 

b 

qk 

qk 

qk 

qk 

Two-Loop ! Born Interference 

Two-Loop 

Two-Loop 

! 

! 

Born 

Born 

Interference 

Interference 

18 

18 

18 

|M (0) 

gjk g ij 

gjk 2Re[M (1) (0)∗ 

2Re[M M (1) (0)∗ 

2Re[M M (1) (0)∗ 

M 

qi g 

qk 

ik 

a 

b X 

c 

qi g 

qk 

ik 

qi 

g 

c 

jk 

g a ij 

qk 

b 

g 

qk 

jkX 

a 

qi g 

qk 

ik 

qi 

g 

c 

jk 

g a ij 

qk 

b 

g 

qk 

jk 

X 

a 

qi g 

qk 

ik 

qi 

g 

c 

jk 

g a ij 

qk 

b 

g 

qk 

jk 

a 

qk 

qk 

X ] 

�� 

� �� 

�� 

� 

+ 

qk 

Two-Loop ! Born 30 

30 

30 Interference 

X ] X ] 

X a ] 

2Re[M (1) 

2Re[M (1) 

2Re[M (1) 

18 

18 

18 

and so on, at each fixed order in αs 

qk 

2Re[M (1) (0)∗ 

(0)∗ 

X+1 M M 

X+1 

18 

X+1 ]+ 

� 

X+1 ]+ 

� 

X+1 ]+ 

� 

(0)∗ 

X+1M Z Z Z→ → →4: 4: 

qi 

qi 

qi 

qi qi qi 

qj qj 

qj 

gik g ij 

b 

gik g 

a 

ij 

b 

gik g 

a 

ij 

b 

qk qk qk 

X+1 ]+ 

� 

|M (0) 

X+2 |2 X+2 |2 |M (0) 

X+2 |2 

|M (0) 

a 

qj Z → 4: 

qi 

gik g 

a 

ij 

b 

gik g 

a 

ij 

b 

gik g 

a 

ij 

b 

qj 

qj 

qk qk 

qi 

qiqj 

qi 

Real Real ! ! Real Real (X+2) (X+2) qk 

Real ! Real (X+2) 

g ij 

qi qi 

b 

g ik 

a 

53 

|M ( 

X 

Real ! Real 

P. Skands


Event generators 

At high energies many gluons are radiated. Then one needs to include 

high orders in αs for which complete, IR regulated matrix elements are not 

available. 

⇒ Shower Monte Carlo methods: 

Include only the log enhanced terms of tree digrams (no loops) 

Use phenomenological hadronization model for Q < µ. 

This gives approximate results for hadron 

distributions in the final state. 

The reliability is tested by including 

next-to-leading logarithms and comparing 

several hadronization models. 

Cluster Fragmentation 

55


QCD is very successful in comparisons with data 

Rate of 3-jet events 

in e + e – annihilations 

e + e – → q q g 

→ 3 jets 

Ex: Estimate the CM 

energy in e+ e– annihilations 

at which 2-jet structure 

emerges. In quark 

fragmentation, pions get an 

average fraction ≈ 0.1 

of the quark energy, and 

≈ 350 MeV. 

S. Bethke, hep-ex/0606035 

56

Angular distribution 

of 4-jet events in 

e + e – annihilations 



57

Measurement of quark 

and gluon color charges 

in e + e – annihilations 


quark 

color 

charge 

e + e – 


gluon charge 

58

(Nonperturbative) 


Jet production in hadron collisions 

(Inclusive sum) 

(log. scaling 

violations) 

(Perturbative) 

(Nonperturbative) 

(log. scaling 

violations) 

arXiv:1002.1708 

(or: h + X) 

(Higher 

order in αs) 

59


The CDF Detector 

60

Fermilab: 

〉 pp 

– �= → 0 jet and + X 〈0|F a µνF µν 

ECM = 1.96 TeV 

Quarks and gluons 

are pointlike down 

q 

to the best resolution 

that has been reached 

Ex: Estimate the maximum 

radius of quarks and gluons, 

given the agreement of 

QCD with the Fermilab jet 

2x 

data. 

ξ = 

Rapidity: 

F2(x) = � 

x = 

1 + 

y = log E + p � 

� m 2 + p 2 ⊥ 


e 2 q xfq(x) 

Q 2 

Q 2 + M 2 X 

� 

1 + 4m 2 px 2 /Q 2 

� − log tan(θ/2) 

ECM = 1960 GeV 

a |0〉 �= 0 

CDF Collab., hep-ex/0701051 

61


pp 

– 

→ jet + X: Jet mass distribution 

62


Concluding remarks 

Perturbative QCD has been successfully applied to hard collision data. 

! 

" 

! $% & ' ()**+, - ()((*( 

# 

QCD effects constitutes the major background in the search for new physics 

at the Tevatron and the LHC. Hence much effort is expended on making the 

calculations as accurate as possible. 

e Standard Model A. Pich - CER 

High intensity electron beams at Jefferson Lab, Mainz,... are mapping out 

hadron structure through Form factors and Generalized Parton Distributions. 

Lattice QCD methods allow fast progress in the calculation of nonperturbative 

quantities, such as the hadron spectrum. 

The simple Quark Model systematics of the hadron spectrum remains an 

encouraging mystery. 

63

Introduction to QCD slides - P.Hoyer.pdf - High Energy Physics Group

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