# Document

```Introduction to QCD and perturbative QCD
ELFT Summer School, May 24, 2005
Three lectures:
1. Symmetries
– exact
– approximate
2. Asymptotic freedom
– renormalisation group
– β function
3. Basics of pQCD
– fixed order
– Resummation
Should be understood by everybody, so will be trivial
When you measure what you are speaking about
and express it in numbers, you know something
about it, but when you cannot express it in
and unsatisfactory kind.
Lord Kelvin
Field content
• Quark fields: qf
f=1,…,6
B=1/3
f
1
2
3
4
5
6
qf
u
d
s
c
b
t
mf
~5MeV ~7MeV
~100MeV
1.2GeV 4.2GeV
174GeV
mf running masses (see later) at =2GeV, approximate
values; Nf is the number of light quarks (e.g. 3)
•Gluon fields: Aɑ
ɑ=1,…,8
The QCD Lagrangian
• QCD is a QFT, part of the SM
• The SM is a gauge theory with underlying
SUc(3) SUL(2) UY(1)
LQCD  L
0
QCD
0
QCD
L
L
sources
QCD
 LC  LGF  LGhost
The Classical Lagrangian

Nf

6
1 a a
LC   F F   q f i  D  q f   q f i  D   m f q f
4
f 1
f  N f 1
  ,  0
  ,  2g
5
D     igs Aa T a
are the SU(Nc) generators with algebra:
Ta
T ,T   if
a
b
abc
T
c
TR  1 / 2
T r T a T b  TR ab
fundamental representation:
1 a
T    ta
2
T a bc  iabc  Fbca
a
 
The Classical Lagrangian
Colour factors (like eigenvalues of J2):
•in the fundamental representation:
t t 
a a
bc
 CF bc
N c2  1
CF  TR
Nc
F F 
a
a
bc
 CAbc
CA  2TR Nc
Fa    Aa   Aa  igs Fbca Ab Ac

source of non-Abelian nature: gluon self coupling
Note: in lattice QCD
gs A  A
a
a
 F
a
1 a
 F
gs
Exact symmetries of the classical Lagrangian
• Quantum effects may violate
(e.g. scale invariance, axial anomaly)
• Continuous: local gauge invariance
(suppress flavour and Dirac indices)
U : qi ( x)  qi( x)  U ( x)ij q j ( x) ,
8
U ( x)  exp i a ( x)T a  SU N c 
a 1
• The covariant derivative transforms as the field
itself U : D q( x)i  D q( x)i  U ( x)ij D q( x) j
D q(x)    ig A U(x) q( x) , A  AT
  U(x)q( x)  U(x) q( x)  ig A U(x) q( x)
 U(x) q( x)  ig U(x) A q( x)
i
A   U  U  UA U
 F  UF U
g
a
a
s





s
1

s
1
s




1
Exact symmetries of the classical Lagrangian
• Almost supersymmetric:
– massless QCD for one flavour
QCD
C
L
1 a a
  F F  q i  D  q
4
– SUSY Yang-Mills :
SYM
C
L
1 a a
  F F   i  D  
4
• q transforms under the fundamental,  under
• Advantageous in deriving matrix elements
Exact symmetries of the classical Lagrangian
• The quark mass term is, the gluon mass term is
not gauge invariant
• Discrete: C, P and T in agreement with observed
properties of strong interactions (C, P and T
violating strong decays are not observed)
• Note:  additional gauge invariant dimensionfour operator, the -term:
g s
~ 
L 
F F ,
2 
32
~  1 
F  
F
2
Conventional normalisation
•violates P and T (corresponds to EB in electrodyn.)
  is small (&lt;10-9 experimentally), set =0 in pQCD
Approximate symmetries of the classical Lagrangian
Related to the quark mass matrix
Introduce:
 u   q1 
   
   d    q2 
 s  q 
   3
P P  P P ,
P2  P ,
…and from Dirac algebra:
Let:
   P
Eigenvector of
5:
1
P  1   5 
2
P  P  1
  P  P 
    0
     P
 52  1   5    
Approximate symmetries of the classical Lagrangian
The quark sector of the Lagrangian can be written:
LChir  i  D   P  P  i  D

P  P 

 P i  D P   P i  D P 
  i  D     i  D    L  L  LL  LR
• would not work if gluons were not vectors (in D)
• the left- and right-handed fields are not coupled
 LChir is invariant under UL(Nf) UR(Nf)
• the group elements can be parametrised in
terms of 2 Nf2 real numbers:
gL , gR   e
i i ia T a
e e
ia T a
e
i -i ia T a
,e e e
-ia T a
e

UV 1 U A 1  SUL N f  SUR N f 
Approximate symmetries of the classical Lagrangian
• This symmetry acts separately on left- and
right-handed fields: chiral symmetry
• Has vector subgroups SUV(Nf) UV(1)
g , g   e
i a T a
,e
i a T a
 e
i a T a 
 SU V N f 
• The axial transformations do not form a
subgroup
h, h   e

ia T a
,e
-ia T a
 e
ia T a  5
T  ,T    if
a
b
5
5
• Chiral symmetry is not observed in the QCD
spectrum, it is spontaneously broken to
SUV(Nf) UV(1)
abc
Tc
Chiral perturbation theory
• In QCD it is believed that the vacuum has a nonzero VEV of the light-quark operator
0 | qq | 0  0 | uu  dd | 0
3


 250 MeV
• This quark condensate breaks chiral symmetry
because it connects left- and right-handed fields
0 | qq | 0  0 | qL qR  qR qL | 0
• The SSB of chiral symmetry implies the existence of
Nf2-1massless Goldstone bosons
• The light quarks are not exactly massless  the
chiral symmetry is not exact, the Goldstone bosons
are not massless: pseudoscalar meson octet
• The mf are treated as perturbation  PT
Topics of QCD (T=0)
• Low-energy
properties (&lt;GeV)
• PT (light
quark masses)
• High energy collisions
(&gt;GeV)
• Perturbative
• Non-perturbative
• Jet physics
• Sum rules,
lattice QCD
Approximate symmetries of the classical Lagrangian
Choose Weyl representation:
 0  1

 0  
 1 0 
 0 i 

 i  
 i 0 
1 0 

 5  
 0  1
 0  ~ two-component
~ 
    
    ~ 
Weyl spinors
 0 
  
~  x   exp i p  x ~  p  helicity

Define
~ x   0    p
~  p  0
i  


~  p   p 
~  p  0
 E

i i

eigenstates
if m=0, g=0
~  p   
~  p
ˆ
σ p


Asymptotic freedom
• At the heart of QCD  Nobel prize 2004
• Consider a dimensionless physical observable
R=R(Q), with Q being a large energy scale,
Q  any other dimensionful parameter (e.g. mf)
 set mf =0 (check later if R (mf =0) is OK)
dR
0
• Classically dimR = 1 
dQ
lim
0 
R

R
Q 2 
• In a renormalized QFT we need an additional
scale:  renormalization scale  R = R (Q2/2)
is not a constant: scaling violation
•  the „small” parameter in the perturbative
expansion of R, s() also depends on the
scale choice

Asymptotic freedom
• But  is an arbitrary, non-physical parameter
(LCl does not depend on it)  physical quantites
cannot depend on 
2
 2 

Q
 
2 d
2 
2  s
 R  0

R 2 , s     

2
2
2
d  
  s 
   
2  s
2
2
• Let t = ln (Q / ), (s)=

2
  s0 fixed
 
s
 
  t
     s 
 R e , s  0
 s 
 t


Asymptotic freedom
• To solve this renormalization-group equtaion,
we introduce the running coupling s(Q 2):
 
s Q2
dx
t 
 x 
  2 
s

 
 s Q
1
1
 s Q2
t
  
2
• If 2 =Q 2  et = 1, the scale-dependence in R
enters through s(Q 2)
• All this was non-perturbative yet
The  function in perturbation theory
• We solve
  s Q
2

 
2


Q
 Q2 s 2
Q
in PT (we analyse the validity of PT a little later)
 s 
  s    s   n  
 4 
n 0

n 1
• known coefficients:
11
4
34 2
20
 0  C A  TR N f 1  C A  4CF TR N f  C ATR N f
3
3
3
3
2857 5033
325 2
2 

Nf 
Nf
2
18
54
3  29243 6946.3N f  405.9N 2f 1.5N 3f
The  function in perturbation theory
• if s(Q 2) can be treated as small parameter, we
can truncate the series, keep the first two terms:
 
2


Q
Q 2 s 2  b0 s2 Q 2 1  b1 s Q 2  O  s2 Q 2
Q
1
0
b1 
with b0 
40
4
 1
1 
• LO: 1  s


 b0t
 b0
2
2 
2
s  
 s t
 s Q
 
 
 
 s 

1

2
2 
s Q
1  b0t  s  
2
 
    
 
1
if
 
 Q  ~
2
t
s
• Relation between s(Q 2) and :s( 2) if both
small
1
b0t
The  function in perturbation theory
• The running coupling resums logs:
if R = R1 s+O(s2) 
    R        b t 
R 1, s Q

2
2
1
s
2
j 1
s
j
0
• NLO (b10):
 s
1
 b0
2
 s 1  b1 s  t
 
 
 
 
s Q
1  b1 s Q
1
1

 b1 ln
 b1 ln
 b0t
2
2
2
2
s Q
s 
s 
1  b1 s 
2
 
 
2
QCD
• A more traditional approach to solving the
renormalization-group equation: introduce 

2
Q
dx
ln 2  
  Q 2   x 
s
•  indicates the scale at which s(Q 2) gets
strong
1
2
• LO (b00, bi=0):  s Q 
Q2
 
b0 ln
• NLO (b10):
2
 
 
b1 s Q 2
1
Q2
 b1 ln
 b0 ln 2
2
2
s Q
1  b1 s Q

 
• The two solutions differ by subleading terms that are
important in present day precision measurements
The running coupling
The quark masses
• Assume one flavour with renormalized mass m:
yet another mass scale
 2 

 
2
2
 











m
R
Q

, s , m Q  0
s
m
s
2

 s
m 
 


• m is the mass anomalous dimension, in PT:
303 10N f
1
2
2
2
 m  c0s Q 1  c1s Q   c0 
c1 

72
• R is dimensionless 
 
  
 2 
2 
2  
2
2
 Q




m
R
Q

, s , m Q  0
2
2
2 

m 
 Q
 2 
 1
 

2
2
 Q












m
R
Q

, s , m Q  0


s
m
s
2

 s  2
 m 
 Q




The running quark mass
• To solve this renormalization-group equtaion,
we introduce the running quark mass m(Q 2):


     
1  mQ   
 R 1, Q ,0   
 R 1, Q ,0 
n! Q
R Q2  2 ,s ,m Q  R 1,s Q2 ,m Q2 Q
n
2
N
2
s
n
n 1


2
s
• the derivative terms (if finite) are suppressed by
at least an inverse power of Q at high Q 2
 dropping the quark masses is justified
 only IR-safe observables can be computed
The running quark mass
• All non-trivial scale dependence of R can be
included in the running of mass and coupling:
m
2


Q




m
Q
m
s
2
Q
 
2
Solution:
Q
2


d
Q
m Q 2  m  2 exp  2  m  s Q 2 
  2 Q

2
  s Q 
 




 m  2 exp  d  s m s   m  s Q 2
  s  
 s  2 
 
 
 
2
  
   1  
cb
• c/b &gt; 0  the running mass vanishes with the
running coupling at high Q 2
2 hard photons in CMS
4 muons in CMS
4 muons in CMS
Basics of Perturbative QCD
• Vast subject – only give the flavour
• Will use a specific example:
 


e
 e  qq 
 e  e   hadrons q
R

 
 
 e e      e  e       
• 2→2 scattering has one free kinematical
parameter, the θ scattering angle
 
• The differential cross section for e e  f f
d
 2 
2

 1  cos 
d cos
2s 



 2

 2 s2

2
2
2
2
   
Q f  Ae  Ve Af  V f
2 2
2
2
s  M Z  Z M Z






The total cross section
• below the Z pole
• on the Z pole




4 2  22 2
Q2f Ae  Ve2 A2f  V f2
0 
3s Z



• LO: the hadronic cross section is obtained by
counting the possible final states:
2
2
A

V
 q q
R  3 Q  R0
2
q
q

RZ  3
q
A
2

 V2


• With q = u,d,s,c,b R =11/3=3.67 and RZ =20.09
• The measured value at LEP is RZ =20.79&plusmn;0.04
• The 3.5% difference is mainly due to QCD effects:
– Real
– virtual
gluon emission
NLO: real gluon emission
• Three-body phase space has 5 independent
variables: 2 energies and 3 angles
• Integrate over the angles and use yij = 2pi&middot;pj/s
scaled two-particle subenergies, y12+y13+y23=1
• The real contribution to the total cross section:
 s  y23 y13 2 y12 

1  y13  y23 
   0 R0  dy13  dy23CF


2  y13 y23 y13 y23 
0
0
• Divergent along the boundaries at yi3 = 0:
1
1
R
• Unphyisical singularities - quarks and gluons
are never on (zero) mass shell: Breakdown of PT
yi 3 s  2Ei E3 1  cosi 3 
• Divergent when E3→0 (soft gluon), or θi3 →0
(collinear gluon)
NLO: the real and virtual contribution in d  4
• To make sense of the real contribution, we use
dimensional regularization:
1

 y23 y13  2 y12
d
y
s 
R
13 dy 23
 
   0 R0 H      C F

 2 
1   
y13 0 y23
2 
 y13 y23  y13 y23
0

 2 3 19

2
  0 R0 H   2      O 
H    1  O 
 2


1
• Has to be combined with the virtual contribution
 2 3

V
2
   0 R0 H   2   8    O 

 

• The sum of the real and virtual contributions is
finite in d = 4:
 s
2 
R  R0 1   O  s 
(same for RZ)


 

The total hadronic cross section at O(αs3)
• The total cross section can be computed more
easily using the optical theorem   Im f    
2




2
2 2
R  R0 1  c1 s   c2  c1b0 ln 2  s  
Q 


 
 
2
2






2
2 3
 c3   2c2b0  c1b1  c1b0 ln 2  ln 2  s   O  s4
Q  Q 


 
c1 
1

c2 
1.409

2
c3  



 
12.85
3
• Satisfies the renormalization-group equation to
order αs4
The total hadronic cross section at O(αs3)
The total hadronic cross section at O(αs4)
(non-singlet contribution)
Jet cross sections
Typical final states in high energy electron-positron
collisions
2 jets
3 jets
Modelling of events with jets
Production probability pattern:
2jets : 3jets : 4jets ~ O(αs0) : O(α s1) : O(α
⇒ jets reflect the partonic structure
s
2)
Jet cross sections
• We average over event orientation ⇒ |M2|2 has
no dependence on the parton momenta

1
LO
 M2
• NLO corrections:
2
 dy  1  y  J  p ,p 
12
12
2
1
2
0
 s  y23 y13 2 y12 

 J 3  p1,p2 ,p3 
  M 2  dy13  dy23CF


2  y13 y23 y13 y23 
0
0
1
1
2
R
s
1  4

  M 2 CF
2 1     s
2
V
2




 2 3

2





8



O

  2 

1
  dy12  1  y12  J 2  p1,p2 
0
• Cannot combine the integrands (like for σtot)
The subtraction scheme
• Process and observable independent solution
 NLO   3NLO   2NLO
d 3NLO  d R J 3  d A J 2


d 2NLO  d V  d A J 2
• Made possible by the process-independent
factorisation properties of QCD matrix elements
y23 y13 2 y12
y
2 y12


 23 
 1  2
y13 y23 y13 y23 y13 y13  y13  y23 
 y12  y13  y23


 2   1  y12  y13   1  2
 2
y13  y23



 1 

y23 
2
  
 1  z1  
  1  2
 1  y13 
 y13  1  z1 1  y13 
1

y13
The subtraction scheme
• The approximate
cross section:

1
1
2


1
4


  dy13  dz1 1  y13 1- 2 ε y13 -ε z1 1  z1 -ε D13, 2
d A 
1     s  0
0



= I(ε) |M2|2
 1  2
• with universal factorisation:

s 1 
2

D13, 2  p1,p2 ,p3   M 2 CF
 1  z1   1  z1 
2 y13  1  z1 1  y13 

2
s
1  4

   M 2 CF
2 1     s
2
2




2 3

2
 O 
 2   10 

3


The subtraction scheme
• The integrated approximate cross section:
 A    dy13, 2 1  y13, 2 J 2  ~p13 ,~p2   1  2
s
1  4

  M 2 CF
2 1     s
2
V
2




 2 3

2
  2    8    O 
1
  dy12  1  y12  J 2  p1,p2 
0
 2NLO
s
2
 M 2 CF

 2
1
 O   dy12  1  y12  J 2  p1,p2 
1 
3

0
Parton showers and resummation
• The universal factorisation
D ij,k  pi ,p j ,pk   M m
2

 s 1 
2
CF
 1  zi   1  zi 

2 yij  1  zi 1  yij 

can be used to describe parton showers
(neglecting colour correlation of soft emissions
and azimuthal correlations of gluon splitting not transparent in the simple example
considered here)
General picture of high-energy collisions
R at low energies
```