1. Introduction to QCD and perturbative QCD ELFT Summer School, May 24, 2005
Three lectures:
Symmetries
exact
approximate
Asymptotic freedom
renormalisation group
β function
Basics of pQCD
fixed order
Resummation
Should be understood by everybody, so will be trivial

2. 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 numbers your knowledge about is of a meagre and unsatisfactory kind.
Lord Kelvin

3. Field content Quark fields: qf f=1,…,6 B=1/3 qf mf2 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

4. 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)

5. The Classical Lagrangian
are the SU(Nc) generators with algebra:
fundamental representation:
adjoint representation:

6. The Classical Lagrangian
Colour factors (like eigenvalues of J2):
in the fundamental representation:
in the adjoint representation:
source of non-Abelian nature: gluon self coupling
Note: in lattice QCD

7. 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)
The covariant derivative transforms as the field itself

8. Exact symmetries of the classical Lagrangian
Almost supersymmetric:
massless QCD for one flavour
SUSY Yang-Mills :
q transforms under the fundamental,  under the adjoint representation of SU(Nc)
Advantageous in deriving matrix elements

9. 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 dimension-four operator, the -term:
Conventional normalisation
violates P and T (corresponds to EB in electrodyn.)
  is small (<10-9 experimentally), set =0 in pQCD

10. Approximate symmetries of the classical Lagrangian
Related to the quark mass matrix
Introduce:
…and from Dirac algebra:
Let:
Eigenvector of 5:

11. Approximate symmetries of the classical Lagrangian
The quark sector of the Lagrangian can be written:
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:

12. 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)
The axial transformations do not form a subgroup
Chiral symmetry is not observed in the QCD spectrum, it is spontaneously broken to
SUV(Nf) UV(1)

13. Chiral perturbation theory
In QCD it is believed that the vacuum has a non-zero VEV of the light-quark operator
This quark condensate breaks chiral symmetry because it connects left- and right-handed fields
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

14. Topics of QCD (T=0) Low-energy properties (<GeV)
High energy collisions (>GeV)
Perturbative
Non-perturbative
PT (light quark masses)
Jet physics
Sum rules, lattice QCD

15. Approximate symmetries of the classical Lagrangian
Choose Weyl representation:
two-component
Weyl spinors
helicity
eigenstates
if m=0, g=0
Define

16. 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)
Classically dimR = 1 
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

17. Asymptotic freedom
But  is an arbitrary, non-physical parameter (LCl does not depend on it)  physical quantites cannot depend on 
Let t = ln (Q 2/2), (s) = 

18. Asymptotic freedom
To solve this renormalization-group equtaion, we introduce the running coupling s(Q 2): 
If 2 =Q 2  et = 1, the scale-dependence in R enters through s(Q 2)
All this was non-perturbative yet

19. The  function in perturbation theory
We solve
in PT (we analyse the validity of PT a little later)
known coefficients:

20. 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:
with
LO:
if t:
Relation between s(Q 2) and s( 2) if both small
Q 2  0  s(Q 2)  0: asymptotic freedom (sign!)

21. The  function in perturbation theory
The running coupling resums logs:
if R = R1 s+O(s2) 
R2 s2 gives one less log in each term
NLO (b10):

22. QCD
A more traditional approach to solving the renormalization-group equation: introduce 
 indicates the scale at which s(Q 2) gets strong
LO (b00, bi=0):
NLO (b10):
The two solutions differ by subleading terms that are important in present day precision measurements

23. The running coupling

24. The quark masses
Assume one flavour with renormalized mass m: yet another mass scale
m is the mass anomalous dimension, in PT:
R is dimensionless 

25. The running quark mass
To solve this renormalization-group equtaion, we introduce the running quark mass m(Q 2):
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

26. The running quark mass
All non-trivial scale dependence of R can be included in the running of mass and coupling:
Solution:
c/b > 0  the running mass vanishes with the running coupling at high Q 2

27. 2 hard photons in CMS

28. 4 muons in CMS

29. 4 muons in CMS

30. Basics of Perturbative QCD
Vast subject – only give the flavour
Will use a specific example:
2→2 scattering has one free kinematical parameter, the θ scattering angle
The differential cross section for
The total cross section
below the Z pole
on the Z pole

31. The total hadronic cross section
LO: the hadronic cross section is obtained by counting the possible final states:
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±0.04
The 3.5% difference is mainly due to QCD effects:
Real
virtual
gluon emission

32. 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·pj/s scaled two-particle subenergies, y12+y13+y23=1
The real contribution to the total cross section:
Divergent along the boundaries at yi3 = 0:
Unphyisical singularities - quarks and gluons are never on (zero) mass shell: Breakdown of PT
Divergent when E3→0 (soft gluon), or θi3 →0 (collinear gluon)

33. NLO: the real and virtual contribution in d  4
To make sense of the real contribution, we use dimensional regularization:
Has to be combined with the virtual contribution
The sum of the real and virtual contributions is finite in d = 4:
(same for RZ)

34. The total hadronic cross section at O(αs3)
The total cross section can be computed more easily using the optical theorem
Satisfies the renormalization-group equation to order αs4

35. The total hadronic cross section at O(αs3)

36. The total hadronic cross section at O(αs4) (non-singlet contribution)

37. Jet cross sections
Typical final states in high energy electron-positron collisions
2 jets 3 jets

38. Modelling of events with jets
Production probability pattern:
2jets : 3jets : 4jets ~ O(αs0) : O(α s1) : O(α s2) ⇒ jets reflect the partonic structure

39. Jet cross sections
We average over event orientation ⇒ |M2|2 has no dependence on the parton momenta
NLO corrections:
Cannot combine the integrands (like for σtot)

40. The subtraction scheme
Process and observable independent solution
Made possible by the process-independent factorisation properties of QCD matrix elements

41. The subtraction scheme
The approximate cross section:
= I(ε) |M2|2
with universal factorisation:

42. The subtraction scheme
The integrated approximate cross section:

43. Parton showers and resummation
The universal factorisation
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)

44. General picture of high-energy collisions

45. R at low energies