1. Short Introduction to QCD
Renormalisation
Quark Parton Model and its improvement by QCD
Factorisation and the Altarelli-Parisi splitting functions
Evolution equations: DGLAP and BFKL
Determination of the proton parton density
Parton shower
NLO calculation principles…
Deep-inelastic scattering
Proton-proton collisions:

2. Principle of Renormalisation - QED
bare charge
bare charge screened
physical charge
Loop momentum can be anything
ultraviolet divergence
In fact, this
integration is rather
complicated
(see Aitchison&Hey
problem 6.13)
Geometrical series
Result depend on unphysical cut-off M
but physically measured coupling contains all orders
and must be independent of M

3. Renormalisation - QED _ Relation between bare and physical charge
has to specified at a
particular value of the
photon momentum _
Infinities removed at
the prise of
renormalisation scale,
but results depends on
arbitrary parameter m
Note: physical observable
do not depend on m
charge experimentalist
measures depends on scale
“running coupling constant”

4. QED: QCD: Loops in QED and QCD Low resolution: charge is screened
by ee-pairs
High resolution :
charge is big

5. The QCD Scale Parameter L
“Confinement region”:
coupling gets very large
Asymptotic freedom:
unique to non-abelian theories
“soft” “hard”
QCD explains confinement of colour and
allows calculations of hard hadronic processes
via perturbative expansion of coupling !

6. Quark-Parton Model and Deep-Inelastic Scattering1 x
1/3
Gluons
Naive picture:
f(x)

7. The Proton Structure Function
But number of parton is stable

8. QCD improved Parton Model
QPM:
QCD improved Parton Model Q2
QCD: Q2
Integral diverges !
Need to introduce artificial
regulator.non-perturbative scale
where pQCD breaks down

9. Factorisation Q2 DGLAP*-equation: From
iteration
Scaling violation of F2 caused by gluon emission !
*DGLAP: Dokshitzer, Gribov, Lipatov, Altarelli, Parisi

10. Summary DGLAP-Equations
Altarelli-Parisi splitting functions:
Given a parton density f at Q0 the DGLAP evolution predicts f at any Q2 !
DGLAP equation are the basis for parton shower model in MC

11. The Parton Shower Approximation
Hard 22 process calculation has all (external leg) partons on mass shell
However, partons can be off-shell for short times (uncertainty principle)
close to the hard interaction
Outgoing partons radiate
softer and softer partons
Incoming partons radiate
harder and harder partons
For more complex
reaction often not
clear which subdiagram
Should be treated as
hardest double counting

12. Scaling Violations Scaling violations Gluon indirectly determined
by scaling violations
Sensitivity logarithmic

13. Determination of the Parton Densities Functions
Parton density
Factorisation
scale
Most events at LHC
In low-x region
LHC is gluon-gluon collider !

14. Scaling violations Scaling F2 F2 LHC is gluon-gluon collider
gluon density
Scaling F2
LHC is gluon-gluon collider
A part of Wilczek’s comments upon the Nobel Prize announcement …

15. Calculation of Hadron-Hadron Cross-Section
Factorisation Theorem:
PDF is universal
Once extracted can
calculate any
cross-section within
same theoretical
scheme
 Diverges for low PT0
Calculation of exact matrix elements: LO, NLO done, NNLO close-by (many processes)
(loops, divergences, cancellations between large positive/negative numbers)

16. Global NLO QCD Analysis
Parton densities are from „global“ fits, i.e. from all available data:
Recently (2002): PDF with uncertainties using phenomenological analysis
Quantifiable uncertainties on PDF and physical predictions
Problem: complexity of global analysis as results from many experiments
from variety of physical processes with diverse characteristics and errors
often mutally not compatible
and theoretical uncertainty can not be rigoursly quantified
Tung (2004):
„PDF-users must
be well informed
about nature of
uncertainties !“

17. Determination of Parton Density Function
Experimental data and errors
e.g. DIS structure functions
Theoretical framework
e.g. NLO DGLAP fit, MS scheme etc.
Theoretical assumptions and prejudices
e.g. omit certain data,
correct for non-perturbative effects
(nucleon shadowing etc) LO
NLO
NNLO

18. Parton-Parton Luminosities at LHC
Example: x2 x1

19. Example: Single Inclusive Cross-section
- test of pQCD in an energy regime never probed!
- validate our understanding of pQCD at high momentum transfers
Rather general
string theory
toy-model
(hep-ph/ )
Large momentum transfers and small-x !
from PDF
from ren+fac scale
LHC reach in the
first year
At very small
distances,
particles disappear
into curled
extra-dimensions
TeVatron reach
ends here
At the LHC the statistical uncertainties on the jet cross-section will be small.
Main systematic errors ?
Theory uncertainty ?

20. BFKL- Evolution Equation
Recursive BFKL equation:

21. Typical Evolution in an Event
DGLAP
BFKL
leads to strong ordering
of transverse momenta
diffusion pattern along the ladder

22. Physical Interpretation of Evolution Equations
At low-x probability that parton radiates becomes large
Struck parton originates most likely from a cascade
initiated by a parton with large longitudinal momentum
DGLAP
describes change
of parton densities with
varying spatial resolution
of the probe
leads to strong ordering
of transverse momenta
from photon to proton end
BFKL
describes how
high momentum parton
in the proton is dressed
by a cloud of gluons
localised in
fixed transverse spatial
region of the proton
diffusion pattern along
the ladder
Non-perturbative region

23. BFKL evolution equation - resummation of log (1/x) terms
Standard picture
Parton collinear in proton
DGLAP evolution, i.e. resummation of log Q2-terms
ordering in parton virtualities
Pythia, Herwig
Sherpa, Alpgen
NLOJET++,MCFM
Alternative picture
BFKL evolution equation
- resummation of log (1/x) terms
-unintegrated gluon distribution
-ordering in rapidity, unordered in virtuality
2) CFFM evolution:
resummation of log Q2-log (1/x) terms
unintegrated parton distribution
angular ordering
Skewed parton distribution
Colour dipole showers (?)
Cascade
Quark is reabsorbed
Second quark is emitted
proton
Mainly for soft and diffractive processes

24. Back-up

25. Renormalisation Group Equation (RGE)
RGE ensures that entire Q2 dependence of R
comes from running of the strong coupling constant
(not shown here)

26. Example to second order
Consider a quantity R:

27. Reminder: Next-To-Leading-Order calculations
Born:
First Order: Virtual
First-Order: Real
Loop diagram
infra-red singularities
cancel each other (KNL-theorem), if
(infra-red safeness)
Real and virtual contributions can be regularised by introducing integral in d=4-2e dim.
In this case:
One can show that for any observable where the NLO prediction is:
where:

28. Subtraction Method The only divergent term has B&V kinematics
Ellis, Ross, Terrano (1981)
Let us look at the real contribution:
regularised
Add and subtract locally a counter-term with same point-wise singular behaviour as R(x):
Since
By construction this integral is finite
Add and subtract counter-term
The only divergent term has B&V kinematics
and gets cancels against as B/2e-term of virtual contribution
 cancellation independent of Observable

29. Quark Parton Model
Interaction of hadrons due to interaction of partons
Structure of hadron describable by distribution of partons at any time
changes in number and momenta of partons
should be small during time they are probed
Infinite momentum frame:
proton is moving with
infinite momentum
(all masses can be
neglected)
flat,
frozen,
unexpected
time dilation:
partons frozen (no interaction)
they can be treated as ‘free’ during the
short time they interact with photon
Photon-Proton interaction can be expressed
as sum of incoherent scattering from point-like quark !

30. QPM
Quarks interact
-> redistribution of
momenta
QCD

32. BFKL Toy Model

33. Relation PDF and cross-section/F2
Parton density
function
Structure Function/cross-section
Physical observable
Theoretical construct
Model independent
Model dependent
Definition depends on:
1) order of alpha_s
2) factorisation scheme
3) factorisation scale
well defined

34. LHC gives access:
to high momentum transfers
at relatively low-x
to high-x
DIS:
o) theoretically well defined
o) experimentally clean
HERA:
o) measure strong coupling
and parton densities
o) verifiy/falsify DGLAP evolution
o) develop techniques to constrain
theory uncertainties from LHC data
DGLAP evolution

35. W-Boson Production at LHC
Huge statistical samples & clean experimental channel.
W and Z production
~105 events containing W (pTW > 400 GeV)
~104 events containing Z (pTZ > 400 GeV)
“Standard candles” at LHC:
Luminosity - detector calibration
- constrain quark and anti-quark densities in the proton.
Precision measurements MW etc.
…after effort of 10 years
a differential NNLO calculation
is available !
Scale dependence at y=0:
LO: 30% NLO: 6% NNLO: 0.6%
No change in shape from NLO->NNLO
Precision from theory
challenge for experiment !
rapidity

36. PDF Impact on W-Boson Cross-section at LHC
NNLO
NLO
PDF uncertainties:
2 NNLO sets by MRST
Mode=4 gives better
description of
Tevatron High Et jet data
At NLO:
PDF uncertainties
are absorbed
in scale dependence
At NNLO:
PDF uncertainties are larger !
1-2% difference
visible/measurable at LHC ?
rapidity