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QCD-2004 Lesson 2 :Perturbative QCD II
1) Preliminaries: Basic quantities in field theory
2) Preliminaries: COLOUR
3) The QCD Lagrangian and Feynman rules
4) Asymptotic freedom from e+ e- -> hadrons
5) Deep Inelastic Scattering
Guido Martinelli Bejing 2004
NO DEPENDENCE ON THE CUTOFF, NON INFRARED DIVERGENCE
Deep Inelastic Scattering
DIS
Guido Martinelli Bejing 2004
hadronic system with invariant mass W and momentum pX
l(k) l=e,,
(q) q=k-k’
k’
proton,neutronof momentum p
pX
l(k) l=e,,
(q) q=k-k’
k’
p
Bjorkendimensionlessvariables
q
Kinematics
pX
l(k) l=e,,
(q) q=k-k’
k’
p
Structure Functions
Scaling limit
CROSS SECTION pX
l(k) l=e,,
(q) q=k-k’
k’
p
Naive Parton Model For electromagnetic scattering processes:
fragments
(q) + q(pi) -> q(pf)
by neglecting parton virtuality and transverse
momenta
pi
pf
strucked quark
Naive Parton Model
pi
pf
Parton cross-section:
From which we find:
longitudinal cross-section
THE LONGITUDINAL STRUCTURE FUNCTION (CROSS-SECTION)IS ZERO FOR HELICITY CONSERVATION:
pi=(Q/2,Q/2,0,0)
pf =(Q/2,-Q/2,0,0)
q=(0,-Q,0,0)
massless spin 1/2 partons
= helicity
longitudinallypolarized photon
spinless partons would give Ftransverse=0
Parton Model:Useful Relations and Flavour Sum Rules
strange quarks in the proton?proton = uud + qq pairs
u
gluon
s
s
photon
GottfriedSum Rule
Neutrino Cross Section
pi
pf
W
y
d
From neutrino-antineutrino cross-sectionwe can distinguish quarks from antiquarks
Parton Model and QCD
q + q´for simplicity let us consider first only the non-singletcase, namely
q + q´ + g
Parton Model and QCD
is a cutoff necessary toregularize collinear divergences
Effective quark distribution
Classic Interpretation
p = z P
z´=(x/z)p = x P
dW is the probability of finding a quark with a fraction x/z of its``parent” quark and a given k2
T<<Q2
The total probability (up to non leading logarithms) is
2
THE EFFECTIVE NUMBER OF QUARKS WITH THE APPROPRIATE X VARIES WITH Q2
z1
z2
z3
x
2 )2 )
2 Q2
THE EFFECTIVE NUMBER OF QUARKS WITH THE APPROPRIATE X VARIES WITH Q2
z1
z2
z3
x
Q2)
t=ln(Q2/2)
Mellin Transform
Differential equation
Solution
It will be shown later as q(n,t0 ) can be related to hadronic matrix elements of local operators which can computed in lattice QCD
GLUON CONTRIBUTION TO THE STRUCTURE FUNCTIONS
THE GLUON DISTRIBUTION IS DIFFICULT TO MEASURE BECAUSE IT ENTERS ONLY AT ORDER
z x/z
SPLITTING FUNCTIONS
By B. Foster (Bristol U.), A.D. Martin (Durham U.), M.G. Vincter (Alberta U.),.On Page 166-171 of the Review of Particle Properties, please cite the entire review Phys.Lett.B592: 1,2004.
By B. Foster (Bristol U.), A.D. Martin (Durham U.), M.G. Vincter (Alberta U.),.On Page 166-171 of the Review of Particle Properties, please cite the entire review Phys.Lett.B592: 1,2004.
NEXT-TO-LEADING CORRECTIONS TO THE STRUCTURE FUNCTIONS
IN THE NAÏVE PARTON MODEL
F3(x) = q(x) - q(x) ˜ qV(x)
IN THE LEADING LOG IMPROVED PARTON MODEL
F3(x Q2) = q(x,Q2) - q(x, Q2) ˜ qV(x, Q2)
Gluoncontribution
Next-to-leading correction
NON UNIVERSAL
REGULARIZATION PRESCRIPTION DEPENDENT
CANNOT HAVE A PHYSICAL MEANING, HOWEVER
What matters is the combination:
regularization independentprocess dependent
NLL EVOLUTION
LET US DEFINE
BY ABSORBING THE ENTIRE NLL CORRECTION INTHE DEFINITION OF
THEN
The Operator Product Expansion
pi
,W
d
X
The Operator Product Expansion
The term at x0 < 0 does not contribute because cannot satisfy the 4-momentum -function
Neglecting the light quark mass (up to a factor i):
the covariant derivative corresponds tomomenta of
order QCD
the covariant derivative corresponds tolarge momenta of order
q >> MN, QCD
Thus, a part a trivial Lorentz structure, we have to compute
Short Distance Expansion
x -> 0Local
operator ôx0
Higher twistSuppressed as
Local operators and Mellin Transforms of the Structure Functions
Renormalization scale
DEFINE:
Moment of the Structure Functions and Operators
Total momentum conservation
Current conservation
(Adler Sum Rule)
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