QCD-2004 Lesson 2 :Perturbative QCD II 1)Preliminaries: Basic quantities in field theory...
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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
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NO DEPENDENCE ON THE CUTOFF, NON INFRARED DIVERGENCE
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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
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pX
l(k) l=e,,
(q) q=k-k’
k’
p
Bjorkendimensionlessvariables
q
Kinematics
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pX
l(k) l=e,,
(q) q=k-k’
k’
p
Structure Functions
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Scaling limit
CROSS SECTION pX
l(k) l=e,,
(q) q=k-k’
k’
p
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Naive Parton Model For electromagnetic scattering processes:
fragments
(q) + q(pi) -> q(pf)
by neglecting parton virtuality and transverse
momenta
pi
pf
strucked quark
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Naive Parton Model
pi
pf
Parton cross-section:
From which we find:
longitudinal cross-section
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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
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Parton Model:Useful Relations and Flavour Sum Rules
strange quarks in the proton?proton = uud + qq pairs
u
gluon
s
s
photon
GottfriedSum Rule
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Neutrino Cross Section
pi
pf
W
y
d
From neutrino-antineutrino cross-sectionwe can distinguish quarks from antiquarks
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Parton Model and QCD
q + q´for simplicity let us consider first only the non-singletcase, namely
q + q´ + g
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Parton Model and QCD
is a cutoff necessary toregularize collinear divergences
Effective quark distribution
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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
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2
THE EFFECTIVE NUMBER OF QUARKS WITH THE APPROPRIATE X VARIES WITH Q2
z1
z2
z3
x
2 )2 )
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2 Q2
THE EFFECTIVE NUMBER OF QUARKS WITH THE APPROPRIATE X VARIES WITH Q2
z1
z2
z3
x
Q2)
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t=ln(Q2/2)
Mellin Transform
Differential equation
Solution
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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
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GLUON CONTRIBUTION TO THE STRUCTURE FUNCTIONS
THE GLUON DISTRIBUTION IS DIFFICULT TO MEASURE BECAUSE IT ENTERS ONLY AT ORDER
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z x/z
SPLITTING FUNCTIONS
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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.
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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.
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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
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NON UNIVERSAL
REGULARIZATION PRESCRIPTION DEPENDENT
CANNOT HAVE A PHYSICAL MEANING, HOWEVER
What matters is the combination:
regularization independentprocess dependent
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NLL EVOLUTION
LET US DEFINE
BY ABSORBING THE ENTIRE NLL CORRECTION INTHE DEFINITION OF
THEN
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The Operator Product Expansion
pi
,W
d
X
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The Operator Product Expansion
The term at x0 < 0 does not contribute because cannot satisfy the 4-momentum -function
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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
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Short Distance Expansion
x -> 0Local
operator ôx0
Higher twistSuppressed as
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Local operators and Mellin Transforms of the Structure Functions
Renormalization scale
DEFINE:
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Moment of the Structure Functions and Operators
Total momentum conservation
Current conservation
(Adler Sum Rule)