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Perturbative Quantum Chromodynamics {QCD}
Von: Adrian Seeger
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Perturbative Quantum Chromodynamics {QCD}
Contents
• Brief Historical Overview
• Experimental results and Perturbative QCD
• Standard Model
• Group Theorie
• Operators in QCD
• Feynman Graphs
• Gauge Theory
• Asymptotic Freedom
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Brief Historical Overview
• 1950s Invention of bubble chambers and spark chambers lead to observation of a large number of different hadrons
• 1961 Gell-Mann and Yuval Ne‘eman invented the „eightfold way“
• 1963 Gell-Mann proposed, hadrons are build from Quarks [Nobel Prize 1969]
• 1965 Moo-Young Han and other solved Violation of the Pauli Principle (d++Barion) by giving the Quarks an additional degree of freedom
• 1969 Parton model was veryfied for deep inelastic scattering at SLAC
• 1973 David Gross, David Politzer and Frank Wilczek discovered „asymptotic freedom“ [Nobel Prize 2004]
• 1979 Evidence of Gluons was discovered in three jet events at PETRA
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Perturbative Quantum Chromodynamics {QCD}
Experimental results and perturbative QCD
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Perturbative Quantum Chromodynamics {QCD}
Experimental results and perturbative QCD
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Standard Model
Is a unified Quantum field theorie consisting of:
• Electroweak theory (Glashow, Salam, Weinberg)
– Electromagnetic theory
– Weak theory
• Quantum chromodynamics (Strong Force)
• A field theory can be described by a Lagrangian density
• Solving the functional S
• Leads to the Euler-Lagrange equations
Perturbative Quantum Chromodynamics {QCD}
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Perturbative Quantum Chromodynamics {QCD}
Standard Model
Two important equations:
• Klein-Gordon equation
– Describes bosons (Spin=0, gauge bosons)
– Scalar field φ
• Dirac equation
– Describes fermions (Spin=1/2, leptons, quarks )
– Spinor field ψ
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Perturbative Quantum Chromodynamics {QCD}
Standard Model
Symmetries:
• Electroweak theory
• Quantum Chromodynamics
Standard model is a gauge theory with symmetry:
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CSU )3(
YI USU )1()2(
9
Perturbative Quantum Chromodynamics {QCD}
Group Theory
U means unitary:
A unitaryl operator can be written as is hermitean
• A unitary transformation leaves the normalization of the state vectors unchanged
• U(1): Group of all unitary linear operators in one dimension
• Example:
• U(1) is abelian:
SU means specific unitary:
• SU(2): Group of all specific unitary linear operators in two dimensions
• Generators of the SU(2) group: ½*Pauli matrices
• SU(2) is non abelian:
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Perturbative Quantum Chromodynamics {QCD}
Group Theory
SU means specific unitary:
• SU(3): Group of all specific unitary linear operators in three dimensions
• Generators of the SU(3) group: ½*Gell-Mann matrices
• SU(3) is non abelian
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Perturbative Quantum Chromodynamics {QCD}
Operators in QCD
• To change the color state of a quark we can construct operators from the Gell-Man matrices
• Two Gell-Mann matrices remain
• They don‘t change color!
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Perturbative Quantum Chromodynamics {QCD}
Operators in QCD
Example:
• Transition from green to red
• The quark colour changes from green to red
– a specific gluon is absorbed
• So, you can relate a gluon to every operator!
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Perturbative Quantum Chromodynamics {QCD}
Gauge Theory
Gauge theory in Electrodynamics:
• Maxwell equations stay invariant under a transformation
Example
• From the second Maxwell equation follows
• Applyed to the third equation yields
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Perturbative Quantum Chromodynamics {QCD}
Gauge Theory
Gauge theory in quantum mechanics:
• Schrödinger equation stays invariant under a transformation
• Schrödinger equation for a particel with charge q in electromagnetic field φ
• We can combine φ and A to the four vector
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Perturbative Quantum Chromodynamics {QCD}
Gauge Theory
Gauge theory in QCD:
• The Lagrangian density is invariant
• under a global transformation
• If the transformation is local,
• the Lagrangian density is not invariant
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Perturbative Quantum Chromodynamics {QCD}
Gauge Theory
To make it invariant:
• is replaced by ,where are vector fields
• Gauge transformation of the vector fields
• With the field-strength tensor
• We obtain the Lagrangian density of QCD
• The Dirac equation of a free particle obtained from the Lagrangian density is invariant under a local transformation
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Perturbative Quantum Chromodynamics {QCD}
Feynman Graphs
• With the ϒ-matrices
• The four gradient
• We can write the Dirac equation of a particle with charge e in the field
• To solve this inhomogeneous differential equation we define a Green‘s function K
• Solution of the Dirac equation
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Perturbative Quantum Chromodynamics {QCD}
Feynman Graphs
• We can add any solution of the homogeneous Dirac equation to the solution
• Perturbative calculation means, to handle the second term as small perturbation
• Now the only thing we need is the Green‘s function K
• We define the Fourier transform of K
• and find the electron propagator
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Perturbative Quantum Chromodynamics {QCD}
Feynman Graphs
• With the electron propagator and integration we find the Green‘s function for a particle moving foreward in time to be
• And for a particle moving backwards in time
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Perturbative Quantum Chromodynamics {QCD}
Scattering in QED
• Defining the scattering matrix as
• We can calculate the matrix elements in first order
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Perturbative Quantum Chromodynamics {QCD}
Scattering in QED
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Perturbative Quantum Chromodynamics {QCD}
Feynman Graphs in QED
• Incoming (outgoing) fermion:
• Incoming (outgoing) photon:
• Virtual photon:
• Virtual electron:
• Lepton-Photon-Vertex
Fermion
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Perturbative Quantum Chromodynamics {QCD}
Feynman Graphs in QED
• Elementary processes
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Perturbative Quantum Chromodynamics {QCD}
Feynman Graphs in QCD
• Incoming (outgoing) fermion:
• Incoming (outgoing) photon:
• Virtual gluon:
• Virtual electron:
• Quark-Gluon-Vertex
Fermion
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Perturbative Quantum Chromodynamics {QCD}
Asymptotic Freedom
• As seen, we can describe processes in QED in low order approximations
• The approximation converges fast, because the coupling constant is small
• In QCD it is not sure, that the perturbative calculation converges.
• Therefore ,a very important discovery was
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Perturbative Quantum Chromodynamics {QCD}
Asymptotic Freedom
• There is also a Q dependence in the QED
– Not as strong
– inverse behaviour
Example:
• Positive charge in dielectric oil
• Following QED there are vacuum fluctuation creating virtual electron positron pairs, that are causing this effect of shielding
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Perturbative Quantum Chromodynamics {QCD}
Asymptotic Freedom
• Experiments at LEP support this theory
Feynman diagram
27
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Perturbative Quantum Chromodynamics {QCD}
Asymptotic Freedom
• The Q dependence in QCD is stronger
– α is called running coupling constant
– Two effects contribute
• Quark-Antiquark production
• Emission of a Gluon
• Quark-Antiquark production would dominate only if we had more than 16 quark flavours!
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Perturbative Quantum Chromodynamics {QCD}
Asymptotic Freedom
• Feynman diagram quark-antiquark production
• Feynman diagram emission of a gluon
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