Kern und Teilchenphysik 2

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Kern- und Teilchenphysik 2
Vorlesung 3

• EM Wechselwirkung
• invarianter WQ
• Feynman-Diagramme

TU Berlin, Konrad Czerski, SS2008
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Dirac-Gleichung mit EM-Feld
Viererpotential
Impuls
Dirac-Gleichung
Für ein Elektron Pauli-Gleichung (n.rel.)
gyromagnetisches Verhältnis
g 2
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Magnetisches Moment
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Fermis goldene Regel
Übergangswahrscheinlichkeit
wobei
Übergangsmatrixelement
Endzustandsdichte
Wirkungsquerschnitt
phase space
Fluss der einfallenden Teilchen
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Relativistische Effekte
für die Teilchendichte
Lorentz invariant space phase
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Feynman-Diagramme
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Richard Feynman
(Baron) Ernest Stückelberg
von Breidenbach zu Breidenstein und Melsbach
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e
e
p1
p3
e
e
p1
p3
q
q
p2
p4
e-
e-
p2
p4
e-
e-
Leading order diagrams for Bhabha Scattering
e e? ? e e?
Some Rules for the Construction Interpretation
of Feynman Diagrams

• Energy momentum are conserved at each vertex
• Charge is conserved
• Straight lines with arrows pointing towards
  increasing time represent fermions. Those
  pointing backwards in time represent
  anti-fermions
• Broken, wavy or curly lines represent bosons
• External lines (one end free) represent real
  particles
• Internal lines generally represent virtual
  particles

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e
e
p1
p3
e
e
p1
p3
q
q
p2
p4
e-
e-
p2
p4
e-
e-
Leading order diagrams for Bhabha Scattering
e e? ? e e?
Some Rules for the Construction Interpretation
of Feynman Diagrams

• Time ordering of internal lines is unobservable
  and, quantum mechanically, all
  possibilities must be summed together. However,
  by convention, only one unordered diagram is
  actually drawn
• Incoming/outgoing particles typically have their
  4-momenta labelled as pn and internal lines as
  qn
• Associate each vertex with the square root of
  the appropriate coupling constant, ??x
  , so when the amplitude is squared to yield a
  cross-section, there will be a factor of ?xn ,
  where n is the number of vertices (also known as
  the ''order" of the diagram)

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e
e
p1
p3
e
e
p1
p3
q
q
p2
p4
e-
e-
p2
p4
e-
e-
Leading order diagrams for Bhabha Scattering
e e? ? e e?
Some Rules for the Construction Interpretation
of Feynman Diagrams

• Associate an appropriate propagator of the
  general form 1/(q2 M2) with each internal
  line, where M is the mass of mediating boson
• Source vertices of indistinguishable particles
  may be re-associated to form new diagrams
  (often implied) which are added to the sum

Thus, the leading order diagrams for pair
annihilation ( e- e ? ? ??) are
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Photon-Austausch
Übergangsmatrixelement
t
Strom-Strom-Kopplung
wobei
x
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Austausch massiver Teilchen
Propagator
Beispiel Elektron-Positron-Vernichtung
auslaufendes Photon
- Polarisation
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Gleiche Teilchen Vertauschungssymmetrie
zwei Amplituden

Bosonen
-
Fermionen
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Teilchen-Antiteilchen-Symmetrie
Die Matrixelemente haben eine Symmetrie gegen
eine Teilchen-Antiteilchen-Vertauschung
(Crossing-Symmetrie)
Prozesse
e- ?- ? e- ?-
e- e ? ? ?-
Durch die Substitution
p2 ? -k3 , u(p2) ? v(k3) p3 ? -k2, u(p3) ? v(k2)
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Vakuum-Polarisation
Thus, we never actually ever see a ''bare"
charge, only an effective charge shielded by
polarized virtual electron/positron pairs. A
larger charge (or, equivalently, ?) will be seen
in interactions involving a high momemtum
transfer as they probe closer to the central
charge.
?????????? ''running coupling constant"
In QED, the bare charge of an electron is
actually infinite !!!
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Renormalisation
At large q the product of the propagators will
go as 1/q4
? logarithmically divergent !!