Interactions - Introduction to Feynman Graphs

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Interactions - Introduction to Feynman Graphs
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Quantum Electrodynamics (QED)

• Basic Vertex
• A rule particle forward is equivalent to
  anti-particle backward in time
• Time direction is arbitrary.

e means e -
Griffiths always uses convention above, but I
often use convention below
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Quantum Electrodynamics (QED)

• All electromagnetic processes are made up from
  various combinations of these.
• Could have any charged particle (e, q, ?, etc.)
  at vertex with photon
• Each vertex has coupling (strength) ge \/4??
  (?1/137)

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Some Examples

• Two simple combinations are
• Moller Scattering Bhabha Scattering
• e- e- ? e- e- e- e ? e- e
• Both processes are of magnitude / ge2 4??
  4?/137

e
e
e
e
g
g
e
e
e
e
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Bhabha Scattering

• Actually, Bhabha scattering can also be
  represented by another diagram.
• Since it is impossible to distinguish, the two
  must interfere.
• Only the external lines are observable.

e- e ? e- e
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More Examples

• Other electromagnetic interactions
• All these processes are of magnitude / ge2 ( ?)

Pair production ? ? ? e e-
Pair annihilation e e- ? ? ?
Compton scattering ? e- ? ? e-
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More Rules

• Charge and spin must be conserved at a vertex
• Energy-momentum must be conserved at each vertex
• At basic, single vertex this means that photon
  has mass!
• If we imagine the vertex viewed from one e- rest
  frame
• The CM momentum of incoming ee- 0
• So the momentum of recoiling ? 0
• So the photon cannot be mass-less.
• Internal lines are virtual i.e. off the mass
  shell.

e-
In ? CMS E?2 m?2 (E E-)2 gt 0
e
?
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Higher Order Graphs (gt 2 vertices)

• Should sum over all orders in number of vertices
• Diagrams with 4 vertices ?2
• Series can converge since ? 1/137 ltlt 1

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Vacuum Polarization

• A bare charge Q in a medium is screened by the
  di-electric effect of the medium
• Effective charge is reduced by halo of opposite
  charge from molecules by a factor e (dielectric
  constant)
• At distances lt molecular separation 5 x 10-8
  cm, the bare charge is seen

Qeff
Q
Inter-molecular spacing
Q/e
distance
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Vacuum Polarization

• In QED, even a vacuum can become polarized by
  production of e-pairs
• The e-pairs become polarized rather than
  molecules
• They shield charge of the electron
• At distances lt Compton wavelength of the e-, the
  full bare charge is seen

aeff
lc h/mec 2.43 x 10-10 cm
1/137
distance
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Feynman Rules for Electrodynamics

• More on this later !

Extracted from Griffiths
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Quantum Chromo-dynamics (QCD)

• Basic vertex (? ? ?s)
• Simplest quark-quark interaction

• Gluons are mass-less like gs
• However
• THREE colors (rgb) replace
• ONE electric charge
• AND as gt 1 (but runs)

q
Time
(b)
g
(r)
(rb)
q
q
q
Time
(b)
g
(r)

• NOTICE
• the gluon transfers color
• un-like the photon

(r)
(rb)
(b)
q
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QCD and Color

• In QCD one electric charge is replaced by 3
  colors (rgb)
• Color is conserved so when q ? q g the gluon
  transfers color from q to q
• So gluons are bi-colored (e.q. rb)
• The colors belong to the group SU(3).
• Instead of 3 x 3 9 types of gluons, there are
  only 8 an octet.
• Since gluons are colored, they attract each other
  too

g
g
g
g
OR
etc
g
g
g
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Asymptotic Freedom

• Asymptotic freedom - as (effective) ltlt 1 at
  short distances
• Vacuum polarization occurs in QCD also
• Increases effective color charge at close
  distances
• Also get polarization from gluon pairs but with
  opposite effect on the effective charge
• Decreases effective color charge at close
  distances

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Asymptotic Freedom (contd)

• In field theory, the renormalisation procedure
  leads to
• where (for nf Fermion loops and nb Boson loops)
• In QED (nf 3, nb 0) b0 p. Therefore
• In QCD (nf 6, nb 3) b0 p. Therefore

aem
1/137
Distance / 1/s
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Confinement

• As as becomes larger as distance grows, it is
  difficult to separate a quark from its hadron
• Difficult to prove, but plausible
• No free quarks have yet been observed
• If a quark is pulled from its hadron, the energy
  generated is sufficient to produce a new q-q pair.

S c
u
d
c
s
s
d
c
u
X c0
K
u
s
s
d
c
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Feynman Rules for QCD
Extracted from Griffiths
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Weak Interactions

• Basic charged current lepton vertex
• Examples

ne,m,t

• No current necessary
• In the standard model, strength
• of vertex related to aem

W
e, m, t
ne
e
nm
m
nm
nt
W
W
m
t
m ? e ne nm
t ? m nm nt
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Neutral Current Vertices

• Neutral current lepton vertex
• Interference observed between g and Z0 at higher
  energies

e, m, t, ne,m,t

• Similar to e/m vertex
• In the standard model,
• Z 0 is related to the g

Z 0
e, m, t, ne,m,t
e
e
e
e
e
e
e
e
g
Z0
g



Z0
e
e
e
e
e
e
e
e
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Quarks in Weak Interactions

• Basic charged current lepton vertex
• So the W can connect quarks and leptons

• Couples quarks with different charges
• NOTE quark flavor can change
• e.g. c ? s, etc..

q-1/3
u
d
u
ne
nt
e
W
W
d
t
b-decay d ? e ne u
t ? p nt
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Quarks in Weak Interactions

• Basic neutral current lepton vertex
• Dominant in ne proton scattering

q

• Couples quarks with same charges
• NOTE quark flavor CANNOT change!

Z 0
q
u
u
ne
e
BUT NOT Why?
Z 0
Z 0
ne
u
e
u
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Flavor Changing Weak Interactions

• Basic charged current lepton vertex
• Provides mechanism for flavor changing
• Recall - in strong interactions, flavor is
  strictly conserved

• Couples quarks with different charges
• NOTE quark flavor can change
• e.g. c ? s, etc..

etc.
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Flavor Changing Weak Interactions

• SM explains non-conservation of flavor in terms
  of quark doublets mixing
• The transformation is through the
  Cabbibo-Kobayashi-Maskawa (CKM) unitary matrix U
• Experimentally, the matrix is found to be
  approximately

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Feynman Rules for GWS Model
Extracted from Griffiths