Particle Physics

1
Particle Physics
3nd Handout

• Feynman Graphs of QFT
• QED
• Standard model vertices
• Amplitudes and Probabilities
• Forces from particle exchange
• QCD

http//ppewww.ph.gla.ac.uk/parkes/teaching/PP/PP.
html
Chris Parkes
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Quantum ElectroDynamics (QED)

• Developed 1948 Feynman, Tomonaga,Schwinger
• Feynman illustrated with diagrams

annhilation
Pair production
Photon emission
e-
e-
e
?
?
e
c.f. Dirac hole theory MS 1.3.1,1.3.2
Anti-particlesbackwards in time.
Time Left to Right.
Process broken down into basic components. In
this case all processes are same diagram rotated
We can draw lots of diagrams for electron
scattering (see lecture)
Compare with
3
Orders of ?

• The amplitude T is the sum of all amplitudes from
  all possible diagrams

Feynman graphs are calculational tools, they have
terms associated with them
Each vertex involves the emag coupling (?1/137)
in its amplitude
So, we have a perturbation series only lowest
order terms needed More precision ? more diagrams
There can be a lot of diagrams! N photons, gives
?n in amplitude c.f. anomalous magnetic moment
After 1650 two-loop Electroweak diagrams -
Calculation accurate at 10-10 level and
experimental precision also!
4
The main standard model vertices
At low energy
Strong All quarks (and anti-quarks) No change of
flavour
Weak neutral current All particles No change of
flavour
Weak charged current All particles Flavour
changes
EM All charged particles No change of flavour
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Amplitude? Probability
The Feynman diagrams give us the amplitude, c.f.
? in QM whereas probability is ?2
??Tfi2
(1)
So, two emag vertices e.g. e-e
??-? amplitude gets factor from each vertex
And xsec gets
amplitude squared for e-e ?qq with quarks of
charge q (1/3 or 2/3)

• Also remember u,d,s,c,t,b quarks and they each
  come in 3 colours

• Scattering from a nucleus would have a Z term

(2)
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Massive particle exchange
Forces are due to exchange of virtual field
quanta (?,W,Z,g..)
E,p conserved overall in the process but not for
exchanged bosons.
You can break Energy conservation - as long as
you do it for a short enough time that you dont
notice!
i.e. dont break uncertainty principle.
Consider exchange of particle X, mass mx, in CM
of A
B
X
A
For all p, energy not conserved
Uncertainty principle
Particle range R
So for real photon, mass 0, range is infinite For
W (80.4 GeV/c2) or Z (91.2 GeV/c2), range is
2x10-3 fm
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Virtual particles
This particle exchanged is virtual (off mass
shell)
?
e.g.
?-
e-
(E,p)
symmetric Electron-positron collider
?
e
(E,-p)
(E ?, p?)
Yukawa Potential
Strong Force was explained in previous course as
neutral pion exchange Consider again

• Spin-0 boson exchanged, so obeys Klein-Gordon
  equation

See MS 1.4.2, can show solution is
Can rewrite in terms of dimensionless strength
parameter
R is range
For mx?0, get coulomb potential
8
Quantum Chromodynamics (QCD)
7.1 MS
QED mediated by spin 1 bosons (photons)
coupling to conserved electric charge QCD
mediated by spin 1 bosons (gluons) coupling to
conserved colour charge
u,d,c,s,t,b have same 3 colours (red,green,blue),
so identical strong interactions c.f. isospin
symmetry for u,d, leptons are colourless so
dont feel strong force

• Significant difference from QED
• photons have no electric charge
• But gluons do have colour charge eight
  different colour mixtures.

Hence, gluons interact with each other.
Additional Feynman graph vertices
Self-interaction
4-gluon
3-gluon
These diagrams and the difference in size of the
coupling constants are responsible for the
difference between EM and QCD
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Running Coupling Constants - QED
Charge Q in dielectric medium Molecules nearby
screened, At large distances dont see full
charge Only at small distances see Q
-
-
-
Q
-
-
-
-
-
Also happens in vacuum due to spontaneous
production of virtual ee- pairs
And diagrams with two loops ,three loops. each
with smaller effect ?,?2.
?
QED small variation
1/128
As a result coupling strength grows with q2 of
photon, higher energy ?smaller wavelength gets
closer to bare charge
1/137
(90GeV)2
q2
0
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Coupling constant in QCD

• Exactly same replacing photons with gluons and
  electrons with quarks
• But also have gluon splitting diagrams

This gives anti-screening effect. Coupling
strength falls as q2 increases
g
g
g
Grand Unification ?
g
Strong variation in strong coupling From ?s? 1 at
q2 of 1 GeV2 To ?s at q2 of 104 GeV2
LEP data

• Hence
• Quarks scatter freely at
• high energy
• Perturbation theory converges very
• Slowly as ?s ? 0.1 at current expts
• And lots of gluon self interaction diagrams

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Range of Strong Force
Gluons are massless, hence expect a QED like
long range force But potential is changed by
gluon self coupling
Qualitatively
QED
Form of QCD potential
QCD
Coulomb like to start with, but on 1 fermi
scale energy sufficient for fragmentation
q

q
-
Field lines pulled into strings By gluon self
interaction
Standard EM field
QCD energy/unit length stored in field
constant. Need infinite energy to separate qqbar
pair. Instead energy in colour field exceeds 2mq
and new q qbar pair created in vacuum This
explains absence of free quarks in
nature. Instead jets (fragmentation) of
mesons/baryons
NB Hadrons are colourless, Force between hadrons
due to pion exchange. 140MeV?1.4fm