Download Transparancies for Feynman Graphs

Particle Physics
2nd Handout
Feynman Graphs of QFT
•Relativistic Quantum Mechanics
•QED
•Standard model vertices
•Amplitudes and Probabilities
•QCD
•Running Coupling Constants
•Quark confinement
http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html
Chris Parkes
• See Advanced QM option
Adding Relativity to QM
p2
Apply QM prescription p  i
E
2m
2 2

   i
Get Schrödinger Equation 
2m
dt
Missing phenomena:
Anti-particles, pair production, spin
Free particle
Or non relativistic
Whereas relativistically
1 2 p2
E  mv 
2
2m
E 2  p 2c 2  m2c 4
Applying QM prescription again gives:
Klein-Gordon Equation

1 
 mc 
2






2
2
c dt
  
2
2
Quadratic equation  2 solutions
One for particle, one for anti-particle
Dirac Equation  4 solutions
particle, anti-particle each with spin up +1/2, spin down -1/2
2
Positron
KG as old as QM, originally dismissed. No spin 0 particles known.
Pion was only discovered in 1948.
Dirac equation of 1928 described known spin ½ electron.
Also described an anti-particle – Dirac boldly postulated existence of positron
Discovered by Anderson in 1933 using a cloud chamber (C.Wilson)
Track curves due to magnetic field F=qv×B
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Transition Probability
reactions will have transition probability
How likely that a particular initial state will transform to a specified final state
Interactions
e.g. decays
 IV
Transition rateProby of decay/unit time
cross-section x incident flux
We want to calculate the transition rate between initial state i and final state f,
We Use Fermi’s golden rule
This tells us that fi (transition rate) is proportional to the
transition matrix element Tfi squared (Tfi 2)
T fi  f H ' i  
k i
f H' k k H'i
Ei  Ek
 ....
This is what we calculate from our QFT, using Feynman graphs
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Quantum ElectroDynamics (QED)
• Developed ~1948 Feynman,Tomonaga,Schwinger
• Feynman illustrated with diagrams
Photon emission e-
Pair production
e-
annhilation
e-
e-

Time: Left to Right.

e+
e+
Anti-particles:backwards in time.

c.f. Dirac hole theory
M&S 1.3.1,1.3.2
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
T fi  f H ' i  
k i
f H' k k H'i
Ei  Ek
 ....
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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
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precision also!
The main standard model vertices
s

W
 s  0 .1
1
At low energy:   137
 
W
1
29
Strong:
All quarks (and
EM:
anti-quarks)
Weak neutral current: Weak charged current:
All charged particles
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No change of
All particles
All particles
No change of flavour
flavour
No change of flavour Flavour changes
Amplitude Probability
|Tfi|2
The Feynman diagrams give us the amplitude,
c.f.  in QM whereas probability is ||2
(1)
So, two emag vertices:
e.g.
e-e+ -+
  
amplitude gets factor from each vertex
And xsec gets amplitude squared
2
for e-e+ qq with quarks of charge q (1/3 or 2/3)  (q   )  q 
•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
(2)
If we have several diagrams contributing to same process,
we much consider interference between them e.g.
(b)
e(a)
eee+
e+

e+
2
2
ee+
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Same final state, get terms for (a+b)2=a2+b2+ab+ba
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 don’t notice!
i.e. don’t break uncertainty principle.
Consider exchange of particle X, mass mx, in CM of A:
B
X
A
Uncertainty principle
Particle range R c  c / E
A(mA ,0)  A( EA , p)  X ( EX ,p)
E  E X  E A  m A
E  2 p : p  
E  m X : p  0
E  m X For all p, energy not conserved
  /mx c
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
(E,-p)
collider
E  2 E
+
e+
(E , p)
p  0
m
*2
 E  p  0
2
2
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 M&S 1.4.2, can show solution is
g 2 er / R
V (r )  
4 r
Can rewrite in terms of dimensionless
strength parameter
g2
X  
2
4c
e 1
For mx0, get coulomb potential V (r )  
10
40 r
R is range
7.1 M&S
Quantum Chromodynamics (QCD)
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 don’t 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
s
3-gluon
4-gluon
These diagrams and the difference in size of the coupling constants are responsible
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for the difference between EM and QCD
Running Coupling Constants - QED
+
-
+Q
-
Charge +Q in dielectric medium
Molecules nearby screened,
At large distances don’t see full charge
Only at small distances see +Q
+
Also happens in vacuum – due to spontaneous production of virtual e+e- pairs
e+


e+ e-
And diagrams with
two loops ,three loops….
each with smaller effect: ,2….


QED – small variation
eAs a result coupling strength grows with |q2| of photon,
1/128
1/137
higher energy smaller wavelength gets closer to bare charge
0
|q2|
(90GeV)2
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Coupling constant in QCD
•Exactly same replacing photons with gluons and electrons with quarks
•But also have gluon splitting diagrams
g
g
g
This gives anti-screening effect.
Coupling strength falls as |q2| increases
Grand Unification ?
g
LEP data
Strong variation in strong coupling
From s 1 at |q2| of 1 GeV2
To s at |q2| of 104 GeV2
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
VQCD  
+
-
Standard EM field
q
q
Field lines pulled into strings
By gluon self interaction
4
3
s
r
 kr
Coulomb like to start with,
but on ~1 fermi scale energy
sufficient for fragmentation
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
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Formation of jets
1. Quantum Field Theory – calculation
2. Parton shower development
3. Hadronisation
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Summary
Add Relativity to QM anti-particles,spin
Quantum Field Theory of Emag – QED
1.
2.
•
•
3.
Feynman graphs represent terms in perturbation series in powers
of α
Couples to electric charge
Standard Model vertices for Emag, Weak,Strong
•
Diagrams only exist if coupling exists
•
e.g. neutrino no electric charge, so no emag diagram
QCD – like QED but..
4.
•
•
5.
Gluon self-coupling diagrams
α strong larger than α emag
Running Coupling Constants
•
α strong varies, perturbation series approach breaks down
QCD potential – differ from QED due to gluon interactions
6.
•
Absence of free quarks, fragmentation into colourless hadrons
Now, consider evidence for quarks, gluons….
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